The Stochastic Perturbation Method for Computational Mechanics

The Stochastic PerturbationMethod for ComputationalMechanics

The Stochastic PerturbationMethod for ComputationalMechanics

Marcin KaminskiDepartment of Structural MechanicsTechnical University of Łodz, Poland

A John Wiley & Sons, Ltd., Publication

This edition first published 2013 2013, John Wiley & Sons Ltd

Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to applyfor permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with theCopyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise,except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission ofthe publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print maynot be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brandnames and product names used in this book are trade names, service marks, trademarks or registeredtrademarks of their respective owners. The publisher is not associated with any product or vendormentioned in this book. This publication is designed to provide accurate and authoritative information inregard to the subject matter covered. It is sold on the understanding that the publisher is not engaged inrendering professional services. If professional advice or other expert assistance is required, the servicesof a competent professional should be sought.

Computations in this book were performed by using Maple. Maple is a trademark of Waterloo MapleInc.

Library of Congress Cataloging-in-Publication Data

Kaminski, M. M. (Marcin M.), 1969-The stochastic perturbation method for computational mechanics / Marcin Kaminski.

pages cmIncludes bibliographical references and index.

ISBN 978-0-470-77082-5 (hardback)1. Engineering–Statistical methods. 2. Perturbation (Mathematics) I. Title.

TA340.K36 2013620.001′51922–dc23

2012029897

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-77082-5

Typeset in 10/12.5 Palatino by Laserwords Private Limited, Chennai, India from an electronic versionsupplied by the Author

Contents

Acknowledgments vii

Introduction ix

1 Mathematical Considerations 11.1 Stochastic Perturbation Technique Basis 11.2 Least-Squares Technique Description 261.3 Time Series Analysis 40

2 The Stochastic Finite Element Method 692.1 Governing Equations and Variational Formulations 69

2.1.1 Linear Potential Problems 692.1.2 Linear Elastostatics 722.1.3 Non-linear Elasticity Problems 752.1.4 Variational Equations of Elastodynamics 762.1.5 Transient Analysis of the Heat Transfer 772.1.6 Thermopiezoelectricity Governing Equations 802.1.7 Navier–Stokes Equations 83

2.2 Stochastic Finite Element Method Equations 872.2.1 Linear Potential Problems 872.2.2 Linear Elastostatics 882.2.3 Non-linear Elasticity Problems 922.2.4 SFEM in Elastodynamics 972.2.5 Transient Analysis of the Heat Transfer 1002.2.6 Coupled Thermo-piezoelectrostatics SFEM Equations 1042.2.7 Navier–Stokes Perturbation-Based Equations 106

2.3 Computational Illustrations 1102.3.1 Linear Potential Problems 110

2.3.1.1 1D Fluid Flow with Random Viscosity 1102.3.1.2 2D Potential Problem with the Response Function

Method 114

vi Contents

2.3.2 Linear Elasticity 1182.3.2.1 Simple Extended Bar with Random Stiffness 1182.3.2.2 Elastic Stability Analysis of the Steel

Telecommunication Tower 1242.3.3 Non-linear Elasticity Problems 1312.3.4 Stochastic Vibrations of the Elastic Structures 135

2.3.4.1 Forced Vibrations with Random Parameters for aSimple 2 DOF System 135

2.3.4.2 Eigenvibrations of the Steel TelecommunicationTower with Random Stiffness 139

2.3.5 Transient Analysis of the Heat Transfer 1422.3.5.1 Heat Conduction in the Statistically Homogeneous

Rod 1422.3.5.2 Transient Heat Transfer Analysis by the RFM 147

3 Stochastic Boundary Element Method 1553.1 Deterministic Formulation of the Boundary Element Method 1563.2 Stochastic Generalized Perturbation Approach to the BEM 1603.3 The Response Function Method in the SBEM Equations 1643.4 Computational Experiments 168

4 The Stochastic Finite Difference Method 1954.1 Analysis of the Unidirectional Problems 196

4.1.1 Elasticity Problems 1964.1.2 Determination of the Critical Moment for the

Thin-Walled Elastic Structures 2094.1.3 Introduction to Elastodynamics with Finite Differences 2144.1.4 Advection–Diffusion Equation 220

4.2 Analysis of Boundary Value Problems on 2D Grids 2254.2.1 Poisson Equation 2254.2.2 Deflection of Thin Elastic Plates in Cartesian Coordinates 2304.2.3 Vibration Analysis of Elastic Plates 239

5 Homogenization Problem 2415.1 Composite Material Model 2435.2 Statement of the Problem and Basic Equations 2495.3 Computational Implementation 2565.4 Numerical Experiments 258

6 Concluding Remarks 297

Appendix 303

References 319

Index 329

Acknowledgments

The author would like to acknowledge the financial support of the Polish Ministryof Science and Higher Education in Warsaw under Research Grant No. 519-386-636entitled ‘‘Computer modeling of the aging processes using stochastic perturbationmethod’’ transferred recently to the Polish National Science Center in Cracow, Poland.This grant enabled me to make most of the research findings contained in this book.Its final shape is thanks to a professor’s grant from the Rector of the TechnicalUniversity of Łodz during the year 2011. Undoubtedly, my PhD students – with theircuriosity, engagement in computer work, and research questions – helped me toprepare the numerical illustrations provided in the chapter focused on the stochasticfinite element method.

Introduction

Uncertainty and stochasticity accompany our life from the very beginning and are stilla matter of interest, guesses, and predictions made by mathematicians, economists,and fortune tellers. Their results may be as dramatic as car or airplane accidents,sudden weather changes, stock price fluctuations, diseases, and mortality in largerpopulations. All these phenomena and processes, although completely unpredictablefor most people, have mathematical models to explain some trends and limitedprognosis. There is a philosophical issue undertaken by various famous scientistswhether the universe has a deterministic nature and some marginal stochasticnoise – some kind of chaos or, in contrast, everything is uncertain – more, less,or fully.

In civil engineering we may observe the most dangerous aspects resulting fromearthquakes, tornadoes, ice covers, and extensive rainfalls. These are the cases whenstochastic fluctuations may also be treated as fully unpredictable, usually havingno mean (expected) value and quantified coefficient of variation, so we are unableto provide any specific computer simulation. Let us recall that engineering codesusually apply the Poisson process to model huge catastrophic failures but they needextended and reliable statistics unavailable in many countries and sometimes evennon-existent due to the enormous technological progress required. On a smallerscale (counting economic disasters and their possible consequences) we notice almostevery day wind-blow variations and their results [158], accidental loading of carsand railways on bridges during rush hours, statistical strength properties of buildingmaterials, corrosion, interface cracks, volumetric structural defects, and a number ofgeometrical imperfections in structural engineering [142]. These are all included inmathematical and computational models with basic statistics coming from observa-tions, engineering experience and, first of all, experimental verification. We need toassume that our design parameters have some distribution function and the mostpractical assumption is that they have Gaussian distributions. This reflects the CentralLimit Theorem, stating that the mixture of different random variables tends to thisparticular distribution when their total number tends to infinity.

x Introduction

We are not interested in analyses and predictions without expectations in thisbook; computational analysis is strictly addressed to engineering and scientificproblems having perfectly known expected values as well as standard deviations andto the case where the initial random dispersion is Gaussian or may be approximatedby a Gaussian distribution with relatively small modeling error. In exceptionalcircumstances it is possible to consider lognormal distributions as they have recursiveequations for higher-order probabilistic moments. From the probabilistic point ofview we provide up to a fourth central probabilistic moments analysis of state func-tions like deformations, stresses, temperatures, and eigenfrequencies, because then itis possible to verify whether these functions really may have Gaussian distributionsor not. The stochastic perturbation technique of course has a non-statistical characterso we cannot engage any statistical hypothesis and we are interested in quantifi-cation of the resulting skewness and kurtosis. Recognition of the Gaussian outputprobability density function (PDF) will simplify further numerical experiments ofsimilar character since these PDFs are uniquely defined by their first two momentsand then the numerical determination of higher moments may be postponed.

From a historical point of view the first contribution to probability theory wasmade by the Italian mathematician Hieronimus Cardanus in the first part of his bookentitled Philologica, Logica, Moralia published more than 100 years after he finished itin seventeenth century. As many later elaborations, it was devoted to the probabilityof winning in random games and had some continuation and extension in the work ofChristian Huygens. It was summarized and published in London, in 1714, under theself-explanatory title The Value of All Chances in Games of Fortune; Cards, Dice, Wagers,Lotteries & C. Mathematically Demonstrated. The main objective at that time was tostudy the discrete nature of random events and combinatorics, as also documentedby the pioneering works of Blaise Pascal and Pierre de Fermat. One of the mostamazing facts joining probability theory with the world of analytical continuousfunctions is that the widely known PDF named after the German mathematicianKarl Friedrich Gauss was nevertheless elaborated by Abraham de Moivre, mostfamous for his formula in complex number theory. The beginnings of modernprobability theory date to the 1930s and are connected with the axioms proposedby Andriei Kolmogorov (exactly 200 years after the normal distribution introducedby de Moivre). However, the main engine of this branch of mathematics was, as inthe previous century, just mechanics and, particularly, quantum mechanics based onthe statistical and unpredictable nature noticed on the molecular scale, especially forgases. Studies slowly expanded to other media exhibiting strong statistical aspects inlaboratory experiments performed in long repeatable series. There is no doubt todaythat a second milestone was the technical development in computer machinery andsciences, enabling large statistical simulations.

Probabilistic methods in engineering and applied sciences follow mathematicalequations and methods [158], however the recent fast progress of computers andrelevant numerical techniques has brought about some new perspectives, a little bitunavailable for broader audience because of mathematical complexity. Historically,it is necessary to mention a variety of mathematical methods, where undoubtedly the

Introduction xi

oldest one is based on straightforward evaluation of the probabilistic moments of theresulting analytical functions on the basis of moments of some input parameters. Thiscan be done using integral definitions of moments or using specific algebraic proper-ties of probabilistic moments themselves; similar considerations may be provided forthe time series defining some random time fluctuations of engineering systems andpopulations as well as related simple stochastic processes. It is possible, of course, toprovide analytical calculations and justification that some structure or system givesa stationary (or not) stochastic response. According to the progress of mathematicaldisciplines after classical probability theory, at the beginning of the twentieth centurywe noticed an elaboration of the theory of stochastic differential equations and theirsolutions for specific cases having applications in non-stationary technical processeslike structural vibrations and signal analysis [158].

Nowadays these methods have brand new applications with the enormous expan-sion of computer algebra systems, where analytical and visualization tools give newopportunities in conjunction with old, well-established mathematical theories. Sincethese systems work as neural networks, we are able to perform statistical reasoningand decision-making based on the verification of various statistical hypothesesimplemented. The successive expansion of uncertainty analysis continued thanksto computers, important for large data set analysis and, naturally, additional sta-tistical estimators. The first of the computer-based methods, following traditionalobservation and laboratory experiments, is of course the Monte Carlo simulationtechnique [5, 25, 53, 71], where a large set of computational realizations of the originaldeterministic problem on the generated population returns through statistical esti-mation the desired probabilistic moments and coefficients. The pros and cons of thistechnique result from the quality and subprocedures of the internal random numbergenerator (generation itself and shuffling routines) as well as the estimators (espe-cially important for higher-order moments) implemented in the computer program.Usually, precise information about these estimator types is not included in commer-cial software guides. An application of this method needs an a priori definition of bothbasic moments and the PDF of the random or stochastic input, however, we usuallyrestrict ourselves to the Gaussian, truncated Gaussian, or lognormal PDF becauseof a difficulty in recovering and analytical processing of the probabilistic moments.The next technique that evolved was fuzzy analysis [132], where an engineer needsprecise information about the maximum and minimum values of a given randomparameter, which also naturally comes from observation or experiments. Then, thismethod operates using interval analysis to show the admissible intervals for theresulting state functions on the basis of the intervals for given input parameters.A separate direction is represented by the spectral methods widely implemented inthe finite element method (FEM), with commercial software like ABAQUS or ANSYS,for instance. These are closely related to vibration analysis, where a structure withdeterministic characteristics is subjected to some random excitation with the firsttwo probabilistic moments given [117, 153]. Application of the FEM system makesit possible to determine the power spectral density (PSD) function for the nodal

xii Introduction

response. General stochastic vibration analysis is still the subject of many works[30, 143], and many problems in that area remain unsolved.

We also have the family of perturbation methods of first, second, and generalorder applied in computational mechanics and, also, the Karhunen–Loeve expan-sion techniques [38, 39] as well as some mixed hybrid techniques, popular especiallyfor multiscale models [176]. These expansion techniques are provided using theeigenfunctions and eigenvectors of the covariance kernel for the input randomfields or processes, both Gaussian and non-Gaussian [168, 174]. They need moreassumptions and mathematical effort to randomize the given physical problem thanthe perturbation methods and, further, determination of higher moments is not sostraightforward. Moreover, there is no commercial implementation in any of thepopular existing FEM systems in this case. There are some new theoretical ideas inrandom analysis for both discrete [55] and continuous variables or processes [33, 52,173], but they have no widely available computational realizations or general appli-cations in engineering. The reader is advised to study [41, 154] for a comprehensivereview of modern probabilistic methods in structural mechanics.

Restricting our overview to the perturbation method we need to mention thatthe first-order technique is useful for the very small random dispersion of inputrandom variables (with coefficient of variation smaller than α < 0.10) to replaceMonte Carlo simulations in simplified first-two-moments analysis. The second-ordertechniques [112, 118] are applicable for α < 0.15 in second-moment analysis also forboth symmetrical distributions (second-order second-moment analysis – SOSM) andfor some non-symmetrical probability functions like the Weibull distribution (the so-called Weibull second-order third-moment approach – WSOTM). The main idea ofthe generalized stochastic perturbation method proposed here is to calculate higher-order moments and coefficients to recognize the resulting distributions of structuralresponse. The second purpose is to allow for larger input coefficients of variation, buthigher moments were initially derived in many numerical experiments containedin this book using fourth- and sixth-order expansions only. Implementation of thegiven general-order stochastic perturbation technique was elaborated first of all tominimize the modeling error [139] and now is based on polynomials of uncertaininput variable with deterministic coefficients. It needs to be mentioned that randomor stochastic polynomials appeared in probabilistic analysis before [50, 147], but werenever connected with the perturbation method and deterministic structural responsedetermination.

It should be emphasized further that the perturbation method was neither strictlyconnected with the stochastic or probabilistic analysis nor developed for theseproblems [135]. The main idea of this method is to make an analytical expansion ofsome input parameter or phenomenon around its mean value thanks to some seriesrepresentation, where Taylor series expansions are traditionally the most popular.Deterministic applications of this technique are known first of all from dynamicalproblems, where system vibrations are frequently found thanks to such an expansionin more complex situations. One interesting application is the homogenizationmethod, where effective material properties tensors of some multi-material systems

Introduction xiii

are found from the solution of the so-called homogenization problem including initialperturbation-based expansions of these effective tensor components with respect tovarious separate geometrical scales [6, 56, 151]. Further, as also demonstrated in thisbook, such a deterministic expansion may be linked with probabilistic analysis, wheremany materials constituting such a structure are separately statistically homogeneous(finite and constant expectations and deviations of physical properties) and resultsin a statistically heterogeneous global system (partially constant expectations anddeviations of physical properties). This is the case when the geometry is perfectlyperiodic and the physical nature of the composite exhibits some random fluctuation.Then such a homogenization procedure returns statistical homogeneity using somemixing procedure and remains clearly deterministic, because expansion deals withgeometric scales that show no uncertainty.

Let us note that the very attractive aspect of the perturbation method is that itincludes sensitivity analysis [35, 44, 83, 91] since first-, second-, and higher-order par-tial derivatives of the objective function with respect to the design parameter(s) mustbe known before the expansions are provided. Therefore, before we start uncertaintyanalysis of some state function in the given boundary value problem, we shouldperform first-order sensitivity analysis and randomize only these parameters whosegradients (after normalization) have dominating and significant values. Further, thestochastic perturbation method is not really associated with any discrete computa-tional technique available [111, 152] like FEM, the Finite Difference Method (FDM),the Finite Volume Method (FVM), the Boundary Element Method (BEM), variousmeshless techniques, or even molecular dynamics simulations. We can use it firstof all to make additional probabilistic expansions of the given analytical solutionsexhibiting some parametric randomness or even to solve analytically some algebraicor differential equations using explicit, implicit, and even symbolic techniques.

The stochastic perturbation technique is shown here in two differentrealizations – with use of the Diret Differentiation Method (DDM) and in conjunctionwith the Response Function Method (RFM). First of them is based on the straightfor-ward differentiation of the basic deterministic counterpart of the stochastic problem,so that we obtain for a numerical solution a system of hierarchical equations withincreasing order. The zeroth-order solution is computed from the first equation andinserted into the second equation, where first-order approximation is obtained and soon, until the highest-order solution is completed. Computational implementation ofthe DDM proceeds through direct implementation with the deterministic source codeor, alternatively, with use of some of the automatic differentiation tools availablewidely as shareware. Although higher-order partial derivatives are calculatedanalytically at the mean values of input parameters, and so that are determinedexactly, the final solution of the system of algebraic equations of increasingorder enlarges the final error in probabilistic moments – the higher order of thesolution, the larger possible numerical error. The complexity of the general-orderimplementation, as well as this aspect, usually results in DDM implementationsof lowest order – as first or the second. Contrary to numerous previous models,

xiv Introduction

now full tenth-order stochastic expansions are used to recover all the probabilisticmoments and coefficients; this significantly increases the accuracy of the final results.

We employ the RFM consecutively, where we carry out numerical determinationof the analytical function for a given structural response like displacement or tempe-rature as the polynomial representation of the chosen random input design param-eter (to determine its deterministic coefficients). Generally, it can be implemented ina global sense, where a single function connects the probabilistic output and inputand, in a more delicate manner – in the local formulation, where the approximatingpolynomial form varies from the mesh or grid node to another node in the discretemodel. It is apparent that global approximation is much faster but may show a largermodeling error; such a numerical error [139] in the local formulation is partially con-nected with the discretization procedure and may need some special adaptivity toolssimilar to these worked out in deterministic analyses. The main advantages of RFMover DDM are that (i) error analysis issues deal with the deterministic approximationproblems and (ii) there is an opportunity for a relatively easy interoperability withcommercial (or any) packages for discrete computational techniques. The RFM proce-dures do not need any symbolic algebra system because we differentiate well-knownpolynomials of random variables, so this differentiation is also of deterministic char-acter. The RFM is used here in the few different realizations starting from classicalpolynomial interpolation with the given order, some interval spline approximations,through the non-weighted least-squares method until more sophisticated weightedoptimized least-squares methods. This aspect is now closely related to the computeralgebra system and this choice also follows enriched visualization procedures, butmay be implemented in classical programming language. The RFM is somewhatsimilar to the response surface method (RSM) applicable in reliability analysis [175]or the response function technique known from vibration analysis. The major andvery important difference is that the RFM uses a higher-order polynomial responserelating a single input random variable with the structural output, whereas the RSMis based on first- or second-order approximations of this output with respect tomultiple random structural parameters. An application of the RSM is impossiblein the current context because the second-order truncation of the response elimi-nates all higher-order terms necessary for reliable computation of the probabilisticstructural response. Furthermore, the RSM has some statistical aspects and issues,while the RFM has a purely deterministic character and exhibits some errors typicalfor mathematical approximation theory methods only.

Finally, let us note that the generalized stochastic perturbation technique wasinitially worked out for a single input random variable but we have some helpfulcomments in this book concerning how to complete its realization in case of a vectorof correlated or not random input sources. The uncorrelated situation is a simpleextension of the initial single-variable case, while non-zero cross-correlations, espe-cially of higher order, will introduce a large number of new components into theperturbation-based equations for the probabilistic moments, even for expectations.

It is clear that stochastic analysis in various branches of engineering does notresult from a fascination with random dispersion and stochastic fluctuations in civil

Introduction xv

or aerospace structures, mechanical as well as electronic systems – it is directly con-nected with reliability assessment and durability predictions [1]. Recently we noticeda number of probabilistic numerical studies in non-linear problems in mechanicsdealing particularly with the design of experiments [45], gradient plasticity [177], andviscoelastic structures [42], summarized for multiscale random media in [140]. Eventhe simplest model of the first-order reliability method is based on the reliabilityindex giving quantified information about the safety margin computed using theexpected values and standard deviations for two or more components of the limitfunction. According to various numerical illustrations presented here, the tenth-orderstochastic perturbation technique is as efficient for this purpose as the MCS methodand frequently does not need further comparative studies. It is also independentof the input random dispersion of the given variable of the problem and should bechecked for correlated variables also. As is known, the second-order reliability meth-ods [128] include some correction factors and/or multipliers like the curvature of thelimit functions usually expressed by the second partial derivatives of the objectivefunction with respect to the random input. The generalized perturbation techniqueserves in a straightforward manner in this situation, because these derivatives areincluded in the Taylor expansions themselves, so there is no need for an additionalnumerical procedure. As has been documented, this stochastic perturbation-basedfinite element method (SFEM) implemented using the RFM idea may be useful atleast for civil engineers following Eurocode 0 statements and making simulations oncommercial FEM software. It is worth emphasizing that the stochastic perturbationmethod may be efficient in time-dependent reliability analysis, where time serieshaving Gaussian coefficients approximate time fluctuations of the given designparameters. There are some further issues not discussed in this book, like the adap-tivity method related to the stochastic finite elements [171], which may need somenew approaches to the computational implementation of the perturbation technique.

This book is organized into five main chapters – Chapter 1 is devoted to themathematical aspects of the stochastic perturbation technique, necessary definitionsand properties of the probability theory. It is also full of computational examplesshowing implementations of various engineering problems with uncertainty intothe computer algebra system Maple [17] supporting all further examples andsolutions. Some of these are shown directly as scripts with screenshots, especiallyonce some analytical derivations have been provided. The remaining case studies,where numerical data has been processed, are focused on a discussion of the resultsvisualized as the parametric plots of probabilistic moments and characteristics, mostlywith respect to the input random dispersion coefficient. They are also illustrated withthe Maple scripts accompanying the book, which are still being expanded bythe author and may be obtained by special request in the most recent versions.Special attention is given to the RFM here, various-order approximations of themoments in the stochastic perturbation technique, some comparisons against theMonte Carlo technique and computerized analytical methods, as well as simpletime-series analysis with the perturbation technique.

xvi Introduction

Chapter 2 is the largest in the book and is devoted entirely to the SFEM. It startswith the statements of various more important boundary-value or boundary-initialproblems in engineering with random parameters, which are then transformed intoadditional variational statements, also convenient for general nth-order stochasticformulations. According to the above considerations, these stochastic variationalprinciples and the resulting systems of algebraic equations are expanded using bothDDM and RFM approaches to enable alternative implementations depending on thesource code and automatic differentiation routines availability; there are multipleMaple source codes for most of the numerical illustrations here, as also in thepreceding chapter. Theoretical developments start from the FEM for the uncou-pled equilibrium problems with scalar and vector state functions and are continueduntil the thermo-electro-elastic couplings as well as Navier–Stokes equations forincompressible and non-turbulent Newtonian fluid flows. The particular key com-putational experiments obey Newtonian viscous unidirectional and 2D fluid flows,linear elastic response and buckling of a spatial elastic system, elasto-plastic behaviorof a simple 2D truss, eigenvibrations analysis of a 3D steel tower, non-stationaryheat transfer in a unidirectional rod, as well as forced vibrations in a 2 DOF sys-tem, all with randomized material parameters. It is demonstrated that the Maplesystem may be used efficiently as the FEM postprocessor, making a visualizationof the mesh together with the desired probabilistic characteristics in vector form;three-dimensional graphics are not so complicated in this environment, but phys-ical interpretation of higher-order moments does not require such sophisticatedtools right now. The discussion is restricted each time to the first four probabilisticmoments and coefficients for the structural response shown as functions of the inputcoefficient of variation and, sometimes, the stochastic perturbation order. Usually,we (i) check the probabilistic convergence of the SFEM together with its order, (ii)detect the influence of an initial uncertainty source, and (iii) verify the output PDF.

Chapter 3 describes the basic equilibrium equations and computational imple-mentation of the Stochastic Perturbation-based Boundary Element Method (SBEM)related to the linear isotropic elasticity of the statistically homogeneous and multi-component domains; numerical work has been completed using the open-sourceacademic BEM code [4]. The basic equations have all been rewritten in the responsefunctions language with numerical illustrations showing uncertain elastic behaviorof a steel plane panel, an analogous composite layered element with perfect inter-face, as well as a composite with some interface defects between the constituents.A comparison of the SBEM implemented using triangular and Dirac distributionsof the weights in Least Squares Method is also given here using the example of thefirst four probabilistic characteristics presented as functions of the input coefficientof variation for the last problem.

Chapter 4 is addressed to anyone who is interested in Stochastic analysis usingthe specially adopted Finite Difference Method (SFDM) and additional source codes.According to the main philosophy of the method we rewrite the particular differentialequations in the difference forms and introduce first of all their DDM versions to carry

Introduction xvii

out computational modeling directly using the Maple program. The example prob-lem with random parameters is the linear elastic equilibrium of the Euler–Bernoullibeam with constant and linearly varying cross-sectional area; further, this structureis analyzed numerically on an elastic single parameter random foundation. Let usnote that stochastic analysis of beams with random stiffness in civil and mechanicalengineering is of significant practical importance and has been many times studiedtheoretically and numerically [31, 112]. Other models include non-stationary heattransfer in a homogeneous rod with Gaussian physical parameters, eigenvibrationanalysis of a simply supported beam and a thin plate, as well as the unidirectionaldiffusion equation. Some examples show the behavior of the probabilistic momentscomputed together with increasing density of the grid, others are shown to make acomparison with the results obtained from the analytical predictions.

Chapter 5 is particularly and entirely devoted to the homogenization procedurepresented as the unique application of the double perturbation method, wheredeterministic expansion with respect to the scale parameter is used in conjunctionwith stochastic expansions of the basic elastic parameters. Homogenization of theperfectly periodic two-component composite is the main objective in this chapter,and its effective elasticity tensor in a probabilistic and stochastic version is studiedfor material parameters of fiber and matrix defined as Gaussian random variables ortime series with Gaussian coefficients. The main purpose is to verify the stochasticperturbation technique and its FEM realization against the Monte Carlo simulation,as well as some novel computational techniques using the RFM based on analyticalintegration implemented in the Maple system. The examples are used to confirmthe Gaussian character of the resulting homogenized tensor components, check theperturbation technique convergence for various approximation orders, show theprobabilistic entropy fluctuations in the homogenization procedure, and providesome perspectives for further development of both SFEM and RFM techniques.

The last part of this book is given as the Appendix, where all more popularprobability distributions are contrasted. Particularly, their up to the tenth centralprobabilistic moments are derived symbolically to serve the Readers in their ownstochastic implementations.

The major conclusion of this book is that the stochastic perturbation techniqueis a universal numerical method useful with any discrete or symbolic, academicor commercial computer programs, and environments. The applicability range forexpectations is practically unbounded, for second moments – extremely large (muchlarger than before) but for third- and fourth-order statistics – limited (but may begiven precisely in terms of an input random dispersion). Mathematical simplicityand time savings are attractive for engineers, but we need to remember that this is nota computational hammer to randomize everything. Special attention is necessary incase of coupled problems with huge random fluctuations, where output coefficients ofvariation at some iteration step (even the first one) can make it practically useless. Thelocal and global response functions are usually matched very well by the polynomialforms proposed here, and, sometimes, resulting moments show no singularities withrespect to the input coefficient of variation. This situation, however, may change in

xviii Introduction

systems with state-dependent physical and mechanical properties (for example, withrespect to large temperature variations).

The book in its present shape took me almost 20 years of extensive work, fromthe very beginning of my career with the second order version of the SFEM at theInstitute of Fundamental Technological Research in Warsaw, Poland [112]. Slowly myinterest in the finite elements domain evolved towards other discrete computationaltechniques and, after that, an idea of any-order Taylor expansion appeared around 10years ago. I would like to express special thanks to my PhD students at the TechnicalUniversity of Łodz for their help in reworking and reorganizing many numericalexamples for this book, but also for their never-ending questions – pushing me tocarefully check many times the same issues. I appreciate the comments of manycolleagues from all around the world who are interested in my work, as well as theanonymous reviewers who took care over the precision of my formulations.

1MathematicalConsiderations1.1 Stochastic Perturbation Technique Basis

The input random variable of a problem is denoted here consecutively by b(ω) andits probability density by gb(x). The expected value of this variable is expressed byFeller [34] and Vanmarcke [165] as

E[b] =+∞∫

−∞b gb(x)dx, (1.1)

while its mth central probabilistic moment is

µm(b) =+∞∫

−∞(b − E[b])m gb(x)dx. (1.2)

Since we are mostly focused on the Gaussian distribution application, we recallnow its probability density function:

gb(x) = 1√2π Var(b)

exp

(− (x − E[b])2

2Var(b)

). (1.3)

The coefficient of variation, skewness, flatness and kurtosis are introduced in theform

α(b) =√

µ2(b)E[b]

=√

Var(b)E[b]

= σ (b)E[b]

, β(b) = µ3(b)σ 3(b)

, γ (b) = µ4(b)σ 4(b)

= κ(b) + 3.

(1.4)Nowadays, computer algebra software is usually employed to provide analytical

formulas following these statements. A symbolic solution provided in the systemMaple for the well-known case of two Gaussian random variables X and Y having

The Stochastic Perturbation Method for Computational Mechanics, First Edition. Marcin Kaminski. 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd.

2 The Stochastic Perturbation Method for Computational Mechanics

defined expectations and standard deviations equal to EX, EY and SIGX, SIGY isgiven below. As is supposed, we can have more variables, combined in all algebraicforms implemented into this system and, finally, random variables not necessarilyGaussian.

>restart: with(plots): with(plottools): with(Statistics):>X:=RandomVariable(Normal(EX,SX)): Y:=RandomVariable(Normal(EY,SY)):>G:=X*Y:>EG:=ExpectedValue(G);

EG := EX EY

>VarG:=Variance(G);

VarG := EY2 SX2 + SY2 EX2 + SY2 SX2

>StdG:=StandardDeviation(G);

StdG :=√

EY2 SX2 + SY2 EX2 + SY2 SX2

>alfaG:=Variation(G);

alfaG :=√

EY2 SX2 + SY2 EX2 − SY2 SX2

EX EY

>skewG:=Skewness(G);

skewG := 6EX EY SX2 SY2

(EY2 SX2 + SY2 EX2 + SY2 SX2)3/2

>kappaG:=Kurtosis(G);

kappaG := (3(EY4 SX4 + 2EY2 SX2 SY2 EX2 + 6EY2 SX4 SY2

+ SY4 EX4 + 6SY4 EX2 SX2 + 3SY4 SX4))/(EY2 SX2

+ SY2 EX2 + SY2 SX2)2

>m3G:=CentralMoment(G,3);

m3G := 6EX EY SX2 SY2

>m4G:=CentralMoment(G,4);

m4G := 3EY4 SX4 + 6EY2 SX2 SY2 EX2 + 18EY2 SX4 SY2

+ 3SY4 EX4 + 18SY4 EX2 SY2 + 9SY4 SX4

The second, less trivial opportunity with this program is recovery of the probabili-stic moments for the other probability distributions widely applied in engineering,whose formulas are not available in the literature or are hard to find (contained in the

Mathematical Considerations 3

Appendix). The cases of lognormal and Gumbel distributions serve as an examplebelow – one can use more sophisticated algebraic combinations of course.

>restart; with(Statistics): a::real, 0 < b: X1:=RandomVariable(Gumbel(a,b)):>EX1:=ExpectedValue(X1); VX1:=Variance(X1); MX1:=Median(X1); KX1:=Kurtosis(X1):SKX1:=Skewness(X1): COVX1:=Variation(X1); CM3X1:=CentralMoment(X1,3):CM4X1:=CentralMoment(X1,4):

EX1 := a + γ b

VX1 := 16

b2 π2

MX1 := a − b ln(ln(2))

COVX1 := 16

√6bπ

a + γ b

>X2:=RandomVariable(LogNormal(a,b)):>EX2:=ExpectedValue(X2); VX2:=Variance(X2); MX2:=Median(X2); KX2:=Kurtosis(X2);SKX2:=Skewness(X2); COVX2:=Variation(X2); CM3X2:=CentralMoment(X2,3);CM4X2:=CentralMoment(X2,4);

EX2 := ea+ 12 b2

VX2 := e2 a+b2(eb2 − 1

)MX2 := ea

KX2 := −−e4 a+8 b2 + 4e4 a+5 b2 − 6e4 a+3 b2 + 3e2 b2+4 a(e2 a+b2)2 (eb2−1

)2SKX2 := e3 a+ 9

2 b2 − 3e3 a+ 52 b2 + 2e

32 b2+3 a(

e2 a+b2 (eb2 − 1))3/2

COVX2 :=√

e2 a+b2 (eb2 − 1)

ea+ 12 b2

CM3X2 := e3 a+ 92 b2 − 3e3 a+ 5

2 b2 + 2e32 b2+3 a

CM4X2 := e4 a+8 b2 − 4 e4 a+5 b2 + 6 e4 a+3 b2 − 3 e2 b2+4 a

Besides the probabilistic moments and coefficients, the entropy of random variablesand processes is also sometimes considered. Probabilistic entropy [155, 156] (contraryto that popular in thermodynamics) illustrates an uncertainty of occurrence of someevent in the next moment, so that entropy equal to 0 accompanies a probability equalto 1 (or 0) for any random experiment showing no randomness at all. If the countableset of random events has n elements associated with the probabilities pi for i = 1, . . . , n,


then the entropy in this space is defined uniquely by the following sum [155, 156]:

H (x) = −n∑

i=1

pi logr

(pi)

, (1.5)

where the logarithm basis r is the entropy unit; computational information theoryis naturally based on bits, where r = 2. This discrete definition restricts the valuesto the non-negative real numbers, where H(x) reaches maximum for two elements’random space with both events having the same probability −1 (like a bit of entropyper single throw with a geometrically regular coin). Its generalization to continuousvariables in case of the Gaussian distribution is

h (x) = −+∞∫

−∞

1

σ√

2πexp

(− (x − m)2

2σ 2

)log(

1

σ√

2πexp

(− (x − m)2

2σ 2

))dx. (1.6)

where m, σ denote traditionally its expectation and standard deviation.An integration process is carried out using classical normalization:

t = x − m√2σ

; dx =√

2σ dt (1.7)

and therefore

h (t) = −+∞∫

−∞

1

σ√

2πexp

(−t2) log(

1

σ√

2πexp

(−t2))√2σdt

= − 1√π

+∞∫−∞

exp(−t2) log

(1

σ√

2πexp

(−t2))dt

= − 1√π

log(

1

σ√

2π

) +∞∫−∞

exp(−t2)dt − 1√

π

+∞∫−∞

(−t2) exp(−t2)dt

= −√

π√π

log(

1

σ√

2π

)+ 1√

π

+∞∫−∞

t2 exp(−t2)dt

= − log(

1

σ√

2π

)+ 1√

π

√π

2= 1

2log(2πeσ 2) . (1.8)

The entropy formula remains unimplemented in most computer algebra systems,so this integral definition may appear useful in some engineering applications,especially with time series or stochastic processes. As could be expected in thecase of Gaussian variables it is entirely affected by the standard deviation, so thatthe proposed stochastic perturbation technique – with its perfect agreement withthe other numerical techniques in determination of the second-order probabilisticmoments – is a reliable computational tool to determine entropy also.


A very interesting problem for any state function and its uncertainty source wouldbe the entropy variation, and this can be defined through initial and final values as

�h = h(f (b))− h(b) = −

+∞∫−∞

gf (b) (x) log(

gf (b) (x))

dx

++∞∫

−∞gb(x) log

(gb(x)

)dx = log

(√2πeσ

(f (b)))− log

(√2πeσ (b)

)

= log

(σ(f (b))

σ (b)

). (1.9)

This entropy change shows whether the uncertainty may be amplified by thegiven boundary value problem, preserved or damped. The most interesting case,from a probabilistic point of view, would be �h = 0, which can be interpreted as noinfluence of the problem solution method or the problem itself on the initial randomdispersion.

Now let us focus on the generalized stochastic perturbation technique – the mainphilosophy of this method is to expand all state parameters and the response functionsin an initially deterministic problem (heat conductivity, heat capacity, temperature,and its gradient as well as the material density) using a given-order Taylor series withrandom coefficients. It is provided by the following representation of the randomfunction u(b) with respect to its parameter b around its mean value [74, 81]:

u(b) = u0 (b0)+ ε∂u(b)∂b

∣∣∣∣b=b0

�b + · · · + εn

n!∂nu(b)∂bn

∣∣∣∣b=b0

�bn, (1.10)

where ε is a given small perturbation (usually taken equal to 1), while the nth-ordervariation is given as follows:

εn�bn = (δb)n = εn (b − b0)n . (1.11)

The expected values can be derived exactly with use of the tenth-order expansionfor Gaussian variables as

E [u(b)] = u0 (b0)+ ε2

2∂2u(b)∂b2 µ2(b)

+ ε4

4!∂4u(b)∂b4 µ4(b) + ε6

6!∂6u(b)∂b6 µ6(b) + ε8

8!∂8u(b)∂b8 µ8(b) + ε10

10!∂10u(b)∂b10 µ10(b)

(1.12)

for any natural m with µ2m being central probabilistic moment of 2mth order. It isobtained via substitution of an expansion (1.10) into the definition (1.1), by droppingoff all odd order terms and integration of all the remaining order variations. It returnseven order central probabilistic moments of variable b as well as deterministic


odd order partial derivatives with respect to this b at its mean value. Usually,according to some previous computational convergence studies, we may limit thisexpansion to tenth order but consecutively for all moments of interest here. Quitesimilar considerations lead to the expressions for higher moments, like the variance,for instance:

Var (u(b)) = µ2 (u(b)) =+∞∫

−∞(u(b) − E [u(b)])2 gb(x)dx

= µ2(b)(

∂u(b)∂b

)2

+ µ4(b)

{14

(∂2u(b)∂b2

)2

+ 13

∂3u(b)∂b3

∂u(b)∂b

}

+ µ6(b)

{1

36

(∂3u(b)∂b3

)2

+ 124

∂4u(b)∂b4

∂2u(b)∂b2 + 1

60∂5u(b)∂b5

∂u(b)∂b

}

+ µ8(b)

{1

576

(∂4u(b)∂b4

)2

+ 1360

∂5u(b)∂b5

∂3u(b)∂b3 + 1

2520∂7u(b)∂b7

∂u(b)∂b

+ 1720

∂6u(b)∂b6

∂2u(b)∂b2

}

+ µ10(b)

{1

14400

(∂5u(b)∂b5

)2

+ 140320

∂8u(b)∂b8

∂2u(b)∂b2 + 1

8640∂6u(b)∂b6

∂4u(b)∂b4

}

+ µ10(b){

115120

∂7u(b)∂b7

∂3u(b)∂b3 + 1

181440∂9u(b)∂b9

∂u(b)∂b

}. (1.13)

One may notice that each component corresponds to the next consecutive order inEquation (1.12), while a linear increase of the components is noticed from each orderto the next one for the variance.

The third probabilistic moment may be recovered from this scheme as

µ3 (u(b)) =+∞∫

−∞(u(b) − E [u(b)])3 gb(x)dx = 3

2µ4(b)

(∂u(b)∂b

)2∂2u(b)∂b2

+ µ6(b)

{18

(∂2u(b)∂b2

)3

+ 12

∂u(b)∂b

∂2u(b)∂b2

∂3u(b)∂b3 + 1

8

(∂u(b)∂b

)2∂4u(b)∂b4

}

+ µ8(b){

124

∂u(b)∂b

∂3u(b)∂b3

∂4u(b)∂b4 + 1

40∂u(b)∂b

∂2u(b)∂b2

∂5u(b)∂b5

+ 1240

(∂u(b)∂b

)2∂6u(b)∂b6

}


+ µ8(b)

{132

(∂2u(b)∂b2

)2∂4u(b)∂b4 + 1

24

(∂3u(b)∂b3

)2∂2u(b)∂b2

}

+ µ10(b){

1480

∂u(b)∂b

∂4u(b)∂b4

∂5u(b)∂b5 + 1

1680∂u(b)∂b

∂2u(b)∂b2

∂7u(b)∂b7

+ 1720

∂u(b)∂b

∂3u(b)∂b3

∂6u(b)∂b6

}

+ µ10(b)

{1

240∂2u(b)∂b2

∂3u(b)∂b3

∂5u(b)∂b5 + 1

13440

(∂u(b)∂b

)2∂8u(b)∂b8

+ 1960

(∂2u(b)∂b2

)2∂6u(b)∂b6

}

+ µ10(b)

{1

384

(∂4u(b)∂b4

)2∂2u(b)∂b2 + 1

288

(∂3u(b)∂b3

)2∂4u(b)∂b4

}(1.14)

while the fourth probabilistic moment computation proceeds with use of the follo-wing formula:

µ4 (u(b)) =+∞∫

−∞(u(b) − E [u(b)])4 gb(x)dx

= µ4(b)(

∂u(b)∂b

)4

+ µ6(b)

{32

(∂u(b)∂b

)2 (∂2u(b)∂b2

)2

+ 23

(∂u(b)∂b

)3∂3u(b)∂b3

}

+ µ8(b)

{1

16

(∂2u(b)∂b2

)4

+ 130

(∂u(b)∂b

)3∂5u(b)∂b5 + 1

6

(∂u(b)∂b

)2 (∂3u(b)∂b3

)2}

+ µ8(b)

{14

(∂u(b)∂b

)2∂2u(b)∂b2

∂4u(b)∂b4 + 1

2∂u(b)∂b

(∂2u(b)∂b2

)2∂3u(b)∂b3

}

+ µ10(b)

{1

1260

(∂u(b)∂b

)3∂7u(b)∂b7 + 1

96

(∂u(b)∂b

)2 (∂4u(b)∂b4

)2

+ 154

∂u(b)∂b

(∂3u(b)∂b3

)3}

+ µ10(b)

{1

48∂4u(b)∂b4

(∂2u(b)∂b2

)3

+ 124

(∂2u(b)∂b2

)2 (∂3u(b)∂b3

)2

+ 112

∂u(b)∂b

∂2u(b)∂b2

∂3u(b)∂b3

∂4u(b)∂b4

}

+ µ10(b)

{1

60

(∂u(b)∂b

)2∂3u(b)∂b3

∂5u(b)∂b5 + 1

120

(∂u(b)∂b

)2∂2u(b)∂b2

∂6u(b)∂b6

}

+ µ10(b)140

∂u(b)∂b

(∂2u(b)∂b2

)2∂5u(b)∂b5 . (1.15)


Of course, the higher probabilistic moment, the larger Taylor expansion and thefaster increase of the components number corresponding to the neighboring ordercentral moments.

The central moments of the Gaussian variable b may obviously be simply recoveredhere as

µp(b) ={

0; p = 2k + 1{σ (b)

}p (p − 1)!!; p = 2k

(1.16)

for any natural k ≤ 1. As one may suppose, the higher-order moments we needto compute the higher-order perturbations need to be included in all formulas, sothat the complexity of the computational model grows non-proportionally togetherwith the precision and size of the output information needed. Once we take the

polynomial f (b) =10∑

i=1cib

i, then its general perturbation-based formula for the tenth-

order expectation equals

E[f (b)] = 1

2c1 + 1

4c2 + 1

8c3 + 1

16c4 + 1

32c5 + 1

64c6 + 1

128c7 + 1

256c8 + 1

512c9 + 1

1024c10

+ α2(b)(

14

c2 + 38

c3 + 38

c4 + 516

c5 + 1564

c6 + 21128

c7 + 764

c8 + 9128

c9 + 451024

c10

)

+ α4(b)(

316

c4 + 1532

c5 + 4564

c6 + 105128

c7 + 105128

c8 + 189256

c9 + 315512

c10

)

+ α6(b)(

105128

c7 + 10564

c8 + 315128

c9 + 1575512

c10

)

+ α8(b)(

105256

c8 + 945512

c9 + 47251024

c10

)+ α10(b)

9451024

c10. (1.17)

The variance, third and fourth probabilistic moments of this function, consideringtheir lengths, are omitted here and may be found in the Maple source files locatedon the book’s website.

It is obvious that the symmetric probability density functions do not requirefull expansions, but for the general distribution and specifically non-symmetricdistributions such as the lognormal, we need to complete them with odd-orderterms. These additional terms are specified below:

E′ [u(b)] = ε∂u(b)∂b

µ1(b) + ε3

3!∂3u(b)∂b3 µ3(b)

+ ε5

5!∂5u(b)∂b5 µ5(b) + ε7

7!∂7u(b)∂b7 µ7(b) + ε9

9!∂9u(b)∂b9 µ9(b) (1.18)

for the variances:

Var′ (u(b)) = µ′2 (u(b))

= µ3(b)∂u(b)∂b

∂2u(b)∂b2 + µ5(b)

{112

∂4u(b)∂b4

∂u(b)∂b

+ 16

∂3u(b)∂b3

∂2u(b)∂b2

}


+µ7(b){

172

∂4u(b)∂b4

∂3u(b)∂b3 + 1

120∂5u(b)∂b5

∂2u(b)∂b2 + 1

360∂6u(b)∂b6

∂u(b)∂b

}

+µ9(b){

15040

∂7u(b)∂b7

∂2u(b)∂b2 + 1

20160∂8u(b)∂b8

∂u(b)∂b

}

+µ9(b){

11440

∂5u(b)∂b5

∂4u(b)∂b4 + 1

2160∂6u(b)∂b6

∂3u(b)∂b3

}(1.19)

in case of the third central probabilistic moment:

µ′3 (u(b)) = 1

3µ3(b)

(∂u(b)∂b

)3

+ µ5(b)

{34

(∂2u(b)∂b2

)2∂u(b)∂b

+ 12

(∂u(b)∂b

)2∂3u(b)∂b3

}

+ µ7(b)

{1

40

(∂u(b)∂b

)2∂5u(b)∂b5 + 1

12∂u(b)∂b

(∂3u(b)∂b3

)2}

+ µ7(b)

{18

∂3u(b)∂b3

(∂2u(b)∂b2

)2

+ 18

∂u(b)∂b

∂2u(b)∂b2

∂4u(b)∂b4

}

+ µ9(b)

{1

216

(∂3u(b)∂b3

)3

+ 11680

(∂u(b)∂b

)2∂7u(b)∂b7 + 1

192∂u(b)∂b

(∂4u(b)∂b4

)2}

+ µ9(b)

{1

160

(∂2u(b)∂b2

)2∂5u(b)∂b5 + 1

48∂2u(b)∂b2

∂3u(b)∂b3

∂4u(b)∂b4

}

+ µ9(b){

1120

∂u(b)∂b

∂3u(b)∂b3

∂5u(b)∂b5 + 1

240∂u(b)∂b

∂2u(b)∂b2

∂6u(b)∂b6

}(1.20)

as well as the fourth one:

µ′4 (u(b)) = 2µ5(b)

(∂u(b)∂b

)3∂2u(b)∂b2

+ µ7(b)

{16

∂4u(b)∂b4

(∂u(b)∂b

)3

+ 12

∂u(b)∂b

(∂2u(b)∂b2

)3

+(

∂u(b)∂b

)2∂2u(b)∂b2

∂3u(b)∂b3

}

+ µ9(b)

{1

180

(∂u(b)∂b

)3∂6u(b)∂b6 + 1

12

(∂2u(b)∂b2

)3 (∂3u(b)∂b3

)2

+ 112

(∂u(b)∂b

)2∂3u(b)∂b3

∂4u(b)∂b4

}+ µ9(b)

∂u(b)∂b

{120

∂u(b)∂b

∂2u(b)∂b2

∂5u(b)∂b5

+18

∂4u(b)∂b4

(∂2u(b)∂b2

)2

+ 16

∂2u(b)∂b2

(∂3u(b)∂b3

)2}. (1.21)


The situation becomes definitely more complicated when we consider a prob-lem with multiple random variables, let’s say p random variables being totallyuncorrelated – we vectorize these variables here as br for r = 1, . . . , p. Then the Taylorexpansion with random coefficients proposed in Equation (1.10) is provided for allthese components as

u(br) = u0 (b0

r)+ ε

∂u(br)

∂br

∣∣∣∣∣br=b0

r

�br + · · · + εn

n!∂nu

(br)

∂bnr

∣∣∣∣∣br=b0

r

�bnr . (1.22)

The most fundamental difference is that the zeroth-order component is calculatedonly once – for the mean values of the design vector components, but higher-orderterms include partial derivatives of the response function with respect to all these pcomponents separately. So, the tenth-order expansion, instead of 11 components forthe single input random variable, will contain 10p + 1 independent terms. In viewof the above, the expectation for the structural response is calculated as (where thesummation convention is replaced for brevity with a classical sum)

E[u(br)] = u0 (b0

r)+

r∑p=1

ε2

2

∂2u(

bp

)∂b2

pµ2

(bp

)+

r∑p=1

ε4

4!

∂4u(

bp

)∂b4

pµ4

(bp

)

+r∑

p=1

ε6

6!

∂6u(

bp

)∂b6

pµ6

(bp

)+

r∑p=1

ε8

8!

∂8u(

bp

)∂b8

pµ8

(bp

)

+r∑

p=1

ε10

10!

∂10u(

bp

)∂b10

pµ10

(bp

). (1.23)

Therefore, following this idea it is relatively easy to extend Equations (1.13)–(1.15)with the additional summation procedure over the independent components of theinput random variables vector to get multi-parametric equations for the variancesas well as the third and fourth central probabilistic moments. As one can realize,the correlation effect in these expansions will result in cross-correlations (of higherorder also) between all the components of the vector br. It yields, for the second-orderexpansion of three random variables after Equation (1.23),

E[u(b1, b2, b3

)] = u0 (b01, b0

2, b03)+ ∂2u

(b1)

∂b21

µ2(b1)+ ∂2u

(b2)

∂b22

µ2(b2)+ ∂2u

(b3)

∂b23

µ2(b3)

+ ∂u(b1)

∂b1

∂u(b2)

∂b2Cov

(b1, b2

)+ ∂u(b1)

∂b1

∂u(b3)

∂b3Cov

(b1, b3

)

+ ∂u(b2)

∂b2

∂u(b3)

∂b3Cov

(b2, b3

)(1.24)


where Cov(b1, b2) stands for the covariance matrix of two random quantities definedclassically as [34]

Cov(b1, b2

) =+∞∫

−∞

(b1 − E

[b1]) (

b2 − E[b2])

gb1b2(x1,x2)dx, (1.25)

replaced frequently with the non-dimensional and normalized correlation coefficientintroduced as

ρ(b1, b2

) = Cov(b1, b2

)σ(b1)σ(b1)

=

+∞∫−∞

(b1 − E

[b1]) (

b2 − E[b2])

gb1 b2(x1, x2)dx

√√√√√+∞∫

−∞

(b1 − E

[b1])2 gb1

(x1)dx ×+∞∫

−∞

(b2 − E

[b2])2 gb2

(x2)dx

(1.26)

taking values − 1 ≤ ρ(b1, b2) ≤ 1 only. Of course, gb1 b2

(x1, x2

)denotes here the joint

probability density function of the variables b1 and b2. The basic problem with higher-order perturbation terms is the necessity of including higher-order cross-correlationsbetween all input random variables. This is not due to the mathematical and/ornumerical level of complexity of equations for all basic random characteristics, butthe lack of practical engineering knowledge about these correlations. Usually, thisknowledge reduces to the ordinary covariance of two or more random parameters,which can be a subject or the result of some statistical regression models.

As seen above, the most important numerical issue is determination of the partialderivatives of the state function u(b) with respect to the input parameter b and,depending on the case study, we can apply an analytical technique – fully imple-mented in the computer algebra system, a semi-analytical approach – a combinationof the symbolic calculus with some other discrete technique implementation soft-ware (like the finite element method (FEM) solver) or, finally, just a typical numericalsolution provided entirely by this solver itself. Nevertheless, we need to consider thefollowing equation system:

L(b)u(b) = f(b), (1.27)

where L(b) usually represents the main system matrix, f(b) includes the boundary con-ditions imposed on the system, while u(b) is the structural response. More advancedproblems, like transient heat transfer or the dynamical equilibrium equations, areconsidered further in this context – see Chapter 2. According to the main philosophy


proposed above, we provide the nth-order expansion of both sides of this statementto get

ε0L0 (b0)+n∑

j=1

εj ∂jL(b)∂bj (�b)j

ε0u0 (b0)+

n∑j=1

εj ∂ju(b)∂bj (�b)j

= f0 (b0)+n∑

j=1

εj ∂jf(b)∂bj (�b)j . (1.28)

After multiplication of the left-hand side (LHS) in Equation (1.28), we collectcomponents of same order of the perturbation parameter ε to arrive at the increasing-order hierarchical equilibrium equations

ε0 : L0(b0)

u0(b0) = f0

(b0)

ε1 :∂L(b)∂b

u0(b) + L0(b)∂u(b)∂b

= ∂f(b)∂b

ε2 :∂2L(b)∂b2 u0(b) + 2

∂L(b)∂b

∂u(b)∂b

+ L0(b)∂2u(b)∂b2 = ∂2f(b)

∂b2

. . .

εn :n∑

j=0

n

j

∂n−jL(b)

∂bn−j

∂ ju(b)∂bj = ∂nf(b)

∂bn .

(1.29)

Further, leaving the highest-order derivative of the solution u(b) on the LHS onlyand using simple algebra, one gets zeroth-, first-, second-, and finally nth-orderequations

u0(b0) = {L0

(b0)}−1 f0

(b0)

∂u(b)∂b

= {L0 (b0)}−1{

∂f(b)∂b

− ∂L(b)∂b

u0(b)}

∂2u(b)∂b2 = {L0 (b0)}−1

{∂2f(b)∂b2 − ∂2L(b)

∂b2 u0(b) − 2∂L(b)∂b

∂u(b)∂b

}

. . .

∂nu(b)∂bn = {L0 (b0)}−1

∂nf(b)

∂bn −n∑

j=1

n

j

∂n−jL(b)

∂bn−j

∂ ju(b)∂bj

.

(1.30)

Hence it is apparent that the zeroth-order equation returns a zeroth-order solution,as in a deterministic problem, which, inserted into the first-order equation next givesthe first-order partial derivative of the function u(b). Then, both-order terms for thisfunction inserted into the second-order equation result in ∂2u(b)

∂b2 , and so on, untilthe highest-order partial derivative of this function is determined. It is characteristic


that during computational implementation only the main matrix of the system (i.e.,stiffness or heat conduction) needs to be inverted and only once during the entiresolution procedure. A second observation is that small modifications of the existingsolvers are necessary since the right-hand side (RHS) vector is modified, while theLHS procedures typical for a solution of the linear algebraic equation system (likevarious decompositions of the main system matrix) remain the same. Usually, whenL = L(b), f may remain deterministic and vice versa, and then we can make somefurther simplifications. The following hold:

1. For f = f(b) and L �= L(b)

u0(b0) = {L0

(b0)}−1 f0

(b0)

∂u(b)∂b

= {L0 (b0)}−1 ∂f(b)∂b

∂2u(b)∂b2 = {L0 (b0)}−1 ∂2f(b)

∂b2

. . .

∂nu(b)∂bn = {L0 (b0)}−1 ∂nf(b)

∂bn .

(1.31)

Calculation of the expectation for u(b) in this case seems to be very straightfor-ward and

E [u(b)] = {L0 (b0)}−1

f0(b) +

n∑j=1

∂ jf(b)∂bj µj(b)

(1.32)

where further modifications result from Equation (1.16) – zeroing of the oddcentral probabilistic moments µj(b) on the RHS if b is Gaussian.

2. For f �= f(b) and L = L(b), Equation (1.21) reduces to

u0(b0) = {L0

(b0)}−1 f0

(b0)

∂u(b)∂b

= − {L0 (b0)}−1 ∂L(b)∂b

u0(b)

∂2u(b)∂b2 = − {L0 (b0)}−1

{∂2L(b)∂b2 u0(b) + 2

∂L(b)∂b

∂u(b)∂b

}

. . .

∂nu(b)∂bn = − {L0 (b0)}−1

n∑j=1

n

j

∂n−jL(b)

∂bn−j

∂ ju(b)∂bj .

(1.33)

The very special case appearing relatively frequently in a number of engineeringproblems is a linear dependence L(b) from b and then we may drop some further


terms from these equations, for example

u0(b0) = {L0

(b0)}−1 f0

(b0)

∂u(b)∂b

= − {L0 (b0)}−1 ∂L(b)∂b

u0(b)

∂2u(b)∂b2 = −2

{L0 (b0)}−1 ∂L(b)

∂b∂u(b)∂b

. . .

∂nu(b)∂bn = −n

{L0 (b0)}−1 ∂L(b)

∂b∂n−1u(b)∂bn−1 .

(1.34)

Embedding the partial derivatives into Equations (1.12)–(1.15) we recover allnecessary probabilistic characteristics of the structural response. This is usuallyverified against the Monte Carlo simulation, which, as is well known, is treated as theexact solution and is based on a series of computational experiments with randomlygenerated random spaces of additional input parameters. We need to rememberthat independently of the implementation type of this method (crude simulationor stratified sampling, for instance), the estimators given below are convergent fornumber of experiments tending to infinity and that the random number generatorsalso have their deficiencies. Therefore, it is very reasonable to compare first the givensimulation tool with the probabilistic analytical results for various types of expectedoutput distributions. The computations according to the Monte Carlo simulationsfollow the well-known, most reliable estimators of the expected value, variance, aswell as the nth-order central probabilistic moment. The following hold [5]:

E [u(b)] = 1M

M∑i=1

u(i)(b), (1.35)

Var (u(b)) = 1M − 1

M∑i=1

(u(i)(b) − E [u(b)]

)2, (1.36)

µn (u(b)) = 1M

M∑i=1

(ui(b) − E [u(b)]

)n, (1.37)

and having estimated these moments we employ Equation (1.4) to calculate thecoefficient of variation, skewness, and kurtosis shown in the next two sections (M isthe total number of random trials). Since most of the verifications are providedvia the simulation tool implemented in the computer algebra system Maple,further discussion on statistical aspects may be found in the literature. Very simpleillustrations are given only to enable readers to carry out a simple verification of theirown statistical simulations in this package – it is confirmed below that the standarderror of higher-moments determination for the same number of random trials isdefinitely larger than for the first two, even for a Gaussian distribution.


>restart: with(Statistics):>X:=RandomVariable(Normal(10,1)): A:=Sample(X,10000):>Bootstrap(Mean,X,replications=10000,output=[‘value’,‘standarderror’]);

[10.00033977, 0.03184353072]

>Bootstrap(Mean,A,replications=10000,output=[‘value’,‘standarderror’]);

[9.991094312, 0.01001230197]

>Mean(X);10

>Bootstrap(Variance,X,replications=10000,output=[‘value’,‘standarderror’]);

[0.9997266603, 0.0451189936]

>Bootstrap(Variance,A,replications=10000,output=[‘value’,‘standarderror’]);

[0.9992068113, 0.01406359292]

>Variance(X);1

>Bootstrap(Skewness,X,replications=10000,output=[‘value’,‘standarderror’]);

[−0.0001756788883, 0.07814788973]

>Bootstrap(Skewness,A,replications=10000,output=[‘value’,‘standarderror’]);

[0.01192136730, 0.02285198363]

>Skewness(X);0

>Bootstrap(Kurtosis,X,replications=10000,output=[‘value’,‘standarderror’]);

[2.992500847, 0.1545298225]

>Bootstrap(Kurtosis,A,replications=10000,output=[‘value’,‘standarderror’]);

[2.981760768, 0.04047992704]

>Kurtosis(X);3


We study an interrelation of the probabilistic techniques described briefly abovefor the two computational examples below – for the relatively simple, well-knownengineering problems where an analytical expression for u(b) does exist, but notalways can be simply integrated according to classical definitions.

Example 1.1: Simple tension of an elastic barWe consider first a linear elastic rod with constant cross-sectional area A, Young’smodulus e, length l under tension on the RHS with force P. Extension of this bar is thestate function dependent on the Gaussian variable e. This problem has a well-knownsolution from the strength of materials, that is, u (e) = Pl

eA and its second-order second-moment (SOSM) solution may be found in [112]. It is clear from the very beginningthat u(e) has an inverse Gaussian distribution, but since usually output distributionscannot be predicted straightforwardly without such an analytical solution, we focuson the expectations, coefficients of variation, skewness, and kurtosis all computedas functions of the input coefficient of variation. The numerical data in this testare taken as E[e] = 210 GPa, A = 0.0001 m2, l = 10 m, P = 10 kN, while the resultsof stochastic perturbation technique (SPT)-based (full tenth-order method) andMonte Carlo simulation (MCS)-based (M = 5 × 105 random trials) computationsare presented in Figures 1.1 and 1.2. We notice a perfect agreement of the first twoprobabilistic characteristics in the entire range of the input coefficient of variation.The worst coincidence is obtained in case of skewness, while kurtosis agrees well upto α(e)<0.15.

Example 1.2: Simply supported elastic beam under uniform transverseloadThe second example concerns a deterministically homogeneous and linear elasticstainless steel beam simply supported at both ends and loaded with a constantdistributed load. Its parameters are defined as e = 210 GPa, J = 0.0001 m4, E[l] = 10 m,

0.00495

0.00490

0 0.05 0.10a(e)

E(u

)

a(u

)

0.15 0.200

0.05

0.10

0.15

0.20

0 0.05 0.10a(e)

0.15 0.20

0.00485

0.00480

SPT solutionMCS solution

(a) (b)

Figure 1.1 Expected values (a) and coefficients of variation (b) for u (e (ω)) = Ple(ω)A


2

1.5

1

0.5

0 0.05 0.10

(a) (b)

a(e)

b(u

)

k(u

)

0.15 0.20 0 0.05 0.10

a(e)

0.15 0.200

2

4

6

8

10

12

14

16

18

SPT solutionMCS solution

Figure 1.2 Skewness (a) and kurtosis (b) for u (e (ω)) = Ple(ω)A

and q = 10 kN/m. Once more, we provide a visualization of the expectations, coeffi-cients of variation (Figure 1.3), skewness, and kurtosis (Figure 1.4) – all computed asfunctions of the input coefficient of variation for the beam length. Naturally, our statefunction is the maximum deflection at the mid-span of this beam widely used in civil

engineering as the basis for reliability analysis, u (l (ω)) = 5

384

q(l(ω))4

eJ. Now, analytical

calculations are relatively easy, so thanks to the additional internal functions of the

0.085

0.080

0.075

0.070

E(u

)

0.065

0.05 0.10 0.15

(a) (b)a(e)

0.20 0.25 0.050.1

0.2

0.3

0.4

0.5SPT solutionMCS solutionAM solution

a(u

)

0.6

0.7

0.8

0.9

0.10 0.15a(e)

0.20 0.25

Figure 1.3 Expectations (a) and coefficients of variation (b) for u(l (ω)

) = 5

384

q(l(ω))4

eJ


2.5

2

1.5

b(u

)

k(u

)

1

0.5

0.05 0.10 0.15

(a) (b)

a(e)

0.20 0.25

SPT solutionMCS solutionAM solution

0.05

1

2

3

4

5

6

7

8

9

0.10 0.15

a(e)

0.20 0.25

Figure 1.4 Skewness (a) and kurtosis (b) for u(l (ω)

) = 5

384

q(l(ω))4

eJ

Maple system, we compare the triples (analytical (AM) versus perturbation (SPT)versus simulation (MCS) methods) of basic probabilistic moments and coefficients.

In this particular case all the methods coincide perfectly for all probabilisticcharacteristics, which leads to the conclusion that the perturbation method seemsto be very efficient in polynomial transforms of random variables; this efficiency ismuch worse once the input random variable is to be inversed.

Let us focus now on the convergence of the generalized stochastic perturbationtechnique implemented in conjunction with the polynomial-based response functionmethod. Using classical theorems on mathematical analysis we will show that thefirst four probabilistic moments and coefficients converge together with the analysisorder. Let us start from the expectations and adopt the notation that the right lowerindex being the natural number stands for the perturbation order. We will omit theabsolute values because of the fact that higher-order formulas are enormously long,so that this operator may be visually lost during additional mathematical derivations.We postpone also truncation of the additional probability density functions in theintegration below, because this mostly does not affect the main idea behind thedemonstration. Partial differentiation of the response function u(b) is provided asbefore at the mean values of the input random parameter b, whose symbol is omittedhere also. It holds that the n + 1 order rest equal to

�n+1 (E [u(b)]) = En+1 [u(b)] − En [u(x)]

=+∞∫

−∞

n+1∑k=0

εk ∂ku(b)∂bk

(�b)k gb(x)dx −+∞∫

−∞

n∑k=0

εk ∂ku(b)∂bk

(�b)k gb(x)dx

=+∞∫

−∞

εn+1 ∂n+1u(b)∂bn+1

(�b)n+1 gb(x). (1.38)


Using as above a unitary perturbation parameter and inserting a definition of thenth central probabilistic moment, one obtains

�n+1 (E [u(b)]) = ∂n+1u(b)∂bn+1 µn+1(b). (1.39)

because n + 1 partial derivative of u(b) with respect to b is evaluated at b = b0 intraditional deterministic way and may be excluded from an integrations process overthe given P.D.F. as simply constant.

So, it remains clear that unconditional convergence of the expected value approx-imation is equivalent to zeroing of the (n + 1)th-order partial derivative of thefunction u(b) (with finite value of the n+1 central moment of b itself), which meansthat it is quite enough to adopt it as the nth-order polynomial, as is exploited infurther computational experiments. A similar derivation for the variance is a littlemore complex – we try to reorganize the first integral to group the nth-order termsand the rest by reducing the zeroth-order components in both integrals with theirexpectations, so that we have

�n+1 (Var (u(b))) = Varn+1 (u(b)) − Varn (u(b))

=+∞∫

−∞

{n+1∑k=0

εk ∂ku(b)∂bk (�b)k − E [u(b)]

}2

gb(x)dx

−+∞∫

−∞

{n∑

k=0


}2

gb(x)dx

=+∞∫

−∞

{n+1∑k=1

εk ∂ku(b)∂bk (�b)k

}2

gb(x)dx −+∞∫

−∞

{n∑

k=1


}2

gb(x)dx

=+∞∫

−∞

{n∑

k=1


}2

gb(x)dx −+∞∫

−∞

{n∑

k=1


}2

gb(x)dx

+ 2

+∞∫−∞

n∑k=1


εn+1 ∂n+1u(b)∂bn+1 (�b)n+1 gb(x)dx

++∞∫

−∞

{εn+1 ∂n+1u(b)

∂bn+1 (�b)n+1}2

gb(x)dx

= ∂n+1u(b)∂bn+1

2

+∞∫−∞

n∑k=1


εn+1 (�b)n+1 gb(x)dx

++∞∫

−∞ε2(n+1) ∂

n+1u(b)∂bn+1 (�b)2(n+1) gb(x)dx

. (1.40)


Using ε = 1 and a definition of the central probabilistic moment, we arrive at thefinal conclusion that

�n+1 (Var (u(b))) = ∂n+1u(b)∂bn+1

2

+∞∫−∞

n∑k=1

∂ku(b)∂bk (�b)k

(�b)n+1 gb(x)dx

+∂n+1u(b)∂bn+1 µn+1 (b)

)

= ∂n+1u(b)∂bn+1

(2

n∑k=1

∂ku(b)∂bk

µk+n+1(b) + ∂n+1u(b)∂bn+1 µn+1 (b)

)(1.41)

which has analogous properties as the error for the expected values – since highercentral probabilistic moments exist and are bounded, the governing factor for theseries convergence is a higher-order partial derivative of the state function withrespect to the random input parameter. Now we proceed with the third centralprobabilistic moment, also from its definition. Denoting by µ3,n (u(b)) the nth-orderstochastic perturbation-based approximation of this moment yields

�n+1(µ3 (u(b))

) = µ3,n+1 (u(b)) − µ3,n (u(b))

=+∞∫

−∞

{n+1∑k=0


}3

gb(x)dx

−+∞∫

−∞

{n∑

k=0


}3

gb(x)dx

=+∞∫

−∞

{n+1∑k=1


}3

gb(x)dx −+∞∫

−∞

{n∑

k=1


}3

gb(x)dx. (1.42)

Further, we employ the Pascal triangle to divide the first series integral into ncomponents and the last one. Substituting for ε = 1 we obtain

�n+1(µ3 (u(b))

)

=+∞∫

−∞

{n∑

k=1


}3

gb(x)dx −+∞∫

−∞

{n∑

k=1


}3

gb(x)dx

+ 3

+∞∫−∞

{n∑

k=1


}2∂n+1u(b)∂bn+1 (�b)n+1 gb(x)dx

+ 3

+∞∫−∞

n∑k=1


{∂n+1u(b)∂bn+1 (�b)n+1

}2

gb(x)dx


++∞∫

−∞

n∑k=1

{∂n+1u(b)∂bn+1 (�b)n+1

}3

gb(x)dx

= ∂n+1u(b)∂bn+1

3

+∞∫−∞

{n∑

k=1


}2

(�b)n+1 gb(x)dx

+3∂n+1u(b)∂bn+1

n∑k=1

∂ku(b)∂bk

µk+2(n+1)(b)

)

+ ∂n+1u(b)∂bn+1

(∂n+1u(b)∂bn+1

)2

µ3(n+1)(b). (1.43)

Once more, independently of the summation and bounded moments includedin the brackets, the zeroing of the n (z + 1)th-order rest may be guaranteed bythe polynomial of order equal or less than the nth one. We need to realize thatan integration of the series consisting of higher-order partial derivatives as wellas perturbations of b itself around the mean value also returns almost the sameneighboring odd- and even-order moments, so that at least for a Gaussian distributionwe will obtain some zero-valued components there. Finally, we consider the fourth-order moment and it holds that

�n+1(µ4 (u(b))

) = µ4,n+1 (u(b)) − µ4,n (u(b))

=+∞∫

−∞

{n+1∑k=0


}4

gb(x)dx

−+∞∫

−∞

{n∑

k=0


}4

gb(x)dx

=+∞∫

−∞

{n+1∑k=1


}4

gb(x)dx −+∞∫

−∞

{n∑

k=1


}4

gb(x)dx. (1.44)

Almost identically as before, taking ε = 1 and reorganizing the components fromthe fourth power of the (n + 1)th order we obtain

�n+1(µ4 (u(b))

)

=+∞∫

−∞

{n∑

k=1


}4

gb(x)dx −+∞∫

−∞

{n∑

k=1


}4

gb(x)dx

+ 4

+∞∫−∞

{n∑

k=1


}3∂n+1u(b)∂bn+1 (�b)n+1 gb(x)dx


+ 6

+∞∫−∞

{n∑

k=1


}2 {∂n+1u(b)∂bn+1 (�b)n+1

}2

gb(x)dx

+ 4

+∞∫−∞

n∑k=1


{∂n+1u(b)∂bn+1 (�b)n+1

}3

gb(x)dx

++∞∫

−∞

{∂n+1u(b)∂bn+1 (�b)n+1

}4

gb(x)dx. (1.45)

Finally, we eliminate the perturbation parameter as not influencing the overallconvergence and try to get the total multiplier whose zeroing may give a desiredresult. It holds that

�n+1(µ4 (u(b))

) = 4∂n+1u(b)∂bn+1

+∞∫−∞

{n∑

k=1


}3

(�b)n+1 gb(x)dx

+ 4∂n+1u(b)∂bn+1

{∂n+1u(b)∂bn+1

}2 +∞∫−∞

n∑k=1


(�b)3(n+1) gb(x)dx

+ 6∂n+1u(b)∂bn+1

∂n+1u(b)∂bn+1

+∞∫−∞

{n∑

k=1


}2

(�b)2(n+1) gb(x)dx

+ ∂n+1u(b)∂bn+1

{∂n+1u(b)∂bn+1

}3 +∞∫−∞

(�b)4(n+1) gb(x)dx

= ∂n+1u (b)∂bn+1

4

+∞∫−∞

{n∑

k=1


}3

(�b)n+1 gb(x)dx

+4{

∂n+1u(b)∂bn+1

}2 n∑k=1

∂ku(b)∂bk

µk+3(n+1)(b)

)

+ ∂n+1u(b)∂bn+1

6

∂n+1u(b)∂bn+1

+∞∫−∞

{n∑

k=1

∂ku(b)∂bk (�b)k+n+1

}2

gb(x)dx

+{

∂n+1u(b)∂bn+1

}3

µ4(n+1)(b)

). (1.46)

Concluding these derivations we see that having the least-squares approximationof a certain order for the state function u(b) it is just enough to apply the perturbationtechnique only a single order higher, quite independent of the probabilistic momentunder consideration. Analysis of the convergence for the probabilistic coefficients is


carried out using the other well-known theorem in mathematical analysis. Let usconsider now the coefficient of variation, its (n + 1)th- combined with its nth-orderapproximation, and the following ratio:

δn+1 (α (u(b))) = αn+1 (u(b))αn (u(b))

=√

Varn+1 (u(b))En+1 [u(b)]

En [u(b)]√Varn (u(b))

. (1.47)

Now we add an artificial zero to the counter of the second fraction in thefollowing form (according to the order n of the response polynomial):

0 =+∞∫

−∞εn+1 ∂n+1u(b)

∂bn+1 (�b)n+1 gb(x)dx (1.48)

to get simply

δn+1 (α (u(b))) = αn+1 (u(b))αn (u(b))

=√


En [u(b)] ++∞∫

−∞εn+1 ∂n+1u(b)

∂bn+1 (�b)n+1 gb(x)dx

√Varn (u(b))

=√


En+1 [u(b)]√Varn (u(b))

=√

Varn+1 (u(b))Varn (u(b))

. (1.49)

This means that the coefficient of variation is convergent, together with a varianceof the random function u(b). Analogously we can proceed to study the skewnesscoefficient; we have

δn+1 (β (u(b))) = βn+1 (u(b))βn (u(b))

= µ3,n+1 (u(b))(Varn+1 (u(b))

) 32

(Varn (u(b))

) 32

µ3,n (u(b))

= µ3,n+1 (u(b))µ3,n (u(b))

{ (Varn (u(b))

)(Varn+1 (u(b))

)} 3

2n→∞−−−−→ 1 (1.50)

since the limits of both components in this product also tend together to 1, asdocumented above. The situation is quite similar for the coefficient of flatness andkurtosis, because

δn+1 (γ (u(b))) = βn+1 (u(b))βn (u(b))

= µ4,n+1 (u(b))(Varn+1 (u(b))

)2(Varn (u(b))

)2µ4,n (u(b))

= µ4,n+1 (u(b))µ4,n (u(b))

{ (Varn (u(b))

)(Varn+1 (u(b))

)}2

n→∞−−−−→ 1 (1.51)


0.12

0.10

0.08

0.06

0.04

E(f

)

0.02

00 0.05 0.10 0.15 0.20 0.25

a(b)

OE(2)OE(3)OE(4)OE(5)OE(6)OE(7)OE(8)OE(9)OE(10)OE(11)OE(12)OE(13)OE(14)OE(15)OE(16)OE(17)OE(18)OE(19)OE(20)

Figure 1.5 Probabilistic convergence of the expected values for f (b) =10∑i=1

bi ,

E[b] = 0.5

0.20

0.15

0.10

Var

(f)

0.05

00 0.05 0.10 0.15 0.20 0.25

a(b)

OV(2)OV(3)OV(4)OV(5)OV(6)OV(7)OV(8)OV(9)OV(10)OV(11)OV(12)OV(13)OV(14)OV(15)OV(16)OV(17)OV(18)OV(19)OV(20)

Figure 1.6 Probabilistic convergence of the variances for f (b) =10∑i=1

bi , E[b] = 0.5


1

0.8

0.6b

(f)

0.4

0.2

00 0.05 0.10 0.15 0.20 0.25

a(b)

OS(3)OS(4)OS(5)OS(6)OS(7)OS(8)OS(9)OS(10)OS(11)OS(12)OS(13)OS(14)OS(15)OS(16)OS(17)OS(18)OS(19)OS(20)

Figure 1.7 Probabilistic convergence of the skewness for f (b) =10∑i=1

bi , E[b] = 0.5

3

2

k(f

)

1

00 0.05 0.10 0.15 0.20 0.25

a(b)

OK(3)OK(4)OK(5)OK(6)OK(7)OK(8)OK(9)OK(10)OK(11)OK(12)OK(13)OK(14)OK(15)OK(16)OK(17)OK(18)OK(19)OK(20)

Figure 1.8 Probabilistic convergence of the kurtosis for f (b) =10∑i=1

bi , E[b] = 0.5


and, therefore, the kurtosis

�n+1 (γ (u(b))) = κn+1 (u(b)) − κn (u(b)) = γn+1 (u(b)) + 3 − γn (u(b)) − 3

= γn+1 (u(b)) − γn (u(b))n→∞−−−−→ 0. (1.52)

Numerical verification of this procedure is provided using as example the polyno-

mial f (b) =10∑

i=1bi, where b is a Gaussian random variable with E[b] = 0.5. Its coefficient

of variation is taken as the parameter in this study since the analyzed convergence isexpected to be faster for smaller values of this parameter (for lower-order approxima-tions, of course). We study all the aforementioned random moments and characteris-tics and verify numerically the differences �n+1 ((u(b))) for all n ≤ 20 using the 20th-order SPT. Computational results of this verification are provided in Figures 1.5–1.8in turn – for the expected values, coefficients of variation, skewness, and kurtosis.

1.2 Least-Squares Technique Description

Consider a set of m data points(

bi, u(i)β

)for β = 1, . . . , N, the non-linear continuous

function to be found: uβ = f (b) and a curve (approximating function) uβ = f(b, Dβ

),

which additionally depends on the parameters D(j)β , j = 1, . . . , n, where m ≥ n [9]. We

define also the residuals ri(uβ ):

ri(uβ

) = u(i)β − f

(bi, D(i)

β

)(1.53)

to finally determine the coefficients D(j)β from the minimization of the following

functional:

S (u) =m∑

i=1

r2i

(uβ

). (1.54)

This is carried out using the gradient method, so that

∂S (u)

∂D(j)β

= 2m∑

i=1

ri(uβ

) ∂f(

bi, D(i)β

)∂D(j)

β

= 0; j = 1, . . . , n. (1.55)

Since the LHS derivatives combine the independent variable and the parameters,so these gradient equations may not have a closed solution and some initial valuesmust be adopted for these parameters. They are finally determined via some iterativeapproximation as

D(j)β

∼= (k+1)D(j)β = (k)D(j)

β + �D(j)β (1.56)

where k denotes an iteration number and �D(j)β is the so-called shift vector. The

following linearization based on a first-order Taylor series representation is appliedat each iteration:


f(

bi, D(i)β

) ∼= f(

bi,(k)D(i)

β

)+

n∑j=1

∂f(

bi, D(i)β

)∂D(j)

β

(D(j)

β − (k)D(j)β

)

∼= f(

bi,(k)D(i)

β

)+

n∑j=1

Jβij �D(j)β . (1.57)

Since ∂ri

∂D(j)β

= −Jβij it holds that

ri(uβ

) = �u(i)β −

n∑s=1

Jβis�D(s)β ; �u(i)

β = u(i)β − f

(bi,

(k)D(i)β

). (1.58)

Substituting these expressions into the gradient Equation (1.55), one gets

−2m∑

i=1

Jβij

(�u(i)

β −n∑

s=1

Jβis�D(s)β

)= 0, (1.59)

which effectively becomes a system of linear equations

m∑i=1

n∑s=1

Jβij Jβ

is�D(s)β =

n∑i=1

Jβij �u(i)β , j = 1, . . . , n. (1.60)

Finally, we obtain(JTJ)�D = JT�u. (1.61)

Further determination of the probabilistic moments and characteristics is based ondetermination of the partial derivatives of the response function with respect to theinput random parameter b using the analytical form

dkfdbk

=k∏

i=1

(n − i) anbn−k +k∏

i=2

(n − i) an−1bn−(k+1) + · · · + an−k. (1.62)

Application of this method to the tensors needs small extensions only; observethe basic equations set on the example of the fourth-order tensor serving further inChapter 5 for a solution of the homogenization problem. Let us consider once morea data set of m trial points

(bi, C(eff )(i)

αβγ δ

)with α, β, γ , δ = 1, 2, 3, some non-linear continu-

ous function C(eff )αβγ δ = f (b) and a curve (approximating function) C(eff )

αβγ δ = f(b, Dαβγ δ

),

which additionally depends on n parameters D(j)αβγ δ , j = 1, . . . , n, where m ≥ n. The

upper index ‘‘eff ’’ relates the tensor under consideration to the homogenizationproblem only. We define the residuals ri

(C(eff )

αβγ δ

):

ri

(C(eff )

αβγ δ

)= C(eff )(i)

αβγ δ − f(

bi, D(i)αβγ δ

)(1.63)


to finally determine the components of the tensor Dαβγ δ from the minimization of

S =m∑

i=1

r2i

(C(eff )

αβγ δ

). (1.64)

We employ the formula

∂S

∂D(j)αβγ δ

= 2m∑

i=1

ri

(C(eff )

αβγ δ

) ∂f(

bi,D(i)αβγ δ

)∂D(j)

αβγ δ

= 0; j = 1, . . . , n. (1.65)

Analogously to the above, an iterative approximation is proposed as

D(j)αβγ δ

∼= (k+1)D(j)αβγ δ = (k)D(j)

αβγ δ + �D(j)αβγ δ (1.66)

where k denotes an iteration number and �D(j)αβγ δ stands for the shift vector. The

following linearization for the tensor is applied:

f(

bi, D(i)αβγ δ

) ∼= f(

bi,(k)D(i)

αβγ δ

)+

n∑j=1

∂f(

bi, D(i)αβγ δ

)∂D(j)

αβγ δ

(D(j)

αβγ δ − (k)D(i)αβγ δ

)

∼= f(

bi,(k)D(i)

αβγ δ

)+

n∑j=1

Jβij �D(j)αβγ δ. (1.67)

because ∂ri

∂D(j)αβγ δ

= −Jαβγ δ

ij and with no summation on α,β,γ ,δ, it holds that

ri

(C(eff )

αβγ δ

)= �C(eff )(i)

αβγ δ −n∑

s=1

Jαβγ δ

is �D(s)αβγ δ ; �C(eff )(i)

αβγ δ = C(eff )(i)αβγ δ − f

(bi,

(k)D(i)αβγ δ

).

(1.68)Substituting these expressions for the gradient Equation (1.65), they become

−2m∑

i=1

Jαβγ δ

ij

(�C(eff )(i)

αβγ δ −n∑

s=1

Jαβγ δ

is �D(s)αβγ δ

)= 0, (1.69)

which finally equals to

m∑i=1

n∑s=1

Jαβγ δ

ij Jαβγ δ

is �D(s)αβγ δ =

n∑i=1

Jαβγ δ

ij �C(eff )(i)αβγ δ , j = 1, . . . , n. (1.70)

We obtain in matrix notation as

(JTJ)

�D = JT�C(eff ). (1.71)


The observations may also be minimized using the weighted sum for the tensor asfollows:

S =m∑

i=1

Wiir2i

(C(eff )

αβγ δ

). (1.72)

Each element of the diagonal weights matrix W equals the reciprocal of themeasurement variance (completely independent of the variance of the input randomparameter in this problem). Then, the equation system Equation (1.71) becomes

(JTWJ)�D = JTW�C(eff ). (1.73)

Of course, some further numerical improvements of this algorithm are availablenow, but according to the numerical results obtained, they do not seem to benecessary. The sense and value of the least-square method approximation behindother, well-described numerical methods can be displayed with the following simplecomputational example.

Example 1.3: Comparison of various approximation techniquesLet us consider an approximation problem consisting of the set of trial pointswith values equal to l = [3.5E−8, 3.6E−8, 3.7E−8, 3.8E−8, 3.9E−8, 4.0E−8, 4.1E−8,4.2E−8, 4.3E−8] resulting in some structural response given by the values vectorf (l) = [7.4568E11, 7.2497E11, 7.0537E11, 6.8681E11, 6.6910E11, 6.5247E11, 6.3656E11,6.2140E11, 6.0695E11]. The goal is to provide a curvilinear continuous approximationof this function and to compare the first- and second-order partial derivatives withrespect to the independent parameter l [95].

The solution to this problem with Maple is visualized in Figures 1.9 and 1.10,showing in turn various approximations for the function f (l), its first and secondderivatives computed on the basis of the functions recovered via Newton, Lagrange,monomial, power, and least-squares methods (using a quadratic form only). Thetrial points set is given in Figure 1.9 using large dots, while the remaining datacorresponds to the comments in the legend.

As is apparent, practically all the methods within the numerical error range returnthe same functions, however the differences are significant for both first- and second-order derivatives. All the methods except the least-squares method (LSM) returnfunctions whose both derivatives demonstrate inacceptable fluctuations at both endsof the given computational domain. Even if the second-order partial derivativescomputed at the mean value of the design parameter l all have almost the same value,the first-order derivatives are totally different at this point; monomial and powermethods return a few percentage points higher value at this point, while Newton andLagrange algorithms agree rather well with the results obtained thanks to the least-squares algorithm. This last technique enables a determination of two very stablederivatives, which furthermore have the same values (for the second-order derivative)or the values with relatively small fluctuations (for the first-order derivative) over theentire variability interval of l, quite contrary to the first four methods. Consideringthis result, we consequently use the LSM technique to determine all the response


7.4 × 1011

7.2 × 1011

7 × 1011

6.8 × 1011

6.6 × 1011

f (l)

6.4 × 1011

6.2 × 1011

6 × 1011

3.5 × 10−8

LSTNewton Lagrange monomial power

3.8 × 10−8 4 × 10−8

l4.3 × 10−8

Figure 1.9 Various approximations for the response function f (l).Reproduced with permission from Springer

−0.90

−0.95

−1.05

f(l)

f(l)

−1.10

3.5 × 10−8

NewtonLST

Lagrange monomial power NewtonLST

Lagrange monomial power

3.9 × 10−83.7 × 10−8 4.1 × 10−8

l l

4.3 × 10−8

1 × 1027

−1 × 1027

−2 × 1027

−3 × 1027

−4 × 1027

−5 × 1027

−6 × 1027

−7 × 1027

−8 × 1027

3.5 × 10−8 3.8 × 10−8 4 × 10−8 4.3 × 10−8

0

−1

Figure 1.10 Various approximations for the first and second partial derivatives of theresponse function f (l) with respect to l. Reproduced with permission from Springer


functions – this is especially advised in case when the general form of structuralresponse cannot be predicted from previous examples or theoretical considerations,otherwise polynomial interpolation implemented in Maple may be recommended.The second issue, shown in Equation (1.73), is the weighting procedure and its generalinfluence on the LSM approximation results for the stochastic perturbation methodand the particular moments. This is explained well within the next example, whereour main goal is to find such a weights distribution over a computational domain ofthe design parameter to assure the best stability of these moments and coefficients aswell as their fastest probabilistic convergence. We expect that a systematic increase ofthe perturbation order applied should result in a decrease of the differences betweenthe results obtained for these increasing order terms.

Example 1.4: Probabilistic moments efficiency in weightedleast-squares techniqueThe main aim of this computational study is to look at the influence of the weightingprocedure on the probabilistic moments computed via the least-squares approxima-tion of a certain set of trial data points; this is done for the homogenization problemdescribed in detail in the last chapter of this book [85]. Let us consider a derivationof the expected values, coefficients of variation, skewness, and coefficients of flatnessfor the set of trial points b = [0.29, 0.30, 0.31, 0.32, 0.33, 0.34, 0.35, 0.36, 0.37, 0.38, 0.39]representing an input random variable, which results in some structural responsegiven by the first components of the fourth-order tensor as C1111(b) = [9.4956E9,9.7022E9, 9.9404E9, 1.0204E10, 1.0500E10, 1.0841E10, 1.1221E10, 1.1664E10, 1.2165E10,1.2744E10, 1.3442E10]. The primary goal is to provide from the first- up to the ninth-order polynomial LSM-based continuous approximation of the function relatingthese two data sets for the unweighted LSM (ULSM) procedure, with triangularweights distribution according to the pattern [1,2,3,4,5,6,5,4,3,2,1,] and, finally, for theDirac-type distribution of these weights given as [1,1,1,1,1,6,1,1,1,1,1]. Secondly, weare going to compare the first four probabilistic characteristics to check the influenceof the weighting procedure and find an optimal perturbation procedure order in thiscontext. Since the LSM implemented in Maple has an opportunity to propose anapproximating polynomial degree, one may use this capability to calibrate an optimalorder and this degree first, and then solve for the probabilistic response instead ofautomatic usage of this procedure for all structural problems.

The response functions are given in Figures 1.11 and 1.12 for the three methods withrespect to the input design parameter b, while the remaining functions – expectations(Figures 1.13 and 1.14), coefficients of variation (Figures 1.15 and 1.16), skewness(Figures 1.17 and 1.18), as well as flatness coefficient (Figures 1.19 and 1.20) – areshown with respect to the input coefficient of variation α ∈ [0.0, 0.2].

It is apparent that all order approximations (except for the first) exhibit reallyvery small differences and it is almost impossible to distinguish between higher-order functions at all. One may notice that the distance between the first-orderapproximation and the final function taken at the expectation of input variable isgreatest for the non-weighted technique and significantly smaller for the weighted

2The Stochastic FiniteElement Method

2.1 Governing Equations and Variational Formulations

2.1.1 Linear Potential Problems

Let us consider the following partial differential equation:

∑i

∂

∂xi

(ki

∂φ

∂xi

)+ g = 0; xi ∈ �, (2.1)

equivalent to the field problem in a certain anisotropic domain, where φ denotesthe potential function, q is equivalent to the source, while kx, ky, and kz denotephysical parameters of the domain governing its behavior for the particular scalarfield considered. This equation is traditionally solved with the following essential

φ = φ; xi ∈ ∂�φ (2.2)

and natural boundary conditions

∑i

ki∂φ

∂xini = q; xi ∈ ∂�q, (2.3)

where ∂�φ ∪ ∂�q = ∂� with ∂�φ ∩ ∂�q = 0. Now let us consider the arbitrary virtualpotential distribution δφ, which allows us to rewrite Equation (2.1) with constantcoefficients in the following integral form:

∫�

{(∑i

ki∂2φ

∂x2i

+ g

)δφ

}d� = 0; xi ∈ �. (2.4)



Application of the classical differentiation rules together with the divergencetheorem leads to

∫�

{∑i

∂

∂xi

(δφki

∂φ

∂xi

)}d� =

∫∂�

(∑i

ki∂φ

∂xini

)δφ d (∂�) . (2.5)

Further usage of the boundary conditions with δφ = 0 for xi ∈ ∂�φ results in thevariational statement

∫�

(∑i

δ

(∂φ

∂xi

)ki

∂φ

∂xi

)d� =

∫�

(g δφ

)d� +

∫∂�q

qini δφ d (∂�), xi ∈ �. (2.6)

The above equation is known as the principle of the virtual potentials, whichcan be the basis of the FEM standard discretization procedure. It starts from theapproximation of the potential function φ = φ(xi) for all nodes in the FEM mesh as

φ(xi) = Nα

(xi)�α , α = 1, . . . , R, (2.7)

where Nα represents the shape functions of the finite elements applied, �α are thenodal values of the potential, and R denotes here the total number of degrees offreedom. Similarly, one discretizes the spatial derivatives of the function φ, that is,

∂φ

∂x= ∂Nα

∂x�α , α = 1, . . . , R. (2.8)

Then, the left-hand side of Equation (2.6) is represented as the additional sumof the finite element contributions (denoted by f = 1, . . . , F), so that the followingdefinition is applied:

k(f)

αβ =∫�f

(∑i

ki∂Nα

∂xi

∂Nβ

∂xi

)d�, (2.9)

which constitutes the system elemental matrix equivalent to the stiffness matrix inthe elasticity problems. Representing similarly the right-hand side of Equation (2.6),it may be written that

Q(f)α =

∫�f

(g Nα

)d� +

∫∂�q

qini Nα d (∂�) . (2.10)

The assemblage and condensation procedures for the system matrices and vectorslead to the final well-known algebraic equations system

Kαβ�β = Qα , (2.11)

The Stochastic Finite Element Method 71

whose solution enables us to compute the nodal solution’s vector �β and toapproximate the potential values between the nodes. Further numerical illustra-tions concerning the problems with random coefficients deal with the unidirectionalviscous flow described by the differential equation

2µ∂2φ

∂x2 + g = 0; (2.12)

the coefficient µ characterizes the fluid viscosity, whereas the potential function isequivalent to the fluid velocity.

The second example is devoted to the 2D torsion of the planar domain describedby the equation (

1G

φ,x

),x

+(

1G

φ,y

),y

+ 2θ = 0; xi ∈ �, (2.13)

where G stands for the Kirchhoff modulus, θ is the twisting angle, and where thestresses are given as

τzx = φ,y, τzy = −φ,x; xi ∈ �. (2.14)

Therefore, the stochastic Taylor expansion is applied to the integral formulation ofthe field problem given by Equation (2.6), so that the following set of increasing-ordervariational statements has been returned [82]:

∫�

(∑i

δ

(∂φ

∂xi

)( n∑m=0

(nm

)∂mki

∂bm

∂n−m

∂bn−m

∂φ

∂xi

))d�

=∫�

(∂ng∂bn δφ

)d� +

∫∂�g

∂nqi

∂bn ni δφ d (∂�) (2.15)

As is known, the FEM does not demand formulation based on the scalar function,like the twisting angle above; we can alternatively use some formulation with scalarfunction gradients like heat fluxes (or stress Airy functions), for instance [63, 64, 125].We consider for this purpose torsion of a linear, isotropic, and homogeneous mediumgiven by Equation (2.13) together with the stress tensor components introduced indifferent notation than for Equation (2.14) as

σ13 = ∂φ

∂x2, σ23 = − ∂φ

∂x1, (2.16)

while the constitutive relation can be rewritten as

∂φ

∂x2= G

(∂u∂x1

− θx2

),

∂φ

∂x2= G

(∂u∂x2

+ θx1

)(2.17)


in the case of an externally applied twisting moment Q(xi) = − 2θ . Finally, thecomplementary energy necessary for finite element discretization can be expressed as

� (φ) =∫�

12G

[(∂φ

∂x1

)2

+(

∂φ

∂x2

)2]

d�−∫

∂�u

Qφd�. (2.18)

As is known, the torsion problem is the specific scalar field problem and isequivalent to heat conduction, seepage, and other related physical phenomenadescribed by the same Laplace partial differential equation; solution of this particularproblem makes it possible to describe at the same time all equivalent problems in thecontext of field analogies [3].

2.1.2 Linear Elastostatics

Let us consider a statistically homogeneous and bounded continuum � ⊂ R withoutany initial stresses and strains. The elastic properties and geometry of � may betreated as design random parameters and they result in a random displacement fieldand random stress tensor satisfying the classical boundary value problem of linearelasticity. Let us assume that there are non-empty subsets of external boundaries of�, namely ∂�σ and ∂�u, where the stress and displacement boundary conditionsare defined. The boundary differential equations system describing this equilibriumproblem can be written as follows:

σij = Cijklεkl, (2.19)

εij = 12

(∂ui

∂xj+ ∂uj

∂xi

), (2.20)

σij,j + ρfi = 0, (2.21)

ui = ui; xi ∈ ∂�u, (2.22)

σij nj = ti; xi ∈ ∂�σ , (2.23)

whereCijkl = δijδkl

eν(1 + ν) (1 − 2ν)

+(δikδjl + δilδjk

) e2 (1 + ν)

(2.24)

for i, j, k, l = 1, 2; e denotes here Young’s modulus, while ν stands for the Poissonratio. Let us note that theoretical foundations and some numerical aspects of ellipticboundary value problems with random coefficients are proposed in [2]. Generally, theequation system posed above is solved using well-established numerical methods.Usually, it should be transformed first to the variational formulation. This yields

∫�

Cijkl ui,jδuk,ld� =∫�

ρfiδuid� +∫

∂�σ

tiδuid (∂�), (2.25)


where the left-hand side of Equation (2.25) corresponds to elastic behavior of thestructure, the first component on the right-hand side includes the body force effects,while the second is equivalent to the stress boundary conditions applied. Thus, thestochastic version of the minimum potential energy principle has the following form[74, 112]:

• zeroth-order equation∫�

δuk,lC0ijklu

0i,jd� =

∫�

ρ0f 0i δuid� +

∫∂�σ

δuit0i d (∂�) (2.26)

• first-order equation

∫�

δuk,lC0ijkl

∂ui,j

∂bd� =

∫�

(∂ρ

∂bf 0i + ρ0 ∂fi

∂b

)δuid� +

∫∂�σ

δui∂ ti

∂bd (∂�)−

∫�

δuk,l

∂Cijkl

∂bu0

i,jd�

(2.27)• second-order equation

∫�

δuk,lC0ijkl

∂2ui,j

∂b2 d� =∫

∂�σ

δui∂2ti

∂b2 d (∂�) −∫�

δuk,l

(2∂Cijkl

∂b

∂ui,j

∂b+ ∂2Cijkl

∂b2 u0i,j

)d�

+∫�

(∂2ρ

∂b2 f 0i + 2

∂ρ

∂b∂fi∂b

+ ρ0 ∂2fi∂b2

)δuid� (2.28)

• nth-order equation

∫�

δuk,lC0ijkl

∂nui,j

∂bn d� =∫

∂�σ

δui∂nti

∂bn d(∂�) −∫�

δuk,l

n∑k=1

(n

n − k

)∂kCijkl

∂bk

∂n−kui,j

∂bn−kd�

+∫�

n∑k=0

(n

n − k

)∂kρ

∂bk

∂n−kfi∂bn−k

δuid� (2.29)

If, for instance, the Young’s modulus of � is considered as a random variable of theproblem, that is, b ≡ e, then the first partial derivatives of the elasticity tensor withrespect to this variable are derived as

Aijkl =∂Cijkl

∂b= δijδkl

ν

(1 + ν) (1 − 2ν)+ (δikδjl + δilδjk)

12 (1 + ν)

; (2.30)

and they are all deterministic quantities. Further, all higher-order partial derivativesof this tensor with respect to the same variable are equal to 0. Neglecting the bodyforce effects and eliminating all partial derivatives of the stress boundary conditions,we can write down the fundamental equations of the problem as:


• zeroth-order equation ∫�

δuk,lC0ijklu

0i,jd� =

∫∂�σ

δuit0i d (∂�) (2.31)

• first-order equation ∫�

δuk,lC0ijkl

∂ui,j

∂bd� = −

∫�

δuk,lAijklu0i,jd� (2.32)

• second-order equation

∫�

δuk,lC0ijkl

∂2ui,j

∂b2 d� = −2∫�

δuk,lAijkl

∂ui,j

∂bd� (2.33)


∫�

δuk,lC0ijkl

∂nui,j

∂bn d� = − (n − 1)

∫�

δuk,lAijkl

∂n−1ui,j

∂bn−1 d�. (2.34)

Alternatively, the compliance tensor may be introduced for � as

cijkl = 1e

((1 + ν) δikδjl − ν (1 + κν) δijδkl

), (2.35)

where κ = 0, 1 denotes the plane stress or plane strain problem, respectively. Theboundary value problem of linear elasticity is defined using Equations (2.19)–(2.23)together with the constitutive relation

εij = cijklσkl; xi ∈ �. (2.36)

A variational formulation for the boundary value problem can be proposed usinga statically admissible stress space consisting of the real-defined symmetric tensorsfulfilling equilibrium and boundary conditions. Introducing the stress variation δσ ij,one can represent the equivalent variational equation in the region � as∫

�

(εij − 1

2

(ui,j + uj,i

))δσijd� = 0. (2.37)

Applying further the Green formula to displacement field components ui, the so-called complementary energy principle is obtained for any δσ ij in the form∫

�

εijδσijd� −∫

∂�u

uiδσijnjd (∂�) = 0 (2.38)


or, alternatively, with use of the following functional:

� (σ) = 12

∫�

cijklσijσkld� −∫

∂�u

uiσijnjd (∂�) , (2.39)

whose minimization leads to the real stress field being a solution to the basic equilib-rium problem. We need to clearly emphasize that probabilistic solution to the givenelastostatic boundary value problem using displacement function differs significantlyfrom that obtained via stress functions. Despite of classical differences resulting frompotential and complementary energies applied in the FEM implementations, theoutput probability density function for the structural response coming from thesame random input may be different. When we randomize Young’s modulus for thestatistically homogeneous medium, an inversion procedure contained in Equation(2.35) essentially affects higher order perturbation terms, which simply vanish indisplacement formulation as a consequence of Equation (2.30).

2.1.3 Non-linear Elasticity Problems

The following boundary value problem is considered now:

�σkl,l + ρ�fk = 0; xi ∈ � (2.40)

�σkl = Cklmn�εmn; xi ∈ � (2.41)

�εkl = 12

[�uk,l + �ul,k + ui,k�ui,l + �ui,kui,l + �ui,k�ui,l]; xi ∈ � (2.42)

with the boundary conditions

�σklnl = �tk; xi ∈ ∂ �σ , k = 1, 2, 3 (2.43)

�uk = �uk; xi ∈ ∂ �u, k = 1, 2, 3 (2.44)

This problem is solved for the displacement vector uk(x), the strain tensor εkl (x),and the stress tensor σkl (x) fulfilling the system [Equations (2.40)–(2.44)]. Let usnote that the tensor functions �σkl (x), �σkl (x) denote here the first and secondPiola–Kirchhoff tensors

�σkl = �Fkm�σml + Fkm�σml + �Fkmσml; xi ∈ �, (2.45)

where�Fkm = �uk,m; xi ∈ �. (2.46)


The following functional defined on �uk is introduced in order to obtain a numericalsolution to this problem:

J(�uk

) =∫�

(12

Cklmn�εkl�εmn + 12σkl�ui,k�ui,l − ρ�fk�uk

)d� −

∫∂�

�tk�ukd (∂�) .

(2.47)This solution is determined from the minimization of the incremental version of

the potential energy stationarity principle

δJ(�uk

) = ∂J∂�uk

δ(�uk

). (2.48)

2.1.4 Variational Equations of Elastodynamics

Let us consider the system of equations representing the elastodynamics problem withdeterministic parameters assuming that all state variables are sufficiently smooth andcontinuous functions of x and τ . It is a simple consequence of Equations (2.19)–(2.23)[112]:

• equations of motion

(Cijkluk,l

),j+ ρfi = ρui, xi ∈ �, τ ∈ [t0, ∞) , i, j, k, l = 1, 2, 3 (2.49)

• constitutive equations

σij = Cijklεkl, xi ∈ �, τ ∈ [t0, ∞), i, j, k, l = 1, 2, 3 (2.50)

• geometrical equations

εij = 12

(ui,j + uj,i

), xi ∈ �, τ ∈ [t0, ∞), i, j, k, l = 1, 2, 3 (2.51)

• displacement boundary conditions

uj = uj, xi ∈ ∂�u, τ ∈ [t0, ∞), j = 1, 2, 3 (2.52)

• stress boundary conditions

σijnj = ti, xi ∈ ∂�σ , τ ∈ [t0, ∞), i, j = 1, 2, 3 (2.53)

• initial conditions

uj = u0j , uj = ˆu

0j , xi ∈ ∂�u, τ = t0, j = 1, 2, 3 (2.54)


Introducing a variation δui (x, t) for any time moment we obtain the above equationsas

−∫�

(σij,j + ρfi − ρui

)δuid� +

∫∂�σ

(σijnj − ti

)δuid (∂�) . (2.55)

Assuming known u(x, t1

) = 0 and u(x, t2

) = 0, this yields

δu(x, t1

) = 0, δu(x, t2

) = 0. (2.56)

Integration by parts with respect to x and τ gives

t2∫t1

δT −

∫�

σij,j δεijd� +∫�

ρfi δuid� +∫

∂�σ

ti δuid (∂�)

dτ = 0, (2.57)

which is used together with

δεij = 12

(δui,j + δuj,i

), xi ∈ �, τ ∈ [t1, t2

],

δui = 0, xi ∈ ∂�u, τ ∈ [t1, t2]. (2.58)

Assuming independence of the vectors fj and tj from ui, Equation (2.57) may bewritten as

δ

t2∫t1

(T − Jp

)dτ = 0 (2.59)

and is known as the Hamilton principle, where

T = 12

∫�

ρuiuid� (2.60)

is the kinetic energy stored in � and the potential energy is given as

Jp = 12

∫�

Cijklεijεkld� −∫�

ρfiuid� −∫

∂�σ

tiuid� = 0. (2.61)

2.1.5 Transient Analysis of the Heat Transfer

Generally, the transient heat flow problem consists of determining the temperaturefield θ = θ (x, τ ) governed by the following differential equation [13, 97]:

ρcθ −(

kijθ,j

),i− g = 0; xi ∈ �; τ ∈ [0, ∞), (2.62)


where c is the heat capacity characterizing the region �, ρ is the density of thematerial contained in �, kij is the thermal conductivity tensor, while g is the rate ofheat generated per unit volume, and the variables θ and τ denote temperature fieldvalues and time, respectively. This equation should fulfill the boundary conditionson the additional subsets of the external boundary ∂�, given as follows:

1. temperature (essential) boundary conditions

θ = θ; xi ∈ ∂�θ (2.63)

2. heat flux (natural) boundary conditions

∂θ

∂ni= qi; xi ∈ ∂�q, (2.64)

where ∂�θ ∪ ∂�q = ∂� and ∂�θ ∩ ∂�q = {∅}.The initial conditions are proposed here as

θ0 = θ(xi; 0

) ; xi ∈ �, τ = 0. (2.65)

Let us consider further some continuous temperature variations δθ (xi) defined inthe interior of the region � and vanishing on ∂�θ . Multiplying Equation (2.62) by thetest function specified and integrating it over the entire �, we obtain [96, 97]

∫�

(ρcθ −

(kijθ,j

),i− g)

δθ d� = 0; xi ∈ �; τ ∈ [0, ∞). (2.66)

Taking into account that the derivative defined on the temperature variation is infact a variation of the respective temperature derivative

∂ (δθ)

∂xi= δ

(∂θ

∂xi

)≡ δθ,i, (2.67)

we can arrive at

∫�

(ρcθ δT −

(kijθ,jδθ

),i−(

kijθ,j

)δθ,i −gδθ

)d� = 0; xi ∈ �; τ ∈ [0, ∞). (2.68)

Introducing the additional heat transfer boundary conditions

∫�

(kijθ,j δθ

)d� =

∫∂�

kij θ,j ni δθd (δ�) =∫

∂�q

qini δθd (∂�) (2.69)


and integrating by parts, we obtain∫�

(ρcθ δT + kijθ,jδθ,i − gδθ

)d� −

∫∂�q

qini δθd (∂�) = 0; xi ∈ �; τ ∈ [0, ∞) .

(2.70)This equation is the transient formulation of the principle of virtual temperature

and is used to provide its stochastic perturbation technique counterpart. As a result,we obtain a set of algebraic equations of systematically increasing order (from zerothup to nth). The following hold:

• zeroth-order partial differential equation∫�

(ρ0c0θ0δT + k0

ijθ0,j δθ,i

)d� =

∫∂�q

q0i ni δθd (∂�) +

∫�

g0δθ d� , (2.71)

• first-order partial differential equation∫�

(ρ0c0 ∂θ

∂bδT + k0

ij

∂θ,j

∂bδθ,i

)d� =

∫∂�q

∂ qi

∂bni δθ d (∂�) +

∫�

∂g∂b

δθd�

−∫�

((∂ρ

∂bc0 + ρ0 ∂c

∂b

)θ0δθ + ∂kij

∂bθ0

,j δθ,i

)d�, (2.72)

• second-order partial differential equation

∫�

(ρ0c0

˙∂2θ

∂b2 δT + k0ij

∂2θ,j

∂b2 δθ,i

)d� =

∫∂�q

∂2qi

∂b2 ni δθd (∂�) +∫�

∂2g∂b2 δθ d� +

−∫�

((∂2ρ

∂b2 c0 + 2∂ρ

∂b∂c∂b

+ ρ0 ∂2c∂b2

)θ0 +

(∂ρ

∂bc0 + ρ0 ∂c

∂b

)∂θ

∂b

)δθd�+

−∫�

(∂2kij

∂b2 θ0,j + 2

∂kij

∂b

∂θ,j

∂b

)δθ,id�, (2.73)


∫�

(n∑

k=0

(nk

)( k∑m=0

(km

)∂kρ

∂bk

∂k−mc∂bk−m

)∂n−kθ

∂bn−k

)δTd� +

∫�

(n∑

k=0

(nk

)∂kkij

∂bk

∂n−kθ,j

∂bn−k

)δθ,id�

=∫

∂�q

∂nqi

∂bn ni δθd (∂�) +∫�

∂ng∂bn δθd� (2.74)


Having solved these equations for θ0 up to the nth-order partial derivative∂nθ∂bn , respectively, we derive the expressions for the expected values and higherprobabilistic moments and the coefficients for the temperature field and its timefluctuations.

2.1.6 Thermopiezoelectricity Governing Equations

Let us consider the free energy of the thermopiezoelectric continuum consisting ofthermal, electric, and mechanical field contributions [95, 116, 137]:

W(εij, Ei, �

)= 1

2Cijklεijεkl − eijkEiεjk − 1

2bijEiEj − λijθεij − diEiθ − 1

2αTθ2. (2.75)

Here εij are the components of the overall strain tensor, Ei the components ofthe electric field vector, and θ the actual temperature change from the referencetemperature θ0 defining the state with no initial stresses or strains. The fourth- andthird-order tensors Cijkl and eijk contain the elastic and piezoelectric constants, whilebij stands for the dielectric permittivity. Further, the tensors λij and di representthermomechanical and thermopiezoelectric coupling constants. Additionally, thedissipation function F is considered here as

F(ei) = 1

2kijeiej, (2.76)

where kij means the heat conductivity tensor and ei is the thermal field vector. Aswe know, the constitutive equations are derived from Equation (2.75), whereas theFourier law is a consequence of the dissipation function. The following hold:

σij =∂W

(εij, Ei, �

)∂εij

= Cijklεkl − eijkEk − λijθ , (2.77)

Di = −∂W

(εij, Ei, �

)∂Ei

= eijkεjk + bijEj + diθ , (2.78)

S = −∂W

(εij, Ei, �

)∂θ

= λijεij + diEi + αTθ , (2.79)

qi = ∂F(ei)

∂ei= kijej. (2.80)

The stress tensor σ ij, the components of the electric displacement Di, the entropyS, and the components of the heat flux vector qi are determined uniquely from theabove equations. The following assumptions accompany this solution:

1. small displacements limitation, where

εkl = 12

(∂uk

∂xl+ ∂ul

∂xk

)(2.81)


2. linear piezoelectricityEi = −�,i (2.82)

where � denotes the scalar potential function3. the following description of the thermal field vector:

ei = −θ,i. (2.83)

Finally, the Hamilton principle is used (see Equation (2.59)):

δ

t2∫t1

(L + W) dt = 0, (2.84)

where L and W include all thermal, electric, and mechanical contributions with allvariations (of displacement, temperature, etc.) vanishing at the beginning t1 and atthe end t2 of the considered time period. The Lagrangian L is defined here as follows:

L =∫�

(T + F − U − Sθ − Sθ0θ

)d�, (2.85)

whereT = 1

2ρuiui (2.86)

as before, is the kinetic energy, while the remaining components become

U = 12σijεij − 1

2DiEi − 1

2Sθ , (2.87)

F = 12

qiei. (2.88)

Determining all variations with respect to displacement, temperature, and electricpotential one can obtain the following set of variational equations having uniquesolution [137]:

δu (L + W) = −t2∫

t1

∫�

(ρuiδui + σijδεij

)d�dt+

t2∫t1

∫∂�σ

tiδui d (∂�) dt = 0, (2.89)

δ� (L + W) = −t2∫

t1

∫�

Diδ�,i d�dt+t2∫

t1

∫∂�e

qeδ� d (∂�) dt = 0, (2.90)

δθ (L + W) =t2∫

t1

∫�

(−qiδθ,i + Sθ0δθ)

d�dt+t2∫

t1

∫∂�S

qSδθ d (∂�) dt = 0. (2.91)


These equations are valid for some continuum � with external boundary ∂�,where the boundary conditions expressed by ti, qe, and qS represent the boundarytractions, the charge density, and the boundary heat fluxes, all uniquely defined.Variational Equations (2.89)–(2.91) are consequently rewritten in the context of thestochastic perturbation technique and, for a computational implementation, theyare presented in the recursive form. This starts traditionally from the zeroth-orderequations, being in fact almost the same as these given above and that is why wesuppress the first- and the second-order relations [95]. The following hold:

t2∫t1

∫�

(ρ0u0

i δui + σ 0ij δεij

)d�dt =

t2∫t1

∫∂�σ

t0i δui d (∂�) dt, (2.92)

t2∫t1

∫�

D0i δ�,i d�dt =

t2∫t1

∫∂�e

q0e δ� d (∂�) dt, (2.93)

t2∫t1

∫�

(q0

i δθ,i − S0θ00 δθ)

d�dt =t2∫

t1

∫∂�S

q0Sδθ d (∂�) dt. (2.94)

Having solved the zeroth-order quantities, the higher-order responses need to becomputed. Therefore, the general nth-order variational equations are introduced as

t2∫t1

∫�

(n∑

k=0

(nk

)∂kρ

∂bk

∂n−ku∂bn−k

δui + ∂nσij

∂bn δεij

)d�dt =

t2∫t1

∫∂�σ

∂nti

∂bn δui d (∂�) dt, (2.95)

t2∫t1

∫�

∂nDi

∂bn δ�,i d�dt =t2∫

t1

∫∂�e

∂nqe

∂bn δ� d (∂�) dt, (2.96)

t2∫t1

∫�

(∂nqi

∂bn δθ,i −n∑

k=0

(nk

)∂kS∂bk

∂n−kθ0

∂bn−kδθ

)d�dt =

t2∫t1

∫∂�S

∂nqS

∂bn δθ d (∂�) dt. (2.97)

where∂nσij∂bn is equivalent to the nth-order partial derivative of the stress tensor with

respect to the input random variable, and so forth. The remaining part of the solutionproceeds relatively easily – zeroth-order solutions are inserted into the first-orderversions of Equations (2.95)–(2.97), where the first-order solutions are extracted tobe included in the second-order equations, and etc. The choice of highest-orderstochastic expansion depends strongly on the desired accuracy of the computationalprocess and needs to be the subject of separate analysis. As we demonstrate further,


the straightforward differentiation technique applied to the Hamilton principle maybe replaced with some alternative techniques to determine partial derivatives of thestructural response with respect to some random variables. Finally, let us mentionthat some analytical solutions to such coupled problems (even with magnetic fieldcontributions) are available, but they are valid for specific domains or componentdistributions in space, so that it is impossible to provide any general probabilisticmethodology on their basis; one may use in these cases the Monte Carlo simulationonly to calculate statistical parameters for the structural response.

2.1.7 Navier–Stokes Equations

Let us consider the phenomenon of a viscous incompressible flow for a homogeneousfluid in the region �. The equations of momentum, of continuity, as well as theconstitutive relations defining our problem can be listed as follows [3, 94]:

ρ

(∂vi

∂t+ vi,jvj

)= σij,j + fi, (2.98)

vi,i = 0, (2.99)

σij = −pδij + 2µεij, (2.100)

ρc(

∂θ

∂t+ θ,ivi

)− (kθ,i

),i = g, (2.101)

whereεij = 1

2

(vi,j + vj,i

). (2.102)

The state variables vi, p, εij, σ ij, and θ denote velocity and pressure of the fluid, com-ponents of velocity strain and stress tensor, as well as temperature, respectively,while fi is a component of the body force vector (force per unit volume of fluid).The variables – that is, ρ, µ, c, and k – denote mass density, fluid (laminar) viscosity,heat capacity, and heat conductivity, respectively. Equations (2.98)–(2.102) representthe standard Navier–Stokes equations governing the motion of a viscous, incom-pressible fluid in laminar flow coupled with heat transfer. The boundary conditionscorresponding to these equations are introduced in the following form:

• fluid velocity boundary conditions

vi = vi; xi ∈ ∂�v , (2.103)

• stress boundary conditions

σijnj = ti; xi ∈ ∂�σ . (2.104)


• temperature boundary conditions

θ = θ; xi ∈ ∂�θ (2.105)

and prescribed heat fluxes

k∂θ

∂x= q; xi ∈ ∂�q. (2.106)

Next, the weak formulation is proposed which makes it possible to introducethe FEM equations of the problem. For this purpose, Equation (2.98) with insertedEquation (2.100) is multiplied by the virtual velocity δvi and integrated over the fluiddomain � as follows:∫

�

δviρ(

vi + vi,jvj

)d� +

∫�

δεij

(2µεij − pδij

)d� =

∫�

δvifid�

+∫

∂�σ

δvitid(∂�σ

). (2.107)

Equation (2.99) is transformed through multiplication by the virtual pressure δp:∫�

δpvi,id� = 0, (2.108)

whereas Equation (2.101) adjacent to the heat transfer is obtained as∫�

δθρc(θ + θ,ivi

)d� +

∫�

kδθ,iθ,id� =∫�

δθgd� +∫

∂�q

δθ qinid (∂�). (2.109)

Equations (2)–(2.109) are next extended using the generalized perturbation tech-nique to formulate and solve the corresponding flow problem. Equating terms ofequal order in the resulting expressions, the zeroth-, first-, second and n-th orderequations for the flow considered we obtained as follows [67, 94]:

• zeroth-order (ε0 terms, one partial differential equation)∫�

δviρ0(

v0i + v0

i,jv0j

)d� +

∫�

δεij

(2µ0ε0

ij − p0δij

)d� =

=∫�

δvi(fi)0 d� +

∫∂�σ

δvi(ti)0 d (∂�), (2.110)

∫�

δpv0i,id� = 0, (2.111)


∫�

δθρ0c0 (θ0 + θ0,i v

0i

)d� +

∫�

k0δθ,iθ0,i d� =

∫�

δθ(g)0 d� +

∫∂�q

δθ(qi)0 nid (∂�);

(2.112)• first-order equations

∫�

δvi

(∂ρ

∂bv0

i + ρ0 ∂ vi

∂b+ ∂ρ

∂bv0

i,jv0j + ρ0

∂vi,j

∂bv0

j + ρ0v0i,j

∂vj

∂b

)d�+

+∫�

δεij

(2∂µ

∂bε0

ij + 2µ0∂εij

∂b− ∂p

∂bδij

)d� =

∫�

δvi∂fi∂b

d� +∫

∂�σ

δvi∂ ti

∂bd (∂�)

(2.113)

∫�

δp∂vi,i

∂bd� = 0, (2.114)

∫�

δθ

(∂ρ

∂bc0θ0 + ρ0 ∂c

∂bθ0 + ρ0c0 ∂θ

∂b+ ∂ρ

∂bc0θ0

,i v0i + ρ0 ∂c

∂bθ0

,i v0i

)d�+

+∫�

δθ

(ρ0c0 ∂θ,i

∂bv0

i + ρ0c0θ0,i∂vi

∂b

)d�

+∫�

δθ,i

(∂k∂b

θ0,i + k0 ∂θ,i

∂b

)d� =

∫�

δθ∂g∂b

d� +∫

∂�q

δθ∂ qi

∂bnid (∂�) (2.115)

• second-order equations

∫�

δviρ0

(∂2vi

∂b2 + ∂2vi,j

∂b2 v0j + v0

i,j

∂2vj

∂b2

)d� + 2

∫�

δεijµ0∂2εij

∂b2 d�

= −∫�

δεij

(4∂µ

∂b

∂εij

∂b− ∂2p

∂b2 δij

)d� +

∫�

δvi∂2fi∂b2 d� +

∫∂�σ

δvi∂2 ti

∂b2 d (∂�)+

− 2∫�

δvi

(∂ρ

∂b∂ vi

∂b+ ∂ρ

∂b

∂vi,j

∂bv0

j + ρ0∂vi,j

∂b

∂vj

∂b+ ∂ρ

∂bv0

i,j

∂vj

∂b

)d� (2.116)

∫�

δp∂2vi,i

∂b2 d� = 0, (2.117)

∫�

δθρ0c0

(∂2θ

∂b2 + ∂2θ,i

∂b2 v0i + θ0

,i∂2vi

∂b2

)d� +

∫�

δθ,ik0 ∂2θ,i

∂b2 d� =∫�

δθ∂2g∂b2 d�


+∫

∂�q

δθ∂2qi

∂b2 nid (∂�) − 2∫�

δθ

(∂ρ

∂bc0 ∂θ,i

∂bv0

i + ∂ρ

∂bc0θ0

,i∂vi

∂b

)d�+

− 2∫�

δθ

(ρ0 ∂c

∂b∂θ,i

∂bv0

i + ρ0 ∂c∂b

θ0,i∂vi

∂b+ ρ0c0 ∂θ,i

∂b∂vi

∂b

)d� − 2

∫�

δθ,i∂k∂b

∂θ,i

∂bd�

(2.118)

Let us observe that this formulation can be generalized to the nth-order perturbationand then, the following coupled equations are obtained thanks to the Leibniz chaindifferentiation rule:

∫�

δvi

(n∑

k=0

(n

k

)∂kρ

∂bk

∂n−kvi

∂bn−k+

n∑k=0

(n

k

)∂kρ

∂bk

(n−k∑m=0

(n − k

m

)∂mvi,j

∂bm

∂n−k−mvj

∂bn−k−m

))d�

+ 2∫�

δεij

(n∑

k=0

(n

k

)∂kµ

∂bk

∂n−kεij

∂bn−k− ∂np

∂bn δij

)d� =

∫�

δvi∂nfi∂bn d� +

∫∂�σ

δvi∂nti

∂bn d (∂�)

(2.119)

∫�

δp∂nvi,i

∂bn d� = 0, (2.120)

∫�

δθ

(n∑

k=0

(nk

)( k∑m=0

(km

)∂kρ

∂bk

∂k−mc∂bk−m

)∂n−kθ

∂bn−k

)d�+

+∫�

δθ

(n∑

k=0

(nk

)( k∑m=0

(km

)∂kρ

∂bk

∂k−mc∂bk−m

)(n−k∑l=0

(n − k

l

)∂ lθ,i

∂bl

∂n−k−lvi

∂bn−k−l

))d�+

+∫�

δθ,i

n∑k=0

(nk

)∂kki

∂bk

∂n−kθ,i

∂bn−kd� =

∫�

δθ∂ng∂bn d� +

∫∂�q

δθ∂nqi

∂bn nid (∂�) (2.121)

To transform these hierarchical equations into a form convenient for some com-putation, it is necessary to multiply each of them by the nth-order probabilisticmoments of the input random variables or fields. Because of the great complexity ofsuch a solution, the second-order perturbation approach is usually preferred, wherecomputing the zeroth-order velocity, pressure, and temperature functions fromEquations (2.110)–(2.112), next their first-order approximations using Equations(2.113)–(2.115), and finally the second-order terms from Equations (2.116)–(2.118),the first two probabilistic moments of these functions are derived [94].


2.2 Stochastic Finite Element Method Equations


Following the traditional FEM discretization statement [3, 180], we introduce thenodal definition of any partial order derivatives of the potential function with respectto the random input parameter as

∂mφ

∂bm = Nβ

∂m�β

∂bm , β = 1, . . . , R, m = 0, . . . , n; (2.122)

and quite similarly for their spatial derivatives

∂m(φ,i)

∂bm = ∂Nβ

∂xi

∂m�β

∂bm , β = 1, . . . , R, m = 0, . . . , n. (2.123)

To provide the direct differentiation method (DDM) formulation of theperturbation-based FEM, we consequently rewrite Equation (2.9) in this context as

∂mk(f)

αβ

∂bm =∫�f

(∑i

∂mki

∂bm

∂Nα

∂xi

∂Nβ

∂xi

)d�,

α, β = 1, . . . , R, m = 0, . . . , n; f = 1, . . . , F. (2.124)

and, analogously using Equation (2.10), one can rewrite

∂mQ(f)α

∂bm =∫�f

(−∂mq

∂bm Nα

)d� +

∫∂�

(f )q

∂mqi

∂bm ni Nα d (∂�) . (2.125)

Finally, the hierarchical algebraic equations of up to nth order are formed and maybe represented as the single recursive statement

n∑m=0

(nm

)∂n−mKαβ

∂bn−m

∂m�β

∂bm = ∂nQα

∂bn . (2.126)

The zeroth-, first-, second-, and highest-order equations are extracted from thisstatement and, then, the first of them is solved for zeroth-order potentials, thissolution is embedded into the second one on the left-hand side, and this equation issolved for first-order potentials. Consequently, this solution is inserted numericallyinto the next-order equations and this recursion proceeds until the highest-orderequation is found. Having solved these equations for ∂m�β

∂bm , m = 0, . . . , n, these nodalsolutions are used to determine the first four probabilistic moments in any degree


of freedom β. Let us note that in most practical applications higher order partialderivatives vanish very fast since the system matrices and vectors depend on randomvariables usually by lower-order polynomials only.

Alternatively, the response function method (RFM) is proposed to implement thenth-order perturbation-based stochastic finite element method (SFEM). The nodalvalues of the potential function and its partial derivatives to the random input arefound here in the approximate sense using polynomial response functions of thisinput as [82]

∂m�β

∂bm = ∂m

∂bm fβ (b) = ∂m

∂bm D(k)β bk, β = 1, . . . , R, m = 0, . . . , n; (2.127)

so that the random field discretization procedure of this system response starts fromthe following representation:

∂mφ

∂bm = Nβ

∂m�β

∂bm = Nβ

∂m

∂bm

(D(k)

β bk)

= NβD(k)β

∂m(bk)

∂bm , β = 1, . . . , R, m = 0, . . . , n;(2.128)

Deterministic coefficients D(k)β were initially numerically determined from several

deterministic solutions to the original matrix equation with random parameter valuevarying around its mean value in the interval b = [b0 − �b, b0 + �b]. The uniformsubdivision of this interval resulting in a set of the values bi, i = 1, . . . , n was madeaccording to the assumed order of the perturbation method (second order analysisrequire three points in such a grid, etc.). The additional nodal equation systems forall degrees of freedom β = 1, . . . , R may be solved independently for this purpose as

bn−11 bn−2

1 . . . b01

bn−12 bn−2

2 b02

. . . . . . . . .

bn−1n bn−2

n . . . b0n

D(n−1)β

D(n−2)β

. . .

D(0)β

=

�(1)β

�(2)β

. . .

�(n)β

(2.129)

The unique solutions for these systems make it possible to calculate analytically upto n-th-order ordinary derivatives of the nodal responses �β (b) with respect to b atthe given b0. It holds that

∂k�β

∂bk=

k∏i=1

(n − i) D(n−1)β bn−k +

k∏i=2

(n − i) D(n−2)β bn−(k+1) + . . . + D(n−k)

β . (2.130)

2.2.2 Linear Elastostatics

Let us introduce the following approximation for the displacement field [180] andits nth-order partial derivatives with respect to the input random variable using the


shape functions ϕiα(x):

u0i (x) = ϕiα (x) q0

α , xi ∈ �, (2.131)

∂kui (x)

∂bk= ϕiα (x)

∂kqα

∂bk, xi ∈ �, (2.132)

i = 1, 2, 3; α = 1, . . . , R (R is the total number of degrees of freedom introduced in�); and k = 1, . . . , n. The strain tensor components are discretized analogously and ityields

ε0ij(x) = Bijα(x)q0

α , xi ∈ �, (2.133)

∂nεij(x)

∂bn = Bijα(x)∂nqα

∂bn , xi ∈ �, (2.134)

where Bijα(xk) is the matrix containing the shape function derivatives

Bklα(xi) = 12

[ϕkα,l(xi) + ϕlα,k(xi)

], xi ∈ �. (2.135)

The FEM approach is finally obtained as the linear algebraic equations system[3, 180]

Kαβqβ = Qα , (2.136)

where qβ is the displacement solution vector. When some random quantities areinserted into the matrix Kαβ and the vector Qα , then Equation (2.136) should berewritten and solved to determine the first consecutive orders of the random structuralresponse.It yields

• zeroth-order equationsK0

αβq0β = Q0

α (2.137)

• first-order equations

K0αβ

∂qβ

∂b= ∂Qα

∂b− ∂Kαβ

∂bq0β (2.138)


K0αβ

∂2qβ

∂b2 = ∂2Qα

∂b2 − 2∂Kαβ

∂b

∂qβ

∂b− ∂2Kαβ

∂b2 q0β (2.139)

• nth-order equationsn∑

k=0

(nk

)∂kKαβ

∂bk

∂n−kqβ

∂bn−k= ∂nQα

∂bn . (2.140)


Let us recall the classical definition of the stiffness matrix in the form

Kαβ =F∑

f=1

∫�f

C(f )ijklBijαBklβd�, (2.141)

where C(f )ijkl denotes the elasticity tensor components for the finite element f and F

is their total number within �. Thus, we can describe the stiffness matrix nth-orderderivatives with respect to Young’s modulus e as follows:

∂Kαβ

∂f= ∂Cijkl

∂f

F∑f=1

∫�f

BijαBklβd� = Aijkl

F∑f=1

∫�f

BijαBklβd� (2.142)

with the tensor Aijkl defined in Equation (2.30) and

∂nKαβ

∂en = 0 (2.143)

for any n ≥ 2. Furthermore, it is seen that all partial derivatives of external load vectoras independent of Young’s modulus are equal to 0. Therefore, we have

• zeroth-order equationsK0

αβq0β = Q0

α (2.144)

• first-order equations

K0αβ

∂qβ

∂e= −∂Kαβ

∂eq0β (2.145)


K0αβ

∂2qβ

∂e2 = −2∂Kαβ

∂e

∂qβ

∂e(2.146)

• nth-order equations

K0αβ

∂nqβ

∂en = −(n − 1)∂Kαβ

∂e

∂n−1qβ

∂en−1 . (2.147)

Finally, from the first equation of this system a zeroth-order solution is determined,which, inserted into the next equations, returns a first-order solution, and so forth,until an nth-order solution is completed. After all the solution vector components havebeen determined, their expected values, variances, and other probabilistic momentscan be extracted. As is demonstrated further, symbolic computations packages,like Maple, for example, having linear algebra options implemented, can be veryefficient for the purpose of perturbation methodology implementations. Finally, itshould be underlined that in most engineering applications, the state function andthe state vector are not Gaussian variables. This makes it necessary to compute higher


and especially odd probabilistic moments to proceed in a similar way (see Appendixfor further equations).

Contrary to the straightforward differentiation methodology given above, let usfollow the idea of the common implementation of the FEM and RFM displayed inChapter 1. The crucial point of this approach is polynomial representation of thestructural response functions – displacements, strains, and stresses in the followingform, similarly to the traditional discretization:

• displacement vector discretization

qβ = D(p)β hp, p = 0, . . . , n − 1; β = 1, . . . , R (2.148)

• strain tensor components discretization

εkl = Bklβqβ = BklβD(p)β hp, p = 0, . . . , n − 1; β = 1, . . . , R; k, l = 1, 2, 3; (2.149)

• stress tensor discretization

σij = Cijklεkl = CijklBklβqβ = CijklBklβD(p)β hp, p = 0, . . . , n − 1;

α, β = 1, . . . , R; i, j, k, l = 1, 2, 3. (2.150)

Hence, partial derivatives of the state functions with respect to the randomparameter b can be derived simply as

• first partial derivative of the displacement vector

∂qβ

∂b= ∂

∂b

(D(p)

β bp)

= pD(p)β bp−1, p = 0, . . . , n − 1; β = 1, . . . , R (2.151)

• first partial derivative of the strain tensor components

∂εkl

∂b= Bklβ

∂

∂b

(qβ

) = BklβpD(p)β bp−1, p = 0, . . . , n − 1; β = 1, . . . , R; k, l = 1, 2, 3;

(2.152)• first partial derivative of the stress tensor components

∂σij

∂b= ∂

∂b

(Cijklεkl

)= ∂Cijkl

∂bBklβqβ + CijklBklβ

∂qβ

∂b

= ∂Cijkl

∂bBklβD(p)

β bp + CijklBklβpD(p)β bp−1,

p = 0, . . . , n − 1; α, β = 1, . . . , R; i, j, k, l = 1, 2, 3. (2.153)


Quite similarly, higher-order derivatives can be determined by simple differen-tiation according to the chain rule. All these derivatives may be embedded intothe additional equations for probabilistic moments of the structural state functionsfollowing the traditional approach.

To obtain the generalized stochastic perturbation approach for the alternativestress-based formulation, we rewrite the complementary energy principle as∫

�

cijklσklδσijd� =∫

∂�u

uiδσijnjd (∂�). (2.154)

and then, by introducing the additional expansions of all random functions, one gets

• zeroth-order equation with ε0 terms:∫�

c0ijklσ

0klδσijd� =

∫∂�u

u0i δσijnjd (∂�), (2.155)

• first-order equation with ε1 terms:

ε

∫�

c0ijkl

∂σkl

∂bδσijd� = ε

∫∂�u

∂ui

∂bδσijnjd (∂�) − ε

∫�

∂cijkl

∂bσ 0

klδσijd�, (2.156)

• second-order equation with ε2 terms:

ε2∫�

c0ijkl

∂2σkl

∂b2 δσijd� = ε2∫

∂�u

∂2ui

∂b2 δσijnjd (∂�) − ε2∫�

(2∂cijkl

∂b∂σkl

∂b+ ∂2cijkl

∂b2 σ 0kl

)δσijd� .

(2.157)

This procedure is repeated until the given nth-order equation is obtained with εn

terms:

εn∫�

c0ijkl

∂nσkl

∂bn δσijd� = εn∫

∂�u

∂nui

∂bn δσijnjd (∂�) − εn∫�

n−1∑

p=0

(np

)∂n−pcijkl

∂bn−p

∂pσkl

∂bp

δσijd�.

(2.158)Of course, the stress function variations δσ ij are deterministic, analogously to thedisplacement variations δui provided in Sec. 2.1.6, for instance. The increasing powersof the perturbation parameter ε are left for example here to show its influence (or itslack) on various orders’ solutions to the perturbation-based equilibrium equations.


Let us start from an incremental formulation of the FEM equations [112, 138, 141] andtheir stochastic counterparts using additional extension of the structural polynomialresponse [79]. So that, instead of the displacement vector, we now need to discretize


• the structural displacement increments [79]

�uζ = aζβ�qβ = aζβD(p)β bp, p = 0, . . . , n − 1; β, ζ = 1, . . . , R (2.159)

• increments of the strain tensor components

�εkl = �εkl + � ¯εkl = Bklζ �uζ + ¯Bklξζ �uζ �uξ

= Bklζ aζα�qα + ¯Bklξζ aζα�qαaξβ�qβ = Bklζ aζαD(p)α bp + ¯Bklξζ aζαD(p)

α aξβD(r)β bp+r

p, r = 0, . . . , n − 1; α, β = 1, . . . , R; k, l = 1, 2, 3; (2.160)

• increments of the second Piola–Kirchhoff stress tensor components

�σij = Cijkl�εkl = Cijkl(�εkl + � ¯εkl

) = Cijkl

(Bklζ �uζ + ¯Bklξζ �uζ �uξ

)= Cijkl

(Bklζ aζα�qα + ¯Bklξζ aζα�qαaξβ�qβ

)= Cijkl

(Bklζ aζαD(p)

α bp + ¯Bklξζ aζαD(p)α aξβD(r)

β bp+r)

p, r = 0, . . . , n − 1; α, β = 1, . . . , R; i, j, k, l = 1, 2, 3; (2.161)

Then, the partial derivatives of the state functions increments with respect to therandom parameter b can be derived analytically and further, be used to determinethe probabilistic moments of the structural response according to the desired orderof the Taylor series expansion. The following hold:

• partial derivatives of the displacements with respect to the random input variableof the first order

∂�uζ

∂b= aζβ

∂�qβ

∂b= aζβ

∂(

D(p)β bp

)∂b

= paζβD(p)β bp−1, p = 0, . . . , n − 1; β = 1, . . . , R

(2.162)as well as the mth-order partial derivatives

∂m�uζ

∂bm = aζβ

∂m�qβ

∂bm = aζβ

∂m(

D(p)β bp

)∂bm = p . . . (p − m)aζβD(p)

β bp−m,

p = 0, . . . , n − 1; β = 1, . . . , R (2.163)

further, we determine the increments of the strain tensor component partialderivatives with respect to the input random variable b as

∂�εkl

∂b= ∂�εkl

∂b+ ∂� ¯εkl

∂b= Bklζ

∂�uζ

∂b+ ¯Bklξζ

∂�uζ

∂b�uξ + ¯Bklξζ �uζ

∂�uξ

∂b


= Bklζ aζα

∂�qα

∂b+ ¯Bklξζ aζα

∂�qα

∂baξβ�qβ + ¯Bklξζ aζα�qαaξβ

∂�qβ

∂b

= Bklζ aζαpD(p)α bp−1 + ¯Bklξζ aζαD(p)

α paξβD(r)β bp+r−1 + ¯Bklξζ aζαD(p)

α aξβD(r)β r

p, r = 0, . . . , n − 1; β = 1, . . . , R; k, l = 1, 2, 3; (2.164)

• increments of the stress tensor component partial derivatives with respect to theinput random variable

∂�σij

∂b= ∂

∂b

(Cijkl�εkl

)= ∂Cijkl

∂b

(�εkl + � ¯εkl

)+ Cijkl

(∂�εkl

∂b+ ∂� ¯εkl

∂b

)

= ∂Cijkl

∂b

(Bklζ �uζ + ¯Bklξζ �uζ �uξ

+ Cijkl

(Bklζ

∂�uζ

∂b+ ¯Bklξζ

∂�uζ

∂b�uξ + ¯Bklξζ �uζ

∂�uξ

∂b

)

= ∂Cijkl

∂b

(Bklζ aζα�qα + ¯Bklξζ aζα�qαaξβ�qβ

)

+ Cijkl

(Bklζ aζα

∂�qα

∂b+ ¯Bklξζ aζα

∂�qα

∂baξβ�qβ + ¯Bklξζ aζα�qαaξβ

∂�qβ

∂b

)

= ∂Cijkl

∂b

(Bklζ aζαD(p)

α bp + ¯Bklξζ aζαD(p)α aξβD(r)

β bp+r)

+ Cijkl

(Bklζ aζαD(p)

α pbp−1 + ¯Bklξζ aζαD(p)α paξβD(r)

β bp+r−1

+ ¯Bklξζ aζαD(p)α aξβD(r)

β rbp+r−1)

= 0, . . . , n − 1; α, β = 1, . . . , R; i, j, k, l = 1, 2, 3. (2.165)

All these derivatives may be embedded into the additional equations for probabilisticmoments of the structural state functions following the traditional approach in thismethod; quite similarly, higher-order derivatives can be determined by a simpledifferentiation according to the chain rule. As one may expect, these derivatives maybe obtained without the RFM. To this purpose we use the initial algebraic equationsin this problem and, by successful formation and solution of the increasing-orderequations, the nth-order derivatives are derived. In this case, the following hold [79,112]:

• zeroth-order equation

K(1)0αβ �q0

β + K(2)0αβγ �q0

β�q0γ + K(3)0

αβγ δ�q0β�q0

γ �q0δ = �Q0

α (2.166)


K(1)0αβ

∂�qβ

∂b+ K(2)0

αβγ

∂�qβ

∂b�q0

γ + K(2)0αβγ �q0

β

∂�qγ

∂b+ K(3)0

αβγ δ

∂�qβ

∂b�q0

γ �q0δ


+ K(3)0αβγ δ�q0

β

∂�qγ

∂b�q0

δ + K(3)0αβγ δ�q0

β�q0γ

∂�qδ

∂b

= ∂�Qα

∂b− ∂K(1)

αβ

∂b�q0

β − ∂K(2)αβγ

∂b�q0

β�q0γ − ∂K(3)

αβγ δ

∂b�q0

β�q0γ �q0

δ (2.167)

• a recursive nth-order formula

n∑k=0

(n

k

)∂kK(1)

αβ

∂bk

∂n−k�qβ

∂n−kb+

n∑k=0

(n

k

)∂kK(2)

αβγ

∂kb

n−k∑l=o

(n − k

l

)∂ l�q0

β

∂bl

∂n−k−l�q0γ

∂bn−k−l

+n∑

k=0

(n

k

)∂kK(3)

αβγ δ

∂bk

n−k∑l=o

(n − k

l

)∂ l�q0

β

∂bl

n−k−l∑m=o

(n − k − l

m

)∂m�q0

γ

∂bm

∂n−k−l−m�q0δ

∂bn−k−l−m

= ∂n�Qα

∂bn . (2.168)

Having determined these partial derivatives, the probability density functionof the input random variable is assumed to compute its moments and finally,all these quantities are embedded into the perturbation-based equations for theresponse probabilistic moments (cf. Equations (1.12)–(1.15)). It is also clear that therandomness modeling within the non-linear boundary value problems is essentiallymuch more complex than for the linear elasticity. The next step toward computationalanalysis would be exploration of the existing constitutive theories in plasticity andviscoplasticity [15] accounting for uncertainty in especially material characteristicshaving large experimental evidence [128].

It is clear that the FEM formulation given above includes the specific case of thestability analysis, so that we present briefly its specific RFM version. The deformationenergy of some linear elastic structure may be rewritten in matrix notation as[111, 112]

U(α) = 12

qT(α)K

(e)(α)q(α) + 1

2qT

(α)K(σ )(α)q(α) (2.169)

where q(α) is the generalized displacement vector. Global matrices are composedfrom the finite element contributions that can be defined for a two-noded 3D barelement fundamental used in further computational experiments as

k(e)(α) = e(α)A(α)

l(α)

1 0 0 −1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

−1 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

, (2.170)


being the elemental stiffness matrix and k(σ )(α) denoting

k(σ )(α) = F(α)

l(α)

0 0 0 0 0 00 0 0 0 0 00 0 1 0 0 −10 0 0 0 0 00 0 0 0 0 00 0 −1 0 0 1

; (2.171)

traditionally, an index α in brackets is necessary to show the RFM iteration number.Stability analysis of the planar or spatial frames as well as plates or shells demandsthe relevant structural element being employed. So that, the potential energy for theentire discrete system may be expressed as

J(α)P = 1

2qT

(α)

(K(e)

(α) + K(σ )(α)

)q(α) − QT

(α)q(α) (2.172)

whose minimization with respect to the generalized displacements leads to

(K(e)

(α) + K(σ )(α)

)q(α) = Q(α). (2.173)

Computational analysis with the FEM in linearized buckling analysis is reallyfocused on the critical load factor λ(α) such that

Q(α) = λ(α)Q(α), (2.174)

where Q(α) is the unit force series directed according to the critical forces to bedetermined. Then, the discrete stability equation rewritten for the entire system inthe RFM version [103] becomes

(K(e)

(α) + λ(α)K(σ )(α)

(F(α)))

q(α) = λ(α)Q(α). (2.175)

Further, a distribution of internal forces F(α) is equivalent to the load Q(α) and thedisplacements q(α) are equivalent to the load λ(α)Q(α). We determine the values of λ(α)from the following equations:

(

K(e)(α) + λ(α)K

(σ )(α)

(F(α)))

q1(α) = λ(α)Q(α)(K(e)

(α) + λ(α)K(σ )(α)

(F(α)))

q2(α) = λ(α)Q(α)

, where q1(α) = q2(α) (2.176)

and, by subtracting, we obtain the basic algebraic equation series representing theelastic stability (

K(s)(α) + λ(α)K

(σ )(α)

(F(α)))

q∗(α) = 0. (2.177)


An introduction of the basic condition for the critical value λ(α) = λcr(α) leads to thefollowing final well known condition:

det(

K(s)(α) + λ(α)K

(σ )(α)

(F(α))) = 0. (2.178)

The minimum positive solution to this eigenproblem allows us to determine thefirst critical value and, in the context of probabilistic analysis, basic characteristics ofthe critical force. It is known that most of the structures do not collapse when theexternal forces reach their first critical value (or its expectation). A definitely moreinteresting (and challenging) problem would be to follow a post-critical path withuncertainty, which is of paramount importance for the skeletal tower structures likethis studied in further computational experiments.

2.2.4 SFEM in Elastodynamics

Let us consider the following approximation of the displacement field ui (x, τ) [51]:

ui(x, τ) = ϕiα (x) qα(τ ), (2.179)

where qα(τ ) is the generalized coordinate vector. The strain tensor is rewritten as

εij(x, τ)=Bijα (x) qα(τ ) . (2.180)

Therefore, an equation of motion is obtained as

δ

t2∫t1

[12

qα Mαβ qβ − 12

qα Kαβ qβ + Qαqα

]dτ = 0, (2.181)

where the matrices of mass and stiffness may be composed from the followingintegrals:

Mαβ =∫�

ρϕiαϕiβd� Kαβ =∫�

BijαCijklBklβd�. (2.182)

Integration with respect to a time variable and variation in Equation (2.181) leads to

qβMαβδqα −t2∫

t1

(qβMαβ + qβKαβ − Qα

)∂qα∂τ = 0 (2.183)

so that, in the view ofδqα

(t1) = 0, δqα

(t2) = 0, (2.184)

leads to the well-known matrix equation of motion and its stochastic RFM counterpart[89] equals to

Mαβ qβ + Kαβqβ = Qα M(i)αβ q(i)

β + K(i)αβq(i)

β = Q(i)α (2.185)


which is frequently proposed, including the damping component as

Mαβ qβ + Cαβ qβ + Kαβqβ = Qα M(i)αβ q(i)

β + C(i)αβ q(i)

β + K(i)αβq(i)

β = Q(i)α . (2.186)

Upper index i in these two equations is needed to underline particular test solutionin the RFM scheme. Modal analysis, being one of many techniques available now fornumerical solution of the equations of motion, is based on the following transform:

q (τ ) =n∑

j=1

ajxj (τ ) = Ax (τ ), (2.187)

where A = [ai]

is composed of the eigenvectors of a structure, while x (τ ) is the vectorof modal coordinates. The global equation of motion is transformed into [51]

n∑j=1

aTi Majxj (τ ) +

n∑j=1

aTi Cajxj (τ ) +

n∑j=1

aTi Kajxj (τ ) = aT

i R (τ ) . (2.188)

The orthogonality condition on the eigenvectors leads to a simplification

aTi Mai =

{0 i = j

mi i = j, aT

i Kai ={

0 i = j

ki i = j(2.189)

where mi, ki stand for the modal masses and stiffnesses. Then

n∑j=1

aTi Majxj (τ ) = mixi (τ ),

n∑j=1

aTi Kajxj (τ ) = kixi (τ ) . (2.190)

Introducing a further modal loading vector and modal damping matrix in case ofproportionality to mass and stiffness components (i.e., C = αM + κK)

pi (τ ) = aTi R (τ ),

n∑j=1

aTi Cajxj (τ ) = cixi (τ ) , (2.191)

the equation of motion (2.188) finally becomes

mixi (τ ) + cixi (τ ) + kixi (τ ) = pi (τ ), i = 1, 2, 3, . . . , n. (2.192)

Normalization of the eigenvectors gives aTi Mai = 1 and with mi = 1, ki = ω2

i , ci =2γiωi, one obtains

xi (τ ) + 2γiωixi (τ ) + ω2i xi (τ ) = pi (τ ) . (2.193)


This transformation is also used to express the boundary and initial conditions,hence

r (0) =n∑

j=1

ajxj (0), r (0) =n∑

j=1

ajxj (0) (2.194)

andx (0) = ATMq (0), x (0) = ATMv (0) . (2.195)

A complete solution of Equation (2.193) is known as the Duhamel integral [3]

xi (τ ) = 1

ωi

√1 − γ 2

i

t∫0

pi (τ ) exp(−γiωiτ

)hi (t − τ) sin

(ωi

√1 − γ 2

i (t − τ)

)dτ

+ exp(−γiωit

) {Ai1 sin

(ωi

√1 − γ 2

i t)

+ Ai2 cos(

ωi

√1 − γ 2

i t)}

(2.196)

The elastic forces vector in the initial system may be determined as

fs (τ ) =n∑

i=1

Kaixi (τ ) +n∑

i=1

ω2i Maixi (τ ) . (2.197)

The DDM, applied in some further numerical experiment, is based on the straight-forward determination of the latter exactly from the given-order state equilibriumequations. If we calculate the eigenfrequencies as a result of

(K − � 2M

)ϕ = 0, (2.198)

then we start from the zeroth-order relation [112](K0 − (� 0)2 M0

)ϕ0 = 0 (2.199)

and after differentiation w.r.t. b, it holds that(∂K∂b

−(

∂

∂b

(� 0)2)M0 − (� 0)2 ∂M

∂b

)ϕ0 = −

(K0 − (� 0)2 M0

) ∂ϕ

∂b. (2.200)

A sequential differentiation up to nth order leads to the following recursive form[104, 105]:

n∑k=0

(nk

)∂n−kK∂bn−k

∂kϕ

∂bk=

n∑k=0

(nk

) k−1∑l=0

(k − 1

l

)∂(k−(l+1))(2� )

∂b(k−(l+1))

∂ l+1�

∂bl+1

×n−k∑m=0

(n − k

m

)∂mM∂bm

∂(n−k−m)ϕ

∂bn−k−m(2.201)


so that we can calculate the partial derivatives of the increasing order for theeigenvalues from Equations (2.198)–(2.201) and then, embed them directly into theparticular probabilistic moment equations to recover their values. Analogously, onecan rewrite the equation for the forced vibrations in the perturbation-based nth-orderversion as

n∑k=0

(nk

){∂n−kM∂bn−k

∂kq∂bk

+ ∂n−kC∂bn−k

∂kq∂bk

+ ∂n−kK∂bn−k

∂kq∂bk

}= ∂nQ

∂bn (2.202)

or the modal equations system

∂nxi (τ )

∂bn +n∑

k=0

(nk

){2∂n−k

(γiωi

)∂bn−k

∂k(xi (τ )

)∂bk

+ ∂n−k(ω2

i

)∂bn−k

∂k(xi (τ )

)∂bk

}= ∂npi (τ )

∂bn ,

(2.203)where usually the right-hand side vector equals 0 except for the zeroth-orderequation, since the main interest is in structural randomness. The complex natureof the perturbation-based equations follows intermediately a chain differentiation ofthe second powers of random state functions with respect to b. Final determinationof the elastic forces in the modal approach proceeds using the differential form

∂pfs (τ )

∂bp =n∑

i=1

∂p

∂bp

{Kaixi (τ )

}+n∑

i=1

∂p

∂bp

{ω2

i Maixi (τ )}. (2.204)


Let us assume that the region � is discretized by use of the set of finite elements andthat the scalar temperature field θ (xi) is described by the nodal temperature vectorTα [113, 145]:

θ(xi) = ϕα

(xi)

Tα; i = 1, 2, 3; α = 1, 2, . . . , R, (2.205)

The temperature derivatives can be written in the form

θ,i = ϕα,i Tα , i = 1, 2, 3. (2.206)

Moreover, let us introduce the heat capacity matrix Cαβ , the heat conductivitymatrix Kαβ , and the vector Pα as follows:

Cαβ =∫�

ρc φα φβ d�, (2.207)

Kαβ =∫�

kij φα,i φβ,j d�, (2.208)

andPα =

∫�

g ϕαd� +∫∂�

qini ϕαd�. (2.209)


Next, let us introduce these matrices into the additional variational formulation(cf. Equation (2.70)) to obtain the following algebraic equations system:

Cαβ Tβ + KαβTβ = Pα. (2.210)

As it is known [3], the main issue in transient problems is the additional time dis-cretization using some time increment �t and the forward finite differencing scheme.Then, we can rewrite the last equation in the following manner for temperatureindependent system’s matrices:

Cαβ

Tβ (t + �t) − Tβ (t)

�t+ KαβTβ (t) = Pα. (2.211)

Considering the second component in this statement we obtain the explicit method,where the nodal temperature vector in this component is taken at the beginning ofthis time step. However, it is possible to introduce the extra coefficient 0 ≤ δ ≤ 1 toinclude in this term the temperature vector after the time step also. It holds that

Cαβ

Tβ (t + �t) − Tβ (t)

�t+ Kαβ

{δTβ (t + �t) + (1 − δ)Tβ (t)

} = Pα , (2.212)

where δ = 0 is equivalent to the explicit method, δ = 1/2 serves for theCrank–Nicholson method, δ = 2/3 stands for the Galerkin method, and lastly δ = 1is used in the implicit method (one can use this algorithm with δ as the extrainput parameter); there are also three level schemes, where the temperatures inthe moments t + �t, t, t − �t are included at once. A further procedure leads toreformulation of this equation to obtain the nodal temperatures for t + �t on theLHS and the temperature vector in the preceding time step on the RHS, so that forthe Crank–Nicholson scheme we proceed as follows:

Cαβ

{Tβ (t + �t) − Tβ (t)

�t

}+ Kαβ

{Tβ (t + �t)

2+ Tβ (t)

2

}= Pα , (2.213)

therefore (Cαβ

�t+ Kαβ

2

)Tβ (t + �t) = Pα +

(Cαβ

�t− Kαβ

2

)Tβ (t). (2.214)

Introduction of the matrix Aαβ instead of a combination of heat conductivity andheat capacity matrices on the LHS as well as the additional matrix Eαβ for the RHSbracket leads to the relation

AαβTβ (t + �t) = Pα + EαβTβ (t). (2.215)

Let us focus first on the DDM version of the SFEM implementation of theproblem. Analogously to the previous considerations (provided also for the second-order analysis in [46, 96]), we obtain the following systems of algebraic equationsdescribing the generalized stochastic formulation of the transient heat flow problem:

• zeroth-order equationC0

αβ T0β + K0

αβT0β = P0

α (2.216)



C0αβ

∂Tβ

∂b+ K0

αβ

∂Tβ

∂b= ∂Pα

∂b−(

∂Cαβ

∂bT0

β + ∂Kαβ

∂bT0

β

)(2.217)


C0αβ

∂2Tβ

∂b2 + K0αβ

∂2Tβ

∂b2 = ∂2Pα

∂b2 − 2

(∂Cαβ

∂b

∂Tβ

∂b+ ∂Kαβ

∂b

∂Tβ

∂b

)

−(

∂2Cαβ

∂b2 T0β + ∂2Kαβ

∂b2 T0β

)(2.218)

and, at last, nth-order equation

C0αβ

∂nTβ

∂bn + K0αβ

∂nTβ

∂bn = ∂nPα

∂bn −n∑

k=1

(nk

)∂kCαβ

∂bk

∂n−kTβ

∂bn−k−

n∑k=1

(nk

)∂kKαβ

∂bk

∂n−kTβ

∂bn−k.

(2.219)We have introduced the following matrix notation in the equations posed above:

• the heat capacity matrix derivatives (after Equation (2.207))

∂nCαβ

∂bn =∫�

n∑k=0

(nk

)∂kρ

∂bk

∂n−kc∂bn−k

ϕαϕβd�, (2.220)

• the heat conductivity matrix derivatives (from Equation (2.208))

∂nKαβ

∂bn =∫�

∂nkij

∂bn ϕα,iϕβ,j d�, (2.221)

• the right-hand vector derivatives (since Equation (2.209))

∂nPα

∂bn =∫�

∂ng∂bn ϕαd� +

∫∂�q

∂nqi

∂bn niϕαd�, (2.222)

where all these expressions are evaluated at the expected values of the inputrandom variables. As is clear now, the DDM version needs the formation andsolution of the increasing-order equations obtained from the initial one bya systematic differentiation with respect to the random input variable provided ina quite deterministic way. Readers interested in the second-order version of thismethodology relevant to composite materials with random physical properties mayfind the necessary details in [97].


The main idea behind the RFM is to recover the polynomial approximation of thetemperature at a given node with respect to the input random variable b of the FEMmesh in the following form:

Tβ = D(m)β bm, m = 0, . . . , n − 1; β = 1, . . . , R (2.223)

so that:

θ(xi) = ϕβ

(xi)

Tβ = ϕβ

(xi)

D(m)β bm; i = 1, 2; α = 1, 2, . . . , R,

m = 0, . . . , n − 1. (2.224)

Therefore, the temperature gradients are similarly determined as

θ,j = ϕβ,j Tβ = ϕβ,j D(m)β bm, i = 1, 2, m = 0, . . . , n − 1. (2.225)

The key feature of this approach is to determine numerically the coefficientsD(m)

β for each node of the initial FEM mesh and each power of the polynomialrepresentations of the nodal temperatures with respect to the random input. Startingfrom a random polynomial representation, it is possible to calculate up to nth-orderordinary derivatives of the nodal responses θβ (b) with respect to b at the given b0 asfollows:

• first-order derivative

dθβ

db= (n − 1) D(1)

β bn−2 + (n − 2) D(2)β bn−3 + . . . + D(n−1)

β , (2.226)

• second-order derivative

d2θβ

db2 = (n − 1) (n − 2) D(1)β bn−3 + (n − 2) (n − 3) D(2)

β bn−4 + . . . + D(n−2)β , (2.227)

• kth-order derivative

dkθβ

dbk=

k∏i=1

(n − i) D(1)β bn−k +

k∏i=2

(n − i) D(2)β bn−(k+1) + . . . + D(n−k)

β . (2.228)

This differentiation has quite an analytical character and the increasing-orderpartial derivatives of the nodal temperatures with respect to the random variable arenot so much affected by the numerical errors from the hierarchical equation solutions.It is also clear that the probabilistic transient problem needs successive polynomialresponses from time increment to the next one. Therefore, for a discrete time momentτ , we have the following approximation [89]:

Tβ (τ ) = Dτ(m)β bm, m = 0, . . . , n − 1; β = 1, . . . , R. (2.229)


where Dτ(m)β denotes the coefficient relevant to the mth power of b computed in the

degree of freedom β at the time moment τ .Hence, it yields

θ(xiτ) = ϕβ

(xi)

Tβ (τ ) = ϕβ

(xi)

Dτ(m)β bm; i = 1, 2; α = 1, 2, . . . , R,

m = 0, . . . , n − 1. (2.230)

Therefore, the temperature gradients are similarly determined as

θ,j (τ ) = ϕβ,j Tβ (τ ) = ϕβ,j Dτ(m)β bm, i = 1, 2, m = 0, . . . , n − 1. (2.231)

Finally, one realizes that the temperature-dependent physical parameters may leadto further significant numerical complications in SFEM implementations to transientproblems even when using the RFM.

2.2.6 Coupled Thermo-piezoelectrostatics SFEM Equations

The SFEM discretization starts from the traditional vectorization of the displacementfield u, the electric potential �, and temperature θ . It is completed using the nodalvalues q, �, T and the shape functions ϕq, ϕ� as well as ϕT as follows [95, 116]:

u = ϕqq, � = ϕ��, θ = ϕTT. (2.232)

Then, the strain tensor ε, the electric field E, and the thermal field F are related tothe vectorized state functions as

ε = ∇φqq = Bqq, (2.233)

E = −∇φ�� = B��, (2.234)

F = −∇φTT = BTT. (2.235)

The constitutive Equations (2.77)–(2.80) become, in this manner, equal to

σ = Cε − PE − κθ , (2.236)

D = PTε + BE + dθ , (2.237)

S = κTε + dTE + αTθ , (2.238)

q = kF, (2.239)

where σ , D are the stress tensor components and the electric displacement vec-tor, and matrices C, B, P, κ , d, and k denote the elasticity, dielectric permittivity,


piezoelectric constant, thermomechanical coupling, thermopiezoelectric coupling,and thermal conductivity tensors. Then, we introduce this discretization into thevariational equations and, as a result, the fundamental of equilibrium equations forthe thermopiezoelectric problem becomes

Mqq 0 0

0 0symm. 0

q�

T

+

0 0 00 0 0

CTq CT� CTT

q�

T

+

Kqq Kq� KqT

K�q K�� K�T

0 0 KTT

q�

T

=Fq

F�

FT

, (2.240)

where Mqq is the mass matrix, CTq, CT� , and CTT are the damping matrices resultingfrom the thermomechanical, thermoelectric couplings, and thermal field, respectively.The matrices Kq� , K�q, KqT, K�T, Kqq, K�� , and KTT are the stiffness matricesrelated to the piezoelectric-mechanical (first two), thermomechanical, thermoelectriccouplings as well as the stiffness adjacent to the mechanical, electric, and thermalfields, correspondingly. Finally, the RHS vectors Fq, F� , as well as FT representmechanical, electric, and thermal boundary conditions. Their definitions are givenas follows:

Mqq =∫�

ϕTq ρ ϕq d�, (2.241)

CTq = −∫�

T0ϕTT kT Bq d� CT� = −

∫�

T0ϕTT dT B� d� CTT = −

∫�

T0ϕTTαTϕT d�

(2.242)

Kqq =∫�

BTq C Bq d� Kq� = −

∫�

BTq P B� d� KqT = −

∫�

BTq k ϕT d� (2.243)

K�q = −∫�

BT�PTBq d� K�� =

∫�

BT�B B� d� K�T = −

∫�

BT� d ϕT d� (2.244)

KTT = −∫�

BTTκ BT d�. (2.245)

Let us also note that

K�q = KTq� CTq = T0 KT

qT CT� = T0 KT�T. (2.246)


On this basis we may write the zeroth- and the nth-order stochastic equationsystems corresponding to the deterministic system given as follows:

M(0)qq 0 0

0 0symm. 0

q(0)

�(0)

T(0)

+

0 0 00 0 0

C(0)Tq C(0)

T� C(0)TT

q(0)

�(0)

T(0)

+

K(0)qq K(0)

q� K(0)qT

K(0)�q K(0)

�� K(0)�T

0 0 K(0)TT

q(0)

�(0)

T(0)

=

F(0)q

F(0)�

F(0)T

(2.247)

and

n∑k=0

(n

k

)∂kMqq

∂bk0 0

0 0

symm. 0

∂n−kq∂bn−k

∂n−k�

∂bn−k

∂n−kT∂bn−k

+n∑

k=0

(n

k

)0 0 00 0 0

∂kCTq

∂bk

∂kCT�

∂bk

∂kCTT

∂bk

∂n−kq∂bn−k

∂n−k�

∂bn−k

∂n−kT∂bn−k

+n∑

k=0

(nk

)

∂kKqq

∂bk

∂kKq�

∂bk

∂kKqT

∂bk

∂kK�q

∂bk

∂kK��

∂bk

∂kK�T

∂bk

0 0∂kKTT

∂bk

∂n−kq∂bn−k

∂n−k�

∂bn−k

∂n−kT∂bn−k

=

∂nFq

∂bn

∂nF�

∂bn

∂nFT

∂bn

(2.248)

which include all the variational statements formulated before. Let us note that,contrary to the initial Equation (2.247), most of the submatrices in Equation (2.248)equal 0, especially for a single random variable, since this variable appears effectivelyin a few of them only, frequently as the lower-order polynomial form.

2.2.7 Navier–Stokes Perturbation-Based Equations

Let us consider a discretization of the fluid volume � by any type of finite elements

�f, such that � =E⋃

f=1

�f , where N nodes are introduced. Further, we consider for

simplicity only the unidirectional incompressible fluid flow including heat transferin Cartesian coordinates along the x-axis. The following description is proposed fortemperatures, velocities, and pressures in �:

θ (x; t) = ϕα (x) Tα (t) ; (2.249)

v (x; t) = ϕα (x) vα (t) ; (2.250)

p (x; t) = ϕα (x) Pα (t) ; (2.251)


where α runs over all degrees of freedom α = 1, . . . , R. It can be shown that in case ofunidirectional flow, the fundamental set of corresponding variational equations canbe reduced to the matrix form as follows [3, 94]:M 0 0

0 0 00 0 C

V

PT

+

KµVXVX

+ KVVXKVXP 0

KTVXP 0 00 0 KVT + KTT

V

PT

=

RB + RS

0QB + QS

.

(2.252)The submatrices used in Equation (2.252) may be represented as

M =∫�

ρϕTϕd�, (2.253)

KµVXVX=∫�

2µϕT,xϕ,xd�, KVVX

=∫�

ρϕTϕvxϕ,xd�, KVxP = −∫�

ϕT,xϕd� (2.254)

C =∫�

ρcϕTϕd�, KTT =∫�

kϕT,xϕ,xd�, KVT =

∫�

ρcϕTϕvxϕ,xd� (2.255)

as well as

RB =∫�

ϕTfd�, RS =∫

∂�σ

ϕST t d (∂�), (2.256)

QB =∫�

ϕTgd�, QS =∫

∂�q

ϕSTqd (∂�). (2.257)

It corresponds directly to the variational formulation given by Equations(2.107)–(2.109).

When the pressure terms can be neglected, the initial equations system may bereduced to rank 2 by an elimination of the second columns and rows, respectively.

As it was demonstrated in Sec. 2.1.7, an application of stochastic perturbationmethodology results in hierarchical equations which can be proposed for the second-order approach as follows:

• zeroth-order equations

M0 0 0

0 0 0

0 0 C0

v0

P0

T0

+

K0µVXVX

+ K0VXV K0

VXP 0

K0 TVXP 0 0

0 0 K0VT + K0

TT

v0

P0

T0

=

R0B + R0

S

0

Q0B + Q0

S

(2.258)


• first-order statements

M0 0 0

0 0 0

0 0 C0

∂v∂b

∂P∂b

∂T∂b

+

K0µVXVX

+ K0VXV K0

VXP 0

K0 TVXP 0 0

0 0 K0VT + K0

TT

∂v∂b∂P∂b∂T∂b

=

∂RB

∂b+ ∂RS

∂b0

∂QB

∂b+ ∂QS

∂b

−

∂M∂b

0 0

0 0 0

0 0∂C∂b

v0

P0

T0

−

∂KµVXVX

∂b+ ∂KVXV

∂b

∂KVXP

∂b0

∂KTVXP

∂b0 0

0 0∂KVT

∂b+ ∂KTT

∂b

v0

P0

T0

(2.259)


M0 0 0

0 0 0

0 0 C0

∂2v∂b2

∂2P∂b2

∂2T∂b2

+

K0µVXVX

+ K0VXV K0

VXP 0

K0 TVXP 0 0

0 0 K0VT + K0

TT

∂2v∂b2

∂2P∂b2

∂2T∂b2

=

∂2RB

∂b2 + ∂2RS

∂b2

0

∂2QB

∂b2 + ∂2QS

∂b2

−

∂2M∂b2 0 0

0 0 0

0 0∂2C∂b2

v0

P0

T0

−

∂2KµVXVX

∂b2 + ∂2KVXV

∂b2

∂2KVXP

∂b2 0

∂2KTVXP

∂b2 0 0

0 0∂2KVT

∂b2 + ∂2KTT

∂b2

v0

P0

T0


− 2

∂M∂b

0 0

0 0 0

0 0∂C∂b

∂v∂b

∂P∂b

∂T∂b

− 2

∂KµVXVX

∂b+ ∂KVXV

∂b

∂KVXP

∂b0

∂KTVXP

∂b0 0

0 0∂KVT

∂b+ ∂KTT

∂b

∂v∂b∂P∂b∂T∂b

.

(2.260)

Furthermore, following Equations (2.258)–(2.260), we introduce the nth-orderrecursive formulas corresponding to the generalized stochastic perturbationtechnique:

n∑k=0

(n

k

)

∂kM∂bk

0 0

0 0 0

0 0∂kC∂bk

∂n−kv∂bn−k

∂n−kP∂bn−k

∂n−kT∂bn−k

+n∑

k=0

(n

k

)

∂kKµVXVX

∂bk+ ∂kKVXV

∂bk

∂kKVXP

∂bk0

∂kKTVXP

∂bk0 0

0 0∂kKVT

∂bk+ ∂kKTT

∂bk

∂n−kv∂bn−k

∂n−kP∂bn−k

∂n−kT∂bn−k

=

∂nRB

∂bn + ∂nRS

∂bn

0

∂nQB

∂bn + ∂nQS

∂bn

(2.261)

which gives the fundamental set of stochastic perturbation method equationsenabling computations of 0 to n partial derivatives of the state variables and theirfinal probabilistic moments. It is necessary to mention that the coupling matricescontaining usually the linear functions of material parameters differentiated twicewith respect to any of them vanish, so that the basic difference usually appears inthis context between the first- and second-order equations. The second reason tohighlight the second-order equations, despite the recursive formulas for the givennth order, is that very often the LSM technique returns an almost linear responsefunction, so that, instead of the higher-order technique one may return to the classicalSOSM technique. A simple numerical illustration of the DDM technique with fewdegrees of freedom is provided for coupled heat and fluid flow in [65].


2.3 Computational Illustrations


2.3.1.1 1D Fluid Flow with Random Viscosity

Numerical analysis deals with viscous fluid flow in the channel, where viscosity isthe random Gaussian input parameter with expected value E[µ] = 1.0 and coefficientof variation α(µ) = 0.15 (or varying in the interval [0.0, 0.20] in further numericalstudies). From the physical point of view, we test the flow between two parallel platessliding with relative velocity in the horizontal direction. The boundary conditionsfor fluid flow at the bottom and top of this channel are purely deterministic,so that theoretically any probabilistic characteristics except the expectation mustequal 0 (which is documented by the computations). The SFEM discretization isprovided using 11 linear stochastic two-noded finite elements and up to tenth-orderperturbations are taken into account (see Figure 2.1 containing a deterministic velocityprofile shown by the dashed line). The main purpose of this illustration is analysisof the probabilistic fluid flow profile (its expectation and higher-order statistics) asa function of the vertical coordinate for the specified α(µ) and, separately, maximumfluid velocity probabilistic moments as functions of the input random fluctuationsas well as the perturbation parameter ε ∈ [0.9, 1.1] (traditionally substituted as 1).This is done to check which one of these parameters is more decisive for the finalprobabilistic characteristics.

The results of numerical analysis are presented in Figures 2.2–2.4 in the formof, in turn, expected values and coefficients of variation (Figure 2.2), third andfourth central probabilistic moments (Figure 2.3), as well as skewness and kurtosis(Figure 2.4) of the fluid velocity profiles obtained for α(b) = 0.15. All these resultsare shown with respect to the second-, fourth-, sixth-, eighth-, and tenth-orderSFEM analyses to verify probabilistic convergence of the method with respect to

H = 2.0

HU

t (y) = m

yHU

u(y) =v

U = 10

U = 0.0

Figure 2.1 FEM discretization of the 1D problem. Reproduced with permission fromElsevier


2

1.5

1

0.5

00 2

2nd-order

8th-order

4th-order10th-order

6th-order 2nd-order

8th-order

4th-order

10th-order

6th-order

4 6

E(u) a(u)

8 10 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

h

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

h

(a) (b)

Figure 2.2 The expected (a) values and (b) coefficients of variation of the fluidvelocity. Reproduced with permission from Elsevier

1.8

1.6

1.4

1.2

1

0.8

h

0.6

0.4

0.2

0

1.8

1.6

1.4

1.2

1

0.8

h

0.6

0.4

0.2

00

4th-order

10th-order

6th-order 8th-order 4th-order

10th-order

6th-order 8th-order

1 2m3 (u) m4 (u)

3

(a) (b)

4 5 5 10 20 30

Figure 2.3 (a) Third and (b) fourth central probabilistic moments of the fluid velocity.Reproduced with permission from Elsevier


1.8

1.6

1.4

1.2

1

0.8

h

0.6

0.4

0.2

0

1.8

1.6

1.4

1.2

1

0.8

h

0.6

0.4

0.2

00

4th-order

10th-order


10th-order

6th-order 8th-order

0.2 0.4b(u) k(u)

0.6 0.8

(a) (b)

1 1.2 1.4 0 1 2 3 4 5

Figure 2.4 (a) Skewness and (b) kurtosis of the fluid velocity. Reproduced with per-mission from Elsevier

0.05

0.04

0.03m3(

q)

0.02

0.900.95

1.001.05

1.10 0.20

(a) (b)

0.150.10

0.050

1.81.71.61.51.41.31.21.1

b(q

)

e a

1086

40.90

0.95

1.001.05

1.10 0.200.15

0.10

0.05

0

ea

108

6

4

Figure 2.5 Parametric variations of the (a) third central probabilistic moments and(b) skewness

its order. Further, a parametric variability of higher central moments (Figure 2.5)and characteristics (Figure 2.6) computed at the top of the channel with respect tothe perturbation parameter and the input coefficient of variation α(b) ∈ [0.0, 0.2] isgiven. Let us mention that the total time of computations in this case is only slightlylarger than 11 times the original deterministic solution plus some small amount forprobabilistic symbolic post-procedures.

As one could expect from theoretical considerations, the expected values have forall orders exactly the same linear profile as the deterministic counterpart; some smalldifferences are noticed for the close neighborhood of the channel bottom. Now, even


0.070.06

0.040.05

m4(

q)

0.02

0.900.95

0.03

1.001.05

1.10 0.200.15

0.100.05

0

6

5

4

3

k(q

)

e a

108

6

4

0.90

0.95

1.00

1.05

1.10 0.20(b)(a)

0.15

0.10

0.05

0

e a

10

8

6

4

Figure 2.6 Parametric variations of the (a) fourth central probabilistic momentsand (b) kurtosis

the second-order stochastic perturbation technique is correct for all values of inputα(b) since all models with increasing perturbation orders return the same valuesalong the channel height.

The coefficients of variation of velocity are constant along the height for allperturbation orders except the same small interval close to the bottom, wherewe noticed some differences between the expectations and deterministic origin.Now, we obtain essential differences for the second- and higher-order perturbation-based results, which all approach the limit almost equal to the input coefficient ofvariation, so that the fourth-order results are quite accurate. We may notice herealso that the higher the perturbation order, the larger the final value of the studiedvariation coefficient. Although the variance profiles are omitted here (as partiallyincluded in the coefficient of variation), they have an almost parabolic distributionbecause the expectations distribution suggests a linear transform between the outputfinal velocity and the vertical coordinate; they demonstrate a similar probabilisticconvergence as this coefficient. Probabilistic convergence of higher-order moments(Figure 2.3) and statistics (Figure 2.4) is very slow – as before, the absolute valuesincrease together with the perturbation order and the most transparent differencesbetween the corresponding orders are detected at the top of the channel. The momentprofiles are highly non-linear (the higher the central probabilistic moment, the greaterthe curvature of the relevant functions), while the coefficients remain constant exceptsome fluctuations at the bottom neighborhood, exactly analogous to the variationcoefficient (Figure 2.2). The third central moment is positive everywhere and for allperturbation orders, so that the final distribution of the velocity profile shows rightasymmetry (a large portion of this distribution is larger than its median). Final valuesof the kurtosis remain clearly positive, enabling us to conclude that the velocityPDF is not Gaussian, so that higher-order statistics computations are necessary and,further, they are distributed more closely to the expectation than is typical for the


normal distribution (leptokurtic distribution). Statistical analysis in this case is moreexpensive than for the Gaussian state functions since higher-order statistics essentiallyneed larger populations to be generated and estimated, so that the efficient stochasticperturbation approach is of special importance in computational engineering.

Contrary to other implementations of the stochastic perturbation method [112],the proposed hybrid methodology based on symbolic analysis allows for a directinclusion of both parameters α(b) and ε in determination of all probabilistic char-acteristics. We study the two-parametric polynomial expressions for the third andfourth probabilistic moments and the corresponding coefficients for a point veryclose to the channel top (cf. Figures 2.5 and 2.6). The expectations and variancesare omitted because for the given variability of the input parameters there arepractically no apparent differences between the perturbation orders, so that theirfluctuations are analogous to these presented above. Both probabilistic moments areclearly non-linear with respect to the perturbation parameter remaining for givenvalues almost constant with respect to the input random fluctuations; skewness andkurtosis are almost linear with respect to both input parameters. As one may expect,the larger the value of the parameter ε, the larger the final value of the probabilisticmoment and the coefficient computed; they also increase together with an order ofthe stochastic perturbation method. From a comparison of the third- and fourth-ordercharacteristics one can conclude that a small difference between the eighth and tenthorders for the third moment increases for the same combination of input parametersin case of the fourth-order probabilistic characteristics. From a practical point of view,the absolute values of differences for the highest perturbation orders applied hereand obtained for kurtosis and the fourth moment do not qualitatively change thePDF of the fluid response. Since these differences are in the range of a few percentonly for the maximum input dispersion taken into account here, we can recommenda full tenth-order approach for an efficient computation of the first four momentsand characteristics. Summing up, a general conclusion that can be drawn from thisexample is that the higher the moment that needs to be computed, the higher thestochastic perturbation approach that is necessary.

2.3.1.2 2D Potential Problem with the Response Function Method

The second numerical illustration deals with the torsion of the isotropic, linear ran-domly elastic prismatic rectangular beam and is carried out using 100 three-nodedtriangular plane finite elements; FEM discretization is shown in Figure 2.7 [82]. Thetwisting moment has unit value here, and is applied at the left bottom corner ofthe rectangle, while the upper and right edges are free from stresses. The expectedvalue of the Gaussian transverse modulus is taken as equal to E[G] = 0.16080E + 06,while its standard deviation corresponds as before to 15% of this value. The entirecomputational procedure is carried out using the academic FEM program FENAPlinked to the symbolic computing platform of Maple, v. 11, where the nodalresponse functions, element fluxes, as well as the probabilistic moments and coef-ficients are computed and visualized together with the original FEM mesh. Theresulting state variables and their functions are compared in Figures 2.8–2.11; the


expected values and standard deviations (Figure 2.8), the variances and coefficientsof variation (Figure 2.9), the third and fourth and fourth probabilistic moments ofthese fluxes (Figure 2.10), as well as their asymmetry coefficients and kurtosis withinall the finite elements (Figure 2.11). A comparison of Figures 2.7 and 2.8 enables usto momentarily conclude that the expected values are practically the same as thecorresponding deterministic results – the differences obtained for extremum valuesare negligible, while the spatial distribution of the fluxes is the same.

It is necessary to mention that the stress vectors have some physical meaning in thecase of the expected values only – the remaining maps inform us about the specificproperties or tendencies of the output probabilistic distributions of local stressesonly. Third-order moments behave quite unsystematically and close to the positivemaxima we notice the local extremum values in the opposite direction, whereas thefourth moments are always positive and the maximum stress vectors are directedaccordingly.

As one could expect, the coefficient of variation of both horizontal and verticalfluxes within the entire plane region is almost constant and equal to 0.15 (exactly theinput coefficient of variation). This result coincides with the results of the first test andconfirms the quality of the output obtained using an application of the local versionof the RFM against these obtained with the well-recognized SFEM based on thedirect differentiation procedure. However, contrary to the Poiseulle flow analyzedbefore, now the random variable should induce planar output random field, which isconfirmed by Figures 2.8–2.11. Figures 2.8 and 2.9 verify the intuitive prediction thatthe locations of extremum values for expectations, standard deviations, and variancesmust coincide (where the last two are always positive). According to the mathematicallimitations only, maxima of expectations are directed accordingly to the input twistingmoment and since this direction is anticlockwise then for the accompanying maximaof variances and standard deviations we obtain an inverse direction. The right bottomedge of the twisted structure only shows the consistent vertical directions and valuesof both moments’ maximum values. As is documented further in Figure 2.10, thethird and fourth central probabilistic moments exhibit almost the same tendencyexcept for the fact that for the rest of the cross-sectional area these moments havealmost zero values. Finally, Figure 2.11 illustrates the distributions of the asymmetrycoefficient and the kurtosis, respectively, demonstrating a completely chaotic disorderin these characteristic distributions. It is impossible to recognize whether the finaldistribution has dominating positive or negative asymmetry, even with respect tosome specific location within this cross-sectional area; the same notice is valid withrespect to the kurtosis – most probably a finer meshing of this cross-section will notgive an answer. The intervals of these parameters’ variability are obtained – for theasymmetry coefficients as [−0.290, 0.205] for the horizontal stresses, and [−0.335,0.328] for their vertical components; the additional intervals for the kurtosis are asfollows: for the horizontal components [−0.390, 0.428] and [−0.398, 0.405] – in thevertical direction. So, one can conclude that some negative asymmetry prevails in bothdirections (larger parts of the distributions below the medians), whether the positivetendency prevails in kurtosis. Therefore, the flux distributions are more concentratedaround their expected values than the input random transverse modulus.


= 0

= 0

∂y∂F

∂F∂x

M

0.05

(a) (b)

0.1

Local Maxima: Q[x] = −.53333E+02,Q[y] = .14309E+04

Figure 2.7 (a) FEM discretization of the 2D problem and (b) its deterministic solution.Reproduced with permission from Elsevier

Local Maxima: Q[x] = −.53370E+02,Q[y] = .14310E+04

(a) (b)

Local Maxima: Q[x] = 0.17043E+03,Q[y] = 0.21449E+03

Figure 2.8 (a) The expected values and (b) standard deviations of the fluxes. Repro-duced with permission from Elsevier


Local Maxima: Q[x] = 0.29046E+05,Q[y] = 0.46004E+05

(a) (b)

Q[x] = 0.15523E+00,Q[y] = 0.15558E+00

Figure 2.9 (a) The variances and (b) coefficients of variation of the fluxes. Reproducedwith permission from Elsevier

Local Maxima: Q[x] = 0.40804E+04,Q[y] = 0.13372E+06Local Maxima: Q[x] = −0.12787E+06,Q[y] = −0.15708E+06

Local Maxima: Q[x] = 0.12285E+05,Q[y] = 0.63491+10

(a)

(b)

Figure 2.10 (a) Third and (b) fourth central probabilistic moments of the fluxes.Reproduced with permission from Elsevier


Local Maxima: Q[x] = 0.20531E–01, Q[y] = 0.32837E–01Local Minima: Q[x] = 0.29041E–01, Q[y] = 0.33468E–01

Local Maxima: Q[x] = 0.42775E+00, Q[y] = 0.40516E+00Local Minima: Q[x] = –0.39048E+00, Q[y] = –0.39770E+01

(a)

(b)

Figure 2.11 (a) The skewness and (b) kurtosis of the fluxes. Reproduced with permissionfrom Elsevier

Finally, let us note that the visualization of higher probabilistic moments in theFEM analysis is a quite separate challenge here, the very exceptional capability inthe commercial systems, and would be very difficult in case of 3D modeling. Theimplementation and presentation method does not depend on the choice of inputrandom variable – multiple sets of partial derivatives need to be incorporated intoanalogous equations for the resulting probabilistic moments for multiple uncertaintysources, and Maple can serve together with its extended procedure options.Analogously to the deterministic FEM modeling, the results obtained for the firsttwo cases may be, according to the well-known field similarities, used to predict thefirst four probabilistic characteristics in seepage analysis, electromagnetic analysis,and other related scalar problems governed by similar equilibrium equations. Sinceautomatic data processing is used in Maple, the size of the original deterministicFEM problem does not result in a dramatic increase of computer power or timeconsumption in the performance of probabilistic analysis.

2.3.2 Linear Elasticity

2.3.2.1 Simple Extended Bar with Random Stiffness

Computational implementation of the generalized perturbation technique in con-junction with finite element analysis is performed in this case study entirely in theMaple symbolic environment using the DDM. How it was documented above, itenables (i) relatively easy extension of the technique to any perturbation order, (ii)convergence analysis for particular probabilistic moments and analysis orders, but


P P

1 11

Figure 2.12 Numerical example discretization. Reproduced with permission from Else-vier

also (iii) integration with FORTRAN codes for other FEM programs, as well as (iv) effi-cient visualization of the results. This code is tested on an example of 10 finite elementdiscretization (1D linear elastic bar elements) for the simple tension of a prismaticbeam with constant cross-section [74]. The following data, consistent with Figure 2.12,have been adopted: E[e] = 200.0E9 (Young’s modulus), A = 1.0E−4 (cross-sectionalarea), l = 1E−1 (finite element basic dimension), P = 10E5 (forces applied).

The test problem is a good illustration of probabilistic convergence because of theinfinite number of non-zero partial derivatives with respect to the random inputparameter, since each new perturbation order introduces some new components intothe probabilistic output of tensioned edge deformation. Next, it is observed thatthe exact analytical solution for this problem consists of inversion of the Gaussianinput quantity, which means that an integral equivalent to the expected valuedoes not converge and its value can be obtained by numerical integration only.The polynomial expression for the expected values in the tenth-order approach isreturned symbolically by Maple for the purpose of illustration at this edge as

E[q11 (ε, α(b))

] = q011 + 0.02392 ε2α2(e) + 0.07177 ε4α4(e) + 0.35885 ε6α6(e)

+ 2.51196 ε8α8(e) + 22.60766 ε10α10(e)

and all further response characteristics are studied at the same location. The compu-tational illustration is summarized first in the form of the expectations and variancesplotted in Figure 2.13, standard deviations and coefficients of variation, cf. Figure 2.14,third and fourth central probabilistic moments in Figure 2.15, as well as skewness andkurtosis, see Figure 2.16, where the perturbation parameter is taken as equal to 1. Theyare all computed as functions of the input coefficient of variation α(b) ∈ [0.00, 0.25]and the perturbation order of the approximation – from second up to tenth for theexpected values and from fourth up to tenth for the remaining characteristics. Asis documented by the first four diagrams, the second-order technique returns reallysatisfactory values for the first- and second-order characteristics, maybe except forthe values obtained for α(b) > 0.20, but it may follow the simplicity of the exampleconsidered. Of course, the higher the order of the SFEM analysis, the larger the valuesof the first two moments and all their functions are undoubtedly convex with slightlydifferent second-order partial derivatives w.r.t. the coefficient α. It is seen that theuncertainty level is remarkably increased by this system since the ratio of output toinput random dispersion is, after right-hand diagram in Figure 2.14, larger than 1.

The third and fourth central probabilistic moments behave similarly to the lower-order moments – also convex, continuous, and monotonously increasing togetherwith α, highly non-linear but with very small absolute values close to 0. Theirnormalization with the additional powers of the standard deviation into the skewness


0.0206

0.00004

0.00003

0.00002

0.00001

00 0.05 0.10 0.15 0.20 0.250 0.05 0.10 0.15

(a) (b)

a a0.20 0.25

0.0204

0.0202

0.0200

0.0198E(u

max

)

Var

(um

ax)

0.0196

0.0194

0.0192

2nd-order

8th-order 10th-order


10th-order

4th-order 6th-order

Figure 2.13 (a) Expected values and (b) variances of the maximum deformation

0.007

0.006

0.005

0.004

s(u

max

)

a(u

max

)

0.003

0.002

0.001

0 0

0.3

0.2

0.1

0 0.05 0.10a

0.15

(a) (b)

0.20 0.25 0 0.05 0.10a

0.15 0.20 0.25

4th-order 6th-order 8th-order10th-order


Figure 2.14 (a) Standard deviations and (b) coefficients of variation of the maximumdeformation

and kurtosis confirm the initial theoretical consideration that the output randomdeformation is never Gaussian and the larger the input α, the larger the distancefrom Gaussian characteristics. It is interesting that while skewness remains alwaysnegative (left asymmetry of the PDF), the kurtosis starts from negative values for thefourth-order SFEM and through moderate values for sixth order tends to significantpositive values in the tenth-order SFEM analysis. A good convergence for both


9. x 10–7

2. x 10–8

1.5 x 10–8

1. x 10–8

5. x 10–9

8. x 10–7

7. x 10–7

6. x 10–7

5. x 10–7

3. x 10–7

2. x 10–7

1. x 10–7

0 0

4. x 10–7m3(

u max

)

m4(

u max

)

0 0.05 0.10a

0.15

(a) (b)

0.20 0.25


0 0.05 0.10a

0.15 0.20 0.25


Figure 2.15 (a) Third and (b) fourth central probabilistic moment coefficients ofvariation of the maximum deformation

–0.5

–1

–1.5

–2

–2.5 –1

0

b(u

max

)

k(u

max

)

1

2

3

4

5

6

0 0.05 0.10a

0.15

(a) (b)

0.20 0.25


0 0.05 0.10a

0.15 0.20 0.25


Figure 2.16 (a) Skewness and (b) kurtosis of the maximum deformation

characteristics is noticed for α < 0.15 and we need to increase the approximationorder in SFEM for larger values.

Further, we study the expectations (Figure 2.17), the coefficient of variation(Figure 2.18), skewness (Figure 2.19), as well as kurtosis (Figure 2.20) as thetwo-parametric functions of both perturbation parameter ε and input coefficientof variation α; the variability interval of the first variable is slightly wider thanbefore and we have ε ∈ [0.80, 1.20]. Apparently, the first two random characteristics


0.02060.02040.02020.02000.01980.0196

E(u

max

)

0.01940.0192

0.8

0.9

1.0

1.1

1.2 0.20

0.15

0.10

0.15

0

ae

108642

Figure 2.17 The expected values of the maximum deformation

0.3

0.2

0.1a(u

max

)

0.8

0.91.0

1.1

1.2 0.20

0.15

0.10

0.05

0

e a

108642

Figure 2.18 The coefficients of variation of the maximum deformation

of a deformation are non-linearly dependent upon both input parameters havingpositive value for the entire computational domain. It is clear that the more influentialparameter is the input coefficient of variation, even for such a wide choice for theinput ε.

The changes between neighboring orders of the SFEM are regular, as the figuresshow, in the sense that the higher the analysis order, the larger the probabilisticfirst- and second-order response but these differences even for the largest valuesof both input variables are practically negligible. The dominating role of the inputcoefficient of variation remains valid for the skewness and kurtosis also, but aninterrelation with respect to the perturbation parameter is even closer to the linear


–0.5

–1

–1.5

–2.50.8

0.9

1.0

a1.1

1.2 0.20

0.15

0.10

0.05

0

–2b(u

max

)

10

86

4

ε

Figure 2.19 Skewness of the maximum deformation

6543210

–1

k(u

max

)

0.8

0.9

1.0

1.1

1.2 0.20

0.15

0.10

0.05

0

a

4

6

8

10

ε

Figure 2.20 Kurtosis of the maximum deformation

function, especially for lower-order SFEM analysis. It should be mentioned thatincreasing the perturbation parameter value increases each time absolute value ofthe probabilistic response, so that for skewness we have effectively the negativeextremum for a combination of maximum values of both input parameters.

The main value of the DDM implementation is a lack of response function determi-nation for all nodes separately, so that the computational effort and time for solution


are almost the same as for the deterministic origin. We also avoid any numericalerror coming from polynomial interpolations or the least-squares technique, havingas a benefit two-parametric polynomials of the stochastic response – with respect tothe perturbation parameter and, independently, the input coefficient of variation.

2.3.2.2 Elastic Stability Analysis of the Steel Telecommunication Tower

The main aim of this example is a determination of the critical load multiplier for thesteel telecommunication tower, whose FEM discretization and additional unit forcesfor stability analysis are shown in Figure 2.21. This 55.20 m high lightweight structurewas initially designed with nine 6.0 m long subsections and surface X stiffeners, itshorizontal cross-section being a regular triangle with basic dimension equal to 7.3 mat the bottom and only 2.5 m at the top [106]. The main structure is formed withsemi-horizontal legs designed as full round cross-sections with diameters 110, 100,90, and 80 mm counted from the bottom. The legs inside the particular subsectionsare treated as continuous elements, connected with additional small, thin steel plates(10 mm thick) via bolts. Practically, this enables very limited rotation of the higherleg with respect to the lower one, but in the computational model we treat thisconnection as infinitely stiff. The stiffening braces under compression/tension areperformed with the use of angular cross-sections L150 × 100 × 10, L120 × 80 × 10,and L90 × 60 × 8 (the higher the location, the lower the cross-section). An additionaltechnological platform 1.2 m high is mounted above the top of the last segment,which is perfectly attached to the upper legs and manufactured with three steelpipes ∅ 82.9 mm × 4 mm with some horizontal barriers. Considering three reinforcedconcrete foundation blocks having dimensions 4.0 × 4.0 m and 3.00 m thick, wemodel this tower as completely clamped at the bottom ends of the leg. Probabilisticcomputational analysis is performed here using the commercial engineering FEMsystem ROBOT in cooperation with the system Maple. The FEM model in ROBOTconsists of 30 3D linear beam finite elements (as the tower legs) and 156 3D barlinear finite elements – connected at 298 nodal points. From a probabilistic pointof view, we contrast the influence of two input Gaussian random variables – theYoung’s modulus of steel as well as the diameter of the lower legs (mostly exhibitedby the corrosion process and any possible mechanical accidents), see Figure 2.22.We calculate the basic probabilistic characteristics of the critical load multipliersto verify how much additional telecommunication equipment we can add at thetop of the tower to keep this structure in a safe exploitation regime, extending itssignal transmission capabilities. Further analyses will obey some sinusoidal forcedvibrations with the dynamic force located at the top also because of the future usageof the towers in global renewable energy production.

The results of computational analysis for the first and fourth critical load magnitudeare given in pairs – for the randomized Young’s modulus on the right and for therandomized leg diameter on the left. The response functions given in Figure 2.23 aswell as the expected values plotted in Figure 2.24 are given in kN. Then, we comparethe coefficients of variation (Figure 2.25), skewness (Figure 2.26), as well as kurtosis


FZ = –1.00FZ = –1.00

Figure 2.21 3D FEM model of the steel tower

(Figure 2.27) – all computed for increasing stochastic perturbation technique orders,from second or fourth up to tenth. A quite analogous visualization is providedin Figures 2.28 and 2.29 (for the uncertain Young’s modulus only) in the contextof the fourth critical value and, similarly to most of the figures, we introduce theinput coefficient of variation on the horizontal axes as the independent parameter.Its upper bound is not driven by the experimental results, more important here isthe computational aspect – to enable precise evaluation of the particular moments’behavior with respect to input random fluctuation levels. The recovery of bothresponse functions having global character is based on 13 FEM iterations of theinitial deterministic problem with fluctuating random input variables accordingto Figure 2.23, and the least-squares approximation by the fifth-order polynomial


Figure 2.22 First critical configuration for Young’s modulus (a) and for the randomdiameter of the structural elements (b)

190 200 210

e

220 230 0.106 0.110

d

0.116

31

32

33

34

35

36

37

38

39

Pcr

Pcr

34.6

34.8

35.0

35.2

35.4

35.6

35.8

Figure 2.23 Polynomial response function for the critical load (first critical value)for the randomized Young’s modulus (a) and leg diameter (b)


0 0.05 0.10

(a) (b)

a(e)0.15 0.20 0 0.05 0.10

a(d)0.15 0.20

4th-order analysis8th-order analysis6th-order analysis

2nd-order analysis

10th-order analysis


2nd-order analysis

10th-order analysis

35.278135.278235.278335.278435.278535.278635.278735.278835.278935.2790

E(P

cr)

E(P

cr)

20

25

30

35

Figure 2.24 The expected values for the critical force (first critical value)for the (a) randomized Young’s modulus and (b) diameter

0 0.05 0.10

(a) (b)

a(e)

0.15 0.20 0 0.05 0.10

a(d)

0.15 0.20


2nd-order analysis

10th-order analysis


2nd-order analysis

10th-order analysis

0

0.05

0.10

0.15

0.20

0

1

2

3

4

5

6

a(P

cr)

a(P

cr)

Figure 2.25 The coefficient of variation for the critical force (first critical value)for (a) the random Young’s modulus and (b) random diameter

function. Let us mention that the window size variation for the polynomial responsefunction reconstruction does not essentially change the final form of this function forthe critical load, but this is not the case for all computational examples provided in thisbook. As one could guess, both response functions are linear (for Young’s modulus)


0 0.05 0.10

(a) (b)

a(e)0.15 0.20 0 0.05 0.10

a(d)0.15 0.20

4th-order analysis8th-order analysis




0

0.0005

0.0010

0.0015

0.0020

–2.5

–2

–1.5

–1

–0.5

0b

(Pcr

)

b(P

cr)

Figure 2.26 The coefficient of skewness for the critical force (first critical value)for (a) the random Young’s modulus and (b) random diameter

0 0.05 0.10

(a) (b)

a(e)0.15 0.20 0 0.05 0.10

a(e)0.15 0.20





–0.10

–0.08

–0.06

–0.04

–0.02

0

0.02

0.04

0.06

0.08

k(P

cr)

k(P

cr)

–2

–1

0

1

2

3

Figure 2.27 Kurtosis for the critical force (first critical value) for (a) the random Young’smodulus and (b) random diameter

or almost linear (for a random diameter), so that the higher-order perturbations arenot so influential on the probabilistic characteristics computed.

The most important conclusion, which partially confirms this model, is that thetower is really very sensitive to its leg diameters and a few orders less sensitiveto the Young’s modulus. It follows directly Figure 2.24, where expectation changeswith respect to the variation coefficient are negligible for Young’s modulus andextremely large in case of the random diameter. The first impression after these


0 0.05 0.10a(e)

0.15 0.20 00

0.05

0.05 0.10a(e)

0.15 0.20

(a) (b)


2nd-order analysis

10th-order analysis


2nd-order analysis

10th-order analysis

49.6360

49.6365

49.6370

49.6375

49.6380

49.6385E

(Pcr

)

a(P

cr)

0.10

0.15

0.20

Figure 2.28 (a) The expected values and (b) coefficient of variation for the fourthcritical force with randomized Young’s modulus

0

0

0.001

0.002

0.003

0.004

b(P

cr)

k(P

cr)

0.05 0.10

(a) (b)

a(e)0.15 0.20 0

–0.02

–0.01

0

0.01

0.05 0.10a(e)

0.15 0.20





0.02

Figure 2.29 (a) The skewness and (b) kurtosis for the fourth critical forcewith randomized Young’s modulus

figures is that the second-order analysis is extremely different from the higher-orderapproximations and, as such, inacceptable in both cases – where the least commonvalue with the remaining functions is for α(b) = 0.075. However, while the left graphshows marginal fluctuations of the critical load magnitude expectations, the rightgraph shows its fluctuations of paramount importance. It is interesting that all


higher-order moments give almost exactly the same values independently from thisinput α(b), and it is a good illustration for the fact that efficient determination of theexpectations for some engineering problems may demand higher than second-orderperturbations even for small input random fluctuations. As one could imagine afterthe basic strength of materials formulas, the higher the variation coefficient for theleg’s diameter, the smaller the critical load expectation. The output coefficient ofvariation in Figure 2.25 shows essentially different behavior – almost linear for theYoung’s modulus (as in the previous examples), and highly non-linear for the legdiameter. We see that the Gaussian randomization of the Young’s modulus of steelpreserves the uncertainty level using the SFEM technique proposed (and also othernumerical techniques) [103]. The spectrum of possible values is also completelydifferent for both input variables being measured in units of d(ω); some comparablebehavior is noticed up to the level of α(b) < 0.1, limiting the range of validity forthe second-order perturbation theories. Apparently, after Figures 2.26 and 2.27 thecritical load magnitude has Gaussian distribution once we randomize the Young’smodulus, while the uncertainty in the cross-sectional area differs completely fromthis PDF, having a spectrum of both negative and positive values of skewness andkurtosis significantly different from Gaussian zero magnitudes. Although the largerthe input dispersion the larger absolute values of the computed characteristics in thiscase, the final extremum values are still counted in tenths, so that the main tendencyremains the same.

The functions describing these coefficients with respect to the initial α(b) are alsodifferent – we notice monotonous behavior for Young’s modulus randomization,resulting in both increasing and decreasing together with an additional increase of theinput coefficient of variation. Analogous curves obtained for diameter randomizationresult in both monotonous functions for smaller α(b) but also clearly non-monotonousand exhibiting few saddle points for intermediate values of the input parameter α(b).Anyway, they all start from 0, which corresponds to the deterministic case and givessimple verification and, further, show no singularities for any value of input variationcoefficient.

The results obtained for the fourth critical value for randomized Young’s modulusonly are in almost perfect agreement with these computed for the first one before. It isseen that independent of the critical value number, the randomness in elastic stabilityanalysis could be preserved (with zero fluctuation of the probabilistic entropy forthe output and input random variables). We obtain clearly Gaussian distributionsfor both critical loads in all perturbation order methods and all values of the inputcoefficient of variation. Considering simple random transforms of the Euler formulathis is a somewhat predictable result since the critical force is linearly dependent onthe Young’s modulus and also on the inertia moment of the bar cross-section, whichmeans the fourth power of the leg’s diameter; this fact may cause the significantdifferences in uncertainty results for both parameters.



A demonstration of the stochastic generalized perturbation-based FEM related to thenon-linear elasticity problems is made using the plane truss structure presented inFigure 2.30, where the Young’s modulus of this structure is considered as the inputGaussian random variable. Its expected value equals 30 GPa, while the coefficientof variation ranges from 0.0 (equivalent to the deterministic test) to 0.20. The entirecomputational procedure is performed using the classical academic FEM softwareFENAP with enabled elastoplastic analysis and the linear 2D truss finite elementsand symbolic computational algebra system Maple.

Using this hybrid computational procedure we examine the maximum verticaldisplacements at node 1 (having maximum absolute value), where the verticalconcentrated force acting downwards has been applied [79]. The set of all input datais enabled for nine increments in the analysis, and each time the response functionof this displacement with respect to a random input is recovered using 11-pointpolynomial interpolation (cf. Figure 2.31(a)). All polynomial interpolations givenin this figure show that the approximants are continuous and smooth functions ofthe input parameter with no singularities; they exhibit monotonous behavior forall increments. As one may expect, the larger the Young’s modulus (marked on thehorizontal axis), the less the vertical displacements computed and they systematicallyincrease together with the increment number.

Next, the probabilistic moments are plotted – now as functions of the input coef-ficient of variation given on the horizontal axis and ranging from 0.0 to 0.20. These

3

8

6

5

4

3

2

1

5

7

1 2

10

4

6

9

0,91

4 m

0,914 m

1,828 m

0,914 m

Figure 2.30 The plane truss structure. Reproduced with permission from Springer


–0.14

–0.12

–0.10

–0.08

–0.06

–0.04

–0.02u(

e)

1. × 1072. × 1073. × 107

(a) (b)

4. × 1075. × 107

e0 0.05 0.10 0.15 0.20

a

–0.05

–0.04

–0.03

–0.02

–0.01

E(u

(b))

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

Figure 2.31 (a) Deterministic and (b) expected values for the vertical displacementsat the first node. Reproduced with permission from Springer

00 0

0.002

0.05 0.10

(a) (b)

a

0.15 0.20 0 0.05 0.10a

0.15 0.20

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

0.00014

0.00016

Var

(u(b

))

s(u

(b))

0.004

0.006

0.008

0.010

0.012

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

Figure 2.32 (a) Variances and (b) standard deviations of the vertical displacementsat the first node. Reproduced with permission from Springer

are in turn: the expected values (Figure 2.31b), the variances and standard deviations(Figure 2.32), the coefficient of variation of the resulting displacements (Figure 2.33),the third and fourth central probabilistic moments (Figure 2.34), as well as skew-ness and kurtosis (cf. Figure 2.35). All probabilistic moments and characteristics aredetermined consecutively using the tenth-order expansion. Now it is apparent fromFigure 2.31 that the higher the increment of the analysis, the larger the expectation


0 0.05 0.10a

0.15 0.20

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

0

0.05

0.10

0.15

0.20

a(u

(b))

Figure 2.33 The coefficients of variation for the vertical displacementsat the first node. Reproduced with permission from Springer

0

–4. × 10–6 5. × 10–8

1. × 10–7

1.5 × 10–7

2. × 10–7

–3. × 10–6

–2. × 10–6

–1. × 10–6

0

0.05 0.10

(a) (b)

a0.15 0.20 0

00.05 0.10

a0.15 0.20

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

m3(

u(b)

)

m4(

u(b)

)

Figure 2.34 (a) The third and (b) fourth central probabilistic moments of verticaldisplacements at the first node. Reproduced with permission from Springer


0 0.05 0.10

(a) (b)

a0.15 0.20 0

00.05 0.10

a0.15 0.20

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

inc 1inc 6

inc 2inc 7

inc 3inc 8

inc 4 inc 5inc 9

–2

–1.8

–1.6

–1.4

–1.2

–1

–0.8

–0.6

–0.4

–0.2b

(u(b

))

k(u

(b))

1

2

3

4

5

Figure 2.35 (a) The skewness and (b) kurtosis of the vertical displacementsat the first node. Reproduced with permission from Springer

of the vertical displacement and, at the same time, the larger the variations of thisexpectation with respect to the coefficient of variation. This expectation is almostinsensitive to this coefficient during the first step of the non-linear incrementingprocedure, whereas for the last one it systematically decreases.

Once more, similar to the response functions, the functions E(u) = E(u(α)) arecontinuous and monotonous for all increments; the particular values increase system-atically together with the increments in the SFEM analysis. The variances (Figure 2.32),although all apparently non-negative according to the definition, are very close atthe beginning of the incrementation process, even for the largest value of the inputcoefficient of variation. Similar to the expectations, they exhibit continuous andmonotonous behavior (everywhere positive derivatives w.r.t. the input coefficient ofvariation) but, unlike the expectations, are convex and start to grow rapidly afterthe seventh increment. Since the standard deviations given in the right-hand graphof Figure 2.32 are determined as square roots of the variances, they have the samegeneral tendencies, although the differences between the particular increments aremore uniform, like in the case of expectations. Contrasting the maximum standarddeviation computed in the last increment with the corresponding expected value, itremains apparent that the maximum output coefficient of variation is in the samerange as the input one but somewhat larger (cf. Figure 2.33). The third and fourth cen-tral probabilistic moments (Figure 2.34) behave similarly to the variances – the onlyexceptions are the negative sign for the third moments here and the largest curvatureof the particular distributions w.r.t. the input coefficient α. Their values for the entireincrementation process remain almost constant and very close to 0, while during thelast two to three steps they grow significantly. All the moments presented are continu-ous and monotonous functions of the input Young’s modulus coefficient of variation;they exhibit no local numerical oscillations as the skewness and kurtosis computedfinally (see Figure 2.35). This last graph shows that the final probability distribution is


non-Gaussian, has more realizations below the median, and is also leptokurtic (moreconcentrated around the mean than the Gaussian bell-shaped curve) and, therefore,higher-order moments and coefficients may be necessary in reliability analysis. Onemay compare the results obtained here thanks to polynomial interpolation with theleast-squares method approximation, however, considering the expected and linearratio between the output and input random variables, these detailed studies havebeen postponed and left for further numerical investigation by the reader.

2.3.4 Stochastic Vibrations of the Elastic Structures

2.3.4.1 Forced Vibrations with Random Parameters for a Simple 2DOF System

Computational experiment for the modal superposition method is prepared usingthe following simple case study illustrating a steel column with partially constantcross-sectional area and subjected to certain longitudinal impulse forced vibrations.The SFEM model consists of two linear finite elements (with three nodes) and isfixed at the left end as well as loaded with the force impulse at the right-hand side(Figure 2.36). This load has zero magnitude in the intervals t ∈ [0, t0] and t ∈ [t1, ∞),while it equals P for t ∈ [t0, t1]. The cross-sectional area A is chosen as the truncatedGaussian input random variable with expected value equivalent to the double HEB1000, so that E[A] = 800 cm2, P = 1 kN, E = 210 GPa, l = 8.0 m; further, t0 = 5 s andt1 = 5.1 s. A choice of random input coincides with the possible stochastic corrosion,which, acting on this element, may significantly decrease its effective cross-sectionalarea. The computational procedure based on the DDM SFEM algorithm has beenentirely coded into the computer algebra system Maple, v. 14, to calculate up to thefirst four probabilistic moments and characteristics of the displacements and elasticforces in this system.

The expectations of the structural response during the loading phase (Figure 2.37)appear to be quite insensitive to the input coefficient of variation, and the dynamicalreactions at the top and half-height of the column have very similar character butdiffer by almost twice their absolute values; local extremum values appear in bothlocations at the same time. The coefficients of variation given in Figure 2.38 are almosttime-insensitive except for larger values of input coefficient α for the very beginningof the loading phase, where the second-order coefficients start from the extremum.They take enormously large values at any time for input variation α ≥ 0.15 very closeto unity rather than the traditional tenths or percents. It is characteristic that theoutput random dispersion at the column top, where P(τ ) is applied, is almost twice

2A, E, r A, E, rx P (t )ˆ

y1 2 3

Figure 2.36 Computational model for the forced vibrations.Reproduced with permission from Tech Science Press


01. × 10–6

–1. × 10–6

1. × 10–6

3. × 10–6

5. × 10–6

2. × 10–6

3. × 10–6

E(r

3)E(r

2)

5.10 0.20(a) (b)

0.150.10

at

0.050

5.085.06

5.045.025

5.10 0.200.15

0.10at

0.050

5.085.06

5.045.025

Figure 2.37 The expected values at (a) the midpoint and (b) the top during loadingphase. Reproduced with permission from Tech Science Press

0.5

1

1.5

2

5.105.08

5.065.04

5.025

0.20(a) (b)

0.15

0.050

0.10at

5.105.08

5.065.04

5.025

1

2

3

0.200.15

0.050

0.10at

a(r

2)

a(r

3)

Figure 2.38 Coefficients of variation at (a) the midpoint and (b) the top duringloading phase. Reproduced with permission from Tech Science Press

that for its half-height. A very interesting structural response is obtained for theskewness (Figure 2.39), where the waviness is detected with respect to both time and,especially, input coefficient α. It starts from almost 0 for both displacements r2 and r3,passes through some extremum values for α = 0.10, to return to 0 for maximum α(A).As far as β(r2) shows some kind of symmetry within the range of the vertical axis (thesame ranges of absolute values), β(r3) is dominated by negative values especially atthe beginning of the loading period.

Kurtosis for the loading phase presented in Figure 2.40 has values totally differentfrom 0, so that neither r2(ω) nor r3(ω) are Gaussian. Further, r3 exhibits some enormousincrease at the beginning of the loading phase and this is the only exception fromthe fact that this parameter is almost insensitive with respect to time within this


5.105.08

5.065.04

5.025

0.20(a) (b)

0.15

0.050

0.10at

5.105.08

5.065.04

5.025

0.200.15

0.050

0.10at

–0.6–2

–1

0

–0.4–0.2

00.20.40.6

b(r

2)

b(r

3)Figure 2.39 Coefficients of skewness at the (a) midpoint and (b) the top duringloading phase. Reproduced with permission from Tech Science Press

–2

–1

0

k(r

2)

k(r

2)

55.02

5.045.06

5.085.10 0.20

(a) (b)

0.150.10 a

t

0.050 5

5.025.04

5.065.08

5.10 0.20

0.150.10 a

t

0.050

–202468

10

Figure 2.40 Kurtosis at (a) the midpoint and (b) the top during loading phase.Reproduced with permission from Tech Science Press

interval. The main difference in the unloading phase expectations (Figure 2.41) isobserved in case of vibrations of the column top, where the amplitude significantlydecreases; still an influence of the input randomness is negligibly small. Now thecoefficients of variation are always highly sensitive to α(A), unlike with respect to τ ,but have almost the same values for both locations within the entire time domain(Figure 2.42). The skewness (Figure 2.43) and kurtosis (Figure 2.44) are very similarto each other for the top and half-height of the column without any sudden jumps orlocal fluctuations.


–3. × 10–6

–1. × 10–6

1. × 10–6

3. × 10–6

E(r

2)

–1.5 × 10–6

–5. × 10–7

5. × 10–7

1.5 × 10–6

E(r

3)

2018

1614

1210

86

0.20(a) (b)

0.150.10 a

0.050

t

2018

1614

1210

86

0.20

0.150.10 a

0.050

t

Figure 2.41 The expected values at (a) the midpoint and (b) the top during unloadingphase. Reproduced with permission from Tech Science Press

06

810

1214

1618

20 0.20(a) (b)

0.150.10

0.050

0.20.40.60.8

1

a(r

2)

at

06

810

1214

1618

20 0.200.15

0.100.05

0

0.20.40.60.8

1

a(r

3)

at

Figure 2.42 Coefficients of variation at (a) the midpoint and (b) the top duringunloading phase. Reproduced with permission from Tech Science Press

Whereas skewness has the same ranges of positive and negative values, thekurtosis variability is dominated by clearly negative values. This results from the factthat higher-order partial derivatives of the structural response have negative valuesand start to dominate for larger values of the input coefficient of variation. So, asa final conclusion, both displacements have non-Gaussian distributions, during bothloading and unloading phases. It should be mentioned that further applications of themethod proposed may be validated through higher-order statistics with theoreticalresults available in stochastic vibration analysis [117, 153] as well as with SOSManalyses in case of the first two structural response moments [112].


–0.6

6810

1214

1618

20 0.20(a) (b)

0.150.10

0.050

0.6

–0.4

0.4

–0.2

0.20

b(r

2)

–0.6

0.6

–0.4

0.4

–0.2

0.20

b(r

3)

at

6810

1214

1618

20 0.200.15

0.100.05

0

at

Figure 2.43 Coefficients of skewness at (a) the midpoint and (b) the top duringunloading phase. Reproduced with permission from Tech Science Press

6810

1214

1618

20 0.20(a) (b)

0.150.10

0.050

k(r

2)

k(r

3)

at

6810

1214

1618

20 0.200.15

0.100.05

0

at

–2.5–2

–1.5–1

–0.50

0.5

–2.5–2

–1.5–1

–0.50

0.5

Figure 2.44 Kurtosis at (a) the midpoint and (b) the top during unloading phase.Reproduced with permission from Tech Science Press

2.3.4.2 Eigenvibrations of the Steel Telecommunication Towerwith Random Stiffness

Next numerical case study is provided for a steel telecommunication tower withheight equal to 52.0 m, very similar to that shown schematically in Figure 2.21. TheFEM model consists of 63 two-noded linear elastic beam elements (legs, designedwith full circular cross-sections) and 120 two-noded bars connected at all 67 nodes(rebars designed as angular profiles). This structure is subjected to its dead loadand the additional weight of the antennas, cables, technological platforms, and their


supports as well as to the technological loadings introduced by the Eurocodes. Theinput random variable of this problem is the Young’s modulus having expectatione = 205 GPa and coefficient of variation belonging to the interval α ∈ [0.0, 0.2]; it isassumed to have truncated Gaussian distribution, restricted to the positive valuesonly. Usually, this coefficient is taken to be smaller than or equal to 0.15. Theengineering FEM system Rama 3D is used to determine the first few eigenfrequenciesusing the subspace iteration algorithm, which are then transferred to the systemMaple for further probabilistic computations. Polynomial approximation of theinterrelations between the particular eigenvalues and the Young’s modulus necessaryto determine partial derivatives of the eigenvalues w.r.t. this modulus is providedusing a nine-point probe with equidistant subdivision of the Young’s modulusvariability interval (±50% of the mean value around it). The final response functionsfor up to the tenth eigenfrequency are obtained as very smooth over the entiredomain, without any local oscillations, so that the partial differentiation of up to thetenth order in Taylor expansions is quite straightforward and reliable. The results ofcomputational experiments are contained in Figure 2.45 for the expected values ofthe first (a) and fourth (b) eigenfrequencies, in Figure 2.46 in case of the coefficientsof variation for both frequencies, in Figure 2.47 for their skewness, as well as inFigure 2.48 for their kurtosis; they are all, of course, shown as functions of inputrandom dispersion and for up to tenth-order stochastic perturbations.

The expected values as well as the variation coefficients for both eigenfrequenciesare independent from the order of the stochastic perturbation method for almostthe entire interval of α variability, so that here the second-order technique is totallyefficient. Further, the expectations decrease non-linearly together with an increase ofthe input coefficient of variation, but the range of these changes is insignificant. The

2.038

2.039

2.040

2.041

2.042

2.043

2.044

2.045

2.046

2.047

E(w

(e))

E(w

(e))

0 0.05 0.10

(a) (b)

a0.15 0.20 0 0.05 0.10

a0.15 0.20

2.548

2.550

2.552

2.554

2.556

2.558

2.560

2nd-order analysis6th-order analysis


10th-order analysis



10th-order analysis

Figure 2.45 Expectations of (a) the first and (b) ninth eigenvalue. Reproduced withpermission from Tech Science Press




10th-order analysis



10th-order analysis

00

0.05 0.10

(a) (b)

a0.15 0.20 0 0.05 0.10

a0.15 0.20

0.02

0.04

0.06

0.08

0.10

0

0.02

0.04

0.06

0.08

0.10

a(w

(e))

a(w

(e))

Figure 2.46 The coefficient of variation of (a) the first and (b) ninth eigenvalue.Reproduced with permission from Tech Science Press

–0.4

–0.3

–0.2

–0.1

0 0.05 0.10

(a) (b)

a0.15 0.20 0 0.05 0.10

a0.15 0.20

b(w

(e))

–0.4

–0.3

–0.2

–0.1

b(w

(e))

6th-order analysis4th-order analysis8th-order analysis 10th-order analysis


Figure 2.47 The coefficient of skewness of (a) the first and (b) ninth eigenvalue.Reproduced with permission from Tech Science Press

coefficients of variation for both eigenfrequencies increase linearly together with theinput coefficient of variation, and finally are half that of the input value.

Having further computed the third and fourth central probabilistic moments, werecover numerically the skewness and kurtosis for both eigenvibrations. Skewnessis somewhat antisymmetric to the variation coefficient with respect to the horizontalaxis, returning negative values for the entire domain and linear dependence on the


0 0.05 0.10

(a) (b)

a0.15 0.20 0 0.05 0.10

a0.15 0.20



–0.1

0

0.1

k(w

(e))

0.2

0.3

–0.1

0

0.1

k(w

(e))

0.2

0.3

Figure 2.48 Kurtosis of (a) the first and (b) ninth eigenvalue.Reproduced with permission from Tech Science Press

input α – the larger the input, the smaller the resulting skewness. However, we noticesome differences for larger input coefficients between the results computed for thefourth and remaining orders equal to each other up to α = 0.20. Kurtosis variabilityis more complex – we notice negative values decreasing with increasing values ofα, while they are positive everywhere for higher-order perturbation SFEM analyses.Probabilistic convergence is relatively slower than before since a difference for theeighth and tenth orders is almost invisible now. The values of both coefficients arevery close to 0, so that one can conclude that the final distribution is close to theGaussian one and, hence, the first two probabilistic moments may give sufficientinformation about the probabilistic eigenfrequencies of this tower. It is necessary topoint out that the left-hand graphs are exactly the same as the right-hand ones (forhigher eigenfrequency) – the only difference is in the values for the expectations.The computational effort is relatively small since the global character of the singleresponse function is recovered separately for each eigenfrequency.


2.3.5.1 Heat Conduction in the Statistically Homogeneous Rod

This computational study is devoted to heating of the statistically homogeneousand isotropic rod with constant cross-sectional area and length l = 2.0 m, where theheat conductivity is the input Gaussian random variable with variation coefficientα(k) = 0.15; its expected value is E[k] = 0.10. The temperature is fixed at the left edgeas T = 0, the heat flux is applied at the right corner, whereas the entire structureis divided into 10 three-noded parabolic finite elements. The entire computationalprocedure is implemented in the symbolic platform of Maple, v. 11, where up


0 0.05 0.10

(a) (b)

a

0.15 0.20 00

0.05 0.10a

0.15 0.20



10th-order analysis



10th-order analysis

20

20.1

20.2

20.3

20.4

20.5

E(T

(b))

a(T

(b))

20.6

20.7

20.8

20.9

0.05

0.10

0.15

0.20

Figure 2.49 (a) Expected values and (b) coefficients of variation at the heated end

to tenth-order equations are formed and solved with analytical differentiation ofall necessary system matrices; these solutions are combined into up to fourth-orderprobabilistic moments and coefficients. Uncertainty in a single physical parameterresults here in the random field of temperatures and that is why we presentprobabilistic moments and coefficients for extremum values as well as probabilisticmoment distributions along the rod. Therefore, we have the expectations andcoefficients of variation at the heated end in Figure 2.49, together with the skewnessand kurtosis at the same location in Figure 2.50, all as functions of the perturbationorder and the input variation coefficient. All these functions’ values increase togetherwith the input parameter α as well as together with the stochastic analysis order(except for the fourth-order approximation for kurtosis). Undoubtedly, the finaltemperature distribution is non-Gaussian considering the remarkable positive valuesof both skewness and kurtosis, whose probabilistic convergence with analysis orderis rather slow. In contrast, we notice as before faster convergence of expectations andcoefficients of variation, where the fourth-order approximants are quite sufficientfor the given combination of input parameters and any value of input randomdispersion. Similarly to the SFEM analysis of the extended elastic bar (cf. Figures 2.13and 2.14), the expectations exhibit convex character and the coefficients of variationare almost linear functions of the input α. The temperature distribution moments andcoefficients along the heated rod are given in Figures 2.51–2.54, where Figure 2.51contains the spatial distribution of the expected values (a) and standard deviations (b).

Now, the spatial coordinate is marked on the horizontal axes of all figures in thisseries, while numerical analysis is repeated for five different input coefficients ofvariation shown in the legends. Then we have in turn: the variances and coefficientsof variation of the temperature field (Figure 2.52), the third and fourth centralprobabilistic moments (Figure 2.53) and, finally, the asymmetry coefficients and


0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0





0.05 0.10

(a) (b)

0.15 0.20a

0 0.05 0.10 0.15 0.20a

–1

0

1

2

3

4

5

b(T

(b))

k(T

(b))

Figure 2.50 (a) Skewness and (b) kurtosis at the heated end

0

5

10

15

20

0 0.5 1

(a) (b)

1.5 2x

0 0.5 1 1.5 2x

0

1

2

3

4

5

6

7

E(T

(x))

s(T

(x))

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

Figure 2.51 Spatial distribution of (a) the expected values and (b) standard deviationsin the heated rod

kurtosis spatial variations (Figure 2.53). As one may expect, the mean values havea linear distribution along this rod – quite similarly to the deterministic case – fromzero to a maximum at the right end.

The standard deviations also have an almost linear distribution, but they haveclearly different values for the various input coefficients α, where of course thehigher the input random fluctuation, the larger the dispersion at the output, butthe increases between particular curves increase together with the value of the


0

10

20

30

40

50

Var

(T(x

))

a(T

(x))

0 0.5 1 1.5 20

x0 0.5 1 1.5 2

xa = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

0.1

0.2

0.3

(a) (b)

Figure 2.52 Spatial distribution of (a) the variances and (b) coefficients of variationin the heated rod

0

00

0.5

0.5 1 1.5 2x

0 0.5 1 1.5 2x

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

200

400

600

800

1000

1

1.5

2

2.5

m3(

T(x

))

b(T

(x))

(a) (b)

Figure 2.53 Spatial distribution of (a) the third central probabilistic momentsand (b) skewness in the heated rod

coefficient α. The distribution of expectations does not show this clearly, but it isapparent from the standard deviations graph that some local fluctuations alongthe curve represent the given characteristic for larger values of α. This appears soespecially close to the right edge, where the heat source is applied, and is partially


smoothened by the polynomial interpolation algorithm used to process these datafrom the corresponding nodal values.

This fact is even more important in Figure 2.52, where the variances close tothe left edge with temperature fixed at T = 0 show some negative values betweenthe nodal values interpolated automatically to the negative curvature, but anywayshow some tendency noticed also in some previous second-order analyses [96, 97].It should be underlined that input coefficients of variation like α ≤ 0.20 do notcause such numerical discrepancies. The graph showing the coefficients of variationfor temperature is naturally also affected by this phenomenon – we observe almostconstant values within the rod except for the close neighborhood of 0 like before forα ≤ 0.20 and its largest value returns a totally different, slightly dispersed distribution.The third moment and skewness in Figure 2.53 show analogous anomalies, howevernow even for α = 0.20 we notice some fluctuations, especially on the right graph.Consistently with all previous figures, the larger the value of the parameter α,the higher the value of this moment and the corresponding coefficient – the onlyexception is seen for a small surrounding of the left edge (in case of the secondand third moments only). Further, the skewnesses obtained for any input coefficientof variation appear to be significantly positive, so that the final distribution of thetemperatures cannot be Gaussian. The fluctuating graphs for larger input variationcoefficient of the heat conductivity are especially transparent in Figure 2.54, wherekurtosis fluctuations make two neighboring curves almost equal in a few locationsand close to the right end this kurtosis suddenly increases almost twice. Apparently,positive values of the kurtosis along this rod confirm a lack of Gaussian characterand show a significantly larger concentration around the mean value than for thenormal distribution.

00

0.5 1 1.5 2x

00

5

0.5 1 1.5 2x

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

a = 0.05a = 0.20

a = 0.10 a = 0.15a = 0.25

5000

10000

15000

20000

25000

m4(

T(x

))

k(T

(x))

10

15

(a) (b)

Figure 2.54 Spatial distribution of (a) the fourth central probabilistic momentsand (b) kurtosis in the heated rod


The results of this test show that careful FEM discretization close to the boundaryconditions location may be necessary for larger values of the input coefficient ofvariation (like α ≥ 0.25) even for the steady-state linear problems of heat transfer,where randomization is applied to physical parameters of the homogeneous media.

2.3.5.2 Transient Heat Transfer Analysis by the RFM

The second illustration is the transient heat transfer problem solution by SFEM forthe same statistically homogeneous and isotropic rod with constant cross-sectionalarea and length l = 2.0 m, where the time increment used was �t = 2 s. Now theheat conductivity and capacity are taken as truncated input Gaussian randomvariables with the same random dispersions, α(k) = 0.15 = α(c); their expected valuesare E[k] = 0.10 and E[c] = 1.0, respectively. The temperature is fixed at the left edgeas T = 0, the heat flux is applied at the right corner, and the entire structure isdivided into 10 three-noded parabolic finite elements. The larger part of the com-putational experiment is conducted in the symbolic environment of Maple, wherethe local response functions for the additional time increments are determined and,further, combined into the output probabilistic moments together with the additionalvisualization presented below; a deterministic solution to this problem has beenprovided externally by the academic FEM code FENAP. The 11-point discretizationin the random space is used to define the numerical probing process around theexpected value of the random heat conductivity and also capacity (the basic incrementis equivalent to 10% of this parameter expectation), and the tenth-order stochasticperturbation method is applied consecutively. These nodal response functions aredetermined numerically for all time increments, which significantly increases the totalcomputational time consumption. The results contained in Figures 2.55–2.62 contrastspatial distributions of the expected values (Figure 2.55), variances (Figure 2.56),standard deviations (Figure 2.57), coefficients of variation (Figure 2.58), third centralprobabilistic moments (Figure 2.59), skewness (Figure 2.60), fourth central probabilis-tic moments (Figure 2.61), and kurtosis (Figure 2.62) for randomized heat capacity(left-hand graphs) with uncertain heat conductivity (right-hand graphs). They areall presented along this rod for time moments corresponding to 5, 10, 20, 30, 40,60, 80, and 100 seconds of two transient processes (the last time corresponds to thesteady-state temperature distribution). Analysis of these pairs shows that the resultsare similar in the case of expectations only (Figure 2.55) – they exhibit really similarbehavior and quite similar values during the entire heating process. They all natu-rally start from 0 at the left edge, undergo zero values within the first quarter of thisstructure, especially at the beginning of the heating process, and frequently reacha maximum at the opposite edge of this bar. They all increase monotonously along thisbar to the linear function corresponding to the steady state of the temperatures – verypredictable after its previous deterministic origin; these expectations are generallyconsistent with similar tests performed before using second-order analysis [97, 112].Contrary to expectations, the variances shown in Figure 2.56 are quite different underuncertainty in heat capacity and conductivity. It is important that the variances forthis second parameter are more than 10 times larger than for the heat capacity.


This contrast remains true for the given combination of input parameters. Secondly,the variances of temperatures all have convex distributions for random k and increasemonotonously from the left edge up to the right one as well as together with thespecified time increments.

The coefficients of variation (Figure 2.58) show that the randomness of the heatcapacity coefficient is very important at the beginning of the heating process, when theheat starts to penetrate the medium considered; the variation coefficient has maximumvalue close to the left unheated end. After the initial extremum distribution thesecoefficients decrease monotonously at particular time moments to reach almost zeroat the end. This is quite natural as the steady-state distribution, even random, does notdepend at all upon the heat capacity of the material. The same coefficient of variationcomputed for a random parameter k has almost sinusoidal shape at the first incrementof the heating simulation and rather accidentally reaches a maximum (around halfthat for random c) close to the left edge. Then, its distribution stabilizes along the rodand becomes very close to the results presented in the previous section, that is, almostconstant everywhere with a large decrease to 0 at the left end. These results are quiteconsistent with these obtained before with the use of the SOSM SFEM implementation.The variance distributions obtained for the randomized c exhibit zero convexity closeto the middle of the rod. Their time evolution starts with zeros corresponding todeterministically given initial temperatures, goes through maximum values for about10–20 s of this process, to finally return to almost zero everywhere for the steadystate. It naturally affects the distributions of the temperature standard deviations,cf. Figure 2.57. These computed for random k are very close, concerning shape andevolution, to the corresponding expectations, whereas uncertainty in c rather leadsto similarity of the standard deviations with variances.

A spatial distribution of the third central probabilistic moments (Figure 2.59) forboth random inputs is very similar to the variances except for some negative valuesobtained for randomized c at the heated edge for the end of the heating process.These moments for uncertain k once more decrease monotonously together with timeand with the spatial coordinate x. The skewness distributions (Figure 2.60) are moreregular for random c forming some kind of envelope with both positive and negativevalues, rather irregular for uncertain k at the beginning of heating, to become clearlypositive and have shape close to the corresponding variation coefficient. Despitethese apparent differences, the final temperature distribution at any moment in timeof the heating process has no chance of being Gaussian after Figure 2.60 because of thelarge absolute values of the skewness parameter. The fourth probabilistic moments(Figure 2.61) are, as the third moments, also similar in shape and convexity to thevariances. Unfortunately, the fourth moments computed for random c exhibit somenumerical discrepancies resulting in negative values in the middle of the heatingprocess. This is most probably caused by the polynomial interpolation of the responsefunctions, where higher-order partial derivatives may have significant negativevalues. Kurtosis (Figure 2.62) shows, for both input variables, large variations at thebeginning of the transient process taking accidentally negative values, even withrandom k. Kurtosis distributions, even for later moments of heating, never equal 0and this is the second reason to treat the temperature distributions as non-Gaussian.


00 0

0.5 1x

1.5 2 0 0.5 1x

1.5 2

2

4

6

8

10

12

14

16

18E

(T(x

))

E(T

(x))

5

10

15

20

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

(a) (b)

Figure 2.55 Expected values of the temperature distribution in the heated rod,(a) random c and (b) random k. Reproduced with permission from Tech Science Press

0 0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Var

(T(x

))

Var

(T(x

))

2

4

6

8

10

(a) (b)

Figure 2.56 The variances of the temperature distribution in the heated rod,(a) random c and (b) random k. Reproduced with permission from Tech SciencePress


0 0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

s(T

(x))

s(T

(x))

0

1

2

3

(a) (b)

Figure 2.57 Standard deviations of the temperature distribution in the heated rod,(a) random c and (b) random k. Reproduced with permission from Tech Science Press

00 0

0.02

0.03

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

0.1

0.2

a(T

(x))

a(T

(x))

0.3

(a) (b)

Figure 2.58 Coefficients of variation of the temperature distribution in the heated rod,(a) random c and (b) random k. Reproduced with permission from Tech Science Press


0

00

0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

0.1

0.2

0.3

0.4

10

20

30

40

50m

3(T

(x))

m3(

T(x

))(a) (b)

Figure 2.59 The third central probabilistic moments of the temperature distribution inthe heated rod, (a) random c and (b) random k. Reproduced with permission fromTech Science Press

0 0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

–2

–1

0

1

–4

–3

–2

–1

0

1

2

b(T

(x))

b(T

(x))

(a) (b)

Figure 2.60 Skewness of the temperature distribution in the heated rod, (a) randomc and (b) random k. Reproduced with permission from Tech Science Press


0

0

0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

0.1

0.2

m4(

T(x

))

m4(

T(x

))

0.3

0.4

0

10

20

30

40

50

(a) (b)

Figure 2.61 The fourth central probabilistic moments of the temperature distributionin the heated rod, (a) random c and (b) random k. Reproduced with permission fromTech Science Press

0

0

0.5 1x x

1.5 2 0 0.5 1 1.5 2

5 s40 s

10 s60 s

20 s80 s

30 s100 s

5 s40 s

10 s60 s

20 s80 s

30 s100 s

1

2

3

4

k(T

(x))

k(T

(x))

–50

–40

–30

–20

–10

0

10

20

(a) (b)

Figure 2.62 Kurtosis of the temperature distribution in the heated rod, (a) random cand (b) random k. Reproduced with permission from Tech Science Press


Finally, let us note that the overall computational effort of this analysis is relativelysmall in comparison to the Monte Carlo method – for 11-point discretization of therandom space it equals 11 times the deterministic solution time plus a relatively smallamount of post-processing time to recover the nodal (time-dependent in transientanalysis) response functions and to calculate the final probabilistic moments withadditional visualization. Thanks to the symbolic computations package one mayalso apply other probability distributions easily to include the extra components tocompute the cross-correlations when heat capacity and conductivity are randomizedat the same time. Further, using such an extended symbolic software like Maplefor instance, we can make a posteriori some correction or better choice of theapproximating polynomial order as well as to increase graphical resolution byappropriate window size on the input coefficient of variation.

3Stochastic BoundaryElement Method

Stochastic realizations of the Boundary Element Method (BEM) have been known formany years [11] and are available mainly in geotechnical problems, where randomdispersion in civil engineering seems to be the largest area of use, for groundwater[19, 29, 108], seawater [57, 134], and porous media flows [149], in heat conduction andtransfer issues [32, 124], or even for some elastodynamic [109] and wave propagation[122, 123] problems. It should be mentioned that the stochastic boundary elementmethod (SBEM) analyses are especially efficient of course in random boundarygeometry modeling, so that it is used in models with random boundary conditionsor in stochastic shape design sensitivity [12]. This is the main reason to propose thenew version of the generalized, perturbation-based SBEM here and test to verify itsnumerical efficiency. This version of SBEM offers both higher perturbation ordersand a computation of higher than second probabilistic moments [61], analogouslyto the first version based also on the RFM [83, 87]. The basic difference is innumerical determination of this response function, which is now carried out usingmore efficient weighted least-squares method. It is also possible to optimize theorder of the approximating polynomial during symbolic processing in the computeralgebra system Maple. Let us remember here that the order of this approximationis limited a priori and cannot be larger than the number of trial points around themean value of the random input parameter. This order optimization undoubtedlyleads in turn to minimization of the approximating error (and, effectively, speedsup probabilistic convergence of the method), since apparent linear interrelations donot need to be approximated using higher-order polynomials. Now, the optimalorder of the approximating polynomials determined via the least-squares methodguarantees perfect matching of the trial set of computational results with the responsefunction. This was not the case in some previous studies, where highly non-linearinterrelations between input and output parameters led to high oscillations of theresponse functions, somewhat against the tendency deduced from the trial points (see



Example 5.3, page 262 and next). Probabilistic moments of the structural response arestudied using the proposed method on the examples adopted after and comparedwith those simulated using the Monte Carlo scheme; independently, they are derivedfrom the analytical formulas using a stochastic perturbation technique. The very goodagreement of these independent methods, as well as an application to the compositestructure, make this strategy attractive and promising.

3.1 Deterministic Formulation of the Boundary Element Method

Let us consider a linear elastic homogeneous body occupying the region � ⊂ R2

with boundary � ≡ ∂� and let us assume that � is the Lyapunov surface. Let us alsoconsider the body forces vector f i with surface loadings pi acting on this region andresulting in the displacement field u(x), strain tensor εij(x), and stress tensor σ ij(x).The linear elastostatic boundary value problem adjacent to the isotropic domain isdefined using the following set of partial differential equations:

div σ + f = 0, (3.1)

σ = Cε (3.2)

where

Cijkl = δijδkl λ +(δikδjl + δilδjk

)µ, (3.3)

ε = 12

(∇u + ∇uT) (3.4)

with boundary conditions

ui(x) = ui(x); x ∈ �u; pi(x) = pi(x); x ∈ �p. (3.5)

Finally, it holds that�u ∪ �p = � (3.6)

and�u ∩ �p = ∅. (3.7)

The elasticity tensor given in Equation (3.4) may be composed for the two-component material with use of the following characteristic function [4]:[

λ(x)µ(x)

]= χ1(x)

[λ1µ1

]+ (1 − χ1(x)

) [λ2µ2

],

χ1 ={

1, x ∈ �1

0, x ∈ �2where � = �1 ∪ �2 and �1 ∩ �2 = {∅} . (3.8)

Stochastic Boundary Element Method 157

The set of Equations (3.1)–(3.7) for the homogeneous domain with regularboundary can be replaced with the following equivalent integral formulation rewrit-ten in vector notation [10, 12, 60, 73]:

c(x)u(x) +∫�

P(x, y)u(y)d�(y)

=∫�

U(x, y)p(y)d�(y) +∫�

U(x, z)b(z)d�(z) (3.9)

whereP(x, y) = [Pij(x, y)] (3.10)

represents the stress vector components for direction j at point y in the unboundedelastic space resulting from a unitary force applied at the point defined by x in thedirection defined by the index i. If the point being considered belongs to the interiorof the domain � then c = I but when it belongs to its boundary then

c(x) = I + limδ→0

∫�δ

P(x, y)d�(y), (3.11)

where �δ denotes a sphere with center at point x and radius equal to δ. It alsoholds that

c = α I ;α ∈ (0, 1) (3.12)

where for a boundary regular in the Lyapunov sense α = 12 . Further, let us introduce

a discretization of the boundary � ∈ R2 using the boundary elements �e such that

� =K∑

e=1

�e. (3.13)

Hence, the displacement field u(x) can be approximated for any element �e in localcoordinates ξ = (ξi) by the nodal values (u)w

e as follows:

u (x (ξ)) ≈ Nw(ξ)(u)we = INw(ξ)(u)w

e (3.14)

where Nw(ξ ) are the interpolation functions analogous to these provided for thefinite elements and their degrees of freedom. Introducing these statements intoEquation (3.9), it is obtained that

c(x)u(x) +K∑

e=1

We∑w=1

(u)we

∫�e

P(x, y (ξ)

)Nw (ξ) J (ξ) d� (ξ)

=K∑

e=1

We∑w=1

(p)w

e

∫�e

U(x, y (ξ)

)Nw (ξ) J (ξ) d� (ξ) + B(x). (3.15)


Variable We in Equation (3.15) denotes the total number of boundary elementswithin the applied boundary element type. The body force operator B(x) is defined as

B(x) =∫�

U(x, y)f(y)d�(y) (3.16)

in the presence of the volumetric boundary forces f(y). The Jacobian J (ξ) is given fortwo-dimensional body by

J (ξ) = d�

dξ=√√√√[(∂x1

∂ξ

)2

+(

∂x2

∂ξ

)2]

. (3.17)

Further, the fundamental solutions included in Equation (3.15) adequate to theisotropic linear elastic body in the plane strain are given as

Uij(x, y) = − 18π (1 − ν)µ

[(3 − 4µ) ln(r)δij − r,ir,j

], (3.18)

Pij(x, y) = − 1

4π (1 − ν) r

{[(1 − 2ν) δij + 2r,ir,j

] ∂r∂n

− (1 − 2ν)(

r,inj − r,jni

)}(3.19)

wherer(x, y) = √ri • rj, ri = xi

(y)− xi (x) (3.20)

andr,i = ∂r

∂xi(y) = ri

r. (3.21)

We can use the parabolic shape functions Nke (ξ) as in Example 3.4, defined as

N1e (ξ) = 1

2ξ (1 − ξ) ,N2

e (ξ) = (1 − ξ) (1 + ξ) ,N3

e (ξ) = 12ξ (1 + ξ) .

(3.22)

Equations (3.22) are valid for the boundary element defined uniquely at the ends innodes 1, 3 and the midpoint in 2, where the local longitudinal coordinate ξ has value0 at the midpoint and ξ = ± 1 at both ends; alternatively, one may use traditionallinear functions as provided in the remaining tests.

Next, let us introduce the global numbering of boundary nodal points xβ ,β = 1, 2, . . . , W, where W is the total number of nodes. Then, we have [12]

r (x) =W∑

β=1

[Pβ (x) uβ − Uβ (x) pβ

]− B (x) = 0 (3.23)


with

Pβ (x) = c (x) +∑

m

∫�m

P(x, y (ξ)

)Nβ (ξ) J (ξ) d� (ξ) , (3.24)

Uβ (x) =∑

m

∫�m

U(x, y (ξ)

)Nβ (ξ) J (ξ) d� (ξ) . (3.25)

The unknown values of displacements u(x) and forces p(x) can be found byminimization of the functional r(x) from the following condition:∫

�e

Tα (x) r (x) d� (x) = 0 (3.26)

where Tα (x) = I Tα (x), x ∈ �e, e = 1, 2, . . . , K is the weighted function matrix. In thecase of the collocation method, this matrix has the form [4]

Tα (x) = I δ(x − xα

)(3.27)

where the xα are collocation nodes. If the collocation nodal points are equivalentto the boundary nodes x = xα , α = 1, 2, . . . , W then this minimization condition takesthe form

r(xα) = 0. (3.28)

Thus, it yields

cαβuβ + Hαβuβ = Gαβpβ + Bα , α,β = 1, . . . ,W (3.29)

whereHαβ =

∑e

∫�e

P[xα , y (ξ)

]Nβ (ξ)J (ξ)d� (ξ) (3.30)

andGαβ =

∑e

∫�e

U[xα , y(ξ)

]Nβ(ξ) J(ξ) d�(ξ) . (3.31)

The fundamental set of BEM algebraic equations is proposed as

Hαβuβ = Gαβpβ + Bα ;α,β = 1, . . . ,W. (3.32)

Further, for the needs of deterministic problem discretization, the following decom-position is usually employed:

A ≡[

H11 −G11

H21 −G21

](3.33)


andF ≡

[−H11 G12

−H21 G22

]{u1

p2

}+{

B1

B2

}, (3.34)

where the matrices Hij, Gij, ui, and Bi for i, j = 1, 2 are submatrices of the matrices H,G, u, and B equivalent to the boundary segments �i, i = 1, 2. The column matrix ofbody forces B is calculated numerically in the region � with use of the formula

Bα ≡ B(xα) =

W∑w=1

f(yw) ∫

�w

U(xα , y

)d�(y)

(3.35)

where the function f (y) has a constant value inside the cell �w. It may be shown thatEquation (3.32) can be rewritten using the following system of linear equations withnon-symmetric stiffness matrix:

AαβXβ = Fα ; α,β = 1, . . . , W. (3.36)

The solution to this system contains the displacements and stresses calculated at theboundary of a domain being modeled. Then we compute in turn, the displacementsand stresses in the interior of the domain � as

u(x) = G(x)p − H(x)u + B(x); x ∈ �

σ (x) = GD(x)p − HS(x)u + BD(x); x ∈ � (3.37)

where the matrices G(x), H(x), GS(x), and HD(x) generally depend on boundaryintegrals with nuclei U(x,y), P(x,y), D(x,y), and S(x,y), respectively, calculated for afixed point x ∈ � and for y ∈ �, while the column matrices B(x) and BD(x) depend onthe integrals defined on the � region with nuclei U(x,y) and D(x,y), respectively. Itholds that

D(x, y) =

∑x

[U(x, y)]

; x ∈ �,y ∈ � (3.38)

S(x, y) =

∑x

[P(x, y)]

; x ∈ �,y ∈ � (3.39)

where U(x,y) is the fundamental solution given by Equation (3.18) and for anyadditional function f ∑(

f) = µ

[∇(f ) + ∇T(f )]+ λ Idiv(f ). (3.40)

3.2 Stochastic Generalized Perturbation Approach to the BEM

Traditionally, the stochastic perturbation approach to all the physical problems isentered by the respective perturbed equations of zeroth-, first-, and successively


higher orders being a modification of the variational integral formulation. Thefollowing hold from Equation (3.15):

• zeroth-order equation

c0 (x) u0 (x) +K∑

e=1

We∑w=1

(u0)w

e

∫�e

P0 (x, y (ξ))

Nw (ξ) J (ξ) d� (ξ)

=K∑

e=1

We∑w=1

(p0)w

e

∫�e

U0 (x, y (ξ))

Nw (ξ) J (ξ) d� (ξ) + B0 (x) (3.41)


∂c (x)

∂bu0 (x) + c0 (x)

∂u (x)

∂b+

K∑e=1

We∑w=1

(∂u∂b

)w

e

∫�e

P0 (x, y (ξ))

Nw (ξ) J (ξ) d� (ξ)

+K∑

e=1

We∑w=1

(u0)w

e

∫�e

∂P(x, y (ξ)

)∂b

Nw (ξ) J (ξ) d� (ξ)

=K∑

e=1

We∑w=1

(∂p∂b

)w

e

∫�e

U0 (x, y (ξ))

Nw (ξ) J (ξ) d� (ξ)

+K∑

e=1

We∑w=1

(p0)w

e

∫�e

∂U(x, y (ξ)

)∂b

Nw (ξ) J (ξ) d� (ξ) + ∂B (x)

∂b(3.42)


∂2c (x)

∂b2 u0 (x) + 2∂c (x)

∂b∂u (x)

∂b+ c0 (x)

∂2u (x)

∂b2

+K∑

e=1

We∑w=1

(∂2u∂b2

)w

e

∫�e

P0 (x, y (ξ))

Nw (ξ) J (ξ) d� (ξ)

+K∑

e=1

We∑w=1

(u0)w

e

∫�e

∂2P(x, y (ξ)

)∂b2 Nw (ξ) J (ξ) d� (ξ)

+ 2K∑

e=1

We∑w=1

(∂u∂b

)w

e

∫�e

∂P(x, y (ξ)

)∂b

Nw (ξ) J (ξ) d� (ξ)

=K∑

e=1

We∑w=1

(∂2p∂b2

)w

e

∫�e

U0 (x, y (ξ))

Nw (ξ) J (ξ) d� (ξ)


+K∑

e=1

We∑w=1

(p0)w

e

∫�e

∂2U(x, y (ξ)

)∂b2 Nw (ξ) J (ξ) d� (ξ)

+ 2K∑

e=1

We∑w=1

(∂p∂b

)w

e

∫�e

∂U(x, y (ξ)

)∂b

Nw (ξ) J (ξ) d� (ξ) + ∂B (x)

∂b(3.43)


n∑k=0

(nk

)∂n−kc (x)

∂bn−k

∂ku (x)

∂bk+

n∑k=o

(nk

) K∑e=1

We∑w=1

(∂n−ku∂bn−k

)w

e

×∫�e

∂kP(x, y (ξ)

)∂bk

Nw (ξ) J (ξ) d� (ξ)

=n∑

k=0

(nk

) K∑e=1

We∑w=1

(∂n−kp∂bn−k

)w

e

∫�e

∂kU(x, y (ξ)

)∂bk

Nw (ξ) J (ξ) d� (ξ) + ∂nB (x)

∂bn . (3.44)

It is well known from the SFEM formulations that the system of linear algebraicequations, which is the basis of the model, can be transformed, due to the generalizedstochastic perturbation-based technique, to the following systems of equations, cf.Equation (3.29):

• zeroth-order equationc0αβu0

β + H0αβu0

β = G0αβp0

β + B0α (3.45)


∂cαβ

∂bu0

β + c0αβ

∂uβ

∂b+ ∂Hαβ

∂bu0

β + H0αβ

∂uβ

∂b= ∂Gαβ

∂bp0

β + G0αβ

∂pβ

∂b+ ∂Bα

∂b(3.46)


∂2cαβ

∂b2 u0β + 2

∂cαβ

∂b

∂uβ

∂b+ c0

αβ

∂2uβ

∂b2 + ∂2Hαβ

∂b2 u0β + 2

∂Hαβ

∂b

∂uβ

∂b+ H0

αβ

∂2uβ

∂b2

= ∂2Gαβ

∂b2 p0β + 2

∂Gαβ

∂b

∂pβ

∂b+ G0

αβ

∂2pβ

∂b2 + ∂2Bα

∂b2 (3.47)


n∑k=0

(nk

)∂n−kcαβ

∂bn−k

∂kuβ

∂bk+

n∑k=0

(nk

)∂n−kHαβ

∂bn−k

∂kuβ

∂bk=

n∑k=0

(nk

)∂n−kGαβ

∂bn−k

∂kpβ

∂bk+ ∂nBα

∂bn .

(3.48)


Further, we proceed with a formulation of the hierarchical equations of increasingorder provided on a basis of Equation (3.36):

A0αβX0

β = F0α ,

A0αβ

∂Xβ

∂b= ∂Fα

∂b− ∂Aαβ

∂bX0

β ,

( . . . )

n∑k=0

n

k

∂kAαβ

∂bk

∂n−kXβ

∂bn−k= ∂Fα

∂b.

(3.49)

Furthermore, it is known from deterministic BEM implementation that the matrixA is represented by the components of the following perturbation orders:

• zeroth-order components

A0 ≡[(

H11)0 (−G11

)0(H21

)0 (−G21)0]

(3.50)

• and nth-order derivatives

∂nA∂bn =

∂nH11

∂bn −∂nG11

∂bn

∂nH21

∂bn −∂nG21

∂bn

. (3.51)

Quite analogously, it is possible to obtain equations describing derivatives of theRHS vector. The following hold:

• zeroth-order equations{(F(1))0 = (−H11

)0 (u1)0 + (G12

)0 (p2)0 + (B1

)0 ,(F(2))0 = (−H21

)0 (u1)0 + (G22

)0 (p2)0 + (B2

)0 (3.52)

• and nth-order equations

∂nF(1)

∂bn = −n∑

k=0

n

k

∂kH11

∂bb

∂n−ku1

∂bn−k+

n∑k=0

n

k

∂kG12

∂bk

∂n−kp2

∂bn−k+ ∂nB1

∂bn ,

∂nF(2)

∂bn = −n∑

k=0

n

k

∂kH21

∂bb

∂n−ku1

∂bn−k+

n∑k=0

n

k

∂kG22

∂bk

∂n−kp2

∂bn−k+ ∂nB2

∂bn .

(3.53)

In order to calculate the expected values and higher-order probabilistic moments ofdisplacements, strains, and stress functions, the same Taylor expansion is employed


to the definitions of probabilistic moments calculated for any state random variablesassuming their continuous character. It is obtained from the first of Equations (3.37)that

u0(x) = G0(x)p0 − H0(x)u0 + B0(x); x ∈ � (3.54)

as well as in the recursive form for n-th order partial derivative

∂nu (x)

∂bn =n∑

k=0

(nk

)∂kG (x)

∂bk

∂n−kp∂bn−k

−n∑

k=0

(nk

)∂kH (x)

∂bk

∂n−ku∂bn−k

+ ∂nB (x)

∂bn ; x ∈ �. (3.55)

Similarly, the stress tensor components are calculated as, cf. Equation (3.37)

σ 0(x) = G0D(x)p0 − H0

S(x)u0 + B0D(x); x ∈ � (3.56)

and its derivatives also as

∂nσ (x)

∂bn =n∑

k=0

(nk

)∂kGD (x)

∂bk

∂n−kp∂bn−k

−n∑

k=0

(nk

)∂kHS (x)

∂bk

∂n−ku∂bn−k

+ ∂nBD (x )

∂bn ; x ∈ �.

(3.57)These equations enable us to calculate any-order approximations for any proba-

bilistic moments of the structural response but since the DDM may produce unknownerror at higher orders the RFM is proposed and implemented below to avoid theseproblems.

3.3 The Response Function Method in the SBEM Equations

In this alternative approach we rewrite variational Equation (3.15) as

c(m) (x) u(m) (x) +K∑

e=1

We∑w=1

(u(m)

)we

∫�e

P(m)(x, y (ξ)

)Nw (ξ) J (ξ) d� (ξ)

=K∑

e=1

We∑w=1

(p(m)

)we

∫�e

U(m)(x, y (ξ)

)Nw (ξ) J (ξ) d� (ξ) + B(m) (x) (3.58)

where m = 1, . . . , M indexes trial solutions around the mean value of the randominput. Then, the following discretization is proposed for the displacement field:

u(

b(

xj

))= u

(bj

)≈ Nw

(u(bj)

)w

e= INw

(u(bj)

)w

e= INw

(a(w)

i bij

)e; i, j ∈ N (3.59)

where Nw(ξ ) are the classical interpolation functions used traditionally in the BEMprograms and a(w)

i bij denotes the polynomial response function. The boundary trac-

tions are discretized in the similar manner as

p(

b(xj))

= p(

bj

)≈ Nw

(p(bj)

)w

e= INw

(p(bj)

)w

e= INw

(d(w)

i bij

)e

(3.60)

where d(w)i bi

j is the nodal response function for the tractions p(x). Following the sameconsiderations as above, the following algebraic system of equations is obtained:


Hαβj

(a(β)

i bij

)= Gαβj

(d(β)

i bij

)+ Bαj; α,β = 1, . . . , W, j = 1, . . . , n (3.61)

the solution of which, together with the procedure of internal node displacementsand tractions, enables the reliable computation of the sensitivity gradients by themodified BEM procedure. We illustrate our methodology with full tenth orderexpansion employed to derive first two probabilistic moments of the boundarydisplacements and forces. There holds for the expectations [83]

E[uβ

] = ε0aβ0 + 12!

ε2 (i2 − i)

aβibi µ2 (b)

b2 + 14!

ε4 (i4 − 6i3 + 11i2 − 6i)

aβibi µ4 (b)

b4

+ ε6 16!

(i6 − 15i5 + 85i4 − 225i3 + 274i2 − 120i

)aβib

i µ6 (b)b6

+ 18!

ε8 (i8 − 28i7 + 332i6 − 1960i5 + 6769i4 − 13132i3 + 13068i2 − 5040i)

aβibi µ8 (b)

b8

+ 110!

ε10 (i10 − 45i9 + 870i8 − 9450i7 + 63273i6 − 269325i5 + 723680i4)

aβibi µ10 (b)

b10

+ 110!

ε10 (−1172700i3 + 1026576i2 − 362880i)

aβibi µ10 (b)

b10 . (3.62)

The specific powers of b appearing in denominators are equivalent to the meanvalues of this input parameter b. Then, by inserting a definition of the coefficient ofvariation and putting ε = 1, one gets

E[uβ

] = aβ0 + 12!

i (i − 1) aβibiα2 (b) + 1

4!i(i3 − 6i2 + 11i − 6

)aβib

iα4 (b)

+ 16!

i(i5 − 15i4 + 85i3 − 225i2 + 274i − 120

)aβib

iα6 (b)

+ 18!

i(i7 − 28i6 + 332i5 − 1960i4 + 6769i3 − 13132i2 + 13068i − 5040

)aβib

iα8 (b)

+ 110!

i4(i6 − 45i5 + 870i4 − 9450i3 + 63273i2 − 269325i + 723680

)aβib

iα10 (b)

+ 110!

i(−1172700i2 + 1026576i − 362880

)aβib

iα10 (b) . (3.63)

Quite analogously, one recovers the expectations of the boundary tractions as

E[pβ

] = dβ0 + 12!

i (i − 1) dβibiα2 (b) + 1

4!i(i3 − 6i2 + 11i − 6

)dβib

iα4 (b)

+ 16!

i(i5 − 15i4 + 85i3 − 225i2 + 274i − 120

)dβib

iα6 (b)

+ 18!

i(i7 − 28i6 + 332i5 − 1960i4 + 6769i3 − 13132i2 + 13068i − 5040

)dβib

iα8 (b)

+ 110!

i4(i6 − 45i5 + 870i4 − 9450i3 + 63273i2 − 269325i + 723680

)dβib

iα10 (b)

+ 110!

i(−1172700i2 + 1026576i − 362880

)dβib

iα10 (b) . (3.64)


The formula for variance of the function f (b) thanks to the tenth-order perturbationin terms of the Gaussian random input b is given as

Var(uβ

) =(

aβibi)2

i2α2 (b) +{

14

(i2 − i

)2 + 13

i(i3 − 3i2 + 2i

)} (aβib

i)2

α4 (b)

+{

136

(i3 − 3i2 + 2i

)2 + 124

(i2 − i

) (i4 − 6i3 + 11i2 − 6i

)} (aβib

i)2

α6 (b)

+ 160

i(i5 − 10i4 + 35i4 − 50i2 + 24i

) (aβib

i)2

α6 (b)

+{

1576

(i4 − 6i3 + 11i2 − 6i

)2 + 1360

(i5 − 10i4 + 35i3 − 50i2 + 24i

)

× (i3 − 3i2 + 2i)} (

aβibi)2

α8 (b)

+ 12520

i(i7 − 21i6 + 175i5 − 735i4 + 1624i3 − 1764i2 + 720i

) (aβib

i)2

α8 (b)

+ 1720

(i2 − i

) (i6 − 15i5 + 85i4 − 225i3 + 274i2 − 120i

) (aβib

i)2

α8 (b)

+ 114400

(i5 − 10i4 + 35i3 − 50i2 + 24i

) (aβib

i)2

α10 (b)

+ 140320

(i2 − i

)(i8 − 28i7 + 322i6 − 1960i5 + 6769i4 − 13132i3

+ 13068i2 − 5040i)(

aβibi)2

α10 (b)

+ 18640

(i4 − 6i3 + 11i2 − 6i

) (i6 − 15i5 + 85i4 − 225i3 + 274i2 − 120i

) (aβib

i)2

α10 (b)

+ 115120

(i3 − 3i2 + 2i

)(i7 − 21i6 + 175i5 − 735i4 + 1624i3

− 1764i2 + 720i)(

aβibi)2

α10 (b)

+ 1181440

i(i9 − 36i8 + 546i7 − 4536i6 + 22449i5 − 67284i4

) (aβib

i)2

α10 (b)

+ 1181440

i(118124i3 − 109584i2 + 40320i

) (aβib

i)2

α10 (b) . (3.65)

The third central probabilistic moment is derived [74] as

µ3(uβ

) = 32

i3 (i − 1)(

aβibi)3

α4 (b)

+{

18

(i − 1)3 + 12

(i2 − 3i + 2

)(i − 1) + 1

8

(i3 − 6i2 + 11i − 6

)}i3(

aβibi)3

α6 (b)


+{

124

(i3 − 6i2 + 11i − 6

) (i2 − 3i + 2

)+ 140

(i4 − 10i3 + 35i2 − 50i + 24

)(i − 1)

}

× i3(

aβibi)3

α8 (b)

+{

1240

(i5 − 15i4 + 85i3 − 225i2 + 274i − 120

)+ 132

i3(i3 − 6i2 + 11i − 6

)(i − 1)2

}

× i3(

aβibi)3

α8 (b)

+ 124

i3(i2 − 3i + 2

)2(i − 1)

(aβib

i)3

α8 (b)

+ 1480

i3(i4 − 10i3 + 35i2 − 50i + 24

) (i3 − 6i2 + 11i − 6

) (aβib

i)3

α10 (b)

+ 11680

i3 (i − 1)(i6 − 21i5 + 175i4 − 735i3 + 1624i2 − 1764i + 720

) (aβib

i)3

α10 (b)

+ 1720

i3(i5 − 15i4 + 85i3 − 225i2 + 274i − 120

) (i2 − 3i + 1

) (aβib

i)3

α10 (b)

+ 1240

i3(i4 − 10i3 + 35i2 − 50i + 24

) (i2 − 3i + 2

)(i − 1)

(aβib

i)3

α10 (b)

+ 113440

i3(i7 − 28i6 + 322i5 − 1960i4 + 6769i3 − 13132i2 + 13068i − 5040

)×(

aβibi)3

α10 (b)

+ 1960

i3(i5 − 15i4 + 85i3 − 225i2 + 274i − 120

)(i − 1)

(aβib

i)3

α10 (b)

+{

1384

i3 (i − 1)(i3 − 6i2 + 11i − 6

)2 + 1288

i3(i3 − 6i2 + 11i − 6

) (i2 − 3i + 2

)2}

×(

aβibi)3

α10 (b) (3.66)

Finally, we derive the fourth central probabilistic moment in the form

µ4(uβ

) = i4(

aβibi)4

α4 (b) +{

32

i4 (i − 1)4 + 23

i4(i2 − 3i + 2

)} (aβib

i)4

α6 (b)

+{

130

i4(i4 − 10i3 + 35i2 − 50i + 24

)+ 16

i4(i2 − 3i + 2

)2 + 116

i4 (i − 1)4}

×(

aβibi)4

α8 (b)

+ 11260

i4(i6 − 21i5 + 175i4 − 735i3 + 1624i2 − 1764i + 720

) (aβib

i)4

α10 (b)

+ 196

i4(i3 − 6i2 + 11i − 6

)2 (aβib

i)4

α10 (b)


+ 154

i4(i2 − 3i + 2

)3 (aβib

i)4

α10 (b)

+ 148

i4(i3 − 6i2 + 11i − 6

)(i − 1)3

(aβib

i)4

α10 (b)

+ 124

i4(i2 − 3i + 2

)2(i − 1)2

(aβib

i)4

α10 (b)

+ 112

i4(i3 − 6i2 + 11i − 6

) (i2 − 3i + 2

)(i − 1)

(aβib

i)4

α10 (b)

+ 14

i4(i3 − 6i2 + 11i − 6

)(i − 1)

(aβib

i)4

α8 (b)

+ 12

i4 (i − 1)2 (i2 − 3i + 2) (

aβibi)4

α8 (b)

+ 160

i4(i4 − 10i3 + 35i2 − 50i + 24

) (i2 − 3i + 2

) (aβib

i)4

α10 (b)

+ 1120

i4(i5 − 15i4 + 85i3 − 225i2 + 274i − 120

)(i − 1)

(aβib

i)4

α10 (b)

+ 140

i4(i4 − 10i3 + 35i2 − 50i + 24

)(i − 1)2

(aβib

i)4

α10 (b) . (3.67)

The very specific case of the linear response function returns the well-knownrelations

E[uβ

] = aβ0, Var(uβ

) = a2β1E2 [b] α2 (b) ,

µ3(uβ

) = 0, µ4(uβ

) = a4β1E4 [b] α4 (b) . (3.68)

The readers interested in the full tenth order expansion are advised to explore thewebpage accompanying this book to find the additional Maple sources. It needs tobe mentioned that having derived these equations we can make a straightforwardimplementation in any programming language for further numerical realization ofpolynomial response functions in probabilistic computations due to all the methodsdispayed in this book. Symbolic realization is necessary when one wants to replacepolynomial response with exponential or harmonic functions as also easily differen-tiable, however then probabilistic convergence is not guaranteed.

3.4 Computational Experiments

Example 3.1: Uniformly loaded steel plate partially simplysupported at lower edgesThe methodology described above is tested using the example of the plane strainanalysis of the structure given schematically in Figure 3.1. The BEM discretizationis performed using 56 linear two-noded boundary elements −18 for the upper andlower edges and 10 for the left and right ends; supported parts of the lower edge arediscretized using five boundary elements. The mean value of the Young’s modulus


q

1.0

0.10 0.100.80

Figure 3.1 Static scheme of the computational example. Reproduced with permissionfrom Elsevier

e is adopted as 210 GPa, the expected value of the Poisson ratio is taken as equal to0.3, the height and length of the plate both equal to 1.0, while the mean value forthe length of both side supports is adopted as 0.1. The response function for all thestate parameters is approximated for ±10% variations from the additional expectedvalues; all the random input quantities, that is, Young’s modulus, Poisson ratio,and support depth are defined as Gaussian. The entire numerical analysis is carriedout using a classical academic BEM solver [4] for the linear plane strain problem inconjunction with the symbolic environment of the system Maple. We analyze herethe probabilistic moments and coefficients of maximum displacement (for a midpointat the lower boundary) and stress in the given plate.

The set of trial points corresponding to the structural response determinationwith respect to varying Young’s modulus is given twice in Figure 3.2 – (a) for linearfitting of the proposed function u(e) and (b) for parabolic fitting; they are bothobtained thanks to the unweighted least-squares method. We study two differentapproximations of the smallest possible order resulting in a function close to the trialpoints, to verify the role of the approximating polynomial order. The case study isnot trivial because an interrelation between u and e is a kind of inverse function,where parabolic fitting seems to be more accurate. This influence, together with theperturbation method order, is studied for the expected values (Figure 3.3), coefficientsof variation (Figure 3.4), skewness (Figure 3.5), as well as kurtosis (Figure 3.6). Thesemoments and coefficients are all given as functions of the input coefficient ofvariation α ∈ [0.00, 0.20]. As one could expect, a linear response results in a constantexpectation independent of both α(e) and the perturbation order as all higher-orderpartial derivatives simply vanish. The larger the input random dispersion, thehigher the expectation for a parabolic function but they are all once more totallyindependent of the perturbation order (as higher than second-order terms do notappear). Independent of the response function order, both output coefficients ofvariation have very similar values, almost equal to the input one and both linearly


0.00175

0.00170

0.00165

0.00160

u(e)

2.02 × 1011 2.1 × 1011 2.2 × 1011

e

0.00175

0.00170

0.00165

0.00160

u(e)

2.02 × 1011 2.1 × 1011 2.2 × 1011

e(a) (b)

Figure 3.2 (a,b) Linear versus parabolic response functions of the maximum verticaldisplacement with respect to Young’s modulus

2nd-order analysis6th-order analysis10th-order analysis


0 0.05 0.10 0.15 0.20a

0.0025

0.0020

0.0015

0.0010

E(u

(e))



0 0.05 0.10 0.15 0.20a

0.00175

0.00174

0.00173

0.00172

0.00171

0.00170

0.00169

E(u

(e))

(a) (b)

Figure 3.3 (a,b) Expectations for the linear versus parabolic response functionsof the maximum vertical displacement with respect to Young’s modulus

depend on the input coefficient. The difference here is observed in some marginaldependence of α(u) for larger values of α(e) on the perturbation order – practically,second-order gives slightly different results than all higher-order terms.

We study further skewness and kurtosis of this displacement (cf. Figures 3.5and 3.6) – these coefficients automatically equal 0 for a linear approximation sincehigher-order partial derivatives both disappear, so they cannot depend on the


0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

a(u

(e))

0 0.05 0.10 0.15 0.20a



a(u

(e))

0 0.05 0.10 0.15 0.20a

0.20

0.15

0.10

0.05

0



(a) (b)

Figure 3.4 (a,b) Coefficients of variation for the linear versus parabolic responsefunctions of the maximum vertical displacement with respect to Young’s modulus

0.5

0

−0.5

−1

1

0 0.05 0.10 0.15 0.20a

b(u

(e))

4th-order analysis 6th-order analysis10th-order analysis8th-order analysis

0 0.05 0.10 0.15 0.20a

b(u

(e))

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

4th-order analysis 6th-order analysis10th-order analysis8th-order analysis

(a) (b)

Figure 3.5 (a,b) Skewness for the linear versus parabolic response functions of themaximum vertical displacement with respect to Young’s modulus

input coefficient α(e). Therefore, the output distribution according to this inter-polation method is Gaussian and this is not the case for the parabolic responsefunction (Figures 3.5(b) and 3.6(b)), where the higher the input α(e), the larger theabsolute values of both coefficients. Skewness obtained from the fourth-order per-turbation method is slightly different from other-order approximations (only forhigher α(e)), but probabilistic convergence of kurtosis with the perturbation order


1

0.5

0

−0.5

−10 0.05 0.10 0.15 0.20

a

k(u

(e))



0



0 0.05 0.10 0.15 0.20a

k(u

(e))

2

1.5

1

0.5

−0.5

(a) (b)

Figure 3.6 (a,b) Kurtosis for the linear versus parabolic response functionsof the maximum vertical displacement with respect to Young’s modulus

0 0.05 0.10 0.15 0.20a

0.001674

0.001672

0.001670

0.001668

0.001666

0.001664

0.001662

0.001660

E(u

(ν))



0 0.05 0.10 0.15 0.20a

a(u

(ν))

0.04

0.03

0.02

0.01

0

2nd-order analysis 4th-order analysis8th-order analysis6th-order analysis

10th-order analysis

(a) (b)

Figure 3.7 (a,b) Expectations and variation coefficients of maximum displacementto Poisson ratio

is apparently slower; both coefficients are almost always positive except the lowestorder approximation for the kurtosis.

Analogous moments and coefficients computed for a single response function withrespect to the input Poisson ratio behave in a different way (see Figures 3.7 and 3.8).Now the expected values (Figure 3.7(b)) decrease together with an increase of theinput α(e), where all perturbation orders return exactly the same values. Looking for


−0.2

−0.4

−0.6

−0.8

−1

−1.2

−1.4

−1.6

0 0.05 0.10 0.15 0.20a a

b(u

(n))





0 0.05 0.10 0.15 0.20

2.5

2

1.5

1

0.5

0

−0.5

k(u

(n))

(a) (b)

Figure 3.8 (a,b) Skewness and kurtosis of maximum displacement to Poisson ratio

the vertical scale of this figure, it is concluded that these differences may be totallydisregarded. The coefficient of variation is linearly proportional to the input one asin Figure 3.4, but now the window size in the vertical axis is almost five times lessthan the horizontal one, so that the input random dispersion has been significantlydamped by this system. Skewness and kurtosis given in Figure 3.8 have negative andpositive values respectively, for the entire domain of input α(ν), therefore the output

0.00137740

0.00137735

0.00137730

0.00137725

0.00137720

0.00137715

0 0.05 0.10 0.15 0.20a

E(u

(h))

0 0.05 0.10 0.15 0.20a



10th-order analysis



10th-order analysis

0.020

0.015

0.010

0.005

0

a(u

(h))

(a) (b)

Figure 3.9 (a,b) Expectations and variation coefficients of maximum displacementwith respect to Poisson ratio


−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

−0.07

−0.08

−0.09

−0.100 0.05 0.10 0.15 0.20

a



b(u

(h))

0 0.05 0.10 0.15 0.20a



k(u

(h))

0.008

0.006

0.004

0.002

−0.002

0

(a) (b)

Figure 3.10 (a,b) Skewness and kurtosis of maximum displacement with respectto support length

displacement cannot be Gaussian for any combination of the input moments for thePoisson ratio. Quite similarly to other results, their absolute values increase togetherwith an input parameter.

We study in turn the same probabilistic characteristics of the maximum displace-ments with respect to the length h of the supporting boundaries at the lower surfaceof this panel. Figure 3.9 – (a) with expectations and (b) containing coefficients ofvariation – shows practically no sensitivity to the perturbation order in this analysis.A ratio of output to input coefficient of variation remains constant for all α(h),but overall probabilistic damping is even larger than before and α(u(h)) is around10 times smaller than the input one. Expectations, although decreasing togetherwith α(h), show differences in the limits of a numerical error here, so practicallyshould be treated as constant. Now, the skewness of displacements being negativehas the same values for all perturbation orders and depends linearly on the inputrandom dispersion. The kurtosis has small positive values, essentially different forthe fourth-order analysis and the remaining orders, but the absolute values of bothhigher moments are very close to 0, so that a Gaussian distribution of the resultingmaximum deflection is detected (Figures 3.8–3.10).

Finally, we study the maximum stresses as a function of randomized Poisson ratioof the structure (Figures 3.11–3.12). Now the situation is more complex because evenexpectations, although computed with the same precision for all perturbation orders,vary non-linearly with input randomness level; these fluctuations are indeed negli-gible, but for the coefficients of variation second-order analysis returns apparentlylower values of α(σ (ν)) than the remaining theories. The ratio of output to inputvariation coefficients is not constant for fourth up to tenth perturbation orders, unlikein the previous cases. Further, the output dispersion is about 20 times smaller and thecoefficients of skewness and kurtosis (counted here in units) show a lack of Gaussian


0 0.05 0.10 0.15 0.20a

6.07 × 107

6.06 × 107

6.05 × 107

6.04 × 107

E(s

(ν))


10th-order analysis

0 0.05 0.10 0.15 0.20a

a(s

(ν))

0.010

0.008

0.006

0.004

0.002

0


10th-order analysis

(a) (b)

Figure 3.11 (a,b) Expectations and variation coefficients of maximum stresseswith respect to support length

3

2.5

2

1.5

1

0.5

0 0.05 0.10 0.15 0.20a

b(s

(ν))



0 0.05 0.10 0.15 0.20a

k(s

(ν))

8

6

4

2

−2

0



(a) (b)

Figure 3.12 (a,b) Skewness and kurtosis of the maximum stresses with respectto Poisson ratio

character in case of stresses. Both higher-order coefficients show additionally a veryslow probabilistic convergence since negligible differences are noticed for the firsttime between the eighth and tenth orders. Therefore, we can conclude that thereliability index computation for stresses may need much more computational effortin application of the stochastic perturbation technique than determination of theanalogous index based on the plate deflections.


5 m

1 3 2 5 4 7 6 9 8 11 10

x

y

22 23 20 21 18 19 16 17 14 15 12

ux=uy=0

24 e, n 1m 13

t=0.10 MN/m

Figure 3.13 Topology and discretization of the stainless steel cantilever beam.Reproduced with permission from Elsevier

Example 3.2: Steel cantilever plateThe next numerical example deals with the static analysis of the linear elasticand isotropic stainless steel cantilever plate with Young’s modulus randomizedaccording to the Gaussian PDF (Figure 3.13). Its expected value equals 210 GPa,whereas the coefficient of variation is the additional parameter in this analysis;Poisson ratio is taken of course as 0.3, while geometrical parameters are provided inFigure 3.13. Let us mention at the very beginning that the elasticity modulus usuallyhas coefficient of variation equal to or even smaller than 0.10, however, it may bereasonable to check larger values for some specific cases like extensively corrodedsteel structures, for instance. The details of computational analysis and discretizationof this structure are given in Figure 3.14 – the darker nodes denote edge nodes for theboundary elements with parabolic shape functions, while the midpoints are markedwhite. The vertical displacements at the 13th node are the subject of this analysisand are recovered using a parabolic function of the Young’s modulus, where theSBEM model results are compared with the following well-known formula frombeam theory: f = 1

3Pl3eJ , randomized using both stochastic perturbation analysis and

the classical Monte Carlo technique (for M = 3 × 105 random trials). Figure 3.14,presenting a comparison of the initially generated Young’s modulus and outputprobability distribution of the deflection, contains theoretical distributions (markedas continuous lines) and simulation-based histograms. It is apparent that the finaldistribution is not Gaussian due to the non-zero asymmetry, and has smaller expectedvalue than the corresponding Gaussian PDF with the same parameters. This meansthat determination of the first two moments is not sufficient to define uniquely themaximum deflection of this structure even for such a simple analytical formula.

The expected values presented in Figure 3.15(a) are computed according to thestochastic perturbation-based BEM, while Figure 3.15(b) shows Monte Carlo analysisand perturbation-based analytical formulas, also from the second to the tenth order.Analytical results start to diverge a little for larger values of α(e), while thesecomputed using BEM are all the same for various orders. The expectations computed


0.000018

0.000016

0.000014

0.000012

0.000010

0.000008

0.000006

0.000004

0.000002

0200000 300000

b

p(b)

b

p(b)

16000

14000

12000

10000

8000

6000

4000

2000

00 0.0001 0.0002 0.0003 0.0004

(a) (b)

Figure 3.14 (a,b) Input and output PDFs of the stainless steel cantilever beam.Reproduced with permission from Elsevier

−0.000223

−0.000224

−0.000225

−0.000226

0 0.05 0.10 0.15 0.20a

2nd-order8th-order

4th-order10th-order

6th-order

E(u

)

−0.000240

−0.000242

−0.000244

−0.000246

−0.000248

0 0.05 0.10 0.15 0.20a

MCS4th-order exact

2nd-order exact6th-order exact10th-order exact8th-order exact

E(u

)

(a) (b)

Figure 3.15 The expected values of the vertical displacement at the cantilever endvia (a) MCS-SBEM and (b) perturbation-based analytical formulas

according to the SBEM second group have absolute values a little smaller thananalytical computations for the entire spectrum of input coefficient variability andthese simulated in Monte Carlo scheme. Of course, the larger the random inputdispersion, the higher absolute value of maximum vertical deflection. Coefficientsof variation divided in an analogous way into two graphs of Figure 3.16 depend


0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

00 0.05 0.10 0.15 0.20

a

a(u

)



10th-order analysis

0 0.05 0.10 0.15 0.20a

a(u

)

0.20

0.15

0.10

0.05

0

MCS4th-order exact8th-order exact

2nd-order exact6th-order exact10th-order exact

(a) (b)

Figure 3.16 Coefficients of variation of the vertical displacement at the cantileverend via (a) MCS-SBEM and (b) perturbation-based analytical formulas

linearly on the input α(e) and show some gradual convexity in analytical calculus.Also now perturbation-based BEM gives somewhat smaller values than the remainingapproaches, being completely insensitive to the perturbation order. This is not thecase for analytical derivations, where second-order approximation is apparently notsufficient in reliability analysis. Almost independent of the analysis type, there is noprobabilistic entropy change between the input and output random variables. Thecomputed displacement is clearly not Gaussian, as documented in Figures 3.17 and3.18. Both skewness and kurtosis obtained with the use of the SBEM have significantlysmaller absolute values than these obtained in an analytical way. It needs to beunderlined that they mainly follow a deterministic difference between the BEM andthe beam theory itself, so they grow together with the central moment number.The differences for perturbation-based analytical calculations of various orders andMonte Carlo simulation of skewness are not so large for smaller input coefficient ofvariation only and they are most transparent for various curvatures at higher α(e).

SBEM determination of this skewness returns the same values for all analysisorders, keeping a straight line with respect to α(e). The resulting kurtosis (seeFigure 3.18) is non-linear in all cases – now fourth-order theory implemented withthe BEM gives significantly smaller results than the remaining higher-order appro-ximations (counted in tenths). Analytical computations return kurtosis measured inunits, which diverges for α(e) > 0.15; close to the maximum value of this coefficient,Monte Carlo simulation gives enormously large values, five times larger than theperturbation analysis. Independent of the analysis type, the resulting displacementhas a concentration around the mean value higher than the Gaussian distribution.

Further, we study expectations of maximum stresses (Figure 3.19(a)) and theircoefficients of variation (Figure 3.19(b)), as well as the skewness (Figure 3.20(a)) andkurtosis (Figure 3.20(b)) of these stresses – all computed according to various orders


−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

0 0.05 0.10 0.15 0.20a

b(u

)



0 0.05 0.10 0.15 0.20a

b(u

)

0

−0.5

−1

−1.5

−2

MCS6th-order exact10th-order exact

4th-order exact8th-order exact

(a) (b)

Figure 3.17 Skewness of the vertical displacement at the cantilever end via(a) MCS-SBEM and (b) perturbation-based analytical formulas

0.6

0.5

0.4

0.3

0.2

0.1

0

−0.1

0 0.05 0.10 0.15 0.20a

k(u

)



0 0.05 0.10 0.15 0.20a

k(u

)

25

20

15

10

5

0

MCS6th-order analysis10th-order analysis


(a) (b)

Figure 3.18 Kurtosis of the vertical displacement at the cantilever end via(a) MCS-SBEM and (b) perturbation-based analytical formulas

of the SBEM technique; it is once more based on a second-order polynomial withrespect to the input random variable which is Poisson ratio ν. Absolute values ofall probabilistic coefficients increase together with the input coefficient α(ν). Theoutput coefficient of variation for stresses has extremely small values increasinglinearly together with α(ν), analogously to the previous example. The expectation


0 0.05 0.10 0.15 0.20a

−3.071

−3.072

−3.073

−3.074

E(s

)

6th-order2nd-order10th-order4th-order

8th-order



a(s

)

0 0.05 0.10 0.15 0.20a

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

(a) (b)

Figure 3.19 (a) The expected values and (b) coefficients of variation for maximumnormal stresses, Gaussian Poisson ratio

−0.2

−0.4

−0.6

−0.8

−1.0

−1.2

0

0 0.05 0.10 0.15 0.20a

b(s

)

a





0 0.05 0.10 0.15 0.20

1.5

1

0.5

0

k(s

)

(a) (b)

Figure 3.20 (a) Skewness and (b) kurtosis for maximum normal stresses, GaussianPoisson ratio

and skewness are totally insensitive to the perturbation order; the coefficient ofvariation α(σ ) shows some marginal differences for larger α(ν) only, whereas thekurtosis shows full convergence starting from eighth-order SBEM analysis. Theexpected values decrease non-linearly together with the input dispersion, but therelative difference between the maximum and minimum values for a given windowis practically negligible.

Like in many cases before, kurtosis is totally negative for the fourth-order analysisand, therefore, gives a false conclusion about smaller concentration around the mean


6th-order2nd-order10th-order4th-order

8th-order



0.20

0.15

0.10

0.05

0

a(t

)

0 0.05 0.10 0.15 0.20a

E(t

)

0 0.05 0.10 0.15 0.20a

0.963

0.962

0.961

0.960

0.959

0.958

0.957

0.956

0.955

(a) (b)

Figure 3.21 (a) The expected values and (b) coefficients of variation for maximumnormal stresses, Gaussian Poisson ratio

0 0.05 0.10 0.15 0.20a

b(t

)



0.4

0.3

0.2

0.1

0.08

0.06

0.04

0.02

0

−0.02

0.10

0.12

0.14

a



0 0.05 0.10 0.15 0.20

k(t

)

(a) (b)

Figure 3.22 (a) Skewness and (b) kurtosis for maximum normal stresses, GaussianPoisson ratio

value than in the case of a Gaussian distribution. For a comparison to the maximumnormal stresses we contrast maximum shear stresses using expectations and coeffi-cients of variation (Figure 3.21) as well as skewness and kurtosis (Figure 3.22). Thesecharacteristics exhibit very similar tendencies concerning probabilistic convergencewith respect to the analysis order (even faster than before). The resulting coefficientof variation depends linearly upon the input one, having values slightly larger than


y10 11 8 9 6 21 19 20 17

1 3 2 5 4 14 13 16 15x

2.5 m 2.5 m

ux=uy=0

12 e1, n1 e2, n27 18

t = 0,10 MN/m

1 m

Figure 3.23 Topology and discretization of the composite cantilever. Reproducedwith permission from Elsevier

α(ν). The skewness β(τ ) also has linear dependence on this coefficient but, contrary toβ(σ ), is positive everywhere and, as before, rather distant from 0 having values twotimes larger than the input coefficient of variation. Kurtosis also shows some anoma-lies for the fourth-order stochastic perturbation and, although it is definitely smallerthan κ(σ ), we can conclude that both stress tensor components have non-Gaussiandistributions.

Example 3.3: Composite structure with random material parametersThe next example with discretization shown in Figure 3.23 deals with a plate ofsimilar rectangular shape, where the stainless steel is replaced by a compositemade of glass and epoxy components having expected values of material para-meters E[e1] = 84.0 GPa, E[e2] = 4.0 GPa, and Poisson ratios taken as E[ν1] = 0.22 andE[ν2] = 0.34. The response functions of the maximum vertical displacement in thisstructure with respect to these parameters are recovered numerically using the least-squares technique [87]. It needs to be mentioned at the beginning that such a sheartest for a two-component layered composite may serve for numerical verification ofthe interface quality and detection of possible interface defects and their influence onthe overall behavior of such a composite.

We collect now basic probabilistic characteristics of maximum vertical displace-ment at node 18 as functions of the input coefficient of variation α(b) ∈ [0.00, 0.20]and for various perturbation orders using consecutively parabolic approximationbetween the output and input random parameters. Additionally, we analyze theseexpectations, coefficients of variation, skewness, and kurtosis as resulting from sepa-rate randomization of the Young’s modulus of the reinforcement and matrix as wellas Poisson ratios of both components. This is done to verify the most decisive inputparameter in view of the reliability assessment and overall durability and integrityof a composite. The general conclusion on probabilistic convergence is that all expe-ctations as before are practically insensitive to the given perturbation order, so that


−0.001980

−0.001985

−0.001990

−0.001995

−0.002000

0 0.05 0.10 0.15 0.20

2nd-order8th-order

4th-order 6th-order10th-order

a(e1)

E(u

)0.05

0.04

0.03

0.02

0.01

00 0.05 0.10 0.15 0.20



a(e1)

a(u

)(a) (b)

Figure 3.24 (a,b) The expectations and coefficients of variation for the cantilever enddeflection for Young’s modulus of the reinforcement

second-order analysis is quite sufficient (Figures 3.24a, 3.26a, 3.28a and 3.30a). Thisorder is invalid for a precise computation of the coefficient of variation, where espe-cially for α(b) close to 0.20 higher orders than the second are all the same and a little bitlarger than second-order results (cf. Figures 3.24b, 3.26b, 3.28b and 3.30b). Then, for theneeds of reliability analysis based on Eurocode 0 and using the Hasofer–Lind index,fourth-order analysis should be preferred. Skewness analysis used to recognize a typeof output probability distribution shows unconditional convergence (independent ofthe input α(b)) together with sixth-order perturbation-based SBEM. As the resultingcoefficients of variation depend almost linearly on the input random dispersion,skewness as a function of α(b) always shows small concavity or convexity, especiallyfor larger values of this parameter (see Figures 3.25a, 3.27a, 3.29a as well as 3.31a).

As could be expected, the lowest convergence is numerically verified for kurtosis(given in Figures 3.25b, 3.27b, 3.29b and 3.31b), where as before, second-orderanalysis returns initially negative values and then, depending on the input variabletype is faster or slower – the fastest obtained during randomization of the Poissonratio of reinforcement, where sixth-order analysis is precise enough. Comparison ofdetailed numerical values of all characteristics shows that the largest absolute valuesof the cantilever maximum deflections are noticed, while the Young’s modulus ofthe matrix is random. The difference between minimum and maximum expectationsis also negligible, as for the steel panel in the previous two numerical examples.Moreover, uncertainty of the Young’s modulus of both components leads to anincrease of this expectation absolute value together with α(e), whereas randomnessof the Poisson ratio gives an inverse effect. A comparison of the variation coefficientsshows that the most influential parameter is the Young’s modulus of the matrix;


0 0.05 0.10 0.15 0.20a(e1)

b(u

)

−0.8

−1

−1.2

−1.4

−1.6

−1.8

−0.6

−0.4

−0.2



3

2

1

0

a(e1)4th-order analysis8th-order analysis


0 0.05 0.10 0.15 0.20

k(u

)(a) (b)

Figure 3.25 (a,b) Skewness and kurtosis for the cantilever end deflection for Young’smodulus of the reinforcement

−0.00204

−0.00205

−0.00206

−0.00207

−0.00208

−0.00209

0 0.05 0.10 0.15 0.20a(e2)

E(u

)

2nd-order8th-order


0.15

0.10

0.05

00 0.05 0.10 0.15 0.20



a(e2)

a(u

)

(a) (b)

Figure 3.26 (a,b) The expectations and coefficients of variation for the cantilever enddeflection for Young’s modulus of the matrix

further we have Young’s modulus od the reinforcement, the Poisson ratio of thereinforcement and, finally, this ratio for the matrix. The difference between the firstand last parameter is dramatic, because α(u) for e2 = e2(ω) is around one-quartersmaller than the initial variation and almost 0 for ν2 = ν2(ω). Finally, we notice thatnone of the studied probabilistic distributions is Gaussian since the skewness andkurtosis differ significantly from 0. The skewness and kurtosis have correspondingly


0 0.05 0.10 0.15 0.20a(e2)

b(u

)

−0.8

−1.0

−1.2

−1.4

−0.6

−0.4

−0.22

1.5

0.5

1

−0.5

0

a(e2)





0 0.05 0.10 0.15 0.20

k(u

)(a) (b)

Figure 3.27 (a,b) Skewness and kurtosis for the cantilever end deflection for Young’smodulus of the matrix

−0.001997

−0.001998

−0.001999

−0.002000

−0.002001

−0.002002

E(u

)

0 0.05 0.10 0.15 0.20a(n1)

2nd-order8th-order


0.025

0.020

0.015

0.010

0.005

0



0 0.05 0.10 0.15 0.20a(n1)

a(u

)

(a) (b)

Figure 3.28 (a,b) The expectations and coefficients of variation for the cantilever enddeflection for Poisson ratio of the reinforcement

very similar values, where negative third moments are obtained with randomizingboth Young’s moduli and positive for both Poisson ratios (Figure 3.31). The kurtosisin all four cases starts from negative values to become positive for higher thanfourth-order models. Considering all the moments at once it is common that thelarger the input coefficient of variation, the higher the absolute value of the givenprobabilistic characteristic.


0 0.05 0.10 0.15 0.20a(n1)

b(u

)

0.6

0.5

0.4

0.3

0.2

0.1

0.7

0.8

0.9





0.8

0.6

0.2

0.4

−0.2

0

a(n1)0 0.05 0.10 0.15 0.20

k(u

)(a) (b)

Figure 3.29 (a,b) Skewness and kurtosis for the cantilever end deflection for Poissonratio of the reinforcement

−0.0019995

−0.0020000

−0.0020005

−0.0020010

−0.0020015

−0.0020020

0 0.05 0.10 0.15 0.20a(n2)

E(u

)

2nd-order8th-order




0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

00 0.05 0.10 0.15 0.20

a(n2)

a(u

)

(a) (b)

Figure 3.30 (a,b) The expectations and coefficients of variation for the cantilever enddeflection for Poisson ratio of the matrix

Example 3.4: SBEM analysis of the composite with interface defectsThe two-component, two-layer (for simplicity) carbon–epoxy composite given inFigure 3.32 is subjected to the SBEM analysis with the following material characte-ristics: e1 = 77.5 GPa, ν1 = 0.24, e2 = 4.1 GPa, and ν2 = 0.34. We use 113 nodal pointslinked with 64 parabolic boundary elements forming two closed regions havinga common, partially continuous interface; this composite specimen is clamped at theleft edge and subjected to the simple shear test realized by the tangent stresses applied


0 0.05 0.10 0.15 0.20a(n2)

b(u

) 0.8

0.6

0.4

0.2

1.0

1.2

1.4



2

1.5

0.5

1

−0.5

0

a(n2)4th-order analysis8th-order analysis


0 0.05 0.10 0.15 0.20

k(u

)(a) (b)

Figure 3.31 (a,b) Skewness and kurtosis for the cantilever end deflection for Poissonratio of the matrix

at the opposite vertical edge. Four semi-circular cavities are inserted at this interfacein the area of the epoxy matrix with uniform spatial distribution and the same radius(E[r] = 0.2), each of them modeled using four curvilinear boundary elements [93].The very detailed theoretical model of such a composite with imperfect interface isproposed in Chapter 5 of this book (see Page 245).

Using the proposed methodology implemented as the hybrid Maple and themulti-domain BEM solver [4], we determine first the response functions betweenthe maximum vertical deflection in the system (Figures 3.33–3.35) and the cavities’radius. These functions are determined numerically for a uniform discretization ofthe given input random variable r = [0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3.1.4, 1.5, 1.6]and shown in Figures 3.35–3.37 (in turn, non-weighted method, triangular weightsdistribution around the mean value of random input parameter [1–6, 5, 4, 3, 2, 1],and Dirac distribution of the weights – [1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1]). Figures 3.33a,3.34a and 3.35a show the variability of the different-order approximations for theentire input parameter domain and Figures 3.33b, 3.34b and 3.35b correspond tothe very narrow neighborhood of the mean value. According to the opportunity ofthe RFM, we compare approximations of various orders – from the first until theninth – to distinguish the differences between them and to determine for the futureapplications the most optimal choice for the specific application. It is apparent fromthe comparison of Figures 3.33–3.35 that while the non-weighted method showssome upper- and lower-end numerical discrepancies, the triangular weighted resultsdemonstrate the same but for lower-order approximations only. The approach basedon the Dirac distribution of the weights returns the most regular approximatingfunctions without any oscillations. It is also clear that each first-order approximationis too weak to be used for reliable determination of any probabilistic moments for thestructural response, whereas the differences between the second and higher orders are


t=0.10 MN/m

2.6

m0.8 m 0.2 m 0.6 m

umax

Figure 3.32 SBEM discretization model of the representative volume element (RVE)with real interface defects. Reproduced with permission from Elsevier

1st-order4th-order7th-order

2nd-order5th-order8th-order 9th-order

3rd-order6th-order

−0.00008

−0.00010

−0.00012

−0.00014

−0.00016

−0.00018

−0.00020

−0.00022

u(b)

0.06 0.08 0.10 0.12 0.14 0.16b



3rd-order6th-order

−0.00012

−0.00013

−0.00014

−0.00015

u(b)

0.100 0.105 0.110 0.115 0.120b

(a) (b)

Figure 3.33 (a,b) The response function of the maximum vertical displacement versuscavities’ radius, unweighted RFM–SBEM. Reproduced with permission from Elsevier




3rd-order6th-order



3rd-order6th-order

−0.00008

−0.00010

−0.00012

−0.00014

−0.00016

−0.00018

−0.00020

−0.00022

−0.000115

−0.000120

−0.000125

−0.000130

−0.000135

−0.000140

−0.000145

u(b)

0.06 0.08 0.10 0.12 0.14 0.16b

u(b)

0.100 0.105 0.110 0.115 0.120b

(a) (b)

Figure 3.34 (a,b) The response function of the maximum vertical displacement ver-sus cavities’ radius, triangular weights RFM–SBEM. Reproduced with permission fromElsevier

practically negligible here. Further, we collect the expected values (Figures 3.36 and3.37), the coefficients of variation (Figures 3.38 and 3.39), the skewnesses (Figures 3.40and 3.41), and the kurtosis (Figures 3.42 and 3.43) of the same vertical displacementsobtained in the lower right corner of this composite specimen. They are all computedand presented as functions of the input coefficient of variation of the cavities’ radius(taken as α ∈ [0.0, 0.10]) for perturbation parameter equal to 1 according to thenon-weighted response function, the response with triangular weights as well as forthe Dirac distribution of these weights. Higher-order probabilistic coefficients arecomputed in order to verify whether the vertical displacement at this specific pointcan have Gaussian distribution or not.

The expected values shown in Figures 3.36 and 3.37 exhibit very similar propertiesto these exposed in the response function graphs. Once more, the closest results forvarious-order approximations are noticed in the case of the weighted version of theSBEM according to the Dirac distribution. There are no specific orders resulting inapparently different results, like these obtained in the case of seventh order for thetriangular weights distribution; of course, the first-order results are unacceptable forall three methods. It is also interesting that the absolute values of these expectationsreach a maximum for the Dirac distribution, while a minimum is reached for thetriangular weights and the non-weighted scheme returns intermediate values – thisfact has no simple explanation. The function between the output expectation andthe input coefficient of variation is entirely non-linear, where the larger the inputα, the larger absolute value of the maximum vertical displacement in the sheartest. Since we are interested in an optimal order of the method, then consideringexpectations only, the second one would be the most recommended as it is almost




3rd-order6th-order

−0.00008

−0.00010

−0.00012

−0.00014

−0.00016

−0.00018

−0.00020

−0.00022

u(b)

0.06 0.08 0.10 0.12 0.14 0.16b



3rd-order6th-order

−0.000115

−0.000120

−0.000125

−0.000130

−0.000135

−0.000140

−0.000145

u(b)

0.100 0.105 0.110 0.115 0.120b

(a) (b)

Figure 3.35 (a,b) The response function of the maximum vertical displacement versuscavities’ radius, Dirac weights RFM–SBEM. Reproduced with permission from Elsevier

−0.000124

−0.000126

−0.000128

−0.000130

−0.000132

−0.000134

−0.000136

−0.000138

−0.000140

E(u

)

0 0.05 0.10 0.15a

1st-order 2nd-order 3rd-order5th-order 6th-order4th-order

7th-order 8th-order

−0.000122

−0.000124

−0.000126

−0.000128

−0.000130

−0.000132

−0.000134

E(u

)

0 0.05 0.10 0.15a

1st-order 2nd-order 3rd-order5th-order 6th-order4th-order

7th-order 8th-order

(a) (b)

Figure 3.36 The expected values of the maximum vertical displacement, unweighted(a) and triangular weights (b) RFM–SBEM. Reproduced with permission from Elsevier

equal to the results of the highest possible order. The coefficients of variationshown in Figures 3.38 and 3.39 demonstrate that the system has neither dampingnor amplifying properties in the context of uncertainty – the randomness at theinput is almost the same as at the output and this interrelation is almost linear;


−0.000126

−0.000128

−0.000130

−0.000132

−0.000134

−0.000136E

(u)

0 0.05 0.10 0.15a

1st-order 2nd-order

5th-order 6th-order

3rd-order

4th-order

7th-order 8th-order

Figure 3.37 The expected values of the maximum vertical displacement, Diracweights RFM–SBEM. Reproduced with permission from Elsevier

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

00 0.05 0.10 0.15

a

a(u

)


2nd-order5th-order8th-order

3rd-order6th-order

a

a(u

)

0.10

0.08

0.06

0.04

0.02

00.100.080.060.040.020



3rd-order6th-order

(a) (b)

Figure 3.38 The variation coefficients of maximum vertical displacement,(a) non-weighted and (b) triangular weights RFM–SBEM. Reproduced with permissionfrom Elsevier

nevertheless various techniques return slightly different values. As in the case ofexpectations, right now we have some lower-order numerical discrepancies for thenon-weighted and for the triangular weights, while the Dirac weighting returns themost stable results. Now, the second order analysis is not sufficient because wenotice significant differences to higher orders even in Figure 3.39. The differences forthe skewness coefficients are more apparent than for the coefficients given before,


0 0.05 0.10 0.15a

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

a(u

)


3rd-order6th-order


Figure 3.39 The variation coefficients of maximum vertical displacement, Diracweights RFM–SBEM. Reproduced with permission from Elsevier

0 0.02 0.04 0.06 0.08 0.10a

0

−0.2

−0.6

−1

−1.2

−1.4

−1.6

−0.8

−0.4

b(u

)



3rd-order6th-order

b(u

)

0 0.02 0.04 0.06 0.08 0.10a



3rd-order6th-order

0

−1

−2

−3

(a) (b)

Figure 3.40 The skewness of the maximum vertical displacement, (a) non-weightedand (b) triangular weights RFM–SBEM. Reproduced with permission from Elsevier

however negative values are obtained for all cases. They all start from 0 for thedeterministic test and decrease systematically together with an increase in value ofthe input coefficient of variation, so that we cannot treat the resulting deflectionas a Gaussian variable and, unfortunately, probabilistic computational analysis andreliability index approximation cannot be limited to the rather simple formulasknown from Eurocode 0. Analogously to the previous figures and to the kurtosis


0

−0.5

−1

−1.5

−2

0 0.05 0.10 0.15a



3rd-order6th-order

b(u

)

Figure 3.41 The skewness of the maximum vertical displacement, Dirac weightsRFM–SBEM. Reproduced with permission from Elsevier

0 0.02 0.04 0.06 0.08 0.10a



3rd-order6th-order

2

1.5

0.5

−0.5

0

1

k(u

)

0 0.02 0.04 0.06 0.08 0.10a



3rd-order6th-order

k(u

)

7

6

5

4

3

2

1

0

(a) (b)

Figure 3.42 The kurtosis of the maximum vertical displacement, (a) non-weightedand (b) triangular weights RFM–SBEM. Reproduced with permission from Elsevier

given in Figures 3.42 and 3.43, the seventh-order analysis in the triangular weightsscheme demonstrates some numerical instabilities and once more, the first-orderresults are not acceptable at all. The fourth-order coefficients also show that the finaldistribution is not Gaussian, but they demonstrate various properties – sometimespositive, sometimes negative – whereas their absolute values increase together withthe additional increase of the input coefficient of variation. It is confirmed byFigures 3.40–3.43 that probabilistic convergence of higher order moments and


0 0.05 0.10 0.15

1st-order 2nd-order5th-order 6th-order4th-order

7th-order 8th-order

3rd-order

a

3

2

1

0

−1

k(u

)

Figure 3.43 The kurtosis of the maximum vertical displacement, Dirac weightsRFM–SBEM. Reproduced with permission from Elsevier

coefficients is decisively slower together with the perturbation order than for theexpectations and variances.

As is demonstrated above, the SBEM is especially valuable for engineers andscientists in boundary value problems with random parameters, where the macro-or microgeometry includes some uncertainty. It is especially important for theResponse Function Method, where the SFEM application would need additionalremeshing for the few models with different geometry. Then, the differences in-between these meshes could deterministically affect the final diagrams with thestructural responses’ probabilistic moments. This conclusion, drawn on the basisof the generalized perturbation SBEM, remains in perfect agreement with someprevious numerical studies [27, 48]. It could especially be of paramount importancein computational fracture mechanics by the SBEM [24], where initiation and crackpropagation have a generally stochastic character [93].

4The Stochastic FiniteDifference Method

The finite difference method (FDM), being probably one of the oldest numericaltechniques in engineering and scientific computations [20, 131, 167], may be stillwidely used in the domain of heat transfer [130], electrodynamics [163], electro-magnetics [115], and hydromechanics [169], even for large-scale problems. Sinceengineers and scientists usually obtain an information about structural and systemparameters in the form of statistical estimators or random processes, the idea ofextending the FDM to modeling of random variables, fields, or even processes seemsquite natural [102, 164]. Owing to such an extension, one can determine how theprobabilistic parameters of the input affect the corresponding parameters of thestructural response; it is known that this can be done using various mathematical andnumerical method including these displayed in the preceeding chapters of this book.Alternative methods of modeling random processes, in the context of FDM stochasticextension, are via application of the Monte Carlo simulation technique or stochasticspectral finite differences (improved Neumann, Karhunen–Loeve, or polynomialchaos expansions), but their usage has a limited number of well-documented real-izations and unresolved accuracy of higher probabilistic moments’ determination.The generalized perturbation technique is used here analogously to its previousapplications in the FEM and BEM stochastic extensions to utilize the FDM for theneeds of probabilistic analysis [76, 90]. This idea follows the previous SOSM versionof the stochastic FDM [66, 68, 102]. The essential difference from the FEM and BEM, asintegral methods, is an assumption that all random functions and fields should be dif-ferentiable with respect to both spatial coordinates and, what is more important, at thesame time to assumed random variables. We analyze the boundary value problems ortransient problems with the use of unidirectional and planar regular grids includingthe closest and also wider neighborhood of difference operators [111]. Computationalanalysis in this section is entirely symbolic, because the system Maple with its pro-gramming language and internal linear algebra tools enables algebraic equation’s



solutions of increasing order. The response function method (RFM) application isalso quite straightforward in this context, having all difference equations coded intothe Maple script. Further implementations and interoperability of Maple withFORTRAN codes give a unique opportunity to employ academic FDM software forstochastic perturbation technique applications in a symbolic environment, which isvery important in view of the powerful visualization procedures.

4.1 Analysis of the Unidirectional Problems

4.1.1 Elasticity Problems

Let us consider the following ordinary fourth-order differential equation for a linearelastic isotropic and statistically homogeneous beam exposed to the transversallydistributed load q(x):

d2

dx2

(e (x) J (x)

d2w (x)

dx2

)= q (x) , (4.1)

fulfilling typical boundary conditions applicable in engineering theories of the elasticbeams, like these idealized in Figure 4.1.

Investigation of the influence of the longitudinal and transversal forces is postponedhere (but also possible), as we reduce the analysis to small deflections, whichsignificantly simplifies the final form of the equilibrium equation subjected to theperturbation procedure. Let us divide the entire domain of length l into n equidistantsubdomains having length �x. We adopt the following notation for the ith point ofthis discretization:

e (i�x) = ei, J (i�x) = Ji, q (i�x) = qi, (4.2)

for the coordinate system introduced in Figure 4.1.A series of approximations follows for derivatives of ascending order with the

central finite differences, such as [23, 115]:

• first derivative (�w�x

)i= wi+1 − wi−1

2�x(4.3)

q (x )

xE (x ), J (x )

l

W

Figure 4.1 Elastic beam subject to the transversal load q(x). Reproduced withpermission from the Journal of Theoretical and Applied Mechanics

The Stochastic Finite Difference Method 197

• second derivative (�2w�x2

)i= wi+1 − 2wi + wi−1

�x2 (4.4)

• third derivative(

�3w�x3

)i= −wi+2 + 2wi+1 − 2wi−1 + wi−2

2�x3 (4.5)

• fourth derivative(

�4w�x4

)i= wi+2 − 4wi+1 + 6wi − 4wi−1 + wi−2

�x4 . (4.6)

Equation (4.3) is an arithmetic average of the forward and backward differenceequation and the next relations contain its simple consequence as a derivative ofthe derivative, and so forth. Introducing further for the fourth-order derivatives inEquation (4.1), and using the statement (4.2), one can obtain for the ith point of thisgrid the recursive difference equation as

ei−1Ji−1wi−2 − 2(ei−1Ji−1 + eiJi

)wi−1 + (

ei−1Ji−1 + 4eiJi + ei+1Ji+1)

wi

− 2(eiJi + ei+1Ji+1

)wi+1 + ei+1Ji+1wi+2 = qi�x4. (4.7)

The particular case of e(x)J(x) = const. = eJ enables us to transform Equation (4.7)into much simpler following formula:

wi−2 − 4wi−1 + 6wi − 4wi+1 + wi+2 = qi�x4

eJ. (4.8)

The situation changes only slightly when the elastic beam rests on a single-parameter elastic foundation (known from geotechnics as the Winkler foundation).Let us assume that this foundation has a homogeneous character and is completelycharacterized by the compliance coefficient k. Therefore, the equilibrium equationincluding the deflection w(x) has a single extra component on the RHS. It holds that

d2

dx2

(e (x) J (x)

d2w (x)

dx2

)= −kw (x) + q (x) . (4.9)

Inserting, as previously, the additional formulas for the derivatives, which includefinite differences expressed by discrete values of the function w(x) in the neighborhoodof the given ith point of the grid, it is obtained that

ei−1Ji−1wi−2 − 2(ei−1Ji−1 + eiJi

)wi−1 + (

ei−1Ji−1 + 4eiJi + ei+1Ji+1 + k�x4)wi

− 2(eiJi + ei+1Ji+1

)wi+1 + ei+1Ji+1wi+2 = qi�x4. (4.10)


A reduction in case of the constant stiffness e(x)J(x) = const. = eJ leads to thefollowing equation:

wi−2 − 4wi−1 +(

6 + k�x4

eJ

)wi − 4wi+1 + wi+2 = qi�x4

eJ. (4.11)

Traditionally, the stochastic perturbation approach to many of the physicalproblems is entered by the respective perturbed equations of the zeroth, firstand, successively, higher orders being a modification of the variational integralformulation. Therefore, we provide an extra differentiation of the existing differen-tial/difference equations with respect to the input random variable of the problemand we obtain:

• zeroth-order partial differential equation

e0J0 d4

dx4

(w0(x)

) = q0 (4.12)

• first-order partial differential equation

∂e∂b

J0 d4

dx4

(w0(x)

) + e0 ∂J∂b

d4

dx4

(w0(x)

) + e0J0 d4

dx4

(∂w(x)

∂b

)= ∂q

∂b(4.13)


∂2e∂b2 J0 d4

dx4

(w0(x)

) + e0 ∂2J∂b2

d4

dx4

(w0(x)

) + e0J0 d4

dx4

(∂2w(x)

∂b2

)

+ 2{

∂e∂b

∂J∂b

d4

dx4

(w0(x)

) + e0 ∂J∂b

d4

dx4

(∂w(x)

∂b

)+ ∂e

∂bJ0 d4

dx4

(∂w(x)

∂b

)}= ∂2q

∂b2 .

(4.14)

Quite similarly, one can derive the ascending-order partial differential equationsfor the elastic beam on the elastic foundation starting from relation (4.9). The followinghold:

• zeroth-order partial differential equation

e0J0 d4

dx4

(w0(x)

) = −k0w0(x) + q0 (4.15)

• first-order partial differential equation

∂e∂b

J0 d4

dx4

(w0(x)

) + e0 ∂J∂b

d4

dx4

(w0(x)

) + e0J0 d4

dx4

(∂w(x)

∂b

)

= ∂q∂b

−(

∂k∂b

w0(x) + k0 ∂w(x)∂b

)(4.16)



∂2e∂b2 J0 d4

dx4

(w0(x)

) + e0 ∂2J∂b2

d4

dx4

(w0(x)

) + e0J0 d4

dx4

(∂2w(x)

∂b2

)

+ 2{

∂e∂b

∂J∂b

d4

dx4

(w0(x)

) + e0 ∂J∂b

d4

dx4

(∂w(x)

∂b

)+ ∂e

∂bJ0 d4

dx4

(∂w(x)

∂b

)}

= ∂2q∂b2 −

(∂2k∂b2 w0(x) + 2

∂k∂b

∂w(x)∂b

+ k0 ∂2w(x)∂b2

). (4.17)

The derivation of higher-order equations proceeds quite similarly – by system-atic differentiation until the nth-order equation is recovered. Having solved theseequations for w0(x) and higher orders, consecutively (specifically for all partialderivatives w.r.t. a random input within all discrete points of the grid), we derive theexpressions for the expected values and other moments of the elastic beam deflec-tions. The discrete equations for the stochastic FDM built upon the above equationsfor the perturbation-based analysis are essentially different in case of an elastic beamwith and without an elastic foundation beneath it. Putting here k = 0 returns variousorder relations derived on the basis of Equation (4.8) such as, for example:

• zeroth-order relation

w0i−2 − 4w0

i−1 + 6w0i − 4w0

i+1 + w0i+2 = q0

i

(�x0

)4

e0J0 (4.18)

• nth-order relation

∂nwi−2

∂bn − 4∂nwi−1

∂bn + 6∂nwi

∂bn − 4∂nwi+1

∂bn + ∂nwi+2

∂bn = ∂n

∂bn

(qi�x4

eJ

)(4.19)

so that the uniformly distributed randomized load gives here non-zero equations ofup to first order only, random length results in up to fourth-order equations, whereascross-sectional and/or material randomness brings an infinite number of equations.Looking for the perturbation-based specially adopted FDM (SFDM) equations forbeams on the elastic foundation, one gets from Equation (4.11):

• zeroth-order equations

w0i−2 − 4w0

i−1 +(

6 + k0(�x0

)4

E0J0

)w0

i − 4w0i+1 + w0

i+2 = q0i

(�x0

)4

e0J0 (4.20)

• nth-order equations

∂nwi−2

∂bn − 4∂nwi−1

∂bn + 6∂nwi

∂bn +n∑

p=1

(np

)∂p

∂bp

(k�x4

eJ

)∂n−pwi

∂bn−p − 4∂nwi+1

∂bn + ∂nwi+2

∂bn

= ∂n

∂bn

(qi�x4

eJ

). (4.21)


q = 1000 N/m

0.03

m

0.06

m

l = 0.6 m0.03 m 0.03 m

−1 0 1 2 3 4 5 6 8

0.1 m 0.1 m 0.1 m 0.1 m 0.1 m 0.1 m 0.1 m 0.1 m 0.1 m

l = 6·∆x = 6·0.1 m

Figure 4.2 FDM discretization of the cantilever with linearly varying cross-section.Reproduced with permission from the Journal of Theoretical and Applied Mechanics

Let us note also that n − 1 of the above equations may be generated automaticallyfrom the zeroth-order formula using relatively easy symbolic procedure, which isdocumented in the examples given in the Author’s webpage accompanying thisbook.

We need to remember that according to the general stochastic perturbation philo-sophy, all partial derivatives with respect to the random input variables are calculatedat its expectation. Therefore, the RFM approach needs to be focused on the responsefunctions’ determination in some close neighborhood of this value.

Example 4.1: Deflection of the linear elastic beam with linearly varyingcross-sectional areaLet us determine the first four probabilistic moments for the cantilever beam withlinearly varying cross-sectional area under the constant distributed load q = 1.0 kN/m[146]. The Young’s modulus of this beam is taken as the input Gaussian randomvariable, where the expected value is given as E[e] = 206.01 GPa; a grid for thisstructure consisting of six equidistant elements having length �x = 0.1 m is adopted(Figure 4.2). Let us note that linear variations in the beam bending stiffness needmuch more complex perturbation-based equations than those proposed above.

The following central finite difference equations hold true in this particular case,after Equation (4.7):

J0w−1 − 2(J0 + J1)w0 + (J0 + 4J1 + J2)w1 − 2(J1 + J2)w2 + J2w3 = q�x4

e,


J1w0 − 2(J1 + J2)w1 + (J1 + 4J2 + J3)w2 − 2(J2 + J3)w3 + J3w4 = q�x4

e,


e,


e,


e,


e, (4.22)

where of course inertia moments deterministically depend on the position of a pointwithin the grid. Introducing fictitious nodes as well as using kinematic boundaryconditions at the left clamped edge, we obtain:

(2J0 + 4J1 + J2)w1 − 2(J1 + J2)w2 + J2w3 = q�x4

e,

−2(J1 + J2)w1 + (J1 + 4J2 + J3)w2 − 2(J2 + J3)w3 + J3w4 = q�x4

e,


e,


e,

J4w3 − 2(J4 + J5)w4 + (J4 + 4J5)w5 − 2J5w6 = q�x4

e,

2J5w4 − 4J5w5 + 2J5w6 = q�x4

e. (4.23)

This system of linear equations may be used for further analytical partial dif-ferentiation with respect to the random input variable and recursive formation ofthe higher-order equations necessary to derive probabilistic characteristics of themaximum deflection.

We compute probabilistic moments and coefficients using the Direct DifferentiationMethod and up to the tenth-order perturbation technique – all as a function ofthe input coefficient of variation α ∈ [0.00, 0.25] first. We obtain the expectationand variances given in Figure 4.3, standard deviation and coefficient of variationpresented in Figure 4.4, third and fourth central probabilistic moments in Figure 4.5,as well as skewness and kurtosis shown in Figure 4.6. This is all done for themaximum deflection at the end of the cantilever analyzed, where the limit functionand reliability index are to be verified. All first two moments and characteristicsincrease together with α – expectations and variances highly non-linearly (Figure 4.7),while standard deviations and coefficients of variation decisively closer to the linearfunction (Figure 4.8). Of course, they are all positive and the higher the perturbationorder, the larger final values of these characteristics. Expected values converge


0.000230

0.000225

0.000220

0.000215

E(u

max

)

0.05 0.10 0.15 0.20 0.250α

0.05 0.10 0.15 0.20 0.250α

6. × 10−9

5. × 10−9

4. × 10−9

3. × 10−9

2. × 10−9

1. × 10−9

0

Var

(um

ax)

2nd order8th order

2nd order8th order

4th order 6th order 6th order10th order

4th order10th order

(a) (b)

Figure 4.3 (a) Expected values and (b) variances (b) of the maximum deflection

0.00007

0.00006

0.00005

0.00004

0.00003

0.00002

0.00001

0

σ(u m

ax)

α(u m

ax)

0.05 0.10 0.15 0.20 0.250α

0.05 0.10 0.15 0.20 0.250α

0.3

0.2

0.1

0

2nd order8th order

2nd order8th order


4th order10th order

(a) (b)

Figure 4.4 (a) Standard deviations and (b) coefficients of variation of the maximumdeflection

probabilistically very fast, so that fourth-order analysis is precise enough even forthe maximum value of the input coefficient of variation, while the remaining second-order characteristics need higher-order approximation and then, at least eighth orderis recommended. Let us underline that the difference for the expectations computedfor α = 0 and 0.25 is almost 10% of its minimum and cannot be simply neglected. Atthe same time the output coefficient of variation is definitely larger than the input α,hence the system clearly amplifies input uncertainty in the Young’s modulus.

As expected after Figure 4.3, the higher the central probabilistic moment considered,the smaller absolute value of this moment and also the weaker probabilistic


1.2 × 10−12

1. × 10−12

8. × 10−13

6. × 10−13

4. × 10−13

2. × 10−13

0

µ 3(u

max

)

µ 4(u

max

)

0.05 0.10 0.15 0.20 0.250α

0.05 0.10 0.15 0.20 0.250α

3. × 10−16

2. × 10−16

1. × 10−16

0

4th order10th order


6th order 8th order

(a) (b)

Figure 4.5 (a) Third and (b) fourth central probabilistic moments of the maximumdeflection

2.5

2

1.5

1

0.5

β(u m

ax)

κ(u m

ax)

0.05 0.10 0.15 0.20 0.250α

0.05 0.10 0.15 0.20 0.250α

6

5

4

3

2

1

0

−1

4th order10th order


6th order 8th order

(a) (b)

Figure 4.6 (a) Skewness and (b) kurtosis of the maximum deflection

perturbation convergence. Exceptionally, we can treat tenth-order approximationas satisfactory for the third moment, especially for smaller input α, but this is com-pletely not the case for the fourth moment (Figure 4.9). It is also apparent that thehigher the output moment of a maximum deflection, the larger the curvature of theinterrelations of µk(umax) versus α(e).

We need to remember that higher-order moments tending to 0 do not automat-ically imply that the skewness and kurtosis will behave in the same way. Nowthese coefficients are both dominated by the positive values (except kurtosis in the


0.00020

0.00015

0.00010

0.00005

0

E(u

max

)

Var

(um

ax)

0 0.1 0.2 0.3 0.4 0.5 0.6x

0 0.1 0.2 0.3 0.4 0.5 0.6x

6. × 10−9

5. × 10−9

4. × 10−9

3. × 10−9

2. × 10−9

1. × 10−9

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.7 (a) Expected values and (b) variances of the maximum deflection

0.00007

0.00006

0.00005

0.00004

0.00003

0.00002

0.00001

0

σ(u

max

)

α(u

max

)

0 0.1 0.2 0.3 0.4 0.5 0.6x

0 0.1 0.2 0.3 0.4 0.5 0.6x

0.3

0.2

0.1

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.8 (a) Standard deviations and (b) coefficients of variation of the maximumdeflection

second-order approach), which clearly states that the output maximum deflectionhas non-Gaussian probabilistic distribution. Skewness and kurtosis do not showany probabilistic convergence, especially for larger values of input parameter α,so that sometimes even tenth-order perturbation may be insufficient to determinehigher-order statistics of the structural response precisely enough.


1.2 × 10−12

1. × 10−12

8. × 10−13

6. × 10−13

4. × 10−13

2. × 10−13

0

µ 3(u

max

)

µ 4(u

max

)

0 0.1 0.2 0.3 0.4 0.5 0.6x

0 0.1 0.2 0.3 0.4 0.5 0.6x

3. × 10−16

2. × 10−16

1. × 10−16

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.9 (a) Third and (b) fourth central probabilistic moments of the maximumdeflection

2.5

2

1.5

1

0.5

0

β(u m

ax)

κ(u m

ax)

0 0.1 0.2 0.3 0.4 0.5 0.6x

0 0.1 0.2 0.3 0.4 0.5 0.6x

6

5

4

3

2

1

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.10 (a) Skewness and (b) kurtosis of the maximum deflection

Further, we analyze the spatial distribution of probabilistic characteristics alongthe cantilever beam. Somewhat similarly to the distributions of the central momentsw.r.t. α(e), now expectations, variances, third and fourth central moments all increasenon-linearly from x = 0.0 up to the end at x = 3.0. Of course, the higher the inputcoefficient, the larger the values of these moments and the greater the curvature ofcomputed distributions. We notice moreover that the central moment values decreaseall together with this moment order. Finally, it is noticed that only the differences


0.0007

0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

0

E(w

)

Var

(w)

0 1 2 3x (m)

0 1 2 3x (m)

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25α – 0.30

1.6 × 10−7

1.4 × 10−7

1.2 × 10−7

1. × 10−7

8. × 10−8

6. × 10−8

4. × 10−8

2. × 10−8

0

(a) (b)

Figure 4.11 (a) Expected values and (b) variances of the beam deflections

between the expectations computed for various input coefficients of variation may betreated as almost negligible. The remaining moments all show increasing incrementsof the values computed for neighboring quantities of input α(e).

The coefficients of variation (Figure 4.8) and skewness as well as kurtosis(Figure 4.10) behave similarly but in an unusual way. They all start from 0 atthe fixed boundary and are constant and positive throughout the rest of the beamlength. Particular values follow the previous visualization with respect to the inputα(e). The difference between the output coefficients obtained for neighboring inputcoefficients of variation shows regular (coefficients of variation and skewness) aswell as irregular increases (for kurtosis). The final main conclusion is that the outputdistribution is nowhere Gaussian and it does not depend at all on the choice of inputparameters’ particular values. This situation may change when some other structuralparameters are randomized according to the Gaussian PDF.

Example 4.2: Deflection of the linear elastic beamresting on the Winkler foundationThis case study is devoted to the cantilever I-beam made of stainless steel e = 205 GPahaving constant cross-sectional area, inertia moment J = 1.71 × 10− 6 m4 (IPE 100),and resting on the elastic foundation characterized by expected value of compliancecoefficient k = 5 × 107 N

m2 . The deterministic counterpart of this example may befound with the few point of the grid only in [146]. The external distributed load isconstant along this beam, q = 10 kN

m for l = 3 m, while its boundary conditions havebeen discretized as follows:

w0 = 0,w−1 = w1,wn−1 = 2wn−2 − wn−3,wn = 4wn−2 − 4wn−3 + wn−4

(4.24)


0.0004

0.0003

0.0002

0.0001

0

σ(w

)

α(w

)

0 1 2 3x (m)

0 1 2 3x (m)

1.2

1

0.8

0.6

0.4

0.2

0

α – 0.20α – 0.15

α – 0.05 α – 0.10α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.12 (a) Standard deviations and (b) coefficients of variation of the beamdeflections

and where the nodes indexed with numbers n and n − 1 are of course fictitious(and correspond to the free edge). The computational grid is partitioned into n = 30equidistant intervals and n is treated here as the parameter of this solution. Theprobabilistic characteristics are algebraically derived in a computational cycle taking∼4200 s for each input random dispersion on the double core i5 processor in Maple.The results of numerical analyses are contained in Figure 4.11 (expectations andvariances of deflections), Figure 4.12 (their standard deviations and coefficientsof variation), Figure 4.13 (third and fourth central probabilistic moments of thedeflections), as well as Figure 4.14 (skewness and kurtosis of the overall deflection).They have all been computed for five various coefficients of variation of the inputsubsoil compliance α = 0.05, 0.10, 0.15, 0.20, and 0.25, correspondingly (also 0.30 forthe expectations).

As one could expect, the higher the input coefficient of variation of the subsoilcompliance, the higher the value of the probabilistic moments. This is not reallyapparent for variances, third as well as fourth central probabilistic moments forsmaller values of input variation coefficient, however for α ≥ 0.20 these momentshave values a few times larger than before. This result is more accurate than the pre-vious combined-order approximations of higher-order moments, since a significantunderestimation of these moments in comparison with the Monte Carlo simulationswas noticed [103].

A contrast of the expected values shows that the subsoil random compliancesignificantly influences the final random response of this beam since a more than50% increase is noticed for the extremum value of α(k), while the uncertainty ingeotechnical parameters is frequently very large and may be even much largerthan that resulting from randomization of mechanical and physical properties of thestructures. Let us also underline that the location of the maximum deflection is slightlydifferent for various input coefficients of variation, although the distributions of the


1.2 × 10−10

1. × 10−10

8. × 10−11

6. × 10−11

4. × 10−11

2. × 10−11

0

µ 3(w

)

µ 4(w

)

0 1 2 3x (m)

0 1 2 3x (m)

6. × 10−14

5. × 10−14

4. × 10−14

3. × 10−14

2. × 10−14

1. × 10−14

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.13 (a) Third and (b) fourth central probabilistic moments of the beamdeflections

7

6

5

4

3

2

1

0

β(w

)

κ(w

)

50

40

30

20

10

0

0 1 2 3x (m)

0 1 2 3x (m)

α – 0.05α – 0.15

α – 0.10 α – 0.25α – 0.20

α – 0.25α – 0.20

α – 0.05 α – 0.10α – 0.15

(a) (b)

Figure 4.14 (a) Skewness and (b) kurtosis of the beam deflections

deflection functions remain very similar for all input coefficients of variation. Theratio of the variances to the expectations shows that the output coefficient of variationis significantly larger than the adjacent input value, so that the random dispersionis amplified during solution of this problem even with just a few times. Further, thethird probabilistic moments are positive everywhere, so that the deflections exhibitapparent skewness in all tests and the larger part of the probability curve is above theexpectation. At the same time, the fourth central moments have very large values and


significantly positive kurtosis proves that the concentration around the expectationis many times higher than for the Gaussian distribution with the same expectationand standard deviation.

All the probabilistic moments start from value 0 at the clamped edge of this beam,increase further up to some extremum, and start to decrease till the very end at l = 3 m.The higher the order of the probabilistic moment analyzed, the smaller the values ofthese moments, the steeper the descents of these curves, and the wider the zone withzero values at the left edge of the beam. Taking into account all these results, one canconclude that the highest values of the reliability index are undoubtedly expectedat the left stiff edge of this beam, whereas the location of the smallest values (withthe largest probability failure) corresponds to the expectation and variance extrema(almost the same location for x ∼= 2.0 m in this case study). Spatial distributions ofprobabilistic characteristics show clearly that they are necessary for an efficient andcorrect computation of the reliability index. Calculation of this index in a location ofextremum values obtained via deterministic analysis may be improper and can leadto some substantial modeling errors.

4.1.2 Determination of the Critical Moment for the Thin-WalledElastic Structures

Let us consider a differential equation describing the critical moment Mcr undera combination of bending and twisting according to the Vlasov’s theory of thethin-walled beam with constant cross-section on its length:

eJωd4φ (x)

dx4 − GJsd2φ (x)

dx2 − M2cr

eJyφ (x) = 0 (4.25)

where φ(x) denotes the angle of twist, e and G traditionally denote elastic parametersof the beam in the form of Young’s and Kirchhoff moduli, while Jω, Js, Jy are basicgeometrical characteristics of the cross-section. In this context Mcr denotes the criticalvalue of the bending moment applied at the considered thin-walled elastic beamstructure. Taking into account an application of the RFM adjacent to material and/orgeometrical randomness along the structural element considered, one may proposethe following extension of Equation (4.25):

e(j)J(j)

ω

d4φ(j) (x)

dx4 − G(j)J(j)

sd2φ(j) (x)

dx2 −(

M(j)cr

)2

e(α)J(j)

y

φ(j) (x) = 0. (4.26)

We propose the following boundary conditions from the most popular sup-port types to solve this equation for a given case study: (i) φ = 0 and dφ

dx = 0 (support

completely stiff for both twist and deplanation); (ii) φ = 0 and d2φ

dx2 = 0 (supportstiffened against twist with free deplanation); and (iii) support stiffened against thetwist with a certain elastic deplanation, that is, φ = 0 and dφ

dx = ek

d2φ

dx2 . It is clear thatthe same boundary conditions, where the randomness is omitted for simplicity only,


concern the entire series of solutions φ(j) where j = 1, . . . , N indexes all deterministicsolutions necessary to recover response polynomial. The FDM discretization of theinitial Equation (4.25) leads to the following difference statement:

eJωφ(xi−2

) − (4eJω + GJs (�x)2)φ

(xi−1

) + (6eJω + 2GJs (�x)2 − λ

)φ(xi)

− (4eJω + GJs (�x)2)φ

(xi+1

) + eJωφ(xi+2

) = 0 (4.27)

where

λi = M2cr,i

eJy(�x)4 (4.28)

and λi denote the additional eigenvalues for Equation (4.27). Additionally, theresponse function determination needs a series of solutions obtained from therecursive difference formula

e(j)J(j)

ω φ(j)(xi−2

) −(

4e(j)J(j)

ω + G(j)J(j)

s

(�x(j)

)2)

φ(j)(xi−1

)

+(

6e(j)J(j)

ω + 2G(j)J(j)

s

(�x(j)

)2 − λ(j))

φ(j)(xi)

−(

4e(j)J(j)

ω + G(j)J(j)

s

(�x(j)

)2)

φ(j)(xi+1

) + e(j)J(j)

ω φ(j)(xi+2

) = 0 (4.29)

where, similarly to Equation (4.28), one shows

λ(j)i = M(j)

cr,i

e(j)J(j)

y

(�x(j)

)4. (4.30)

Of course, dependence of the basic dimension of a grid �x on the actual solutionindexed here with j is necessary if and only if some global geometrical parametersare randomized – as length for 1D models; otherwise, it remains constant throughoutall numerical tests.

Further, the finite difference notation for various support types leads to thefollowing conditions:

1. φ = 0 anddφ

dx= 0: φ(xi) = 0 and φ(xi + 1) = − φ(xi − 1),

2. φ = 0 andd2φ

dx2 = 0: φ(xi) = 0 and φ(xi + 1) = φ(xi − 1), and

3. φ = 0 anddφ

dx= e

kd2φ

dx2 : φ(xi) = 0 and φ(xi+1

) = k�x+2ek�x−2eφ

(xi−1

).

Solving for the eigenvalues λi from Equation (4.27), we find consecutively thecritical moments after additional inversion of Equation (4.28).


Example 4.3: Critical moments for the random thin-walled elasticbeamA linear elastic thin-walled steel straight beam is analyzed with Young’smodulus expectation E[e] = 210.0 GPa, Poisson ratio ν = 0.30, expectation ofthe grid distribution E[�x] = 0.20 corresponding to its equidistant subdivisionwith n = 20 [146]. Geometrical cross-sectional parameters are all deterministicand taken as Jω = 0.000265 × 10− 6m6, Js = 0.0172 × 10− 6m4, Jy = 0.122 × 10− 6m4.External boundary conditions imposed on this structure show, after Figure 4.15,that both edges are fully restrained. The RFM version of the perturbationmethod is used here with nine trial points in case of randomized Young’smodulus equal to e = [190, 195, 200, 205, 210, 215, 220, 225, 230] GPa, while�x = [0.18, 0.185, 0.19, 0.195, 0.20, 0.205, 0.21, 0.215, 0.22] m. A quadratic approxima-tion function is used in the non-weighted least-squares algorithm here. Figure 4.15shows for validation a comparison of the first three eigenmodes for the Mapleimplementation of the FDM (left-hand graph) with an analytical solution – both foran initial deterministic situation and, of course, practically no difference is noticedhere. Further, we compare the expected values, coefficients of variation, skewness,and kurtosis for two independently randomized systems – with respect to Young’smodulus (left-hand graphs) and overall length of the structural element (right-handgraphs). This is done to validate which parameter has more influence in terms ofanalogous input uncertainty level. The computational effort is very similar for bothinput parameters as they appear explicitly in the governing difference equation; this

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

φ(b)

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

φ(b)

0 1 2 3 4l

0 1 2 3 4l

first eigenfunction second eigenfunctionthird eigenfunction

first eigenfunction second eigenfunctionthird eigenfunction

(a) (b)

Figure 4.15 (a) First three eigenfunctions and (b) their polynomial interpolationscorresponding to the critical moments


is not the case for the SFEM, where geometrical parameter randomization demandsusually remesh in the RFM approach, while the Young’s modulus of course does not.

The very important general result here is that most computed probabilistic coeffi-cients are totally insensitive to the perturbation order, so that second-order analysisis quite satisfactory. The only exception is kurtosis, where sixth order guarantees thesame results as higher-order analyses. Moreover, skewness and kurtosis very close to0 for randomized elastic modulus enable us to conclude a Gaussian distribution of theresulting critical moment and this is definitely not the case for geometrical uncertaintyin this problem. Generally, all probabilistic characteristics given in Figures 4.16–4.19increase together with an increase of the input random dispersion given by the coef-ficient α (except for lower orders of kurtosis for the Young’s modulus). It is chara-cteristic that the expectation and kurtosis grow non-linearly according to convexcurves, whereas the coefficients of variation and skewness increase linearly by only.Comparing carefully the expectations given in Figure 4.16, we see that randomizationof the Young’s modulus returns almost constant expectations of Mcr, while the meanvalues obtained for the random length change significantly for α ∈ [0.0, 0.20].

The output coefficients of variation show that uncertain length has much moreinfluence on the critical moment than is obtained for Young’s modulus, whichcoincides well with an engineering intuition. The resulting coefficients are about 50%larger than the input ones for the second random variable, while the uncertainty levelis totally preserved during randomization of the elastic modulus. All the remainingcharacteristics are positive and have values measured in tenths for random length,which excludes a Gaussian PDF as the critical moment in this case. The very interestingresult is obtained in Fig. 4.19a, where kurtosis of the critical moment differs from 0at α(e) = 0. It reflects a numerical error of the applied approximating method only,

1.56

1.55

1.54

1.53

1.52

E(M

1, c

r)

E(M

1, c

r)

0.05 0.10 0.15 0.200α

0.05 0.10 0.15 0.200α

1.514889600

1.514889595

1.514889590

1.514889585

1.514889580

1.514889575

2nd-order analysis


4th-order analysis

8th-order analysis

2nd-order analysis


4th-order analysis

8th-order analysis

(a) (b)

Figure 4.16 (a) Expected values of the critical moment for random Young’s modulusand (b) length of the steel element


0.20

0.15

0.10

0.05

0

α(M

1, c

r)

α(M

1, c

r)

0.05 0.10 0.15 0.200α

0.05 0.10 0.15 0.200α

0.3

0.2

0.1

0


4th-order analysis

8th-order analysis2nd-order analysis6th-order analysis10th-order analysis

4th-order analysis

8th-order analysis

(a) (b)

Figure 4.17 Coefficients of variation of the critical moment for (a) random Young’smodulus and (b) length of the steel element

β(M

1, c

r)

β(M

1, c

r)

8. × 10−7

7. × 10−7

6. × 10−7

5. × 10−7

4. × 10−7

3. × 10−7

2. × 10−7

1. × 10−7

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.05 0.10 0.15 0.200α

0.05 0.10 0.15 0.200α





(a) (b)

Figure 4.18 Skewness of the critical moment for (a) random Young’s modulusand (b) length of the steel element


1.5896 × 10−9

1.5895 × 10−9

1.5894 × 10−9

1.5893 × 10−9

1.5892 × 10−9

1.5891 × 10−9

1.589 × 10−9

κ(M

1, c

r)

κ(M

1, c

r)

0.05 0.10 0.15 0.200α

0.05 0.10 0.15 0.200α

0.5

0.4

0.3

0.2

0.1

0





(a) (b)

Figure 4.19 Kurtosis of the critical moment for (a) random Young’s modulusand (b) length of the steel element

because perturbation based formulas return 0 in this case (cf. Equation (3.68), forinstance). It needs to be mentioned also that further computational analysis in thatarea using the same apparatus should be focused on stochasticity within the cross-sectional dimensions, which influences Jω, Js, Jy at the same time and also corrosionsimulation via the time series analysis proposed and carried out in Chapter 1.

4.1.3 Introduction to Elastodynamics with Finite Differences

The solution to the eigenproblems with random parameters for the unidirectionalproblem will be displayed for the example of the Euler–Bernoulli beam. As is known,the equation of motion for the forced lateral vibration of a non-uniform beam has thefollowing form:

∂2

∂x2

[eJ(x)

∂2w∂x2 (x, t)

]+ ρA(x)

∂2w∂t2 (x, t) = f (x, t), (4.31)

where f (x,t) is the external force per unit length of a beam, ρ is the mass density of thebeam, A(x) is the cross-sectional area of the beam. Assuming a constant cross-sectionof the beam along its length, that is, eJ(x) = eJ = const. and A(x) = A = const., Equation(4.31) can therefore be rewritten as

eJ∂4w∂x4 (x, t) + ρA

∂2w∂t2 (x, t) = f (x, t) (4.32)

while, for free vibration, it is obtained that

c2 ∂4w∂x4 (x, t) + ∂2w

∂t2 (x, t) = 0 (4.33)


where

c =√

eJρA

. (4.34)

After traditional separation of space and time variables one gets

w(x, t) = W(x)T(t) (4.35)

for the free vibrations, two equations are obtained:

∂4W(x)∂x4 − β4W(x) = 0,

∂2T(t)∂t4 + ω2T(t) = 0,

(4.36)

where

β4 = ω2

c2 = ρAω2

eJ. (4.37)

The characteristic function of the beam can be obtained as

W(x) = C1 exp (βx) + C2 exp (−βx) + C3 exp (iβx) + C4 exp (−iβx) , (4.38)

while ω being the natural frequency of vibration can be expressed as

ω = β2

√EJρA

. (4.39)

Applying the stochastic perturbation methodology, first of Equations (4.36) can berewritten in the following form:

• zeroth-order equation∂4W0 (x)

∂x4 − (β0)4

W0 (x) = 0 (4.40)


∂5W0 (x)

∂b∂x4 − ∂(β4)

∂bW0 (x) − (

β0)4 ∂W (x)

∂b= 0 (4.41)


∂6W0 (x)

∂b2∂x4 −{

∂2(β4)

∂b2 W0 (x) + 2∂(β4)

∂b∂W (x)

∂b+ (

β0)4 ∂2W (x)

∂b2

}= 0 (4.42)


where the probabilistic derivatives of β4 may be calculated analytically. Finally, werewrite the recursive nth-order formula as

∂n+4W0 (x)

∂bn∂x4 −n∑

k=0

(nk

)∂n−k

(β4)

∂bn−k

∂kW (x)

∂bk= 0. (4.43)

In case of β = β(b) we can calculate the partial derivatives of the deflection functionW(x) from Equation (4.38) as

∂nW (x)

∂bn = C1∂n

∂bn

{exp (βx)

} + C2∂n

∂bn

{exp (−βx)

} + C3∂n

∂bn

{exp (iβx)

}+ C4

∂n

∂bn

{exp (−iβx)

}. (4.44)

Example 4.4: Eigenvibrations of the elastic beam with randomparametersA numerical illustration will be given here for the fixed/pinned beam, where theprobabilistic moments of the free vibration frequencies are found. The analyticalsolution for natural frequencies and modes for the fixed/pinned beam can beobtained as

ω2n = (

βnl)4(

eJρAl4

), n = 1, 2, . . . (4.45)

If the Young’s modulus of a beam is taken as a random variable, then the firsttwo probabilistic moments of the natural frequencies square can be calculated usingfundamental properties of random variables as

E[ω2

n] = Jβ4

n

ρAE[e], n = 1, 2, . . . (4.46)

and

Var(ω2

n) = J2β8

n

ρ2A2 Var(e), n = 1, 2, . . . (4.47)

Starting from these formulas, the expected values and variances of natural fre-quencies can be derived as

E [ω] =√

E2[ω] =√

E[ω2] − Var(ω) (4.48)

andVar

(ω2) = 2Var (ω)

(Var (ω) + 2E2 [ω]

). (4.49)

Using a simplified stochastic second-order perturbation approach, the sameprobabilistic characteristics can be calculated as

E[ω2

n] = (

ω2n)0 + 1

2∂

2(ω2

n)

∂b2 Var (e) = Jβ4n

ρAE [e] , n = 1, 2, . . . (4.50)


as well as

Var(ω2

n) =

[∂(ω2

n)

∂b

]2

Var (e) = J2β8n

ρ2A2 Var (e) , n = 1, 2, . . . (4.51)

The simplicity of Equations (4.48) and (4.49) follows a linear character of thetransform in-between structural output and input, but similar equations may beobtained for higher order polynomials also. These exact formulas are very important,because the Monte-Carlo simulation frequently employed in many comparativestudies is only statistically convergent to the relevant moments being estimated.

As can be observed, the analytical solutions for probabilistic moments calculatedfrom the definition and the second-order perturbation approach are exactly thesame and that is why there is no need to apply the higher-order perturbationapproach, which generally holds true for all outputs being linear functions ofinput random variables. For other type of relations between output frequencies andsystem random parameters, the expected values obtained from the second-orderperturbation approach are higher than these derived explicitly from the definition.The simplicity of Equations (4.48) and (4.49) follows a linear character of the transformin-between structural output and the input, but similar equations may be obtainedfor higher order polynomials also. These exact formulas are very important, becausethe Monte Carlo simulation frequently employed in many comparative studies isonly statistically convergent to the relevant moments being estimated.

Example 4.5: Longitudinal eigenvibrations for the elastic beamThis computational example deals with determination of the eigenfrequency squaresfor longitudinal vibrations of the linear elastic prismatic bar. It is carried outfor the structural element having cross-sectional area A = 113.0 × 10− 4m2 (profileHEA 340), mass density ρ = 88.3 kN

m3 ; it is partitioned into 100 equidistant intervals.This bar is simply supported at both ends and we randomized independentlythe Young’s modulus with mean value E[e] = 210 GPa and element length havingexpectation E[l] = 10.0 m using the RFM approach with 11 trials computed for thefollowing sets of discrete values: ei = [205, 206, 207, 208, 209, 210, 211, 212, 213,214, 215] GPa and li = [9.5, 9.6, 9.7, 9.8, 9.9, 10, 10.1, 10.2, 10.3, 10.4, 10.5] m; thesecond-order polynomial is used to recover the global response functions for theeigenfrequency squares. Probabilistic moments and coefficients are collected forthe first eight eigenfrequency squares – the left-hand graphs correspond to Young’smodulus uncertainty, while randomization of the element length results in theright-hand graphs of Figures 4.20–4.24 – we compare here expectations (Figure 4.20),standard deviations (Figure 4.21), coefficients of variation (Figure 4.22), skewness(Figure 4.23), as well as kurtosis (Figure 4.24).

The general observation is that, as in Example 4.3, all resulting eigenfrequencysquares are distributed according to the Gaussian PDF for randomized elasticmodulus, which follows negligibly small skewness and kurtosis; this is not thecase for random length, where these characteristics are positive, measured in units,and significantly increasing together with an increase of the input coefficient α.Further, this increase is noticed for almost all resulting characteristics except some


1.6 × 108

1.4 × 108

1.2 × 108

1. × 108

8. × 107

6. × 107

4. × 107

2. × 107

1.6 × 108

1.8 × 108

2. × 108

1.4 × 108

1.2 × 108

1. × 108

8. × 107

6. × 107

4. × 107

2. × 107

E(ω

)

E(ω

)

0.05 0.10 0.15 0.250.200α

0.05 0.10 0.15 0.250.200α

ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6 ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6

(a) (b)

Figure 4.20 Expected values of the eigenfrequencies for (a) random Young’s modulusand (b) the bar length

4. × 107

3. × 107

2. × 107

1. × 107

0

σ(ω

)

σ(ω

)

0.05 0.10 0.15 0.250.200α

0.05 0.10 0.15 0.250.200α

ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6 ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6

1. × 108

8. × 107

6. × 107

4. × 107

2. × 107

0

(a) (b)

Figure 4.21 Standard deviations of the eigenfrequencies for (a) random Young’smodulus and (b) the bar length

eigenfrequency skewness negative distributions (Figure 4.22(a)) as well as a completeinsensitivity of kurtosis upon input coefficient of variation. Small values of these twoparameters result from the RFM approximation error and may be simply postponed.

Once more the output randomness level is exactly the same as the input duringrandomization of the Young’s modulus and amplified twice while the element lengthis treated as uncertain; both coefficients are linear w.r.t. the input one, however, the


0.25

0.20

0.15

0.10

0.05

0

α(ω

)

α(ω

)

0.5

0.4

0.3

0.2

0.1

00.05 0.10 0.15 0.250.200

α0.05 0.10 0.15 0.250.200

αω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6 ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6

(a) (b)

Figure 4.22 Coefficients of variation of the eigenfrequencies for (a) random Young’smodulus and (b) the bar length

β(ω

)

β(ω

)

4. × 10−7

−4. × 10−7

−6. × 10−7

−8. × 10−7

−1. × 10−6

−1.2 × 10−6

−1.4 × 10−6

2. × 10−7

−2. × 10−7

02

1.5

1

0.5

0.05 0.10 0.15 0.250.200α

0.05 0.10 0.15 0.250.200α

ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6 ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6

(a) (b)

Figure 4.23 Skewness of the eigenfrequencies for (a) random Young’s modulusand (b) the bar length

expectations behavior is much more complex. The mean values computed for randomelastic modulus are constant, where of course the higher the eigenfrequency square,the larger the expectation. Analogous expected values from the right-hand graphincrease moderately with increasing input α, where the higher the eigenvalue, thelarger the increase; this interrelation is convex everywhere without any singularitiesand discontinuities. The last eigenfrequency square changes by about 25% fromminimum α = 0 to maximum, so that these variations cannot be neglected.


1. × 10−9

−1. × 10−9

5. × 10−10

−5. × 10−10

0κ(ω

)

κ(ω

)

5

4

3

2

1

00.05 0.10 0.15 0.250.200

α0.05 0.10 0.15 0.250.200

αω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6 ω1 ω2

ω7 ω8

ω3 ω4 ω5 ω6

(a) (b)

Figure 4.24 Kurtosis of the eigenfrequencies for (a) random Young’s modulusand (b) the bar length

Finally, it is very characteristic that the resulting skewness and kurtosis are allthe same for various eigenfrequency squares while randomizing the element length,which means that these quantities all have the same probability distribution. Thisdistribution, according to our method, is more and more distant from the GaussianPDF when input random dispersion increases.

4.1.4 Advection–Diffusion Equation

We consider the following advection–diffusion partial differential equation appea-ring frequently in physics and engineering [47], also in the context of temperaturefluctuations within some unidirectional domain. It has the following form:

∂T∂t

+ u∂T∂x

= α∂2T∂x2 , (4.52)

where u and α represent certain physical parameters of the system, considered hereas state independent (α = k/cρ, where k is the heat conductivity coefficient, c standsfor the heat capacity, and ρ denotes mass density). This specific equation has beenchosen as a combination of the parabolic and hyperbolic partial differential equationand the forward-time and central-space (FTCS) method is engaged to provide FDMdiscretization of this equation. It holds for the ith point of the grid that

Tn+1[i] − Tn

[i]

�t= α

Tn[i+1] − 2Tn

[i] + Tn[i−1]

�2x

− uTn

[i+1] − Tn[i−1]

2�x, (4.53)

and with α = 0 we obtain the classical difference notation of the parabolic equationfor unidirectional heat transfer. Upper indices denote here the given discrete time


moment marked with n, while n + 1 is the next one to be determined. Usually, it isrewritten as

Tn+1[i] = Tn

[i] + α�t

�2x

(Tn

[i+1] − 2Tn[i] + Tn

[i−1]

)− u�t

2�x

(Tn

[i+1] − Tn[i−1]

). (4.54)

The perturbation-based stochastic counterpart in the DDM implementation con-sists of an interrelation between partial derivatives computed at neighboring timemoments and has the following general form convenient for the nth order:

∂mTn+1[i]

∂bm =∂mTn

[i]

∂bm + �t

�2x

m∑k=0

(mk

)∂kα

∂bk

(∂m−kTn

[i+1]

∂bm−k− 2

∂m−kTn[i]

∂bm−k+

∂m−kTn[i−1]

∂bm−k

)

− �t

2�x

m∑k=0

(mk

)∂ku∂bk

(∂m−kTn

[i+1]

∂bm−k−

∂m−kTn[i−1]

∂bm−k

). (4.55)

So, having partial derivatives for the given time moment we are able to computethem all after the next step and compose them into the final equations for theprobabilistic moments. Since we apply more frequently the RFM version of theSFDM, more useful would be the following statement:

Tn+1(k)[i] = Tn(k)

[i] + α(k)�t

�2x

(Tn(k)

[i+1] − 2Tn(k)[i] + Tn(k)

[i−1]

)− u(k)�t

2�x

(Tn(k)

[i+1] − Tn(k)[i−1]

), (4.56)

where k = 1, . . . , M indexes deterministic tests with varying input physical parameterto be treated as random. We never treat �t as random here because time parameterin all scientific problems is always deterministic.

Example 4.6: Transient heat transfer in homogeneous rodThis computational illustration deals with the structure having length l = 1.0, subjectto the following boundary conditions: T(0, t) = 0 and T(l, t) = 100, while the initialconditions are adopted as T (x, 0) = 100x

L for x ∈ [0, l]. The constants appearing inthe differential equation are taken as u = 0.10, k = 43, c = 490, and d = 7850. Thecoefficient of heat conduction is chosen here as the input Gaussian random variableand numerical analysis based on the RFM 11-trials cycle is carried out twice – forα(k) = 0.10 (left-hand graphs of Figures 4.25–4.29) and for α(k) = 0.30 (right-handgraphs of Figures 4.25–4.29). We present basic probabilistic characteristics of thetemperature evolution in this structure as functions of the spatial coordinate forthe few time moments τ = [0.0, 0.5, 2.5, 5.0, 10.0, 50.0] s. Therefore, it is impossible toinclude here continuous variations with respect to α(k) as in the previous tests and,further, to present the few surfaces corresponding to various perturbation analysisorder to maintain brevity of presentation. Time-dependent local response functions(at each point of the grid) have been recovered using second-order polynomialsinserted into the unweighted least-squares algorithm. We derive expectations fortemperature histories using tenth-order perturbation, the variances, third and fourth


100

80

60

40

20

0

E(T

)100

80

60

40

20

0

E(T

)

0 0.2 0.4 0.6 0.8 1x (cm)

0 0.2 0.4 0.6 0.8 1x (cm)

5.0 s0.5 s

2.5 s50.0 s

10.0 s0.0 s

5.0 s0.5 s

2.5 s50.0 s

10.0 s0.0 s

(a) (b)

Figure 4.25 Temperature expected values for the advection-diffusion problemfor (a) α = 0.1 and (b) α = 0.30

Var

(T)

250

200

150

100

50

0

Var

(T)

2500

2000

1500

1000

500

00 0.2 0.4 0.6 0.8 1

x (cm)0 0.2 0.4 0.6 0.8 1

x (cm)

50.0 s2.5 s

10.0 s0.5 s

5.0 s0.0 s

50.0 s2.5 s

10.0 s0.5 s

5.0 s0.0 s

(a) (b)

Figure 4.26 Temperature variances for the advection–diffusion problemfor (a) α = 0.1 and (b) α = 0.30

central probabilistic moments according to the sixth-order approximations. As onemay recognize from Figure 4.25, the boundary conditions applied at the left edgeare matched in both cases very well by the expected value, but the temperatureexpectations at the right edge are slightly below the value T(l,t) = 100. Therefore,we need to remember that the boundary condition is imposed on T0(x, t), only,which may have different values from E[T(x, t)], especially when the resulting


0.24

0.22

0.20

0.18

0.16

0.14

0.12

0.10

α(T

)

α(T

)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x (cm)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x (cm)

0.6

0.5

0.4

0.3

0.5 s2.5 s

50.0 s10.0 s

5.0 s0.0 s

0.5 s5.0 s

10.0 s2.5 s

50.0 s0.0 s

(a) (b)

Figure 4.27 Temperature coefficients of variation for the advection–diffusion problemfor (a) α = 0.1 and (b) α = 0.30

0.41.2

1

0.8

0.6

0.4

0.2

0

0.3

0.2

0.1

0

β(T

)

β(T

)

0 0.2 0.4 0.6 0.8 1x (cm)

0 0.2 0.4 0.6 0.8 1x (cm)

0.0 s5.0 s

0.5 s10.0 s

2.5 s50.0 s

0.0 s5.0 s

0.5 s10.0 s

2.5 s50.0 s

(a) (b)

Figure 4.28 Temperature skewness for the advection–diffusion problem for (a) α = 0.1and (b) α = 0.30

distribution is not Gaussian; it can be verified directly in Figures 4.28 and 4.29 forx = l. Comparison of both expectation spatial distributions shows moreover thatfor α(k) = 0.30 the temperature mean values are apparently closer to the right edgeboundary condition. In both cases we notice that the numerical method is stable as


0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

κ(T

)

κ(T

)

0 0.2 0.4 0.6 0.8 1x (cm)

0 0.2 0.4 0.6 0.8 1x (cm)

1.2

1

0.8

0.6

0.4

0.2

0

0.5 s10.0 s

2.5 s50.0 s

5.0 s0.0 s

0.5 s10.0 s

2.5 s50.0 s

5.0 s0.0 s

(a) (b)

Figure 4.29 Temperature kurtosis for the advection–diffusion problem for (a) α = 0.1and (b) α = 0.30

the resulting temperature expectations have no singularities in space and time andthe differences between the distributions presented decrease together with a timeincrease. Generally, these expected values decrease in time unlike the variances,whose decrease for x ∈ [0.6, 1.0] is a few times larger than their increase in time forx ∈ [0.0, 0.6]. The vertical scales in Figure 4.26 demonstrate that increasing three timesan input coefficient of variation, we enlarge about 10 times the temperature variance.

The coefficients of variation given in Figure 4.27 prove that the variances at x = 0do not equal 0, although they are very small in comparison to the variances at theother locations. Then, in a close neighborhood of that point, we notice sudden decayof the variance coefficient having a global maximum for x = 0. After the minimum inthe interval x ∈ [0.20, 0.30] all time distributions stabilize, appearing almost constantfor larger x, especially at the last moments of this simulation. It seems naturalthat an uncertainty starts from 0 at the initial point, where all the conditions aredeterministic and systematically increase in each location to reach maximum randomdispersion in the last moment of the advection–diffusion process. The coefficientsof skewness as well as kurtosis are positive everywhere and increase together withboth x and t. Skewness, analogously to the coefficient of variation, increases almostthree times while changing α(k) from 0.10 to 0.30, but the kurtosis variation isdecisively larger and almost nine times higher. The final probability distributions ofT(x,t) after Figures 4.28 and 4.29 are not Gaussian and the larger α(k), the greaterthe distance to the combination of these parameters typical to this particular PDFtype. It would be interesting indeed to extend this analysis toward the problem withtemperature-dependent and random physical properties of the system as well as touse this methodology to provide stochastic reliability analysis in terms of criticaltemperatures for many of the engineering systems.


4.2 Analysis of Boundary Value Problems on 2D Grids

4.2.1 Poisson Equation

The Poisson equation gives an illustration of the solution of an elliptic partialdifferential equation with random parameters on the 2D regular grid. We considerthe following problem:

∂2u(x1, x2

)∂x2

1

+ ∂2u(x1, x2

)∂x2

2

= −q(x1, x2

), (4.57)

which may represent a steady-state temperature distribution in the isotropic body,steady-state invisible fluid in the bounded domains (with sources), electrostatic fieldpotential within a closed domain with charges, Newtonian gravity field potential,torsion of a prismatic elastic beam, or static deformation of a membrane underpressure and tension. The rectangular net is provided for this purpose with twodifferent lower indices i and j numbering the given point in the directions x1 andx2, independently. The basic smallest cell of this grid is uniquely defined by thedistances �1 and �2 with respect to the given coordinates’ axes. Then, we discretizeall necessary partial derivatives in the above equation using central finite differencesand values of the function u(x1, x2) to be determined in the closest neighborhood ofthe point defined uniquely by the indices [i,j]. It holds that

∂u[i,j]∂x1

=u[i+1,j] − u[i−1,j]

2�1+ O

(�2

1)

, (4.58)

∂u[i,j]∂x2

=u[i,j+1] − u[i,j−1]

2�2+ O

(�2

2)

(4.59)

and, analogously, for the second-order partial derivatives

∂2u[i,j]∂x2

1

=u[i+1,j] − 2u[i,j] + u[i−1,j]

�21

+ O(�3

1)

, (4.60)

∂2u[i,j]∂x2

2

=u[i,j+1] − 2u[i,j] + u[i,j−1]

�22

+ O(�3

2)

(4.61)

with mixed partial derivative of second order given as

∂2u[i,j]∂x1∂x2

=u[i+1,j+1] + u[i−1,j−1] − u[i−1,j+1] − u[i+1,j−1]

4�1�2+ O

(�3

1)

. (4.62)


1

1

1 1−4

h h

hh

Figure 4.30 Finite difference method Laplace operator in a graphical form

An application of these formulas in the Poisson equation yields (postponinghigher-order rests)

u[i+1,j] − 2u[i,j] + u[i−1,j]�2

1

+u[i,j+1] − 2u[i,j] + u[i,j−1]

�22

+ O(�3

1, �32) = −q[i,j] (4.63)

which, with net equidistant in both directions, that is, �1 = �2 = a and by neglectinghigher-order rests returns

4u[i,j] −{

u[i+1,j] + u[i,j+1] + u[i−1,j] + u[i,j−1]}

= q[i,j]a2. (4.64)

A graphical representation of this difference formula is given in Figure 4.30.When the response function recovery is involved in a solution of this equation

in the presence of some uncertainty sources, we solve for the deterministic seriesindexed with upper index m and we have

4u(m)

[i,j] −{

u(m)

[i+1,j] + u(m)

[i,j+1] + u(m)

[i−1,j] + u(m)

i,j−1

}= q(m)

[i,j](a(m)

)2. (4.65)

Much faster is the DDM scheme, where the recursive formula for the nth-orderperturbation-based solution equals

4∂nu[i,j]

∂bn −{

∂nu[i+1,j]∂bn +

∂nu[i,j+1]∂bn +

∂nu[i−1,j]∂bn +

∂nu[i,j−1]∂bn

}

=n∑

k=0

(nk

)∂n−kq[i,j]∂bn−k

∂k(a2)

∂bk. (4.66)

Example 4.7: Solution to the Poisson equation on a rectangular domainThis computational illustration is devoted to the solution of the Poisson equation ontwo different grids − 10 × 10 and 20 × 20 – having randomized basic grid dimensiona = �x = �y. Effectively, considering fictitious nodes in the grids necessary to define


given boundary conditions, these are 12 × 12 and 22 × 22 uniform quadratic nets.The global membrane dimensions equal 10 m × 10 m; it is simply supported alongits external edges and the uniform unitary pressure is applied on the plate. Sincedeterministic values and expectations for the coarser grid appear to be larger, allprobabilistic moments and coefficients in Figures 4.31–4.34 for the needs of reliabilityanalysis are restricted to this finer grid. The RFM analysis is based on nine numericalFDM experiments in Maple provided that a = [0.42, 0.44, 0.46, 0.48, 0.50, 0.52, 0.54,0.56, 0.58] m with quadratic interrelation between the resulting midspan deflectionand the parameter a.

All the resulting moments and coefficients presented in Figures 4.32–4.34 are givenas functions of the input coefficient of variation α(a) = [0.05, 0.10, 0.15, 0.20, 0.25]and for the cross-section parallel to the given edges provided at the midpoint. Thisarrangement is proposed instead of the surface plot in Figure 4.31 to keep betterresolution and for brevity of the final spatial distributions to the varying α(a).A distribution of the expected values is in its shape very similar to the deterministicorigin-preserving zero-boundary conditions at both ends and exhibiting a maximumat the midpoint. This expectation is moderately sensitive to the input coefficientof variation, where of course the larger α(a), the greater the maximum deflectionexpectation. Spatial distributions of variances and standard deviations are similarin the context of shape and extremum locations to expectations, however now theinfluence of the input α(a) is decisive. The variances grow faster than the inputcoefficient of variation, while the standard deviations increase proportionallyto this coefficient, which perfectly agrees with theoretical considerations. Thisnon-proportional increase is much more apparent for higher central probabilistic

5

43

21

02

46

810 8

76

54

32

10 0

24

68

10 108

64

20

87654321

(a) (b)

Figure 4.31 Solutions with two different grids – (a) 10 × 10 and (b) 20 × 20


6 10

8

6

4

2

0

5

4

3

2

1

0

E(u

max

)

Var

(um

ax)

0 2 4 6 8 10x

0 2 4 6 8 10x

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.32 (a) Expected values and (b) variances for coarser grid 20 × 20

3

2

1

0

σ(u m

ax)

α(u m

ax)

0 2 4 6 8 10x

0 2 4 6 8 10x

0.4

0.3

0.2

0.1

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.33 (a) Standard deviations and (b) coefficients of variationfor coarser grid 20 × 20

moments (cf. Figure 4.34) and we can conclude that the higher the moment, thefaster the increase with respect to the input coefficient of variation (according to thehigher exponent in the additional analytical expression). The output dispersion level(right-hand graph of Figure 4.33) is constant along the given cross-section except forthe edges, where according to the deterministic zero-boundary conditions there is norandom fluctuation. After previous experiments with randomized grid size we notice


30

20

10

0

µ 3(u

max

)

µ 4(u

max

)

0 2 4 6 8 10x

0 2 4 6 8 10x

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

400

300

200

100

0

(a) (b)

Figure 4.34 (a) Third and (b) fourth central probabilistic momentsfor coarser grid 20 × 20

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

β(u m

ax)

κ(u m

ax)

0 2 4 6 8 10x

0 2 4 6 8 10x

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.35 (a) Skewness and (b) kurtosis for coarser grid 20 × 20

that the ratio of output to input coefficients of variation equals almost 2; this ratioholds true for all α(a) and means that the perturbation-based expansions cannot bereduced to lower orders only and must be given using at least tenth-order expressions;the second-order technique in particular could be quite inefficient here. Coefficientsof skewness and kurtosis have very similar spatial variations to the coefficient of


variation α(umax), but while skewness increases are proportional to changes of α(a),kurtosis increases much faster. These coefficient values show clearly that the maxi-mum deflection is not distributed according to the Gaussian distribution for any α(a).Furthermore, the larger the input coefficient of variation, the greater the concentrationaround the mean value and the larger right asymmetry of the resulting PDF.

4.2.2 Deflection of Thin Elastic Plates in Cartesian Coordinates

Let us consider further partial differential equation in Cartesian coordinates descri-bing a deflection w(x1, x2) of the thin linear elastic and homogeneous plate as

∂4w(x1, x2)∂x4

1

+ 2∂4w(x1, x2)

∂x21∂x2

2

+ ∂4w(x1, x2)∂x4

2

= q(x1, x2

)D

, (4.67)

where the plate stiffness D is traditionally defined as

D = eh3

12(1 − ν2

) , (4.68)

and e, h, and ν stand for the Young’s modulus, thickness, and Poisson ratio ofthis structure. Having determined this deflection we derive the bending moments,twisting moment, as well as the transverse forces in the plate with use of the followingformulas:

M1 = −D

(∂2w(x1, x2)

∂x21

+ ν∂2w(x1, x2)

∂x22

), (4.69)

M2 = −D

(∂2w(x1, x2)

∂x22

+ ν∂2w(x1, x2)

∂x21

), (4.70)

Ms = − (1 − ν) D∂2w(x1, x2)

∂x1∂x2, (4.71)

T1 = ∂M1

∂x1+ ∂Ms

∂x2, T2 = ∂M2

∂x2+ ∂Ms

∂x1. (4.72)

As can be expected, this procedure also needs the RFM extension for both theinitial differential equation and further determination of the resulting internal forcesindexed with m in further analysis. It holds that

∂4w(m)(x1, x2)∂x4

1

+ 2∂4w(m)(x1, x2)

∂x21∂x2

2

+ ∂4w(m)(x1, x2)∂x4

2

= q(m)(x1, x2

)D(m)

, (4.73)

as we also look for the series of solutions with varying design input randomparameter. Hence

D(m) = e(m)(h(m)

)3

12(

1 − (ν(m)

)2) (4.74)


and, finally:

M(m)

1 = −D(m)

(∂2w(m)(x1, x2)

∂x21

+ ν(m) ∂2w(m)(x1, x2)

∂x22

),

M(m)

2 = −D(m)

(∂2w(m)(x1, x2)

∂x22

+ ν(m) ∂2w(m)(x1, x2)

∂x21

), (4.75)

M(m)s = − (

1 − ν(α))

D(m) ∂2w(m)(x1, x2)

∂x1∂x2, (4.76)

T(m)

1 = ∂M(m)

1

∂x1+ ∂M(m)

s

∂x2, T(m)

2 = ∂M(m)

2

∂x2+ ∂M(m)

s

∂x1. (4.77)

We consider three types of boundary conditions: (i) completely free boundary,(ii) simply supported edge, and (iii) fully restrained boundary. So, considering thestraight boundary parallel to the given x2 axis, one may introduce a zero internalforces condition in case (i) as

M(m)

1 = −D(m)

(∂2w(m)(x1, x2)

∂x21

+ ν(m) ∂2w(m)(x1, x2)

∂x22

)= 0, (4.78)

T(m)

1 + ∂M(m)s

∂x1= ∂M(m)

1

∂x1+ ∂M(m)

s

∂x2+ ∂M(m)

s

∂x1= 0. (4.79)

The boundary conditions relevant to case (ii) are given as

w(m) = 0, M(m)

1 = −D(m)

(∂2w(m)(x1, x2)

∂x21

+ ν(m) ∂2w(m)(x1, x2)

∂x22

)= 0, (4.80)

whereas case (iii) uses

w(m) = 0,∂w(m)

∂x1≡ ϕ1 = 0. (4.81)

For the needs of the FDM we employ a rectangular grid with basic axial distances�x1, �x2 looking for the set of discrete deflections denoted for a point indexed with[i,j] as w[i,j]. So, the following holds for the distributed load concentrated in the point[i,j] denoted as q[i,j]:

w[i−2,j] − 4w[i−1,j] + 6w[i,j] − 4w[i+1,j] + w[i+2,j]

�x41

− 4w[i,j−1] + 4w[i−1,j] − 8w[i,j] + 4w[i+1,j] + 4w[i,j+1]

�x21�x2

2


+ 2w[i−1,j−1] + 2w[i−1,j+1] + 2w[i+1,j+1] + 2w[i+1,j−1]

�x21�x2

2

+ w[i−2,j] − 4w[i−1,j] + 6w[i,j] − 4w[i+1,j] + w[i+2,j]

�x42

= q[i,j]

D. (4.82)

When the grid is equidistant in both directions, that is, �x1 = �x2 = a, one obtains

20w[i,j] − 8(

w[i−1,j] + w[i,j−1] + w[i+1,j] + w[i,j+1]

)+ 2

(w[i−1,j−1] + w[i−1,j+1] + w[i+1,j+1] + w[i+1,j−1]

)+ w[i−2,j] + w[i+2,j] + w[i,j−2] + w[i,j+2] = q[i,j]

Da4. (4.83)

Analogously to the Poisson equation, one can propose a graphical form of thedifference operator multipliers as in Figure 4.36.

Finally, we introduce a modification corresponding to the response functionrecovery by indexing all necessary values once more with upper index m. This yields

20w(m)

[i,j] − 8(

w(m)

[i−1,j] + w(m)

[i,j−1] + w(m)

[i+1,j] + w(m)

[i,j+1]

)+ 2

(w(m)

[i−1,j−1] + w(m)

[i−1,j+1] + w(m)

[i+1,j+1] + w(m)

[i+1,j−1]

)

+ w(m)

[i−2,j] + w(m)

[i+2,j] + w(m)

[i,j−2] + w(m)

[i,j+2] =q(m)

[i,j]

D(m)

(a(m)

)4. (4.84)

Alternatively, one may prefer the DDM and then the stochastic perturbation-basedversion of this finite difference scheme yields

n∑k=0

(nk

)∂kD∂bk

{20

∂n−kw[i,j]

∂bn−k−8

(∂n−kw[i−1,j]

∂bn−k+ ∂n−kw[i,j−1]

∂bn−k+ ∂n−kw[i+1,j]

∂bn−k+ ∂n−kw[i,j+1]

∂bn−k

)}

1

2 −8 2

2 −8

1

2

−81 20 −8 1

a a a a

aa

aa

Figure 4.36 Finite difference method biharmonic operator in a graphical form


+ 2n∑

k=0

(nk

)∂kD∂bk

(∂n−kw[i−1,j−1]

∂bn−k+ ∂n−kw[i−1,j+1]

∂bn−k+ ∂n−kw[i+1,j+1]

∂bn−k+ ∂n−kw[i+1,j−1]

∂bn−k

)

+n∑

k=0

(nk

)∂kD∂bk

{∂n−kw[i−2,j]

∂bn−k+ ∂n−kw[i+2,j]

∂bn−k+ ∂n−kw[i,j−2]

∂bn−k+ ∂n−kw[i,j+2]

∂bn−k

}

=n∑

m=0

(nm

)∂mq[i,j]

∂bm

∂n−m(a4)

∂bn−m . (4.85)

The boundary conditions expressed in finite difference notation for the completelyfree boundary (i) parallel to the given x2 axis become

w(m)

[i+1,j] = −w(m)

[i−1,j] + 2(1 + ν(m)

)w(m)

[i,j] − ν(m)(

w(m)

[i,j−1] + w(m)

[i,j+1]

), (4.86)

w(m)

[i+2,j] = −w(m)

[i−2,j] − 4(3 − ν(m)

)w(m)

[i−1,j] + 6(

2 + 2ν(m) − (ν(m)

)2)

w(m)

[i,j]

+ 2(2 − ν(m)

) (w(m)

[i−1,j−1] + w(m)

[i−1,j+1]

)− 4

(1 + 2ν(m) − (

ν(m))2) (

w(m)

[i,j−1] + w(m)

[i,j+1]

)+ ν(m)

(2 − ν(m)

) (w(m)

[i,j−2] + w(m)

[i,j+2]

). (4.87)

Case (ii) usesw(m)

[i,j] = 0, w(m)

[i+1,j] = −w(m)

[i−1,j], (4.88)

while case (iii) needsw(m)

[i,j] = 0, w(m)

[i+1,j] = w(m)

[i−1,j]. (4.89)

Example 4.8: Bending of the elastic thin rectangular plate with randomparametersThe stainless steel plate (e = 210 GPa, ν = 0.30) with dimensions 5.0 m × 5.0 m, thick-ness h = 0.025 m subject to uniformly distributed load q = 15.0 kN/m2 is discretizedwith a regular grid of 26 × 26 points. The plate is fully clamped at all external edgesand we study separately an uncertainty in the structure thickness and Poisson ratio.The RFM version of the SFDM is employed in the computations, so we use an11-point cycle for randomized thickness taking the discrete values h = [0.020, 0.021,0.022, 0.023, 0.024, 0.025, 0.026, 0.027, 0.028, 0.029, 0.030] m and the best least-squaresfit with smallest order is obtained for the fourth-order polynomial. The RFM dis-cretization for uncertainty in Poisson ratio has the same number of realizationsequal to ν = [0.25, 0.26, 0.27, 0.28, 0.29, 0.30, 0.31, 0.32, 0.33, 0.34, 0.35] and has thesame polynomial order. A comparison of basic probabilistic characteristics for bothrandom inputs, for varying input coefficients α = [0.05, 0.10, 0.15, 0.20, 0.25] and forthe midpoint cross-section is given in Figures 4.38–4.45 (all probabilistic momentsand coefficients of up to fourth order); the deterministic basic solution is presentedin Figure 4.37.


0.025

0.020

0.015

0.010

0.005

0

u max

0 1 2 3 4x

Figure 4.37 FDM solution for the grid 22 × 22

E(u

max

)

E(u

max

)

0.03

0.02

0.01

0

0.025

0.020

0.015

0.010

0.005

00 1 2 3 4

x0 1 2 3 4

x

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.10α – 0.20

α – 0.05 α – 0.15α – 0.25

(a) (b)

Figure 4.38 The expected values for (a) random thickness and (b) Poisson ratio

The expected values for both input random variables have similar shapes, butwhen the right-hand graph in Figure 4.37 is totally insensitive to input random fluc-tuations, the left-hand one shows significant sensitivity to the thickness uncertaintyand maximum values significantly exceed the deterministic origin. This sensitivityis of course more apparent close to the midpoint and less clear close to the clampededges of a structure.

The influence of the input parameter α affects all second-order characteristics – oncemore the shapes of variances and standard deviations are really very similar but


0.0016

0.0014

0.0012

0.0010

0.0008

0.0006

0.0004

0.0002

0

Var

(um

ax)

Var

(um

ax)

0 1 2 3 4x

0 1 2 3 4x

2. × 10−6

1.5 × 10−6

1. × 10−6

5. × 10−7

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.39 The variances for (a) random thickness and (b) Poisson ratio

0.04

0.03

0.02

0.01

0

σ(u

max

)

σ(u

max

)

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

0 1 2 3 4x

0 1 2 3 4x

0.0014

0.0012

0.0010

0.0008

0.0006

0.0004

0.0002

0

(a) (b)

Figure 4.40 The standard deviations for (a) random thickness and (b) Poisson ratio

the values obtained with randomized plate thickness are many times larger than foruncertainty in its Poisson ratio. The largest fluctuations area in the input randomdispersion reduces for variances at the midpoints’ neighborhood much closer than forthe expectations, while the region close to the edges with negligible values becomeswider. This tendency is especially clear after analysis of Figures 4.42 and 4.43 withthird and fourth central probabilistic moments. Analogously to the expectations, the


α(u

max

)

α(u

max

)

1

0.8

0.6

0.4

0.2

00 1 2 3 4

x0 1 2 3 4

x

0.10

0.08

0.06

0.04

0.02

0

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

α – 0.05α – 0.20

α – 0.10 α – 0.15α – 0.25

(a) (b)

Figure 4.41 The coefficients of variation for (a) random thickness and (b) Poisson ratio

α – 0.05α – 0.20 α – 0.25

α – 0.10 α – 0.15

0.00015

0.00010

0.00005

00 1 2 3 4

x

µ 3(u

max

)

α – 0.25α – 0.05 α – 0.15

α – 0.20 α – 0.10

0 1 2 3 4x

0

µ 3(u

max

)

−5. × 10−10

−1. × 10−9

−1.5 × 10−9

−2. × 10−9

(a) (b)

Figure 4.42 The third moments for (a) random thickness and (b) Poisson ratio

variances and standard deviations for random thickness increase much faster thanthe input coefficient of variation; this is not the case for the same characteristicswith random Poisson ratio, which are more proportional to the input coefficientsof variation. The coefficients of variation (Figure 4.41) of the maximum deflectionsprofile are totally incomparable for both random input parameters. The first casestudy results in constant α(umax) throughout the entire profile except at both edges,


0.00003

0.00002

0.00001

00 1 2 3 4

x

µ 4(u

max

)

α – 0.05α – 0.20 α – 0.25

α – 0.10 α – 0.15

00

1 2 3 4x

µ 4(u

max

)

α – 0.05α – 0.20 α – 0.25

α – 0.10 α – 0.15

1.2 × 10−11

1. × 10−11

8. × 10−12

6. × 10−12

4. × 10−12

2. × 10−12

(a) (b)

Figure 4.43 The fourth moments for (a) random thickness and (b) Poisson ratio

2

1.5

1

0.5

00 1 2 3 4

x

β(u m

ax)

α – 0.05α – 0.15 α – 0.20

α – 0.10 α – 0.25

0

−0.5

−1

−1.5

0 1 2 3 4x

β(u m

ax)

α – 0.25α – 0.10 α – 0.15

α – 0.20 α – 0.05

(a) (b)

Figure 4.44 The skewness for (a) random thickness and (b) Poisson ratio

where we observe no randomness at all. These profiles for randomized Poisson ratiosare mostly also constant, but close to the fixed edges they show some sudden largedeviations (especially for the largest proposed input α(ν)); thereafter, they decreaseto 0 at both edges. All the resulting coefficients of course increase together with anadditional increase of the input coefficient α, but for randomized thickness this isalmost four times larger than the input 1 and 10 times larger than the maxima reachedby α(umax) in the second case study.


10

8

6

4

2

00 1 2 3 4

x

κ(u m

ax)

α – 0.05α – 0.20

α – 0.15α – 0.10α – 0.25

10

8

6

4

2

00 1 2 3 4

x

α – 0.05α – 0.25

α – 0.15α – 0.10α – 0.20

κ(u m

ax)

(a) (b)

Figure 4.45 The kurtosis for (a) random thickness and (b) Poisson ratio

Spatial variations of the third central probabilistic moments are very similar to eachother in spatial distribution (except the sign) and, accidentally, to the Gaussian bellcurve, but these computed for a random Poisson ratio are almost negligible – being sixorders less than in the case of uncertain thickness. Randomization of the parameter hresults in a positive third moment (and effectively skewness), contrary to the Poissonratio uncertainty, when it is totally negative (with skewness also). Similarly to thevariances and fourth central moments (given in Figure 4.43) we notice that the largerthe input coefficient α, the greater the absolute values of the moments analyzed.

The fourth central probabilistic moments both grow much faster than the inputcoefficient of variation; now the intervals with zero values take up almost half ofthe plate span (Figure 4.43). The coefficients of skewness (Figure 4.44) and kurtosis(Figure 4.45) exhibit similar properties to the coefficient of variation – these obtainedfor random thickness are constant throughout the plate, reaching zero at the edges.Although we notice that the values of both coefficients stabilize with increasing α(h),their positive values differ significantly from 0. The same coefficients computed forrandomized Poisson ratio have some enormously large local variations close to bothedges, where their values are almost equal to these computed in case of h(ω). For theremaining part of the plate cross-section, skewness is slightly negative and stabilizeswith an increase of α(ν), whereas kurtosis keeps close to 0. Therefore, considering thereliability condition on a maximum deflection for the plate midspan, one can assumewith the relatively small error that this deflection has Gaussian distribution.


4.2.3 Vibration Analysis of Elastic Plates

Starting from the Love’s equations for a plate equilibrium and applying zero curva-tures, the resulting expression can be rewritten as [159]

D∇4w + ρhw = q, (4.90)

where ρ denotes the mass density of the plate. The relation (4.90) is usually rewrittenin rectangular coordinates in the following manner:

D(

∂4w∂x4 + 2

∂4w∂x2∂y2 + ∂4w

∂y4

)+ ρhw = q. (4.91)

Looking for the natural frequencies, the following transformation is applied:

w = W exp(jωt

), (4.92)

which results in

D(

∂4W∂x4 + 2

∂4W∂x2∂y2 + ∂4W

∂y4

)+ ρhω2W = 0. (4.93)

Further solution can be carried out if only detailed boundary conditions arespecified. If, for instance, we have a plate with dimensions a × b corresponding to theaxes x and y and it is simply supported along the axes parallel to direction y, then thesolution can be expressed as

W(x, y

) = Y(y) sinmπx

a, (4.94)

which gives an equation of motion in the following form:

∂4Y∂y4 − 2

(mπ

a

)2 ∂2Y∂y2 +

[(mπ

a

)4− ρh

Dω2]

Y = 0 (4.95)

where

Y(y) =4∑

i=1

Ci exp(λi(y/b)

). (4.96)

Further, it is obtained that

λi = ±ba

mπ√

1 ± K (4.97)


andK = ω(

m2π2

a2

)√Dρh

. (4.98)

Application of the generalized stochastic perturbation technique in Equation (4.91)results in:

• zeroth-order relation

D0(

∂4W0

∂x4 + 2∂4W0

∂x2∂y2 + ∂4W0

∂y4

)+ ρ0h0 (ω0)2

W0 = 0 (4.99)

• first-order relation

D0(

∂5W∂b∂x4 + 2

∂5W∂b∂x2∂y2 + ∂5W

∂b∂y4

)+ ∂

(ρhω2

)∂b

W0 + ρ0h0 (ω0)2 ∂W∂b

= −∂D∂b

(∂4W0

∂x4 + 2∂4W0

∂x2∂y2 + ∂4W0

∂y4

)(4.100)

• nth-order relation

n∑k=0

(nk

)∂kD∂bk

(∂n−k+4W∂bn−k∂x4 + 2

∂n−k+4W∂bn−k∂x2∂y2 + ∂n−k+4W

∂bn−k∂y4

)

+n∑

k=0

(nk

)∂k

∂bk

(ρhω2) ∂n−kW

∂bn−k= 0. (4.101)

Calculating zeroth-order terms from the first equation and next, sequentially, first-and second-order terms, and so on, all the probabilistic moments of the dynamicstructural response may be obtained. It is also seen that prior to determination ofprobabilistic characteristics for the final solution W, we need to determine partialderivatives of the plate eigenfrequencies as they appear explicitly in Equation (4.101).

5Homogenization Problem

The variability of elastic characteristics and geometrical dimensions of homogeneousas well as composite elements is a frequent problem in designing new and inspectingexisting structures. This variability in the design process is included during theoptimization phase, where, thanks to non-gradient or gradient techniques, themost optimal distribution (in functional graded material – FGM applications, forinstance [7]) or the best choice and contrast (for composites with two or more con-stituents) is to be found. Such variability in design parameters is taken into accounta priori and almost always has a clearly deterministic character. In contrast, experi-mental testing and a posteriori inspection of engineering structures return statisticalor sometimes even stochastic information about spatial or spatio-temporal [178] ran-dom distributions of their material and/or geometrical parameters. Therefore, onecan notice that sensitivity analysis [136] and probabilistic modeling can be very simi-lar from a quantitative point of view [77]. This similarity is even more transparent innumerical analysis when the discussed perturbation method is applied to determinesensitivity coefficients or gradients with respect to some design parameters as well assome basic probabilistic moments of structural response; these design parameters aresimply treated as input random variables or fields. The relation between sensitivityand probabilistic analysis appears to be even closer when we realize that usinghigher than second-order perturbation methodology [99] it is possible to calculateboth higher-order sensitivities and, after minor changes of algebraic formulas in therelevant definitions, higher-order probabilistic moments.

The problem raised above is of special importance in the area of composite materi-als [21], where not only single particular parameters but multiple characteristics of thesame type and, furthermore, their composition may be the subject of such analysis.Even if phenomena like delamination or soft matrix penetration by rigid fibers, typi-cal for specific composites only, are neglected, the sensitivity or probabilistic analysisstill has a number of parameters. As is known, the homogenization method has beendiscovered and extended to reduce significantly the number of composite designparameters by the introduction of effective characteristics. Although this technique in

The Stochastic Perturbation Method for Computational Mechanics, First Edition. Marcin Kaminski.© 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd.


its modern view is almost 40 years old [129], there are still some new ideas and applica-tions in the food industry [107], some composites made of wood, metallic components[100], superconductors [101], fully and partially saturated heterogeneous solids [148],and so forth. After fundamental discoveries concerning elastic, thermal [120], andelectric effective properties [114], now thermodynamic wave propagation, variousmultiscale problems [179], even for time-dependent cases with an ‘‘equation-free’’approach as well as a variety of materially non-linear multi-component composites[14, 37, 54, 121] are homogenized. Following numerous engineering applications thestrength of composites by a homogenization method can be estimated [36]; atomisticand nano levels [22, 160] appear to be the smallest resolutions, some new issuesare explored [150] as well as well-known methods recently revisited [119, 157, 170].Much attention is obviously paid to random composites because of the uncertaintyin reinforcement location/shape and/or pore spatial distribution in matrices, andrandomness in the components’ physical and mechanical characteristics [71, 176].

The homogenization method is sometimes connected with sensitivity analysis andoptimization (see the problem of a homogeneous plate with holes), used in con-junction with probabilistic analysis using Monte Carlo simulation [59, 98] or someexpansion methods. Let us note that even if the mathematical apparatus has strictlydeterministic character, some issues like the basic correlation dimension or repre-sentativeness of the basic cell subject to the homogenization process are neverthelessdiscussed. The observations collected above make it negligible to formulate a singleuniversal and general approach to calculate the effective characteristics, where sensi-tivity gradients and probabilistic moments can be extracted from the same equationsusing similar algebra. Let us note here that an analogous homogenization approachcan be provided starting from the complementary energy principle.

The perturbation technique based on the nth-order Taylor expansion displayed inChapter 1 and particularly for the FEM throughout Chapter 2 is applied here in con-junction with the effective modules method to determine practically any-order partialderivatives of the homogenized constitutive tensor. The composite subjected to thisprocedure is linearly elastic and transversely isotropic, where elastic characteristicsof the components are assumed as truncated Gaussian random variables and they aredefined uniquely by the first two probabilistic moments. The composite remains peri-odic in all cases in the sense that long round fibers with constant radius have periodicdistribution in the plane transversal to the fiber directions; all the fibers are perfectlyparallel. The material characteristics are the same in each cell for the sensitivity com-putations and they have the same first two moments in each cell for random analysisto maintain this perfect periodicity in a composite. The key feature here is the responsefunction reconstruction, where using multiple solutions of the deterministic homo-genization problem each effective tensor component is represented as a polynomialfunction of the input random variables. As we know, this is in significantcontradiction to previous applications of the perturbation technique, where zeroth-and higher-order equilibrium equations were derived and solved numerically. Now,an nth-order polynomial function is proposed and implemented without any mixedterms, which reflects the case of a single random input variable or the series of

Homogenization Problem 243

uncorrelated random parameters. The symbolic computations package Maple™ isused here to efficiently solve for the coefficients of this polynomial expression aswell as to process normalization of the sensitivity gradients and/or determinationof probabilistic moments of the homogenized tensor. Thanks to the application ofsymbolic computations, it is possible to insert the perturbation parameter ε intothe polynomial expansion of the random structural response (homogenized tensorhere) and, further, to calculate higher-order partial derivatives analytically. As canbe seen at this point, the approach enables us to eliminate the limitations of thesecond-order perturbation technique, to join sensitivity analysis and probabilisticmodeling, to shorten the entire computational process (related to the Monte Carlosimulation procedure), as well as to provide the computational process with a givena priori accuracy. Since the FEM-based plane strain numerical analysis is the core fora solution of the homogenization problem, all error analysis procedures includingh, p, or hp adaptivity may also be further adopted here [126].

5.1 Composite Material Model

Let us consider a periodic random composite in plane strain in the unstressed andundeformed state and assume that the section of this structure is constant along thex3 axis, and finally consider the section Y ⊂ R2 in the plane x3 = 0. Let us assumethat the representative volume element (RVE) [40, 71] � of Y has a rectangularexternal boundary. Further, let � contain n (n < ∞) coherent regions satisfying thefollowing conditions:

� =n⋃

a=1

�a ∪n⋃

a=2

�(a−1,a), for a = 1,2, . . . ,n (5.1)

�a ∩ �b = ∅, for a, b = 1,2, . . . ,n (5.2)

where �(a − 1,a) is the boundary (interface) between the regions �a−1 and�a. Let us consider such a class of � that for every a the interior ofthe contour �(a − 2,a − 1) is contained in �(a − 1,a) and these contours are dis-joint (Figure 5.1). Considering further aspects of this model it is assumedthat, although �a = �a(ω; t) as well as �(a − 1,a) = �(a − 1,a)(ω; t) for anya = 1, . . . , n are random processes, these two relations hold true for anytime t ∈ [0, ∞) and the external geometrical dimensions of � remain constant.

Let every component of this composite contain linear elastic and transverselyisotropic material and their elastic parameters (Young’s modulus and Poisson ratio)are given using the following processes:

ea(ω; t) = e0a(ω) − ea(ω)t, (5.3)

νa(ω; t) = ν0a(ω) − νa(ω)t. (5.4)

The quantities e0a(ω) and ν0a(ω) are the Gaussian random variables given by theirfirst two probabilistic moments and truncated according to their physical limitations.


Ωn

Ωn–1

Ω2

Ω1

Figure 5.1 The macroscale and microscale of the periodic composite with defects.Reproduced with permission from Elsevier

The two additional variables ea(ω) and νa(ω) denote for any composite componenta = 1, . . . , n the aging rates and are non-negative Gaussian variables defined by theirexpectations and standard deviations. Therefore, the elasticity tensor Cijkl(x; ω; t) isdefined as

Cijkl(x;ω; t) = χa (x) C(a)ijkl (ω; t)

= χa (x) ea(ω; t)

{δijδkl

νa(ω; t)(1 + νa(ω; t)

) (1 − 2νa(ω; t)

)+(δikδjl + δilδjk

) 12(1 + νa(ω; t)

)}

for i, j, k, l = 1,2 (5.5)

where χ a(x) is the characteristic function for the composite component indexed by aand equals

χa (x) ={

1, x ∈ �a

0, elsewhere.(5.6)

Further, let us consider some material microdefects [133] having random shape andlocation distributed along the interfaces inside the specimen � [69, 71]. According tothe engineering evidence such defects in the reinforcement are decisively smaller, sothat they are neglected in this approach.

Considering the matrix defects (microcavities) idealized as bubbles (cf. Figure 5.2versus Figure 5.3), it is assumed that (i) the stochastic nucleation rule for the defectsalong the interface �(a − 1,a) is defined as

n�(a−1,a)(ω; t) = n0�(a−1,a)(ω) + n�(a−1,a)(ω)t, (5.7)

where both Gaussian variables n0�(a−1,a)(ω) (initial defects number) and n�(a−1,a)(ω)

(aging rate) are truncated to positive values only. The second of these parameters


Ωa–1

Ωa

Figure 5.2 The real defects at the composite interface. Reproduced with permissionfrom Springer

Ωa–1

Ωa

rb

bubble

Figure 5.3 Defects approximation by the bubble. Reproduced with permission fromSpringer

is relevant to random nucleation of the new defects inside the existing interphase;(ii) these defects are geometrically approximated by the semicircles (bubbles) lyingwith their diameters on the relevant interfaces; the stochastic growth of the bubblesis described by the following similar law:

r�(a−1,a)(ω; t) = r0�(a−1,a)(ω) + r�(a−1,a)(ω)t. (5.8)

It is possible to consider these defects in a more general form, that is, as semi-ellipses lying with their main axis on the interface (major or minor [72]). We denotehere r0

�(a−1,a)(ω) as the initial average radius of the bubbles at the interface �(a − 1,a) inthe weaker material �a(t), while r�(a−1,a)(ω) is its aging rate (they are both Gaussianvariables with additional truncation). Finally we assume that (iii) the geometricdimensions of every defect belonging to any �a(t) are small in comparison withthe minimal distance between �(a − 2,a − 1) and �(a − 1,a) boundaries for a = 3, . . . , nand also with �1 geometric dimensions; (iv) all elastic characteristics equal to 0 ifx ∈ Da(t), for a = 1, 2, . . . , n. The signs of the aging rates are defined dependingon the dominating external loading character and other physical phenomena – wemay assume non-negative values of the aging velocity if the composite is subject totension and aggressive corrosion, for instance. It should be underlined that the modelintroduced approximates the real defects rather precisely; however, the cavitiesshould be appropriately averaged over the corresponding materials, which theybelong to, to build up a reliable numerical procedure.

Let us consider in particular stochastic material non-homogeneities located inany �a(t) ⊂ � for t ∈ [0, ∞). The set Da(t) of all the defects considered for any


Ωa+1

Ωa

Da(j)

Γ(a, a+1)Γ(a–1, a)

Da(k)Ωa–1

Δ'a

Ba

BaΔa Da(i )

"' "'"'

"'

'

Figure 5.4 The interface and volumetric material defects. Reproduced with permissionfrom Springer

component can be divided into three disjoint subsets D′a(t), D′′

a (t), and D′′′a (t), where

D′a(t) contains all the defect areas having non-zero intersection with the boundary

�(a − 1,a), D′′a (t) having zero intersection with �(a − 1,a) and �(a,a + 1), and D′′′

a (t) havingnon-zero intersection with �(a,a + 1) only. Further, all the defects belonging to subsetsD′

a(t) and D′′′a (t) are called the stochastic interface defects, and these belonging to

D′′a (t) – the volumetric stochastic defects, respectively. Let us consider such �′

a(t),�′′

a (t), and �′′′a (t), where �a(t) = �′

a(t) ∪ �′′a (t) ∪ �′′′

a (t), such that the relations D′a(t) ⊂

�′a(t), D

′′a (t) ⊂ �′′

a (t), and D′′′a (t) ⊂ �′′′

a (t) (cf. Figure 5.4) hold true with probabilityequal to 1.

The subsets �′a(t), �′′

a (t), and �′′′a (t) can be geometrically constructed using prob-

abilistic moments of the defect parameters (their geometric dimensions). Let usintroduce for this purpose the random processes ′

a(ω; t) and ′′′a (ω; t) as upper

bounds on norms of normal vectors defined on the boundaries �(a − 1,a) and �(a,a + 1)

and the boundaries of the defects belonging to D′a(t) and D

′′′a (t), respectively. Next, let

us consider the upper bounds of probabilistic distributions of ′a(ω; t) and

′′′a (ω; t),

respectively, given as follows:

′a(t) = E

[′

a(ω; t)]+ 3σ

(′

a(ω; t))

, (5.9)

′′′a (t) = E

[

′′′a (ω; t)

]+ 3σ

(

′′′a (ω; t)

). (5.10)

Thus, �′a(t) and �′′′

a (t) can be expressed in the following form:

�′a(t) = {P(xi) ∈ �a(t) : d(P, �(a−1,a)) ≤ ′

a(t)}

, (5.11)

�′′′a (t) =

{P(xi) ∈ �a(t) : d(P, �(a,a+1)) ≤

′′′a (t)}

, (5.12)

where i = 1, 2 and d(P, �) denotes the distance from the given point P to theappropriate interface. Let us note that �′′

a (t) can be obtained as

�′′a (t) = �a(t) − {�′

a(t) ∪ �′′′a (t)

}. (5.13)


The deterministic spatial averaging of properties Ya(ω; t) on the continuous anddisjoint subsets �a(t) of � is employed to formulate the stochastic averagingmethod. The effective process Y(eff )(ω; t) characterizing � is given by the followingequation:

Y(eff ) (ω; t) =

n∑a=1

Ya (ω; t)∣∣�a(t)

∣∣|�| ;x ∈ �, (5.14)

where |�| is the two-dimensional Lebesgue measure of � (its area or length). Usingthe equations given above, the mechanical and physical properties are stochasti-cally averaged within the regions �a(t) and their subsets �′

a(t), �′′a (t), �′′′

a (t) fora = 1, . . . , n. Finally, the primary stochastic geometry of the composite consideredis replaced by the deterministic one – in this way the n-component composite withstochastic interface defects on both sides of every phase boundary and with volumenon-homogeneities is replaced with an artificial 3n − 2-component structure. Letus consider spatial averaging of the elastic characteristics for �a(t) containing thebubbles defined by the parameters na(ω; t) and ra(ω; t). Let us denote by Ym(a)(ω; t) theelastic parameters of the solid in �a(t) and by Yd(a)(ω; t) – within the defects. Then,we can calculate

Y(eff )a(ω; t) = π

2�ana(ω; t)r2

a(ω; t)Yd(a)(ω; t) +(

1 − na(ω; t)πr2a(ω; t)

2�a(t)

)Ym(a)(ω; t)

=(

1 − na(ω; t)πr2a(ω; t)

2�a(t)

)Ym(a)(ω; t) (5.15)

according to the remark (iv) above. So, the effective Young’s modulus in the athcomponent subjected to the stochastic aging equals

e(eff )a (ω; t) =

(1 −

(n0

a(ω) + na(ω)t)π(r0

a(ω) + ra(ω)t)2

2�a(t)

)(e0

a(ω) + ea(ω)t)

. (5.16)

Next, we can rewrite the effective elasticity tensor (EET) components in thismaterial as [62]

Ca(eff )ijkl (ω; t) = e(eff )

a (ω; t) δijδkl

⎧⎨⎩ ν

(eff )a (ω; t)(

1 + ν(eff )a (ω; t)

) (1 − 2ν

(eff )a (ω; t)

)


) 1

2(

1 + ν(eff )a (ω; t)

)⎫⎬⎭ (5.17)


which equals (time and random space dependence is omitted for clarity of presenta-tion only)

C(eff )aijkl =

(1 − naπr2

a

2�a

)e(eff )

a

⎧⎪⎪⎪⎨⎪⎪⎪⎩

δijδkl

(1 − naπr2

a

2�a

)ν

(eff )a(

1 +(

1 − naπr2a

2�a

)ν

(eff )a

)(1 − 2

(1 − naπr2

a

2�a

)ν

(eff )a

)


)2(

1 +(

1 − naπr2a

2�a

)ν

(eff )a

)⎫⎪⎪⎪⎬⎪⎪⎪⎭

. (5.18)

Moreover, we can observe that for any a = 2, . . . , m the total area of �a(t) can bedescribed as

�a(t) = π[(

Ra + a(t))2 − R2

a

](5.19)

where Ra denotes the internal radius identifying �a(t), while a(t) denotes the thick-ness of the given interphase. Further, the elasticity tensor of �a may be described usingthe geometry model parameters na, ra, Ra, a, and input elastic parameters as [62]

C(eff )aijkl =

⎛⎝1 − nar2

a

2[(

Ra + a)2 − R2

a

]⎞⎠ e(eff )

a

×

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

δijδkl

(1 − naπr2

a2�a

)ν

(eff )a(

1 +(

1 − nar2a

2[(Ra+a)

2−R2a

])

ν(eff )a

)(1 − 2

(1 − nar2

a

2[(Ra+a)

2−R2a

])

ν(eff )a

)


)

2

(1 +

(1 − nar2

a

2[(Ra+a)

2−R2a

])

ν(eff )a

)⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (5.20)

Let us note that the denominator does not include multiplier 2 when we averagethe stochastic volumetric defects for all composite components, it appears only forthe interphases. Unfortunately, it seems to be very complicated or even impossibleto describe analytically the first two probabilistic moments of the elasticity tensorcomponents as functions of expected values and variances of Poisson ratios, whileit is quite straightforward for the Young’s modulus using fundamental propertiesintroduced in Chapter 1. Then, quite similarly to Equation (5.20) one can derive therandom process of Cijkl(ω;t) having defined time series of geometrical and material


parameters as proposed in Equation (5.8). Some alternative way of stochastic materialdefects FEM analysis for linear elasticity and viscoplasticity may be found in [43].

5.2 Statement of the Problem and Basic Equations

Let us introduce a geometrical scaling parameter ζ > 0 between the micro- andmacroscale of the composite (see Figure 5.5) and introduce two coordinate systemsy = (y1, y2, y3) on the microscale of the composite and x = (x1, x2, x3) on the macroscale.

Next, let us express any state function G defined on composite region Y as

Gζ (x) = G(

xζ

)= G

(y)

. (5.21)

The linear elasticity problem for the periodic composite structure is given asfollows: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂σζ

ij∂xj

+ fi = 0

σζ

ij nj = ti; x ∈ ∂Yσ

uζ

i = 0; x ∈ ∂Yu

σζ

ij = Cζ

ijkl (x) εζ

kl

εζ

kl = 12

(uζ

k,l + uζ

l,k

)i, j, k, l = 1,2. (5.22)

Assuming perfect interfaces between matrix and the fibers as well as no cracks andother defects (after stochastic averaging) in these constituents we solve the problem

x2 ∂Ys

∂Yu

x1

L

Y

l

y1

y2Ω1

Ω2

Ω

l

Lz =

Figure 5.5 Periodic composite structure Y . Reproduced with permission from IFTR PAS


(Equation (5.22)) using a bilinear form

aζ (u, v) =∫�

Cijkl

(xζ

)εij(u)εkl(v)d� (5.23)

and the linear oneL (v) =

∫�

fivid� +∫

∂�σ

tivid (∂�) (5.24)

in the following Hilbert space of admissible displacements defined on Y:

V ={

v| v ∈ (H1 (Y))2

, v|∂�u = 0}

, (5.25)

‖v‖2 =∫�

εij(v)εij(v)d�. (5.26)

Then, the variational statement equivalent to the equilibrium problem (Equation(5.22)) is to find uζ ∈ V, being a solution of the following equation:

aζ(uζ , v

) = L (v) , v ∈ V. (5.27)

Let us define the additional space of the admissible displacement functions P(�) ={v, v ∈ (H1 (�)

)2}periodic on the composite cell �. So, we introduce the new bilinearform for any u, v ∈ P(�):

ay (u, v) =∫�

Cijkl(y)εij(u)εkl(v)d� (5.28)

and the homogenization function χ (ij)k ∈ P(�) (also of displacement type) as a solutionfor the so-called local problem on a periodicity cell

ay

((χ(ij)k + yjδki

)nk, w

)= 0 (5.29)

for any w ∈ P(�), where nk is the additional versor. The elasticity coefficients can bedefined according to Equation (5.21) as

Cζ

ijkl (x) = Cijkl(y)

. (5.30)

The elasticity tensor components fulfill the following conditions:

Cijkl ∈ L∞(R2), for i, j, k, l = 1,2,3 (5.31)

Cijkl = Cklij = Cjikl, (5.32)

∃C0 > 0; Cijklξijξkl ≥ C0ξijξij ∀i,j ξij = ξji. (5.33)


Moreover, for any of the composite constituents this tensor is defined after Equation(4.24) as

Cijkl (x) = e (x)

{δijδkl

ν (x)

(1 + ν (x)) (1 − 2ν (x))+(δikδjl + δilδjk

) 12 (1 + ν (x))

}. (5.34)

If, analogously to Equation (2.30), it is introduced that

Aijkl(x) = δijδklν(x)

(1 + ν(x))(1 − 2ν(x))+ (δikδjl + δilδjk)

12(1 + ν(x))

, (5.35)

then the partial derivatives of the elasticity tensor with respect to the Young’smodulus in homogeneous material is equal to

∂Cijkl

∂e= Aijkl, (5.36)

and any higher-order partial derivatives equal 0:

∂nCijkl

∂en = 0. (5.37)

The case of differentiation w.r.t. Poisson ratio is essentially more complex andreturns

∂Cijkl

∂ν= e

{δijδkl

1(1 + ν) (1 − 2ν)

(1 − ν

(1 + ν)+ 2

ν

(1 − 2ν)

)

−(δikδjl + δilδjk

) 1

2 (1 + ν)2

}, (5.38)

and further successive differentiation enlarges its algebraic expression complexity.Next, we find the EET components from the following theorem.

The solution uζ of Equation (5.22) converges weakly in space V:

uζ → u as ζ → 0 (5.39)

for the �-periodic tensor Cζ

ijkl (x), where the solution u is the unique one for theproblem:

u ∈ V, S (u, v) = L(v) (5.40)

for any v ∈ V and

S (u, v) =∫Y

Sijklεij(u)εkl(v)dY, (5.41)

whereSijkl = 1

|�| ay

((χ(ij)p + yiδpj

)np,(χ(kl)r + ylδrk

)nr

). (5.42)


Now we consider the boundary value problem in its differential form

∂

∂xj

(Cijkl

(xζ

)εkl(uζ))+ fi = 0; uζ

i = 0 for y ∈ ∂Y (5.43)

and the following representation for the displacement using the parameter ζ :

uζ (x) =∞∑

m=0

ζ mu(m) (x, y)

, (5.44)

where any u(m)(x, y) is periodic in y on � and m = 0, 1, 2, . . . Let us note thatcontrary to the stochastic perturbation methods, this expansion is provided with asmall positive-scale perturbation parameter but the general idea remains the same.A separation of the variables x and y leads to the statement

εij(v) = εxij(v) + 1

ζε

yij(v), (5.45)

where, for instance.

εxij(v) = 1

2

(∂vi

∂xj+ ∂vj

∂xi

). (5.46)

Thus, relation (5.43) is expanded as

(ζ−2L1 + ζ−1L2 + L3

)×∞∑

m=1

ζ mu(m) (x, y)+ f = 0, (5.47)

where

L1u = ∂

∂yi

(Cijkl(y)

)ε

ykl(u), (5.48)

L2u = Cijkl(y)∂

∂xj

(ε

ykl(u)

)+ ∂

∂yi

(Cijkl(y)εx

kl(u))

, (5.49)

L3u = Cijkl(y)∂

∂xj

(εx

kl(u))

. (5.50)

Next, equating the terms with the same order of ζ to 0, the following equations ofzeroth, first, and second order are obtained:

L1u(0) = 0, (5.51)

L1u(1) + L2u(0) = 0, (5.52)

L1u(2) + L2u(1) + L3u(0) + f = 0 (5.53)


to determine the solutions u(0), u(1), and u(2). Let us also note that the equation

L1u + P = 0 (5.54)

with u a �-periodic function has a unique solution for

〈P〉 = 1|�|

∫�

P dy = 0. (5.55)

Starting from the above equations yields

u(0)(x, y) = u(x), (5.56)

while Equation (5.52) takes the following form:

L1u(1)(x, y) + ∂

∂yi

(Cijkl(y)

)εx

kl

(u(0)(x)

)= 0. (5.57)

The separation process of the variables x and y leads to

u(1)i (x, y) = χ(kl)i(y)εx

kl

(u(0))

+ ui (x) . (5.58)

The last two equations give the formulation for the �-periodic functions χ(kl)i(y) as

∂

∂yi

(Cijkl(y)

∂χ(kl)m(y)

∂ym

)+ ∂

∂yiCijkl(y) = 0. (5.59)

So, the local problem of the homogenization function χ(kl)i(y) reduces to theequations posed above and the following conditions hold true:

χ(kl)i periodic on � (5.60)

andσij

(χ(pq)

)nj =

[Cpqij

]|�12

nj = F(pq)i|�12= C(2)

pqij − C(1)pqij on �12. (5.61)

Hence, the variational formulation necessary for the displacement version of theFEM analysis of the local problem can be written for two-component composite as

C(1)ijkl

∫�1

εkl

(χ (pq)

)εij (v) d� + C(2)

ijkl

∫�2

εkl

(χ (pq)

)εij (v) d� = −

∫�12

σij

(χ (pq)

)njvid�.

(5.62)After solving for χ (pq)i from Equation (5.62) one computes the EET as

C(eff )ijpq = 1

|�|∫�

(Cijpq + Cijklεpq

(χ (kl)

))d�. (5.63)


Further, it is clear that if the second component of the RHS integrand function isomitted, well-known upper bounds for the EET for the composite are returned (assimple spatial averages of the elasticity tensor). The experimental and computationalanalyses prove that these bounds are easy to calculate even for the case of randomspaces of composite constituent elastic characteristics, but their values significantlyoverestimates the real effective properties. Let us finally note that since our proba-bilistic method based on the RFM uses several deterministic solutions to completethe entire homogenization of a random composite, this mathematical apparatus doesnot need further improvement and the homogenization theorem remains valid (thesame as for the Monte Carlo simulation).

We consider the special case of the multi-component unidirectional compositebeam with prismatic cross-section constant along the beam length, which has perfectlyperiodic structure and material properties varying along the spatial macro-coordinatex3 and constant with respect to the remaining axes. Now, the RVE consists of thefew layers with deterministically defined thicknesses corresponding to the compositeconstituents and having perfect interfaces perpendicular to x3. The following systemof partial differential equations is employed to calculate the effective properties [146]:(

Cijkl

(x3

ζ

)uζ

k,l

),j = fi (x) , uζ (x) = u0 (x) , x ∈ ∂�. (5.64)

Similarly to the procedure displayed above (cf. Equation (5.59)), the periodichomogenization functions χ (mn)

(y)

are employed and determined as the solution tothe local problem on the RVE:

∂

∂yj

(Cijkl

(y3) ∂

∂yl

(χ(mn)k

)+ Cijmn(y3)) = 0. (5.65)

Obviously, a solution is expected in the form χ (mn)

(y) = χ (mn)

(y3)

. It yields

∂

∂y3

(Ci3k3

(y3) ∂

∂y3

(χ(mn)k

)+ Ci3mn(y3)) = 0. (5.66)

for any periodic χ (mn)

(y3). Therefore, Equation (5.66) is transformed into the form

Ci3k3(y3)χ(mn)k,3 + Ci3mn

(y3) = Aimn (5.67)

and may be solved explicitly, so that

χ(mn)k,3 = −{

Ck3j3

}−1Cj3mn +

{Ck3j3

}−1Ajmn. (5.68)

One obtains from the periodicity condition⟨χ (mn),3

⟩�

= 0:

−⟨{

Ck3j3

}−1Cj3mn

⟩�

+⟨{

Ck3j3

}−1⟩�

Ajmn = 0. (5.69)


ThereforeAimn =

⟨{Ci3k3

}−1⟩−1

�

⟨{Ck3j3

}−1Cj3mn

⟩�

(5.70)

and it holds that

χ(mn)k,3 = −{

Ck3j3

}−1Cj3mn +

{Ck3j3

}−1⟨{

Cj3q3

}−1⟩−1

�

⟨{Cq3p3

}−1Cp3mn

⟩�

. (5.71)

Taking into account that the state functions depend only on the y3 axis, thefollowing holds:

C(eff )ijkl =

⟨Cijkl + Cijm3χ(kl)m,3

⟩�

. (5.72)

Finally, the homogenized elasticity tensor components are given by

C(eff )ijkl =

⟨Cijkl

⟩�

−⟨Cijm3

{Cm3p3

}−1Cp3kl

⟩�

+⟨Cijm3

{Cm3p3

}−1⟩�

⟨{Cp3n3

}−1⟩−1

�

⟨{Cn3q3

}−1Cq3kl

⟩�

. (5.73)

In case of isotropic and linear elastic constituent materials of this composite, it isobtained after some algebraic manipulation that [58]

C(eff )1111 = C(eff )

2222 =⟨

(1 − ν) e(1 + ν) (1 − 2ν)

⟩�

−⟨

e (1 − 2ν)

1 − ν2

⟩�

+

⟨1 − 2ν

1 − ν

⟩2

�⟨(1 + ν) (1 − 2ν)

(1 − ν) e

⟩�

, (5.74)

C(eff )3333 =

⟨(1 + ν) (1 − 2ν)

(1 − ν) e

⟩−1

�

, (5.75)

C(eff )1133 = C(eff )

3311 = C(eff )2233 = C(eff )

3322 =

⟨1 − 2ν

1 − ν

⟩�⟨

(1 + ν) (1 − 2ν)

(1 − ν) e

⟩�

, (5.76)

C(eff )1122 = C(eff )

2211 =⟨

e1 − ν

⟩�

−⟨

e (1 − 2ν)

1 − ν2

⟩�

+

⟨1 − 2ν

1 − ν

⟩�⟨

(1 + ν) (1 − 2ν)

(1 − ν) e

⟩�

, (5.77)

C(eff )1212 = C(eff )

2121 =⟨

e1 + ν

⟩�

, (5.78)


C(eff )1313 = C(eff )

3131 = C(eff )2323 = C(eff )

3232 =⟨

11+ν

e

⟩�

, (5.79)

with the remaining components of the EET equal to 0. The equations given aboveenable both analytical derivations of the probabilistic moments for the given ran-dom variable in the system, Monte Carlo simulation of the homogenized tensorand generalized stochasticperturbation-based computational analysis; all these tech-niques are contrasted in computational experiments provided later in this chapter.Another model of effective in-plane characteristics of the laminates to be consideredin the presence of some uncertainty may be found in [162].

5.3 Computational Implementation

Let us introduce the following approximation of homogenization functions χ (rs)i atany point of the considered continuum � in terms of a finite number of generalizedcoordinates q(rs)α and shape functions ϕiα [71, 92]:

χ(rs)i = ϕiαq(rs)α , i, r, s = 1,2, α = 1, . . . ,N, (5.80)

and the strain εij(χ (rs)) as well as stress σ ij(χ (rs)) tensors

εij(χ (rs)) = Bijαq(rs)α , (5.81)

σij(rs) = σij(χ (rs)) = Cijklεkl(χ (rs)) = CijklBklαq(rs)α , (5.82)

where Bklα is the conventional FEM strain–nodal displacement operator provided inEquation (2.135). Therefore, a variational statement for the homogenization problemis proposed as

∫�

δχ(rs)i,jCijklχ(rs)k,l d� =n∑

a=2

∫�(a−1,a)

δχ(rs)i[F(rs)i

] |�(a−1,a)d� (no sum on r, s) . (5.83)

Next, let us define the global stiffness matrix as

Kαβ =E∑

f=1

K(f )αβ =

E∑f=1

∫�f

C(f )ijklBijαBklβd�. (5.84)

Introducing this matrix into Equation (5.83) together with the discretization madein Equation (5.80) and minimizing it with respect to the generalized coordinates wearrive at

Kαβq(rs)α = Q(rs)α , (5.85)

where Q(rs)α is the external load vector which includes the stress boundary conditionsapplied along the interface (if only the neighboring components are perfectly bonded)


Table 5.1 The components of the forces F(pq)i

χ (11) χ (12) χ (22)

F(pq)1 C(2)1111 − C(1)

1111 C(2)1212 − C(1)

1212 C(2)1122 − C(1)

1122

F(pq)2 C(2)2211 − C(1)

2211 C(2)1212 − C(1)

1212 C(2)2222 − C(1)

2222

in the following form:

σij

(χ (pq)

)nj =

[Cijpq

]nj = F(pq)i; x ∈ ∂�12, (5.86)

where nj is the component of the unit vector normal to the given interface anddirected to the RVE interior, while [f ] denotes the difference of the function f values:[

f] = f (2) − f (1). (5.87)

The stress boundary conditions corresponding to different homogenization prob-lems are specified in Table 5.1.

It should be underlined that taking into account the interface phenomena inengineering composites, the fiber and matrix boundaries may be partially differentcontours (lack of contact between the components caused by a delamination), whichmay be the result of composite processing thermal stresses or extensive fatigueprocesses. Finally let us note that to assure the symmetry conditions on periodicitycell edges, the displacements perpendicular to the external RVE boundary are fixedfor every nodal point.

As could be noticed during derivation of the equations for the generalizedperturbation-based approach, one of the most complicated issues is numerical deter-mination of up to nth-order partial derivatives of the structural response functionwith respect to the randomized parameter. As we demonstrated in the preceedingchapters, it is possible to determine this function by a multiple solution of theboundary value problem around the expectation of the random parameter. Theresponse function for each component of the homogenized tensor is built up fromuniform symmetric discretization of this expectation in its close neighborhood withequidistant intervals. A set of ordinary deterministic computations of the homoge-nized tensor components leads to the final formation of the response function for allC(eff )

ijkl . That is why we consider further the problem of unknown response functionapproximation with the following polynomial of (n − 1)th order [80]:

C(eff )ijkl = D(n−1)

ijkl bn−1 + D(n−2)

ijkl bn−2 + · · · + D(0)

ijkl (5.88)

it is solved using the least-squares method in both weighted and non-weightedschemes displayed in Chapter 1. Having determined the coefficients in Equation (5.88)it is possible to calculate up to nth-order ordinary derivatives of the homogenizedelasticity tensor with respect to the random input parameter b at the given mean b0as follows:


• first-order derivative

∂C(eff )ijkl

∂b= (n − 1) D(n−1)

ijkl bn−2 + (n − 2) D(n−2)

ijkl bn−3 + · · · + D(1)

ijkl (5.89)

• second-order derivative

∂2C(eff )ijkl

∂b2 = (n − 1) (n − 2) D(n−1)

ijkl bn−3 + (n − 2) (n − 3) D(n−2)

ijkl bn−4 + · · · + D(2)

ijkl

(5.90)• kth-order derivative

∂kC(eff )ijkl

∂bk=

k∏i=1

(n − i) D(n−i)ijkl bn−k +

k∏i=2

(n − i) D(n−2)

ijkl bn−(k+1) + · · · + D(n−k)ijkl . (5.91)

Providing that the response function of the EET has a single independent argumentbeing the input random variable of the problem, it is possible to employ the stochasticperturbation technique based on the Taylor representation to compute up to the mthcentral probabilistic moments μm(C(eff )

ijkl ).

5.4 Numerical Experiments

Example 5.1: Aging analysis of the interphase parametersThe main aim of the first example is to study the effective Young’s modulus ofthe interphase using some combination of the input probabilistic moments usingEquation (5.15) [78]. The output probabilistic moments from this approximationmay be used in further homogenization of the entire periodicity cell – the onlymodification in relation to the traditional RVE meshing would be an additionaldiscretization of the interphase between the fiber and the matrix. The followingcombination of the interface defects input data is used to check the time fluctuationsof the first two probabilistic moments for the homogenized Young’s modulus:(i) for the expected values E

[r0

a(ω)] = 0.004, E

[ra(ω)

] = 4.0E − 5/year, E[n0

a(ω)] =

100 and (ii) for all the coefficients of variation for the defect parameters takenequal to 0.2, so that Var

(n0

a(ω)) = (0.2 E

[n0

a(ω)])2, Var

(na(ω)

) = (0.2 E[na(ω)

])2,Var

(r0

a(ω)) = (0.2 E

[r0

a(ω)])2, Var

(ra(ω)

) = (0.2 E[ra(ω)

])2. The internal radius of theboundary, where these defects are located, is adopted as R = 0.4, and material data forthe original matrix are taken as E

[e0

a(ω)] = 200 × 1010, E[ea(ω)] = 0.015 × 1010/year,

whereas the variances are equal to Var(e0

a(ω)) = (0.1 × E

[e0

a(ω)])2 and Var(ea(ω))

= (0.2 × E[ea(ω)])2; we assume finally a lack of any cross-correlations between allthese variables.

The results of numerical analysis performed in the symbolic system Maple™are presented in Figure 5.6 (expected values) and Figure 5.7 (coefficient ofvariation) – they are determined as functions of time (given in years) and expectedvalue of the cavities nucleating at the considered interface. A computer script


02

4

E(n(v))68

102520

15t

105

08 × 1011

1 × 1012

1.2 × 1012

1.4 × 1012

E(e

)

1.6 × 1012

Figure 5.6 The expected values for time-dependent effective Young’s modulus.Reproduced with permission from Begell House

00

510

15

t

2025

0.140.160.180.20α(

e)

0.220.240.26

2 4 6

E(n(v))

8 10

Figure 5.7 The coefficients of variation for time-dependent effective Young’smodulus. Reproduced with permission from Begell House

written in the Maple™ internal language enables analytic determination of the firsttwo moments of the homogenized parameter random process given on the LHSof Equation (5.15) and may be expanded further toward determination of highermoments and characteristics also. These analytic derivations are possible mainlydue to the fact that the interphase area appearing in the RHS denominator is givenas deterministically dependent on time, so that the first two central moments areobtained as algebraic combinations of the first two moments for the defect variationsas well as the Young’s modulus of the virgin material (the truncation effect of theseprocesses is not included in this study).


Analyzing Figure 5.6 it is clear that (as one may expect) even for expectationequal to 0 and resulting from some initial non-zero expectations of these defects,the primary Young’s modulus is significantly reduced even at the beginning of theaging process – for t = 0. An increase of time for the same value E

[n0

a(ω)] = 0 leads to

some further small reduction of the interphase effective Young’s modulus, where thischange is almost linear. The general tendency demonstrated by this surface is thatthe larger the number of defects nucleated and the larger the time of the compositehistory, the smaller the expectation of the effective parameter being studied. It isseen that the total impact of the stochastic interface defects is very significant becausethe expectation of the interphase longitudinal modulus is reduced here three timeswithin the first 10 years of composite material service. Quite an inverse observationcan be made in case of the coefficient of variation for this homogenized parameter. Asone can predict, the larger the number of defects nucleated and the longer the time ofthe composite history, the higher the coefficient of variation for α

(e(eff )

a (ω; t))

. As iseasy to detect, the output coefficient of variation is not proportional and rather largerthan the input coefficient of variation, so that considering a decrease in expectedvalue for the same time moment of this process means the random dispersion of theYoung’s modulus in the interphase cannot be disregarded in further computationalanalysis of the interface phenomena modeled using the interphase concept.

Example 5.2: Material sensitivity analysis of the EETThe main purpose of this analysis is to determine the sensitivity coefficients forthe in-plane EET components. It is completed using the RFM displayed aboveand, for a comparison, the central difference method (CFD) implemented for thisproblem before [77]. The combined approach is based on the Maple™ computationsof the sensitivity gradients of the spatially averaged elasticity tensor componentswith respect to various design parameters. The spatially averaged stress tensorcomponents coming from the homogenization function are approximated using theresponse function approach and differentiation is provided symbolically in Maple™also. A full straightforward numerical technique is implemented in this system usingthe RFM technique applied to both spatially averaged elasticity and homogenizingstress tensor components. The separate solver for the RFM computations basedon the least-squares method is implemented in the Maple™ symbolic environmenttogether with normalization procedures for all sensitivity coefficients computed. Itis necessary to notice that the core of the homogenization process is carried out withuse of the FEM-based program MCCEFF, where plane strain elements are employed.

We consider a composite with quarter of the periodicity cell (Figure 5.8) – the fiberhas a round cross-section and the entire cell is square; the reinforcement ratio isequal to 50% of the RVE total area. Material characteristics for the computationalanalysis are as follows: e1 = 84.0 GPa, ν1 = 0.22 as well as e2 = 4.0 GPa, ν2 = 0.34; theFEM discretization using 62 four-noded elements with 76 nodal points for the planestrain analysis implemented in the system MCCEFF is presented below.

The normalized sensitivity coefficients for the EET are collected in Table 5.2, and forthe spatially averaged over the RVE elasticity tensor components (AET) in Table 5.3;the design variables are taken separately as Young’s moduli of the fiber and the


Figure 5.8 FEM discretization of the RVE quarter. Reproduced with permission fromElsevier

Table 5.2 Sensitivity coefficients for the effective elasticity tensor

h∂C(eff )

1111

∂h∂C(eff )

1122

∂h∂C(eff )

1212

∂h

e1 0.110404 0.025965 0.9548750.110171 0.029391 0.954875

ν1 0.011347 0.141546 −0.1780310.012469 0.138449 −0.178211

e2 0.880843 0.932087 0.0394810.882215 0.939607 0.040033(0.867) (0.926) (0.044)

ν2 1.081097 2.505468 −0.0086921.101308 2.556652 −0.009940(1.205) (2.814) (−0.011)

matrix together with the additional Poisson ratios. The first table is so arranged thatthe first value in both tables corresponds to the RFM method, the second to the CFDmethod (for the quarter of the cell), whereas the CFD results for the entire RVE areincluded in the brackets below. The next table collects the analytical results for theAET components, the RFM implemented and, finally, the CFD computational results.Let us note that all the results computed using the central finite difference scheme areobtained for the increment of the perturbed parameter equal to 1%, which followsthe conclusions from other numerical models.

The first and most important conclusion which can be drawn from these resultsis almost perfect agreement of various numerical approaches for the homogenizedcharacteristics’ response function approach, the finite difference technique in the firsttable as well as both of them with pure analytical differentiation implemented in thesystem Maple™ (provided in Table 5.3). In practical terms it means that for the needs


Table 5.3 Sensitivity coefficients for the averaged elasticitytensor

h∂C(eff )

1111

∂h∂C(eff )

1122

∂h∂C(eff )

1212

∂h

e1 0.940271 0.896041 0.9588660.941264 0.894144 0.9579080.941028 0.894261 0.957907

ν1 0.059728 0.103959 0.0411340.060429 0.105011 0.0416600.060439 0.105028 0.041667

e2 0.304023 1.438492 −0.1729100.304494 1.436554 −0.1731950.303889 1.434796 −0.173158

ν2 0.080999 0.298495 −0.0104370.080488 0.296135 −0.0102600.082141 0.302260 −0.010470

of the homogenization method, computational implementation of the sensitivityanalysis with all these methods is accurate and can be used equivalently dependingon the engineering software employed for a simulation. It should be clearly exposedhere that the usage of the RFM is independent of any further numerical parameterslike the increments in the CFD computations, no closed formulas are necessarylike in the analytical approach, and no technical interventions are really needed inany of the source codes for FEM homogenization-oriented programs. Therefore, theapparent efficiency of this technique compared with the remaining methodologiesgives a new modeling tool for sensitivity analysis as well as for further randommodeling, as will be seen in the next subsection.

In further numerical simulation it would be interesting to check the influence ofthe number of fibers of the RVE discretized on the values of these gradients, also incase of random distribution of these fibers in the computational domain. Accordingto previous studies in this area, it can be confirmed on the RVE quarter that themost important role in such a composite is played by the elastic characteristics of thematrix, with a smaller influence of the fiber Young’s modulus, whereas its Poissonratio can be practically neglected (during the optimization process). Some of theseparameters, namely the Poisson ratio, can result in negative sensitivity coefficients,so that its increase will decrease some homogenized elasticity tensor components.

Example 5.3: Computations of expected values of the EET usingpolynomial interpolationThe probabilistic RFM-related technique is implemented here in two quite sepa-rate ways. The first approach can be classified as a combined analytical–numericalmethodology, where the zeroth-order spatially averaged elasticity tensor togetherwith its higher-order derivatives with respect to the input random variable are


determined all using the system Maple™. The second part, consisting of the spa-tially averaged homogenizing stresses over the RVE, is partially computed in thefinite element-based system MCCEFF and then included in the Maple™ system toapproximate the response functions of the spatially averaged homogenizing stresstensor components w.r.t. random input quantities. The procedure has been pro-grammed with use of the homogenization-oriented computer program MCCEFFused previously for computations of the EET component probabilistic moments viathe Monte Carlo simulation technique.

Let us consider as an illustration a composite with quarter of the periodicitycell – the fiber has a round cross-section and the entire cell is square; the reinforcementratio is equal to 50% of the total area of the RVE as before. Elastic properties of the glassfiber and epoxy matrix are adopted as follows: the Young’s moduli expected valuesE[e1] = 84.0 GPa, E[e2] = 4.0 GPa, while the deterministic Poisson ratios are taken asequal to E[ν1] = 0.22 for the fiber and E[ν2] = 0.34 for the matrix (each parameter israndomized separately and then the expectations of the remaining properties becomesimply their deterministic values) [77].

The preliminary results of the computational analysis are presented in Figure 5.9as the response functions of all the homogenized elastic tensor components, wherethe Poisson ratio of the matrix is taken as the input random variable. This primarychoice was justified by the fact that all previous computational studies show thatthis particular composite is the most sensitive to this ratio’s variations. As is clearfrom all the graphs, the very smooth function is obtained at the expectation of thisparameter but at both edges of the computational domain the resulting polynomialrepresentation of the tenth order returns some fluctuations of the response function.It results here from the fact that simple polynomial interpolation is used to recoverall necessary response functions.

Next, Figures 5.10–5.13 show the expected values of the first component for the EET,E[C(eff )

1111

], as functions of the perturbation order of the method – from the second

until the tenth – as well as of the input coefficient of variation of the random input

15.6C111115.4

15.2

15

14.8

14.6

14.4

14.2

14

13.80 0.005 0.01 0.015 0.02 0.025 0.03

5.6

5.4

5.2

5

4.8

4.6

4.4

4.20 0.005 0.01 0.015 0.02 0.025 0.03

18.015

18.01

18.005

18

17.995

17.99

17.9850 0.005 0.01 0.015 0.02 0.025 0.03

v2 v2 v2

(eff )C1122

(eff )C1212

(eff )

Figure 5.9 The probabilistic response functions around the expectations for thehomogenized tensor components. Reproduced with permission from Elsevier


14.70408

14.70406

14.70404

14.70402

14.70400

14.70398

14.70396

14.70394

14.70392

14.70390

0.10 0.12 0.14 0.16 0.18 0.20α

2nd order analysis 6th order analysis8th order analysis10th order analysis

4th order analysis

EC

(eff

)ijk

l⎡ ⎢ ⎣

⎤ ⎥ ⎦

Figure 5.10 The expected values of C(eff )1111 for randomized Young’s modulus of the

fiber. Reproduced with permission from Elsevier

14.7048

14.7047

14.7046

14.7045

14.7044

14.7043

2nd order analysis6th order analysis8th order analysis10th order analysis4th order analysis

0.10 0.12 0.14 0.16 0.200.18α

Cijk

l

(eff

)E


matrix. Reproduced with permission from Elsevier


14.7042

14.7040

14.7038

14.7036

14.7034

14.7032

0.10 0.12 0.14 0.16 0.200.18α


Cijk

l

(eff

)E

Figure 5.12 The expected values of C(eff )1111 for randomized Poisson ratio of the fiber.

Reproduced with permission from Elsevier

14.706

14.704

14.702

14.700

14.698

14.696

14.694

14.692

14.690


0.10 0.12 0.14 0.16 0.200.18α

Cijk

l

(eff

)E


matrix. Reproduced with permission from Elsevier


parameter (each test contains only a single random input); the Young’s modulus ofthe fiber, next of the matrix; then the Poisson ratio of the fiber and finally the Poissonratio for the matrix. First, the most general observation is that even for the largestvalue of the input coefficient of variation the method implemented is convergent, sothat there is practically no difference between the expectations computed accordingto the eighth- and tenth-order perturbation formulations. Since large coefficients ofvariation of the input random variables are very rare in solid mechanics applications,this coefficient has been bounded here by the value 0.2. Incidentally, one can notice thedifferences between the second-order method known from the literature and higher-order results, even if this coefficient does not exceed the recommended 0.1 value [112].It is very characteristic that probabilistic convergence of all these expected values hasa different type depending strongly on the random input type but generally has adefinitely non-linear character (with respect to the coefficient of variance).

Example 5.4: Computations of probabilistic moments of the EETusing LSMNumerical analysis of homogenization of the periodic random fiber composite isprovided now using the least-squares approximation technique and performedusing the program MCCEFF and the symbolic computing environment of systemMaple™. The internal automatic generator of this program for the square RVE withcentrally located round fiber occupying 34% of the RVE is used to prepare the meshconsisting of 144 four-noded rectangular plane strain finite elements and 153 nodes(see Figure 5.14). Elastic parameters of the fiber material are taken as e1 = 84.0 GPa,ν1 = 0.22 and for the matrix e2 = 4.0 GPa, ν2 = 0.34 (its expected value). According tomany previous studies in that area, Poisson ratio of the matrix has been detectedfor this composite as the most influential parameter. A set of 11 trial equidistantpoints is used to make the simplest non-weighted least-squares approximation ofthe response function between homogenized tensor C(eff )

ijkl and ν2 [84]. The discretevalues of this input parameter are symmetrically chosen around its expected valueand the basic length of this subdivision equals 0.01 (about 3% of the basic value).Now we randomize this Poisson ratio of the matrix using coefficient of variation α

as the additional input parameter of this analysis, which is given each time on thehorizontal axis. The expected values and standard deviations for all the homogenizedtensor components are computed using the first few perturbation orders to verify theprobabilistic convergence of this method (see Figures 5.15–5.20, correspondingly).Since full analytical expansion is available here, the perturbation parameter may alsobe included in the numerical analysis, so that the separate results (cf. Figures 5.21and 5.22) demonstrate its influence on the output probabilistic moments. Theyshow coefficients of asymmetry and concentration of the single homogenized tensorcomponent to prove the applicability of the proposed method for computing ofhigher probabilistic moments.

As is clear from Figures 5.15–5.17, the second-order approach is acceptable forthe very small input coefficient of variation (according to the previous predictions),but for α > 0.10 higher-order terms really need to be included. Higher-order analysisleads immediately to the conclusion that for α < 0.15 a tenth-order approach has


Ω1

Ω2

Figure 5.14 Discretization of the RVE with a single fiber

0

1.09 × 1010

1.1 × 1010

1.11 × 1010

1.12 × 1010

1.13 × 1010

1.14 × 1010

0.05a

0.10 0.15

2nd order8th order

4th order 6th order10th order

C11

11

(eff

)E

Figure 5.15 Various order expectations of C(eff )1111, ν2 = ν2(ω). Reproduced with permis-

sion from Begell House


0

4.5 × 109

4.6 × 109

4.7 × 109

4.8 × 109

4.9 × 109

5. × 109

5.1 × 109

0.05a

0.10 0.15

2nd order8th order


C11

22

(eff

)E



0

1.24728 × 1010

1.2473 × 1010

1.24732 × 1010

1.24734 × 1010

1.24736 × 1010

1.24738 × 1010

1.2474 × 1010

2nd order8th order


0.05a

0.10 0.15

C12

12

(eff

)E




00

5 × 108

1 × 109

1.5 × 109

2 × 109

2.5 × 109

2nd order 4th order 6th order

0.05a

0.10 0.15

C11

11

(eff

)σ

⎛ ⎝⎞ ⎠

Figure 5.18 Various order standard deviations of C(eff )1111, ν2 = ν2(ω). Reproduced with

permission from Begell House

00

5 × 108

1 × 109

1.5 × 109

2 × 109

2.5 × 109


0.05a

0.10 0.15

C11

22

(eff

)σ

⎛ ⎝⎞ ⎠




00

1 × 107

2 × 107

3 × 107


0.05a

0.10 0.15

C12

12

(eff

)σ

⎛ ⎝⎞ ⎠



sufficient accuracy for the expected values of all components of the homogenizedtensor. Contrasting Figures 5.16 and 5.17 shows that tenth-order analysis not alwaysreally does result in the largest magnitude of expectations – sometimes probabilisticconvergence has an asymptotic character; the differences between the neighboringorder approximations decrease systematically anyway. As one could expect afterdeterministic sensitivity analysis, the largest differences are noticed in Figure 5.16because this particular component demonstrates the largest sensitivity coefficientsw.r.t. ν2. These coefficients are also computed in this approach and can be extractedfrom the first-order partial derivatives of C(eff )

ijkl . Let us underline that here, contrary tothe statistical estimation methods [71], the expected values demonstrate some smallvariability with respect to the input coefficient of variation, which is the inherentaspect of the entire stochastic perturbation technique.

It is obvious that the standard deviations show significantly slower probabilisticconvergence and have parameter variability significantly closer to the linear function.Now even for α > 0.10 the differences between the lowest orders of these deviationsare apparent and should not be neglected. Contrary to the expectations, now allnew orders increase systematically the final approximation results. Let us rememberthat real engineering materials do not exhibit such a large standard deviation as 0.2(except for geotechnical engineering), however, considering other input parameters,it shows possible range of the homogenized tensor random fluctuations.

Next, we examine in Figures 5.21 and 5.22 the coefficients of asymmetry andconcentration, parametrized also with the perturbation parameter ε = 0.9, . . . , 1.1.It confirms that the fourth-order characteristic is almost entirely influenced by this


00.90

0.95

1.00

1.05

e

1.10

0.1

0.2

0.3

0.4

0.050.10

a

0.15

C12

12

(eff

)β

⎛ ⎝⎞ ⎠

Figure 5.21 Coefficients of asymmetry for C(eff )1212, ν2 = ν2(ω). Reproduced with permis-


00.05

0.100.90

0.95e1.00

1.051.10

2

2.5

3

3.5

4

4.5

a

C12

12

(eff

)γ

⎛ ⎝⎞ ⎠

Figure 5.22 Coefficients of concentration for C(eff )1212, ν2 = ν2(ω). Reproduced with


parameter choice and completely insensitive to the input coefficient of variation.Computer analysis returns here γ = 3 for ε = 1, which is typical for the Gaussiandistribution. The third-order coefficient is dominated by α and less influenced byε, however for all combinations it appears to be positive and nowhere exactlyequal to 0 as for the Gaussian variables. The results obtained for the generalizedstochastic perturbation technique are contrasted in Figure 5.22 with the Monte Carlosimulation results obtained for 104 random trials marked here with the asterisks.All the simulations have been provided for α = 0.0, 0.01, . . . , 0.15 and ε = 1 and, asis apparent, the simulation results are each time slightly larger than the stochasticperturbation technique results. So, the output PDFs are recognized thanks to bothmethods as very close to the Gaussian distributions, which remains in perfectagreement with the previous Monte Carlo simulations.


Example 5.5: Comparison of three various probabilistichomogenization methodsWe analyze the same case study as before to study the basic four probabilisticcharacteristics of the homogenized tensor resulting from randomization of the fiberYoung’s modulus e1 assumed as the Gaussian quantity with given expected value; itscoefficient of variation belongs to the interval [0.0, 0.2]. They are collected in turn – theexpectations (Figure 5.23), the coefficient of variation (Figure 5.24), the coefficientof skewness (Figure 5.25), and the coefficient of concentration (Figure 5.26) – allas functions of α(e1), and computed according to three various methods: (i) semi-analytical approach, (ii) Monte Carlo simulations (M = 104), and (iii) generalizedstochastic perturbation technique. The results of the first two methods are shown ina discrete mode, whereas the last approach enables continuous approximation of thebasic random characteristics of C(eff)

ijkl . This is due to the fact that the computer algebrasystem uses automatic differentiation to process the response functions, which ofcourse preserves that continuity during the computations. The expected values ofC(eff)

1111 decrease together with an increase of α(e1) in the Monte Carlo simulation andremain the same in the other methods; this underestimation is nevertheless negligibleas comparable to a numerical error of the homogenization method itself. However,the difference between the maximum and minimum values of this expectation forthe simulation method is around one or two promiles, and so can be neglectedin practice. The resulting coefficients of variation (Figure 5.24) represent differenttrend – the higher the value of α(e1), the larger α

(C(eff)

1111

). Semi-analytical method

returns here the same values as both Monte Carlo simulation and the stochasticperturbation method. The homogenization may be treated as linear transform on the

0

1.0815 × 1010

1.082 × 1010

1.0825 × 1010

1.083 × 1010

1.0835 × 1010

1.084 × 1010

semi-analytical methodMonte-Carlo simulationgeneralized stochastic perturbation technique

0.05 0.10a

0.15 0.20

Pa

C11

11

(eff

)E

Figure 5.23 The expected values of the component C(eff )1111 with randomized e1.

Reproduced with permission from John Wiley & Sons Ltd.


0.020

0.015

0.010

0.005

00.10 0.15 0.200.050

α

Semi-analytical methodMonte-Carlo simulationStochastic perturbation method

a(C

1111

)(e

ff)

Figure 5.24 The coefficients of variation of the component C(eff )1111 with randomized e1.


basis of these results which perfectly agrees with other results obtained using MonteCarlo simulation for different volumetric ratios of the reinforcement.

The coefficient of skewness (Figure 5.25) is essentially different now for thesimulation-based method, where we obtain clearly negative values decreasing for anadditional increase of α(e1), while for the other methods β

(C(eff)

1111

)= 0. It is apparent

that this decrease is also somewhat non-linear, but it does not influence furthercomputations devoted to the entropy variations, since β

(C(eff)

1111

)never appears in the

additional equations. Even more apparent differences between statistical and non-statistical models are shown in Figure 5.26 – in case of the concentration coefficient.The non-statistical methods clearly return here γ

(C(eff)

1111

)= 3, which is typical for the

Gaussian distribution, whereas Monte Carlo simulation gives a convex non-linearfunction of this coefficient with respect to α(e1). Effectively, the simulation methodalmost doubles the coefficient γ for the input α taken from the interval [0.0, 0.2]; oncemore, however, these differences do not influence the final entropy computations.One may conclude at this point that non-statistical methods need to be improved,because the Monte Carlo simulation is theoretically a precise approximation ofthe probabilistic moments. We need to recall the asymptotic convergence of thistechnique, which may be efficient in determination of the first two moments forM = 104, but for higher moments essentially a larger number of random trials isrequired (in the crude version of this simulation). It is clear that the very longexpansion is not really necessary because of the perfect agreement with the semi-analytical method here. We need to emphasize that the dominating part of thenumerical error in this technique is in deterministic least-squares approximation of


0

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

semi-analytical methodMonte-Carlo simulation

0.05 0.10a

0.15 0.20

generalized stochastic perturbation technique

b(C

1111

)(e

ff)

Figure 5.25 The coefficients of skewness of the component C(eff )1111 with randomized e1.


03

3.5

4.5

4

5

5.5


0.05 0.10a

0.15 0.20

g(C

1111

)(e

ff)

Figure 5.26 The coefficients of concentration of the component C(eff )1111 with random-

ized e1. Reproduced with permission from John Wiley & Sons Ltd.


the analytical function between random input and output; the probabilistic part isprovided straightforwardly from the definitions.

Further, we discuss the entropy fluctuations (difference between output and inputprobabilistic entropies) of the components C(eff)

1111 (Figure 5.27), C(eff)1122 (Figure 5.28),

and C(eff)1212 (Figure 5.29) with respect to α(e1) and, separately, with respect to α(e2)

(Figures 5.30–5.32). They are calculated all from Equation (1.9) having a prioriconfirmed that the output PDF is Gaussian. As expected after previous probabilisticmodels, the entropy variations with respect to e1 have larger absolute values (butare all negative) than the entropies computed for the variable e2, because e1, as theinput random variable, is more influential than e2 in the homogenization problemand, at the same time, e1 itself is associated with significantly larger uncertainty thanC(eff)

ijkl . The largest absolute values within Figures 5.27–5.29 are noticed for C(eff)1122 , then

for C(eff)1111 , while the smallest are in case of C(eff)

1212 . The detailed comparison of threenumerical methods shows that the semi-analytical method always returns constantentropy loss and that the generalized stochastic perturbation method gives once morealmost identical values. The exception from this rule is in the case of C(eff)

1212 , where theperturbation method results increase slowly non-linearly together with an increaseof α(e1). The Monte Carlo simulation results in the largest absolute values each time,always increasing together with α(e1) (not really apparently for C(eff)

1212 ), but entropyvariations together with α(e1) are really in the range of the few percents, so that theyare not qualitatively different from the other methods. We need to remember thatthe Young’s modulus for most of the structural components exhibits a randomnessequivalent to α(e) ≤ 0.10, so that practically we obtain each time really almost the sameresults; the larger window for α(e) is taken to verify numerical efficiency here only.

Finally, we study the entropy fluctuations with respect to the variable e2 = e2(ω)and we have three different situations – all positive for C(eff)

1111 , almost equal to 0 (or

tending to 0) in the case of C(eff)1122 , and apparently negative – as shown in Figure 5.32 for

C(eff)1212 . The perfect agreement between the semi-analytical method and the generalized

stochastic perturbation technique remains the same and, further, they both returnconstant values, totally independent of α(e2). Contrary to the previous case ofe1 = e1(ω), the Monte Carlo simulations start each time from essentially differentvalues (for α(e2) close to 0) than these returned by the first two methods. All theymeet each other for about α(e2) = 0.15 and keep very close for larger values of the initialrandom dispersion of e2. Looking for the vertical window sizes of Figures 5.30–5.32,one notices that now the differences are given in tenths (contrary to Figures 5.27–5.29)and cannot be simply postponed; the main reason could be a numerical errorpresent in even a deterministic evaluation of the response functions of C(eff)

ijkl . Thehomogenizing stresses (first-order corrector in the EET formula) are constant withinthe fiber and exhibit large variations in the matrix area. So, a randomization of thematrix properties should increase those fluctuations a few times and decisively affectthe graphs analyzed here. The second reason, of almost the same importance, is thatthe sensitivity gradients of C(eff)

ijkl are essentially smaller with respect to e2 than to e1.


0

−4.34

−4.33

−4.32

−4.31

−4.30

−4.29

−4.28

−4.27

−4.26

semi-analytical methodMonte-Carlo simulationstochastic generalized perturbation technique

0.05 0.10a

0.15 0.20

Δh(C

1111

)(e

ff)

Figure 5.27 Entropy fluctuations of the component C(eff )1111 with randomized e1.


0

−5.57

−5.56

−5.54

−5.55

−5.53

−5.52

−5.51

−5.50

−5.49

−5.48


0.05 0.10a

0.15 0.20

Δh(C

1111

)(e

ff)




0

−1.990

−1.989

−1.988

−1.987


0.05 0.10a

0.15 0.20

Δh(C

1111

)(e

ff)



0

−1

−0.5

0

0.5


0.05 0.10a

0.15 0.20

Δh(C

1111

)(e

ff)




0

−2.5

−1.5

−2

−0.5

−1

0


0.05 0.10a

0.15 0.20

Δh(C

1111

)(e

ff)



0

−1.38

−1.36

−1.34

−1.32

−1.30

−1.28

−1.26


0.05 0.10a

0.15 0.20

Δh(C

1111

)(e

ff)



Example 5.6: Validation of the stochastic perturbation methodin homogenization of 1D compositeA computational illustration is provided for the two-component composite withmean values of elastic parameters e1 = 84.0 GPa, ν1 = 0.22 (for reinforcement) and forthe matrix taken as e2 = 4.0 GPa, ν2 = 0.34 (both having the same volume fractions).Now, the Young’s modulus of the stronger material is taken as the input random


4 × 10−11

3 × 10−11

2 × 10−11

1 × 10−11

05 × 1010 7 × 1010 9 × 1010

e1(w)

p(e 1

)

1.2 × 1011

Figure 5.33 Initial histogram and theoretical PDF of the reinforcement Young’smodulus. Reproduced with permission from IFTR PAS

variable in numerical simulation, where the standard deviation corresponds to its 10%random dispersion – the theoretical PDF and simulated counterpart (total number ofrandom trials equal to 105) are shown in Figure 5.33. The entire analysis is providedusing the symbolic environment of the system Maple™, where (i) direct numericalintegration following classical definitions of the probability theory, (ii) Monte Carlosimulation and statistical estimation, together with (iii) the response functions and thegeneralized stochastic perturbation technique equations are implemented [85]. Thefirst strategy needs some attention since general integration in the system Maple™ isunavailable in this case for the unbounded real domain, so that bounded numericalintegration is provided within the limits 0 and double expectation. These bounds aredetermined after some a posteriori error-based analysis, where minimization of thecomputational domain width is carried out with respect to some a priori given errorlevel (less than a single percent). The results for all homogenized tensor componentsare collected in Table 5.4 in the form of expected values (Pa), variances (Pa2), standarddeviations (Pa), fourth central probabilistic moments (Pa4), as well as coefficients ofvariation, skewness, and concentration (all dimensionless).

The general conclusion is that all the methods return almost the same results – theagreement in the expected values as well as in the coefficients of concentration isperfect, some extremely small differences appear for the second-order characteristics,while the largest differences are noticed in case of skewness. The agreement ofthese techniques mainly follows the fact that the effective tensor components comefrom the algebraic transformations during the smearing of the original materialswithin the RVE, but the homogenization function has an analytical form unlike inmost 2D problems, where some small differences caused by the SFEM itself areobserved. Quite naturally, the largest variations between the methods are noticed for


Table 5.4 A comparison of analytical (AM), statistical (MC), and perturbation-based(PM) probabilistic characteristics of the homogenized tensor

Probabilisticparameters

C1111 = C2222 C3333 C1133 =C3311 =C3322 =

C2233

C1122 = C2211 C1212 = C2121 C1313 =C3131 =C2323 =

C3232

E[X] 2.9929E10 (AM) 1.1564E10 6.9545E9 3.8391E10 3.5919E10 5.7197E92.9927E10 (MC) 1.1564E10 6.9545E9 3.8389E10 3.5916E10 5.7197E92.9929E10 (PM) 1.1564E10 6.9546E9 3.8391E10 3.5919E10 5.7198E9

Var(X) 5.5343E18 (AM) 5.2451E15 1.8970E15 8.7387E18 1.1852E19 6.1053E145.5260E18 (MC) 5.2453E15 1.8971E15 8.7414E18 1.1855E19 6.1054E145.5211E18 (PM) 5.0237E15 1.8170E15 8.7324E18 1.1852E19 5.8366E14

σ (X) 2.3504E9 (AM) 7.2423E7 4.3555E7 2.9561E9 3.4426E9 2.4709E72.3507E9 (MC) 7.2424E7 4.3556E7 2.9566E9 3.4431E9 2.4709E72.3497E9 (PM) 7.0878E7 4.2626E7 2.9551E9 3.4426E9 2.4159E7

μ4(X) 9.1634E37 (AM) 1.0343E32 1.3530E31 2.2936E38 4.2138E38 1.4152E309.1348E37 (MC) 1.0260E32 1.3421E31 2.2864E38 4.2008E38 1.4035E309.1451E37 (PM) 9.0917E31 1.1893E31 2.2878E38 4.2138E38 1.2350E30

α(X) 0.0785 (AM) 0.0063 0.0063 0.0770 0.0958 0.00430.0785 (MC) 0.0063 0.0063 0.0770 0.0959 0.00430.0785 (PM) 0.0061 0.0061 0.0770 0.0958 0.0042

β(X) −0.0066 (AM) −0.6053 −0.6053 −0.0085 9.0630E−9 −0.6193−0.0057 (MC) −0.6000 −0.6000 −0.0075 8.6663E−4 −0.6139−0.0094 (PM) −0.8259 −0.8259 −0.0120 2.1499E−8 −0.8416

κ(X) 3.0026 (AM) 3.7596 3.7596 3.0034 3.0000 3.79672.9915 (MC) 3.7292 3.7292 2.9921 2.9889 3.76523.0000 (PM) 3.6024 3.6024 3.0001 3.0000 3.6254

the homogenized tensor components indexed with ‘‘3,’’ since the random variableis smeared in this direction together with the deterministic quantity – the Young’smodulus of the weaker material. This tensor also shows some probabilistic dampingsince the output coefficients of variation are generally smaller than for the inputrandom variable. Also, generally one can conclude that quite independently from thenumerical strategy, the effective tensor appears to be a Gaussian random variable,since its components have higher-order characteristics typical for this distribution.The generalized stochastic perturbation method seems to be efficient but some furtherimplementation needs to be provided to eliminate small discrepancies in third-ordercharacteristic numerical determination.

Example 5.7: Semi-analytical probabilistic determination of thehomogenized tensorThe main issue in this example is to verify the influence of random dispersion in com-posite constituent’s material parameters on the first four probabilistic characteristicsof the homogenized tensor according to the semi-analytical method. It consists ofdetermination of the response functions for the homogenized tensor w.r.t. the given


1.105 × 1010

1.1 × 1010

1.095 × 1010

1.09 × 1010

1.085 × 1010

0.05

v1 v2

0.10 0.15

a(b)

0.20

e1 e2

E(C

1111

)(e

ff)

Figure 5.34 The expected values of C(eff )1111. Reproduced with permission from John

Wiley & Sons Ltd.

random input variable and further symbolic recovery of probabilistic moments fromEquations (1.1), (1.2) and (1.4). We assume that the resulting tensor’s components areall distributed according to the Gaussian PDF given in Equation (1.3). It is assumedthat the Young’s moduli as well as the Poisson ratios are independent Gaussianrandom variables with a priori given expectations; these parameters are randomizedindependently. Their coefficients of variation belong each time to the interval α ∈ [0.0,0. 25] – the lower bounds correspond to a deterministic problem, where an upperbound was taken before as the extreme random dispersion. Generally, we would liketo verify how random dispersion of the given input variable influences the dispersionof the homogenized tensor as well as its remaining basic probabilistic characteris-tics. The results of this analysis – the expectations E

[C(eff )

ijkl

], standard deviations

σ(

C(eff )ijkl

), asymmetry β

(C(eff )

ijkl

), and flatness γ

(C(eff )

ijkl

)coefficients – are contained in

Figures 5.34–5.37 for C(eff )1111, Figures 5.38–5.41 for C(eff )

1122, and Figures 5.42–5.45 forC(eff )

1212. The horizontal axes of these graphs include the input coefficient of variationof the following four separate input random variables: e1 – Young’s modulus of thefiber, e2 – Young’s modulus of the matrix, ν1 – Poisson ratio of the fiber, ν2 – Poissonratio of the matrix. The first observation, quite consistent with the previous analyses,is that the expected values of all effective tensor components (Figures 5.34, 5.38, and5.42) are quite insensitive to the input coefficient of variation. Let us mention thatthis is not the case for a generalized stochastic perturbation technique, where somevery small variations may be noticed each time.

The differences between the expected values obtained during randomization of e1,e2, ν1, and ν2 for C(eff )

1111, C(eff )1122, and C(eff )

1212 separately are almost negligible. One may


3 × 109

2.5 × 109

2 × 109

1.5 × 109

1 × 109

5 × 108

0.05

v1 v2

0.10 0.15 0.20

a(b)

e1 e2

s(C

1111

)(e

ff)

Figure 5.35 The standard deviations of C(eff )1111. Reproduced with permission from John

Wiley & Sons Ltd.

1 × 10−7

2 × 10−7

3 × 10−7

4 × 10−7

5 × 10−7

0

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

b(C

1111

)(e

ff)

Figure 5.36 The asymmetry coefficient of C(eff )1111. Reproduced with permission from

John Wiley & Sons Ltd.


3

3.000002

3.000004

3.000006

3.000008

3.000010

3.000012

3.000013

3.000016

0.05

v1 v2

0.10 0.15

a(b)

0.20

e1 e2

g(C

1111

)(e

ff)

Figure 5.37 The flatness coefficient of C(eff )1111. Reproduced with permission from John

Wiley & Sons Ltd.

4.5 × 109

4.55 × 109

4.6 × 109

4.65 × 109

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

E(C

1111

)(e

ff)


Wiley & Sons Ltd.


5 × 108

1 × 109

1.5 × 109

2.5 × 109

2 × 109

0.05

v1 v2

0.10 0.15

a(b)

0.20

e1 e2

s(C

1111

)(e

ff)


Wiley & Sons Ltd.

1 × 10−9

2 × 10−9

0

3 × 10−9

4 × 10−9

5 × 10−9

6 × 10−9

9 × 10−9

8 × 10−9

7 × 10−9

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

b(C

1111

)(e

ff)




3.000001

3.000002

3.000003

30.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

g(C

1111

)(e

ff)


Wiley & Sons Ltd.

1.2473 × 1010

1.2474 × 1010

1.2475 × 1010

1.2476 × 1010

1.2477 × 1010

1.2478 × 1010

1.2479 × 1010

1.248 × 1010

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

E(C

1111

)(e

ff)


Wiley & Sons Ltd.


5 × 108

1 × 109

2 × 109

1.5 × 109

2.5 × 109

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

s(C

1111

)(e

ff)


Wiley & Sons Ltd.

−2.5 × 10−6

−1.5 × 10−6

−2 × 10−6

−1 × 10−6

−5 × 10−7

0

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

b(C

1111

)(e

ff)




2.999991

2.999992

2.999993

2.999994

2.999995

2.999996

2.999997

2.999998

2.999999

3

0.05

e1 e2 v1 v2

0.10 0.15

a(b)

0.20

g(C

1111

)(e

ff)


Wiley & Sons Ltd.

conclude that E[C(eff )

1111

]and E

[C(eff )

1122

]reach a maximum when ν2 = ν2(ω) and

E[C(eff )

1212

]in the case of ν1 = ν1(ω). Further, analogously to the previous simulation-

and perturbation-based studies, all the interrelations between the output standarddeviations of C(eff )

ijkl and the input coefficients of variation remain strictly linear (cf.Figures 5.35, 5.39, and 5.43). It is apparent that the random dispersion in matrix elasticcharacteristics is decisive for C(eff )

1111 and C(eff )1122; the randomness in fiber properties is

of marginal importance. This changes in case of C(eff )1212, where uncertainty in the fiber

characteristics prevails but also a difference between the fibers’ Poisson ratio effectand matrix characteristic effect is not so apparent as before. The asymmetry andflatness coefficients of C(eff )

ijkl are analyzed and compared with each other to confirmthe thesis that these tensor components all have Gaussian distributions; this was firstdetected thanks to the Monte Carlo simulation-based homogenization method [71].Now the results are almost the same – all flatness coefficients are clearly equal to3; some extremely small numerical discrepancies are observed for input coefficientof variation equal to 0. The coefficients of asymmetry do not equal 0 perfectly forthe entire parametric variability of the input, but with relatively small error all theoutput distributions may be treated as fully symmetric. An interesting observationhere would be that the larger the input coefficient of variation, the smaller theabsolute value of the coefficient β. This asymptotic behavior is observed throughthe positive values for C(eff )

1111 and C(eff )1122, while for C(eff )

1212 it is through the very smallnegative numbers only.


The last group of numerical tests was completely devoted to determination ofthe computer time consumption during determination of the first four probabilisticmoments for the variable X, which is determined as the function of the polynomialexpansion order. This study has been performed on a 64-bit version of the Windows7 operating system installed on a computer with Intel Core i5 M430 2.27 GHz with3.86 GB RAM space available and using the Maple™, v. 13. The results given inFigure 5.46 show clearly that the expansion of up to 15th order has no significanttime cost, even for the highest-order moments. Most engineering cases are fromelastostatics or elastodynamics, where the interrelation between the random inputand the random structural response may be guessed according to the so-called‘‘engineering intuition.’’ This, however, may not be the case for elastoplasticity orgeneral non-linear problems, where higher-order expansions could be important.As one could expect, the higher the probabilistic moment computed, the largerthe computational time cost and this interrelation remains highly non-linear. Thisdifference is most apparent for the analysis of 25th order, where the expected value isstill determined within a few seconds, the variance needs about 103 seconds, while thethird- and fourth-order moment computations are so expensive (even more than 104

seconds) that they are omitted here for brevity of presentation (one must rememberthat the CPU time usage is not directly the same as the real computer time necessary tocomplete the specific task, especially for larger values from this graph). The graph iscompared with Figure 5.46 showing analogous CPU time cost, but for the lognormalvariable being an argument of the polynomial expansion. It is quite apparent now thatthe expected values and variances are calculated a few times faster for the lognormalpolynomials, whereas the third and higher moments, inversely, need incomparably

CP

U ti

me

[s]

6000

5000

4000

3000

2000

10

E [X] Var[X ] m3 m4

15 20

Expansion order

25 30

1000

Figure 5.46 Computer cost of the Gaussian polynomial expansion procedure.Reproduced with permission from John Wiley & Sons Ltd.


CP

U ti

me

[s]

4000

3000

2000

1000

10

E [X] Var[X ] m3(x) m4(x)

20Expansion order

30 40

Figure 5.47 Computer cost of the lognormal polynomial expansion procedure.Reproduced with permission from John Wiley & Sons Ltd.

more time-consuming calculations than is necessary for polynomial expansions ofGaussian variables. Since higher than fourth probabilistic moments may not havea direct application for interpretation of the structural response, this computationalcost seems to be complete for further structural numerical analyses. Independentlyfrom the expansion order chosen for the particular homogenization problem solution,the total computational time cost is always many times smaller than in the MonteCarlo simulation-based analysis.

Example 5.8: Stochastic fluctuations of the round fiber radiusin homogenizationAccording to the engineering evidence [128], degradation of the fibers sometimesresults in the uncertain loss of their diameters, so that the main aim of this compu-tational example is to verify quantitatively this effect [80]. Let us consider for thispurpose a composite with square periodicity cell and unitary dimensions discretizedas in Example 5.4; the fiber has a round cross-section and the expected value of itsinitial radius is taken as E[R] = 0.40. The elastic properties of the glass fiber and epoxymatrix are adopted as follows: Young’s moduli e1 = 84 GPa and e2 = 4.0 GPa, Poissonratios ν1 = 0.22 for the fiber and ν2 = 0.34 for the matrix; for brevity, we postpone in thisinitial study the aging rate of the Young’s moduli of the components. The first partof the computations consists of hybrid FEM–symbolic determination of the responsefunctions of the homogenized tensor components with respect to the fiber radiusranging from 0.40 to 0.35 to be used in further stochastic perturbation-based analysis.The results of this procedure are given in Figure 5.48 – a continuous line representsC(eff )

1111 (R), the dash-dot line is for the component C(eff )1122 (R), while the space-dash line


16

14

12

10

8

0.32 0.33 0.34 0.35 0.36 0.37 0.38

6

R

Cijk

l(e

ff)

Figure 5.48 The approximations of C(eff )1111

(R), C(eff )

1122

(R), C(eff )

1212

(R)

(in GPa).Reproduced with permission from Thomas Telford

corresponds to C(eff )1212 (R). It is apparent that the polynomial interpolations obtained

are smooth, continuous, and monotonous within the given range of variability (thevertical axes are given in gigapascals).

Having determined these functions it is possible to compute partial derivativesof the EET components with respect to the fiber radius and, next, to include themin the general relation for probabilistic moments of any order. Assuming finally thespecific form of the aging process we can analyze the expected values of the EETcomponents with respect to the additional time changes in expectations and standarddeviations of the fiber radius subject to stochastic decay [86]. The results of furthernumerical experiments are presented in Figure 5.49 for C(eff )

1111 (R) (expectations (a) andstandard deviations (b)), in Figure 5.50 for C(eff )

1122 (R) (same order as before), and forC(eff )

1212 (R) (expectations (a) and variances (b) in Figure 5.51). The variability range forthe expected values corresponds to its decrease from initial value 0.40 to final value0.35, while its coefficient of variation ranges from 0.0 to 0.25. Such a presentationassures the maximum generality of the analysis since the continuous spectrum ofthe aging process is collected here and, furthermore, the aging interdependence withtime remains implicit (to finally fit this time scale using a combination of the first twomoments into the given level of an environment with aggressive behavior).

All the probabilistic moments given in Figures 5.49–5.51 show the sametendencies – they decrease together with a decrease of both expectations andcoefficients of variation of the fiber radius (a maximum is obtained for the upperlimits of these variables). Analyzing these figures one may notice that having evena linear decay in expectations for the simplest aging rule, the additional decay instandard deviations (or variances) remains quadratic. Therefore, the grids given onthe surfaces presented are not adequate to the specific aging mechanisms. Further,a decrease of the expected values for the fiber from 0.40 to 0.34 corresponds (for most


310

210

110

100.35

0.37

0.39 0.0

0.1 a

0.2

E[R]

0.35

100

200

0

0.370.39

0.0

0.1a

0.2

E[R]

C11

11

(eff

)E

C11

11

(eff

)σ

⎛ ⎝⎞ ⎠

Figure 5.49 The expected values and standard deviations for C(eff )1111

(ω; t)

(in GPa andGPa2). Reproduced with permission from Thomas Telford

0.34

14

4

24

34

44

0.360.38 0.0

0.4

0.1a

0.2

E[R]

0.38

0.340

10

20

0.0

0.1a

0.2

E[R]

C11

22

(eff

)E

C11

22

(eff

)σ

⎛ ⎝⎞ ⎠

Figure 5.50 The expected values and standard deviations for C(eff )1122

(ω; t)

(in GPa andGPa2). Reproduced with permission from Thomas Telford

engineering fibers) to the period of time from 50 to 100 years rather than few-yearintervals only. The second important observation is that the expected values of allhomogenized tensor components decrease more than 10 times from their initialvalues in the case of deterministic analysis (when the parameter is close to 0), whilethe analogous decrease for the maximum value of α is not so significant. Comparingthe influence of the expected value of the fiber radius and its coefficient of variation(for the given variability intervals) it is clear that they are of almost the sameimportance in the case of final standard deviations (or variances); the coefficientof variation is essentially more important for the changes in the homogenized


320

420

220

120

200.34

0.380.36

0.4 0.0

0.1a

0.2

20 000

40 000

00.34

0.380.0

0.1a

0.2

e

C12

12

(eff

)E

C12

12

(eff

)σ

⎛ ⎝⎞ ⎠

E[R]

Figure 5.51 The expected values and variances for C(eff )1212

(ω; t)

(in GPa and GPa2 andsquare gigapascals). Reproduced with permission from Thomas Telford

tensor component expectations than the relevant expected value of the fiber radius.The next computational analyses will allow for a more precise comparison of theaging process effects applied to various composite parameters like geometrical and,separately, material characteristics of the composite constituents. The methodologyapplied guarantees that the analysis shown above may be extended to other types ofcomposites, different constitutive relations, as well as other stochastic aging rules.

Example 5.9: Homogenization for aging material characteristicsof the constituentsThe main issue in this study is to verify how much stochastic aging fluctuationsof the material properties affect probabilistic moments of the EET time variations[75]. The material parameters taken here are adequate for the glass fibers embeddedperiodically in the epoxy matrix, so that their initial values are E[e1] = 84.0 GPa,E[e2] = 4.0 GPa, ν1 = 0.22, ν2 = 0.34. The influence of the aging phenomenon issimulated numerically in two different scenarios: the first one assumes that thematrix only is subject to a decrease of its Young’s modulus according to the equatione2 = e20 − 0.02 GPa

year × t. The second scenario obeys the aging of both the matrix andthe fiber, where a decrease of the reinforcement Young’s modulus is described bya quite analogous equation like e1 = e10 − 0.02 GPa

year × t, where both e10 and e20 areequal to the expectations given above. The fiber has a round shape, is centrally locatedin the RVE, and occupies 50% of its area; both components are perfectly connected.The initial determination of the simulation input probabilistic moments together withthe post-processing stage consisting of the probabilistic moments of the homogenizedtensor time evolution are prepared using Maple™. The results of these computationswithin the first 20 years of the aging process are demonstrated in Figures 5.52–5.58(the time parameter is introduced on all horizontal axes, while the vertical axescorrespond to the probabilistic moment values). The first two figures show the


0

1.42 × 107

1.44 ×107

1.46 × 107

1.48 × 107

1.5 × 107

1.52 × 107

5 10t

15 20

C11

11

(eff

)E

Figure 5.52 Evolution of the expected values for the component C(eff )1111(ω; t).

Reproduced with permission from Begell House

0

0.101

0.102

0.103

0.104

0.105

0.106

0.107

0.108

0.109

0.110

5 10t

15 20

C11

11

(eff

)α

⎛ ⎝⎞ ⎠

Figure 5.53 Evolution of the coefficients of variation for the component C(eff )1111(ω; t).


expected values and the coefficients of variation for the first component of thehomogenized tensor, Figures 5.54 and 5.55 show the same parameters for C(eff )

1122(ω; t).The evolution of the PDFs for all components is given in Figures 5.56–5.58, whereparts (a) correspond to the first scenario and parts (b) contain the results for thesecond scenario with aging of both composite components.

As is apparent in the first four figures, the expected values of the first twohomogenized tensor components decrease almost linearly with time, where the firstscenario given by the higher lines results (lighter line) in a slower decrease of thehomogenized expectations. The coefficients of variation of these effective tensor


0

4.7 × 106

4.8 × 106

4.9 × 106

5 × 106

5 10t

15 20

C11

22

(eff

)E

Figure 5.54 Evolution of the expected values for the component C(eff )1122(ω; t).


0

0.102

0.104

0.106

0.108

0.110

5 10t

15 20

C11

22

(eff

)α

⎛ ⎝⎞ ⎠

Figure 5.55 Evolution of the coefficients of variation for the component C(eff )1122(ω; t).


components behave in a quite opposite way – they increase almost linearly with timein such a way that the continuous line in lighter color representing the first scenarioas for the mean values demonstrates a somewhat slower increase in comparison withthe darker diamonds reflecting the second aging scenario. Neglecting the scenariocharacter, we observe almost 10% overall decrease of the expectations for the first 20years of the aging stochastic process and a very similar increase in the ratio of thestandard deviation to these expectations.

The time fluctuations of the PDFs (histograms) for the homogenized tensor com-ponents are shown in Figures 5.56–5.58, where all possible values of the analyzedeffective tensor components are given on the horizontal axes. The essential differences


1.2 × 107

5 × 10−8

1 × 10−7

1.5 × 10−7

2.5 × 10−7

2 × 10−7

1.6 × 107 2 × 107 1.2 × 107

5 × 10−8

1 × 10−7

1.5 × 10−7

2.5 × 10−7

2 × 10−7

1.6 × 107

(a)

2 × 107

C11

11

(eff

)p

⎛ ⎝⎞ ⎠

C1111

(eff )

(b)

C1111

(eff )

C11

11

(eff

)p

⎛ ⎝⎞ ⎠

Figure 5.56 Evolution of the density function distributions for the component C(eff )1111(ω; t):

(a) first scenario; (b) second scenario. Reproduced with permission from Begell House

4 × 1063 × 106

1 × 10−7

0

2 × 10−7

3 × 10−7

4 × 10−7

5 × 10−7

6 × 10−7

7 × 10−7

6 × 1065 × 106 7 × 106 4 × 106

1 × 10−7

0

2 × 10−7

3 × 10−7

4 × 10−7

5 × 10−7

6 × 10−7

7 × 10−7

6 × 1065 × 106

(a)

7 × 106

C11

22

(eff

)p

⎛ ⎝⎞ ⎠

C11

22

(eff

)p

⎛ ⎝⎞ ⎠

C1122

(eff )

(b)

C1122

(eff )



between the first and second scenarios are apparent for the component C(eff )1212(ω; t)

only (the final plot is thinner (a) and thicker (b) in Figure 5.58). These outputs comefrom the systematic plots of the Gaussian curve corresponding to the homogenizedtensor for exploitation time varying from 0 to 20 years with a 2-year increase likebefore. The first bell-shaped curve is drawn and next, moving from the right to theleft of the horizontal axis values, we plot these curves after the next time incrementin Figures 5.56–5.58; this evolution in probability densities for various time moments


1.6 × 1071.2 × 107

5 × 10−8

1 × 10−7

1.5 × 10−7

2 × 10−7

2 × 107 2.4 × 107 1.6 × 1071.2 × 107

5 × 10−8

1 × 10−7

1.5 × 10−7

2 × 10−7

2 × 107

(a)

2.4 × 107

C12

12

(eff

)p

⎛ ⎝⎞ ⎠

C12

12

(eff

)p

⎛ ⎝⎞ ⎠

C1212

(eff )

(b)

C1212

(eff )



follows of course the additional decrease of the expectations noticed before. It isnecessary to point out that the aging laws for the fiber and the matrix have beenadopted almost in the same form, however in practice the fibers demonstrate sig-nificantly larger resistance to environmental influences, so that the aging must belimited more strongly than in the case of matrices. Therefore, without any doubt wecan conclude that for composites with a large contrast between Young’s moduli likethese taken here, the matrix aging is the most influential mechanism for compositestructural safety and reliability.

6Concluding Remarks

1. The stochastic perturbation method known mainly from its previous first- andsecond-order realizations has been presented here in its generalized versionassuming that practically any order of the common fundamental Taylor expan-sion is allowable. This was invented to significantly increase the overall accuracyof probabilistic analysis, especially for larger coefficients of variation of inputrandom variables. It was possible due to the response function idea, which isbased upon continuous and differentiable function (including higher derivativesalso with preferably recursive formulas) between desired structural responses,like a deflection at some point of the mesh or grid and the given random inputparameter. They are proposed here as the polynomials of random variables withconstant coefficients but may be harmonic or exponential also. This response func-tion is recovered using here simple polynomial interpolation or advanced leastsquares techniques made on several numerical solutions of the initial boundaryvalue problem, where the deterministic value of the randomized parameter fluc-tuates in a deterministic sense in the certain neighborhood of its mean value. So,we search for some kind of an analytical solution to this problem in a global or ina local sense using classical discrete numerical techniques. Usually the set of 10–11experimental input values for randomized parameter has symmetric distribution(most frequently uniform in this interval) with respect to its mean. The main sourceof computational error in this method is deterministic polynomial interpolationprocedure, since further probabilistic analysis is based on classical integral defini-tions, unlike the direct differentiation procedure applied before, where successivesolutions of increasing-order algebraic equations were responsible for this error.Of course, an error following the Taylor expansion with random parameters ofthe certain order remains common for both methods. Computational implemen-tation’s effort and overall time of numerical analysis of the technique proposed issignificantly lower than for traditional Monte Carlo simulations with comparableprecision, especially for expectations and standard deviations. It does not dependon the choice between the global response determination (a single-state function



vs. random input) or the local search (state functions at any point of the gridor mesh).

2. The generalized stochastic perturbation method is verified here as suitable forengineering and scientific problems with response functions being lower-orderpolynomials, while a little bit less efficient for higher-order responses and largeinput random deviations. Indeed, we have no evidence on highly nonlinearproblems with random parameters, particularly state-dependent (relying ona temperature or actual strain, for instance). Especially, linear response func-tions appearing frequently in elastic systems with randomized Young’s modulusalways return results extremely close to the Monte Carlo simulation method withlarge populations. However, we can expect some numerical instabilities for cou-pled analyses, even with a single random Gaussian parameter, where the finalefficiency may depend upon the partial differential equations system being solved.The perturbation method has been verified theoretically and computationally asprobabilistically convergent for the polynomial response functions of practicallyany degree. A choice of optimal degree in symbolic recovery of this response pro-ceeds as minimization within the set of satisfactory interpolations fitting well thediscrete data and showing no local oscillations. Further, stochastic perturbationtechnique should be of a single order higher than the degree of the polynomialresponse function. Of course, higher orders do not give any extra components ashigher-order partial derivatives for random inputs simply vanish; at the same timecomputational error of lower-order expansions cannot be neglected, especially forthird and fourth central moments. It must be underlined that the least-squaresmethod employed to recover the response functions in both global and localformulations in its non-weighted version shows decisively larger variations ofespecially for higher central probabilistic moments and additional coefficients.The best results are noticed for the weighted LSM, where the spatial distributionof the weights has a similar shape to the Dirac function – it shows a maximumequivalent to the input parameter mean value and a few times smaller weightselsewhere. We can apply a triangular distribution of these weights within the trialinterval of random parameter variability also but with worse efficiency.

3. The stochastic perturbation technique displayed in this book is applicable in itsRFM version with any commercial and academic software realizing some discretetechnique, except for these displayed in this book we can additionally modify thefinite volume method [8, 26, 28], the discrete element method, as well as a varietyof meshless techniques as well as even molecular dynamics matrix formulations.Further application in reliability analysis (in both FORM and SORM version)depends strongly on whether the structural output verified a posteriori is Gaussianor not; the final values of the reliability index may be computed straightforwardlyaccording to Eurocode 0 while our verification is positive. Recalculation of thisindex in SORM is not complicated as the limit function shape factor (its curvature) isobtained with additional symbolic double differentiation of the response functionwith respect to the random input parameter. Unfortunately, the DDM alternativeto the RFM needs each time a brand new computer implementation of the discrete

Concluding Remarks 299

computations code or direct access to the existing well-documented source code ofthe deterministic original program. The only exception is when symbolic computeralgebra is used – then we can make use of the entire implementation of the specificboundary value problems in this environment or build a macro-procedure thatreads system matrices from a deterministic procedure, calculates their derivativesfor the random input and, after some rearrangement, inserts them once more as aninput to the same program. This is possible since algebraic equations for successiveorders have essentially modified RHS vectors using lower-order equations, whilethe LHS system matrix remains always the same.

4. Computational experiments carried out using the SBEM and contained inChapter 3 show that this technique seems to be the most efficient for analysisof boundary value problems with random shape and/or boundary conditions[16], especially in the RFM version, where small modifications of the few nodesare quite straightforward and sufficient (that can be automatically provided bythe computer program itself). Of course, a similar situation will occur when onerandomizes the boundary conditions imposed on displacement and/or boundarytractions (stresses) [16]. The SFEM application in this context usually demandssome remeshing of the entire computational domain, except for the global meshconsisting of a few macro-elements, where geometrical uncertainty appearswithin the only one. The SFDM is also not attractive in this situation except foruncertainty in global dimensions, because randomization of boundary wavinessand/or some regular holes inside larger domains results each time in a changefrom regular to the irregular grid. Computational implementation of the DDMversion in this context is not automatic in a symbolic environment and mustproceed in a way typical for the SFEM. It would be indeed interesting to developstochastic perturbation-based implementation of the hybrid FBEM [172], whichcan link the pros of uncertainty analysis coming from both methods or somenumerical experiments carried out with complex fundamental solutions typicalfor electromagnetics [49].

5. It needs to be emphasized that the SFDM application presented in Chapter 4 tothe regular grids in Cartesian coordinates is relatively easy in both DDM andRFM versions, because algebraic difference relations may be symbolically par-tially differentiated with respect to the random input variable successively manytimes, and then, immediately solved. As the solution procedure for linear algebraicequation systems no restriction on the system matrix dimension and, further, thesesystems can be automatically generated, even large-scale systems may be modeledin this way. Usually, difference equations may be reorganized algebraically so asto have random input quantities on the RHS only, which speeds up the entirealgorithm – the only problem may occur when random fields are considered, likean elastic beam or plate with varying thickness, where some correcting factorsneed to be inserted even in the deterministic case. Symbolic implementations ofdifference equations rewritten for polar or cylindrical coordinates are more com-plicated during implementation of perturbation analysis. The other unresolvedproblems are – the SFDM for the irregular grids and 3D visualization to show


variations of the basic probabilistic characteristics coming from the SFDM studies(common for all the methods in this book).

6. The homogenization method displayed in Chapter 5 is a very specific area ofapplication of the perturbation technique, where deterministic geometrical per-turbation is used to introduce the effective elasticity tensor components and,separately, the stochastic perturbation technique to randomize this problem. Theircommon usage enables determination of this tensor basic probabilistic charac-teristics, where material or some geometrical composite parameters are random(not violating periodicity with respect to the single periodicity cell). We used herefour different probabilistic techniques – analytical, Monte Carlo simulation, andsemi-analytical contrasted with the stochastic perturbation approach – and theygive very similar results, where the Gaussian input in material parameters resultsin the Gaussian homogenized tensor. The homogenization problem solution wasan occasion to demonstrate symbolic computations of the probabilistic entropyvariations in boundary value problems with uncertain parameters. It is naturallyprovided for continuous variables, unlike classical applications in informationtheory, where discrete distributions were analyzed in this context, especiallyin computer science (data processing and cryptology applications). The chapterdevoted to this issue also contains computational analysis of material parameters’,fiber radius’ as well as interface defects’ stochastic aging according to the simplelinear decay with two Gaussian coefficients. As one could expect, numerical resultsconfirm that such a decay results in a linear decrease of the homogenized tensorexpectations as well as in almost parabolic increase of variances for this tensorcomponents. It could be extended further with exponential degradation of theYoung’s modulus as well as Poisson ratios and/or exponential increase of thestochastic interface defects appearing between the composite constituents.

7. One of the unexpected opportunities discovered during various implementationsof the perturbation-based SFEM is its semi-analytical formulation introducedin the last chapter – to homogenize periodic composite materials by only. Itfollows directly from the fact that after global or local response function recoverywe have polynomial analytical interrelations between random structural outputand input variable. It is not necessary at this point to employ a perturbationtechnique to provide a Taylor expansion with random parameters to derive finalprobabilistic moments and characteristics of an output. Thanks to the symbolicintegration procedures implemented in widely used mathematical software, wecan follow traditional integral definitions of these moments. Contrary to thegeneralized perturbation method, a semi-analytical approach cannot be usedautomatically – we need to detect the final distribution type first, after somecomputational experiments, which is indeed relatively easy for the Gaussianvariables only. Then we are able to carry out symbolic integration of the momentsusing the bell-shaped distribution curve. That is why this new computationaltool has been used with a deterministic homogenization procedure with the well-documented fact that the resulting tensor is also distributed according to theGaussian function [71].

Concluding Remarks 301

8. The generalized stochastic perturbation technique undoubtedly has a lot ofpotential applications and offers new research directions. Although we pre-sented a single-variable analysis shown consecutively in all numerical examples,a multivariable formulation with at least first-order cross-correlations has beenintroduced briefly in the first chapter. It is of course possible to extend thismethod toward many correlated random input variables, however, we need torealize that such an analysis needs precise numerical knowledge about any (order)cross-correlations inserted in the perturbation-based equations. Usually, expectedvalues, standard deviations, and sometimes skewness and kurtosis for uncertainparameters are experimentally established only, which in this case is not satisfac-tory at all. The only opportunity realistic from an engineering point of view is toanalyze uncorrelated random inputs via simple extension of the equations pro-vided here that sometimes (with reach and adequate data sets) may be expandedwith first-order covariances. Further, we can usually assume that the dead load ofthe structure, its main geometrical parameters, as well as the strength and physicalproperties are usually independent of each other, but some material parametersthemselves may and should have some statistical correlation. The second impor-tant perspective is an analysis of various probability density functions, like thelognormal, for instance, but also Fisher–Trippett, Weibull, or Poisson, all havingmany well-documented applications in science and engineering. It is clear thatthey will need full Taylor expansions with even and odd central probabilisticmoments and, further, symbolic integral derivation of higher central probabilisticmoments as their recursive formulas most frequently are not available – this willallow for the first time a validation of the generalized stochastic perturbationtechnique for non-symmetric probability density functions on a large scale. Thatis why the Readers will find necessary formulas in the Appendix to includederived probabilistic moments for the probability distribution of their choice intotheir own computer programs. Finally, considering large-scale [144] or multi-scale systems modeling and increasing computer power, one may invent someparallelization procedure for the perturbation-based programs analogous to thatworked out before in [110] for instance, and relevant to other stochastic methods.Undoubtedly, the RFM approach should be many times faster than the DDM andapplicable also for the commercial programs or the black box type academic soft-ware. Another advantage of multiscale analysis is the multi-resolution technique,where random variables and processes may be transferred between neighboringscales using some specially modified wavelet transforms [70, 161]. This can bethe basis for the reliability-based structural optimization [127] including certainmulti-resolutional features [18, 71].

Appendix

Distribution name ChiSquare

PDF

0 t < 0t

12 ν−1 e− 1

2 t

212 ν�

( 12 ν

) otherwise

E[b] ν

σ (b)√

2√

ν

α(b)

√2√ν

β(b)2√

2√ν

κ(b)3 (ν + 4)

ν− 3

µ1(b) 0µ2(b) = Var(b) 2 ν

µ3(b) 8 ν

µ4(b) 12 ν (ν + 4)

µ5(b) 32 ν (5 ν + 12)

µ6(b) 40 ν(52 ν + 3 ν2 + 96

)µ7(b) 96 ν

(480 + 35 ν2 + 308 ν

)µ8(b) 112 ν

(5760 + 4176 ν + 15 ν3 + 680 ν2

)µ9(b) 256 ν

(40320 + 315 ν3 + 6608 ν2 + 32112 ν

)µ10(b) 288 ν

(645120 + 134960 ν2 + 105 ν4 + 9800 ν3 + 554112 ν

)


304 Appendix

Distribution name Erlang

PDF

0 t < 0(tb

)c−1

e− tb

b� (c)otherwise

E[b] b cσ (b) b

√c

α(b)1√c

β(b)2√c

κ(b)3 (2 + c)

c− 3

µ1(b) 0µ2(b) = Var(b) b2cµ3(b) 2 b3 cµ4(b) 6 b4c + 3 b4 c2

µ5(b) 24 b5 c + 20 b5 c2

µ6(b) 120 b6 c + 130 b6 c2 + 15 b6 c3

µ7(b) 720 b7 c + 924 b7 c2 + 210 b7 c3

µ8(b) 5040 b8 c + 7308 b8 c2 + 2380 b8 c3 + 105 b8 c4

µ9(b) 64224 b9 c2 + 26432 b9 c3 + 2520 b9 c4 + 40320 b9 cµ10(b) 303660 b10 c3 + 44100 b10 c4 + 945 b10 c5 + 362880 b10 c + 623376 b10 c2

Distribution name Exponential

PDF

0 t < 0e− t

b

botherwise

E[b] bσ (b) bα(b) 1β(b) 2κ(b) 6µ1(b) 0µ2(b) = Var(b) b2

µ3(b) 2 b3

µ4(b) 9 b4

µ5(b) 44 b5

µ6(b) 265 b6

µ7(b) 1854 b7

µ8(b) 14833 b8

µ9(b) 133496 b9

µ10(b) 1334961 b10

Appendix 305

Distribution name Gamma

PDF

0 t < 0(tb

)c−1

e− tb

b � (c)otherwise

E[b] bc

σ (b) b√

c

α(b)1√c

β(b)2√c

κ(b)3 (2 + c)

c− 3

µ1(b) 0µ2(b) = Var(b) b2 cµ3(b) 2 b3 cµ4(b) 6 b4 c + 3 b4 c2

µ5(b) 24 b5 c + 20 b5 c2

µ6(b) 120 b6 c + 130 b6 c2 + 15 b5 c3

µ7(b) 720 b7 c + 924 b7 c2 + 210 b7 c3

µ8(b) 5040 b8 c + 7308 b8 c2 + 2380 b8 c3 + 105 b8 c4

µ9(b) 64224 b9 c2 + 26432 b9 c3 + 2520 b9 c4 + 40320 b9 cµ10(b) 303660 b10 c3 + 44100 b10 c4 + 945 b10 c5 + 362880 b10 c + 623376 b10 c2

Distribution name Poisson

PDF∞∑

k=0

λk e−λ Dirac (t − k)k!

E[b] λ

σ (b)√

λ

α(b)1√λ

β(b)1√λ

κ(b)1λ

µ1(b) 0µ2(b) = Var(b) λ

µ3(b) λ

µ4(b) 3 λ2 + λ

µ5(b) 10 λ2 + λ

µ6(b) 15 λ3 + 25 λ2 + λ

µ7(b) 105 λ3 + 56 λ2 + λ

306 Appendix

µ8(b) 105 λ4 + 490 λ3 + 119 λ2 + λ

µ9(b) 1260 λ4 + 1918 λ3 + 246 λ2 + λ

µ10(b) 945 λ5 + 9450 λ4 + 6825 λ3 + 501 λ2 + λ

Distribution name Inverse Gaussian

PDF

0 t < 012

√2

√λ

π t3e

− 12

λ (t − µ)2

µ2 t otherwise

E[b] µ

σ (b) µ

√µ

λ

α(b)√

µ

λ

β(b)3 µ2

λ2(µ

λ

)3/2

κ(b)3 (λ + 5 µ)

λ− 3

µ1(b) 0

µ2(b) = Var(b)µ3

λ

µ3(b)3 µ5

λ2

µ4(b)3 µ6 (λ + 5 µ)

λ3

µ5(b)15 µ8 (2 λ + 7 µ)

λ4

µ6(b)15 µ9

(21 λµ + 63 µ2 + λ2

)λ5

µ7(b)315 µ11

(33 µ2 + λ2 + 12 λµ

)λ6

µ8(b)105 µ12

(495 λµ2 + λ3 + 54 µλ2 + 1287 µ3

)λ7

µ9(b)945 µ14

(2145 µ3 + 858 λµ2 + 110 µλ2 + 4λ3

)λ8

µ10(b)945 µ15

(36465 µ4 + 15015 λµ3 + 110 λ3 µ + 2145 λ2 µ2 + λ4

)λ9

Appendix 307

Distribution name Laplace

PDF12

e− |−t+a|b

bE[b] aσ (b)

√2 b

α(b)

√2 ba

β(b) 0κ(b) 3µ1(b) 0µ2(b) = Var(b) 2 b2

µ3(b) 0µ4(b) 24 b4

µ5(b) 0µ6(b) 720 b6

µ7(b) 0µ8(b) 40320 b8

µ9(b) 0µ10(b) 3628800 b10

Distribution name Log-Normal

PDF

0, t < 0

√2

2t σ√

πexp

(− (ln(t) − µ)2

2σ 2

)otherwise

E[b] eµ+ 12 σ 2

σ (b)√

e2 µ+σ 2 (eσ 2 − 1

)

α(b)

√e2 µ+σ 2

(eσ2 − 1

)

eµ+

12

σ 2

β(b)e3 µ+ 9

2 σ 2 − 3 e3 µ+ 52 σ 2 + 2 e

32 σ 2+3 µ(

e2µ+σ 2 (eσ 2 − 1

))3/2

κ(b)e4 µ+8 σ 2 − 4 e4 µ+5 σ 2 + 6 e4 µ+3 σ 2 − 3 e2 σ 2+ 4 µ(

e2 µ+σ 2)2 (eσ 2 − 1

)2 − 3

µ1(b) 0µ2(b) = Var(b) e2 µ+2 σ 2 − e2 µ+σ 2

308 Appendix

µ3(b) e3 µ+ 92 σ 2 − 3 e3 µ+ 5

2 σ 2 + 2 e32 σ 2+ 3 µ

µ4(b) e4 µ+8 σ 2 − 4 e4 µ+5 σ 2 + 6 e4 µ+3 σ 2 − 3 e2 σ 2+ 4 µ

µ5(b) e5 µ+ 252 σ 2 − 5 e5 µ+ 17

2 σ 2 + 10 e5 µ+ 112 σ 2 − 10 e5 µ+ 7

2 σ 2 + 4 e52 σ 2+5 µ

µ6(b) e6µ+18σ 2 − 6e6µ+13σ 2 + 15e6µ+9σ 2 − 20e6µ+6σ 2 + 15e6µ+4σ 2 − 5e3σ 2+6µ

µ7(b) e7µ+ 492 σ 2 + 6e

72 σ 2+7µ − 7e7µ+ 37

2 σ 2 + 21e7µ+ 272 σ 2 − 35e7µ+ 19

2 σ 2

+35e7µ+ 132 σ 2 − 21e7µ+ 9

2 σ 2

µ8(b) e8µ+32σ 2 − 7e4σ 2+8µ − 8e8µ+25σ 2 + 28e8µ+19σ 2 − 56e8µ+14σ 2

+70e8µ+10σ 2 − 56e8µ+7σ 2 + 28e8µ+5σ 2

µ9(b) e9µ+ 812 σ 2 − 36e9µ+ 11

2 σ 2 − 9e9µ+ 652 σ 2 + 8e

92 σ 2+9µ + 36e9µ+ 51

2 σ 2

−84e9µ+ 392 σ 2 + 126e9µ+ 29

2 σ 2 − 126e9µ+ 212 σ 2 + 84e9µ+ 15

2 σ 2

µ10(b) e10 µ+50 σ 2 − 120 e10 µ+8 σ 2 − 10 e10 µ+41 σ 2 + 45 e10 µ+6 σ 2 + 45 e10 µ+33 σ 2

−9e5σ 2+10µ − 120e10µ+26σ 2 + 210e10µ+20σ 2 − 252e10µ+15σ 2 + 210e10µ+11σ 2

Distributionname

Maxwell

PDF

0 t < 0√

2√

1π

t2 e− 12

t2

α2

α3otherwise

E[b]2√

2 α√π

σ (b) α

√3 π − 8

π

α(b)14

√3 π − 8

π

√2√

π

β(b) − 2√

2 (5 π − 16)

π 3/2

(3 π − 8

π

)3/2

κ(b)16 π − 192 + 15 π 2

(3 π − 8)2 − 3

Appendix 309

µ1(b) 0

µ2(b) = Var(b)α2 (3 π − 8)

π

µ3(b) −2√

2 α3 (5 π − 16)

π 3/2

µ4(b)α4

(16 π − 192 + 15 π2

)π 2

µ5(b) −2 α5√

2(−256 + 51 π2 − 80 π

)π 5/2

µ6(b)α6

(648 π 2 − 2240 π − 2560 + 105 π 3

)π 3

µ7(b) −2√

2 α7(−3072 + 543 π3 + 168 π 2 − 4928 π

)π 7/2

µ8(b)α8

(−18816 π 2 − 28672 − 71680 π + 11232 π 3 + 945 π 4)

π 4

µ9(b) −2√

2 α9(−32768 + 6585 π 4 + 15264 π 3 − 72576 π 2 − 116736 π

)π 9/2

µ10(b)3α10

(−494592π 2 + 62200π 4 − 471040π − 21120π 3 − 98304 + 3465π 5)

π 5

Distributionname

Gaussian

PDF12

√2 e

− 12

(t − µ)2

σ 2

√π σ

E[b] µ

σ (b) σ

α(b)σ

µβ(b) 0κ(b) 0µ1(b) 0µ2(b) = Var(b) σ 2

µ3(b) 0µ4(b) 3 σ 4

µ5(b) 0

µ6(b) 15 σ 6

µ7(b) 0µ8(b) 105 σ 8

µ9(b) 0µ10(b) 945 σ 10

310 Appendix

Distributionname

Power

PDF

0 t < 0c tc − 1

bct ≤ b

0 otherwise

E[b]b c

c + 1

σ (b)b√

c2 + c

c + 1

α(b)

√c

2 + cc

β(b) − 2 c (c − 1)(c

2 + c

)3/2 (5 c + 6 + c2

)

κ(b)3 (2 + c)

(−c + 3 c2 + 2)

(c2 + 7 c + 12

)c

− 3

µ1(b) 0

µ2(b) = Var(b)b2 c

4 c2 + c3 + 5 c + 2

µ3(b) − 2 b3 c (c − 1)

8 c4 + 24 c3 + c5 + 34 c2 + 23 c + 6

µ4(b)3 b4 c

(−c + 3 c2 + 2)

13 c6 + 68 c5 + 186 c4 + c7 + 289 c3 + 257 c2 + 122 c + 24

µ5(b) − 4b5c(−6 + c + 11c3 − 6c2

)151c7 + 659c6 + 19c8 + 1745c5 + c9 + 2921c4 + 3109c3 + 2041c2 + 754c + 120

µ6(b)(5 b6 c

(19 c2 + 2c + 53c4 − 26c3 + 24

))/(7264c7 + 19090c6 + 1830c8

+ 290c9 + 26c10 + c11 + 34026c5 + 41330c4 + 33695c3 + 17644c2 + 5364c + 720)

µ7(b) − (6 b7 c

(−120 − 155 c4 + 85 c3 − 85 c2 − 34 c + 309 c5))

/(5040 + 237594 c7

+ 43308c + 447168c6 + 382381c3 + 576514c4 + 34c12 + 505c11 + 4332c10

+ 23934c9 + 90012c8 + c13 + 167140c2 + 602517c5)

µ8(b)(7 b8 c

) (720 + 625 c4 − 345 c3 + 324 c + 496c2 − 1059c5 + 2119c6

))/(40320

+ 8220681 c4 + 4606616c3 + 819c13 + 43c14 + c15 + 391824c + 6527781c7

+9149 c12 + 66962 c11 + 340074 c10 + 1239174c9 + 3305610c8 + 1731932c2

+10391143c5 + 9576511c6)

Appendix 311

µ9(b) − (8b9 c (−5040 − 2988c + 4858c5 + 1603c3 − 3556c2 + 16687c7 − 8344c6

−3220c4))/(362880 + 50736447c9 + 17726c14 + 198320710c6 + 164906282c7

+ 53c16 + c17 + 121784221c4 + 180333713c5 + 59170688c3 + 3929616c

+ 1092035c12 + 5242572c11 + 18758016c10 + 165823c13 + 1258c15

+104604811c8 + 19545948 c2)µ10(b) (9 b10 c (29996c2 + 148329c8 + 43274c6 + 40320 + 20321c4 + 28944c

− 7756c3 − 27664c5 − 74164c7))/(3628800 + 43287840c + 309500343c11

+ 239048136c2 + 3058337922c8 + 4010924343c7 + 4088662164c6 + 373389c15

+ 3093348c14 + 18913187c13 + 1822923673c9 + 810641924c3 + 1888265726c4

+ 3202134249c5 + 850285888c10 + 87346658c12 + 64c18 + c19 + 1851c17

+ 32094c16)

Distributionname

Rayleigh

PDF

0 t < 0

t e− 12

t2

b2

b2otherwise

E[b]12

b√

2√

π

σ (b)

√2 − 1

2π b

α(b)

√2 − 1

2 π√

2√

π

β(b)12

√2

√π (−3 + π)(

2 − 12 π

)3/2

κ(b) −−32 + 3 π 2

(−4 + π)2 − 3

µ1(b) 0µ2(b) = Var(b) − 1

2 b2 π + 2b2

µ3(b) − 32 b3

√2

√π + 1

2 b3√

2 π 3/2

µ4(b) 8b4 − 34

b4 π 2

µ5(b)12

b5√

2 π 5/2 − 252

b5√

2√

π + 52

b5√

2 π 3/2

µ6(b) −58

b6 π 3 − 152

b6 π 2 + 48 b6 + 15 b6 π

µ7(b) −2312

b7√

2√

π + 354

b7√

2 π 3/2 + 638

b7√

2π 5/2 + 38

b7√

2π 7/2

µ8(b) 252 b8 π + 384b8 − 14b8 π 3 − 716

b8 π 4 − 70 b8 π 2

µ9(b) − 2511

2b9

√2√

π − 63 b9√

2π 3/2 + 1

4b9

√2π 9/2 + 441

4b9

√2π 5/2 + 45

4b9

√2π 7/2

µ10(b) 3840 b10 − 1358

b10 π 4 + 3915 b10 π − 630 b10 π 2 − 5252

b10 π 3 − 932

b10 π 5

312 Appendix

Distributionname

Triangular

PDF

0 t < a2 (t − a)

(b − a) (c − a)t ≤ c

2 (b − t)(b − a) (−c + b)

t ≤ b

0 otherwise

E[b]13

a + 13

b + 13

c

σ (b)16

√2a2 + 2b2 + 2c2 − 2ab − 2ac − 2bc

α(b)

√2

(a2 + b2 + c2 − ab − bc − ac

)2 (a + b + c)

β(b)45

2b3 − 3b2 c − 3 ab2 + 12abc − 3a2 b − 3bc2 + 2a3 − 3ac2 + 2c3 − 3a2 c(2a2 + 2b2 + 2c2 − 2ab − 2ac − 2bc

)3/2

κ(b) −35

µ1(b) 0

µ2(b) =Var(b)

118

a2 + 118

b2 + 118

c2 − 118

ab − 118

ac − 118

bc

µ3(b)1

135b3 − 1

90b2 c − 1

90ab2 + 2

45abc − 1

90a2 b − 1

90bc2 + 1

135a3

+ 1135

c3 − 190

ac2 − 190

a2 c

µ4(b)1

135b4 − 2

135ab3 − 2

135b3 c + 1

45a2 b2 + 1

45b2 c2 − 2

135bc3 − 2

135a3 b

+ 145

a2 c2 − 2135

ac3 + 1135

c4 + 1135

a4 − 2135

a3 c

µ5(b)4

1701b5 − 10

1701ab4 − 10

1701b4 c + 4

1701a2 b3 + 4

1701b3 c2 + 32

1701ab3 c

− 8567

a2 b2 c − 8567

ab2 c2 + 41701

a3 b2 + 41701

b2 c3 + 321701

abc3 − 101701

bc4

− 101701

a4 b + 321701

a3 bc − 8567

a2 bc2 + 41701

a5 + 41701

a3 c2 − 101701

ac4

+ 41701

a2 c3 + 41701

c5 − 101701

a4 c

Appendix 313

µ6(b)31

20412c6 + 31

20412a6 − 7

972ab3 c2 − 7

972ab2 c3 + 7

972abc4 + 7

972a4 bc

− 7972

a3 b2 c − 7972

a3 bc2 − 7972

a2 b3 c + 7324

a2 b2 c2 − 7972

a2 bc3

− 316804

b5 c + 7972

ab4 c − 16320412

b3 c3 + 536804

b2 c4 − 316804

bc5 − 316804

a5 b

− 316804

a5 c + 536804

a4 b2 + 536804

a4 c2 − 16320412

a3 b3 − 16320412

a3 c3 + 536804

a2 b4

+ 536804

a2 c4 − 316804

ab5 − 316804

ac5 + 3120412

b6 + 536804

b4 c2

µ7(b)1

1458c7 − 7

2916ab6 − 7

2916a6 b + 1

324a5 b2 − 5

2916a4 b3 − 7

2916a6 c

+ 1324

a5 c2 − 52916

a4 c3 − 52916

a3 c4 − 52916

b4 a3 + 1324

b5 a2 + 1324

a2 c5

− 72916

ac6 − 52916

b4c3 − 52916

b3 c4 + 1324

b2 c5 − 72916

bc6 − 5486

b4 a2 c

+ 2243

a5 bc − 5486

a4 b2 c − 5486

a4 bc2 − 5486

a2c4 b + 10729

ab3 c3

− 5486

ab2 c4 + 2243

abc5 + 2243

ab5 c − 5486

ab4 c2 + 10729

a3 b3 c + 1324

b5 c2

+ 10729

a3 c3 b + 11458

a7 − 72916

b6c + 11458

b7

µ8(b) − 7910935

a2 b5 c + 15810935

a2 b4 c2 + 15832805

ab6 c − 7910935

ab5 c2

+ 7932805

b4 c3 a − 31632805

b3 c3 a2 + 7932805

b3 c4 a − 7910935

a5 b2 c + 7932805

a4 b3 c

+ 15810935

a4 b2 c2 − 31632805

a3 b3 c2 − 31632805

a3 b2 c3 − 7910935

a5 bc2 + 15832805

c6 ab

+ 15832805

a6 bc + 7932805

a4 bc3 + 7932805

a3 bc4 + 15810935

a2 b2 c4 + 1332805

a8

+ 1332805

b8 + 13932805

a4 b4 − 12732805

a3b5 + 10332805

a2 b6 − 5232805

ab7

− 12732805

b5 c3 + 13932805

b4 c4 − 12732805

b3 c5 + 13932805

c4 a4 − 12732805

c5 a3

+ 10332805

c6 a2 − 5232805

c7 a + 10332805

c6 b2 − 5232805

c7 b − 5232805

a7 b

+ 10332805

a6 b2 − 12732805

a5 b3 − 5232805

a7 c + 10332805

a6 c2 − 12732805

a5 c3

+ 10332805

b6 c2 + 1332805

c8 − 7910935

a2bc5 − 7910935

ab2 c5 + 7932805

a3 b4 c

− 5232805

b7 c

314 Appendix

µ9(b)224

1082565b9 − 112

120285ab8 − 112

120285ac8 + 218

120285b7 a2 + 218

120285c7 a2

− 112120285

a8 b − 112120285

a8 c − 721360855

a3 b6 − 721360855

a3 c6 + 97120285

a4 b5

+ 97120285

a4 c5 + 97120285

a5 b4 + 97120285

a5 c4 + 218120285

a7 b2 + 218120285

a7 c2

− 721360855

a6 b3 − 721360855

a6 c3 − 112120285

bc8 + 218120285

c7 b2 − 721360855

b3 c6

+ 97120285

b4 c5 + 97120285

b5 c4 + 1054120285

a3 b5 c − 16124057

ab6 c2 + 1054120285

a b5 c3

+ 9224057

ac7b − 16124057

ab2 c6 + 1054120285

a b3 c5 − 19424057

ab4 c4 − 13924057

a3 b4 c2

+ 111272171

a3 b3 c3 + 9224057

a7 bc − 16124057

a6 b2 c + 27840095

a2 b2 c5 − 16124057

a2 bc6

− 13924057

a2 b3 c4 − 16124057

a2 b6 c + 27840095

a2 b5 c2 − 13924057

a2 b4 c3

− 16124057

a6 bc2 + 27840095

a5 b2 c2 + 1054120285

a5 bc3 + 1054120285

a5 b3 c

− 19424057

a4 b4 c − 13924057

a4 b3 c2 − 13924057

a4 b2 c3 − 19424057

a4bc4

− 13924057

a3 b2 c4 + 1054120285

a3 bc5 − 112120285

b8c + 2241082565

a9 − 721360855

b6c3

+ 2241082565

c9 + 9224057

ab7 c + 218120285

b7 c2

µ10(b) − 85144342

b9 c + 114374

a8 bc + 6748114

a8 b2 − 4924057

a7 b3 + 3716038

a6 b4

− 10948114

a5 b5 + 3716038

a4b6 − 85144342

a9b − 4924057

a3b7 + 6748114

a8 c2

− 4924057

a7 c3 + 3716038

a6 c4 − 10948114

a5 c5 + 3716038

a4c6 − 85144342

a9c

− 4924057

a3 c7 − 4924057

b3c7 − 85144342

bc9 − 85144342

ac9 + 3716038

b6 c4

− 10948114

b5 c5 + 3716038

b4c6 + 6748114

a2b8 + 6748114

a2c8 + 6748114

b2 c8

− 85144342

ab9 + 17144342

a10 + 6748114

b8 c2 + 17144342

b10 − 112187

a7 bc2

+ 112187

a6 bc3 − 114374

a5 bc4 − 114374

a4 b5 c + 554374

a4 b4 c2 + 554374

a4 b2 c4

+ 112187

ac6 b3 − 112187

ac7 b2 + 114374

ac8 b + 222187

a2 c6 b2 − 112187

a2 c7 b

+ 114374

ab8 c − 112187

ab7 c2 + 112187

ab6 c3 − 114374

ab5 c4 − 114374

ab4 c5

Appendix 315

− 112187

a7 b2 c + 112187

a6b3c + 222187

a6 b2 c2 − 114374

a5 b4 c − 222187

a5 b2 c3

− 222187

a5 b3 c2 − 114374

a4 bc5 + 112187

a3 b6 c − 222187

a3 b5 c2 − 222187

a3 b2 c5

+ 112187

a3 c6 b − 112187

a2 b7 c + 222187

a2 b6 c2 − 222187

a2 b5 c3 + 554374

a2 b4 c4

− 222187

a2 b3 c5 − 4924057

b7 c3 + 17144342

c10

Distributionname

Uniform

PDF

0 t < a

1b − a

t < b

0 otherwise

E[b]12

a + 12

b

σ (b)16

√3 (b − a)

α(b)

√3 (b − a)

3 (a + b)

β(b) 0

κ(b) −65

µ1(b) 0

µ2(b) = Var(b)1

12a2 − 1

6a b + 1

12b2

µ3(b) 0

µ4(b)1

80a4 − 1

20a3 b + 3

40a2 b2 − 1

20a b3 + 1

80b4

µ5(b) 0

µ6(b)1

448a6 − 3

224a5 b + 15

448a4 b2 − 5

112a3 b3 + 15

448a2 b4 − 3

224ab5

+ 1448

b6

µ7(b) 0

µ8(b)1

2304a8 − 1

288a7b + 7

576a6 b2 − 7

288a5 b3 + 35

1152a4 b4 − 7

288a3 b5

+ 7576

a2 b6 − 1288

ab7 + 12304

b8

µ9(b) 0

316 Appendix

µ10(b)1

11264a10 − 5

5632a9b + 45

11264a8 b2 − 15

1408a7 b3 + 105

5632a6 b4

− 632816

a5 b5 + 1055632

a4 b6 − 151408

a3 b7 + 4511264

a2 b8 − 55632

a b9 + 111264

b10

Distributionname

Weibull

PDF

0 t < 0

c tc−1 e−

t

b

c

bcotherwise

E[b] b �

(c + 1

c

)

σ (b)

√�

(2 + c

c

)− �

(1 + c

c

)2

b

α(b)

√�

(2 + c

c

)− �

(c + 1

c

)2

�

(c + 1

c

)

β(b)−3 �

(c + 1

c

)�

(2 + c

c

)+ �

(3 + c

c

)+ 2 �

(c + 1

c

)3

(�

(2 + c

c

)− �

(c + 1

c

)2)3/2

κ(b)

6 �

(c + 1

c

)2

�

(2 + c

c

)− 4 �

(c + 1

c

)�

(3 + c

c

)+ �

(4 + c

c

)− 3 �

(c + 1

c

)4

(�

(2 + c

c

)− �

(c + 1

c

)2)2

−3µ1(b) 0

µ2(b)= Var(b)

b2

(�

(2 + c

c

)− �

(c + 1

c

)2)

µ3(b) b3

(−3 �

(c + 1

c

)�

(2 + c

c

)+ �

(3 + c

c

)+ 2 �

(c + 1

c

)3)

µ4(b) b4

(6 �

(c + 1

c

)2

�

(2 + c

c

)− 4 �

(c + 1

c

)�

(3 + c

c

)+ �

(4 + c

c

)

−3 �

(c + 1

c

)4)

Appendix 317

µ5(b) b5

(− 10 �

(c + 1

c

)3

�

(2 + c

c

)+ 10 �

(c + 1

c

)2

�

(3 + c

c

)

−5 �

(c + 1

c

)�

(4 + c

c

)+ �

(5 + c

c

)+ 4 �

(c + 1

c

)5)

µ6(b) −b6

(− 15 �

(c + 1

c

)4

�

(2 + c

c

)+ 20 �

(c + 1

c

)3

�

(3 + c

c

)

−15 �

(c + 1

c

)2

�

(4 + c

c

)+ 6 �

(c + 1

c

)�

(5 + c

c

)− �

(6 + c

c

)

+ 5 �

(c + 1

c

)6)

µ7(b) −b7

(21 �

(c + 1

c

)5

�

(2 + c

c

)− 35 �

(c + 1

c

)4

�

(3 + c

c

)

+ 35 �

(c + 1

c

)3

�

(4 + c

c

)− 21 �

(c + 1

c

)2

�

(5 + c

c

)+ 7 �

(c + 1

c

)�

(6 + c

c

)

−�

(7 + c

c

)− 6 �

(c + 1

c

)7)

µ8(b) −b8

(− 28 �

(c + 1

c

)6

�

(2 + c

c

)+ 56 �

(c + 1

c

)5

�

(3 + c

c

)

−70 �

(c + 1

c

)4

�

(4 + c

c

)+ 56 �

(c + 1

c

)3

�

(5 + c

c

)

−28 �

(c + 1

c

)2

�

(6 + c

c

)+ 8 �

(c + 1

c

)�

(7 + c

c

)− �

(8 + c

c

)

+ 7 �

(c + 1

c

)8)

µ9(b) −b9

(36 �

(c + 1

c

)7

�

(2 + c

c

)− 84 �

(c + 1

c

)6

�

(3 + c

c

)

+ 126 �

(c + 1

c

)5

�

(4 + c

c

)− 126 �

(c + 1

c

)4

�

(5 + c

c

)

+ 84 �

(c + 1

c

)3

�

(6 + c

c

)− 36 �

(c + 1

c

)2

�

(7 + c

c

)+ 9 �

(c + 1

c

)�

(8 + c

c

)

−�

(9 + c

c

)− 8 �

(c + 1

c

)9)

318 Appendix

µ10(b) −b10

(− 45 �

(c + 1

c

)8

�

(2 + c

c

)+ 120 �

(c + 1

c

)7

�

(3 + c

c

)

−210 �

(c + 1

c

)6

�

(4 + c

c

)+ 252 �

(c + 1

c

)5

�

(5 + c

c

)

−210 �

(c + 1

c

)4

�

(6 + c

c

)+ 120 �

(c + 1

c

)3

�

(7 + c

c

)

− 45 �

(c + 1

c

)2

�

(8 + c

c

)+ 10 �

(c + 1

c

)�

(9 + c

c

)− �

(10 + c

c

)

+ 9 �

(c + 1

c

)10)

References

[1] Ahammed, M. and Melchers, R.E. (2009) A convenient approach for estimating time-dependent structural reliability in the load space. Probabilistic Engineering Mechanics,24(3), 467–472.

[2] Babuska, I., Tempone, R., and Zouraris, G.E. (2005) Solving elliptic boundary value prob-lems with uncertain coefficients by the finite element method: the stochastic formulation.Computer Methods in Applied Mechanics and Engineering, 194, 1251–1294.

[3] Bathe, K.J. (1996) Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ.[4] Beer, G. (2001) Programming the Boundary Element Method. An Introduction for Engineers,

Wiley-VCH Verlag GmbH, New York.[5] Bendat, J.S. and Piersol, A.G. (1971) Random Data: Analysis and Measurement Procedures,

Wiley-VCH Verlag GmbH, New York.[6] Bensoussan, A., Lions, J.L., and Papanicolaou, G. (1978) Asymptotic Analysis for Periodic

Structures, North-Holland, Amsterdam.[7] Bhangale, R.K. and Ganesan, N. (2006) Static analysis of simply supported functionally

graded and layered magneto-electro-elastic plates. International Journal of Solids andStructures, 43, 3230–3253.

[8] Bijelonja, I., Demirdzic, I., and Muzaferija, S. (2006) A finite volume method for incom-pressible linear elasticity. Computer Methods in Applied Mechanics and Engineering, 195,6378–6390.

[9] Bjorck, A. (1996) Numerical Methods for Least Squares Problems, SIAM, Philadelphia.[10] Brebbia, C.A. and Dominguez, J. (1996) Boundary Elements. An Introductory Course,

Computational Mechanics Publications, Southampton.[11] Breitung, K., Casciati, F., and Faravelli, L. (1997) A random field description for stochastic

boundary elements. Engineering Analysis with Boundary Elements, 19(3), 223–229.[12] Burczynski, T. (1995) Boundary element method in stochastic shape design sensitivity

analysis and identification of uncertain elastic solids. Engineering Analysis with BoundaryElements, 15(2), 151–160.

[13] Carlsaw, H.S. and Jaeger, J.C. (1986) Conduction of Heat in Solids, Clarendon Press, Oxford.[14] Castaneda, P.P. and Suquet, P.M. (1998) Nonlinear composites, in Advances in Applied

Mechanics, Vol. 34 (eds E. Van Der Giessen and T.Y. Wu), Academic Press, New York.[15] Chaboche, J.L. (2008) A review of some plasticity and viscoplasticity constitutive theories.

International Journal of Plasticity, 24, 1642–1693.


320 References

[16] Chantasiriwan, S. (2005) Solutions of partial differential equations with random Dirichletboundary conditions by multiquadric collocation method. Engineering Analysis withBoundary Elements, 29(12), 1124–1129.

[17] Char, B.W. et al. (1992) First Leaves: A Tutorial Introduction to Maple V, Springer-Verlag,Berlin.

[18] Chellappa, S., Diaz, A.R., and Bendsoe, M. (2004) Layout optimization of structureswith finite-sized features using multiresolution analysis. Structural and MultidisciplinaryOptimization, 26, 77–91.

[19] Cheng, H.D. and Lafe, O.E. (1991) Boundary element solution for stochastic groundwaterflow: random boundary conditions and recharge. Water Resources, 27(2), 231–242.

[20] Chou, Y.F. (1991) Applications of Discrete Functional Analysis to the Finite Difference Method,Elsevier, Amsterdam.

[21] Christensen, R.M. (1979) Mechanics of Composite Materials, Wiley-VCH Verlag GmbH,New York.

[22] Clayton, J.D. and Chung, P.W. (2006) An atomistic-to-continuum framework for non-linear crystal mechanics based on asymptotic homogenization. Journal of Mechanics andPhysics of Solids, 54, 1604–1639.

[23] Collatz, L. (1966) The Numerical Treatment of Differential Equations, 3rd edn, Springer-Verlag, New York.

[24] Cruse, T.A. (1988) Boundary Elements Analysis in Computational Fracture Mechanics, KluwerAcademic Publishers, Dordrecht.

[25] Cruz, M.E. and Patera, A.T. (1995) A parallel Monte-Carlo finite element procedurefor the analysis of multicomponent media. International Journal for Numerical Methods inEngineering, 38, 1087–1121.

[26] Cueto-Felgueroso, L., Colominas, I., Nogueira, X., and Casteleiro, M. (2007) Finite volumesolvers and moving least-squares approximations for the compressible Navier–Stokesequations on unstructured grids. Computer Methods in Applied Mechanics and Engineering,196, 4712–4736.

[27] Dasgupta, G., Papusha, A.N., and Malsch, E. (2001) First order stochasticity in boundarygeometry: a computer algebra BE environment. Engineering Analysis with BoundaryElements, 25(9), 741–751.

[28] Durany, J., Pereira, J., and Varas, F. (2006) A cell-vertex finite volume method forthermohydrodynamic problems in lubrication theory. Computer Methods in AppliedMechanics and Engineering, 195, 5949–5961.

[29] El Harrouni, K., Ouazar, D., Wrobel, L.C., and Cheng, A.H.D. (1997) Uncertainty analysisof groundwater flow with DRBEM. Engineering Analysis with Boundary Elements, 19(3),217–221.

[30] Elishakoff, I. (1995) Random vibration of structures: a personal perspective. AppliedMechanics Review, 48(12), 809–825.

[31] Elishakoff, I. and Ren, Y.J. (1995) Some exact solutions for the bending of beamswith spatially stochastic stiffness. International Journal of Solids and Structures, 32(16),2315–2327.

[32] Emery, A.F. (2004) Solving stochastic heat transfer problems. Engineering Analysis withBoundary Elements, 28(3), 279–291.

[33] Falsone, G. (2005) An extension of the Kazakov relationship for non-Gaussian randomvariables and its use in the non-linear stochastic dynamics. Probabilistic EngineeringMechanics, 20, 45–56.

References 321

[34] Feller, W. (1965) An Introduction to Probability Theory and its Applications, Wiley-VCHVerlag GmbH, New York.

[35] Fish, J. and Ghouali, A. (2001) Multiscale analytical sensitivity analysis for compositematerials. International Journal for Numerical Methods in Engineering, 50, 1501–1520.

[36] Florence, C. and Sab, K. (2006) A rigorous homogenization method for the determinationof the overall ultimate strength of periodic discrete media and an application to generalhexagonal lattices of beams. European Journal of Mechanics A/Solids, 25, 72–97.

[37] Friebel, C., Doghri, I., and Legat, V. (2006) General mean-field homogenization schemesfor viscoelastic composites containing multiple phases of coated inclusions. InternationalJournal of Solids and Structures, 43, 2513–2541.

[38] Ghanem, R.G. and Spanos, P.D. (1991) Stochastic Finite Elements. A Spectral Approach,Springer-Verlag, Berlin.

[39] Ghanem, R.G. and Spanos, P.D. (1997) Spectral techniques for stochastic finite elements.Archives of Computer Methods in Engineering, 4(1), 63–100.

[40] Gitman, I.M., Gitman, M.B., and Askes, H. (2006) Quantification of stochastically stablerepresentative volumes for random heterogeneous media. Archives of Applied Mechanics,75, 79–92.

[41] Grigoriu, M. (1999) Stochastic mechanics. International Journal of Solids and Structures, 37,197–214.

[42] Guedri, M., Lima, A.M.G., Bouhaddi, N., and Rade, D.A. (2010) Robust design ofviscoelastic structures based on stochastic finite element models. Mechanical SystemsSignal Processing, 24(1), 59–77.

[43] Gutierrez, M.A. and de Borst, R. (1999) Numerical analysis of localization usinga viscoplastic regularization: influence of stochastic material defects. International Journalfor Numerical Methods in Engineering, 44, 1823–1841.

[44] Haftka, R.T. and Gurdal, Z. (1992) Elements of Structural Optimization, Kluwer AcademicPublishers, Dordrecht.

[45] Harth, T., Schwan, S., Lehn, J., and Kollmann, F.G. (2004) Identification of materialparameters for inelastic constitutive models: statistical analysis and design of experi-ments. International Journal of Plasticity, 20, 1403–1440.

[46] Hien, T.D. and Kleiber, M. (1997) Stochastic finite element modelling in linear transientheat transfer. Computer Methods in Applied Mechanics and Engineering, 144, 111–124.

[47] Hoffman, J.D. (1992) Numerical Methods for Engineers and Scientists, McGraw-Hill, NewYork.

[48] Honda, R. (2005) Stochastic BEM with spectral approach in elastostatic and elasto-dynamic problems with geometrical uncertainty. Engineering Analysis with BoundaryElements, 29(5), 415–427.

[49] Hromadka, T. and Lai, Ch. (1986) The Complex Variable Boundary Element Method inEngineering Analysis, Springer-Verlag, New York.

[50] Huang, B., Li, Q.S., Tuan, A.Y., and Zhu, H. (2009) Recursive approach for ran-dom response analysis using non-orthogonal polynomial expansions. ComputationalMechanics, 44, 309–320.

[51] Hughes, T.J.R. (2000) The Finite Element Method – Linear Static and Dynamic Finite ElementAnalysis, Dover Publications, New York.

[52] Hurtado, J.E. (2002) Reanalysis of linear and nonlinear structures using iterated Shankstransformation. Computer Methods in Applied Mechanics and Engineering, 191, 4215–4229.

[53] Hurtado, J.E. and Barbat, A.H. (1998) Monte-Carlo techniques in computational stocha-stic mechanics. Archives of Computer Methods in Engineering, 5(1), 3–30.

322 References

[54] Idiart, M.I., Moulinec, H., Ponte Castaneda, P., and Suquet, P. (2006) Macroscopicbehavior and field fluctuations in viscoplastic composites: second-order estimates versusfull-field simulations. Journal of Mechanics and Physics of Solids, 54, 1029–1063.

[55] Jefferson, G., Garmestani, H., Tannenbaum, R., Gokhale, A., and Tadd, E. (2005)Two-point probability distribution function analysis of co-polymer nano-composites.International Journal of Plasticity, 21, 185–198.

[56] Jeulin, D. and Ostoja-Starzewski, M. (eds) (2001) Mechanics of Random and MultiscaleStructures, CISM Courses and Lectures, Vol. 430, Springer-Verlag, NewYork.

[57] Jianting, Z. and Satish, M.G. (1997) A boundary element method for stochastic flowproblems in a semiconfined aquifier with random boundary conditions. EngineeringAnalysis with Boundary Elements, 19(3), 199–208.

[58] Kalamkarov, A.L., and Kolpakov, A.G. (1997) Analysis, Design and Optimization ofComposite Structures, Wiley.

[59] Kaminski, M. (1996) Homogenization in elastic random media. Computer AssistedMechanics and Engineering Science, 3(1), 9–21.

[60] Kaminski, M. (1999) Boundary element method homogenization of the periodic linearelastic composites. Engineering Analysis with Boundary Elements, 23(10), 815–823.

[61] Kaminski, M. (1999) Stochastic second order perturbation BEM formulation. EngineeringAnalysis with Boundary Elements, 2, 123–130.

[62] Kaminski, M. (2001) On stochastic microstructural finite elements. International Journalof Fracture, 110(1), L3–L8.

[63] Kaminski, M. (2001) Stochastic hybrid stress-based finite element method. Hybrid Methodsin Engineering, 3(1), 25–51.

[64] Kaminski, M. (2001) Stochastic perturbation approach to the stress-based finite elementmethod. International Journal of Solids and Structures, 38(21), 3831–3852.

[65] Kaminski, M. (2001) Stochastic problem of the viscous incompressible fluid flow withheat transfer. Zeitschrift fur Angewandte Mathematik und Mechanik, 81(12), 827–837.

[66] Kaminski, M. (2001) The stochastic second order perturbation technique in the finitedifference method. Communications in Numerical Methods in Engineering, 17(9), 613–622.

[67] Kaminski, M. (2002) On probabilistic viscous incompressible flow of some compositefluids. Computational Mechanics, 28(6), 505–515.

[68] Kaminski, M. (2002) Stochastic perturbation approach to engineering structures vibra-tions by the finite difference method. International Journal of Sound and Vibration, 251(4),651–670.

[69] Kaminski, M. (2002) Stochastic problem of fiber-reinforced composite with interfacedefects. Engineering Computations: International Journal of Computer-Aided Engineering,19(7), 854–868.

[70] Kaminski, M. (2004) Stochastic perturbation approach to wavelet-based multiresolu-tional analysis. Numerical Linear Algebra with Engineering Applications, 11(4), 355–370.

[71] Kaminski, M. (2005) Computational Mechanics of Composite Materials. Sensitivity, Random-ness and Multiscale Behaviour, Springer-Verlag, New York.

[72] Kaminski, M. (2005) Multiscale homogenization of n-component composites withsemi-elliptical stochastic interface defects. International Journal of Solids and Structures,42(11&12), 3751–3590.

[73] Kaminski, M. (2007) Application of the generalized perturbation-based stochasticboundary element method to elastostatics. Engineering Analysis with Boundary Elements,31(6), 514–527.

References 323

[74] Kaminski, M. (2007) Generalized perturbation-based stochastic finite element method inelastostatics. Computers and Structures, 85(10), 586–594.

[75] Kaminski, M. (2008) Homogenization of the fiber-reinforced composites under stocha-stic ageing processes. International Journal of Multiscale Computational Engineering, 6(4),361–370.

[76] Kaminski, M. (2009) A generalized stochastic finite difference method. Journal ofTheoretical and Applied Mechanics, 47(4), 957–975.

[77] Kaminski, M. (2009) Sensitivity and randomness in homogenization of periodic fiber-reinforced composites via the response function method. International Journal of Solidsand Structures, 46(3&4), 923–937.

[78] Kaminski, M. (2009) Stochastic interface defects in composite materials subjected tothe aging processes. International Journal of Multiscale Computational Engineering, 7(4),371–380.

[79] Kaminski, M. (2010) An introduction to the plasticity problems using the generalizedstochastic perturbation technique. Computational Mechanics, 45(4), 349–361.

[80] Kaminski, M. (2010) Effective elastic parameters of the periodic fibrous compositeswith random fibers according to the perturbation approach with response function.Communications in Numerical Methods in Engineering, 26(21), 1548–1558.

[81] Kaminski, M. (2010) Generalized stochastic perturbation technique in engineering com-putations. International Journal of Mathematical and Computer Modelling, 51(3&4), 272–285.

[82] Kaminski, M. (2010) Potential problems with random parameters by the generalizedperturbation-based stochastic finite element method. Computers and Structures, 88,437–445.

[83] Kaminski, M. (2010) Structural design sensitivity using boundary elements and polyno-mial response function. Structural and Multidisciplinary Optimization, 41, 107–116.

[84] Kaminski, M. (2011) Homogenization of fiber-reinforced composites with randomproperties using the least squares response function approach. International Journalof Multiscale Computational Engineering, 9(3), 257–270.

[85] Kaminski, M. (2011) Homogenization of fiber-reinforced composites with randomproperties using the weighted least squares response function approach. Archives ofMechanics, 63(5&6), 479–505.

[86] Kaminski, M. (2011) Homogenization of fibrous composites under stochastic ageing. ICEProceedings Engineering and Computational Mechanics, 164(3), 155–162.

[87] Kaminski, M. (2011) Least squares stochastic boundary element method. EngineeringAnalysis with Boundary Elements, 35(5), 776–784.

[88] Kaminski, M. (2011) On semi-analytical probabilistic finite element method forhomogenization of the periodic fiber-reinforced composites. International Journal forNumerical Methods in Engineering, 86(9), 1144–1162.

[89] Kaminski, M. (2011) Probabilistic analysis of transient problems by the least squaresstochastic perturbation-based finite element method. CMES: Computer Modelling inEngineering and Sciences, 80(1), 113–140.

[90] Kaminski, M. (2011) Steady state reaction–diffusion problems with random parametersusing the generalized stochastic finite difference method. Journal of Applied ComputerScience, 19(2), 31–45.

[91] Kaminski, M. (2011) Structural sensitivity analysis in nonlinear and transient problemsusing the local response function technique. Structural and Multidisciplinary Optimization,43(2), 261–274.

324 References

[92] Kaminski, M. (2012) Probabilistic entropy in homogenization of the periodic fiber-reinforced composites with random elastic parameters. International Journal for NumericalMethods in Engineering, 90(8), 939–954.

[93] Kaminski, M. (2012) Stochastic boundary element method analysis of the interfacedefects in composite materials. Composite Structures, 94, 394–402.

[94] Kaminski, M. and Carey, G.F. (2005) Stochastic perturbation-based finite elementapproach to fluid flow problems. International Journal of Numerical Methods in Heatand Fluid Flow, 15(7), 671–697.

[95] Kaminski, M. and Corigliano, A. (2012) Sensitivity, probabilistic and stochastic analysis ofthe thermo-piezoelectric phenomena in solids by the stochastic perturbation technique.Meccanica, 47, 877–891.

[96] Kaminski, M. and Hien, T.D. (1999) Stochastic finite element modeling of transient heattransfer in layered composites. International Communications in Heat and Mass Transfer,26(6), 791–800.

[97] Kaminski, M. and Hien, T.D. (1999) Stochastic structural defects in heat conductionproblems for composite materials. Archives of Mechanics, 51(3&4), 269–288.

[98] Kaminski, M. and Kleiber, M. (2000) Numerical homogenization of n-component com-posites including stochastic interface defects. International Journal for Numerical Methodsin Engineering, 47, 1001–1027.

[99] Kaminski, M. and Kleiber, M. (2000) Perturbation based stochastic finite element methodfor homogenization of two-phase elastic composites. Computers and Structures, 78(6),811–826.

[100] Kaminski, M. and Lesniak, M. (2012) Homogenization of metallic fiber-reinforcedcomposites under stochastic ageing. Composite Structures, 94, 386–393.

[101] Kaminski, M. and Schrefler, B.A. (2000) Probabilistic effective characteristics of cables forsuperconducting coils. Computer Methods in Applied Mechanics and Engineering, 188(1–3),1–16.

[102] Kaminski, M. and Spanos, P.D. (2001) Stochastic second order perturbation approach tofinite difference method in vibration analysis. International Journal of Applied Mechanicsand Engineering, 2(6), 369–393.

[103] Kaminski, M. and Swita, P. (2011) Generalized stochastic finite element method in elasticstability problems. Computers and Structures, 89, 1241–1252.

[104] Kaminski, M. and Szafran, J. (2009) Eigenvalue analysis for high telecommunicationtowers with lognormal stiffness by the response function method and SFEM. ComputerAssisted Mechanics and Engineering Science, 16, 279–290.

[105] Kaminski, M. and Szafran, J. (2009) Random eigenvibrations of elastic structures bythe response function method and the generalized stochastic perturbation technique.Archives of Civil and Mechanical Engineering, 9(4), 5–32.

[106] Kaminski, M. and Szafran, J. (2012) Stochastic analysis and reliability of the steeltelecommunication towers. CMES: Computer Modelling in Engineering and Sciences, 83(2),143–168.

[107] Kanit, T. (2006) Apparent and effective physical properties of heterogeneous materials:representativity of samples of two materials from food industry. Computer Methods inApplied Mechanics and Engineering, 195, 3960–3982.

[108] Karakostas, C.Z. and Manolis, G.D. (1998) A stochastic boundary solution applied togroundwater flow. Engineering Analysis with Boundary Elements, 21(1), 9–21.

References 325

[109] Karakostas, C.Z. and Manolis, G.D. (2002) Dynamic response of tunnels in stochasticsoils by the boundary element method. Engineering Analysis with Boundary Elements,26(8), 667–680.

[110] Keese, A. and Matthies, H.G. (2005) Hierarchical parallelisation for the solution ofstochastic finite element equations. Computers and Structures, 83, 1033–1047.

[111] Kleiber, M. (ed.) (1998) Handbook of Computational Solid Mechanics, Springer-Verlag,Berlin.

[112] Kleiber, M. and Hien, T.D. (1992) The Stochastic Finite Element Method, Wiley-VCH VerlagGmbH, New York.

[113] Krishnamoorthy, C.S. (1994) Finite Element Analysis, 2nd edn, McGraw-Hill, New York.[114] Kumar, A. and Chakraborty, D. (2009) Effective properties of thermo-electro-

mechanically coupled piezoelectric fiber reinforced composites. International Journal ofMaterials and Design, 30(4), 1216–1222.

[115] Kunz, K.S. and Luebbers, R.J. (1993) The Finite Difference Time Domain Method forElectromagnetics, CRC Press, Boca Raton, FL.

[116] Lepage, D. (2006) Stochastic finite element method for the modeling of thermoelasticdamping in micro-resonators. PhD dissertation, University of Liege.

[117] Lin, Y.K. and Cai, G.Q. (1995) Probabilistic Structural Dynamics, McGraw-Hill, New York.[118] Liu, W.K., Belytschko, T., and Mani, A. (1986) Random field finite elements. International

Journal for Numerical Methods in Engineering, 23, 1831–1845.[119] Luciano, R. and Willis, J.R. (2006) Hashin–Shtrikman based FE analysis of the elastic

behaviour of finite random composite bodies. International Journal of Fracture, 137,261–273.

[120] Lux, J., Ahmadi, A., Gobbe, C., and Delisee, C. (2006) Macroscopic thermal propertiesof real fibrous materials: volume averaging method and 3D image analysis. InternationalJournal of Heat and Mass Transfer, 49, 1958–1973.

[121] Ma, W. and Hu, G. (2006) Influence of fiber’s shape and size on overall elastoplasticproperty for micropolar composites. International Journal of Solids and Structures, 43,3025–3043.

[122] Manolis, G.D. and Karakostas, C.Z. (2003) A Green’s function method to SH-wavemotion in a random continuum. Engineering Analysis with Boundary Elements, 27(2),93–100.

[123] Manolis, G.D. and Shaw, R.P. (1995) Wave motions in stochastic heterogeneous media:a Green’s function approach. Engineering Analysis with Boundary Elements, 15(3), 225–234.

[124] Manolis, G.D. and Shaw, R.P. (1996) Boundary integral formulation for 2D and 3Dthermal problems exhibiting a linearly varying stochastic conductivity. ComputationalMechanics, 17, 406–417.

[125] Markovic, D. and Ibrahimbegovic, A. (2006) Complementary energy based FE modelingof coupled elasto-plastic and damage behavior for continuum microstructure computa-tions. Computer Methods in Applied Mechanics and Engineering, 195, 5077–5093.

[126] Matache, A.M., Babuska, I., and Schwab, C. (2000) Generalized p-FEM in homogeniza-tion. Numerische Mathematik, 86, 319–375.

[127] Maute, K., Weickum, G., and Eldred, M. (2009) A reduced-order stochastic finite elementapproach for design optimization under uncertainty. Structural Safety, 31(6), 450–459.

[128] Melchers, R.E. (2002) Structural Reliability Analysis and Prediction, John Wiley & Sons,New York.

[129] Milton, G. (2002) The Theory of Composites, Cambridge University Press, Cambridge.

326 References

[130] Minkowycz, W.J., Sparrow, E.H., Schneider, G.E., and Fletcher, R.H. (1988) Handbook ofNumerical Heat Transfer, Wiley-Interscience, New York.

[131] Mitchell, A.R. and Griffiths, D.F. (1980) The Finite Difference Method in Partial DifferentialEquations, Wiley-VCH Verlag GmbH, New York.

[132] Moller, B. and Beer, M. (2004) Fuzzy Randomness – Uncertainty in Civil Engineering andComputational Mechanics, Springer-Verlag, Berlin.

[133] Mura, T. (1982) Micromechanics of Defects in Solids, Sijthoff and Noordhoff, Netherlands.[134] Naji, A., Cheng, A.H.D., and Ouazar, D. (1999) BEM solutions of stochastic seawater

intrusion problems. Engineering Analysis with Boundary Elements, 23(7), 529–537.[135] Nayfeh, A. (1973) Perturbation Method, Wiley-VCH Verlag GmbH, New York.[136] Noor, A.K. and Shah, R.S. (1993) Effective thermoelastic and thermal properties of

unidirectional fiber-reinforced composites and their sensitivity coefficients. CompositeStructures, 26, 7–23.

[137] Nowacki, W. (1975) Thermodiffusion in solids. Journal of Theoretical and Applied Mechanics,13(2), 143–158.

[138] Oden, J.T. (1972) Finite Elements of Nonlinear Continua, McGraw-Hill, New York.[139] Oden, J.T., Babuska, I., Nobile, F., Feng, Y., and Tempone, R. (2005) Theory and

methodology for estimation and control of errors due to modeling, approximation anduncertainty. Computer Methods in Applied Mechanics and Engineering, 194, 195–204.

[140] Ostoja-Starzewski, M. (2005) Scale effects in plasticity of random media: status andchallenges. International Journal of Plasticity, 21, 1119–1160.

[141] Owen, D.R.J. and Hinton, E. (1980) Finite Elements in Plasticity – Theory and Practice,Pineridge Press, Swansea.

[142] Papadopoulos, V. and Papadrakakis, M. (2005) The effect of material and thicknessvariability on the buckling load of shells with random initial imperfections. ComputerMethods in Applied Mechanics and Engineering, 194, 1405–1426.

[143] Papadopoulos, V., Papadrakakis, M., and Deodatis, G. (2006) Analysis of mean and meansquare response of general linear stochastic finite element systems. Computer Methods inApplied Mechanics and Engineering, 195, 5454–5471.

[144] Papadrakakis, M. (ed.) (1993) Solving Large-Scale Problems in Mechanics. The Developmentand Application of Computational Solution Methods, Wiley-VCH Verlag GmbH, New York.

[145] Pepper, D.W. and Heinrich, J.C. (1992) The Finite Element Method, Computational andPhysical Processes in Mechanics and Thermal Sciences, Hemisphere, New York.

[146] Pietrzak, J., Rakowski, G. and Wrzesniowski, K. (1986), Matrix Analysis of Structures (inPolish), Polish Scientific Publishers, Warsaw.

[147] Rahman, S. (2008) A polynomial dimensional decomposition for stochastic computing.International Journal for Numerical Methods in Engineering, 76, 2091–2116.

[148] Rohan, E., Cimrman, R., and Lukes, V. (2006) Numerical modeling and homogenizedconstitutive law of large deforming fluid saturated heterogeneous solids. Computers andStructures, 84, 1095–1114.

[149] Roy, R.V. and Grilli, S.T. (1997) Probabilistic analysis of flow in random porous mediaby stochastic boundary elements. Engineering Analysis with Boundary Elements, 19(3),239–255.

[150] Samaey, G., Kevrikidis, I.G., and Roose, D. (2006) Patch dynamics with buffers forhomogenization problems. Journal of Computational Physics, 213, 264–287.

[151] Sanchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory, Lecture Notesin Physics, Vol. 127, Springer-Verlag, Berlin.

References 327

[152] Schafer, M. (2006) Computational Engineering – Introduction to Numerical Methods,Springer-Verlag, Berlin.

[153] Schenk, C.A., Pradlwarter, H.J., and Schueller, G.I. (2005) Non-stationary responseof large, non-linear finite element systems under stochastic loading. Computers andStructures, 83, 1086–1102.

[154] Schueller, G.I. (2007) On the treatment of uncertainties in structural mechanics andanalysis. Computers and Structures, 85, 235–243.

[155] Shannon, C.E. (1948) A mathematical theory of communication. The Bell System TechnicalJournal, 27, 379–423.

[156] Shannon, C.E. (1948) A mathematical theory of communication. The Bell System TechnicalJournal, 27, 623–656.

[157] Sigmund, O. (1994) Materials with prescribed constitutive parameters: an inversehomogenization problem. International Journal of Solids and Structures, 31, 2313–2329.

[158] Sobczyk, K. (1991) Stochastic Differential Equations. Applications in Physics, Engineering andMechanics, Kluwer Academic Publishers, Dordrecht.

[159] Soedel, W. (1993) Vibrations of Shells and Plates, Marcel Dekker, New York.[160] Song, Y.S. and Youn, J.R. (2006) Evaluation of effective thermal conductivity for carbon

nanotube/polymer composites using control volume finite element method. Carbon, 44,710–717.

[161] Spanos, P.D., Tezcan, J., and Tratskas, P. (2005) Stochastic processes evolutionaryspectrum estimation via harmonic wavelets. Computer Methods in Applied Mechanics andEngineering, 194, 1367–1383.

[162] Steeves, C.A. and Fleck, N.A. (2006) In-plane properties of composite laminates withthrough-thickness pin reinforcement. International Journal of Solids and Structures, 43,3197–3212.

[163] Taflove, A. (1998) Advances in Computational Electrodynamics. The Finite-Difference Time-Domain Method, Artech House, London.

[164] To, C.W.S. and Kiu, M.L. (1994) Random responses of discretized beams and plates bythe stochastic central difference method with time co-ordinate transformation. Computersand Structures, 53(3), 727–738.

[165] Vanmarcke, E. (1983) Random Fields. Analysis and Synthesis, MIT Press, Cambridge, MA.[166] Van Noortwijk, J.M. and Frangopol, D.M. (2004) Two probabilistic life-cycle maintenance

models for deteriorating civil infrastructures. Probabilistic Engineering Mechanics, 19,345–359.

[167] Wah, T. and Calcote, L.R. (1970) Structural Analysis by Finite Difference Calculus, VanNostrand Reinhold, New York.

[168] Wan, X. and Karniadakis, G.E. (2009) Solving elliptic problems with non-Gaussianspatially-dependent random coefficients. Computer Methods in Applied Mechanics andEngineering, 198(21–26), 1985–1995.

[169] Wang, H.F. and Anderson, M.P. (1995) Introduction to Groundwater Modeling: FiniteDifference and Finite Element Methods, Academic Press, New York.

[170] Wang, W.X., Luo, D., Takao, Y., and Kakimoto, K. (2006) New solution method forhomogenization analysis and its application to the prediction of macroscopic elasticconstants of materials with periodic microstructures. Computers and Structures, 84,991–1001.

[171] Witteveen, J.A.S. and Bijl, H. (2009) Effect of randomness on multifrequency aeroelasticresponses resolved by unsteady adaptive stochastic finite elements. Journal of Computa-tional Physics, 228(18), 7025–7045.

328 References

[172] Wolf, J.P. (2003) The Scaled Boundary Finite Element Method, Wiley-VCH Verlag GmbH,New York.

[173] Xiu, D. (2007) Efficient collocational approach for parametric uncertainty analysis.Communications in Computational Physics, 2(2), 293–309.

[174] Xiu, D. and Karniadakis, G.E. (2003) A new stochastic approach to transient heatconduction modeling with uncertainty. International Journal of Heat and Mass Transfer, 46,4681–4693.

[175] Xiu-Li, G. and Melchers, R.E. (2002) Effect of response surface parameter variation onstructural reliability estimates. Structural Safety, 23, 429–444.

[176] Xu, X.F. and Graham-Brady, L. (2005) A stochastic computational method for evaluationof global and local behavior of random elastic media. Computer Methods in AppliedMechanics and Engineering, 194, 4362–4385.

[177] Zaiser, M. and Aifantis, E.C. (2006) Randomness and slip avalanches in gradientplasticity. International Journal of Plasticity, 22, 1432–1455.

[178] Zhang, H., Zhang, S., Guo, X., and Bi, J. (2005) Multiple spatial and temporal scalesmethod for numerical simulation of non-classical heat conduction problems: one dimen-sional case. International Journal of Solids and Structures, 42, 877–899.

[179] Zhang, H.W., Zhang, S., Bi, J., and Schrefler, B.A. (2007) Thermo-mechanical analysisof periodic multiphase materials by a multiscale asymptotic homogenization approach.International Journal for Numerical Methods in Engineering, 69(1), 87–113.

[180] Zienkiewicz, O.C. and Taylor, R.L. (2005) Finite Element Method for Solid and StructuralMechanics, 6th edn, Elsevier, Amsterdam.

Index

Advection-diffusion equation 220Airy functions 71Ageing process 43, 48, 292

Beam on elastic foundation 197, 206Boundary Element Method (BEM) 156, 159

Central probabilistic moment 1, 8, 20, 21–22,38–40, 48, 166–167

Chi-Square distribution 303Coefficient of thermal expansion 62Coefficient of variation 1, 23, 38, 40Compliance tensor 74Complementary energy 74Correlation coefficient 11Corrosion rate 49Covariance matrix 11Crank-Nicholson scheme 101Critical load 97, 124Critical moment 210–211

Dielectric permittivity 62Difference operator 232, 226Dirac-type weights 31Direct Differentiation Method 87, 89, 90, 92,

99, 101, 106, 109, 161, 198, 215, 226, 232,240

Dissipation function 80Duhamel integral 99

Effective elasticity tensor 255, 257, 258Eigenfrequency 99, 139, 216–217


Elasticity tensor 72, 80, 244, 247–248,250–251, 255

Engesser theory 45Entropy 4, 276–278Erlang distribution 304Euler-Bernoulli beam 196, 214Euler force 46Expected value 1, 5, 19, 37, 39–40, 48, 165Exponential distribution 304

Fiber-reinforced composite 249, 260, 266, 272,280, 289, 292

Finite Element Method (FEM) 70–89, 256Finite Difference Method (FDM) 196–197,

221, 225, 232Flatness coefficient 1, 23, 26, 40Forced vibrations 98, 100, 135, 214Fourier law 80Fundamental solution 158

Gamma distribution 305Gamma function 39Gaussian distribution 1, 309

Hamilton principle 77, 81Heat capacity 62, 77, 102Heat conduction 142Heat conductivity 62, 77, 102Hilbert space 250Histogram 279, 177Homogenization function 253Homogenization theorem 251Hooke’s law 45–46, 62, 72, 76, 156

330 Index

Interface defect 245, 186, 188Inverse Gaussian distribution 306

Johnson-Ostenfeld theory 46

Kirchhoff modulus 71, 209Kinetic energy 77Kurtosis 1

Lagrangian 81Laplace distribution 307Laplace equation 72Layered composite 182, 254, 278Least Squares Method (LSM) 26, 28, 266Log-normal distribution 39, 307Love’s equation 239Lyapunov boundary 157

Mass matrix 97Maxwell distribution 308Modal analysis 98Monte-Carlo simulation 15–18, 177, 272

Navier-Stokes equations 83, 106

Periodicity cell 244, 249, 261, 267Perturbation parameter 5, 112, 122,

271Perturbation order 5, 18–26, 252Piezoelectric constant 62Piola-Kirchhoff tensor 75Poisson distribution 305Poisson equation 225–226Polynomial interpolation 262, 30Potential energy 76–77, 96Potential function 69, 110, 114Power distribution 310

Probabilistic convergence 18–26Probability density function 1, 303–316

Rayleigh distribution 311Residual 26–27Response Function Method 88, 91, 103, 164,

226, 230

Scale parameter 249Semi-analytical probabilistic method 37, 280,

272Sensitivity analysis 260–262Shape function, BEM 158Shape function, FEM 70, 87Skewness 1, 38, 23, 40Standard deviation 1, 38Statistical estimators 14Stiffness matrix 90, 94–95, 97Stochastic Boundary Element Method 155Stochastic Finite Difference Method 195Stochastic Finite Element Method 87

Taylor series 5Tetmajer-Jasinski theory 46Thermo-piezo-electric pulsation 62Time series 40, 48Transient heat transfer 77, 100, 147, 220–221Triangular distribution 312Triangular weights 31

Uniform distribution 315

Variance 1, 6, 19–20, 37, 39–40, 48, 166Vlasov equation 209

Weibull distribution 316Weighted Least Squares Method (WLSM) 29

Documents

The Stochastic Perturbation Method for Computational Mechanics