Reliability Engineering and System SafetyReliability Engineering and System Safety 149 (2016) 204–217 dynamic responses of medium/large ﬁnite element models. Spe-ciﬁcally, a

Reliability Engineering and System Safety 149 (2016) 204–217

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Reliability Engineering and System Safety

http://d0951-83

n CorrE-m

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Model-reduction techniques for reliability-based design problemsof complex structural systems

H.A. Jensen a,n, A. Muñoz a, C. Papadimitriou b, E. Millas a

a Department of Civil Engineering, Santa Maria University, Valparaiso, Chileb Department of Mechanical Engineering, University of Thessaly, GR-38334 Volos, Greece

a r t i c l e i n f o

Article history:Received 2 July 2015Received in revised form29 December 2015Accepted 1 January 2016Available online 11 January 2016

Keywords:Advanced simulation techniquesFirst excursion probabilityModel reduction techniquesReliability analysis in high dimensionReliability-based design

x.doi.org/10.1016/j.ress.2016.01.00320/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail address: [email protected] (H.A. Jensen

a b s t r a c t

This work presents a strategy for dealing with reliability-based design problems of a class of linear andnonlinear finite element models under stochastic excitation. In general, the solution of this class ofproblems is computationally very demanding due to the large number of finite element model analysesrequired during the design process. A model reduction technique combined with an appropriate opti-mization scheme is proposed to carry out the design process efficiently in a reduced space of generalizedcoordinates. In particular, a method based on component mode synthesis is implemented to define areduced-order model for the structural system. The re-analyses of the component or substructure modesas well as the re-assembling of the reduced-order system matrices due to changes in the values of thedesign variables are avoided. The effectiveness of the proposed model reduction technique in the contextof reliability-based design problems is demonstrated by two numerical examples.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Structural design via deterministic mathematical programmingtechniques has been widely accepted as a viable tool for engi-neering design [1]. However, in most structural engineeringapplications response predictions are based on models involvinguncertain parameters. This is due to a lack of information aboutthe value of system parameters external to the structure such asenvironmental loads or internal such as system behavior. Underuncertain conditions the field of reliability-based optimizationprovides a realistic and rational framework for structural optimi-zation which explicitly accounts for the uncertainties [2–4]. In thepresent work, structural design problems involving finite elementmodels under stochastic loading are considered. The design pro-blem is formulated as the minimization of an objective functionsubject to multiple design requirements including standard andreliability constraints. The probability that any response of interestexceeds in magnitude some specified threshold level within agiven time duration is used to characterize the system reliability.This probability is commonly known as the first excursion prob-ability [5]. The corresponding reliability problem is expressed interms of a multidimensional probability integral involving a largenumber of uncertain parameters. Reliability-based design for-mulations require advanced and efficient tools for structural

).

modeling, reliability analysis and mathematical programming.Modeling and analysis techniques of structural systems are wellestablished and sufficiently well documented in the literature [6].On the other hand, several tools for assessing structural reliabilityhave lately experienced a substantial development providingsolution of involved systems [7–9]. In the field of reliability-basedoptimization of stochastic dynamical systems several procedureshave been recently developed allowing the solution of problemsdealing with finite element models of relatively small number ofdegrees of freedom [10–14]. However, the application ofreliability-based optimization to stochastic dynamical systemsinvolving medium/large finite element models remains somewhatlimited. In fact, the solution of reliability-based design problems ofstochastic finite element models requires a large number of finiteelement analyses to be perform during the design process. Theseanalyses correspond to finite element re-analyses over the designspace (required by the optimizer), and system responses over theuncertain parameter space (required by the simulation techniquefor reliability estimation). Consequently, the computationaldemands depend highly on the number of finite element analysesand the time taken for performing an individual finite elementanalysis. Thus, the computational demands in solving reliability-based design problems may be large or even excessive.

In this context, it is the main objective of this work to presenta framework for integrating a model reduction technique intothe reliability-based design formulation of a class of stochasticlinear and nonlinear finite element models. The goal is to reducethe time consuming operations involved in the re-analyses and

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H.A. Jensen et al. / Reliability Engineering and System Safety 149 (2016) 204–217 205

dynamic responses of medium/large finite element models. Spe-cifically, a model reduction technique based on substructure cou-pling for dynamic analysis is considered in the present imple-mentation [15]. The proposed method corresponds to a general-ization of substructure coupling applicable to systems with loca-lized nonlinearities. The technique includes dividing the linearcomponents of the structural system into a number of sub-structures obtaining reduced-order models of the substructures,and then assembling a reduced-order model for the entire struc-ture. In summary, the novel aspect of this contribution involves astrategy for integrating a model reduction technique into thereliability-based design formulation of medium/large finite ele-ment models under stochastic excitation. This represents anadditional area of application of substructure coupling which hasbeen already used for uncertainty management in structuraldynamics with applications in areas such as uncertainty analysis,finite element model updating, and reliability sensitivity analysis[16–19]. The organization of this work is as follows. The for-mulation of the reliability-based design problem is presented inSection 2. Next, the characterization of the structural systems ofinterest is considered in Section 3. Implementation issues such asreliability estimation, optimization strategy and model reductionare discussed in Section 4. The mathematical background of themodel reduction technique is outlined in Section 5. The integrationof the model reduction technique into the design process is dis-cussed in Section 6. The effectiveness of the proposed strategy isdemonstrated in Section 7 by the reliability-based design of twostructural systems. The paper closes with some conclusions andfinal remarks.

2. Problem formulation

The reliability-based design problem is characterized in termsof the following constrained non-linear optimization problem

Minθ CðθÞs:t: giðθÞr0 i¼ 1;…;ncPFi ðθÞ�PnFi r0 ; i¼ 1;…;nrθAΘ ð1Þ

where θ;θi; i¼ 1;…;nd is the vector of design variables with sideconstraints θlirθirθ

ui , CðθÞ is the objective function, giðθÞr0; i¼ 1

;…;nc are standard constraints, and PFi ðθÞ�PnFi r0 are the reliabilityconstraints which are defined in terms of the failure probabilityfunctions PFi ðθÞ and target failure probabilities PnFi ; i¼ 1;…;nr . It isassumed that the objective and constraint functions are smoothfunctions of the design variables. The objective function CðθÞ can bedefined in terms of initial, construction, repair or downtime costs,structural weight, or general cost functions. The standard constraintsare related to general design requirements such as geometric condi-tions, material cost components, and availability of materials. On theother hand, the reliability constraints are associated with design spe-cifications characterized through the use of reliability measures givenin terms of failure probabilities with respect to specific failure criteria.For structural systems under stochastic excitation the probability thatdesign conditions are satisfied within a particular reference period Tprovides a useful reliability measure [5]. Such measure is referred asthe first excursion probability and quantifies the plausibility of theoccurrence of unacceptable behavior (failure) of the structural system.In this context, a failure event Fi can be defined as Fiðθ; zÞ ¼diðθ; zÞ41, where di is the so-called normalized demand functiondefined as diðθ; zÞ ¼maxj ¼ 1;…;lmaxtA ½0;T�∣rijðt;θ; zÞ∣=ri

n

j , where zAΩz� Rnz is the vector of uncertain variables involved in the problem(characterization of the excitation), rijðt;θ; zÞ; j¼ 1;…; l are theresponse functions associated with the failure event Fi, and ri

n

j is the

acceptable response level for the response rij. It is clear that theresponses rijðt;θ; zÞ are functions of time (due to the dynamic nature ofthe excitation), the design vector θ, and the random vector z. Theseresponse functions are obtained from the solution of the equation ofmotion that characterizes the structural model (see next Section). Theuncertain variables z are modeled using a prescribed probabilitydensity function pðzÞ. This function indicates the relative plausibility ofthe possible values of the uncertain parameters zAΩz. The probabilityof failure evaluated at the design θ is formally defined as

PFi ðθÞ ¼ P maxj ¼ 1;…;l maxtA ½0;T�∣rijðt;θ; zÞ∣

rin

j

41

" #ð2Þ

where P½�� is the probability that the expression in parenthesis is true.Equivalently, the failure probability function evaluated at the design θcan be written in terms of the multidimensional probability integral

PFi ðθÞ ¼Zdiðθ;zÞ41

pðzÞ dz ð3Þ

It is noted that the above formulation can be extended in adirect manner if the cost of partial or total failure consequences isalso included in the definition of the objective function. It is alsonoted that constraints related to statistics of structural responses(i.e. mean value and/or higher-order statistical moments) can beincluded in the formulation as well. Thus, the above formulation isquite general in the sense that different reliability-based optimi-zation formulations can be considered.

3. Mechanical modeling

A quite general class of structural dynamical systems can becast into the following equation of motion

M €uðtÞþC _uðtÞþKuðtÞ ¼ kðuðtÞ; _uðtÞ; τðtÞÞþfðtÞ ð4Þwhere uðtÞ denotes the displacement vector of dimension n, _uðtÞthe velocity vector, €uðtÞ the acceleration vector, kðuðtÞ; _uðtÞ; τðtÞÞthe vector of non-linear restoring forces, τðtÞ the vector of a set ofvariables which describes the state of the nonlinear components,and fðtÞ the external force vector. The matrices M, C, and Kdescribe the mass, damping, and stiffness, respectively. Note thatsome of the matrices and vectors involved in the equation ofmotion depend on the vector of design variables θ and/or theuncertain system parameters z and therefore the solution is also afunction of these quantities. The explicit dependence of theresponse on these quantities is not shown here for simplicity innotation. The evolution of the set of variables τðtÞ is described by afirst-order differential equation

_τ ðtÞ ¼ κðuðtÞ; _uðtÞ; τðtÞÞ ð5Þwhere κ represents a non-linear vector function. This character-ization allows to model different types of nonlinearities includinghysteresis and degradation [20,21]. From Eq. (5) it is seen that theset of variables τðtÞ is a function of the displacements uðtÞ and thevelocities _uðtÞ, i.e. τðuðtÞ; _uðtÞÞ. Therefore, Eqs. (4) and (5) con-stitute a system of coupled non-linear differential equations foruðtÞ and τðtÞ. The previous formulation is particularly well suitedfor cases where most of the components of the structural systemremain linear and only a small part behaves in a nonlinear manner.Such cases are of particular interest in the present work, that is,linear finite element models with localized nonlinearities. Theexternal force vector fðtÞ is modeled as a non-stationary stochasticprocess. Depending on the application under consideration dif-ferent methodologies are available for generating these types ofprocesses. Such methodologies include filtered Gaussian whitenoise processes, stochastic processes compatible with powerspectral densities, point source-based models, and record-based

H.A. Jensen et al. / Reliability Engineering and System Safety 149 (2016) 204–217206

models [5,22–25]. A common aspect of the aforementionedmethodologies is that the generation of the corresponding sto-chastic processes samples involves in general a large number ofrandom variables, e.g. of the order of hundreds or thousands.Therefore, the evaluation of the reliability constraints for a givendesign constitutes a high-dimensional problemwhich is extremelydemanding from a numerical point of view.

4. Implementation issues

4.1. Reliability estimation

The reliability constraints of the nonlinear constrained opti-mization problem (1) are defined in terms of the first excursionprobability functions PFi ðθÞ; i¼ 1;…nr . As previously pointed out,these reliability measures are given in terms of high-dimensionalintegrals. The difficulty in estimating these quantities favors theapplication of simulation techniques to cope with the probabilityintegrals [26]. It is important to note that each sample implies thesolution of the equation of motion that characterizes the structuralmodel (a dynamic finite element model analysis). Therefore anefficient simulation technique is required in the context of thepresent formulation. A general applicable method named subsetsimulation is adopted here [7]. In this well known advancedsimulation technique the failure probabilities are expressed as aproduct of conditional probabilities of some chosen intermediatefailure events, the evaluation of which only requires simulation ofmore frequent events. The intermediate failure events are chosenadaptively using information from simulated samples so that theycorrespond to some specified values of conditional failure prob-abilities. Therefore, a rare event simulation problem is convertedinto a sequence of more frequent event simulation problems. Themethod uses a Markov chain Monte Carlo method based on theMetropolis algorithm for sampling from the conditional prob-abilities [27]. This is the most widely applicable simulation tech-nique because it is not based on any geometrical assumption aboutthe topology of the failure domain. In fact, validation calculationshave shown that subset simulation can be applied efficiently to awide range of dynamical systems including general linear andnon-linear systems [28,29]. In addition, subset simulation is very-well suited for parallel implementation in a computer cluster [30].For a detailed description and numerical implementation of subsetsimulation see references [7,29,31].

4.2. Optimization strategy

The solution of the reliability-based optimization problemdefined in Eq. (1) can be obtained in principle by a number oftechniques such as standard deterministic optimization schemesor stochastic search algorithms [1,12,32]. In particular, a class ofinterior point algorithms based on the solution of the first-orderoptimality conditions is considered for solving the numericalexamples presented in this work [33]. In general, the abovescheme has proved to be quite effective for a wide range ofapplications in the context of deterministic and stochastic opti-mization problems [34,35]. Since the focus of this work is not onthe optimization strategy but on the integration of a modelreduction technique into the reliability-based design formulationof medium/large stochastic finite element models, the reader isreferred to [33,35] for a detailed description and implementationof the aforementioned algorithm.

4.3. Model reduction

The solution of the reliability-based optimization problem (1) iscomputationally very demanding due to the large number ofdynamic analyses required during the design process. In fact thereliability estimation at each design requires the evaluation of thesystem response at a large number of samples in the uncertainparameter space (of the order of hundreds or thousands). Inaddition, the iterative nature of the optimization strategy mayimpose additional computational demands. Consequently, thecomputational cost may become excessive when the computa-tional time for performing a dynamic analysis is significant. Tocope with this difficulty, a model reduction technique is con-sidered in the present formulation. In particular, a method basedon substructure coupling or component-mode synthesis isimplemented in order to define a reduced-order model for thestructural system [15,17]. The general idea is to divide the linearcomponents of the structural system into a number of linearsubstructures obtaining reduced-order models and then assem-bling a reduced-order model of the entire structural system.

4.4. Synopsis of proposed methodology

The mathematical background of the model reduction techni-que and its integration into the design process are discussed inSections 5 and 6. In Section 5 the transformation matrix thatdefine the set of generalized coordinates of the reduced-ordermodel is first defined. Based on the new set of coordinates thereduced-order model matrices are derived. Then, the response ofthe system in terms of the reduced-order model is discussed. Next,the design process based on the reduced-order model is outlinedin Section 6. First, a particular finite element model parametriza-tion scheme that allows to represent the substructure matricesexplicitly in terms of the design variables is introduced. Therepresentation of the substructure matrices is then used to con-struct an expansion of the reduced-order model matrices withrespect to the design variables. Based on this expansion it isdemonstrated that the reduced-order model matrices need to becomputed and assembled once during the entire design process.This in turn implies a drastic reduction in the computationaldemands involved in the design of complex structural systems.Finally, some advantages and limitations of the proposed for-mulation are discussed.

5. Component mode synthesis technique

5.1. Substructure modes

In the present formulation fixed-interface normal modes andinterface constraint modes are considered in order to define thereduced-order model [15]. To this end, the following partitionedform of the mass matrix MsARn

s�ns and stiffness matrix KsARns�ns

of the substructure s; s¼ 1;…;Ns, is considered

Ms ¼Msii M

sib

Msbi Msbb

" #; Ks ¼

Ksii Ksib

Ksbi Ksbb

" #ð6Þ

where the indices i and b are sets containing the internal andboundary degrees of freedom of the substructure s, respectively.The boundary degrees of freedom include only those that arecommon with the boundary degrees of freedom of adjacent sub-structures, while the internal degrees of freedom are not sharedwith any adjacent substructure. In this framework, all boundarycoordinates are kept as one set usbðtÞARn

sb and the internal coordi-

nates in the set usi ðtÞARnsi . The displacement vector of physical


coordinates of the substructure s is given by usðtÞT ¼ 〈usi ðtÞT ;usbðtÞT 〉ARn

swhere ns ¼ nsi þnsb, and the symbol 〈 〉 represents a row vector.

The fixed-interface normal modes are obtained by restraining allboundary degrees of freedom and solving the eigenproblem

KsiiΦsii�MsiiΦsiiΛsii ¼ 0 ð7Þ

where the matrixΦsii contains the complete set of nis fixed-interfacenormal modes, and Λsii is the corresponding matrix containing theeigenvalues. The fixed-interface normal modes are normalized withrespect to the mass matrix Msii satisfying ΦsiiTMsiiΦsii ¼ Isii, andΦsiiTKsiiΦsii ¼Λ

sii. On the other hand, the interface constraint modes

are defined by setting a unit displacement on the boundary coor-dinates usbðtÞ and zero forces in the internal degrees of freedom.Then, the set of interface constraint modes is given by

Ksii Ksib

Ksbi Ksbb

" #ΨsibIsbb

" #¼

0sibRsbb

" #ð8Þ

from where the interface constraint-mode matrix Ψsc can bedefined as

Ψsc ¼ΨsibIsbb

" #¼ �K

sii�1KsibIsbb

24

35 ð9Þ

where ΨsibARnsi�nsb is the interior partition of the interface

constraint-mode matrix.

5.2. Reduced-order model

Based on the previous fixed-interface normal modes andinterface constraint modes a displacement transformation matrixis introduced to define a set of generalized coordinates. Suchtransformation corresponds to the Craig–Bampton method [15],and it takes the form:

usðtÞ ¼usi ðtÞusbðtÞ

( )¼ Φ

sik Ψ

sib

0sbk Isbb

" #vskðtÞvsbðtÞ

( )¼Ψs vsðtÞ ð10Þ

where ΦsikARnsi�nsk is the interior partition of the matrix Φsii of the

nks kept fixed-interface normal modes, vsðtÞ represents the sub-

structure generalized coordinates composed by the modal coor-dinates vskðtÞ of the kept fixed-interface normal modes and theboundary coordinates vsbðtÞ ¼ usbðtÞ, Ψ

sARns�n̂s is the Craig–

Bampton transformation matrix with n̂s ¼ nskþnsb, and all otherterms have been previously defined. The substructure mass matrixM̂

sARn̂

s�n̂s and stiffness matrix K̂sARn̂

s�n̂s in generalized coordi-nates vsðtÞ are given by

M̂s ¼ΨsTMsΨs; K̂s ¼ΨsTKsΨs ð11ÞNext, the vector of generalized coordinates for all the Ns sub-

structures

vðtÞT ¼ 〈v1ðtÞT ;…; vNs ðtÞT 〉ARnv ð12Þwhere nv ¼

PNss ¼ 1 n̂

s is introduced. Based on this vector, a newvector qðtÞ that contains the independent generalized coordinatesconsisting of the fixed-interface modal coordinates vskðtÞ for eachsubstructure and the physical coordinates vlbðtÞ; l¼ 1;…;Nb at theNb interfaces is defined as

qðtÞT ¼ 〈v1k ðtÞT ;…; vNsk ðtÞT ; v1bðtÞT ;…; vNbb ðtÞT 〉ARnq ð13Þ

where nq ¼PNs

s ¼ 1 nskþ

PNbl ¼ 1 n

lb, and nb

l is the number of degreesof freedom at the interface l (l¼ 1;…;NbÞ. These two vectors arerelated by the transformation:

vðtÞ ¼ TqðtÞ ð14Þwhere the matrix TARnv�nq is a matrix of zeros and ones thatcouples the independent generalized coordinates qðtÞ of the

reduced system with the generalized coordinates of each sub-structure. The assembled mass matrix M̂ARnq�nq and the stiffnessmatrix K̂ARnq�nq for the independent reduced set qðtÞ of gen-eralized coordinates take the form:

M̂ ¼ TTM̂

10 0

0 ⋱ 0

0 0 M̂Ns

2664

3775T; K̂ ¼ TT

K̂1

0 00 ⋱ 0

0 0 K̂Ns

2664

3775T ð15Þ

The number of fixed-interface normal modes to be used in thereduced-order model can be established by using different criteria.One particular criterion is discussed in the Numerical ExamplesSection. Finally, it is also noted that further reduction in the gen-eralized coordinates can be achieved by replacing the interfacedegrees of freedom by a reduced number of interface modes. Thatis, the displacement coordinates vlbðtÞARn

lb at the interface l can be

expressed in terms of a set of generalized coordinates [17]. Forsimplicity and clarity all interface degrees of freedom are kept inthe present implementation.

5.3. Reduced-order model response

In order to write the equation of motion of the structural sys-tem in terms of the reduced set of generalized coordinates qðtÞ, itis first observed that the independent physical coordinates uðtÞ ofthe original unreduced structural model can be written directly interms of qðtÞ asuðtÞ ¼ TΨTqðtÞ ð16Þwhere TARn�nu is a constant matrix that map the vector uðtÞT ¼ou1ðtÞT ;…;uNs ðtÞT 4ARnu ;nu ¼

PNss ¼ 1 n

s of the physical coordi-nates of all substructures to uðtÞ, andΨARnu�nv is a block diagonalmatrix defined in terms of the Craig–Bampton transformationmatrices of all substructures, that is, Ψ¼ blockdiagðΨ1;…;ΨNs Þ.Based on this transformation, the equation of motion of thereduced-order system can be written as

M̂ €qðtÞþ Ĉ _qðtÞþK̂qðtÞ ¼ TTΨTTTkðuðtÞ; _uðtÞ; τðtÞÞþTTΨTTT fðtÞð17Þ

where ĈARnq�nq is the assembled damping matrix for the inde-pendent reduced set qðtÞ, which can be defined in terms of thesubstructures damping matrices in generalized coordinates vsðtÞ asĈs ¼ΨsTCsΨs; s¼ 1;…;Ns, and all other terms have been pre-

viously defined. The assembled damping matrix has a similarstructure as M̂ and K̂ (see Eq. (15)). It is noted that the dimensionof the matrices involved in the equation of motion of the reduced-order model can be substantially smaller than the dimension ofthe unreduced matrices, e.g. nq⪡n. It is also noted that the vector ofnon-linear restoring forces can be characterized in local compo-nent specific coordinates, due to the localized non-linearities, witha minimal number of variables. These local variables are related tothe vector of independent physical coordinates uðtÞ by a standardlinear transformation matrix. In this manner, the set of nonlinearEqs. (5) and (17) can be integrated efficiently by an appropriatestep-by-step integration scheme [6] or by modal analysis[19,36,37]. Finally, it should be pointed out that the matricesinvolved in the equation of motion of the original system maydepend on the vector of design variables θ, and therefore anumber of the quantities that appear in this section are alsofunction of θ. The explicit dependence of those quantities on θ isexamined in the following section.

6. Design based on the reduced-order model

The previous model reduction technique is quite general in thesense that dividing the structure into substructures and reducing


the number of physical coordinates to a much smaller number ofgeneralized coordinates certainly alleviates part of the computa-tional effort. However, the generation of the reduced-order modelat each design implies the solution of the fixed-interface normalmodes eigenproblem and the computation of the interface con-straint modes for each substructure. This procedure can be com-putationally very expensive due to the substantial computationaloverhead that arises at substructure level. In order to make themodel reduction technique more efficient a particular para-metrization scheme in terms of the design variables is consideredin the present formulation. Specifically, it is assumed that thestiffness and mass matrix of a substructure s; s¼ 1;…;Ns, dependon only one of the design variables. Such dependency can be linearor nonlinear. Clearly, the case in which the stiffness and massmatrix of a substructure s do not depend on the design variables isalso included in this parametrization scheme. It should be pointedout that the previous parametrization is often encountered in anumber of practical applications [17,19]. The analysis of thereduced-order model based on this particular parametrizationscheme is presented in this section. Some remarks regarding themore general case for which the stiffness and mass matrices of asubstructure depend on more than one design variable are pro-vided at the end of the section.

6.1. Normal and constraint modes

Let S0 be the set of substructures that do not depend on thevector of design variables θ. In this case the substructure stiffnessand mass matrices are written as Ks ¼Ks0 and Ms ¼Ms0. The sub-structure fixed-interface normal modes and interface constraintmodes are independent of the design variables value. Thus, only asingle analysis is required to estimate the interface modes for theparticular substructure s. The substructure modes of these sub-structures are computed once during the design process. In otherwords, the eigenvalue problem to compute the eigenvalues andmode shapes of the kept fixed-interface normal modesΦsik as wellas the solution of the linear system to compute the interfaceconstraint modes Ψsib for a component sAS0 are not repeated ateach iteration of the design process.

On the other hand, let Sj be the set of substructures that dependon the design variable θj. It is assumed that the stiffness and massmatrices take the general form

Ks ¼KshjðθjÞ; Ms ¼MsgjðθjÞ ð18Þ

where the reference matrices KsandM

sare independent of θj, and

hjðθjÞ and gjðθjÞ are linear or nonlinear functions of the designvariable θj. It is clear that the partitions of the stiffness matrix Ks

and mass matrix Ms (see Eq. (6)) admit the same parametrization.Then, the eigenvalues and eigenvectors associated with the keptfixed-interface normal modes can be expressed as

Λskk ¼ΛskkhjðθjÞgjðθjÞ

; Φsik ¼Φsik

1ffiffiffiffiffiffiffiffiffiffiffiffigjðθjÞ

q ð19Þwhere the matrices Λskk and Φ

sik are the solution of the eigen-

problem

KsiiΦ

sik�M

siiΦ

sikΛ

skk ¼ 0 ð20Þ

where the matrices Φsik and Λskk of the reference eigenvectors and

eigenvalues, respectively, are independent of θj. Since the fixed-interface normal mode are normalized with respect to the massmatrix Msii (see Section 5.1), it is easily shown that the referencemode shapes Φsik satisfy the orthogonal conditions

ΦsT

ikMsiiΦ

sik ¼ Iskk, and Φ

sT

ikKsiiΦ

sik ¼Λ

skk. Furthermore, the interface

constraint modes are also independent of θj since

Ψsib ¼ �Ks�1ii Ksib ¼ �Ks�1ii h

j�1ðθjÞKsib h

jðθjÞ ¼ �Ks�1ii K

sib ¼Ψ

sib

ð21ÞUsing (Eqs. (19) and (20)), a single eigen-analysis of each sub-

structure is required to provide the exact estimate of the normaland constraint modes for any value of the design variable θj. This isa very important result in the context of the proposed formulationsince the computationally intensive re-analyses for estimating thefixed-interface constrained modes and the interface constraintmodes of each substructure for different values of θj requiredduring the design process are completely avoided. In addition, theprevious result allows to express the reduced-order modelmatrices explicitly in terms of the design variables (see next sec-tions). This in turn has a significant implication in terms of thecomputational efforts involved during the design process (seeSection 7). Finally, it is emphasized that the restriction regardingthe matrices dependence on one design variable is at substructurelevel. Of course different substructures can depend on differentdesign variables.

6.2. Substructure matrices

The substructure mass matrix M̂sARn̂

s�n̂s and stiffness matrixK̂

sARn̂

s�n̂s in generalized coordinates vsðtÞ are given in Eq. (11).Based on the previous parametrization, it can be shown that thepartitions for the substructure stiffness matrices K̂

skkAR

nsk�nsk , K̂skb

ARnsk�nsb ; K̂

sbbAR

nsb�nsb and mass matrices M̂skkAR

nsk�nsk , M̂skbAR

n sk �n sb

M̂sbbAR

nsb�nsb of a substructure sASj follow the parametrization

K̂skk ¼Λ

skkhjðθjÞgjðθjÞ

; K̂skb ¼ K̂

sT

bk ¼ 0skb; K̂sbb ¼ ðK

sbbþK

sbiΨ

sibÞhjðθjÞ

ð22Þand

M̂skk ¼ Ikk; M̂

skb ¼ M̂

sT

bk ¼ ðΦsT

ikMsiiΨ

sibþΦ

sT

ikMsibÞ

ffiffiffiffiffiffiffiffiffiffiffiffigjðθjÞ

q

M̂sbb ¼ ½ðΨ

sT

ibMsiiþM

sbiÞΨ

sibþΨ

sT

ibMsibþM

sbb�gjðθjÞ ð23Þ

The partitions in Eqs. (22) and (23) have been derived from (11)using the form of Ψs defined in (10), after substituting thematrices Φsik, Λ

skk, and Ψ

sib from the expressions (19) and (21).

6.3. Parametrization of reduced-order matrices

Substituting into Eq. (15) the previous characterization of thesubstructure matrices partitions, the stiffness matrix of thereduced-order system can be written as

K̂ ¼ K̂0þXndj ¼ 1

K̂ 1jhjðθjÞgjðθjÞ

þK̂ 2jhjðθjÞ

9>=>;

8><>: ð24Þ

where nd is the number of independent design variables, and thematrices K̂0, K̂ 1j and K̂ 2j; j¼ 1;…;nd are constant matrices.Similarly, the mass matrix of the reduced-order system can becharacterized as

M̂ ¼ M̂0þXndj ¼ 1

fM̂1jþM̂2jffiffiffiffiffiffiffiffiffiffiffiffigjðθjÞ

qþM̂ 3jgjðθjÞg ð25Þ

where the matrices M̂0, M̂ 1j, M̂2j, and M̂3j, j¼ 1;…;nd are alsoconstant matrices. The definition of the above matrices is provided

Fig. 1. Three-span eight-story two dimensional frame structure.


in Appendix A. It is important to emphasize that the assembledmatrices K̂0, K̂ 1j, K̂ 2j, M̂0, M̂1j, M̂2j and M̂3j in the expansions (24)and (25) are independent of the values of the vector of designvariables θ. To save computational time these matrices are com-puted and assembled once for a reference model obtained fromthe original model by setting hjðθjÞ ¼ 1, and gjðθjÞ ¼ 1. Thereforethere is no need for this computation to be repeated during theiterations of the design process due to the changes in value of thedesign variables. This feature results in substantial computationalsavings since it avoids re-computing the fixed-interface and con-straint modes for each substructure and also avoids assemblingthe reduced matrices from these substructures. The formulationguarantees that the reduced model is based on the exact sub-structure modes for all values of the design variables. In summary,the reduced-order substructure matrices are computed once dur-ing the design process.

6.4. Parametrization of global transformation matrix

Finally, it is noted that the transformation matrix TΨT thatrelates the independent physical coordinates uðtÞ with the gen-eralized coordinates of the reduced-order model qðtÞ (see Eqs. (16)and (17)) can also be written explicitly in terms of the designvariables. In fact, using the structure of Ψs and the dependence ofΦsik and Ψ

sib on θj derived in (19) and (21) respectively, this

transformation matrix can be expressed as

TΨT¼ T blockdiag ðΨ1δ10;…;ΨNsδNs0ÞT

þXndj ¼ 1

½T blockdiag ðΨ11δ1j;…;ΨNs1 δNsjÞT

1ffiffiffiffiffiffiffiffiffiffiffiffigjðθjÞ

qþT blockdiag ðΨ12δ1j;…;Ψ

Ns2 δNsjÞT� ð26Þ

where Ψs is the Craig–Brampton transformation matrix of a sub-structure s that does not depend on the design variables, Ψs1 andΨs2 are matrices associated with the Craig–Brampton transfor-mation matrix of a substructure s that depends on the designvariable θj, with

Ψs1 ¼Φsik 0sib0sbk 0

sbb

" #; Ψs2 ¼

0sik Ψsib

0sbk Isbb

" #ð27Þ

Note that the matricesΨs1 andΨs2 do not depend on the design

variables. Thus, from the previous expressions it is clear thatthe products involved in the transformation matrix TΨT arecomputed once during the design process. Furthermore, dueto the structure of the matrices T and T it can be shown thatthe previous products involve just permutations of rows and col-

umns of the block diagonal matrices blockdiagðΨ1δ10;…;ΨNsδNs0Þ,

blockdiagðΨ11δ1j;…;ΨNs1 δNsjÞ and blockdiagðΨ

12δ1j;…;Ψ

Ns2 δNsjÞ.

6.5. Final remarks

It is stressed that the efficiency of the above formulation interms of the number of substructure analyses required is based onthe assumption that the substructure matrices depend only on onedesign variable. For the more general case, the normal and con-straint modes have to be recomputed in each iteration of thedesign process. In fact, when the substructure matrices depend ontwo or more design variables, a representation similar to Eqs. (24)and (25) is no longer applicable and the reduced substructurematrices of the reduced-order model should be re-assembled fromthe substructure stiffness and mass matrices for new values of thevector of design variables θ. This repeated computation, however,is usually confined to a small number of substructures in manypractical applications. So, even in the more general case a

significant saving may still arise since the estimation of the fixed-interface modes and the interface constraint modes for most of thesubstructures need not to be repeated during the design process.For the general case (dependence on two or more design variables)it is also interesting to note that interpolation schemes can beadopted to avoid re-analyses at the substructure level by approx-imating the fixed-interface normal modes and the interface con-straint modes at various values of the design variables in terms ofthe corresponding modes of a family of models defined at anumber of design points [38]. The use of approximation schemesis, however, outside the scope of the present work. This aspect ofthe implementation is left for future research efforts (see Con-clusions). Finally, it is noted that parallelization techniques are alsopossible at the model level. In fact, the definition of all sub-structure matrices in generalized coordinates can be carried out inparallel, reducing the computational time of the proposed imple-mentation even further.

7. Numerical examples

Two numerical examples are presented in this section. The firstexample, which is considered as a test problem, deals with asimple linear model while the second example, which is con-sidered as an application problem, considers a bridge structurewith localized nonlinearities. The objective of the test problem isto evaluate the effectiveness of the proposed model reductiontechnique in a relatively simple model, while the second exampleexamines the effect of the proposed technique in a moreinvolved model.

7.1. Example 1: test problem

7.1.1. Model descriptionThe structural model shown in Fig. 1 is considered as a test

problem. It consists of a three-span two dimensional framestructure. The structure has a total length of 30.0 m and a constantfloor height of 5.0 m, leading to a total height of 40.0 m. The finite


element model comprises 160 two dimensional beam elements ofsquare cross section with 140 nodes and a total of 408 degrees offreedom. The dimension of the square cross section of the beamelements is equal to 0.4 m. The axial deformation of the columnelements is assumed to be small and they are neglected in themodel. Material properties of the beam elements have beenassumed as follows: Young's modulus E¼ 2:0� 1010 N=m2, andmass density ρ¼ 2500 kg=m3. A 5% of critical damping for themodal damping ratios is introduced in the model. The structuralsystem is excited horizontally by a ground acceleration modeled asa non-stationary stochastic process. In particular, a stochasticpoint-source model characterized by a series of seismicity para-meters such as the moment magnitude and rupture distance isconsidered in the present implementation [23,24]. The model is asimple, yet a powerful means for simulating ground motions withhigh and low frequency components. The methodology, which wasinitially developed for generating synthetic ground motions, hasbeen reinterpreted to form a stochastic model for ground excita-tion [39].

The input for the stochastic excitation model involves a whitenoise sequence and a series of seismological parameters as pre-viously pointed out. Details of the entire procedure can be found in[23,40]. The duration of the excitation is equal to T¼30 s with asampling interval equal to ΔT ¼ 0:01 s. Based on the character-ization of the point source model, the generation of the stochasticground motions involves more than 3000 random variables for theduration and sampling interval considered. Thus, the vector ofuncertain parameters z involved in the problem (see Section 2) hasmore than 3000 components. The actual excitation generated forthe test problem corresponds to ground motions of low intensityso that the structural response is expected to be dominated bylinear elastic behavior, which is compatible with the model underconsideration. For illustration purposes, Fig. 2 shows a syntheticexcitation sample generated by the stochastic point-source model.

7.1.2. Design problem formulationThe cost C represented by the total volume of the column

elements is chosen as the objective function for the design pro-blem. The design variables comprise the dimension of the columnelements square cross section of the different floors. In particular,the dimension of the column elements square cross section arelinked into two design variables in this example problem. To bemore specific, the dimension of the column elements square crosssection a is parameterized as a¼ θa, where a ¼ 0:4 m denotes thenominal dimension and θ is a normalized parameter that repre-sents the design variable. Design variable number one (θ1) isrelated to the dimension of the column elements square crosssection of floors 1–4, while the second design variable (θ2)

Fig. 2. Excitation time history sample.

controls the dimension of the column elements square cross sec-tion of floors 5–8. To control serviceability, the design criteria aredefined in terms of the relative displacements of the first, fifth andeighth floor with respect to the ground. The failure probabilityfunctions are defined as

PFi ðθÞ ¼ P maxtA ½0;T �∣δiðt;θ; zÞ∣

δin 41

" #; i¼ 1;2;3 ð28Þ

where δiðt;θ; zÞ; i¼ 1;2;3 are the relative displacements of thefirst, fifth and eighth floor with respect to the ground, respectively,and δi

n

; i¼ 1;2;3 are the corresponding critical threshold levels.The threshold levels are defined in terms of a percentage of thetotal height of the frame structure. In particular, the followingvalues are considered: 0.05%; 0.3%; and 0.5% for the thresholdlevels corresponding to the failure events associated with the first,fifth and eighth floor, respectively. The design problem is for-mulated as

Minθ CðθÞs:t: PFi ðθÞr10�4 i¼ 1;2;3θ1Zθ20:75rθir2:00 i¼ 1;2 ð29ÞNote that there are three reliability constraints plus one geo-

metric constraint in addition to the side constraints of the designvariables. It is also noted that the estimation of the probability offailure for a given design θ represents a high-dimensional relia-bility problem. In fact, as previously indicated, more than threethousands random variables are involved in the correspondingmultidimensional probability integral (see Eq. (3)).

7.1.3. Reduced-order modelThe structural model is subdivided into sixteen substructures

as shown in Fig. 3. Substructures 1–8 are composed by the columnelements of the different floors, while substructures 9–16 corre-spond to the beam elements of the different floors. Based on thissubdivision it is noted that substructures 1–4 depend on thedesign variable θ1, substructures 5–8 depend on the design vari-able θ2, while substructures 9–16 are independent of the designvariables. In connection with Section 6, the nonlinear functions ofthe design variables θj; j¼ 1;2 are given by hjðθjÞ ¼ θ4j (related tothe area second moment of inertia term in the stiffness matrices)and gjðθjÞ ¼ θ2j (related to the area term in the mass matrices),respectively.

For each substructure it is selected to retain all fixed-interfacenormal modes that have frequency ω such that ωrαωc , with αbeing a multiplication factor and ωc is a cut-off frequency which istaken equal to 41.0 rad/s in this case (4th modal frequency of theunreduced reference model). It is noted that the value of the factorα affects the computational efficiency and accuracy of the modelreduction technique. The multiplication factor is selected to be10.0 for substructures 1–8, and 2.9 for substructures 9–16. Withthis selection of parameters only 1 fixed-interface normal mode iskept for each substructure. Thus, the reduced-order model com-prises a total of 16 generalized coordinates out of the 312 internaldegrees of freedom of all substructures (95% reduction in terms ofthe internal degrees of freedom). Note that the total number ofinterface degrees of freedom is equal to 96 in this case. Table 1shows the error (in percentage) between the modal frequenciesusing the unreduced reference finite element model and themodal frequencies computed using the reduced-order model. It isobserved that the errors are quite small. In fact, the error for thelowest 4 modes fall bellow 0.1%. The comparison with the lowest4 modes seems to be reasonable since validation calculations showthat the contribution of the higher order modes (higher than the4th mode) in the dynamic response of the model is negligible.

Fig. 3. Substructures of the finite element model for design purposes.

Table 1Modal frequency error between the modal frequencies of the unreduced referencemodel and the reduced-order model. Test problem.

Frequency number Error (%)

1 1.3�10� 32 1.1�10� 23 3.4�10� 24 8.8�10� 2

Fig. 4. Iso-probability curves associated with the failure events of the first, fifth,and eighth floor (unreduced model).

Fig. 5. Iso-probability curves associated with the failure events of the first, fifth,and eighth floor (reduced-order model).


It is important to note that the selection of the number of fixed-interface modes per substructure, necessary to achieve a prescribedaccuracy, can be done off-line, before the design procedure takesplace. This calibration analysis can be carried out not only for thereference model but for other configurations as well. Validationcalculations have shown that the reduced-order model is adequatefor the entire design space defined in (29). In summary, it isobserved that even for this simple model an important reduction inthe number of generalized coordinates is obtained with respect tothe number of the degrees of freedom of the original unreducedfinite element model. In fact, an almost 75% reduction is obtained inthis test problem. It is expected that a more significant reductionwill be obtained for more involved finite element models (see nextexample problem).

7.1.4. ResultsThe effectiveness of the model reduction technique in the

context of the design problem is investigated in this section.Figs. 4 and 5 show some iso-probability curves in the design spaceconstructed by using the original unreduced model and thereduced-order model, respectively. For clarity, the design space isgiven directly in terms of the actual dimension of the columnelements square cross section. The iso-probability curves corre-spond to the three failure events defined in Section 7.1.2. Thesecurves are constructed by using a set of failure probability esti-mates distributed over the design space. Each of the estimate isobtained by subset simulation as indicated before. The resultingcurves, which are somewhat rugged because of the variability ofthe probability estimates, have been smoothed for presentationpurposes. It is observed that the iso-probability curves obtained bythe full and reduced-order model are almost identical. There isonly small differences for some iso-probability curves corre-sponding to low probability events (r10�5), but they follow thesame trend. The small differences at low probability levels are dueto the simulation scheme used to estimate such probabilities(subset simulation in this case), but they are not due to the per-formance of the reduced model which is very accurate. The pre-vious result implies that the final design can be obtained withsufficient accuracy by using the reduced-order model instead ofthe full model.


From the design point of view it is seen that the iso-probabilitycurves associated with the relative displacement of the first floor(PF1 ) show a weak interaction between the dimension of the col-umn elements square cross section of the lower and upper floorsspecially for flexible lower floors columns (i.e. a1r0:4 m). Underthis condition, the iso-probability curves are controlled by thedimension of the column elements square cross section of thelower floors, as expected. For more rigid lower floors columns theinteraction between the design variables is somewhat more pro-nounced. On the other hand, the iso-probability curves related tothe relative displacement of the fifth floor (PF2 ) show a clearinteraction between the dimension of the column elements squarecross section of the lower and upper floors. Finally, the iso-probability curves associated with the roof displacement (eighthfloor) (PF3 ) show a strong non-linear interaction between thedimension of the column elements square cross section of allfloors, as anticipated. These results give a valuable insight into theinteraction and effect of the design variables on the reliability ofthe structural model.

Fig. 6 shows the feasible domain, some normalized objectivecontours and some iso-probability curves as well as the finaldesign. It is observed that the geometric and side constraints areinactive at the final design. The reliability constraints related to therelative displacement of the first and eighth floor with respect tothe ground are active at the final design. Starting from the initialfeasible design θ1 ¼ 2:0 (a1 ¼ 0:8 m) and θ2 ¼ 1:625 (a2 ¼ 0:65 m),the final design is obtained in about ten iterations by using theinterior point algorithm described in [33,35] (see Section 4.2). Interms of the computational cost, the number of finite element runsinvolved during the design process depends on the number of

Fig. 6. Feasible domain and optimum design.

Fig. 7. Finite element mod

iterations and the number of simulations necessary to estimate thefailure probability and its sensitivity for the different designsrequired by the optimizer. Thus, the computational effort forassembling the finite element model and obtaining its dynamicresponse for a given design is the key factor for comparison pur-poses. The results indicate that the speedup achieved by the pro-posed formulation is about 4, where the speedup is defined as theratio of the execution time of the design process by using theunreduced original model and the execution time of the designprocess by using the reduced-order model. As previously pointedout, it is expected that even a more significant reduction in com-putational cost will be achieved for the cases of more involvedmodels (see next example problem). Finally, it is important to notethat this reduction in computational effort is achieved withoutcompromising the accuracy of the final design. In other words, thefinal design obtained from the reduced-order model is analogousto the one obtained from the unreduced model.

7.2. Example 2: Application problem

7.2.1. Structural systemThe bridge structural model shown in Fig. 7, which has been

borrowed from [18], is used in this section as an application pro-blem. The bridge is curved in plan and has a total length of 119 m.It has 5 spans of lengths equal to 24.0 m, 20.0 m, 23.0 m, 25.0 m,and 27.0 m, respectively, and four piers of 8 m height that supportthe girder monolithically. Each pier is founded on an array of fourpiles of 35 m height. The piers and piles are modeled as columnelements of circular cross-section with 1.6 m and 0.6 m diameter,respectively. The deck cross section is a box girder which ismodeled by beam and shell elements. It rests on each abutmentthrough two rubber bearings that consist of layers of rubber andsteel plates, with the rubber being vulcanized to the steel plates. Aschematic representation of a rubber bearing is also shown in thefigure, where Dr represents the external diameter, Di¼0.1 m is theinternal diameter, and Hr ¼ nrtr is the total height of rubber in thebearing, where tr is the layer thickness and nr is the number ofrubber bearings. The force-displacement characteristics of therubber bearings are modeled by a biaxial hysteretic behavior. Ananalytical model based on a series of experimental tests conductedfor real-sized rubber bearings is used in the present application.For a detailed description of the analytical model that describe thenonlinear behavior of the bearings the reader is referred to [41].The interaction between the piles and the soil is modeled by aseries of translational springs along the height of the piles withstiffnesses varying from 11,200 T/m at the base to 0.0 T/m at thesurface.

el of bridge structure.

Table 2Nominal values of structural parameters.

Structural parameter Nominal value (m)

Dpi ; i¼ 1;2;3;4 1.6DLr 0.8

HLr 0.17

DRr 0.8

HRr 0.17

Fig. 8. Substructures of the finite element model used for design purposes.


The net effect of these elements is to increase the translationalstiffness in the x and y direction of the column elements thatmodel the piles. The following values of the material properties ofthe concrete structure are considered. The Young's modulus istaken to be E¼ 2:0� 1010 N=m2, the Poisson ratio ν¼ 0:2, andmass density ρ¼2500 kg/m3. Finally, a 3% of critical damping isadded to the model. The selected finite element model for thebridge structure has 10,068 degrees of freedom. In what follows, itis assumed that the previous model is representative of the actualbehavior of the bridge structure and it is referred to as the nominalmodel. In this context, it is assumed that the structural compo-nents such as the piers, piles and the deck girder remain linearduring the analysis while the nonlinearities are localized in therubber bearings response. In addition, the axial deformation of thepiers is neglected with respect to their bending deformation. Thebridge structure is subjected to a ground acceleration in a direc-tion defined at 25° with respect to the x-axis as shown in Fig. 7.Such ground acceleration is modeled as in the previous example,that is, by a stochastic point-source model. The duration of theexcitation is equal to T¼20 s with a sampling interval equal toΔT ¼ 0:01 s. In this case, the generation of the stochastic groundmotions comprise more than 2000 random variables and thereforethe vector of uncertain parameters z involved in the problem hasmore than 2000 components.

7.2.2. Reliability-based design formulationThe reliability-based design problem is defined in terms of the

following constrained non-linear optimization problem

Minθ CðθÞPFi ðθÞrPnFi ; i¼ 1;2θAΘ ð30Þ

where θ;θi; i¼ 1;…;8 is the vector of design variables, CðθÞ is theobjective function, PFi ðθÞ; i¼ 1;2 are the failure probability func-tions, and PnFi ; i¼ 1;2 are the corresponding target failure prob-abilities which are taken equal to PnFi ¼ 10

�4; i¼ 1;2. The designvariables include the diameter of the piers circular cross section,and the external diameter and total height of rubber in the bear-ings. Design variables θ1;θ2;θ3 and θ4 are related to the diameterof the circular cross section of the four piers, while design vari-ables θ5;θ6;θ7 and θ8 are associated with the external diameterand the total height of rubber of the bearings located at theabutments. The relationship between the design variables and theactual structural parameters is given by Dpi ¼ θiDpi; i¼ 1;2;3;4,DLr ¼ θ5DLr , HLr ¼ θ6HLr , DRr ¼ θ7DRr , and HRr ¼ θ8HRr , where Dpi;i¼ 1;2;3;4 are the diameters of the circular cross section of thepiers, DLr and DRr are the external diameters of the bearingslocated at the left and right abutments, respectively, HLr and HRrare the total heights of rubber of the bearings located at the leftand right abutments, respectively, and Dpi, DLr , HLr , DRr , HRr are thecorresponding nominal values of the parameters which are givenin Table 2.

The side constraints for the design variables are given by:0:75rθir1:25; i¼ 1;2;3;4; 0:75rθir1:25; i¼ 5;7, and0:88rθir1:47; i¼ 6;8. The objective function CðθÞ represents a

cost function which is assumed to be proportional to the totalvolume of rubber in the bearings and to the total volume of thepiers. Failure, that is unacceptable performance, is defined interms of the relative displacement of piers and the relative dis-placement of the rubber bearings. Thus, the corresponding failureprobability functions are given by

PF1 ðθÞ ¼ P maxtA ½0;T �∣ubmaxðt;θ; zÞ∣

0:10 m41

� �;

PF2 ðθÞ ¼ P maxtA ½0;T �∣δmaxðt;θ; zÞ∣

0:07 m41

� �ð31Þ

where ubmaxðt;θ; zÞ represents the maximum relative displacementbetween the deck girder and the base of the rubber bearings ateach abutment (in the x or y direction), and δmaxðt;θ; zÞ denotesthe maximum relative displacement between the top of the piersand their connection with the piles foundations (in the x or ydirection). Note that the estimation of the failure probabilityfunctions for a given design θ represents a high-dimensionalreliability problem as in the test problem.

7.2.3. Substructures characterizationConsidering the previous design formulation the bridge struc-

ture is divided into a number of substructures. The division isguided by a parametrization scheme so that the substructurematrices for each one of the introduced substructures depend ononly one of the design variables. In particular, the structural modelis subdivided into six linear substructures and two nonlinearsubstructure as shown in Fig. 8. Substructure S1 is composed bythe pile elements, substructures S2, S3, S4 and S5 include the dif-ferent pier elements, and substructure S6 corresponds to the deckgirder. Finally, substructures S7 and S8 are the nonlinear sub-structures composed by the rubber bearings located at the left andright abutment, respectively. With this subdivision substructuresS1 and S6 do not depend on the design variables, while sub-structures S2, S3, S4 and S5 depend on the design variables θ1, θ2,θ3, and θ4, respectively, and design variables θ5, θ6, θ7, and θ8 areassociated with the nonlinear substructures S7 and S8. The non-linear functions of the design variables θj; j¼ 1;2;3;4 corre-sponding to the linear substructures are given by hjðθjÞ ¼ θ4j , andgjðθjÞ ¼ θ2j , respectively.

Based on an analysis similar to the one performed in the pre-vious example problem it is concluded that retaining ten general-ized coordinates for substructure S1, two for each one of sub-structures S2, S3, S4 and S5, and ten for substructure S6 is adequate inthe context of this application. The error (in percentage) betweenthe modal frequencies using the full nominal reference finite ele-ment model and the modal frequencies computed using thereduced-order model is shown in Table 3. The modal frequencies forboth models are computed by considering only the linear compo-nents of the structural system. It is seen that with this number ofgeneralized coordinates the error fall bellow 0.5% for the lowest

Fig. 10. Design space in terms of the design variables associated with the rubberbearings ðDr ;HrÞ. PF1 : iso-probability curves of failure event F1. PF2 : iso-probabilitycurves of failure event F2. C: normalized objective function contours.

Table 3Modal frequency error between the modal frequencies of the full model and thereduced-order model.

Frequency number Unreduced model Reduced-order model Error

ω (rad/s) ω (rad/s) (%)

1 4.214 4.216 0.052 4.282 4.284 0.063 4.569 4.572 0.064 12.197 12.249 0.435 15.424 15.462 0.256 23.419 23.421 0.01

Fig. 9. MAC-values between the mode shapes computed from the unreduced finiteelement model and from the reduced-order model.


6 modes. The corresponding matrix of MAC-values (modal assur-ance criterion) between the first six modal vectors computed fromthe unreduced finite element model and from the reduced-ordermodel is shown in terms of a 3-D representation in Fig. 9. It isobserved that the values at the diagonal terms are close to one andalmost zero at the off-diagonal terms. Thus, the modal vectors ofboth models are consistent. The comparison with the lowest6 modes is based on the fact that the contribution of the higherorder modes (higher than the 6th mode) in the dynamic responseof the model is negligible. In summary, a total of 28 generalizedcoordinates, corresponding to the fixed interface normal modes ofthe linear substructures, out of 10,008 internal DOF of the originalmodel, are retained for the six linear substructures. On the otherhand, the number of interface degrees of freedom is equal to 60 inthis case. With this reduction, the total number of generalizedcoordinates of the reduced-order model represents a 99% reductionwith respect to the unreduced model. Thus, a drastic reduction inthe number of generalized coordinates is obtained with respect tothe number of the degrees of freedom of the original unreducedfinite element model. Based on the previous analysis, it is concludedthat the reduced-order model and the full unreduced model areequivalent in the context of this application problem. Therefore, thedesign process of the bridge structural model is carried out by usingthe reduced-order model. As previously pointed out the calibrationof the reduced-order model can be done off-line, before the designprocedure takes place.

7.2.4. Design resultsTaking advantage of the reduced-order model a couple of

design scenarios are investigated in detail in order to get insight

into the reliability and general performance of the bridge structureunder consideration. First, the design of the rubber bearings(isolators) located at the abutments is considered. In particular, theeffect of the external diameter and the total height of rubber in thebearings on the design of such elements is studied. To this end, thedesign variables θ1, θ2, θ3, and θ4, which control the diameters ofthe circular cross sections of the piers, are kept constant and equalto their upper bound values θi ¼ 1:25; i¼ 1;2;3;4 (Dpi ¼ 2:0 m).This is done in order to isolate the effect of the design variablesassociated with the rubber bearings. Moreover, these variables arelinked into two design variables, one related to the external dia-meter (θDr ¼ θ5 ¼ θ7) and the other related to the total height ofrubber (θHr ¼ θ6 ¼ θ8). In other words all rubber bearings areassumed to have the same geometrical properties. With the pre-vious setting Fig. 10 shows some iso-probability curves andobjective function contours as well as the final design.

The design space is shown in terms of the actual values of theexternal diameter (Dr ¼ θDrDr) and the total height of rubber(Hr ¼ θHrHr). The objective contours are normalized by a costfactor and the iso-probability curves are constructed by using a setof failure estimates distributed over the design space. As in theprevious example these curves have been smoothed for pre-sentation purposes. It is important to note that these curves wereconstructed by the reduced-order model in a reasonable compu-tational time. The construction of these curves from the full finiteelement model is not practical due to the excessive computationaltime required to estimate the failure probabilities over the designspace. From the figure it is seen that the probability of the failureevent F1 decreases as the external diameter of the isolatorsincreases. This is reasonable since the isolation system becomesstiffer with rubber bearings having larger external diameters andthus the relative displacements between the deck girder and thebase of the rubber bearings at each abutment, which control thefailure event F1, are expected to decrease. On the contrary, thefailure probability increases as the height of rubber increases. Inthis case the isolation system becomes more flexible and thereforethe relative displacements between the deck girder and the baseof the rubber bearings increase. A similar effect is observed withrespect to the failure event related to the relative displacementbetween the top of the piers and their connection with the pilesfoundations (F2). That is, an increase of the external diameter ofthe isolators decreases the probability of failure event F2 since theoverall system becomes stiffer. On the other hand, an increase ofthe total height of rubber in the isolators increases the probabilityof failure event F2 since in this case the structural system becomesmore flexible. The corresponding final design is given byDr¼0.75 m and Hr¼0.15 m. The side constraint associated withthe height of rubber and the reliability constraint related to the

Fig. 11. Design space in terms of the diameter of the circular cross sections of thepiers (Dp) and the external diameter of the rubber bearings (Dr). PF1 : iso-probabilitycurves of failure event F1. PF2 : iso-probability curves of failure event F2. C: nor-malized objective function contours.

Table 4On-line computational cost of different tasks for a given design.

Task Full finite elementmodel

Reduced-ordermodel

Time (s) Time (s) Speedup

Finite elementgeneration

52.9 0.013 4069

Modal analysis 0.74 0.052 14Dynamic response 1.7 0.51 3

Sum of different tasks 55.34 0.58 95


maximum relative displacement between the deck girder and thebase of the rubber bearings at each abutment are active at the finaldesign. Contrarily, the reliability constraint associated with themaximum relative displacement between the top of the piers andtheir connection with the piles foundations is inactive.

Next, the interaction between bridge structural componentsand rubber bearing parameters is considered. Specifically, thedesign space in terms of the diameter of the circular cross sectionsof the piers and the external diameter of the rubber bearings isconstructed. For this purpose the design variables associated withthe diameters of the circular cross sections of the piers are linkedto one design variable θDpier , i.e. θDpier ¼ θ1 ¼ θ2 ¼ θ3 ¼ θ4, while thedesign variables θ5 and θ7 associated with the external diametersof the rubber bearings are linked to one design variable θDr , i.e,(θDr ¼ θ5 ¼ θ7). Design variables related to the total heights ofrubber in the bearings are kept constants and equal to their lowerbound values, i.e, θ6 ¼ θ8 ¼ 0:88 (Hr¼0.15 m). Fig. 11 shows someobjective contours and iso-probability curves as well as the finaldesign. As in the previous figure, the design space is shown interms of the actual values of the diameter of the circular crosssections of the piers (Dp ¼ θDpierDp) and the external diameter ofthe rubber bearings (Dr ¼ θDrDr). From the figure it is observedthat the probability of failure event F2 decreases as the diameter ofthe circular cross sections of the piers increase. In this case thepiers become stiffer and therefore the relative displacementsbetween the top of the piers and their connection with the pilesfoundations decrease. It is also seen that the failure event F2 iscontrolled by the diameter of the circular cross sections of thepiers for values of this quantity close to its lower bound, i.e.Dpr1:45 m. In this range of values the iso-probability curves arealmost perpendicular. So, the effect of the external diameter of theisolator is negligible. In other words, the flexibility of the pierelements controls the relative displacements between the top ofthe piers and their connection with the piles foundations, asexpected. Contrarily, for values of this quantity close to its upperbound, i.e. DpZ1:70 m a strong interaction between the diameterof the circular cross sections of the piers and the external diameterof the rubber bearings is observed. Thus, for rigid pier elementsthe relative displacements between the top of the piers and theirconnection with the piles foundations is controlled by both designvariables, that is, Dp and Dr. In fact, the iso-probability curvesindicate that for example an increase in the diameter of the cir-cular cross sections of the piers is compensated by a decrease inthe external diameter of the bearings. In other words, for suchcombination of the design variables Dp and Dr the probability offailure remains invariant. On the other hand it seen that the failureevent F1 is mainly controlled by the external diameter of the

rubber bearings. Actually, the iso-probability curves associatedwith failure event F1 show a relatively weak interaction betweenthe diameter of the circular cross sections of the piers and theexternal diameter of the rubber bearings. The probability of failureof this event decreases as the external diameter of the isolatorsincreases, which is the same behavior observed in Fig. 10. The finaldesign for this scenario is given by Dp¼1.64 m and Dr¼0.67 m(point B in the figure) where both reliability constraints are active.

The results shown in Fig. 11 can also be used to demonstratethe benefits of designing the isolators and the bridge structuresimultaneously. For example, if the design process involves onlythe isolators and the diameter of the circular cross sections of thepiers are kept constant at their upper bound values (Dp¼2.0 m),the optimal design is given by Dr¼0.75 m (point A in the figure)with a corresponding normalized cost equal to C¼1.4. On theother hand, if the diameter of the circular cross sections of thepiers is also considered as design variable the final design movesfrom point A to point B, with a decrease of the normalized cost inabout 30%. Thus, taking into account the interaction between thedesign variables associated with the bridge structure and theisolators during the design process is quite beneficial in terms ofthe cost of the final design. It is interesting to note that the finaldesign previously obtained can be improved just marginally if theeight design variables are considered as independent during thedesign process. It can be shown that in this case the nominal costis decreased in about 5% with respect the final design shown inFig. 11. The corresponding design process converges in less thanfifteen iterations starting from the initial feasible design given byDpi ¼ 2:0 m; i¼ 1;2;3;4, DLr ¼DRr ¼ 1:0 m and HLr ¼HRr ¼ 0:25 m.

Finally, it is noted that the above observations and remarks givea valuable insight into the complex interaction of the designvariables on the performance and reliability of the bridge struc-tural system.

7.2.5. Computational costTable 4 shows the on-line computational costs involved in the

assemblage of the finite element model and the computation of thedynamic response for a given design considering the full finite ele-ment model and the reduced-order model. These operations andprocedures are performed at each iteration of the design process. Thelast column of the table indicates the speedup (round to the nearestinteger) achieved by the reduced-order model for the different tasks.As previously pointed out, the speedup is the ratio of the executiontime by using the unreduced model and the execution time by usingthe reduced-order model. It is seen that the difference is quite sig-nificant. In fact, the overall speedup for the online calculations ismore than 90 in this case (last row of the table).

The off-line computational cost, that is the cost of calculationsrelated to the definition of the reduced-order model and thecharacterization of the transformation matrix that maps the gen-eralized coordinates to the physical coordinates of the unreducedmodel, which are preformed once during the design process,corresponds to approximately two full analyses (finite element


model generation and dynamic response) of the unreduced modelin this case. Considering this cost an overall speedup value ofabout 10 is obtained by the proposed methodology in solving thisparticular design problem. However, once the reduced-ordermodel has been defined several design scenarios, i.e. in terms ofdifferent objective functions and reliability constraints, can beexplored and solved efficiently. So, high speedup values (20, 40 oreven more) can be obtained for the design process as a whole. Aspreviously pointed out this reduction in computational effort isachieved without compromising the accuracy of the final design.

8. Conclusions

A general strategy for dealing with a class of reliability-baseddesign problems of stochastic finite elements has been presented.It consists in the integration of a model reduction technique withan appropriate optimization scheme. The design process is carriedout in a reduced space of generalized coordinates. In particular, amodel reduction technique based on fixed interface normal modesand interface constraint modes is considered in the presentimplementation. The reduction technique, which is applied to thelinear components of the structural systems, produces highlyaccurate models with relatively few substructure modes. Numer-ical examples indicate that the integration of the proposed modelreduction technique in the framework of reliability-based opti-mization is applicable for a class of linear and nonlinear finiteelement models under stochastic excitation. The results alsodemonstrate that the computational effort involved during thedesign process is reduced significantly with respect to the processconsidering the unreduced finite element model. In fact, goodspeedup values were obtained. On the other hand, the reduction incomputational effort is achieved without compromising theaccuracy of the design process. Based on the results of this study itis concluded that the proposed approach is potentially an effectivetool for solving a class of reliability-based design problems invol-ving complex structural systems under stochastic excitation.Future research directions aim at considering more involved pro-blems from the optimization point of view (formulations withmore design variables) as well as the implementation of designprocess in a fully parallel environment. In addition, the imple-mentation of interpolation schemes for approximating fixed-interface normal modes and interface constraint modes for casesin which the substructure matrices depend on more than onedesign variable is another research effort. Work in these directionsare currently under consideration.

Acknowledgments

The research reported here was supported in part by CONICYTunder grant number 1150009 which is gratefully acknowledged bythe authors. Also this research has been implemented under the“ARISTEIA” Action of the “Operational Programme Education andLifelong Learning” and was co-funded by the European Social Fund(ESF) and Greek National Resources.

Appendix A

The matrices K̂0, K̂ 1j and K̂ 2j; j¼ 1;…;nd are defined as

K̂0 ¼ TTK̂

10δ10 0 00 ⋱ 0

0 0 K̂Ns0 δNs0

2664

3775T;

K̂ 1j ¼ TTK̂

1

1δ1j 0 00 ⋱ 0

0 0 K̂Ns

1 δNsj

26664

37775T

K̂ 2j ¼ TTK̂

1

2δ1j 0 00 ⋱ 0

0 0 K̂Ns

2 δNsj

26664

37775T ð32Þ

where K̂s0 represents the stiffness matrix in generalized coordi-

nates vsðtÞ of a substructure that does not depend on the designvariables, i.e. δs0 ¼ 1 if sAS0, and δs0 ¼ 0 otherwise, and

K̂s1 ¼

Λskk 0skb0sbk 0

sbb

" #; K̂

s2 ¼

0skk 0skb

0sbk KsbbþK

sbiΨ

sib

" #ð33Þ

where δsj ¼ 1 if sASj and δsj ¼ 0 otherwise. On the other hand, thematrices M̂0, M̂1j, M̂2j, and M̂3j, j¼ 1;…;nd are given by

M̂0 ¼ TTM̂

10δ10 0 00 ⋱ 0

0 0 M̂Ns0 δNs0

2664

3775T;

M̂1j ¼ TTM̂

1

1δ1j 0 00 ⋱ 0

0 0 M̂Ns

1 δNsj

26664

37775T

M̂2j ¼ TTM̂

1

2δ1j 0 00 ⋱ 0

0 0 M̂Ns

2 δNsj

26664

37775T;

M̂3j ¼ TTM̂

1

3δ1j 0 00 ⋱ 0

0 0 M̂Ns

3 δNsj

26664

37775T ð34Þ

where M̂s0 represent the mass matrix in generalized coordinates

vsðtÞ of a substructure that does not depend on the design vari-ables, and

M̂s1 ¼

Ikk 0skb

0sbk 0sbb

" #;

M̂s2 ¼

0skk ΦsT

ikMsiiΨ

sibþΦ

sT

ikMsib

ðΦsT

ikMsiiΨ

sibþΦ

sT

ikMsibÞT 0sbb

264

375

M̂s3 ¼

0skk 0skb

0sbk ðΨsT

ibMsiiþM

sbiÞΨ

sibþΨ

sT

ibMsibþM

sbb

24

35 ð35Þ

where all terms have been previously defined.

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