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RELIABILITY-BASED DESIGN OPTIMIZATION AND ROBUST DESIGN
OPTIMIZATION USING UNIVARIATE DIMENSION REDUCTION METHOD
by
Ikjin Lee
An Abstract
Of a thesis submitted in partial fulfillment of the requirements for the Doctor of
Philosophy degree in Mechanical Engineering in the Graduate College of
The University of Iowa
August 2008
Thesis Supervisor: Professor Kyung K. Choi
1
ABSTRACT
The objective of this study is to propose a new method for inverse reliability
analysis and robust design optimization using the univariate dimension reduction method
(DRM). The current research effort involves: (1) reliability-based robust design
optimization (RBRDO) method using the mean-based DRM; (2) reliability-based design
optimization (RBDO) method using the inverse reliability analysis and the most probable
point (MPP)-based DRM; (3) design sensitivity analyses for RBDO using the MPP-based
DRM; and (4) system inverse reliability analysis and RBDO using the MPP-based DRM
and Ditlevsen’s second order upper bound.
In the RBRDO formulation, the product quality loss function is minimized subject
to the probabilistic constraints. Since the quality loss function is expressed in terms of
the first two statistical moments, mean and variance, it is necessary to accurately estimate
the statistical moments expressed using a multi-dimensional integral, which is
computationally very expensive. To resolve the shortcoming, an RBRDO method is
developed in this study using the mean-based DRM and compared to existing methods.
This study also proposes an inverse reliability analysis method using MPP-based
DRM for highly nonlinear or multi-dimensional systems by obtaining accurate
probability of failure and an RBDO method using the proposed inverse reliability
analysis. Using the MPP-based DRM, a new MPP is obtained, which estimates the
probability of failure of the performance function more accurately than first order
reliability method (FORM) and more efficiently than second order reliability method
(SORM). The new MPP is then used for RBDO to obtain an accurate optimum design
even. Since a gradient-based design optimization is used for RBDO, it is necessary to
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obtain sensitivities of the probabilistic constraints. This study presents rigorous design
sensitivity analyses for both FORM- and DRM-based PMA.
Finally, the MPP-based DRM is extended to the system inverse reliability analysis
and RBDO. This study proposes to use the MPP-based DRM and Ditlevsen’s second
order upper bound for accurate system probability of failure calculation. For the system
RBDO, two efficiency strategies are proposed to save the computational cost.
Abstract Approved:
Thesis Supervisor
Title and Department
Date
3
RELIABILITY- BASED DESIGN OPTIMIZATION AND ROBUST DESIGN
OPTIMIZATION USING UNIVARIATE DIMENSION REDUCTION METHOD
by
Ikjin Lee
A thesis submitted in partial fulfillment of the requirements for the Doctor of
Philosophy degree in Mechanical Engineering in the Graduate College of
The University of Iowa
August 2008
Thesis Supervisor: Professor Kyung K. Choi
Graduate College
The University of Iowa
Iowa City, Iowa
CERTIFICATE OF APPROVAL
PH.D. THESIS
This is to certify that the Ph. D thesis of
Ikjin Lee
has been approved by the Examining Committee
for the thesis requirement for the Doctor of Philosophy
degree in Mechanical Engineering at the August 2008 graduation.
Thesis Committee:
Kyung K. Choi, Thesis Supervisor
Jia Lu
Yong Chen
Shaoping Xiao
Olesya Zhupanska
ii
TABLE OF CONTENTS
LIST OF TABLES ...............................................................................................................v
LIST OF FIGURES ......................................................................................................... viii
CHAPTER
I. INTRODUCTION ............................................................................................1 1.1 Background and Motivation .......................................................................1
1.1.1 Reliability-Based Robust Design Optimization ...............................1 1.1.2 Reliability-Based Design Optimization Using DRM .......................3 1.1.3 System Inverse Reliability Analysis and RBDO ..............................6
1.2 Objectives of the Proposed Study ...............................................................8 1.3 Organization of Thesis ..............................................................................10
II. FUNDAMENTAL CONCEPTS IN DESIGN UNDER
UNCERTAINTY ............................................................................................12 2.1 Introduction ...............................................................................................12 2.2 Reliability Analysis ..................................................................................12
2.2.1 Transformation ...............................................................................13 2.2.2 First Order Reliability Analysis (FORM) .......................................15 2.2.3 Second Order Reliability Analysis (SORM) ..................................16 2.2.4 System Reliability Analysis ...........................................................17
2.3 Inverse Reliability Analysis ......................................................................18 2.4 Reliability-Based Design Optimization (RBDO) .....................................21 2.5 Dimension Reduction Method (DRM) .....................................................22
2.5.1 Mean Value-Based Dimension Reduction Method ........................23 2.5.2 MPP-Based Dimension Reduction Method ....................................25 2.5.3 Rotated Standard Normal V-Space ................................................25
2.6 Reliability-Based Robust Design Optimization........................................26
III. ROBUST DESIGN OPTIMIZATION (RDO) ...............................................30 3.1 Introduction ...............................................................................................30 3.2 Mean Value-Based Dimension Reduction Method ..................................31
3.2.1 Computational Efficiency ...............................................................32 3.2.2 Sensitivity of Statistical Moments ..................................................33
3.3 Performance Moment Integration (PMI) ..................................................35 3.3.1 Derivation of PMI ...........................................................................35 3.3.2 Sensitivity of Statistical Moments ..................................................38
3.4 Percentile Difference Method (PDM) ......................................................39 3.5 Comparison ...............................................................................................42 3.6 Numerical Examples .................................................................................43
3.6.1 Comparison of PMI and DRM for Computation of Moments and Sensitivities .......................................................................................43 3.6.2 Comparison of PMI, DRM and PDM for Identification of Robust Optimum Design .........................................................................47
iii
IV. A NEW DRM-BASED INVERSE RELIABILITY ANALYSIS ..................51
4.1 Introduction ...............................................................................................51 4.2 Error in FORM-Based Reliability Analysis .............................................52 4.3 Inverse Reliability Analysis Using MPP-Based DRM .............................53
4.3.1 Probability of Failure Calculation Using Constraint Shift .............53 4.3.2 Reliability Index Update .................................................................58 4.3.3 MPP Update Method ......................................................................59
4.4 Numerical Examples .................................................................................60 4.4.1 Comparison of FORM, SORM and DRM ......................................60 4.4.2 Inverse Reliability Analysis Using DRM .......................................63
V. SENSITIVITY ANALYSES OF FORM AND DRM-BASED
PERFORMANCE MEASURE APPROACH FOR RBDO ...........................66 5.1 Formulation of FORM and DRM-Based PMA for RBDO ......................66 5.2 Sensitivity Analyses for FORM-Based PMA ...........................................67 5.3 Sensitivity Analyses for DRM-Based PMA .............................................71
5.3.1 Sensitivity of Probabilistic Constraint at True DRM-Based MPP .........................................................................................................72 5.3.2 Sensitivity of Probabilistic Constraint at Approximate DRM-Based MPP ..............................................................................................75 5.3.3 Convergence Study Using Taylor Series Expansion ......................76
5.4 Numerical Examples .................................................................................78 5.4.1 Sensitivities for FORM-based PMA ..............................................79 5.4.2 Sensitivities for DRM-based PMA .................................................81 5.4.3 Convergence Study .........................................................................83
VI. RBDO AND RBRDO USING DRM-BASED INVERSE
RELIABILITY ANALYSIS ...........................................................................85 6.1 Introduction ...............................................................................................85 6.2 Algorithm of DRM-Based RBDO ............................................................85 6.3 Strategy for Efficiency of DRM-Based RBDO ........................................87 6.4 Numerical Examples for DRM-Based RBDO ..........................................90
6.4.1 Effectiveness of Reduced Rotation Matrix .....................................90 6.4.2 Comparison of Various RBDO Methods .......................................91 6.4.3 RBDO for Side Impact Crashworthiness .......................................95 6.4.4 Tracked Vehicle Roadarm Problem .............................................100
6.5 Numerical Examples for RBRDO ..........................................................106 6.5.1 RBRDO for 2-D Mathematic Example ........................................106 6.5.2 RBRDO for Side Impact Crashworthiness ...................................109
VII. SYSTEM INVERSE RELIABILITY ANALYSIS AND RBDO ................112
7.1 Introduction .............................................................................................112 7.2 System Inverse Reliability Analysis .......................................................113
7.2.1 Component Probability of Failure Calculation .............................114 7.2.2 Joint Probability of Failure Calculation Using FORM .................115 7.2.3 System Probability of Failure Calculation ...................................118
7.3 System Reliability-Based Design Optimization .....................................119 7.3.1 Formulation of System RBDO .....................................................119 7.3.2 Sensitivity Analyses .....................................................................120
iv
7.3.3 Efficiency Strategies .....................................................................124 7.4 Numerical Examples ...............................................................................126
7.4.1 Accuracy of Sensitivity ................................................................126 7.4.2 Comparison of Critical Constraint Identification Methods ..........127 7.4.3 Comparison of System RBDO Using FORM and MPP-Based DRM ......................................................................................................130
VIII. CONCLUSION AND FUTURE RECOMMENDATION ...........................134 8.1 Conclusions.............................................................................................134
8.1.1 Reliability-Based Robust Design Optimization (RBRDO) ..........134 8.1.2 DRM-Based Inverse Reliability Analysis and RBDO .................135 8.1.3 Sensitivity Analyses for RBDO using FORM and MPP-Based DRM ...........................................................................................136 8.1.4 System Inverse Reliability Analysis and RBDO ..........................137
8.2 Future Recommendation .........................................................................138
REFERENCES ................................................................................................................139
v
LIST OF TABLES
Table
2.1. Probability Distribution and Its Transformation between X and U-space ................14
2.2. Gaussian Quadrature Points and Weights .................................................................24
3.1. Comparison of First and Second Moments of Eq. (3.29) .........................................44
3.2. Sensitivity of Mean Value Using PMI and DRM for Eq. (3.29) ..............................45
3.3. Sensitivity of Variance Using PMI and DRM for Eq. (3.29). ..................................45
3.4. Comparison of First and Second Moments of Eq. (3.30) .........................................46
3.5. Sensitivity of Mean Value Using PMI and DRM for Eq. (3.30) ..............................46
3.6. Sensitivity of Variance Using PMI and DRM for Eq. (3.30).. .................................47
3.7. Position and Value of Optimum Using Three Methods for Eq. (3.31) .....................49
4.1. PF by MCS When N=2 (Highly Nonlinear) ..............................................................52
4.2. PF by MCS When a=0.2 (Multi-dimensional) ..........................................................53
4.3. Calculation of FP by Various Methods for 2-D Example. .......................................62
4.4. Calculation of FP by Various Methods for 4-D Example ........................................62
4.5. Iterative Way of Finding DRM-Based MPP Using Approximation.........................64
4.6. Iterative Way of Finding DRM-Based MPP Using New MPP Search.....................64
5.1. Comparison of Sensitivities Using Analytic and FDM Results ...............................79
5.2. Properties of Random Variables for Side Impact Problem .......................................80
5.3. Comparison of Sensitivities Using Analytic and FDM Results ...............................80
5.4. Comparison of Sensitivities Using Analytic and FDM Results ...............................81
5.5. Comparison of Sensitivities at True DRM-based MPP ............................................82
5.6. Comparison of Sensitivities at Approximated DRM-based MPP ............................82
5.7. Convergence History of Sensitivities for Second Constraint ...................................84
6.1. Effectiveness of Reduced Rotation Matrix ...............................................................90
6.2. DRM-Based RBDO (3pts) with New Tolerances ....................................................92
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6.3. Various RBDO Results with Target Probability of Failure Tar 5.0%iFP . ...............93
6.4. Updated Reliability Index at the Optimum ...............................................................94
6.5. Properties of Random Variables for Side Impact Problem .......................................96
6.6. Design History for Side Impact Example Using FORM-Based RBDO ...................97
6.7. Constraint History for Side Impact Example Using FORM-Based RBDO..............97
6.8. Probability of Failure at Optimum Using FORM-Based RBDO ..............................98
6.9. Design History for Side Impact Example Using DRM-Based RBDO .....................99
6.10. Constraint History for Side Impact Example Using DRM-Based RBDO ................99
6.11. Probability of Failure at Optimum Using DRM-Based RBDO ..............................100
6.12. Comparison of Various RBDOs .............................................................................100
6.13. Properties of Input Random Variables for Roadarm ..............................................103
6.14. Design History for Roadarm Using DRM-Based RBDO .......................................104
6.15. Constraint History for Roadarm Using DRM-Based RBDO..................................104
6.16. Comparison of Design Optimizations for Roadarm ...............................................105
6.17. Properties of Random Variables of Eq. (6.17)........................................................107
6.18. Variance Estimation Using PMI at Initial and Optimum Design ...........................107
6.19. Variance Estimation Using DRM at Initial and Optimum Design .........................107
6.20. Optimum Design Using FORM-based RBRDO .....................................................108
6.21. Optimum Design Using DRM-based RBRDO .......................................................109
6.22. Properties of Design and Random Parameters for Side Impact Problem ...............109
6.23. Variance Using PMI at Initial and Optimum Design. ............................................110
6.24. Variance Using Mean-Based DRM at Initial and Optimum Design ......................110
6.25. Optimum Design Comparison. ...............................................................................110
6.26. Constraint Comparison at Optimum Design ...........................................................111
7.1. Comparison of Sensitivities Using Analytic and FDM Results .............................127
7.2. Comparison of Analytic and Approximate Sensitivity ...........................................127
7.3. Comparison of Critical Constraint Identification Methods ( 2.275%all
FP ) ........129
vii
7.4. Comparison of FORM and MPP-Based DRM ( 2.275%all
FP ) ...........................132
viii
LIST OF FIGURES
Figure
2.1. MPP and Reliability Index HL in U-Space .............................................................15
2.2. Difference between Reliability Analysis and Inverse Reliability Analysis ..............19
2.3. Standard Normal U-space and Rotated Standard Normal V-space ..........................26
2.4. Comparison of Conventional and Robust Design Optimization. .............................28
3.1. Approximation of CDF Using MPP Locus ..............................................................36
3.2. Basic Concept of Robust Design Using PDM ..........................................................41
3.3. Shape and Variance of Performance Function .........................................................48
3.4. Accuracy of PMI and DRM with Five Quadrature Points .......................................50
4.1. DRM-based MPP for Concave and Convex Functions ............................................56
4.2. Performance Function in X-space and V-space for 2-D problem ............................61
5.1. Comparison of Approximated and True DRM-based MPP......................................72
6.1. Algorithm of DRM-Based RBDO ............................................................................86
6.2. Feasibility Identification Using PMA+ and New Tolerances ...................................92
6.3. Updated Reliability Index at Optimum for 1( )G X and 2 ( )G X ................................94
6.4. Side Impact Model. ...................................................................................................95
6.5. Finite Element Model of Roadarm. ........................................................................101
6.6. Fatigue Life Contour and Critical Nodes of Roadarm ............................................101
6.7. Shape Design Variables for Roadarm. ....................................................................102
6.8. Optimum Design of RBRDO for Eq. (6.17) ...........................................................108
7.1. True and FORM-Base Joint Failure Region ...........................................................116
7.2. Three Cases of Joint Probability of Failure Calculation .........................................119
7.3. Performance Functions for Eq. (7.43) ....................................................................129
7.4. Performance Functions for Eq. (7.45). ...................................................................131
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CHAPTER I
INTRODUCTION
This study presents new methods in reliability-based robust design optimization
(RBRDO) and reliability-based design optimization (RBDO) with independent random
variables. First, a new statistical moment estimation method, which is essential for robust
design optimization (RDO), using the mean value-based dimension reduction method
(DRM), is proposed. Second, a new stochastic method to solve highly nonlinear and
multi-dimensional problems using the most probable point (MPP)-based DRM is
presented for the inverse reliability analysis and subsequent design optimization.
Sensitivity analyses for RBDO using FORM and MPP-based DRM are also carried out.
In the last, a new system reliability analysis and system RBDO using the inverse
reliability analysis through the MPP-based DRM and Ditlevsen’s second order upper
bound are proposed.
Section 1.1 presents background and motivation of the proposed research, and
Section 1.2 provides objectives of the proposed research, and Section 1.3 describes the
thesis organization.
1.1 Background and Motivation
1.1.1 Reliability-Based Robust Design Optimization
RBDO is a method to achieve the confidence in product reliability at a given
probabilistic level, while RDO is a method to improve the product quality by minimizing
variability of the output performance function. Since both design methods make use of
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uncertainties in design variables and other input parameters, it is very natural for the two
different methodologies to be integrated to develop an RBRDO method. Hence, several
approaches to integrate robust design (Su and Renaud, 1997; Kalsi et al., 2001) and
reliability-based design (Du and Chen, 2001; Youn et al., 2005a; Youn et al., 2005b)
have been proposed (Du et al., 2004; Mourelatos and Liang, 2005; Youn et al., 2005c).
The product quality in robust design can be described by use of the first two
statistical moments of a performance function: mean and variance (Chandra, 2001). Thus,
it is necessary to develop methods that estimate the first two statistical moments of the
performance function and their sensitivities accurately and efficiently. The statistical
moments can be analytically expressed using a multi-dimensional integral. However, it is
practically impossible to calculate the statistical moments of the performance function
using the multi-dimensional integral. Hence, there are various numerical methods
proposed to estimate the statistical moments in literature: experimental design (Taguchi
et al., 1989), first order Taylor series expansion (Su and Renaud, 1997; Kalsi et al., 2001;
Buranathiti et al., 2004), Monte Carlo simulation (MCS) (Rubinstein, 1981; Lin et al.,
1997), importance sampling method (Rubinstein, 1981; Schueller and Stix, 1987;
Engelund and Rackwitz, 1993; Denny, 2001), and Latin hypercube sampling method
(Mckay, 1979; Walker, 1986; Stein, 1987; Olsson and Sandberg, 2002).
MCS could be accurate for the statistical moment estimation; however it requires
a very large number of function evaluations. Therefore, in many large-scale engineering
applications, it is not practical to use MCS due to its expensive computational cost. The
experimental design also needs a large amount of computation when the number of
design variables is large. The first order Taylor series expansion has been widely used to
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estimate the first and second statistical moments for robust design due to its simplicity
and easiness. However, the first order Taylor series expansion results in a large error
especially when the input random variables have large variations. This is because the first
order Taylor series expansion does not use all information of the probability density
functions (PDF) of the input random variables.
To overcome the shortcomings explained above, three methods have been
recently proposed: the univariate DRM (Xu and Rahman, 2003; Xu and Rahman, 2004a;
Xu and Rahman, 2004b), performance moment integration (PMI) (Youn et al., 2005c),
and percentile difference method (PDM) (Du et al., 2004; Mourelatos and Liang, 2005).
Hence, a comparison study using three methods for RDO will be needed in terms of how
accurately and efficiently the methods can estimate the statistical moments of the
performance function and thus how accurately these methods can find an optimum design
to minimize the variance of the performance function.
1.1.2 Reliability-Based Design Optimization Using
Dimension Reduction Method
In recent years, there have been various attempts to develop enhanced reliability
analysis methods to accurately compute the probability of failure of a performance
function. The most widely used reliability analysis methods are (1) analytical methods
and (2) simulation or sampling methods. The analytical methods include the MPP-based
method and PDF approximation method. Furthermore, the MPP-based method includes
the First Order Reliability Method (FORM) (Hasofer and Lind, 1974; Palle and Michael,
1982; Madsen et al., 1986; Haldar and Mahadevan, 2000) and the Second Order
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Reliability Method (SORM) (Breitung, 1984; Hohenbichler and Rackwitz, 1988). FORM
or SORM computes the probability of failure by approximating the performance function
( )G X using the first or second order Taylor series expansion at MPP. Since the FORM
or SORM-based method requires MPP search, the sensitivity analysis is usually used for
the methods. When the sensitivities are not available, the response surface method can be
used for the reliability analysis and design optimization (Jin et al., 2003; Youn and Choi,
2004). The PDF approximation method (Rosenblueth, 1975; Du and Huang, 2006; Youn
et al., 2006) evaluates PDF of the performance function by assuming a general
distribution type and then, using the approximated PDF, the method evaluates the
probability of failure of the performance function. The simulation or sampling methods,
such as MCS (Lin et al., 1997; Rubinstein, 1981), importance sampling method
(Rubinstein, 1981; Schueller and Stix, 1987; Engelund and Rackwitz, 1993), and Latin
hypercube sampling method (Mckay, 1979; Walker, 1986; Stein, 1987; Olsson and
Sandberg, 2002), can be readily used for the probability of failure calculation since these
methods do not require any analytical formulation.
Among these methods, the MPP-based method including FORM and SORM is
still a common approach. However, the reliability analysis using FORM could be very
well erroneous if the performance function is highly nonlinear or multi-dimensional or
both. This is because FORM approximates the performance function using a linear
function, which cannot reflect complicity of nonlinear or multi-dimensional functions.
Although the reliability analysis using SORM may be accurate, it is not easy to use since
SORM requires the second-order sensitivities, which are difficult and very expensive to
obtain in practical engineering problems. The accuracy of the response surface method is
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still challenging, especially for the highly nonlinear and multi-dimensional performance
function, even though the method could be efficient. The simulation or sampling-based
method could be accurate. However they require a very large number of function
evaluations at high computational cost. The PDF approximation method can yield
accurate results only if the output probability distribution type is known or accurately
assumed. Moreover, the method should be combined with the response surface method
for the design optimization (Youn et al., 2006), which may have accuracy problem.
DRM has been recently proposed to represent a multi-dimensional function using
the sum of lower dimensional functions. Because of its wide applicability, DRM has been
used for RDO (Lee et al., 2006; Lee et al., 2007), RBDO (Wei, 2006), and PDF
approximation (Du and Huang, 2006; Youn et al., 2006). For the robust design, the mean-
based DRM is used to calculate the statistical moments of the performance function. This
moment calculation using the mean-based DRM is also used to approximate PDF of the
performance function for reliability analysis (Du and Huang, 2006; Youn et al., 2006).
The reliability analysis using the PDF approximation method and mean-based DRM
requires accurate moment calculation, PDF approximation, and response surface
generation, which could be limitations of the method. That is, it is inherently difficult to
accurately estimate the tail end of PDF (i.e., reliability) when PDF is approximated using
numerically obtained moments from the mean-based DRM.
Thus, a new reliability analysis method using MPP-based DRM is needed with
greater accuracy and/or better efficiency than existing methods. Such a method could be
very effective for RBDO.
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RBDO using MPP-based reliability analysis is a gradient-based design
optimization, which requires the sensitivity of the probabilistic constraint at MPP with
respect to the design variable, which is the mean value of input random variables. Many
works have been conducted to study the sensitivities of the probabilistic constraints
(Haldar and Mahadevan, 2000; Tu and Choi, 1999; Ditlevsen and Madsen, 1996; Hou,
2004; Gumbert et al., 2003; Hohenbichler and Rackwitz, 1986; Rahman and Wei, 2007).
The sensitivity using the Reliability Index Approach (RIA) for the FORM-based RBDO
(Haldar and Mahadevan, 2000; Tu and Choi, 1999; Ditlevsen and Madsen, 1996; Hou,
2004; Gumbert et al., 2003; Hohenbichler and Rackwitz, 1986) and the DRM-based
RBDO (Rahman and Wei, 2007) is derived in detail. However, the rigorous analytical
sensitivity of the probabilistic constraints using the Performance Measure Approach
(PMA) for the inverse reliability analysis for both FORM-based and DRM-based RBDO
has not yet been explained in detail in the literature. Thus, one of the main goals of this
thesis is to derive the analytic sensitivities of the probabilistic constraints at MPP
obtained from the inverse reliability analysis using both FORM and MPP-based DRM.
1.1.3 System Inverse Reliability Analysis and RBDO
Not only the estimation of the component probability of failure but also the
estimation of the system probability of failure has been the main concern in structural
reliability analysis for over three decades. According to the logical relationship of the
failure modes of structures, structural systems can be divided into three types: series
systems, parallel systems, and hybrid systems (Zhao et al., 2007). In this thesis, the
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reliability analysis with the series system will be explained since it is most frequently
encountered in practical engineering applications.
Since the analytic estimation of the system probability of failure involves multi-
dimensional integration over the overall failure domain, it is numerically difficult to
evaluate. Hence, several approaches to resolve the numerical difficulty have been
proposed including the narrow bound estimation (Ditlevsen, 1979) and MCS. As
explained in the previous section, MCS requires very expensive computational cost,
which is not acceptable for practical engineering applications. For the narrow bound
method, Ditlevsen’s first order upper bound, which is the summation of component
failure probabilities, can be used as the system probability of failure (Ba-abbad et al.,
2006) or Ditlevsen’s second order upper bound by considering the joint probability of
failure can be used (Ang and Tang, 1984; Liang et al., 2007). However, these narrow
bound methods will only work for linear or mildly nonlinear performance functions since
they approximate performance functions using the first order Taylor series expansion,
i.e., FORM.
Thus, more accurate system reliability analysis method than the traditional
methods is needed for the system with highly nonlinear and multi-dimensional
performance functions. Furthermore, the system RBDO using the accurate system inverse
reliability analysis needs to be proposed to solve highly nonlinear and multi-dimensional
problem accurately and efficiently.
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1.2 Objectives of the Proposed Study
The first objective of this study is to propose RBRDO using the mean-based
DRM, which includes the accurate estimation of the statistical moments and their
sensitivities. To see advantages of the proposed method, comparison studies will be
carried out. For the comparison study, PMI and PDM for RBRDO are introduced and
explained. Both DRM and PMI directly estimate the statistical moments. On the other
hand, in PDM, the robustness is achieved through a design objective in which the
variation of the design performance is approximately evaluated through the percentile
performance difference between the right and left tails of the performance distribution
(Du et al., 2004). Thus, three methods can be compared in terms of how accurately these
methods can find an optimum design to minimize the variance of the performance
function. Hence, in this study, three methods are evaluated by comparing the variances at
the optimum designs. PMI and DRM are also compared in terms of how accurately and
efficiently these methods can estimate the statistical moments of the performance
function. The comparison study through numerical examples illustrates that mean-based
DRM is the most accurate and efficient method when the number of design variables is
small and PMI is a better option when the number of design variables is relatively large.
The second objective of the study is to develop a new inverse reliability analysis
method and subsequent RBDO using the MPP-based DRM, which is known as a more
accurate method than the mean-based DRM for the reliability analysis (Wei, 2006). The
MPP-based DRM is used to accurately calculate the probability of failure after finding
the FORM-based MPP, which is then used to develop an enhanced inverse reliability
analysis method (i.e., MPP search) that is accurate for highly nonlinear and multi-
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dimensional problems. Since the DRM-based inverse reliability analysis still requires
MPP search, it can be categorized as an MPP-based method. A three-step computational
procedure is proposed in this study to carry out the inverse reliability analysis accurately
and efficiently using the MPP-based DRM: the probability of failure calculation using
constraint shift, reliability index update, and MPP update method. Using the three-step
procedure, a new DRM-based MPP is obtained, which is used for the next design
iteration in RBDO. Since the proposed RBDO using the DRM-based MPP requires more
function evaluations, the enhanced hybrid mean value (HMV+) (Youn et al., 2005a) is
used for the efficient inverse reliability analysis, and the enriched performance measure
approach (PMA+) (Youn et al., 2005b) with efficient methods, which are new tolerances
for constraint activeness and a reduced rotation matrix, are proposed for efficient design
optimization in this study. For RBDO using FORM and MPP-based DRM, it is necessary
to carry out a rigorous sensitivity analyses to obtain the optimal design efficiently.
The last objective of the study is to propose more accurate system reliability
analysis method than the traditional methods for the system with highly nonlinear and
multi-dimensional performance functions and subsequent system RBDO. For the series
system with highly nonlinear and multi-dimensional performance functions, Ditlevsen’s
second order upper bound is adapted as the system probability of failure. MPP-based
DRM will be used for the accurate estimation of the component probability of failure and
FORM-based joint probability of failure calculation depending on the convexity or
concavity of the performance functions will be used. The result is also compared with the
system reliability analysis using FORM and MCS. For the system RBDO, two efficiency
strategies, which are the Mean Value (MV) method for identification of critical
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constraints and new design closeness concept, are proposed to save the computational
cost.
1.3 Organization of Thesis
Chapter 2 presents fundamental concepts for the reliability analysis, robust design
optimization, and reliability-based design optimization, which are helpful to better
understand the proposed methods.
Chapter 3 presents an RDO method using the mean-based DRM. First, the
statistical moment estimation and its sensitivity calculation using DRM are explained.
Second, the statistical moment estimation and its sensitivity calculation using PMI and
PDM are explained for the comparison purpose. Last, three methods are compared
through several numerical examples.
Chapter 4 proposes new inverse reliability analysis and system reliability analysis
methods using the MPP-based DRM, which consists of three numerical procedures: the
probability of failure calculation using constraint shift, reliability index update, and MPP
update. Comparison study with FORM and SORM is carried out to evaluate how
accurately the proposed method can estimate the probability of failure.
Chapter 5 carries out rigorous sensitivity analyses for RBDO using FORM and
MPP-based DRM. For RBDO using MPP-based DRM, an approximate sensitivity is also
proposed to save the computational cost. Comparison study between analytic and
approximate sensitivity is carried out in this chapter.
11
11
Chapter 6 proposes an RBDO and RBRDO method using the MPP-based DRM.
The proposed RBDO and RBRDO involve two efficiency strategies: new tolerances for
constraint activeness and reduced rotation matrix.
Chapter 7 proposes an accurate system inverse reliability analysis method using
MPP-based DRM and Ditlevsen’s second order upper bound. Using the accurate system
inverse reliability analysis, the system RBDO is also proposed. For the system RBDO,
sensitivity analysis is also carried out in a similar way as Chapter 5.
Chapter 8 provides conclusions from the proposed work.
12
12
CHAPTER II
FUNDAMENTAL CONCEPTS IN DESIGN UNDER UNCERTAINTY
2.1 Introduction
This chapter presents review of fundamental concepts in design under uncertainty.
Section 2.2 and 2.3 discuss the basic idea of reliability analysis and inverse reliability
analysis which are necessary for RBDO that will be explained in Section 2.4. Section 2.5
illustrates two Dimension Reduction Methods (DRM) based on the reference point used:
the mean value-based DRM and MPP-based DRM. The mean value-based DRM is an
accurate and efficient tool for the statistical moment calculation which is required for
robust design optimization explained in Section 2.6 and Chapter 3. The MPP-based DRM
will be used for more accurate reliability analysis and design optimization in Chapter 4
and Chapter 6, respectively.
2.2 Reliability Analysis
A reliability analysis entails calculation of probability of failure, denoted by FP ,
which is defined using a multi-dimensional integral (Madsen et al., 1986)
( ) 0
[ ( ) 0] ( )FG
P P G f d
XX
X x x (2.1)
where T
1 2={ , , , }NX X XX is an N-dimensional random vector, ( )G X is the
performance function such that ( ) 0G X is defined as failure, and ( )fX x is a joint
probability density function (PDF) of the random variable X . In most real engineering
applications, the exact evaluation of Eq. (2.1) is very difficult or often impossible to
13
13
obtain since ( )fX x is generally non-Gaussian and ( )G X is highly nonlinear. To handle
the non-Gaussian ( ),fX x a transformation from the original X-space into the independent
standard normal U-space is introduced. In addition, ( )G X is approximated using first
order Taylor series expansion in the First Order Reliability Method (FORM) or second
order Taylor series expansion in the Second Order Reliability Method (SORM) if ( )G X
is highly nonlinear.
2.2.1 Transformation
Consider an N-dimensional random vector X with a joint cumulative distribution
function (CDF) ( )FX x . Let :T X U denote a transformation from X-space to U-space
that is defined by Rosenblatt transformation (Rosenblatt, 1952) as
1
2
1
1 1
1
2 2 1
1
1 2 1
:
, , ,N
X
X
N X N N
u F x
u F x xT
u F x x x x
(2.2)
where 1 2 1, , ,iX i iF x x x x is the conditional CDF given by
1 2
1 2 1
1 2 1
1 2 1
1 2 1
( , , , , ), , ,
( , , , )
i
i
i
i
x
X X X i
X i i
X X X i
f x x x dF x x x x
f x x x
(2.3)
and ( ) is the standard normal CDF given by
21 1( ) exp
22
u
u d
(2.4)
The inverse transformation can be obtained from Eq. (2.2) as
14
14
1
2
1
1 1
1
2 2 11
1
1 2 1
( ):
( , , , )N
X
X
N X N N
x F u
x F u xT
x F u x x x
(2.5)
If the N-dimensional random vector X is independent, that is, the joint PDF is given by
1 21 2( ) ( ) ( ) ( )
NX X X Nf f x f x f x X x (2.6)
where ( )iX if x are the marginal PDFs, then Rosenblatt transformation and the inverse
transformation are simplified as
1 1 and i ii X i i X iu F x x F u (2.7)
where ( )iX iF x are the marginal CDFs. Table 2.1 shows five representative distributions
and their transformations assuming random variables are independent.
Table 2.1. Probability Distribution and Its Transformation between X and U-space
Parameters PDF Transformation
Normal mean
standard deviation
20.5[ ]1( )
2
x
f x e
X U
Log-
normal
2 2ln[1 ( ) ]
,
2ln( ) 0.5
2ln0.5[ ]1
( )2
x
f x ex
exp( )X U
Weibull
1(1 )v
k ,
2 2 22 1[ (1 ) (1 )]v
k k
( )1( ) ( )
kx
k vk x
f x e
1
[ ln( ( ))]kX v U
Gumbel 0.577
,6
( )( )( )xx ef x e
1ln[ ln( ( ))]X U
Uniform 2
a b
,
12
b a
1( ) ,f x a x b
b a
( ) ( )X a b a U
15
15
In this study, the input random variables are assumed to be independent for the
simplicity of calculation and the study with dependent input random variables will be the
future research topic.
2.2.2 First Order Reliability Method (FORM)
To calculate the probability of failure of the performance function ( )G x using
FORM and SORM, it is necessary to find the most probable point (MPP), which is
defined as the point *u on the limit state function ( ( ) 0g u ) closest to the origin in the
standard normal U-space as shown in Figure. 2.1.
Figure 2.1. MPP and Reliability Index HL in U-space
Source: Wei, D., “A Univariate Decomposition Method For Higher-Order Reliability
Analysis And Design Optimization,” Ph. D. Thesis, University of Iowa, 2006.
16
16
In this thesis, the performance function in U-space is defined as
( ) ( ( )) ( )g G G u x u x using the Rosenblatt transformation. Hence, MPP can be found
by solving the following optimization problem to
minimize
subject to g( ) 0
u
u (2.8)
After finding MPP, the distance from MPP to the origin is commonly called the
Hasofer-Lind reliability index (Hasofer and Lind, 1974) and denoted by HL , that is,
*
HL u . Using the reliability index HL , FORM can approximate the probability of
failure using a linear approximation of the performance function as
FORM
HL( )FP (2.9)
2.2.3 Second Order Reliability Method (SORM)
MPP obtained by solving Eq. (2.8) is also used for the probability of failure
calculation using SORM. Using a quadratic approximation of the performance function in
U-space and the rotational transformation from U-space to V-space which will be
explained in Section 2.5.3, the probability of failure can be obtained using SORM as
(Breitung, 1984; Hohenbichler and Rackwitz, 1988; Rahman and Wei, 2006)
1
2SORM HL
HL 1 1
HL
( )( ) 2
( )F N NP
I A (2.10)
17
17
where 1 1 T
1
1
2
N N
UN NNg
A AA R HR
A A, H is the Hessian matrix evaluated at the MPP
in X-space, R is the rotation matrix such that u Rv , and ( ) is the PDF of a standard
Gaussian random variable.
2.2.4 System Reliability Analysis
When there are more than one performance function, the system probability of
failure is defined as
sys
1
( ) 0m
F i
i
P P G
X (2.11)
where m is the number of performance functions and a performance function is defined
as failure if ( ) 0iG X . However, since the right hand side of Eq. (2.11) is not easy to
compute numerically, the system probability of failure is conservatively approximated
using Ditlevsen’s first-order upper bound (Ditlevsen, 1979) by the sum of the probability
of failures as
sys
1i
m
F F
i
P P
(2.12)
where iFP is the probability of failure for i
th performance function or using Ditlevsen’s
second-order upper bound (Ditlevsen, 1979) as
sys
1 2
max( )i ij
m m
F F Fj i
i i
P P P
(2.13)
where ijFP is the joint probability of failure when the i
th and j
th failure modes occur
simultaneously.
18
18
2.3 Inverse Reliability Analysis
The reliability analysis presented in Section 2.2 is called the reliability index
approach (RIA) (Tu and Choi, 1999) since it finds the reliability index HL using Eq.
(2.8). The advantage of RIA is that the probability of failure for the performance function
can be calculated at a given design, for example, using Eqs. (2.9) and (2.10).
However, it is well known that the inverse reliability analysis in the performance
measure approach (PMA) (Tu and Choi, 1999; Tu et al., 2001; Choi et al., 2001; Youn et
al., 2003) is much more robust and efficient than the reliability analysis in RIA. PMA
does not calculate the probability of failure directly. Instead, PMA judges whether or not
a given design satisfies the probabilistic constraint for a given target probability of failure
Tar
FP . Using FORM, the target reliability index t can be calculated as 1 Tar
t ( )FP
using Eq. (2.9), and then the feasibility of the given design can be checked by solving the
following optimization problem to
t
maximize g( )
subject to
u
u (2.14)
Since Eq. (2.14) is the inverse problem of Eq. (2.8), this is called the inverse reliability
analysis. The optimum point of Eq. (2.14) is also called the MPP and denoted by *u . If
the constraint function value at the MPP, *( ),g u is less than zero ( ( ) 0G X is defined as
safe), then the probabilistic constraint is satisfied for the given target reliability t and
target probability of failure. The inverse reliability analysis using SORM is much more
difficult and has not been developed yet. Moreover, it requires the second order
sensitivity. We can compare the difference between the reliability analysis and inverse
reliability analysis graphically using Figure. 2.2.
19
19
To find the MPP using the inverse reliability analysis with the given target
reliability index t , some methods have been developed including the mean value (MV)
method, advanced mean value (AMV) method (Wu et al., 1990; Wu, 1994), hybrid mean
value (HMV) method (Youn et al., 2003), and enhanced hybrid mean value (HMV+)
method (Youn et al., 2005a).
Figure 2.2. Difference between Reliability Analysis and Inverse Reliability Analysis
The MV method linearly approximates the performance function using the
function and gradient information at the mean value in U-space as
1
( ) ( ) ( )i
N
i U
i i
gg g U
U
U
U
U=μ
U μ (2.15)
Then, MPP of the inverse reliability analysis using MV can be obtained as
20
20
*
MV t ( ) Uu α μ (2.16)
where ( )Uα μ is the normalized gradient vector evaluated at the mean value and written
as
( )
( )( )
U
U
g
g
UU
U
μα μ
μ (2.17)
where T
1
{ , , }U
NU U
. This MV method is a crude method to find MPP of the
inverse reliability analysis.
However, since it does not require further function evaluation and sensitivity
analysis, MPP by MV method can be a good approximation to judge which constraint is
active or not when a constraint function is far from the design point.
The MPP obtained by the MV method can be considered as the first iteration of
the AMV method. AMV uses the gradient at MPP obtained by the MV method to find the
next MPP candidate and the iteration will continue to perform until the approximate MPP
converges to the correct MPP. Hence, the AMV method can be formulated as
(1) * ( 1) ( )
AMV MV AMV t AMV, ( )k k u u u α u (2.18)
This AMV method is known as an efficient method when the constraint function is
convex. A constraint function is defined as convex around the MPP if FORM-based
reliability analysis underestimates the probability of failure and vice versa for concave.
For example, the constraint function in Figure. 2.2 is concave around the MPP since
FORM-based reliability analysis overestimates the probability of failure.
To resolve the weakness of AMV for a concave function, the HMV method uses
AMV method when a constraint function is convex and the conjugate mean value (CMV)
21
21
(Youn et al., 2003) method when a constraint function is concave. HMV+ method uses an
interpolation between two previous MPP candidate points if the constraint function is
concave instead of using the CMV method.
2.4 Reliability-Based Design Optimization (RBDO)
In general, the RBDO model can be formulated to
Tar
minimize Cost( )
subject to [ ( ) 0] , 1, ,
, R and R
ii F
L U ndv nrv
P G P i nc
d
X
d d d d X
(2.19)
where T{ }id d μ(X) is the design vector; T{ }iXX is the random vector; and nc ,
ndv and nrv are the number of probabilistic constraints, design variables, and random
variables, respectively. Using the inverse reliability analysis, the ith
probabilistic
constraint can be rewritten as
Tar *[ ( ) 0] 0 ( ) 0ii F iP G P G X x (2.20)
where *( )iG x is the thi probabilistic constraint evaluated at the MPP *x in X-space.
Using FORM, Eq. (2.19) can be reformulated to
Tar
minimize Cost( )
subject to [ ( ) 0] ( ), 1, ,
, R and R
i ii F t
L U ndv nrv
P G P i nc
d
X
d d d d X
(2.21)
where it
is the target reliability index for the ith
constraint and the probabilistic
constraint can be changed into
*
FORM[ ( ) 0] ( ) 0 ( ) 0ii t iP G G X x (2.22)
22
22
where *
FORMx is the FORM-based MPP which can be obtained by solving Eq. (2.14) and
transformation ( )T * *u x in
Eq. (2.7). For the simplicity, *
x means the FORM-based
MPP hereafter.
To solve the formulation in Eq. (2.19), it is required to calculate the sensitivity of
the probabilistic constraint in Eq. (2.20) with respect to a design parameter ( )i id X .
The sensitivity of the probabilistic constraint with respect to the design parameter is
written using the chain rule as
* * ***
T*
1
( ) Ni
i i
xG G G G
x
x x x x x xx xx x
x x
d d d d x (2.23)
and Eq. (2.23) can be further simplified as (Gumbert et al., 2003; Hou, 2004)
* * *
T*( )G G G
x x x x x x
x x
d d x x (2.24)
2.5 Dimension Reduction Method (DRM)
The dimension reduction method (Xu and Rahman, 2004a; Xu and Rahman,
2004b) is a newly developed technique to approximate the multi-dimensional integration
of a performance function using a function with reduced dimension. There are several
DRMs depending on the level of dimension reduction: (1) univariate dimension
reduction, which is an additive decomposition of N-dimensional performance function
into one-dimensional functions; (2) bivariate dimension reduction, which is an additive
decomposition of N-dimensional performance function into at most two-dimensional
functions; (3) multivariate dimension reduction, which is an additive decomposition of N-
dimensional performance function into at most S-dimensional functions, where S N . In
23
23
this study, the univariate DRM is used for approximation of the performance function due
to its simplicity and efficiency. Computational efficiency of the univariate DRM will be
discussed in Chapter 3. The univariate DRM can be categorized into two different DRMs
depending on which point is used as a reference point to approximate the performance
function: mean value-based DRM and MPP-based DRM.
2.5.1 Mean Value-Based Dimension Reduction Method
In the mean value-based univariate DRM, any N-dimensional performance
function ( )h X can be additively decomposed into one-dimensional functions as
1 1 1 1
1
( ) ( ) ( , , , , , , ) ( 1) ( , , )Nk
k k k
i i i N N
i
h h h x N h
X X (2.25)
where i is the mean value of a random variable iX and N is the number of design
variables. For example, if 1 2( ) ( , )h h X XX , that is 2 and 1N k , then the univariate
additive decomposition of ( )h X at the mean value is
1 2 1 2 1 2( ) ( ) ( , ) ( , ) ( , )h h h X h X h X X (2.26)
The mean value-based univariate DRM can be used to approximate the multi-
dimensional integration for the statistical moment calculation given by
ˆ({ ( )} ) { ( )} ( ) { ( )} ( )k k kE h h f d h f d
X X
X X x x X x x (2.27)
Then, using the mean value-based univariate DRM, one N-dimensional integration in Eq.
(2.27) becomes a summation of N one-dimensional integrations, which will reduce the
number of function evaluations significantly when the number of design variables is
24
24
large. This reduction of the number of function evaluations will be explained in Chapter
3.
The one-dimensional numerical integration can be calculated using the moment-
based integration rule (MBIR) (Xu and Rahman, 2003), which is similar to Gaussian
quadrature (Atkinson, 1989). According to MBIR, the kth
statistical moment of a one-
dimensional function can be obtained as
1
({ ( )} ) ( )n
k k
i i
i
E h X w h x
(2.28)
where iw are weights, ix are quadrature points and n is the number of weights and
quadrature points. If PDF of the design variable is given, then these weights iw and
quadrature points ix can be obtained using MBIR. For the standard normal input random
variable, the weights and quadrature points are shown in Table 2.2 (Atkinson, 1989).
This mean value-based univariate DRM for the statistical moment calculation is
used for robust design optimization, which will be explained in detail in Section 2.6 and
Chapter 3.
Table 2.2. Gaussian Quadrature Points and Weights
n Quadrature Points Weights
1 0.0 1.0
3 3 0.166667
0.0 0.666667
5
2.856970 0.011257
1.355626 0.222076
0.0 0.533333
25
25
2.5.2 MPP-Based Dimension Reduction Method
In the univariate DRM, an N-dimensional performance function ( )G X can be
additively decomposed into one-dimensional functions at the MPP of the random vector
X as
* * * * *
1 1 1
1
ˆ( ) ( ) ( , , , , , , ) ( 1) ( )N
i i i N
i
G G G x x X x x N G
X X x (2.29)
where * * * T
1 2={ , , , }Nx x x*x is the FORM-based MPP of the performance function ( )G X
obtained from Eq. (2.14) and N is the number of random variables. For example, if
1 2( ) ( , )G G X XX with 2N , then the univariate additive decomposition of ( )G X is
* * * *
1 2 1 2 1 2ˆ( ) ( ) ( , ) ( , ) ( , )G G G X x G x X G x x X X (2.30)
This MPP-based univariate DRM will be used for more accurate reliability analysis than
FORM in Chapter 4 and RBDO in Chapter 6.
2.5.3 Rotated Standard Normal V-Space
Consider a performance function ( )G X that depends on T
1 2={ , , , }NX X XX
and whose MPP is denoted by * * * * T
1 2={ , , , } .Nx x xx Since the reliability analysis is
performed in the standard normal U-space obtained using Rosenblatt transformation in
Eq. (2.7), MPP in -spaceU is denoted by * * * T
1 2={ , , , }Nu u u*u .
To obtain the rotated standard normal V-space from U-space, construct an
orthonormal matrix RN NR whose Nth
column is *
*
HL
,
u
α i.e., *
1[ ],R R α where
1
1 RN N R satisfies * T 1 1
1( ) R N α R 0 and N is the number of random variables
26
26
(Wei, 2006). Using the orthonormal transformation u Rv , v represents the rotated
standard normal V-space with the MPP * T
HL{0, ,0, } .v The orthonormal matrix R
can be found, for example, by Gram-Schmidt orthogonalization. However, the
orthonormal matrix R is not uniquely determined. Figure 2.3 shows U-space and V-
space for 2N .
Figure 2.3. Standard Normal U-space and Rotated Standard Normal V-space
2.6 Reliability-Based Robust Design Optimization
In general, a conventional design optimization problem can be formulated to
minimize ( )
subject to ( ) 0, 1, ,
,
i
L U ndv
h
G i nc
R
X
X
X X X X
(2.31)
27
27
where ( )h X is the cost function, iG is the ith
constraint, and X is the design variable
vector; and nc and ndv are the number of constraints and design variables, respectively.
The optimum design of the conventional optimization problem is the deterministic
optimum that could be sensitive to the variation of input design variables and other
parameters.
Due to the variation of the input design variables and other parameters, the output
performance function ( )h X also has variation. Thus, in the robust design, the robustness
of a design objective can be achieved by simultaneously “optimizing the mean
performance H and minimizing the performance variance 2
H ” (Du et al., 2004). In
other words, the goal of the robust design is to find the most insensitive design to the
variation of the design variables and other parameters. Since the robust design is
fundamentally considering the variations of the design variables and other parameters, it
is very natural to integrate the robust design and reliability-based design in one
formulation. This design optimization is called reliability-based robust design
optimization (RBRDO) and can be formulated to
2
Tar
minimize ( , )
subject to ( ( ; ) 0) , 1, ,
, and
i
H H
i F
L U ndv nrv
f
P G P i nc
R R
X d
d d d d X
(2.32)
where 2( , )H Hf is the cost function, d μ(X) is the design vector, X is the random
vector, and iG is the ith
probabilistic constraint. Quantities nc , ndv , and nrv are the
number of probabilistic constraints, design variables, and random variables, respectively.
Figure 2.4 compares a conventional design optimization with a robust optimum
design for a one-dimensional performance function. With the same variabilities of design
28
28
variables, the robust optimum design shows less variation of the performance function
( )h X than the conventional optimum design.
Figure 2.4. Comparison of Conventional and Robust Design Optimum
Source: Kalsi, M., Hacker, K., and Lewis, K., “A Comprehensive Robust Design
Approach for Decision Trade-Offs in Complex Systems Design,” ASME Journal of
Mechanical Design, Vol. 123, No. 1, pp. 1-10, 2001.
Since the cost function in Eq. (2.32) depends on H and 2
H for the robust
optimum design in RBRDO, it is a bi-objective optimization problem. The optimum of
the bi-objective optimization depends on the weight on each term in the cost function.
However, since the main goal of this research is not focused on determination of the
weights, this topic is referred to Marler (Marler and Arora, 2004) for more details. The
cost function 2( , )H Hf in Eq. (2.32) can be formulated in various ways based on
engineering application types (Chandra, 2001; Youn et al., 2005c). The following are
three important cost function types for reliability-based robust design.
29
29
(1) Nominal-the-Best Type
0 0 0
2 2 2
1 2( , ) ( ) ( )H t HH H
H t H
hf w w
h
(2.33)
where th and 0t
h are the target nominal value and the initial target nominal value of the
performance function ( )h X respectively, and 1 2andw w are weights to be determined by
the designer. To reduce the dimensionality problem of two objectives, each term is
normalized by the initial value 0H and
0H .
(2) Smaller-the-Better Type
0 0
2 2 2
1 2( , ) sgn( ) ( ) ( )H HH H H
H H
f w w
(2.34)
where sgn( )H is the signum function of H and has a value of 1 or −1 depending on
the sign of H .
(3) Larger-the-Better Type
0
0
2 2 2
1 2( , ) sgn( ) ( ) ( )H H
H H H
H H
f w w
(2.35)
30
30
CHAPTER III
ROBUST DESIGN OPTIMIZATION (RDO)
Equation Chapter 3 Section 1
3.1 Introduction
As explained in Section 2.6, major concern of RDO is how to estimate the second
statistical moments and their sensitivities accurately and efficiently. Analytically, the kth
statistical moment of the performance function ( )h X can be obtained using the following
integration
({ ( )} ) { ( )} ( )k kE h h f d
X
X X x x (3.1)
where ( )fX x is the joint PDF of the random parameter X . As stated before, it is
practically impossible to calculate the statistical moments of the performance function
using Eq. (3.1) especially when the dimension of the problem is relatively large. The first
order Taylor series expansion has been widely used to estimate the statistical moments
due to its simplicity. Using the first order Taylor series expansion, the mean value and
variance of the performance function can be estimated (Huang and Du, 2006)
respectively as
( )H h μ (3.2)
and
2
2 2
1i
N
H X
i i
h
X
x μ
(3.3)
where T
1{ , , }N μ is the mean of the input random vector X .
31
31
However, Taylor series expansion results in a large error especially when input
random variables have large variations. This is because the first order Taylor series
expansion does not use all information of PDF of input random variables. To overcome
the shortcoming of Taylor series expansion, three numerical methods have been recently
proposed: the mean value-based DRM, performance moment integration (PMI), and
percentile difference method (PDM).
3.2 Mean Value-Based Dimension Reduction Method
Using Eqs. (2.25), (2.27) and (2.28), the mean value and variance of the
performance function ( )h X can be obtained as
1 1 1 1
1
1 1 1 1
1 1
[ ( )]
{ ( , , , , , , ) ( 1) ( , , )}
( , , , , , , ) ( 1) ( , , )
H
N
i i i N N
i
n Nj j
i i i i N N
j i
E h
E h X N h
w h x N h
X
(3.4)
and
2 2 2 2
2 2 2
1 1
1
2 2 2
1 1
1 1
[( ( ) ) ] [ ( )]
{ ( , , , , ) ( 1) ( , , )}
( , , , , ) ( 1) ( , , )
H H H
N
i N N H
i
n Nj j
i i N N H
j i
E h E h
E h X N h
w h x N h
X X
(3.5)
The estimation of statistical moments using the univariate DRM involves two
approximations. As shown in Eqs (3.4) and (3.5), the univariate DRM approximates the
performance function ( )h X using the sum of one-dimensional functions. If
1
( ) ( )N
i i
i
h h X
X where ( )i ih X is any function of iX only, then the approximation is
32
32
exact. However, if there are off-diagonal or mixed terms, then there is some error that
results from approximating off-diagonal terms using sum of one-dimensional functions.
To reduce this error, the bivariate DRM or multivariate DRM can be used (Xu and
Rahman, 2004). The second approximation involves the numerical integration using the
weights and quadrature points. Based on Gaussian quadrature theory (Atkinson, 1989), n
quadrature points and weights give a degree of precision of 2 1.n Hence, if three
quadrature points and weights for each variable are used, the numerical integration error
for a quadratic performance function will disappear. If the performance function is highly
nonlinear, then three quadrature points may not be sufficient to estimate the moments of
the performance function. In this case, the error can be reduced if the number of
quadrature points is increased.
3.2.1 Computational Efficiency
Even though the accuracy is the most important concern, it is also important to
efficiently estimate the statistical moments of the performance function for large-scale
problems. In general, when the output moments are estimated using the univariate DRM
and MBIR, the number of function evaluations required is
number of F.E. 1n N (3.6)
where n is the number of quadrature points and N is the number of design variables. If
the distributions of all input design variables are symmetric, e.g. normal distribution or
uniform distribution, and the number of design variables is odd, then the required number
of function evaluations can be reduced to
number of F.E. ( 1) 1n N (3.7)
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33
Therefore, when the number of design variables is large, the reduction becomes
significant compared to the number of function evaluation required for directly
integrating Eq. (3.1) numerically, which is Nn . However, although the reduction
becomes significant when N is large, the number of function evaluations is still
increasing proportionally to the number of design variables as shown in Eqs. (3.6) and
(3.7).
If bivariate DRM is used to estimate the first and second output moments, then
the number of function evaluations will increase exponentially to
2( 1)
number of F.E. 12
N Nn n N
(3.8)
For example, if the number of design variables is 5 and the number of quadrature points
is 3, then the number of function evaluations by the univariate DRM is 16 from Eq. (3.7)
and the number of function evaluations by bivariate DRM is 106 from Eq. (3.8). Both
numbers are less than 53 243 , which is the required number of function evaluations for
the numerical integration of Eq (3.1) by including the mixed variable terms. However, the
number of function evaluations by the univariate DRM is significantly less than the
number of function evaluations for bivariate DRM. For this reason, the univariate DRM
will be used to estimate the statistical moments in this study.
3.2.2 Sensitivity of Statistical Moments
To obtain a robust design, not only the values of the first and second statistical
moments but also the sensitivities of these moments are needed. Using Eq. (3.1) and
Rosenblatt transformation from the design space (X-space) to the standard Gaussian
34
34
space (U-space), which can be described as ( ) ( )XF x u as explained in Section 2.2.1,
sensitivities of the mean and variance of the performance function with respect to the
design variable i can be derived as
( )[ ( ) ( ) ]
[ ( ( ; )) ( ) ]
( ( ; ))( )
( ; )( ( ; ))( )
H
i i
U
i
U
i
i i iU
i i
h f d
h d
hd
x uhd
x
X
μx x x
x u μ u u
x u μu u
x u μu u
(3.9)
and
2 22
22
22
22
( )[ ( ) ( ) ]
[ ( ( ; )) ( ) ]
( ( ; ))( )
( ; )( ( ; ))( )
H H
i i i
HU
i i
HU
i i
i i i HU
i i i
h f d
h d
hd
x uhd
x
X
μx x x
x u μ u u
x u μu u
x u μu u
(3.10)
where u is the standard normal variable. The input variables are assumed to be
independent for derivations of Eqs. (3.9) and (3.10).
To calculate ( ; )i i i
i
x u
in Eqs. (3.9) and (3.10), Rosenblatt transformation shown
in Table 2.1 is used. For example, if the input variable is normally distributed, then Table
2.1 shows that ix can be expressed as i i i ix u . Since i is fixed and iu is
independent of an input mean i , ( ; )
1i i i
i
x u
is obtained. For Gumbel and uniform
35
35
distributions, the same result ( ; )
1i i i
i
x u
is obtained from Rosenblatt transformation.
For the Lognormal and Weibull distributions, ( ; )i i i
i
x u
can be approximated to be 1.
By using the inverse transformation from U-space to X-space, the assumption
( ; )1i i i
i
x u
, and Eqs. (3.4) and (3.5), Eqs. (3.9) and (3.10) can be further approximated
as
11 1 ( , , , , )
( ) ( ) ( )( 1)
ji N
n NjH
i
j ik k kx
h hw N
x x
x x μ
μ x x (3.11)
and
1
2 22 2
1 1 ( , , , , )
( ) ( ) ( )( 1)
ji N
n NjH H
i
j ik k k kx
h hw N
x x
x x=μ
μ x x (3.12)
Since the univariate DRM does not use sensitivities of the performance function
evaluated at the quadrature points to estimate the moments, additional function
evaluations are needed for the sensitivity analysis using Eqs. (3.11) and (3.12).
3.3 Performance Moment Integration (PMI)
3.3.1 Derivation of PMI
The multi-dimensional integral in Eq. (3.1) for the statistical moments can be
rewritten using Rosenblatt transformation as
( ( )) ( ) ( ; ) ( ( ; )) ( )k k k
UE h h f d h d
X
X x x μ x x u μ u u (3.13)
which can also be written in terms of the output distribution as
36
36
( ( )) ( ( ; )) ( ) ( ; )k k k
U HE h h d h f h dh
X x u μ u u μ (3.14)
where ( )Hf h is the PDF of the output performance function ( )h X . Since CDF of the
performance function can be expressed in terms of the standard normal CDF using the
following transformation ( ) ( )HF h t , Eq. (3.14) becomes
( ( )) ( ; ) ( ; ) ( )k k k
HE h h f h dh h t t dt
X μ μ (3.15)
where the parametric variable t is the distance from the origin in U-space to MPP as
shown in Figure 3.1.
Figure 3.1. Approximation of CDF Using MPP Locus
Source: Du, X., and Chen, W., “A Most Probable Point-Based Method for Efficient
Uncertainty Analysis,” Journal of Design and Manufacturing Automation, Vol. 4, No. 1,
pp. 47-66, 2001.
Hence, the multi-dimensional integral can be rewritten by a one-dimensional
integral. Similar to the univariate DRM, PMI makes use of three quadrature points and
weights to approximate the one-dimensional integration in Eq. (3.15). A difference
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37
between the two methods is that quadrature points of the univariate DRM lie on the ix -
axis, whereas quadrature points of PMI lie on the MPP locus (Du and Chen, 2001; Youn
and Choi, 2002). Therefore, the number of quadrature points in the univariate DRM
increases as the number of design variables increases as shown in Eqs. (3.6) and (3.7),
whereas the number of quadrature points in PMI does not change since the integration is
performed in the output space.
Since t follows the standard normal distribution, the weights and quadrature
points in Table 2.2 can be used to discretize Eq. (3.15) as
3 0 3
1 4 1( ( )) ( ; ) ( ) ( ; ) ( ; ) ( ; )
6 6 6
k k k k k
t t tE h h t t dt h t h t h t
X μ μ μ μ (3.16)
By changing the order of calculation, Eq. (3.16) becomes
3 0 3
( ( )) ( ; ) ( )
1 4 1{ ( ; )} { ( ; )} { ( ; )}
6 6 6
1 4 1( 3; ) (0; ) ( 3; )
6 6 6
k k
k k k
t t t
k k k
E h h t t dt
h t h t h t
h h h
X μ
μ μ μ
μ μ μ
(3.17)
Using FORM and MPP locus illustrated in Figure 3.1, each term in Eq. (3.17) can be
approximated as two function values at two MPPs and a function value at the design
point. The function values at MPPs can be obtained using the inverse reliability analysis
PMA to
maximize ( )
subject to 3
h
U
U (3.18)
The optimum result of Eq. (3.18) is denoted as max
3t
h
, which can be used to
approximate ( 3; )h μ in Eq. (3.17). The term ( 3; )h μ in Eq. (3.17) can be
38
38
approximated by the optimum result obtained by minimizing ( )h U in Eq. (3.18), which is
denoted as min
3t
h
. The term (0; )h μ in Eq. (3.17) can be approximated by ( )h Xμ , which
is the performance function value at the design point. Hence, using these function values
and Eq. (3.17), the statistical moments of the performance function can be calculated as
min max
3 3
1 4 1( ( )) ( ) ( ) ( )
6 6 6t t
k k k kE h h h h
XX μ (3.19)
Consequently, the mean value and variance can be estimated by
min max
3 3
1 4 1( )
6 6 6t tH h h h
Xμ (3.20)
and
2 min 2 2 max 2 2
3 3
1 4 1( ) ( ) ( )
6 6 6t tH Hh h h
Xμ (3.21)
Thus, PMI is very efficient when the number of design variables is relatively large.
3.3.2 Sensitivity of Statistical Moments
Similar to the sensitivity calculation in DRM, from Eqs. (3.15), (3.17), and (3.19),
the sensitivities of the mean and variance of the performance function with respect to a
the design variable i can be derived as
min max
3 3
* *
( ; )( ; ) ( ) ( )
( )1 4 1
6 6 6
1 ( ) 4 ( ) 1 ( )
6 6 6
t t
H
i i i
i i i
i i
i i i i i
h th t t dt t dt
h hh
x xh h h
x x x
* *min X max
X
x=x x=μ x=x
μμ
μ
x x x
(3.22)
and
39
39
2 2 222
min 2 max 22 2
3 3
* * 22 2 2
( ; )( ; ) ( ) ( )
( ) ( )( )1 4 1
6 6 6
1 ( ) 4 ( ) 1 ( )
6 6 6
t t
H H H
i i i i i
H
i i i i
i i H
i i i i i i
h th t t dt t dt
h hh
x xh h h
x x x
* *min X max
X
x=x x=μ x=x
μμ
μ
x x x
(3.23)
By using the approximation *
1i
i
x
, which is a similar to
( ; )1i i i
i
x u
in DRM,
sensitivities of the mean and variance of the performance function with respect to i in
PMI can be obtained as
1 ( ) 4 ( ) 1 ( )
6 6 6
H
i i i i
h h h
x x x
* *min X maxx=x x=μ x=x
x x x (3.24)
and
2 22 2 21 ( ) 4 ( ) 1 ( )
6 6 6
H H
i i i i i
h h h
x x x
* *min X maxx=x x=μ x=x
x x x (3.25)
Since the sensitivities of the performance function on the right hand side of Eqs.
(3.24) and (3.25) are used during the inverse reliability analysis described in Eq. (3.18),
no additional function evaluations are required to calculate sensitivities using Eqs. (3.24)
and (3.25).
3.4 Percentile Difference Method (PDM)
Like PMI, PDM uses results of the inverse reliability analysis. As explained in the
previous section, PMI utilizes the function values at two MPPs ( max
3t
h
and min
3t
h
)
obtained from the inverse reliability analysis and the function value at the mean X to
40
40
approximate the multi-dimensional integration in Eq. (3.1), whereas PDM “uses the
difference between the function values at two MPPs to represent the variation of the
performance function” (Du et al., 2004). Hence, the RBRDO formulation using PDM is
to
1 2minimize ( ( ), )
subject to ( ( ; ) 0) ( ), 1, ,
,
i
p p
i t
L U ndv nrv
f h h h
P G i nc
R and R
Xμ
X d
d d d d X
(3.26)
where 1p is a right-tail percentile, 2p is a left-tail percentile and, in general, 1 2 1p p .
When 1 0.95p and 2 0.05p (Du et al., 2004; Mourelatos and Liang, 2005), 1ph and
2ph in Eq. (3.26) are calculated from the inverse reliability analysis with a target
reliability index of 1.645 (1
1( ) 1.645t p ), that is, 1
max
1.645tph h and 2
min
1.645tph h .
As shown in Figure 3.2, the idea of PDM is simple and could be viewed as
meaningful. However, it has rather serious shortcomings. If the performance function is
not monotonic, it may not be possible to use 1 2p ph h as a measurement for robustness.
For a non-monotonic performance function, two MPPs obtained from the inverse
reliability analysis may not approximate the left-tail and right-tail percentile accurately
because the inverse reliability analysis searches MPPs on the surface of the hyper-sphere
in -spaceU . For example, if 2( )h X X and ~ (0,1)X N and the target reliability t is
1.645, then two MPPs become 1.645 and –1.645. Thus, two percentile performances 1ph
and 2ph are identical. In contrast to PDM, PMI and mean-based DRM show the correct
moment estimation of the performance function 2( )h X X . Thus, PDM-based RBRDO
may identify a wrong global minimum when there are several local minima, as will be
41
41
demonstrated in Section 3.6.2. More significantly, there is no one percentile that can be
used in PDM to identify all local optima correctly as will be shown in Section 3.6.2.
Figure 3.2. Basic Concept of Robust Design Using PDM
Source: Du, X., Sudjianto, A., and Chen, W., “An Integrated Framework for
Optimization Under Uncertainty Using Inverse Reliability Strategy,” ASME Journal of
Mechanical Design, Vol. 126, No. 4, pp. 562-570, 2004.
Sensitivity of the cost function with respect to a design variable i can be
calculated using a similar procedure with PMI as
( ) ( )
i i
h h
x
X
X
x=μ
μ x (3.27)
and
1 2 1 2
* *
* *
* *
( ) ( )p p p pi i
i i i i i i i i
h h h hx x h h
x x x x
p p1 2
x=x x=x
x x (3.28)
42
42
3.5 Comparison
Two criteria to identify the effectiveness of RDO are computational efficiency
and accuracy of the moment and its sensitivity estimation. In terms of computational
efficiency, both PMI and PDM will require the same number of function evaluations if
the same inverse reliability analysis method is used. In general, if the number of design
variables is large, Eqs. (3.7) and (3.8) show that the univariate DRM requires more
function evaluations than PMI and PDM. However, an advantage of using the univariate
DRM is that the univariate DRM does not require sensitivity information (i.e., no MPP
search) in estimating the moments. Hence, the univariate DRM can reduce the number of
function evaluations during line searches.
The objective of mean-based DRM and PMI is to approximate the multi-
dimensional integration in Eq. (3.1). That is, both methods attempt to transform the
multi-dimensional integration into a readily computable numerical integration. However,
PDM does not use any numerical integration; instead it uses the difference of percentile
performances. Thus, PDM may yield wrong results when the performance function is
non-monotonic. Both PMI and PDM may have a difficulty to find MPPs when the
performance function is non-monotonic and highly nonlinear. On the other hand, DRM
may accurately estimate the moments of the performance function regardless of the
performance function type.
In terms of accuracy of the moment estimation, the mean-based DRM yields
better results in most cases than PMI. If the performance function is highly nonlinear,
then the mean-based DRM with three quadrature points may not accurately estimate the
second moment. In this case, the error can be reduced if more quadrature points are used
43
43
in the mean-based DRM. However, PMI with more quadrature points than three may not
necessarily yield more accurate results. This is because function values at quadrature
points, which are obtained using FORM and MPP search, are approximations. More
details of comparison with numerical examples are given in the following section.
3.6 Numerical Examples
In this section, four cases of comparisons are carried out using numerical
examples. In Section 3.6.1, the mean-based DRM and PMI are compared in terms of
accuracy and efficiency in estimation of the moments and their sensitivities of a
performance function. PDM is excluded in Section 3.6.1 since it cannot estimate the
moments of the performance function. In Section 3.6.2, DRM, PMI, and PDM are
compared using a one-dimensional fourth order polynomial for identification of correct
robust optimum design. In this one-dimensional problem, PMI and the mean-based DRM
with three quadrature points can be considered to be the same method.
3.6.1 Comparison of PMI and DRM for Computation of
Moments and Sensitivities
For the first example, the performance function is
2
1 21( ) 1
20
X Xh X (3.29)
where ~ (5,1)iX N for 1,2i . As shown in Table 3.1, both DRM and PMI provide good
estimation of the mean value and standard deviation in comparison with the exact
numerical integration results. The reason that DRM has a larger error in estimation of
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44
standard deviation is because the performance function in Eq. (3.29) has an off-diagonal
term only. As mentioned in Section 3.2, if the performance function has off-diagonal
terms only, then the univariate additive decompositions of the moments in Eqs. (3.4) and
(3.5) may contain more errors.
Table 3.1. Comparison of the First and Second Moments of Eq. (3.29)
Mean ( H ) Variance (2
H )
PMI DRM NI* PMI DRM NI
1h -5.6500 -5.5000 -5.5000 8.4623 7.9355 8.3175
Error, % 2.73 0 1.74 4.58
No. of F.E. 7+7** 2 2 1 7+7 2 2 1 * NI means numerical integration.
** 7+7 means 7 function evaluations and 7 sensitivity calculations.
For this example, PMI yields reasonable estimation of the moments because the
design variables are normally distributed, which means that the inverse reliability
analysis does not require the nonlinear transformation from X-space to U-space, and the
performance function is monotonic at the given design. In the same token, the
sensitivities in Tables 3.2 and 3.3 have similar errors as Table 3.1.
The total number of function evaluations for PMI to evaluate the mean and
standard deviation is 7+7 as shown in Table 3.1, where the first 7 is the number of
function evaluation for MPP search and the second 7 is the number of sensitivity
calculation for the MPP search. The number of function evaluations for DRM is 5. Since
the design variables are normally distributed and the number of quadrature points is odd,
Eq. (3.7) is used for the total number of function evaluations.
45
45
PMI does not require additional function evaluations for the sensitivity analysis
of moments because PMI uses the sensitivity information in MPP search. However, DRM
does require additional function evaluations for sensitivity analysis, thus the total number
of function evaluations needs to be doubled in DRM as shown in Table 3.2.
Table 3.2. Sensitivity of Mean Value Using PMI and DRM for Eq. (3.29)
PMI DRM Analytic
1
( )H
d
d
2
( )H
d
d
1
( )H
d
d
2
( )H
d
d
1
( )H
d
d
2
( )H
d
d
Sensitivity -2.5475 -1.2823 -2.5000 -1.3000 -2.5000 -1.3000
Error, % 1.90 1.36 0.00 0.00
Additional
No. of F.E. 0 2 2 1
Table 3.3. Sensitivity of Variance Using PMI and DRM for Eq. (3.29)
PMI DRM Analytic
2
1
( )H
d
d
2
2
( )H
d
d
2
1
( )H
d
d
2
2
( )H
d
d
2
1
( )H
d
d
2
2
( )H
d
d
Sensitivity 3.9211 2.5894 3.7500 2.5500 3.9000 2.5500
Error, % 0.54 1.54 3.85 0.00
Since the first example contains an off-diagonal term only and the design
variables are normally distributed, the second example is modeled as
2 2
1 2 1 22
( 5) ( 12)( ) 1
30 120
X X X Xh
X (3.30)
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46
where ~ (5,1)iX Gumbel for 1,2i . The performance function in Eq. (3.30) contains
both off-diagonal terms and diagonal terms, and the degree of the polynomial
performance function is 2. Therefore, it can be expected that DRM may yield better
results for this example. As expected, Tables 3.4, 3.5 and 3.6 illustrate that DRM is
accurate in estimation of the moments and their sensitivities.
On the other hand, PMI yields somewhat larger errors in estimation of the
moments and their sensitivities. This is because the design variables follow Gumbel
distribution. In such a case, the inverse reliability analysis requires nonlinear
transformation, which makes the performance function become highly nonlinear and the
FORM error become larger.
Since the Gumbel distribution is not symmetric, Eq. (3.6) is used for the total
number of function evaluations for mean-based DRM.
Table 3.4. Comparison of the First and Second Moments of Eq. (3.30)
Mean ( H ) Variance (
2
H )
PMI DRM NI PMI DRM NI
2h -1.0594 -1.1167 -1.1167 0.3357 0.3774 0.3833
Error, % 5.13 0 12.42 1.54
No. of
F.E. 5+5 3 2 1 5+5 3 2 1
Table 3.5. Sensitivity of Mean Value Using PMI and DRM for Eq. (3.30)
PMI DRM Analytic
1
( )H
d
d
2
( )H
d
d
1
( )H
d
d
2
( )H
d
d
1
( )H
d
d
2
( )H
d
d
Sensitivity -0.1209 -0.5254 -0.1333 -0.5333 -0.1333 -0.5333
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47
Error, % 9.30 1.49 0.00 0.00
Additional
No. of F.E. 0 3 2 1
Table 3.6. Sensitivity of Variance Using PMI and DRM for Eq. (3.30)
PMI DRM Analytic
2
1
( )H
d
d
2
2
( )H
d
d
2
1
( )H
d
d
2
2
( )H
d
d
2
1
( )H
d
d
2
2
( )H
d
d
Sensitivity 0.0724 0.1033 0.0883 0.1149 0.0884 0.1162
Error, % 18.10 11.10 0.11 1.12
3.6.2 Comparison of PMI, DRM and PDM for
Identification of Robust Optimum Design
In this section, three methods are compared in detail for proper identification of
the robust optimum design, using a one-dimensional example. RDO can be formulated to
2minimize
subject to 0 5
H
X
(3.31)
where 3 4
3 ( ) ( 4) ( 3) 10 and ~ ( ,0.4)h X X X X N . Again, note that in one-
dimensional problem, PMI and the mean-based DRM with three quadrature points can be
considered to be the same method since the design variable is normally distributed and
there is no FORM error in a one-dimensional function. Figure 3.3 (a) illustrates the shape
of the performance function and Figure 3.3 (b) illustrates the variances obtained from
DRM and PMI and percentile differences from PDM.
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48
(a) Performance Function ( 3( )h X ) (b) Measure for Variance (2
H )
Figure 3.3. Shape and Variance of Performance Function
As shown in Figure 3.3 (b), PMI and DRM with three quadrature points can
approximate the variance of the performance function very well. On the other hand, PDM
with various percentiles cannot estimate the moments. More significantly, the location of
the optimum point changes depending on the percentiles used. In fact, there is no one
percentile that can be used to accurately identify the location of both local minima
simultaneously in Figure 3.3 (b). Table 3.7 shows that the best percentile should be
located between 2 and 3 for the left local minimum; and the best percentile should be
located between 1.645 and 2 for the right local minimum. In Figure 3.3 (b), „Measure
for Variance‟ is used instead of variance. This is because PDM cannot estimate the
variance of the performance function and uses percentile differences as the measure for
the variance.
Another problem of using PDM for a highly nonlinear performance function
such as Eq. (3.31) is that PDM might not be able to identify which local minimum is the
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49
global minimum when there is more than one local minimum. As shown in Table 3.7, the
results of PDM with three different percentiles indicate that the value of the cost function
at the left minimum in Figure 3.3 (b) is less than the value at the right minimum, which is
wrong.
Table 3.7 also shows that PMI and DRM with three quadrature points yields
some errors in finding the location of the optimum and estimating the value of the
optimum. This is because the performance function is a polynomial of degree 4, thus
three quadrature points may not be sufficient. In this case, DRM and PMI with five
quadrature points are good options to achieve accuracy. The accuracy of DRM and PMI
with five quadrature points is illustrated in Table 3.7 and Figure 3.4.
Table 3.7. Position and Value of Optimum Using Three Methods for Eq. (3.31)
PMI and DRM PDM
NI 3 pts 5 pts 1 1.645 2 3
Left
Min.
minx 1.463 1.483 1.236 1.315 1.376 1.622 1.485 2
H or 1 2p ph h 3.397 4.361 0.000 0.000 0.000 0.000 4.403
Right
Min.
minx 3.405 3.359 3.464 3.397 3.341 3.037 3.358 2
H or 1 2p ph h 1.075 1.220 1.375 3.239 4.759 10.645 1.234
50
50
Figure 3.4. Accuracy of PMI and DRM with Five Quadrature Points
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51
CHAPTER IV
A NEW DRM-BASED INVERSE RELIABILITY ANALYSIS
4.1 Introduction
FORM has been widely used for the reliability analysis and inverse reliability
analysis due to its simplicity and efficiency, and FORM shows a good approximation for
the mildly nonlinear system or linear system. However, since FORM uses a linear
approximation of the performance function, the estimation of the probability of failure
could include large errors when the system is highly nonlinear and/or multi-dimensional.
These errors can be improved by SORM since SORM uses a quadratic approximation of
the performance function. However, the Hessian matrix is required to calculate the
probability of failure in Eq. (2.10) using SORM, which is very difficult or sometimes
impossible to estimate accurately in real engineering applications. For this reason, use of
SORM in engineering applications has been very limited.
To resolve these disadvantages of FORM and SORM, a new DRM-based inverse
reliability analysis is proposed in this chapter. Section 4.2 demonstrates the weaknesses
of FORM for a highly nonlinear and multi-dimensional system and Section 4.3 illustrates
how to obtain the DRM-based MPP using three computational steps. The DRM-based
MPP will be used for the next design iteration of RBDO and thus yield an accurate
optimum design even for highly nonlinear system. The RBDO with the MPP-based DRM
will be explained in Chapter 6. Section 4.4 compares FORM, SORM, and DRM through
numerical examples.
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52
4.2 Error in FORM-Based Reliability Analysis
Although FORM has been widely used for the reliability analysis and inverse
reliability analysis due to its simplicity and efficiency, FORM could be erroneous if the
performance function is highly nonlinear and/or multi-dimensional as shown in the
following example. Consider
1
2
1
( )N
i N
i
G a X X
X (0.1)
where ~ (0,1) for 1, .iX N i N The performance function has an MPP at
* T{0, ,0, }x and the probability of failure by FORM is FORM ( )FP regardless
of a and N. If 2, then the probability of failure by FORM becomes
FORM ( 2) 2.2750%FP . This probability of failure can be compared with the
probability of failure obtained using MCS for different a and N, respectively, as shown in
Tables 4.1 and 4.2. From Tables 4.1 and 4.2, it can be seen that the probability of failure
obtained using FORM has significant error when a performance function is highly
nonlinear (i.e. larger a ) and especially when a performance function is multi-
dimensional (i.e. larger N ).
Table 4.1. PF by MCS When N=2 (Highly Nonlinear)
0.2a 0.5a 1.0a 2.0a MCS
FP 1.5915% 1.1829% 0.9145% 0.6649%
53
53
Table 4.2. PF by MCS When a=0.2 (Multi-dimensional)
2N 3N 5N 10N MCS
FP 1.5915% 1.1309% 0.5426% 0.0790%
4.3 Inverse Reliability Analysis Using MPP-Based DRM
The objective of the DRM-based inverse reliability analysis is to find a new
DRM-based MPP, denoted by *
DRMx , using the MPP from the FORM-based inverse
reliability analysis denoted by *
FORMx . As stated in Section 2.3, the inverse reliability
analysis does not calculate the probability of failure directly, instead, it judges whether a
given design satisfies the probabilistic constraint by checking the performance function
value at MPP. However, the probabilistic constraint may not be satisfied even though the
constraint value at the FORM-based MPP *
FORM( )G x is less than zero, which means the
MPP is safe. This is because the probability of failure calculated by FORM may have
significant error especially for the highly nonlinear multi-dimensional performance
function. In this section, a new method is proposed to find a DRM-based MPP *
DRMx .
Finding the new DRM-based MPP consists of three steps: constraint shift, reliability
index update, and MPP update.
4.3.1 Probability of Failure Calculation Using Constraint
Shift
In the rotated standard normal V-space, the probability of failure in Eq. (2.1) can
be rewritten as
54
54
( ) 0
[ ( ) 0] ( )FG
P P G f d
VV
V v v (0.2)
where the performance function in V-space is defined as ( ) ( ( ))G Gv x v . Since the
inverse reliability analysis does not calculate the probability of failure, a constraint shift
concept is introduced for the probability of failure calculation such that
*( ) ( ) ( )sG G G v v v (0.3)
where ( )sG v is a shifted performance function and * T{0, ,0, }v is the FORM-based
MPP in V-space with a given reliability index . By applying the MPP-based DRM
explained in Section 2.5.2 to ( )sG v , ( )sG v can be approximated at the MPP *v as
*
1
ˆ( ) ( ) ( ) ( 1) ( )
Ns s s s
i i
i
G G G v N G
v v v (0.4)
where * * * *
1 1 1( ) ( , , , , , , )s s
i i i i i i NG v G v v v v v . By the definition of ( )sG v in Eq. (4.3),
*( )sG v is zero, thus, we obtain
1
1 1
ˆ( ) ( ) ( ) ( ) ( )
N Ns s s s s
i i N N i i
i i
G G G v G v G v
v v (0.5)
Due to the rotational transformation of the coordinates as shown in Fig. 2.3, the
Nth
univariate component ( )s
N NG v can be linearly approximated (Wei, 2006). This linear
assumption of ( )s
N NG v along -axisNv is also used for the probability of failure
calculation in SORM (Breitung, 1984; Hohenbichler, 1988). Using the linear assumption,
Eq. (4.2) can be written as
1
0 1
1
[ ( ) 0] [ ( ) 0]N
s s
F N i i
i
P P G P b bV G V
V (0.6)
55
55
Since function value and gradient at MPP are obtained during the inverse reliability
analysis using FORM, 0 1 Nb bV can be rewritten using the first order Taylor series
expansion at the MPP as
* *
* *
0 1
( ) ( )( ) ( ) ( ) ( )
ss s
N N N N N N N N
N N
G GG v b b v G v v v v
v v
v v v v
v v (0.7)
where * *
* *
1
( ) ( )( ) 0, , and .
ss
N N N
N N
G GG v b v
v v
v v v v
v v Inserting Eq. (4.7) into Eq.
(4.6) yields
1 1
1 1 1
1 1
[ ( ) ( ) 0] [ ( ) ]N N
s s
F N i i N i i
i i
P P b V G V P bV b G V
(0.8)
Since the gradient 1b at MPP is always positive due to maximization of the
inverse reliability analysis in Eq. (2.14), Eq. (4.8) can be rewritten, by dividing both sides
by 1b and using the symmetry of the standard normal distribution since ~ (0,1)V N , as
1
11
1 1
1 11 1
1[ ( ) ]
1 1[ ( ) ] [ ( ( ) )]
Ns
F N i i
i
N Ns s
N i i i i
i i
P P V G Vb
P V G V E G Vb b
(0.9)
where E is the expectation operator. Since Eq. (4.9) is an N−1 dimensional integration,
Eq. (4.9) can be further simplified by applying DRM to the integrand of Eq. (4.9) as
1
DRM 1 1
2
( )( ) ( )
( )
sNi i
i i
i
F N
G vv dv
bP
(0.10)
where 1 Ub g . Detailed derivation of Eq. (4.10) can be found in Ref. (Wei, 2006).
Using the moment-based integration rule (MBIR) (Xu and Rahman, 2003), which is
explained in Section 2.5.1, Eq. (4.10) is further approximated as
56
56
1
11DRM 1
2
( )( )
( )
s jN ni i
j
ji
F N
G vw
bP
(0.11)
where j
iv are quadrature points, jw are weights, and n is the number of quadrature
points and weights. Since iv are standard normal random variables, quadrature points
and weights in Table 2.3 can be used to calculate Eq. (4.11) and locations of quadrature
points are illustrated in Figure 4.1. The coordinates for the quadrature points j
iv in Figure
4.1 are 1,3 ( )(0, , 3, , )k
iv and 2 ( )(0, ,0, , )k
iv , where 3 and 0 are at ith
positions.
(a) Concave Function (b) Convex Function
Figure 4.1 DRM-based MPP for Concave and Convex Functions
For a special case of Eq. (4.11), if 1n , which means one quadrature point and
weight, then Eq. (4.11) can be written as
57
57
11 1
1
DRM 1 11
2 2
( )( ) ( )
( )( ) ( )
sN Ni i
i iF N N
G vw
bP
(0.12)
where 1 1w and 1 0iv by Table 2.3, and 1( ) (0) 0s s
i i iG v G . Equation (4.12) is the
same as the probability of failure by FORM. Therefore, we can say that the probability of
failure calculation by FORM is a special case of the probability of failure calculation by
DRM when one quadrature point and weight is used.
The number of additional function evaluations needed to evaluate Eq. (4.11),
besides the MPP search using FORM, is ( 1) ( 1)N n . Hence, the total number of
function evaluations necessary for Eq. (4.11) is
# of F.E. for MPP search + ( 1) ( 1)N n (0.13)
Since the probability of failure calculation using DRM requires integration in Eq.
(4.10), accuracy of the probability of failure estimation can be easily achieved by
increasing the number of quadrature points and weights in Eq. (4.11). In this case, the
probability of failure by DRM requires only function values at the quadrature points,
which are ( )s j
i iG v in Eq. (4.11). Consequently, the accuracy of the DRM result can be
improved by increasing the number of quadrature points if necessary, which does not
require any sensitivity. The comparison of the number of function evaluations with the
FORM-based reliability analysis will be discussed in detail using numerical examples in
Section 4.5.
Using the MPP-based DRM to use Eq. (4.11) for the probability of failure
calculation, in contrast to the mean-based DRM, is like using the importance sampling
method near the MPP in contrast to the Monte Carlo simulation (MCS) method for the
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58
failure probability estimation. However, even though the importance sampling method
requires less number of samplings than MCS, it still requires quite amount of samplings
to obtain the accurate probability of failure (Denny, 2001). The MPP-based DRM
requires significantly less function evaluations than the importance sampling for the
accurate estimation of the failure probability. Hence, this proposed method can be easily
applied to the practical engineering problems.
4.3.2 Reliability Index Update
After computing the probability of failure DRM
FP using the MPP-based DRM for
the shifted performance function ( )sG v , the corresponding reliability index DRM is
obtained using
1 DRM
DRM ( )FP (0.14)
It is likely that DRM from Eq. (4.14) is not the same as the target reliability index
1 Tar( )t FP . Hence, a new updated reliability index ( 1)k is obtained using the
difference between two reliability indices as
( 1) ( )
DRM( )k k
t (0.15)
where ( )k is the reliability index at the current step, with (0)
t at the initial step.
If the performance function is concave as shown in Figure 4.1 (a), then ( 1)k will
be smaller than ( )k because DRM t , which means that a smaller reliability index
should be used to correctly update MPP using DRM for the concave performance
function, and vice versa for the convex performance function in Figure 4.1 (b). In this
59
59
study, a performance function is defined as concave near the MPP if the FORM-based
reliability analysis overestimates the probability of failure, and convex near the MPP if
the FORM-based reliability analysis underestimates the probability of failure.
4.3.3 MPP Update Method
Using this updated reliability index, we can carry out a new inverse reliability
analysis to find a better MPP which satisfies the given target probability of failure. After
finding a new MPP, constraint shift is again used to compute the probability of failure by
DRM. After iteratively doing this procedure until converged, the DRM-based MPP can
be obtained where DRM
FP is the same as Tar
FP . However, it will be computationally
expensive if a new MPP search is carried out every time an updated reliability index is
obtained. For efficiency, the updated MPP corresponding to the probability of failure by
DRM is obtained without carrying out a new MPP search as (Ba-abbad et al., 2006)
(k+1) (k+1)
* * * *
k+1 k k+1 k( ) ( ) or
k k
u u v v (0.16)
That is, it is assumed that the updated MPP *
k+1u is located along the same radial direction
as the current MPP *
ku in U-space. Figure 4.1 illustrates that the updated DRM-based
MPP in V-space is located along the vN-axis. The same iterative procedure explained
above can be performed until DRM
FP converges to Tar
FP . To verify the MPP approximation
in Eq. (4.16), a comparison test to find the DRM-based MPP using two methods is
carried out in Section 4.4.2 through a numerical example.
60
60
However, this iterative procedure using Eq. (4.16) still requires additional
function evaluations. To reduce the number of function evaluations, the MPP update can
be carried out only once at a given design. The updated MPP obtained through the
procedure is also called the DRM-based MPP and will be used to evaluate whether the
design satisfies the probabilistic constraint or not. Even though the MPP update is carried
out only once at the given design, the probability of failure by DRM will converge to the
target probability of failure as the design moves near the reliability-based optimum
design.
4.4 Numerical Examples
Accuracy of the DRM-based probability of failure is verified by comparing it with
the FORM and SORM-based probabilities of failure in Section 4.4.1. For this purpose,
the probability of failure obtained using MCS is used as a benchmark data. Section 4.4.2
compares two iterative methods, a method using a new MPP search and using Eq. (4.16)
to find the DRM-based MPP.
4.4.1 Comparison of FORM, SORM and DRM for FP Calculation
For the first example, a highly nonlinear fourth-order polynomial function
2 3 4
1 ( ) 0.7361 ( 6) ( 6) 0.6 ( 6)sG Y Y Y Z X (0.17)
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61
where 1
2
0.9063 0.4226
0.4226 0.9063
XY
Z X
, 1 ~ (4,0.3)X N and 2 ~ (3,0.3)X N , is used for
the probability of failure computation. The reliability index of 1.645 is used for the
FORM-based inverse reliability analysis.
Figure 4.2 (a) shows the shifted and original performance functions and Figure
4.2 (b) shows the approximated functions by FORM, SORM, and DRM at MPP in V-
space. In Table 4.3, DRM with three and five quadrature points are used to evaluate Eq.
(4.11). From Table 4.3, it can be seen that DRM with five-quadrature points is the most
accurate method for this example. In fact, this result is even more accurate than the
SORM result, compared with the MCS result with 1 million samples, which can be
considered accurate. In terms of efficiency, FORM shows the best efficiency, which is
always true since SORM and DRM require the FORM-based MPP. However, the
additional number of function evaluations for DRM besides the MPP search does not
require sensitivity analysis. Hence, DRM can estimate the probability of failure as
accurately as SORM without requiring the second-order sensitivity calculation and as
efficiently as FORM without loss of accuracy - the error of the FORM result is about
52% as shown in Table 4.3.
62
62
(a) X-space (b) V-space
Figure 4.2. Performance Function in X-space and V-space for 2-D Example
Table 4.3. Calculation of FP by Various Methods for 2-D Example
FORM SORM DRM
MCS 3 pts 5 pts
FP , % 5.0000 3.4081 3.5844 3.3676 3.2791
F.E. 7* 7*+Hessian 7*+2** 7*+4** * 7 means the number of function and sensitivity analysis for MPP search
** 2 and 4 function evaluations for DRM do not require sensitivity analysis
For the second example, a four dimensional quadratic function
2 2 2 2
2 1 2 3 4 1 2 3 4 ( ) 9 11 11 11 95.75sG X X X X X X X X X (0.18)
where ~ (5,0.4), i=1~ 4iX N , is used for the probability of failure computation. The
reliability index of 1.645 is used for the FORM-based inverse reliability analysis. As
described in Eq. (4.13), the total number of function evaluations for the DRM-based
reliability analysis will increase as the number of random variables increases. Since the
63
63
performance function in Eq. (4.20) has four random variables, (4 1) (3 1) 6 function
evaluations are required for DRM with three quadrature points; and (4 1) (5 1) 12
function evaluations are required for DRM with five quadrature points as shown in Table
4.4. Again, these function evaluations do not require sensitivity analysis. Table 4.4 shows
DRM yields the best accuracy compared with the MCS results.
Table 4.4. Calculation of FP By Various Methods for 4-D Example
FORM SORM DRM
MCS 3 pts 5 pts
FP , % 5.0000 14.0396 11.8049 11.8976 11.9064
F.E. 2 2+Hessian 2+6 2+12
4.4.2 Inverse Reliability Analysis Using DRM
The two-dimensional performance function in Eq. (4.17) is again used for the
convergence test in this section. For the given target probability of failure Tar 2.275%,FP
the corresponding reliability index t is obtained from 1(0.02275) 2t , which is
the initial reliability index in Tables 4.5 and 4.6. This reliability index is used for the
FORM-based inverse reliability analysis to find MPP. After finding the FORM-based
MPP, the probability of failure is calculated using DRM and compared with the target
probability of failure. Since the estimated probability of failure by DRM, which is
1.4109% as shown in Table 4.5, is smaller than the target probability of failure,
Tar 2.275%,FP the following reliability index should be smaller than the initial reliability
index. Using Eq. (4.15), the updated reliability index is obtained as
1
up cur DRM t( ) 2 ( (0.014109) 2) 1.8058 (0.19)
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64
where cur t 2 since it is the initial iteration and 1
DRM (0.014109) . The
updated reliability index is shown in Tables 4.5 and 4.6. Using the MPP update in Eq.
(4.16), the first DRM-based MPP candidate is obtained as (4.5009, 2.7936) in Table 4.5.
By iteratively performing this procedure, finally the DRM-based MPP can be obtained as
(4.5029, 2.7928) where the probability of failure estimation by DRM, 2.2751%, is
almost the same as the target probability of failure 2.2750%. In this example, the updated
reliability index is decreasing since the performance function is concave near the MPP,
which means that the FORM-based reliability analysis overestimates the probability of
failure.
Table 4.6 shows iterative way of finding the DRM-based MPP using the new
MPP search, which means that the new MPP search is carried out using FORM after
obtaining the updated reliability index. Since this requires the MPP search at every design
iteration, it becomes expensive to find the DRM-based MPP as shown in Tables 4.5 and
4.6. The total number of function evaluations and sensitivity analysis needed for Table
4.5 is 16 and 5, respectively. Whereas, the total number of function evaluations and
sensitivity analysis needed for Table 4.6 is 21 and 15, respectively, which is more
expensive than the MPP update using Eq. (4.16).
Table 4.5. Iterative Way of Finding DRM-Based MPP Using Approximation
Iter. *
Approxx DRM
FP , % Tar
FP , % F.E.
0 2.0000 (4.5547, 2.7714) 1.4109
2.2750
7*+5**
1 1.8058 (4.5009, 2.7936) 2.3163 10*+5**
2 1.8134 (4.5030, 2.7927) 2.2726 13*+5**
3 1.8129 (4.5029, 2.7928) 2.2751 16*+5**
65
65
* means the number of function evaluations
** means the number of sensitivity analysis
Table 4.6. Iterative Way of Finding DRM-Based MPP Using New MPP Search
Iter. *
FORMx DRM
FP , % Tar
FP , % F.E.
0 2.0000 (4.5547, 2.7714) 1.4109
2.2750
7*+5**
1 1.8058 (4.5108,2.8192) 2.2612 14*+10**
2 1.8034 (4.5102,2.8199) 2.2749 21*+15**
Both methods converge very fast within three iterations. In addition, two MPPs
obtained using MPP approximation and new MPP search are close to each other as shown
in Tables 4.5 and 4.6, which means that the MPP approximation method in Eq. (4.16) can
be effectively used to find the DRM-based MPP without requiring further MPP search.
Also, the fact that the approximated MPP at the first iteration, (4.5009, 2.7936) , is very
close to the approximated MPP, (4.5029, 2.7928) , at the last iteration in Table 4.5
verifies that the MPP update using Eq. (4.16) can be carried out only once at a given
design without loss of accuracy. This reduction of the number of function evaluations
plays a significant role when it is applied to RBDO.
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66
CHAPTER V
SENSITIVITY ANALYSES OF FORM AND DRM-BASED
PERFORMANCE MEASURE APPROACH FOR RBDO
5.1 Formulation of FORM and DRM-Based PMA for
RBDO
As explained in Section 2.4, the FORM-based PMA for RBDO is formulated to
*
minimize Cost( )
subject to ( ) 0, 1, ,
, R and R
i
L U ndv nrv
G i nc
d
x
d d d d X
(5.1)
where *
x is the FORM-based MPP which can be obtained by solving the inverse
reliability analysis in Eq. (2.14). In a manner similar to FORM-based PMA, the DRM-
based PMA for RBDO is formulated to
minimize Cost( )
subject to ( ) 0, 1, ,
, R and R
i
L U ndv nrv
G i nc
*
DRM
d
x
d d d d X
(5.2)
where *
DRMx is the DRM-based MPP.
For optimizations given by the formulation (5.1) for FORM-based PMA and the
formulation (5.2) for DRM-based PMA, sensitivities of the objective function and
constraints with respect to the design variables are required. In both formulations, it is
straightforward to obtain the sensitivities of the objective function with respect to design
variables since the objective is a function of the design variables, which are the mean
values of the input random variables. However, it is not straightforward to obtain the
67
67
sensitivities of the probabilistic constraint at MPP with respect to the design variables
since MPP of the perturbed design is involved in evaluation of the probabilistic constraint
at the perturbed design. Hence, it is required to analytically derive the sensitivities of the
probabilistic constraints with respect to design variables given by
* * ***
T*
1
( ) Ni
i i
xG G G G
x
x x x x x xx xx x
x x
d d d d x (5.3)
for the FORM-based PMA and
* * ***DRM DRM DRMDRMDRM
T*
DRM
1
( ) Ni
i i
xG G G G
x
x x x x x xx xx x
x x
d d d d x (5.4)
for the DRM-based PMA.
5.2 Sensitivity Analyses for FORM-Based PMA
The FORM-based MPP using PMA is defined in U-space as
( )
( )
Ut t
U
g
g
**
*
uu α
u (5.5)
where α is the normalized gradient vector at the FORM-based MPP. By taking
derivatives on both sides of Eq. (5.5) with respect to jth
design variable dj, which is the
mean value of the jth
random variable, we have
112
1
U t Ut U
j j U j j
g g bb g
d d g b d d
*u (5.6)
where 1 Ub g and all derivatives are evaluated at MPP. The left side of Eq. (5.6) is
rewritten using the Rosenblatt transformation as
68
68
*
T
j j jd d d
*
*
j
u=u x x
u u xT e , (5.7)
assuming that the transformation from U-space to X-space is given by
( )j jx d f u . (5.8)
The transformation matrix T in Eq. (5.7) is given by
1 2
1 1 1
1 2
N
N
N N N
xx x
u u u
xx x
u u u
T . (5.9)
The assumption in Eq. (5.8) works for general distributions whose contour shape
of the joint input PDF does not change when the design point moves; for example,
normal, uniform, Gumbel, exponential, Rayleigh, 3-parameter lognormal, 3-parameter
Weibull distribution, etc. If all input random variables are independent, T becomes a
diagonal matrix, and if the random variables are dependent, then T becomes a triangular
matrix (Noh et al., 2007).
U
j
g
d
and 1
j
b
d
in Eq. (5.6) are calculated in X-space as
*
U
j j
g
d d
x x
xTH (5.10)
using U g G T where ix
and
*
T T
1
1j j
b G
d b d
x x
T TH x, (5.11)
69
69
respectively, where G and H are the gradient vector and Hessian matrix evaluated at
MPP in X-space. Substituting Eqs. (5.7), (5.10), and (5.11) into Eq. (5.6) yields
* * *
T T T T
3
1 1
t t
j j j
G Gd b d b d
j
x x x x x x
x x xe T TH T TH T T . (5.12)
After rearranging Eq. (5.12), we obtain
*
T T T T
3
1 1
t t
j
G Gb b d
j
x x
xI T TH T T T TH e (5.13)
and, in a matrix form,
*
T T T T
3
1 1
t t G Gb b
x x
xI T TH T T T TH I
d. (5.14)
Thus,
*
1
T T T T
3
1 1
t t G Gb b
x x
xI T TH T T T TH
d, (5.15)
which are the sensitivities of MPP in X-space with respect to design variables.
From Eqs. (5.3) and (5.15), the sensitivities of the probabilistic constraint in Eq.
(5.1) with respect to design variables are
* * *
TT
T T T T
3
1 1
t tG GG G G
b b
x x x x x x
xI T TH T T T TH
d d x. (5.16)
To further simplify the right-hand side of Eq. (5.16), consider the following equation:
T
T T T T
3
1 1
T T T T
3
1 1
T T 2
13
1 1
,
t t
t t
t t
G G Gb b
G G Gb b
G G G b Gb b
I T TH T T T TH
I HT T HT T T T
HT T HT T
(5.17)
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70
since H is symmetric and 22 T T
1 Ub g G G T T . Using Eq. (5.17), Eq. (5.16) can
be rewritten as
* * * *
TG G G
x x x x x x x x
x
d d x x, (5.18)
which is the sensitivity of the probabilistic constraint with respect to the design variables
in Eq. (5.3) for the FORM-based PMA. A number of papers that use the probabilistic
constraint assumed Eq. (5.18). However, this relation is exact and there is no need of
assuming it.
When constraints are black box type, which means the design sensitivity is not
available, to evaluate the sensitivities in Eq. (5.3) using FDM, additional MPP searches at
the perturbed designs are required, which is computationally very expensive. In addition,
since the design perturbation is required for each design variable, the sensitivity
calculation using FDM will become very expensive when the number of design variables
increases. However, to evaluate the sensitivities in Eq. (5.18) using FDM, no additional
MPP search is required, and thus very efficient with the same accuracy.
The sensitivities in Eq. (5.18) can be shown in a different way. Using the
definition of MPP in Eq. (5.5) and 1Tα α , the target reliability index is written as
(Ditlevsen and Madsen, 1996)
t T *α u . (5.19)
Taking derivative of Eq. (5.19) with respect to dj yields
t
j j jd d d
T ** Tα u
u α . (5.20)
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71
Since 0jd
Tα
α from 1Tα α and t is constant, Eq. (5.20) can be written as
T T
0j jd d
*
*
u=u
u uα α , (5.21)
which yields
T
0U
j
gd
*u=u
u. (5.22)
Inserting Eq. (5.7) and U g G T into Eq. (5.22) yields
* *
T TT
1 0U
j j j
g G Gd d d
*
j j
u=u x x x x
u x xe T T e (5.23)
and, in a matrix form,
*
T
G
x x
xI 0
d. (5.24)
Hence, the same sensitivities with Eq. (5.18) are obtained.
5.3 Sensitivity Analyses for DRM-Based PMA
Since the DRM-based MPP can be found using either a new MPP search or an
approximation as explained in Section 4.3.3, the sensitivities of the probabilistic
constraints at both true DRM-based MPP ( *
DRMu ) and approximated DRM-based MPP (
a
DRMu ) are derived in Sections 5.3.1 and 5.3.2, respectively. Section 5.3.3 illustrates that
these sensitivities converge to each other as the design point approaches the optimum
design.
72
72
Figure 5.1. Comparison of Approximated and True DRM-based MPP
5.3.1 Sensitivity of Probabilistic Constraint at True DRM-
Based MPP
The DRM-based MPP obtained from a new MPP search with the updated
reliability index up is written from Figure. 5.1 as
up up
( )
( )
U
U
g
g
** DRMDRM DRM*
DRM
uu α
u (5.25)
and
up T *
DRM DRMα u, (5.26)
where DRMα is the normalized gradient vector at the true DRM-based MPP. In a way
similar to that explained in Section 5.2, taking derivative of Eq. (5.26) yields
T T T
up DRM
DRM
( )
( )
U U
j j j j UU
g g
d d d d gg
** * *
DRMDRM DRM DRMDRM *
DRM
uu u uα
u (5.27)
73
73
using the orthogonality of DRMα and jd
DRMα
(Ditlevsen and Madsen, 1996). Hence,
T
up
DRM DRMU U
j j
g gd d
*
DRMu. (5.28)
Since up is not constant for the DRM-based PMA, it is required to derive up
jd
from the definition of the updated reliability index, as shown in Eq. (4.15). In the current
design, since cur and t are constant, the sensitivity of the updated reliability index with
respect to dj can be written using 1 DRM
DRM ( )FP as
DRM
up DRM
DRM
1
( )
F
j j j
P
d d d
. (5.29)
Substituting Eq. (5.29) into Eq. (5.27) yields
TDRM
DRM
DRM
DRM( )
U FU
j j
g Pg
d d
*
DRMu. (5.30)
Using the transformation in Eq. (5.7) and DRM DRMU g G T , the sensitivity of the
probabilistic constraint in Eq. (5.4) at the true DRM-based MPP is obtained as
DRM
DRM
DRM
DRM( )
U Fg PG G
G
* * *
DRM DRM DRM
T
x=x x=x x=x
x
d d x d. (5.31)
The sensitivity of DRM
FP with respect to dj in Eq. (5.31) can be calculated using
Eq. (4.10). Assuming a 2-D performance function for the simplicity of calculation,
74
74
DRM
1 1 1 11 1 1 1
1 1
1 1 1 11 1
1 1
1 1 1 1 11 1 1
1
( ) ( )( ) ( ) ( )
( ) ( )( )
( ) ( ) (( )
s s
F
j j j
s s
j
s ss
j j
P G v G vv dv v dv
d d b d b
G v G vv dv
b d b
G v G v bb G v
b d d
112
1
1 1 1 1 11 1 12
1 1 1
)
( ) ( )( ) .
s k s kns kk
k j j
vdv
b
w G v G v bb G v
b b d d
(5.32)
In Eq. (5.32), 1
j
b
d
is obtained from Eq. (5.11) and 1 1( )s k
j
G v
d
is calculated as
1 1( ) ( ( ) ( ))s k
j j
G v G G
d d
k *v v
(5.33)
where T
1 ,kv kv and T
0,*v . In a 2-D problem, the sensitivity of the kth
quadrature point with respect to dj is given by
* *
1
1
1
1
k
k
v
v
k
x x x x
x x xI I A I I
d d d. (5.34)
Thus, Eq. (5.33) is written as
*
T
1 1( )s kG v G G
*kx x x=x
x=x
xA A I
d d x x (5.35)
where *
x x
x
d is given by Eq. (5.15). Inserting Eqs. (5.35) and (5.11) into Eq. (5.32), we
have
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75
* *
DRM
1 1
21 1 1
T T T
1 1 1
1
( )
( ) ( )
s knkF
k
s k
wP G v
b b
Gb G G G v
b
k
x x x x
d
x x HT TA A I x
d d
. (5.36)
By inserting Eq. (5.36) into Eq. (5.31), the sensitivity of the probabilistic constraint in Eq.
(5.4) at the true MPP is analytically obtained. Unlike the sensitivity of the probabilistic
constraint at the FORM-based MPP given in Eq. (5.18), the sensitivity in Eq. (5.31) of
the probabilistic constraint at the true DRM-based MPP requires the Hessian as shown in
Eq. (5.36), and thus very expensive to use for PMA of RBDO
5.3.2. Sensitivity of Probabilistic Constraint at
Approximate DRM-Based MPP
The DRM-based MPP obtained from the approximation in Eq. (4.16) with the
updated reliability index up is defined from Figure. 5.1 as
up up*
cur up up
cur cur
= .U U
U U
g g
g g
a
DRMu u α (5.37)
Taking derivatives of Eq. (5.37), we have
up
upU U
j j U j U
g g
d d g d g
a
DRMu. (5.38)
After rearranging Eq. (5.38), we obtain
*
T TDRMT
1 DRM
T
Tup 2 T T T T
13
1
( )
.
aDRM
T
j
x=x
x=x
x T Te
xT TH T T T TH
F
j j
j
PG
d b d
b G Gb d
(5.39)
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76
Substituting Eq. (5.39) into
TG G
a a aDRM DRM DRMx=x x=x x=x
x
d d x, (5.40)
the sensitivity of the performance function at the approximated DRM-based MPP is
analytically calculated. However, since the sensitivity requires the Hessian matrix as
shown in Eq. (5.38) like Eq. (5.36), it is also very expensive to obtain the sensitivity in
Eq. (5.40).
Using the assumption a
DRM
a
DRM
G G
GG
where a
DRM ( )G G a
DRMx , the equation
T DRM
DRM( )
U Fg PG
G
a *
DRMx=x x=x
x
d x d (5.41)
can be obtained from Eq. (5.31) as
T
a aT DRMDRM DRMa
DRM
DRM( )
U F
G G
G g PG G
G
a a aDRM DRM DRM
aDRM
x=x x=x x=x
x=x
x
d d x
x
d d
(5.42)
where the same DRM
FP
d in Eq. (5.36) is used for Eq. (5.42). This is the sensitivity of the
probabilistic constraint in Eq. (5.4) at the approximated DRM-based MPP.
5.3.3 Convergence Study Using Taylor Series Expansion
As the design approaches the optimum design, the updated reliability index βup
converges to the current reliability index βcur. This is because Δβ is getting smaller as the
design approaches the optimum. Hence, the sensitivity of the updated reliability index
with respect to dj converges to zero as the design approaches the optimum because
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77
up
cur
j j jd d d
. (5.43)
In conclusion, even though the analytic sensitivities for up
jd
and
DRM
F
j
P
d
are given in
Eqs. (5.29) and (5.36), they can be ignored because the sensitivities are very small near
the optimum and the Hessian matrix is required for the analytic sensitivities.
Using Taylor series expansion, the gradient of the performance function at the
approximated DRM-based MPP can be expressed as
( )
( ) ( ) ( )G
G G
O (5.44)
where a
DRM( )G G and ( )G G . Substituting Eq. (5.44) into Eq. (5.40)
and using Eq. (5.42), the sensitivity of the performance function at the approximated
DRM-based MPP is given by
T T
TDRM
DRM
( )
( ) .( )
U F
j
G G GG
g P GG
d
a a a aDRM DRM DRM DRM
aDRM
x=x x=x x=x x=x
x=x
x x
d d x d
x
d
O
O
(5.45)
As the design approaches the optimum, which means Δβ converges to zero, Eq. (5.45) is
approximated as
G
G
a
DRMx=xd. (5.46)
In addition, since the gradient at the approximated DRM-based MPP converges to
the gradient at the FORM-based MPP as shown in Eq. (5.46), the gradient at the true
DRM-based MPP will converge to the approximated DRM-based MPP and FORM-based
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78
MPP. Hence, without loss of accuracy near the optimum, the sensitivity of the
probabilistic constraint for the DRM-based RBDO in Eq. (5.4) can be approximated as
G G
a a
DRM DRMx=x x=xd x (5.47)
to save the computational cost for RBDO. However, if the initial design is far from the
RBDO optimum, there could be some error in the approximation of the sensitivities in
Eq. (5.47), especially when the performance function is highly nonlinear. To avoid this
situation, PMA+ is used for RBDO, which uses the deterministic optimum design as the
initial design for RBDO because the deterministic optimum design is usually close to the
RBDO optimum design. The RBDO examples using PMA+ and analytical sensitivities in
Eq. (5.47) are presented in Chapter 6.
5.4 Numerical Examples
Numerical studies are carried out in this section to verify the analytic sensitivities
derived in Sections 5.2 and 5.3 using the FDM with various perturbation sizes. For that
purpose, a two-dimensional highly nonlinear performance function, which was studied in
Refs. (Lee et al., 2006; Lee et al., 2008a; Lee et al., 2008b), is used. Analytic sensitivities
derived for the FORM-based RBDO are compared with the FDM results in Section 5.4.1,
and analytic sensitivities derived for the DRM-based RBDO using the true and
approximated DRM-based MPP are compared with the FDM results in Section 5.4.2.
Section 5.4.3 illustrates how the sensitivity at the approximated DRM-based MPP shown
in Eq. (5.47) converges to the sensitivity at the true DRM-based MPP in Eq. (5.31) as the
design approaches the optimum design.
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79
Since the analytic sensitivities derived in this study do not require additional
function evaluations unlike sensitivities using FDM, the analytic calculation is always
significantly more efficient than the FDM. Hence, the efficiency comparison between the
analytic method and FDM is excluded in this study.
5.4.1 Sensitivities for FORM-based PMA
Consider a highly nonlinear performance function
2 3
1 2 1 2
4
1 2 1 2
( ) 1 (0.9063 0.4226 6) (0.9063 0.4226 6)
0.6 (0.9063 0.4226 6) 0.4226 0.9063
G X X X X
X X X X
X (5.48)
where 1 ~ (4.0,0.4)X N , 2 ~ (3.0,0.3)X Uniform , and 2t . Table 5.1 compares the
analytic sensitivity obtained from Eq. (5.18), which was listed in the second column and
labeled “Analytic,” and the sensitivities obtained from FDM with various perturbation
sizes listed in the subsequent columns. From the table, it can be shown that sensitivities
obtained by using two methods agree very well.
Table 5.1. Comparison of Sensitivities Using Analytic and FDM Results
Analytic Finite Difference Method with step size
5% 1% 0.5% 0.1% 0.05% 0.01%
1
G
d
MPP
0.8475 0.6822 0.8115 0.8293 0.8438 0.8457 0.8473
2
G
d
MPP
-0.7082 -0.7370 -0.7141 -0.7111 -0.7088 -0.7085 -0.7081
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80
As demonstrated in Table 5.3, this agreement between the FDM results and the
analytic results is also very good for a multi-dimensional function, which is one of the
constraints in the side impact problem (Youn et al., 2004), given by
3 7 5 6
29 10 9 11 11
( ) 1.35 0.489 0.843
0.0432 0.0556 0.000786
G X X X X
X X X X X
X (5.49)
where the properties of the random variables are listed in Table 5.2.
Table 5.2. Properties of Random Variables for Side Impact Problem
Random Variable Std
Dev.
Distr.
Type L
d d Ud
1. B-pillar inner (mm) 0.050 Normal 0.500 0.5318 1.500
2. B-pillar reinforce (mm) 0.050 Normal 0.450 1.3500 1.350
3. Floor side inner (mm) 0.050 Normal 0.500 1.5000 1.500
4. Cross member (mm) 0.050 Normal 0.500 1.4261 1.500
5. Door beam (mm) 0.050 Normal 0.875 1.4718 2.625
6. Door belt line (mm) 0.050 Normal 0.400 1.2000 1.200
7. Roof rail (mm) 0.050 Normal 0.400 0.4000 1.200
8. Mat. B-pillar inner (GPa) 0.006 Uniform 0.192 0.3450 0.345
9. Mat. Floor side inner (GPa) 0.006 Uniform 0.192 0.1920 0.345
10. Barrier height (mm) 10.00 Uniform 10th
and 11th
variables are
not design variables 11. Barrier hitting (mm) 10.00 Uniform
Table 5.3. Comparison of Sensitivities Using Analytic and FDM Results
Analytic Finite Difference Method with step size
5% 1% 0.1%
3
G
d
MPP
-0.1354 -0.1352 -0.1354 -0.1354
6
G
d
MPP
-0.8972 -0.8968 -0.8971 -0.8972
7
G
d
MPP
-0.5410 -0.5409 -0.5409 -0.5410
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81
5.4.2 Sensitivities for DRM-based PMA
Consider the same performance function in Eq. (5.48), and the properties of
random variables are 1 ~ (4,0.3)X N , 2 ~ (3,0.3)X N , and 2t . Table 5.4 compares the
analytic sensitivity of the DRM-based probability of failure and the FDM results. The
second column, labeled “Analytic,” indicates the sensitivity obtained from Eq. (5.36), and
the subsequent three columns indicate the FDM results with three different perturbation
sizes. As illustrated in Table 5.4, both sensitivity results agree very well; however, the
magnitude is very small and the value will be smaller as the design approaches the
optimum, which will be shown in Section 5.4.3. This is the reason we can approximate
the analytic sensitivity for DRM-based RBDO using Eq. (5.47).
Table 5.4. Comparison of Sensitivities Using Analytic and FDM Results
Analytic Finite Difference Method with step size
1% 0.1% 0.01% DRM
1
FP
d
-0.002349 -0.001956 -0.002305 -0.002345
DRM
2
FP
d
-0.001095 -0.001027 -0.001088 -0.001097
Tables 5.5 and 5.6 compare the FDM and analytic sensitivities of the probabilistic
constraint at the true and approximated DRM-based MPP, respectively. The second
columns of the tables indicate the sensitivities obtained from Eqs. (5.31) and (5.42),
respectively. Table 5.5 demonstrates the good agreement between the analytic and FDM
sensitivities. However, the agreement between the analytic and FDM sensitivity in Table
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82
5.6 is not very good, which is attributed to the assumption that the direction of the
gradients at the FORM-based MPP and approximated DRM-based MPP is the same.
Again, the inaccuracy in Table 5.6 will disappear as the design approaches the optimum
design.
Table 5.5. Comparison of Sensitivities at True DRM-based MPP
Analytic Finite Difference Method with step size
1% 0.1% 0.01%
*1
G
d
DRMx=x
1.2949 1.2565 1.2909 1.2945
*2
G
d
DRMx=x
-0.4996 -0.5059 -0.5002 -0.4997
Table 5.6. Comparison of Sensitivities at Approximated DRM-based MPP
Analytic Finite Difference Method with step size
1% 0.1% 0.01%
1
G
d
aDRMx=x
1.3871 1.2597 1.2939 1.2975
2
G
d
aDRMx=x
-0.4566 -0.5045 -0.4989 -0.4983
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83
5.4.3 Convergence Study
For convergence study of the sensitivities, a two-dimensional mathematical
RBDO problem is formulated using Eq. (5.2) where
2 2
1 2 1 2
2
1 21
2 3
2 1 2 1 2
4
1 2 1 2
3 2
1 2
L T U
( 10) ( 10)( )
30 120
( ) 120
( ) 1 (0.9063 0.4226 6) (0.9063 0.4226 6)
0.6(0.9063 0.4226 6) 0.4226 0.9063
80( ) 1
8 5
[0,0] and [10,
d d d df
X XG
G X X X X
X X X X
GX X
d
X
X
X
d d T initial T10] , [4,3] , ~ ( ,0.3) for =1,2i iX N d id
(5.50)
and the target probability of failure for each constraint is Tar ( ) ( 2), i=1~3iF tP .
Table 5.7 illustrates the current design in the second column, the updated
reliability index in the third column, the probability of failure by DRM in the fourth
column, and the sensitivities at the FORM-based MPP, approximated DRM-based MPP,
and true DRM-based MPP of the second constraint in the subsequent columns,
respectively. The sensitivity at the true DRM-based MPP in the seventh column is
obtained by carrying out a new MPP search at the current design with the updated
reliability index in the third column. The last column shows the sensitivity
DRMup DRM
1
1 DRM 1( )
U Fg P
bd d
, which is used in Eqs. (5.31) and (5.42). From Table 5.7, we
can see that up
1
1
bd
is a very small value and converges to zero as the design approaches
the optimum, which is the reason we can ignore the term. In addition, both sensitivities at
the true and approximated DRM-based MPP converge to the sensitivity at the FORM-
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84
based MPP. At the optimum design, the probability of failure by DRM should be the
target probability of failure, which is Tar ( 2) 2.2750FP . However, due to the
optimization tolerance, there is some difference.
Table 5.7. Convergence History of Sensitivities for Second Constraint
Iter. design
up DRM
FP *
2
1
G
x
x x
aDRM
2
1
G
x
x x
*DRM
2
1
G
x
x x
up
1
1
bd
0,1* 4.000, 3.000 1.8500 1.5777 1.2480 1.4134 1.3199 -0.02410
0,2 4.571, 1.106 1.9238 2.7035 2.4485 2.2884 2.3276 0.00713
1,1 4.608, 1.603 1.8519 1.9137 1.3804 1.4700 1.4229 -0.00227
2,1 4.709, 1.566 1.8429 2.2271 1.2325 1.2418 1.2366 -0.00369
3,1 4.719, 1.559 1.8423 2.2712 1.2236 1.2242 1.2238 -0.00140 * 0,1 means 0
th iteration and 1
st line search.
These numerical results indicate that the sensitivities of the FORM-based PMA in
Eq. (5.18) and DRM-based PMA in Eq. (5.47) are suitable to use for the gradient-based
design optimization. Furthermore, the sensitivity in Eq. (5.47) for the DRM-based PMA
for RBDO is very effective because it does not require an additional MPP search and the
second-order derivatives as it shows very good accuracy near the optimum design.
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85
CHAPTER VI
RBDO AND RBRDO USING DRM-BASED INVERSE RELIABILITY
ANALYSIS
6.1 Introduction
The new algorithm for DRM-based RBDO will be explained in Section 6.2. As
explained in Section 4.3 using Eq. (4.13), the number of function evaluations for the
DRM-based RBDO increases as the number of design variables increases. To reduce the
number of function evaluations, two numerical strategies, which are integrated with
PMA+, will be explained in Section 6.3. The proposed DRM-based RBDO is explained
using numerical examples in Section 6.4. Furthermore, RBRDO combined with the
proposed DRM-based RBDO is demonstrated using numerical examples in Section 6.5.
6.2 Algorithm of DRM-Based RBDO
As explained in Section 5.1, the DRM-based PMA for RBDO is formulated to
minimize Cost( )
subject to ( ) 0, 1, ,
, R and R
i
L U ndv nrv
G i nc
*
DRM
d
x
d d d d X
(6.1)
The detailed algorithm of the proposed DRM-based RBDO in Eq. (6.1) is shown in
Figure 6.1. Note that, if the DRM-based MPP identified the probabilistic constraints in
Eq. (6.1) are very close to linear so that the reliability index update in Eq. (4.15) is not
required, then the FORM-based RBDO is used during the design optimization. A
reliability analysis using the MPP-based DRM will be again used at the FORM-based
86
86
optimum design to verify whether the FORM-based optimum is the true optimum or not.
This procedure will reduce the computational cost and at the same time give users
confidence that the correct optimum design is obtained.
Figure 6.1 Algorithm of DRM-Based RBDO
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87
6.3 Strategies for Efficiency of DRM-Based RBDO
As shown in Eq. (4.13), the number of function evaluations to calculate the
probability of failure using DRM will increase as the number of design variables
increases. However, certain design variables may not affect some performance functions
even if they affect other performance functions. In that case, it is proposed to use a
reduced rotation matrix to reduce the number of required function evaluations identified
in Eq. (4.13). In addition, since the initial MPP is searched using FORM, the enhanced
hybrid mean value (HMV+) is used in this paper for efficient inverse reliability analysis,
and for the design optimization problem in Eq. (6.1), the enriched performance measure
approach (PMA+) is used for efficiency, which includes three key ideas: launching
RBDO at the deterministic optimum design, feasibility checking using constraint
activeness, and design closeness. In this study, new tolerances for constraint activeness
are introduced for additional numerical efficiency of DRM-based RBDO. Finally, the
deterministic optimization with shifted constraints (Wu et al., 2001) is used. This section
describes two new strategies: the reduced rotation matrix and new tolerances for
constraint activeness.
A. Reduced Rotation Matrix
As explained in Section 2.5.3, an N N rotation matrix is used to transform U-
space to V-space. This rotated standard normal variable v is used to compute the
probability of failure in Eq. (4.11). If the random variable iX does not affect the
performance function, i.e.,
* 0 and 0i
i
gu
u
u=0
(6.2)
88
88
then 1 *( ) ( ) ( ) 0s s n s
i i i i i iG v G v G v since iv is a function of iu (i.e., function of ix )
only and ix does not affect the performance function. Hence, the integral along thi axis
can be expressed as
1 1
( )( ) ( )
s jni i
j
j
G vw
b
(6.3)
If there are 0N random variables that do not affect the performance function, then Eq.
(4.11) can be rewritten as
11
1 11 1DRM 1 1
22
( ) ( )( ) ( )
( ) ( )
e
e
Ns j s jN n ni i i i
j j
j ji i
F NN
G v G vw w
b bP
(6.4)
where eN is the effective number of random variables defined by 0eN N N . Since the
reduced number of random variables does not change the probability of failure as shown
in Eq. (6.4), we can use e eN N rotation matrix, which will reduce the number of
function evaluations required to compute the probability of failure from ( 1)( 1)n N to
( 1)( 1)en N , in addition to the FORM-based MPP search. Hence, the total number of
function evaluations using the reduced rotation matrix becomes
# of F.E. for MPP search + ( 1) ( 1)eN n (6.5)
The reduced rotation matrix, which has full rank, can be generated using the Gram-
Schmidt orthogonalization.
This reduced rotation matrix strategy is useful when the problem is multi-
dimensional and contains a number of constraints. This is because some design variables
that affects some constraints may not affect other constraints. The computational
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89
efficiency obtained by the reduced rotation matrix is demonstrated in Section 6.4 using
the side impact example.
B. New Tolerances for Constraint Activeness
The PMA+ provides an efficient feasibility identification using the mean value
(MV) method as explained in Eqs. (2.15) and (2.16), which does not require the MPP
search. In PMA+, a constraint function is identified as active or violated if
( ) 0i fG *
MVX , where the tolerance f is a small positive number and ( )iG *
MVX is
the function value at the MV-based MPP obtained using Eq. (2.16). After the feasibility
identification, if a constraint is identified as active or violated, then an accurate MPP
search is carried out using HMV+. However, a single tolerance f may not be effective
to identify the feasibility of all constraint functions, since the magnitude of constraint
gradients could be rather different.
To avoid this difficulty, it is proposed in this section to adaptively identify
feasibility of the constraint functions using the sensitivities at a given design as
2( ) 0i
i f
GG
N
*
MVX (6.6)
where L2-norm of the sensitivity of thi constraint is normalized using N to eliminate
dimensionality of the norm. However, in case that 2iG
N
is large, the feasibility
identification using Eq. (6.6) may be too conservative, which makes the constraint
activeness strategy ineffective. Hence, in this study, a constraint is identified as active or
violated based on the normalized L2-norm of sensitivities if
90
90
2 2( ) 0, if 1
( ) 0, otherwise
i i
i f
i f
G GG
N N
G
*
MV
*
MV
X
X
(6.7)
6.4 Numerical Examples for DRM-Based RBDO
6.4.1 Effectiveness of Reduced Rotation Matrix
One of the constraint functions from the automotive side impact problem (Youn et
al., 2004)
2
2 3 8 3 10 7 9 2( ) 0.4511 0.61 0.163 0.001232 0.166 0.227G X X X X X X X X X (6.8)
is used to test the effectiveness of the reduced rotation matrix. Since only six random
variables out of eleven are included in the constraint function, 6 6 rotation matrix is
used to compute the probability of failure in Eq. (4.11). This reduced rotation matrix
reduces the number of function evaluations from (3 1) (11 1) 20 to
(3 1) (6 1) 10 when three quadrature points are used, while maintaining the
accuracy as shown in Table 6.1. The reliability index of 2 is used for the FORM-
based inverse reliability analysis.
Table 6.1. Effectiveness of Reduced Rotation Matrix
FORM DRM with
Reduced Matrix MCS
FP , % 2.2750 2.4317 2.4257
F.E. 3* 3*+10** 1 million * 3 means the number of function and sensitivity analysis for MPP search.
** 10 function evaluations for DRM do not require sensitivity analysis.
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91
6.4.2 Comparison of Various RBDO Methods
Consider the following two-dimensional mathematic model for RBDO. The
RBDO problem is formulated to
Tar
2 2
1 2 1 2
2
1 21
minimize ( )
subject to ( ( ( )) 0) , 1, ,
, R and R
( 10) ( 10)where ( )
30 120
( ) 120
ii F
L U ndv nrv
C
P G P i nc
d d d dC
X XG
d
X d
d d d d X
d
X
2 3 4
2
3 2
1 2
initial
( ) 1 ( 6) ( 6) 0.6 ( 6)
80 ( ) 1
8 5
[0,0] and [10,10] , [5,5]
~ ( ,0.5) for =1,2
L T U T T
i i
G Y Y Y Z
GX X
X N d i
X
X
d d d
(6.9)
where 1
2
0.9063 0.4226
0.4226 0.9063
XY
Z X
, and the target probability of failure for each
constraint is Tar 5.0%, i=1~3iFP . Since the given target probability of failure is 5.0%,
the initial reliability index 1 Tar( ) 1.645
iFP is selected, and during the design
iteration the reliability index is updated using the DRM-based probability of failure.
Figure 6.2 (a) shows the approximated feasible region for PMA+ with the
tolerance 0.5f and Figure 6.2 (b) shows the approximated feasible region when the
new tolerance for constraint activeness in Eq. (6.7) is used with the same 0.5f . From
Figure 6.2 (a), we can see that even though the true MPP is far from the third constraint,
the PMA+ would identify the third constraint as active because
3
MV ( ( )) 0.26 0.5 0p fG X d (6.10)
92
92
as shown in the initial design of RBDO in Table 6.2, whereas PMA+ with the new
tolerance identified it as inactive. Table 6.2 shows the history of DRM-based RBDO with
three quadrature points when new tolerances for constraint activeness are used.
Table 6.2. DRM-Based RBDO (3pts) with New Tolerances
Iter. Cost Design 1( )G X 2 ( )G X 3( )G X # of F.E.
D.O.* -1.28 4.621, 3.091 0.00 0.00 -0.39 11+11
0,1**
0,2
1,1
1,2
2,1
2,2
3,1
-1.28
-2.11
-1.83
-1.83
-1.77
-1.77
-1.77
4.621, 3.091
4.925, 1.123
4.604, 1.730
4.606, 1.727
4.678, 1.859
4.674, 1.853
4.682, 1.849
-0.83
0.65
0.14
0.14
-0.01
0.00
0.00
-0.94
0.71
0.00
0.00
-0.01
-0.01
0.00
-0.26
-0.52
-0.50
-0.50
-0.45
-0.45
-0.45
11+11
26+22
41+33
50+38
65+49
74+54
83+59
Opt. -1.77 4.682, 1.849 Active Active Inact. 83+59 * D.O. means deterministic optimum.
** 0,1 means 0th
iteration and 1st line search.
(a) Using PMA+ (b) New Tolerances
Figure 6.2. Feasibility Identification Using PMA+ and New Tolerances
93
93
In this example, we can see that 0.5f is rather large for the third constraint
compared to the other two constraints. In Figure 6.2 (b), since the third constraint is
identified as inactive by Eq. (6.7) with
3 2
3 ( ) 02
f
GG
*
MVX (6.11)
The MPP search for the third constraint is not carried out at the deterministic optimum
design with shift, which saves the number of function evaluations from 109+73 to 83+59
as shown in Table 6.3. Table 6.3 also shows that even though the optimum design of
FORM-based RBDO seems to be close to the optimum design of DRM-based RBDO, the
probability of failure computed at the optimum shows significant difference, especially
for the highly nonlinear second constraint. For the second constraint, three quadrature
points may not be sufficient to detect the nonlinearity of the constraint. In this case, five
quadrature points can be used to enhance the accuracy as shown in Table 6.3.
Table 6.3. Various RBDO Results with Target Probability of Failure Tar 5.0%iFP
Method Cost Optimum
Design
MCS # of F.E.
1FP , % 2FP , %
FORM* -1.77 4.580,1.863 5.8128 2.5794 103+103
DRM** 3 pts -1.77 4.682,1.849 4.9857 3.8030 109+73
5 pts -1.78 4.717,1.833 4.9616 4.5010 129+69
DRM*** 3 pts -1.77 4.682,1.849 4.9857 3.8030 83+59
5 pts -1.78 4.717,1.833 4.9616 4.5010 97+57 * means FORM without PMA+.
** means DRM with PMA+.
*** means DRM with PMA+ and new tolerances for constraint activeness.
94
94
Table 6.4. Updated Reliability Index at the Optimum
DRM with 3 pts DRM with 5 pts Initial
1 2 1 2
1.717 1.462 1.715 1.379 1.645
(a) For 1( )G X (b) For 2 ( )G X
Figure 6.3. Updated Reliability Index at Optimum for 1( )G X and 2 ( )G X
Since the DRM-based RBDO updates the reliability index of active constraints at
each design iteration, the reliability index at the optimum design will be different from
the initial reliability index, which is 1 Tar( ) 1.645
iFP for this example. As shown
in Table 6.4, when DRM with five quadrature point is used, the reliability index for the
second constraint at the optimum design is 1.379, which is significantly reduced from the
initial reliability index since the second constraint is highly nonlinear and concave near
MPP, and the reliability index for the first constraint at the optimum is 1.715, which is
95
95
increased slightly since the first constraint is mildly nonlinear and convex near MPP as
shown in Figure 6.3.
6.4.3 RBDO for Side Impact Crashworthiness
Consider the vehicle side impact crashworthiness (Youn et al., 2004) shown in
Figure 6.4. The design objective is to minimize vehicle weight while enhancing the side
impact crash performances.
Figure 6.4 Side Impact Model
The RBDO for vehicle side impact is formulated to
Tar
minimize Cost( )
subject to ( ( ( )) 0) , 1, ,
ii F
L U
P G P i nc
d
X d
d d d
(6.12)
where the cost function is the weight of vehicle given by
1 2 3 4 5 7Cost( ) 1.98 4.9 6.67 6.98 4.01 1.78 2.73d d d d d d d (6.13)
and constraints are
96
96
1 2 1 8 3 10
2 3 10 1 2 2 8 5 10
7 8 8 9
3 3 1 2 5 10 6 9 7 8 9 10
4
( ) 14.36 9.9 12.9 0.1107
( ) 1.86 2.95 0.17921 5.057 11.0 0.0215
9.98 22.0
( ) 3.02 3.818 4.2 0.0207 6.63 7.7 0.32
( ) 0.059 0.0159
G X X X X X
G X X X X X X X X
X X X X
G X X X X X X X X X X X
G
X
X
X
X 1 2 1 8 2 7 3 5
5 10 6 9 8 11 10 11
5 5 1 8 1 9 2 6 2 7
3 8 3 9 5 6
0.188 0.019 0.0144
0.0008757 0.08045 0.00139 0.00001575
( ) 0.106 0.00817 0.131 0.0704 0.03099 0.018
0.0208 0.121 0.00364 0.0007715
X X X X X X X X
X X X X X X X X
G X X X X X X X X X
X X X X X X
X
5 10
6 10 8 11
26 2 3 8 3 10 7 9 2
27 4 2 3 4 10 6 10 11
8 1 2 2 8 3 10
0.0005354 0.00121
( ) 0.42 0.61 0.163 0.001232 0.166 0.227
( ) 0.72 0.5 0.19 0.0122 0.009325 0.000191
( ) 0.68 0.674 1.95 0.02054
X X
X X X X
G X X X X X X X X
G X X X X X X X X
G X X X X X X
X
X
X 4 10 6 10
29 3 7 5 6 9 10 9 11 11
10 2 4 2 10 3 9 6 10
0.0198 0.028
( ) 1.35 0.489 0.843 0.0432 0.0556 0.000786
( ) 0.16 0.3717 0.00931 0.484 0.01343
X X X X
G X X X X X X X X X
G X X X X X X X X
X
X
Table 6.5. Properties of Random Variables for Side Impact Problem
Random Variable Std
Dev.
Distr.
Type L
d d Ud
1. B-pillar inner (mm) 0.050 Normal 0.500 1.000 1.500
2. B-pillar reinforce (mm) 0.050 Normal 0.450 1.000 1.350
3. Floor side inner (mm) 0.050 Normal 0.500 1.000 1.500
4. Cross member (mm) 0.050 Normal 0.500 1.000 1.500
5. Door beam (mm) 0.050 Normal 0.875 2.000 2.625
6. Door belt line (mm) 0.050 Normal 0.400 1.000 1.200
7. Roof rail (mm) 0.050 Normal 0.400 1.000 1.200
8. Mat. B-pillar inner (GPa) 0.006 Normal 0.192 0.300 0.345
9. Mat. Floor side inner (GPa) 0.006 Normal 0.192 0.300 0.345
10. Barrier height (mm) 10.00 Normal 10th
and 11th
variables are
not design variables 11. Barrier hitting (mm) 10.00 Normal
The target probability of failure is given by Tar 5.0%iFP , and thus the target
reliability index for FORM-based RBDO is 1(0.05) 1.645t . The random
variables 1 11~X X are listed in Table 6.5. As shown in Table 6.5, 9 random variables out
97
97
of 11 are regarded as design variables. Tables 6.6 and 6.7 are design and constraint
history obtained using the FORM-based RBDO and PMA+ with new tolerances for
constraint activeness. The shadowed cells in Table 6.7 show the active constraints at each
design iteration.
Table 6.6. Design History for Side Impact Example Using FORM-Based RBDO
Cost d1 d2 d3 d4 d5 d6 d7 d8 d9 NFE
Initial 30.83 1.000 1.000 1.000 1.000 2.000 1.000 1.000 0.300 0.300 0+0
D.O.* 24.23 0.500 1.226 0.500 1.207 1.238 1.200 0.400 0.345 0.300 8+8
0,1** 24.23 0.500 1.226 0.500 1.207 1.238 1.200 0.400 0.345 0.300 26+26
0,2 26.20 0.520 1.350 0.500 1.377 1.443 1.200 0.400 0.345 0.192 37+37
0,3 26.15 0.519 1.347 0.500 1.372 1.438 1.200 0.400 0.345 0.195 41+41
1,1 26.23 0.510 1.350 0.500 1.391 1.453 1.200 0.400 0.345 0.192 45+45
1,2 26.22 0.512 1.349 0.500 1.387 1.450 1.200 0.400 0.345 0.193 49+49
1,3 26.21 0.513 1.349 0.500 1.385 1.449 1.200 0.400 0.345 0.193 53+53
Opt. 26.20 0.511 1.350 0.500 1.384 1.451 1.200 0.400 0.345 0.192 57+57
* D.O. means deterministic optimum.
** 0,1 means 0th
iteration and 1st line search.
Table 6.7. Constraint History for Side Impact Example Using FORM-Based RBDO
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10
Initial 0.04 -2.45 -1.23 -1.65 -1.13 -0.15 0.04 -0.85 -0.61 -2.23
D.O. -0.00 -1.89 -0.78 -1.23 -0.97 -0.08 -0.00 -0.82 0.00 -2.89
0,1 0.09 -0.47 -0.09 -0.87 -0.81 -0.05 0.10 -0.31 0.22 -2.30
0,2 -0.00 -1.43 -0.65 -1.04 -0.86 -0.05 0.00 -0.59 0.00 -3.09
0,3 -0.00 -1.41 -0.63 -1.04 -0.86 -0.05 0.00 -0.59 0.01 -3.07
1,1 0.00 -1.40 -0.63 -1.03 -0.86 -0.05 -0.00 -0.59 -0.00 -3.13
1,2 0.00 -1.40 -0.63 -1.03 -0.86 -0.05 -0.00 -0.59 -0.00 -3.12
1,3 0.00 -1.40 -0.63 -1.03 -0.86 -0.05 -0.00 -0.59 0.00 -3.11
2,1 0.00 -1.40 -0.63 -1.03 -0.86 -0.05 0.00 -0.59 -0.00 -3.11
Opt. Act. Inact. Inact. Inact. Inact. Inact. Act. Inact. Act. Inact.
Table 6.7 shows that three constraints, G1, G7, and G9, are active, which means
that the failure probability of three constraints using FORM is the target probability of
98
98
failure, Tar 5.0%iFP . However, Table 6.8 illustrates that true failure probability for each
constraint obtained using MCS is not the same as the target probability, especially, for G7
and G9, which means FORM has significant error in estimation of failure probability for
those two constraints. Hence, for more accurate RBDO results, the DRM-based RBDO is
used for the vehicle side impact problem.
Table 6.8. Probability of Failure at Optimum Using FORM-Based RBDO
Tar
FP , % MCS at Optimum
1FP , % 7FP , %
9FP , %
5.000
5.0075
11.4905
2.5117
Tables 6.9 and 10 demonstrates design and constraint history obtained using the
DRM-based RBDO with three quadrature points and PMA+ with new tolerances for
constraint activeness. The shadowed cells in Table 6.10 show the active constraints,
which is identified using PMA+ with new tolerances, at each design point. As illustrated
in Table 6.11, the failure probability estimation at the optimum of the DRM-based RBDO
is closer to the target probability of failure than the FORM-based RBDO optimum.
However, there exist some errors in estimation of the probability of failure using
DRM with three quadrature points. This can be resolved if five quadrature points are used
for the estimation of failure probability. Table 6.12 compares various RBDOs. From
Table 6.12, it can be seen that new tolerances for constraint activeness reduce the number
of function evaluations for both FORM and DRM-based RBDO considerably. This is
because the sixth constraint is identified as inactive using the new tolerance because of its
small sensitivity. Whereas, using PMA+ with the fixed tolerance 0.5f , the sixth
99
99
constraint is identified as active since 6 ( ) 0.05 0.5 0fG *
MVX . In addition, the
DRM-based RBDO shows indeed different optimum design, especially d4 and d5,
compared with the FORM-based RBDO because of the highly nonlinear active
constraints G7 and G9.
Table 6.9. Design History for Side Impact Example Using DRM-Based RBDO
Cost d1 d2 d3 d4 d5 d6 d7 d8 d9 NFE
Initial 30.83 1.000 1.000 1.000 1.000 2.000 1.000 1.000 0.300 0.300 0+0
D.O.* 24.23 0.500 1.226 0.500 1.207 1.238 1.200 0.400 0.345 0.300 8+8
0,1** 24.23 0.500 1.226 0.500 1.207 1.238 1.200 0.400 0.345 0.300 100+32
0,2 26.28 0.519 1.350 0.500 1.412 1.411 1.200 0.400 0.345 0.192 142+44
0,3 26.23 0.518 1.347 0.500 1.407 1.406 1.200 0.400 0.345 0.195 179+51
1,1 26.53 0.509 1.350 0.500 1.488 1.405 1.200 0.400 0.345 0.192 217+59
1,2 26.47 0.511 1.349 0.500 1.472 1.406 1.200 0.400 0.345 0.193 254+66
1,3 26.44 0.511 1.349 0.500 1.465 1.406 1.200 0.400 0.345 0.193 291+73
2,1 26.43 0.510 1.350 0.500 1.462 1.405 1.200 0.400 0.345 0.192 328+80
3,1 26.43 0.510 1.350 0.500 1.464 1.405 1.200 0.400 0.345 0.192 364+86
3,2 26.43 0.510 1.350 0.500 1.463 1.405 1.200 0.400 0.345 0.192 400+92
Opt. 26.43 0.510 1.350 0.500 1.463 1.405 1.200 0.400 0.345 0.192 436+98
Table 6.10. Constraint History for Side Impact Example Using DRM-Based RBDO
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10
Initial 0.04 -2.45 -1.23 -1.65 -1.13 -0.15 0.04 -0.85 -0.61 -2.23
D.O. -0.00 -1.89 -0.78 -1.23 -0.97 -0.08 -0.00 -0.82 0.00 -2.89
0,1 0.09 -0.47 -0.09 -0.87 -0.81 -0.05 0.12 -0.31 0.19 -2.30
0,2 -0.00 -1.43 -0.65 -1.05 -0.87 -0.05 0.02 -0.61 -0.00 -3.19
0,3 -0.00 -1.40 -0.64 -1.05 -0.87 -0.05 0.03 -0.60 0.00 -3.17
1,1 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.01 -0.63 -0.00 -3.43
1,2 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.00 -0.63 -0.00 -3.37
1,3 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.00 -0.62 -0.00 -3.35
2,1 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.00 -0.62 -0.00 -3.35
3,1 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.00 -0.62 0.00 -3.35
3,2 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.00 -0.62 0.00 -3.35
3,3 0.00 -1.39 -0.63 -1.04 -0.86 -0.05 -0.00 -0.62 0.00 -3.35
Opt. Act. Inact. Inact. Inact. Inact. Inact. Act. Inact. Act. Inact.
100
100
Table 6.11. Probability of Failure at Optimum Using DRM-Based RBDO
Tar
FP , % MCS
1FP , % 7FP , %
9FP , %
5.000
5.0270 5.5903 5.4002
Table 6.12. Comparison of Various RBDOs
FORM-based RBDO DRM-based RBDO
Without
PMA+ PMA+
PMA+ with
New f Without
PMA+ PMA+
PMA+ with
New f
d1 0.511 0.511 0.511 0.510 0.510 0.510
d2 1.350 1.350 1.350 1.350 1.350 1.350
d3 0.500 0.500 0.500 0.500 0.500 0.500
d4 1.384 1.384 1.384 1.463 1.463 1.463
d5 1.451 1.451 1.451 1.405 1.405 1.405
d6 1.200 1.200 1.200 1.200 1.200 1.200
d7 0.400 0.400 0.400 0.400 0.400 0.400
d8 0.345 0.345 0.345 0.345 0.345 0.345
d9 0.192 0.192 0.192 0.192 0.192 0.192
Cost 26.20 26.20 26.20 26.43 26.43 26.43
Active
Constraints 1,7,9 1,7,9 1,7,9 1,7,9 1,7,9 1,7,9
NFE 185+185 66+66 57+57 1700+336 557+119 436+98
6.4.4 Tracked Vehicle Roadarm Problem
The roadarm of a tracked vehicle is used to demonstrate applicability of the
DRM-based RBDO. The roadarm is modeled using 1572 eight-node isoparametric finite
elements (SOLID45) and four beam elements (BEAM44) of Ansys (Swanson Analysis
System Inc., 1989), as shown in Figure 6.5, and is made of S4340 steel with Young’s
modulus E=3.0×107 psi and Poisson’s ratio ν=0.3. The durability analysis of the roadarm
is carried out using Durability and Reliability Analysis Workspace (DRAW) (CCAD,
101
101
1999a-b), to obtain the fatigue life contour as shown in Figure 6.6. The fatigue lives at
the critical nodes shown in Figure 6.6 are chosen as the design constraints of RBDO.
Figure 6.5. Finite Element Model of Roadarm
Figure 6.6. Fatigue Life Contour and Critical Nodes of Roadarm
102
102
The shape design variables are shown in Figure 6.7. Eight shape design variables
characterize four cross sectional shapes of the roadarm. Widths ( 1 directionx ) of the
cross-sectional shapes are defined by the design variables d1, d2, d5, and d6 at the
intersections 1 to 4, respectively, and heights ( 3 directionx ) of the cross sectional
shapes are defined using the remaining four design variables. Eight shape design random
variables and six random variables for the fatigue material properties are listed in Table
6.13.
Figure 6.7. Shape Design Variables for Roadarm
103
103
Table 6.13. Properties of Input Random Variables for Roadarm
Random
Variables
Lower Bound L
d
Initial Design 0d
Upper Bound Ud
Standard
Deviation
Distribution
Type
d1 1.3500 1.7500 2.1500 0.0175 Normal
d2 2.6496 3.2496 3.7496 0.0325 Normal
d3 1.3500 1.7500 2.1500 0.0175 Normal
d4 2.5703 3.1703 3.6703 0.0317 Normal
d5 1.3563 1.7563 2.1563 0.0176 Normal
d6 2.4377 3.0377 3.5377 0.0304 Normal
d7 1.3517 1.7517 2.1517 0.0175 Normal
d8 2.5085 2.9085 3.4085 0.0291 Normal
Fatigue Material Properties
Non-design Uncertainties Mean Standard
Deviation
Distribution
Type
Cyclic Strength Coefficient, K 197000 5910 Normal
Cyclic Strength Exponent, n 0.1200 0.0036 Normal
Fatigue Strength Coefficient, 177000 5310 Normal
Fatigue Strength Exponent, b -0.0730 0.00219 Normal
Fatigue Ductility Coefficient, f 0.4100 0.0123 Normal
Fatigue Ductility Exponent, c -0.6000 0.0180 Normal
The RBDO problem for the roadarm can be formulated to
Tar
minimize Cost( )
subject to ( ( ) 0) , 1, ,
ii F
L U
P G P i nc
d
d
d d d
(6.14)
where
Tar 1 1
Cost( ) : Weight of Roadarm
( )( ) 1 , 1, ,
( ) : Crack Initiation Fatigue Life,
: Crack Initiation Target Fatigue Life (=8 years)
( ) ( 2) , 1, ,i
i
t
t
F t
LG i nc
L
L
L
P i nc
d
dd
d (6.15)
and number of constraints nc = 13 as shown in Figure 6.6. The DRM-based RBDO
results are shown in Tables 6.14 and 6.15. After finding the deterministic optimum design
104
104
first, the DRM-based RBDO is launched starting at the deterministic optimum design for
the active constraints only, which require more accurate DRM-based failure probability
estimation. The shadowed cells in Table 6.15 indicate the active constraints at each
design point. After performing the DRM-based inverse reliability analysis at the
deterministic optimum design, all fatigue life constraints in this problem turn out to be
very close to linear. Hence, the FORM-based RBDO is carried out and the DRM-based
inverse reliability analysis is used to validate the RBDO optimum design.
Table 6.14. Design History for Roadarm Using DRM-Based RBDO
Cost d1 d2 d3 d4 d5 d6 d7 d8 NFE
Initial 515.09 1.750 3.250 1.750 3.170 1.756 3.038 1.752 2.908 0+0
D.O. 464.56 1.588 2.650 1.922 2.570 1.476 3.292 1.630 2.508 11+11
0,1 464.56 1.588 2.650 1.922 2.570 1.476 3.292 1.630 2.508 206+24
0,2 490.33 1.789 2.650 1.993 2.570 1.587 3.480 1.871 2.508 220+38
0,3 474.79 1.669 2.650 1.951 2.570 1.521 3.367 1.727 2.508 234+52
0,4 476.11 1.679 2.650 1.954 2.570 1.527 3.377 1.739 2.508 248+66
1,1 476.28 1.681 2.650 1.954 2.570 1.541 3.365 1.730 2.508 262+80
1,2 476.27 1.681 2.650 1.954 2.570 1.541 3.366 1.730 2.508 276+94
Opt. 476.21 1.681 2.650 1.954 2.570 1.541 3.366 1.729 2.508 394+108
Table 6.15. Constraint History for Roadarm Using DRM-Based RBDO
G1 G2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 G 13
Initial -11.2 -17.1 -12.8 -312 -836 -115 -96.8 -394 -426 -468 -542 -55.1 -45.6
D.O. 0.00 -1.03 0.00 -6325 0.00 -115 -159 0.00 -1.06 -3.92 -0.89 0.00 -0.75
0,1 0.75 0.47 0.73 -999 0.72 -23.4 -32.7 0.73 0.45 -0.22 0.50 0.82 0.71
0,2 -5.44 -5.93 -2.61 -431 -4.93 -85.4 -81.2 -2.11 -3.25 -35.7 -11.8 -9.89 -10.1
0,3 0.13 -0.48 0.23 -692 0.17 -40.0 -47.5 0.27 -0.27 -3.59 -0.75 0.04 -0.35
0,4 -0.02 -0.69 0.13 -663 0.03 -42.8 -49.8 0.17 -0.41 -4.46 -1.06 -0.19 -0.63
1,1 -0.00 -0.87 -0.01 -581 -0.00 -42.0 -47.3 -0.02 -0.70 -4.66 -1.14 -0.02 -0.36
1,2 -0.00 -0.86 -0.01 -584 -0.00 -42.0 -47.3 -0.01 -0.69 -4.65 -1.13 -0.02 -0.37
2,1 -0.00 -0.86 0.00 -586 -0.00 -42.0 -47.4 -0.00 -0.67 -4.64 -1.13 -0.00 -0.33
Opt. Act. Inact. Act. Inact. Act. Inact. Inact. Act. Inact. Inact. Inact. Act. Inact.
105
105
Table 6.16 shows the comparison of optimum designs obtained using two
methods. To verify the optimum result of the proposed DRM-based RBDO method, the
full DRM-based RBDO method, which does not take advantage of the FORM-based
RBDO during the optimization, is used because MCS cannot be used due to its
computational cost. As shown in Table 6.16, result of the proposed DRM-based RBDO
using the algorithm in Figure 6.1 is very close to the benchmark results since all fatigue
life constraints are linear, and yet reduced the number of function evaluations
significantly. Furthermore, the proposed method gives users confidence that the optimum
design is indeed a correct one.
Table 6.16. Comparison of Design Optimizations for Roadarm
Initial Deterministic
Optimization
DRM-based
RBDO*
DRM-based
RBDO**
d1 1.7500 1.588 1.681 1.681
d2 3.2496 2.650 2.650 2.650
d3 1.7500 1.922 1.954 1.954
d4 3.1703 2.570 2.570 2.570
d5 1.7563 1.476 1.541 1.541
d6 3.0377 3.292 3.366 3.366
d7 1.7517 1.630 1.729 1.729
d8 2.9085 2.508 2.508 2.508
Cost 515.09 464.56 476.19 476.21
Active
Constraints 1,3,5,8,12 1,3,5,8,12 1,3,5,8,12
NFE 11+11 394+108 864+108 * uses algorithm in Figure 6.1.
** uses full DRM-based RBDO for verification.
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106
6.5 Numerical Examples for RBRDO
Since RDO can be naturally combined with RBDO for the constraint evaluation,
which becomes RBRDO, this section deals with two numerical examples for RBRDO.
For the cost function, which is variance of the performance function, PMI and mean-
based DRM, which are explained in Chapter 3, will be used and, for the constraint
evaluation, the FORM-based RBDO and DRM-based RBDO will be used. A 2-D
numerical example for RBRDO is used in Section 6.5.1 and side impact problem is used
in Section 6.5.2.
6.5.1 RBRDO for 2-D Mathematic Example
The RBRDO model of a 2-D mathematic problem is formulated to
2
Tar
minimize
subject to ( ( ) 0) , 1, ,i
H
i FP G P i nc
X (6.16)
where the performance function for robustness and constraints are, respectively,
2 2
1 2 1 2
2
1 21
2 3 4
2
3 2
1 2
( 6) ( 6)( )
3 12
( ) 120
( ) 1 ( 6) ( 6) 0.6 ( 6)
80( ) 1
8 5
X X X Xh
X XG
G Y Y Y Z
GX X
X
X
X
X
(6.17)
where 1
2
0.9063 0.4226
0.4226 0.9063
XY
Z X
, and the target probability of failure for each
constraint is Tar ( 2), i=1~3iFP . Two random variables are listed in Table 6.17.
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107
Table 6.17. Properties of Random Variables of Eq. (6.17)
Random
Variable
Std
Dev.
Distr.
Type
Lower Bound L
d
Initial Design 0d
Upper Bound Ud
1X 0.500 Normal 0.000 5.000 10.000
2X 0.500 Normal 0.000 5.000 10.000
Table 6.18. Variance Estimation Using PMI at Initial and Optimum Design
Constraint
Evaluation
Variance at Initial Design Variance at Optimum Design
PMI N.I. PMI N.I.
FORM 4.1214 4.1147
0.3179 0.3231
MPP-Based DRM 0.3389 0.3395
Table 6.19. Variance Estimation Using DRM at Initial and Optimum Design
Constraint
Evaluation
Variance at Initial Design Variance at Optimum Design
DRM N.I. DRM N.I.
FORM 4.0773 4.1147
0.2859 0.3232
MPP-Based DRM 0.3013 0.3389
Tables 6.18 and 6.19 are the cost values at the initial and optimum designs using
PMI and the mean-based DRM, respectively. When the MPP-based DRM is used for the
probabilistic constraint evaluation, the cost at the optimum is slightly increased compared
to when FORM is used. This is because the first constraint is convex near MPP at the
optimum design and thus the reliability index at the optimum design is increased as
shown in Figure 6.8. Figure 6.8 (a) and Figure 6.8 (b) show the RBRDO optimum
designs when the MPP-based DRM and FORM are used for the constraint evaluation,
respectively. Tables 6.20 and 6.21 show the optimum design and active constraints.
Because of the highly nonlinear active constraint 2G , optimum designs obtained using
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108
two different constraint evaluation methods, FORM and MPP-based DRM, are indeed
different.
(a) MPP-based DRM (b) FORM
Figure 6.8 Optimum Design of RBRDO for Eq. (6.17)
Table 6.20. Optimum Design Using FORM-based RBRDO
Variance
Calculation 1d 2d 1G 2G 3G
DRM 4.5356 2.0947 -0.00 -0.00 -0.45
PMI 4.5357 2.0941 -0.00 -0.00 -0.45
Table 6.21. Optimum Design Using DRM-based RBRDO
Variance
Calculation 1d 2d 1G 2G 3G
DRM 4.6552 2.0719 -0.00 -0.00 -0.42
PMI 4.6226 2.0914 -0.00 -0.04 -0.43
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109
6.5.2 RBRDO for Side Impact Crashworthiness
The RBRDO model of crashworthiness of the vehicle side impact which is used
in Section 6.4.3 is formulated to
2
Tar
minimize
subject to ( ( ) 0) , 1, ,i
H
i FP G P i nc
X (6.18)
where nc is 10, the target probability of failure for each constraint is Tar ( 2)iFP , and
the performance function ( )h x for robustness is the upper rib deflection given by
3 1 2 5 10 6 9 7 8 9 10( ) 3.818 4.2 0.0207 6.63 7.7 0.32h X X X X X X X X X X X X (6.19)
11 random variables for the side impact problem are listed in Table 6.22. Note that input
standard deviations for d1~d7 are increased from 0.050 to 0.100.
Table 6.22. Properties of Design and Random Parameters for Side Impact Problem
Random Variable Std
Dev.
Distr.
Type L
d d Ud
1. B-pillar inner (mm) 0.100 Normal 0.500 1.000 1.500
2. B-pillar reinforce (mm) 0.100 Normal 0.450 1.000 1.350
3. Floor side inner (mm) 0.100 Normal 0.500 1.000 1.500
4. Cross member (mm) 0.100 Normal 0.500 1.000 1.500
5. Door beam (mm) 0.100 Normal 0.875 2.000 2.625
6. Door belt line (mm) 0.100 Normal 0.400 1.000 1.200
7. Roof rail (mm) 0.100 Normal 0.400 1.000 1.200
8. Mat. B-pillar inner (GPa) 0.006 Normal 0.192 0.300 0.345
9. Mat. Floor side inner (GPa) 0.006 Normal 0.192 0.300 0.345
10. Barrier height (mm) 10.000 Normal 10th
and 11th
random variables
are not regarded as design
variables 11. Barrier hitting (mm) 10.000 Normal
Tables 6.23 and 6.24 show the cost at the initial and optimum designs using PMI
and the mean-based DRM. As shown in Tables 6.23 and 6.24, both methods estimated
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110
variance very accurately compared to MCS. In this example, when the MPP-based DRM
is used for constraint evaluation, the optimum cost is smaller than when FORM is used.
This is because the highly nonlinear active constraint G9 is a concave function.
Furthermore, because of the highly nonlinear constraint G9, optimum designs of two
different constraint evaluation methods, FORM and MPP-based DRM, are indeed
different as shown in Table 6.25.
Table 6.23. Variance Using PMI at Initial and Optimum Design
Constraint
Evaluation
Variance
Initial Design Optimum Design
PMI N.I. PMI N.I.
FORM 2.4831 2.4852
1.4010 1.4024
MPP-Based DRM 1.3878 1.3871
Table 6.24. Variance Using Mean-Based DRM at Initial and Optimum Design
Constraint
Evaluation
Variance
Initial Design Optimum Design
DRM N.I. DRM N.I.
FORM 2.4830 2.4852
1.4008 1.4024
MPP-Based DRM 1.3877 1.3871
Table 6.25. Optimum Design Comparison
Cost Constraint d1 d2 d3 d4 d5 d6 d7 d8 d9 F.E.
DRM FORM
1.087 1.350 1.413 1.167 0.875 1.200 1.200 0.345 0.192 630+630
PMI 1.086 1.350 1.413 1.171 0.875 1.200 1.200 0.345 0.192 197+197
DRM MPP-based
DRM 1.050 1.350 1.331 1.181 0.878 1.200 1.200 0.345 0.192 998+546
PMI 1.050 1.350 1.331 1.181 0.878 1.200 1.200 0.345 0.192 565+211
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111
Table 6.26. Constraint Comparison at Optimum Design
Cost
Evaluation G1 G2 G3 G4 G5 G6 G7 G8 G9 G10
DRM* -0.001 -2.757 -1.021 -2.031 -1.080 -0.167 -0.121 -0.527 -0.000 -2.532
PMI* -0.000 -2.754 -1.019 -2.029 -1.079 -0.167 -0.124 -0.528 -0.000 -2.544
DRM** -0.000 -2.759 -1.058 -1.993 -1.082 -0.161 -0.095 -0.544 -0.000 -2.523
PMI** -0.000 -2.759 -1.058 -1.993 -1.082 -0.161 -0.095 -0.544 -0.000 -2.523
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112
CHAPTER VII
SYSTEM INVERSE RELIABILITY ANALYSIS AND RBDO
7.1 Introduction
Estimations of not only the component probability of failure but also the system
probability of failure have been the main concern in structural reliability analysis for over
three decades. According to the logical relationship of the failure modes of structures,
structural systems can be divided into three types: series system, parallel system, and
hybrid system (Zhao et al., 2007). The series system is also referred to weakest link or
chain system because the system failure is caused by the failure of any one component.
The parallel system is also referred to as a redundant system because the system fails
only if all components fail. The hybrid system is a mixed system of the series and parallel
system. In this study, the reliability analysis of the series system will be discussed since it
is the most frequently encountered in practical engineering applications.
Since the analytic estimation of the system probability of failure involves multi-
dimensional integration over the overall failure domain, it is numerically very difficult to
evaluate. Hence, several approaches to resolve the numerical difficulty have been
proposed including the narrow bound estimation (Ditlevsen, 1979). For the narrow
bound method, Ditlevsen’s first order upper bound, which is the summation of
component failure probabilities, can be used as the system probability of failure (Ba-
abbad et al., 2006) or Ditlevsen’s second order upper bound by considering the joint
probability of failure can be used (Ang and Tang, 1984; Liang et al., 2007). However, if
FORM is used, these narrow bound methods will only work for linear or very mildly
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113
nonlinear performance functions since FORM approximates performance functions using
the first order Taylor series expansion.
Thus, more accurate system reliability analysis method is needed for the system
with nonlinear and multi-dimensional performance functions. In this study, the MPP-
based dimension reduction method (DRM) and Ditlevsen’s second order upper bound are
used to propose a system reliability analysis method. In addition, using the accurate
system reliability analysis, a system Reliability-Based Design Optimization (RBDO) is
proposed.
Section 7.2 demonstrates the accurate system reliability analysis using the
Ditlevsen’s second order upper bound and MPP-based DRM. Section 7.3 illustrates the
system RBDO. For the system RBDO, sensitivity analyses are carried out and two
efficiency strategies are proposed to save the computational burden of the system RBDO.
7.2 System Inverse Reliability Analysis
When there are more than one performance function, the system probability of
failure of the series system is obtained by
sys
1
( ) 0m
F i
i
P P G
X (7.1)
where m is the number of performance functions and the performance function is defined
as failure if ( ) 0iG X . However, since the right side of Eq. (7.1) is not easy to compute
numerically, the system probability of failure is conservatively approximated using
Ditlevsen’s first-order upper bound (Ditlevsen, 1979) by the sum of the component
probabilities of failures as
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114
sys
1i
m
F F
i
P P
(7.2)
where iFP is the component probability of failure for the i
th performance function. A
more refined method to approximate the system probability of failure using Ditlevsen’s
second-order upper bound (Ditlevsen, 1979) is given by
sys
1 2
max( )i ij
m m
F F Fj i
i i
P P P
(7.3)
where ijFP is the joint probability of failure when the i
th and j
th failure modes occur
simultaneously. It is noted that the error in estimating the system probability of failure is
much more significant from the component probability of failure iFP than the correction
from the joint probability of failure ijFP in Eq. (7.3).
7.2.1 Component Probability of Failure Calculation
Since it is shown in Chapter 4 that FORM is not acceptable for the component
probability of failure calculation for a highly nonlinear and/or multi-dimensional system,
the component probability of failure using the MPP-based DRM,
1
11DRM 1
2
( )( )
( )
s jN ni i
j
ji
F N
G vw
bP
(7.4)
which is derived in Section 4.3.1, is used for Eq. (7.3). Hence, this study proposes the
conservative but accurate system probability of failure calculation using the DRM-based
component probability of failure in Eq. (7.4) and Ditlevsen’s second order upper bound
in Eq. (7.3) as
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115
sys DRM
1 2
max( )i ij
m m
F F Fj i
i i
P P P
(7.5)
which will be very accurate even for a highly nonlinear and/or multi-dimensional system.
In this study, Ditlevsen’s second order upper bound is used for the system reliability
analysis since it does not require further function evaluation. The joint probability of
failure in Eq. (7.5) will be explained in the next section.
7.2.2 Joint Probability of Failure Calculation Using FORM
Based on FORM, the joint probability of failure between the ith
and jth
performance function in Eq. (7.5), ijFP , is approximated as (Ang and Tang, 1984;, Zhao et
al., 2007; Liang et al., 2007)
( , ; ) ( , ; )i j
ijF i j ij ijP x y dxdy
(7.6)
where ( , ; ) is the PDF of a bivariate standard normal variable given as
2 2
22
1 1 2( , ; ) exp
2 12 1
x y xyx y
(7.7)
and ρ is the correlation coefficient.
Let two linearly approximated constraints of gi(u) and gj(u) at MPPs be
1 1
( ) and ( )n n
L L
i i ir r j j js s
r s
g u g u
u u (7.8)
where andi jα α are normalized vector from the origin to the MPP of each constraint
and u is the standard normal variable. Let the angle between andi jα α be as shown in
Figure 7.1, then
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116
1
cosn
ir jr
r
(7.9)
Figure 7.1 True and FORM-Base Joint Failure Region
The correlation coefficient between two constraints is defined as
2 2
1 1
Cov[ , ] Cov[ , ]Cov[ , ]
i j
i j i j
ij i jn n
g gir js
r s
g g g gg g
(7.10)
where the covariance of two constraints is given by
1 1
1 1
Cov[ , ] E[( )( )] E[ ]
E[ ] E[ ]
E[( ) ( )]
i j i j i j
i j j i i j i j
i j
i j
i j i g j g i j g j g i g g
i j g g g g g g i j g g
n n
i ir r j js s g g
r s
n n
i j ir jr g g ir jr
r r
g g g g g g g g
g g g g
u u
(7.11)
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117
where E is the expectation operator, hence, the correlation coefficient is given by
1
cos .n
ij ir jr
r
(7.12)
From Eq. (7.7), it can be analytically shown that
2
.x y
(7.13)
Then, the derivative of the bivariate standard normal CDF with respect to ρ is
2
( , ; )
( , ; )
( , ; )
x y x y
yx y x y x
x
d d d d
d d d d d
yd
x y
(7.14)
and ( , ; )x y is equal to 2
x y
by the definition of the bivariate standard normal CDF
given as
( , ; ) ( , ; )x y
x y d d
(7.15)
Hence, the bivariate standard normal CDF has the property
2
x y
(7.16)
Using Eq. (7.16), the joint probability of failure in Eq. (7.6) can be expressed as (Liang et
al., 2007)
0
0
( , ; )( , ; ) ( , ;0)
( ) ( ) ( , ; )
ij
ij
ij
i j
F i j ij i j
i j i j
P d
d
(7.17)
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118
7.2.3 System Probability of Failure Calculation
Since Eq. (7.17) uses the linear approximation of the performance functions at
MPPs, it could underestimate or overestimate the joint probability of failure depending
on convexity or concavity of the performance functions near MPPs. The definition of the
convexity is the same as the definition in Section 4.3.2, that is, if DRM FORM
F FP P , then the
function is “convex” around the MPP and vice versa for “concave”. One example that
FORM overestimates the true joint probability of failure is shown in Figure 7.1.
As explained in Section 7.2.1, the MPP-based DRM is used for the accurate
component probability of failure in Eq. (7.4), and the joint probability of failure in Eq.
(7.17) is conservatively obtained using the type of the performance functions as the
following cases.
Case (a). Ignore the joint probability of failure if both constraints are concave because
the FORM-based joint probability of failure will overestimate the true failure
as shown in Figure 7.2 (a). If all constraints are concave, then the system
probability of failure using Ditlevsen’s second order upper bound will be
identical with the system probability of failure using Ditlevsen’s first order
upper bound.
Case (b). For a highly correlated case shown in Figure 7.2 (b), that is, 0.95 , then
choose the minimum of two constraints as the joint probability of failure.
Case (c). Otherwise, use the FORM-based joint probability of failure calculation in Eq.
(7.17) because it can approximate the true joint failure reasonably as shown in
Figure 7.2 (c).
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119
Case (a) Case (b) Case (c)
Figure 7.2 Three Cases of Joint Probability of Failure Calculation
7.3 System Reliability-Based Design Optimization
7.3.1 Formulation of System RBDO
Using the system inverse reliability analysis described above, the system RBDO
is formulated to
0
find ,
min Cost( )
s.t ( ( , )) 0, 1, ,
1 0
i i
sys
F
all
F
G i nc
PG
P
*
d β
d
x d (7.18)
where d μ(X) is the mean value of the input random variable X, *x is the FORM-based
MPP, all
FP is the allowable system probability of failure, and sys
FP is the system
probability of failure calculated from Eq. (7.5). For the formulation in Eq. (7.18), no
MPP update, which is explained in Section 4.3.3, is used since the reliability indices are
also design variables.
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120
7.3.2 Sensitivity Analyses
Since d and β are both design variables for the formulation in Eq. (7.18), for
sensitivity analyses, it is required to derive the sensitivity of the component probabilistic
constraints at MPP with respect to d and β and the sensitivity of the system probability
of failure with respect to d and β . *( )G
x
d is identical with the FORM-based component
sensitivity in Eq. (5.18) as
*
*( )G G
x x
x
d x (7.19)
*( )G
x can be obtained using the chain rule as
* * *
T( ) ( )U
G gg
x u u (7.20)
From the definition of MPP in U-space in Eq. (5.5), *
u is obtained as
11*
2
1 1
UU
U
g bb g
g
b b
u (7.21)
where 1 Ub g and 2 T
1 U Ub g g . Hence, 1b
is given as
T
1
1
1 UU
gbg
b
(7.22)
By substituting Eq. (7.22) into Eq. (7.21), *
u can be obtained as
2 T*
1
3
1 1
U U U Ug b g g g
b b
Iu (7.23)
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121
and inserting Eq. (7.23) into Eq. (7.20) yields
2 T 2*
T 1 113
1 1 1
( ) U U U UU
g b g g g bGg b
b b b
Ix (7.24)
Since Ditlevsen’s second order upper bound is used for the system inverse
reliability analysis, sensitivities of the system probability of failure constraint G0 with
respect to the design variables involve two terms: sensitivity of the component
probability of failure with respect to the design variables, and sensitivity of the joint
probability of failure with respect to the design variables. The sensitivities of the joint
probability of failure with respect to the design variables can be analytically obtained
using Eq. (7.17). Since the joint probability of failure is a function of reliability indices
only, sensitivities of the joint probability of failure with respect to d is zero and
sensitivities of the joint probability of failure with respect to i is
20
( ) ( ) ( , ; )1
ijijF i j
i j i j
i
Pd
(7.25)
The sensitivity of the component probability of failure with respect to d is derived in Eq.
(5.36) and the numerical example in Section 5.4.2 shows that DRM
FP
d is very small and
even smaller when the design approaches the optimum and, hence, the sensitivity is
approximated as
DRM
.FP
0
d (7.26)
To derive DRM
FP
, let us assume a two-dimensional performance function for the ease of
derivation. The component probability of failure by the MPP-based DRM is given by
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122
DRM 1
1 1
1
( )( ) ( )
s
F
G vP v dv
b
(7.27)
Hence, DRM
FP
for 2-D performance function can be obtained as
DRM
1 11 1
1 1
1 1
1 1 1
1 1 1 1
21 1 1 1
( ) ( )1 ( )
( ) ( )1
( ) ( ) ( )12
s s
F
s k s kn
k
k
s k s k s kn
k
k
P G v G vv dv
b b
G v G vw
b b
G v G v G v bw
b b b
(7.28)
1( )s kG v
in Eq. (7.28) can be obtained using the definition of the shifted performance
function in Eq. (4.3) as
1 1( ) ( ) ( )s k kG v G v G
*v
(7.29)
( )G
*v
is identical with Eq. (7.24) and 1( )kG v
is given by
2
T1 1
1
( ) ( )( )
kk k
iUk
i i
uG v g ug
u
kk u
u (7.30)
where
ku
is given by
1 1
2
1 1
2
1 0
1 0
k k
k k
v v
v v
k **u u
u (7.31)
using the transformation from U-space to V-space. Eq. (7.23) can be rewritten as
123
123
2 T 2 T* *
1 1
3 3
1 1 1 1
U U U U U U Ug b g g g g b g g
b b b b
I Iu uH (7.32)
where H is the Hessian matrix evaluated at MPP in X-space. Hence, *
u is obtained as
12 T 2 T*
1 1
3 3
1 1 1 1
U U U U U Ug b g g b g g g
b b b b
I H H HuI (7.33)
and by inserting Eq. (7.33) into Eq. (7.31) and Eq. (7.31) into Eq. (7.30), finally, we can
obtain the sensitivity of the component probability of failure by the MPP-based DRM
with respect to in Eq. (7.28).
However, as shown in Eq. (7.33), the Hessian matrix is required to accurately
calculate the sensitivity and the Hessian matrix is very difficult and numerically
expensive to accurately estimate in engineering applications. Hence, the sensitivity in Eq.
(7.28) is approximated by
DRM
11 1
1
( )( )
s
FP G vv dv
b
(7.34)
assuming that 1
1
( )sG v
b
is very small. The verification of the assumption is shown in
Section 7.4.1 using numerical examples. Using the same assumption, the sensitivity of
the component probability of failure by the MPP-based DRM with respect to for a
general performance function can be obtained as
DRM 11
1
1
1
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
NNF
k k k j j j
k j k
N
j j j
j
Pf v dv f v dv
f v dv
(7.35)
where ( )jf is defined as
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124
1
( )
( )( )
s
j j
j
G v
bf
(7.36)
7.3.3 Efficiency Strategies
As can be seen in the formulation in Eq. (7.18), since the system RBDO involves
more design variables than the component RBDO, the system RBDO will take more
iteration to converge to the optimum. Hence, two efficiency strategies for the system
RBDO are proposed in this study.
A. Identification of Critical Constraints
Theoretically, all constraints must be considered for the calculation of the system
probability of failure calculation. However, since some constraints may not contribute to
the system probability of failure and it is numerically expensive to consider all
constraints for the system probability of failure calculation, it is necessary to find out
critical constraints which will contribute to the system failure. If RIA is used for the
system RBDO, then, the reliability indices can be used to identify the critical constraints
(Ba-abbad et al., 2006; Liang et al., 2007). However, since PMA is used in this study, it is
required to develop a new method.
Based on PMA+ (Youn et al., 2005b), the system RBDO directly finds the
deterministic optimum, and active constraints at the deterministic optimum will most
probably affect the system failure. But, there is possibility that certain constraints, which
are not active at the deterministic optimum, may affect the system failure because the
system RBDO optimum design is away from the deterministic optimum design. Hence,
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125
the new tolerance for constraint activeness explained in Section 6.3 is reused to identify
the critical constraints. That is, a constraint is identified as critical if
2 2( ) 0, if 1
( ) 0, otherwise
i i
i f
i f
G GG
N N
G
*
MV
*
MV
X
X
(7.37)
After identifying the number of critical constraints denoted as mc, the system probability
of failure calculation using Ditlevsen’s second order upper bound is expressed as
sys DRM
1 2
max( )c c
i ij
m m
F F Fj i
i i
P P P
(7.38)
and β becomes 1cm vector.
B. New Design Closeness Concept
Design closeness concept was first proposed in PMA+ for the FORM-based
component RBDO (Youn et al., 2005b). The design closeness concept is that the previous
MPP in U-space will be used as the current starting MPP if the current design is very
close to the previous design, that is,
(0) * *
( ) ( )*
( )
i i i
i
i k k-1 k-1
k-1
u u uu
(7.39)
where (0)
iku is the 0th
MPP candidate point at the kth
design iteration for ith
constraint and
*
( )ik-1u is the MPP at the (k−1)th
design for ith
constraint. In this case, since the reliability
index is constant, *
( )ii k-1u .
However, since the reliability indexes are changing during the system RBDO
process, it is necessary to modify Eq. (7.39) to take advantage of the design closeness
concept. The modified design closeness is similar with the MPP update in DRM-based
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RBDO. If two designs are very close, then the current starting MPP in U-space is
obtained as
(0) *
( )*
( )
i i
i
k
ik k-1
k-1
u uu
(7.40)
where k
i is the reliability index at the kth
design iteration. Using this new design
closeness concept, the number of function evaluation for MPP search can be reduced.
7.4 Numerical Examples
7.4.1 Accuracy of Sensitivity
The analytic sensitivities derived in Section 7.3.2 are compared with the
sensitivities obtained using FDM. For the comparison, consider a highly nonlinear
performance function used in Section 5.4.1,
2 3
1 2 1 2
4
1 2 1 2
( ) 1 (0.9063 0.4226 6) (0.9063 0.4226 6)
0.6 (0.9063 0.4226 6) 0.4226 0.9063
G X X X X
X X X X
X (7.41)
where 1 ~ (4.0,0.3)X N , 2 ~ (3.0,0.3)X N , and 2 . Table 7.1 shows the accuracy of
the derived sensitivities comparing with FDM. In the table, the analytic sensitivities are
obtained using Eq. (7.24) and Eq. (7.28), respectively. From the table, it can be shown
that the derived sensitivities are exact.
However, as mentioned in Section 7.3.2, the Hessian matrix is required to obtain
the analytic sensitivities in Table 7.1, hence, it is numerically very expensive and
impractical. Table 7.2 compares the analytic sensitivity in Eq. (7.28) and approximate
sensitivity in Eq. (7.35) using the same performance function in Eq. (7.41). The
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approximate sensitivity shows very good accuracy even if the performance function is
highly nonlinear. Furthermore, the approximate sensitivity does not require additional
function evaluation, which means that it is very efficient and accurate to use the
approximate sensitivity for the system RBDO.
Table 7.1 Comparison of Sensitivities Using Analytic and FDM Results
Analytic
Sensitivity
Finite Difference Method with step size
1% 0.1% 0.01% *( )G
x
0.4058 0.4048 0.4056 0.4059
FP
-0.0384 -0.0376 -0.0383 -0.0384
Table 7.2 Comparison of Analytic and Approximate Sensitivity
Analytic
Sensitivity
Approximate
Sensitivity
Relative
Error,%
FP
-0.0384 -0.0379 1.36
7.4.2 Comparison of Critical Constraint Identification
Methods
Using the approximate sensitivity in Eq. (7.35), the system RBDO with the MPP-
based DRM is carried out. In this section, two methods to identify critical constraints are
compared. One method is to use active constraints at the deterministic optimum as critical
constraints and the other method is the proposed method in Section 7.3.3 that uses the
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MV method to identify critical constraint. For the comparison, the system RBDO for a
two-dimensional problem is formulated to
1 2
0
min Cost( )
s.t ( ( , )) 0, 1, ,
1 0
i i
sys
F
all
F
d d
G i nc
PG
P
*
d
x d (7.42)
where the performance functions as shown in Figure 7.3 are
2
1 21
2
2 1 2 1 2
3 4
1 2 1 2
3 2
1 2
( 2) ( 0.4)( ) 1
20
( ) 1 ( 0.4226 0.9063 ) (0.9063 0.4226 6)
(0.9063 0.4226 6) 0.6 (0.9063 0.4226 6)
80( ) 1
8 5
X XG
G X X X X
X X X X
GX X
X
X
X
, (7.43)
~ ( ,0.3) for =1,2i iX N d i , initial T[5,5]d , initial T[2,2,2]β , and the allowable system
probability of failure is 2.275%all
FP .
At the deterministic optimum, G2(X) is identified as active as shown in Figure
7.3. If the active constraint at the deterministic optimum is used as a critical constraint,
then mc=1, β is reduced to 2[ ]β and β1, β3 are unchanged during the optimization. If
the MV method is used, G1(X) and G2(X) are identified as the critical constraints, hence,
mc=2, β is reduced to T
1 2[ , ] β . Table 7.3 compares two critical constraint
identification methods and three component probability of failure calculation methods. A
in the first column of the table means that the active constraint at the deterministic
optimum is used as the critical constraint, whereas B means that the MV method is used
to identify the critical constraints. From the table, it can be shown that the method A
gives us very unreliable design because the system probabilities of failure in the last
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129
column are significantly larger than the allowable system probability of failure. This
difficulty cannot be resolved even if the accurate MPP-based DRM with 5 integration
points (DRM5) is used. This is because the method A does not identify the first constraint
as a critical constraint, even though it affects the system failure as shown in Table 7.3.
Figure 7.3 Performance Functions for Eq. (7.43)
Table 7.3. Comparison of Critical Constraint Identification Methods ( 2.275%all
FP )
Method Optimum Design Component (%) Joint (%)
12
MCS
FP System (%)
(MCS) Mean 1 2[ , ] β 1FP
1
MCS
FP 2FP
2
MCS
FP
A FORM 5.576, 2.231 2.000, 2.000 0.0000 1.0458 2.2750 3.9103 0.0604 4.8957 DRM3 5.553, 2.267 2.000, 2.100 0.0000 1.0372 2.2736 2.9906 0.0402 3.9876 DRM5 5.552, 2.268 2.000, 2.101 0.0000 1.0330 2.2705 2.9904 0.0404 3.9829
B FORM 5.533, 2.319 2.391, 2.188 0.8408 0.9386 1.4342 2.2446 0.0247 3.1584 DRM3 5.516, 2.364 2.422, 2.260 0.8572 0.8623 1.4159 1.7420 0.0141 2.5902 DRM5 5.514, 2.364 2.418, 2.264 0.8681 0.8657 1.4053 1.7355 0.0140 2.5872 A means that active constraints at the deterministic optimum are used as the critical constraints.
B means that the MV method is used to identify the critical constraints.
DRM3 and DRM5 mean the MPP-based DRM with 3 and 5 integration points, respectively.
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130
However, if the method B is used, then FORM and MPP-based DRM show better
results than the method A. Still, FORM even with the method B shows an unreliable
optimum design. This is because FORM has error in computation of the component
probability of failure, especially for highly nonlinear constraint G2(X), as shown in Table
7.3. Since G1(X) and G2(X) are both convex around MPP, the FORM-based joint
probability of failure should be used for the system probability of failure calculation. In
this problem, the FORM-based joint probability of failure is almost zero because ρ is
close to −1. Also, the joint probability of failure by Monte Carlo Simulation (MCS) is
very small as shown in Table 7.3. Thus, it does not affect the system RBDO for Eq.
(7.43).
7.4.3 Comparison of System RBDO Using FORM and
MPP-Based DRM
Since the FORM-based joint probability of failure is very small in the previous
example, it was hard to see the effect of the joint probability of failure. To see the effect
of the joint probability of failure, consider the following system RBDO formulation to
2 2
1 2 1 2
0
( 8) ( 15)min Cost( )
30 120
s.t ( ( , )) 0, 1, ,
1 0
i i
sys
F
all
F
d d d d
G i nc
PG
P
*
d
x d (7.44)
where the performance functions as shown in Figure 7.4 are
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131
2
1 21
2
2 1 2 1 2
3 4
1 2 1 2
3 2
1 2
2 2
1 2 1 24
( 0.3)( ) 1
20
( ) 1 ( 0.4226 0.9063 ) (0.9063 0.4226 6)
(0.9063 0.4226 6) 0.6 (0.9063 0.4226 6)
80( ) 1
8 5
( 2.5) ( 4.5)( ) 1
30 120
X XG
G X X X X
X X X X
GX X
X X X XG
X
X
X
X
, (7.45)
~ ( ,0.3) for =1,2i iX N d i , initial T[2,7]d , initial T[2,2,2,2]β , and the allowable system
probability of failure is 2.275%all
FP .
Figure 7.4 Performance Functions for Eq. (7.45)
As shown in Figure 7.4, since G2(X) and G4(X) are concave around the MPP, the
joint probability of failure 24FP is ignored. In addition, since
12 is close to −1, the
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FORM-based joint probability of failure 12FP
is also almost zero. Hence, the joint
probability of failure 14FP is the only one which affects the system probability of failure
significantly. Using the MV method, G1(X), G2(X) and G4(X) are identified as the critical
constraints. Among these constraints, G1(X) is not active at the deterministic optimum.
Since three constraints are critical, mc=3 and T
1 2 4[ , , ] β .
Table 7.4. Comparison of FORM and MPP-Based DRM ( 2.275%all
FP )
Optimum Design Component (%) System
(%)
(MCS) Mean
T
1 2 4[ , , ] β 1FP
1
MCS
FP 2FP
2
MCS
FP 4FP
4
MCS
FP
FORM 3.764, 3.288 3.078, 2.088, 2.692 0.1042 0.1152 1.8398 1.1354 0.3551 0.3070 1.5333
DRM3 3.808, 3.270 3.170, 1.931, 2.677 0.0843 0.0802 1.8930 1.6929 0.3199 0.3179 2.0727
DRM5 3.814, 3.262 3.192, 1.904, 2.653 0.0789 0.0790 1.8738 1.8270 0.3451 0.3416 2.2279
Table 7.4 compares FORM and MPP-based DRM for the system RBDO in Eq.
(7.44). As expected from Figure 7.4, since G2(X) is highly nonlinear and concave around
the MPP, FORM overestimates the component probability of failure as shown in Table
7.4, which affects the system probability of failure significantly. However, MPP-based
DRM with 3 and 5 integration points can accurately estimate the highly nonlinear
constraint G2(X). Hence, both yield very good estimation of the system probability of
failure as shown in the last column of Table 7.4. The joint probability of failure 14FP
by
MCS is 0.0183% for DRM5 and FORM-based joint probability of failure is 0.0227%
which is close to the MCS result. Both results are very small compared with the system
probability of failure, which means that the accurate component probability of failure
calculation is more important than the joint probability of failure calculation. However, it
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133
should be noted that there may be a case that the joint probability of failure dominates,
for example, when ρ is close to 1 as shown in Figure 7.2. (b).
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CHAPTER VIII
CONCLUSIONS AND FUTURE RECOMMENDATION
8.1 Conclusions
8.1.1 Reliability-Based Robust Design Optimization
(RBRDO)
Three methods (PMI, PDM, and mean-based univariate DRM) are compared in
terms of efficiency and accuracy for computation of the statistical moments and their
sensitivities. To compare the accuracy in estimation of the statistical moments of the
performance function, two polynomial performance functions with two design variables
are employed. In this comparison, PDM is excluded since PDM cannot estimate the
moments of the performance function. The comparison shows that DRM can accurately
estimate the statistical moments of the performance function for the design variables with
both non-normal and normal distributions. On the other hand, PMI can accurately
estimate the statistical moments of the performance function for the design variables with
normal distributions. For non-normally distributed design variables, PMI shows some
errors since nonlinear transformations make the performance function become highly
nonlinear.
For RBRDO, a highly nonlinear performance function was used for comparison
purposes. Both one-dimensional and two-dimensional examples show that, in most cases,
PMI and DRM can identify the optimum design and estimate the cost function accurately,
whereas the optimum design of PDM varies depending on the percentile used, and PDM
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135
has identified a wrong global minimum. To achieve better accuracy, DRM with five
quadrature points can be used.
PMI and PDM yield the same efficiency if the same inverse reliability analysis is
used to find MPPs. Nonlinearity of the performance function affects the total number of
function evaluations most significantly in RBRDO using PMI and PDM. In estimation of
the statistical moments using DRM, the number of design variables affects the total
number of function evaluations most significantly. Hence, if the number of design
variables is large, it is recommended to use PMI, compared to DRM, for RBRDO.
8.1.2 DRM-Based Inverse Reliability Analysis and RBDO
Three methods of evaluating the probability of failure using FORM, SORM, and
MPP-based DRM are compared in terms of efficiency and accuracy. In terms of
efficiency, the probability of failure calculation by FORM is the best since the probability
of failure calculation by SORM and DRM uses the MPP of the FORM-based inverse
reliability analysis. However, as shown using the examples in this study, the probability
of failure calculation by FORM could be very erroneous in particular when the multi-
dimensional performance function is highly nonlinear. Even though SORM can evaluate
the probability of failure more accurately than FORM, SORM has limited application
since SORM requires the second-order sensitivities. On the other hand, the probability of
failure calculation by DRM is as accurate as SORM, and sometimes even better than
SORM, without requiring the second-order sensitivities. For the system probability of
failure calculation, DRM-based reliability analysis shows better results than FORM-
based one since the component probability of failure affects the system probability of
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136
failure significantly for highly nonlinear and/or multi-dimensional performance
functions.
DRM-based inverse reliability analysis is used to find accurate MPP, which can
identify the failure region of the performance function better than the FORM-based MPP.
A three-step computational procedure is proposed to find the DRM-based MPP using the
inverse reliability analysis: the probability of failure calculation using constraint shift,
reliability index update, and MPP update. The DRM-based MPP is used in the design
iteration of RBDO. Since the DRM-based RBDO requires a number of function
evaluations, especially when the number of design variables is large, PMA+ with new
tolerances for constraint activeness and reduced rotation matrix is used to enhance the
efficiency. The design examples show that the optimum design of DRM-based RBDO is
indeed different from the optimum design of FORM-based RBDO, and the probability of
failure by FORM at the optimum is significantly erroneous compared to the probability
of failure by DRM.
8.1.3 Sensitivity Analyses for RBDO using FORM and
MPP-Based DRM
The sensitivities of the probabilistic constraints with respect to design variables
for the FORM-based PMA and DRM-based PMA are analytically derived in this study.
The analytic sensitivities for the FORM-based PMA are verified using the converging
sensitivities obtained by finite differences. The analytic sensitivities of the probabilistic
constraint at the true DRM-based MPP are also verified using the FDM results. However,
since it is computationally very expensive to find the true DRM-based MPP, the
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137
probabilistic constraint at the approximated DRM-based MPP is proposed for the DRM-
based PMA. Although the analytic sensitivities of the probabilistic constraint at the
approximated DRM-based MPP yield some inaccuracy at the initial design, the
sensitivities converge to the sensitivities of the probabilistic constraint at the true DRM-
based MPP as the design approaches the optimum design. In conclusion, it is very
desirable to use the sensitivities of the probabilistic constraint at the approximated DRM-
based MPP for the DRM-based PMA and RBDO because the computational cost can be
reduced significantly while maintaining accuracy near the optimum design.
8.1.4 System Inverse Reliability Analysis and RBDO
The system probability of failure estimation based on two methods, the MPP-
based DRM and FORM, is compared through numerical examples. For the highly
nonlinear problem, the effect of accurate component probability of failure is more
significant than the estimation of the joint probability of failure. Hence, in this case, the
system reliability analysis using the MPP-based DRM yields better accuracy than the
FORM-based system reliability analysis since the MPP-based DRM can accurately
estimate the component probability of failure. Consequently, the system RBDO using
MPP-based DRM shows better results than the system RBDO using FORM. However, it
is also important to use the correct method for critical constraint identification. Numerical
examples show that the system probability of failure estimation could be wrong even if
MPP-based DRM is used for the component probability of failure calculation. Thus, it is
recommended to use the MV method at the deterministic optimum to identify critical
constraints, which affect the system failure.
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8.2 Future Recommendation
In the study for the DRM-based RBRDO and RBDO in the literature, all random
variables are assumed to be statistically independent to each other due to its simplicity for
numerical calculation. However, in practical engineering applications, input random
variables might be correlated, especially for a fatigue problem, input material properties
are correlated. In that case, many researches recommended a transformation such as
Rosenblatt transformation to transform the correlated space to the independent space.
Even though Rosenblatt transformation is theoretically order-independent, it is found that
RBRDO and RBDO show different results for different orders when input random
variables follow non-normal distributions. This is because the nonlinear transformation
significantly violates the assumption used in FORM. In addition, the nonlinear
transformation makes the performance function more nonlinear, which demand the use of
MPP-based DRM.
Hence, for the future research recommendation, the DRM-based RBRDO and
RBDO with correlated random variables will be developed to reduce the order effect
caused by FORM and to enhance the accuracy of the probability of failure calculation.
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139
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