Assessment of high dimensional model representation techniques for reliability analysis

Probabilistic Engineering Mechanics 24 (2009) 100–115www.elsevier.com/locate/probengmech

Assessment of high dimensional model representation techniques forreliability analysis

Rajib Chowdhury, B.N. Rao∗

Structural Engineering Division, Department of Civil Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Received 31 May 2007; received in revised form 26 January 2008; accepted 19 February 2008Available online 4 March 2008

Abstract

This paper presents an assessment of efficient response surface techniques based on the High Dimensional Model Representation (HDMR) andthe Factorized High Dimensional Model Representation (FHDMR). The HDMR is a general set of quantitative model assessment and analysistools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation ofthe system response, if higher order variable correlations are weak and if the response function is dominantly of an additive nature, allowingthe physical model to be captured by the first few lower order terms. But, if the multiplicative nature of the response function is dominant,then all the right hand side components of the HDMR must be used to be able to obtain the best result. However, if the HDMR requires allcomponents, which means 2N of them, to get a desired accuracy, making the method very expensive in practice, then the FHDMR can be used.The component functions of the FHDMR are determined by using the component functions of the HDMR. This paper presents the formulation ofthe FHDMR based response surface approximation of a limit state/performance function which is dominantly multiplicative in nature. It is a giventhat conventional methods for reliability analysis are computationally very demanding, when applied in conjunction with complex finite elementmodels. This study aims to assess how accurately and efficiently HDMR/FHDMR based response surface techniques can capture complex modeloutput uncertainty. As a part of this effort, the efficacy of the HDMR, which is recently applied to reliability analysis, is also demonstrated. Theresponse surface is constructed using the moving least squares interpolation formula by including constant, first-order, and second-order terms ofthe HDMR and the FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation. Results ofseven numerical examples involving structural/solid-mechanics/geo-technical engineering problems indicate that the failure probability obtainedusing the FHDMR based response surface method for a limit state/performance function that is dominantly multiplicative in nature, provides asignificant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.c© 2008 Elsevier Ltd. All rights reserved.

Keywords: Structural reliability; High dimensional model representation; Factorized high dimensional model representation; Response surface method; Movingleast squares; Failure probability

1. Introduction

The estimation of the time-invariant reliability of a systemor component entails the computation of multidimensionalprobability integrals [1–3]

PF ≡ P (g(x) ≤ 0) =∫

g(x)≤0pX (x)dx, (1)

where x = {x1, x2, . . . , xN }, represent the N -dimensionalrandom variables of the model under consideration; g(x) is

∗ Corresponding author. Tel.: +91 44 2257 4285; fax: +91 44 2257 5286.E-mail address: [email protected] (B.N. Rao).

0266-8920/$ - see front matter c© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.probengmech.2008.02.001

the limit state/performance function, such that g(x) ≤ 0represents the failure domain; and pX(x) is the joint probabilitydensity function of the input random variables. In reality,inherent difficulties in reliability estimation arise due to theimplicit nature and high nonlinearity of g(x). Therefore, adetailed finite element (FE) modeling of the structure isnecessary in combination with reliability analysis tools. FEmethods for linear and nonlinear structures in conjunction withfirst- or second-order reliability method (FORM/SORM) [1–4] have been successfully applied for structural reliabilitycomputations [5]. But, such methods are effective inevaluating very small probabilities of failure for small-scaleproblems. In regard to the large-scale problems, the merging

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R. Chowdhury, B.N. Rao / Probabilistic Engineering Mechanics 24 (2009) 100–115 101

of FORM/SORM, with commercial FE programs is notstraightforward [6] especially when the nonlinear problems areaddressed.

The simulation methods [7–10] involve the samplingof basic random variables and simulating the actual limitstate/performance function repeated times. But the simulationmethods seem impractical when each simulation involvesexpensive FE or meshfree calculations. However, simulationmethods do not exhibit the limitations of approximate reliabilitymethods, such as FORM/SORM; they generally require moreextensive calculations than the former methods. Consequently,the simulation methods have been traditionally employed asa measuring stick for the approximate methods. Therefore,there is a need to develop the approximate methods to achievethe computational efficiency in evaluating Eq. (1) withoutcompromising on the accuracy of result. Within the contextof this paper, an assessment of the efficient response surfaceapproximation schemes based on the HDMR and the FHDMRis conducted which enables a fairly accurate representation ofthe systems behavior. The response surface method (RSM) is astatistical technique [11–13], broadly used in several scientificareas for a long time. Wong [14,15] and Faravelli [16,17] usedfactorial designs and regression methods to obtain the leastsquare estimates of the unknown coefficients. To improve theaccuracy of the RSM, Bucher and Bourgund [18] suggestedan alternative process of selecting the sample points. But,estimates of the failure probability obtained by the suggestedprocedure [18] have been shown to be sensitive [19,20]to the sample point position, and may not always providean acceptable approximation to the true failure probability.Kim and Na [21] proposed a vector projection samplingtechnique, in which the points for generating the linear responsesurface are located near the failure surface. Subsequently,Zheng and Das [22] developed a method to account forthe higher order effect in the response function usingvector projection sampling. A complete quadratic responsesurface with resampling was proposed by Gayton et al. [23],which considers the knowledge of the structural behaviorand reduces the cost of the computation using a statisticalformulation of the RSM. Gupta and Manohar [24] attemptedto combine the two approaches provided by Faravelli [16,17]and Bucher and Bourgund [18], and proposed a global responsesurface approach. But the sampling strategy adopted in thisstudy is complicated and requires considerably many samplepoints [25].

It can be observed from the findings in the literature that,several issues like (1) choice of the order of polynomials,(2) positions of sample points, etc. are still questionable inregard to the existing RSMs. Recently, the authors adoptedHDMR concepts to find an equivalent continuous functionto replace a univariate or multivariate piecewise continuousfunction, rather than seeking an exact continuous function [26],and extended HDMR concepts for response surface generationto predict the failure probability of structural or mechanicalsystems subjected to random loads and material properties [27].The major advantage of HDMR-based RSM is the higher orderapproximation of the limit state/performance function, and it

requires fewer original model simulations as compared to fullscale simulation methods.

This paper presents an assessment of two novel RSMs forpredicting the failure probability of structural or geo-technicalsystems subjected to random loads and material properties. Themethod involves the HDMR and the FHDMR techniques inconjunction with the moving least squares (MLS) techniqueto approximate the original implicit limit state/performancefunction with an explicit function. The paper is organized asfollows. Section 2 presents a brief overview of the HDMRand its applicability to reliability analysis. Section 3 presentsthe formulation of the FHDMR. Section 4 and Section 5present response surface generation using the HDMR and theFHDMR, respectively. Section 6 entails the simulation methodfor the evaluation of the reliability using the response surfacegenerated by the HDMR and the FHDMR. Section 7 presentsseven numerical examples to illustrate the performance of thepresent methods. Comparisons have been made with the directMonte Carlo simulation (MCS) method to evaluate the accuracyand the computational efficiency of the present methods.

2. Concept of HDMR and its application to reliabilityanalysis

In recent years, there have been efforts to develop efficientmethods to approximate multivariate functions in such away that the component functions of the approximation areordered starting from a constant and gradually approachingto multivariance as we proceed along the terms like first-order, second-order, and so on. One such method is theHDMR [28–36]. The HDMR is a general set of quantitativemodel assessment and analysis tools for capturing the high-dimensional relationships between sets of input and outputmodel variables. It is a very efficient formulation of thesystem response, if higher order variable correlations are weak,allowing the physical model to be captured by the first fewlower order terms. Practically for most well-defined physicalsystems, only relatively low order correlations of the inputvariables are expected to have a significant effect on theoverall response [28–36]. The HDMRs expansion utilizes thisproperty to present an accurate hierarchical representation ofthe physical system.

The degree of accuracy of a reliability estimation dependson the accurate representation of the limit state/performancefunction. The computational complexity for the generation ofthe response surface arises due to the increase in the number ofinput variables, while using the conventional response surfacein conjunction with the design of the experiments. The conceptof the HDMR expansions is introduced here for the purposeof representing the response function most accurately andefficiently, when the number of input variables is large.

Let the N -dimensional vector x = {x1, x2, . . . , xN },represent the input variables of the model under consideration,and the response variable as g(x). Since the influence of theinput variables on the response variable can be independentand/or cooperative, the HDMR expresses the response g(x) as ahierarchical correlated function expansion in terms of the inputvariables as

102 R. Chowdhury, B.N. Rao / Probabilistic Engineering Mechanics 24 (2009) 100–115

g(x) = g0 +

N∑i=1

gi (xi )+∑

1≤i1<i2≤N

gi1i2

(xi1 , xi2

)+ · · ·

+

∑1≤i1<···<il≤N

gi1i2...il

(xi1 , xi2 , . . . , xil

)+ · · ·

+ g12...N (x1, x2, . . . , xN ) , (2)

where g0 is a constant term representing the zeroth-ordercomponent function or the mean response of g(x). Thefunction gi (xi ) is a first-order term expressing the effect ofthe variable xi acting alone, although generally nonlinearly,upon the output g(x). The function gi1i2

(xi1 , xi2

)is a second-

order term which describes the cooperative effects of thevariables xi1 and xi2 upon the output g(x). The higher orderterms give the cooperative effects of increasing numbers ofinput variables acting together to influence the output g(x).The last term g12,...,N (x1, x2, . . . , xN ) contains any residualdependence of all the input variables locked together in acooperative way to influence the output g(x). Once all therelevant component functions in Eq. (2) are determined andsuitably represented, then the component functions constitutethe HDMR, thereby replacing the original computationallyexpensive method of calculating g(x) by the computationallyefficient model. Usually the higher order terms in Eq. (2) arenegligible [28–31] such that the HDMR with only low ordercorrelations to second-order [31], amongst the input variablesare typically adequate in describing the output behavior andthis has been verified in a number of computational studies [33]where the HDMR expansions up to second-order are oftensufficient to describe the outputs of many realistic systems.Therefore, it is expected that the HDMR expansion convergesvery rapidly.

Depending on the method adopted to determine thecomponent functions in Eq. (2) there are two particular HDMRexpansions: ANOVA-HDMR and cut-HDMR. The ANOVA-HDMR is useful for measuring the contributions of the varianceof individual component functions to the overall variance ofthe output. On the other hand, the cut-HDMR expansion isan exact representation of the output g(x) in the hyperplanepassing through a reference point in the variable space.

In this work, the cut-HDMR procedure is used to develop anovel RSM for predicting the failure probability of structuralor mechanical systems subjected to random loads and materialproperties. With the cut-HDMR method, first a reference pointc = {c1, c2, . . . , cN } is defined in the variable space. In theconvergence limit, the cut-HDMR is invariant to the choiceof the reference point c. In practice, c is chosen within theneighborhood of interest in the input space. The expansionfunctions are determined by evaluating the input–outputresponses of the system relative to the defined reference point calong associated lines, surfaces, subvolumes, etc. (i.e. cuts) inthe input variable space. This process reduces to the followingrelationship for the component functions in Eq. (2)

g0 = g(c), (3)

gi (xi ) = g(

xi , ci)− g0, (4)

gi1i2

(xi1 , xi2

)= g

(xi1 , xi2 , ci1i2

)− gi1

(xi1

)− gi2

(xi2

)− g0, (5)

where the notation g(xi , ci ) = g(c1, c2, . . . , ci−1, xi , ci+1, . . . ,

cN ) denotes that all the input variables are at their referencepoint values except xi . The g0 term is the output response ofthe system evaluated at the reference point c. The higher orderterms are evaluated as cuts in the input variable space throughthe reference point. Therefore, each first-order term gi (xi ) isevaluated along its variable axis through the reference point.Each second-order term gi1i2

(xi1 , xi2

)is evaluated in a plane

defined by the binary set of input variables xi1 , xi2 through thereference point, etc. The process of subtracting off the lowerorder expansion functions removes their dependence to assurea unique contribution from the new expansion function.

Considering terms up to first- and second-order in Eq. (2)yields, respectively

g(x) = g0 +

N∑i=1

gi (xi )+ R2, (6)

and

g(x) = g0 +

N∑i=1

gi (xi )+∑

1≤i1<i2≤N

gi1i2

(xi1 , xi2

)+ R3. (7)

Substituting Eqs. (3)–(5) into Eqs. (6) and (7) leads to

g(x) =N∑

i=1

g (c1, . . . , ci−1, xi , ci+1, . . . , cN )

− (N − 1) g(c)+ R2, (8)

and

g(x) =N∑

i1=1,i2=1i1<i2

g(c1, . . . , ci1−1, xi1 , ci1+1, . . . , ci2−1,

xi2 , ci2+1, . . . , cN)

− (N − 2)N∑

i=1

g (c1, . . . , ci−1, xi , ci+1, . . . , cN )

+(N − 1) (N − 2)

2g(c)+ R3, (9)

respectively. Now consider the first- and second-orderapproximation of g(x), denoted respectively by

g(x) ≡ g (x1, x2, . . . , xN )

=

N∑i=1

g (c1, . . . , ci−1, xi , ci+1, . . . , cN )

− (N − 1) g(c), (10)

and

g(x) =N∑

i1=1,i2=1i1<i2

g(c1, . . . , ci1−1, xi1 , ci1+1, . . . , ci2−1,

xi2 , ci2+1, . . . , cN)


− (N − 2)N∑

i=1

g (c1, . . . , ci−1, xi , ci+1, . . . , cN )

+(N − 1) (N − 2)

2g(c). (11)

The comparison of Eqs. (8) and (10) indicates that thefirst-order approximation leads to the residual error g(x) −g(x) = R2, which includes contributions from terms of two andhigher order component functions. Similarly the second-orderapproximation leads to the residual error g(x) − g(x) = R3,which includes contributions from terms of three and higherorder component functions.

The notion of 0th, 1st, 2nd order, etc. in the HDMRexpansion should not be confused with the terminology usedeither in the Taylor series or in the conventional least-squaresbased response surface model. It can be shown that the firstorder component function gi (xi ) is the sum of all the Taylorseries terms which contain and only contain the variable xi .Similarly, the second order component function gi1i2

(xi1 , xi2

)is the sum of all the Taylor series terms which contain andonly contain the variables xi1 and xi2 . Hence the first- andsecond-order HDMR approximations should not be viewed asthe first- or second-order Taylor series expansions, nor do theylimit the nonlinearity of g(x). Furthermore, the approximationscontain contributions from all input variables. Thus, theinfinite number of terms in the Taylor series are partitionedinto finite different groups and each group corresponds toone cut-HDMR component function. Therefore, any truncatedcut-HDMR expansion provides a better approximation andconvergent solution of g(x) than any truncated Taylor series,because the latter only contains a finite number of termsof the Taylor series. Furthermore, the coefficients associatedwith higher dimensional terms are usually much smaller thanthose with one-dimensional terms. As such, the impact ofhigher dimensional terms on the function is less, and therefore,can be neglected. If the first-order HDMR approximation isnot sufficient the second-order HDMR approximation may beadopted at the expense of additional computational cost.

3. Factorized HDMR

In the previous section, the response function g(x) isrepresented as a few lower order component functionsof the HDMR in an additive form. However, when theresponse function g(x) is dominantly of a multiplicativenature, the HDMR approximation may not be sufficient toaccurately estimate the probabilistic characteristics of thesystem. The basic purpose of this work is to obtain the generalstructure of a multiplicative type response function g(x). Themultiplicative form of the HDMR for a given multivariate limitstate/performance function g(x) can be represented as [37,38]

g(x) = r0

[N∏

i=1

(1+ ri (xi ))

]

×

N∏i1,i2=1i1<i2

(1+ ri1i2

(xi1 , xi2

))× · · ·

× [1+ r12...N (x1, x2, . . . , xN )] , (12)

where r0 is a constant term, ri (xi ) is a first-order termexpressing the effect of variable xi acting alone, althoughgenerally nonlinearly, upon the output g(x). The functionri1i2

(xi1 , xi2

)is a second-order term which describes the

cooperative effects of the variables xi1 and xi2 upon the outputg(x) and so on. The constant term, the first-order term, and thehigher order terms can be found by comparing Eq. (12) withEq. (2) [38]. This process reduces to the following relationshipfor the component functions in Eq. (12),

r0 = g0, (13)

ri (xi ) =gi (xi )

g0, (14)

ri1i2

(xi1 , xi2

)=

g0gi1i2

(xi1 , xi2

)− gi1

(xi1

)gi2

(xi2

)(g0 + gi1

(xi1

)) (g0 + gi2

(xi2

)) , (15)

where, g0, gi (xi ) and gi1i2

(xi1 , xi2

)are defined in Eqs. (3)–(5)

of Section 2. The component functions defined in Eqs. (13)–(15) can be further simplified as follows:

r0 = g(c), (16)

ri (xi ) =g(xi , ci

)g(c)

− 1, (17)

ri1i2

(xi1 , xi2

)=

g(c)g(xi1 , xi2 , ci1i2

)g(xi1 , ci1

)g(xi2 , ci2

) − 1. (18)

Once all the relevant component functions in Eq. (12) aredetermined and suitably represented, then the componentfunctions constitute the factorized form of the HDMR calledthe FHDMR [38]. Therefore, the first- and second-orderapproximation g(x) of the original implicit response functiong(x) can be represented as

g(x) = g(c)

[N∏

i=1

g(xi , ci

)g (c)

], (19)

g(x) = g(c)

[N∏

i=1

g(xi , ci

)g (c)

]

×

N∏i1,i2=1i1<i2

g(c)g(xi1 , xi2 , ci1i2

)g(xi1 , ci1

)g(xi2 , ci2

) . (20)

4. HDMR-based response surface generation

The HDMR in Eq. (2) is exact along any of the cuts and theoutput response g(x) at a point x off of the cuts can be obtainedby following the procedure in step 1 and step 2 below:Step 1: Interpolate each of the low dimensional HDMRexpansion terms with respect to the input values of the pointx. For example, consider the first-order component functiong(xi , ci

)= g (c1, c2, . . . , ci−1, xi , ci+1, . . . , cN ). If for xi =

x ji , n function values


g(

x ji , ci

)= g

(c1, . . . , ci−1, x j

i , ci+1, . . . , cN

);

j = 1, 2, . . . , n, (21)

are given at n (=3, 5, 7 or 9) uniformly distributed samplepoints µi − (n − 1) σi/2, µi − (n − 3) σi/2, . . . , µi , . . . , µi +

(n − 3) σi/2, µi + (n − 1) σi/2 along the variable axis xi withmean µi and standard deviation σi , the function value for anarbitrary xi can be obtained by the MLS interpolation [26] as

g(

xi , ci)=

n∑j=1

φ j (xi ) g(

c1, . . . , ci−1, x ji , ci+1, . . . , cN

).

(22)

Similarly, consider the second-order component functiong(xi1 , xi2 , ci1i2) = g(c1, . . . , ci1−1, xi1 , ci1+1, . . . , ci2−1, xi2 ,

ci2+1, . . . , cN ). If for xi1 = x j1i1

, and xi2 = x j2i2, n2 function

values

g(

x j1i1, x j2

i2, ci1i2

)= g

(c1, . . . , ci1−1, x j1

i1, ci1+1, . . . , ci2−1,

x j2i2, ci2+1, . . . , cN

);

j1 = 1, 2, . . . , n, j2 = 1, 2, . . . , n, (23)

are given on a grid formed by taking n (= 3, 5, 7 or 9)uniformly distributed sample points µi1 − (n − 1)σi1/2, µi1 −

(n − 3)σi1/2, . . . , µi1 , . . . , µi1 + (n − 3)σi1/2, µi1 + (n −1)σi1/2 along xi1 axis with mean µi1 and standard deviationσi1 , and n (= 3, 5, 7 or 9) uniformly distributed sample pointsµi2 − (n − 1) σi2/2, µi2 − (n − 3) σi2/2, . . . , µi2 , . . . , µi2 +

(n − 3) σi2/2, µi2 + (n − 1) σi2/2 along xi2 axis with meanµi2 and standard deviation σi2 , the function value for arbitrary(xi1 , xi2

)can be obtained by MLS interpolation [26] as

g(

xi1 , xi2 , ci1i2)=

n∑j1=1

n∑j2=1

φ j1 j2

(xi1 , xi2

)× g

(c1, . . . , ci1−1, x j1

i1, ci1+1, . . . , ci2−1,

x j2i2, ci2+1, . . . , cN

), (24)

where the interpolation functions φ j (xi ) and φ j1 j2

(xi1 , xi2

)can

be obtained using the MLS interpolation scheme [26].By using Eq. (22), gi (xi ) can be generated if n function

values are given at corresponding sample points. Similarly, byusing Eq. (24), gi1i2

(xi1 , xi2

)can be generated if n2 function

values at corresponding sample points are given. The sameprocedure shall be repeated for all the first-order componentfunctions, i.e., gi (xi ); i = 1, 2, . . . , N and the second-ordercomponent functions, i.e., gi1i2

(xi1 , xi2

); i1, i2 = 1, 2, . . . , N .

Step 2: Sum the interpolated values of the HDMR expansionterms from the zeroth-order to the highest order retained inkeeping with the desired accuracy. This leads to the first-orderHDMR approximation of the function g(x) as

g(x) =N∑

i=1

n∑j=1

φ j (xi )g(

c1, . . . , ci−1, x ji , ci+1, . . . , cN

)− (N − 1) g0, (25)

and second-order HDMR approximation of the function g(x) as

g(x) =N∑

i1=1,i2=1i1<i2

n∑j1=1

n∑j2=1

φ j1 j2

(xi1 , xi2

)× g

(c1, . . . , ci1−1, x j1

i1, ci1+1, . . . , ci2−1, x j2

i2,

ci2+1, . . . , cN

)− (N − 2)

N∑i=1

n∑j=1

φ j (xi )

× g(

c1, . . . , ci−1, x ji , ci+1, . . . , cN

)+(N − 1) (N − 2)

2g0. (26)

5. FHDMR-based response surface generation

Similar to the HDMR, the FHDMR in Eq. (12) is also exactalong any of the cuts and the output response g(x) at a point xoff of the cuts can be obtained by following the procedure instep 1 and step 2 below:Step 1: Interpolate each of the low dimensional HDMRexpansion terms with respect to the input values of the pointx. For example, consider the first-order component functiong(xi , ci

)= g (c1, c2, . . . , ci−1, xi , ci+1, . . . , cN ). If for xi =

x ji , n function values

g(

x ji , ci

)= g

(c1, . . . , ci−1, x j

i , ci+1, . . . , cN

);

j = 1, 2, . . . , n, (27)

are given at n (= 3, 5, 7 or 9) uniformly distributed samplepoints µi − (n − 1) σi/2, µi − (n − 3) σi/2, . . . , µi , . . . , µi +

(n − 3) σi/2, µi + (n − 1) σi/2 along the variable axis xi withmean µi and standard deviation σi , the function value for anarbitrary xi can be obtained by the MLS interpolation [26] as

g(

xi , ci)=

n∑j=1

φ j (xi ) g(

c1, . . . , ci−1, x ji , ci+1, . . . , cN

).

(28)

Similarly, consider the second-order component functiong(xi1 , xi2 , ci1i2) = g(c1, . . . , ci1−1, xi1 , ci1+1, . . . , ci2−1, xi2 ,

ci2+1, . . . , cN ). If for xi1 = x j1i1

, and xi2 = x j2i2, n2 function

values

g(

x j1i1, x j2

i2, ci1i2

)= g

(c1, . . . , ci1−1, x j1

i1, ci1+1, . . . , ci2−1,

x j2i2, ci2+1, . . . , cN

);

j1 = 1, 2, . . . , n, j2 = 1, 2, . . . , n, (29)

are given on a grid formed by taking n (= 3, 5, 7 or 9)uniformly distributed sample points µi1 − (n − 1) σi1/2,µi1 − (n − 3) σi1/2, . . . , µi1 , . . . , µi1 + (n − 3) σi1/2, µi1 +

(n − 1) σi1/2 along xi1 axis with mean µi1 and standard de-viation σi1 , and n (= 3, 5, 7 or 9) uniformly distributed samplepointsµi2−(n − 1) σi2/2,µi2−(n − 3) σi2/2, . . . , µi2 , . . . , µi2

+ (n − 3) σi2/2, µi2 + (n − 1) σi2/2 along xi2 axis with mean


µi2 and standard deviation σi2 , the function value for an arbi-trary

(xi1 , xi2

)can be obtained by the MLS interpolation [26]

as

g(

xi1 , xi2 , ci1i2)=

n∑j1=1

n∑j2=1

φ j1 j2

(xi1 , xi2

)×g

(c1, . . . , ci1−1, x j1

i1, ci1+1, . . . , ci2−1, x j2

i2,

ci2+1, . . . , cN), (30)

where the interpolation functions φ j (xi ) and φ j1 j2

(xi1 , xi2

)can

be obtained using the MLS interpolation scheme [26].By using Eq. (22), gi (xi ) can be generated if n function

values are given at the corresponding sample points. Similarly,by using Eq. (24), gi1i2

(xi1 , xi2

)can be generated if n2 function

values at the corresponding sample points are given. The sameprocedure shall be repeated for all the first-order componentfunctions, i.e., gi (xi ); i = 1, 2, . . . , N and the second-ordercomponent functions, i.e., gi1i2

(xi1 , xi2

); i1, i2 = 1, 2, . . . , N .

Step 2: Multiply the interpolated values of the HDMRexpansion terms from the zeroth-order to the highest orderretained in keeping with the desired accuracy. This leads to thefirst-order FHDMR approximation of the function g(x) as

g(x) = g(c)

×

N∏i=1

n∑j=1

φ j (xi )g(

c1, . . . , ci−1, x ji , ci+1, . . . , cN

)g(c)

,(31)

and the second-order FHDMR approximation of the functiong(x) as

g(x) = g(c)

N∏i=1

n∑j=1

φ j (xi )g(

x ji , ci

)g(c)

×

N∏

i1,i2=1i1<i2

g(c)n∑

j1=1

n∑j2=1

φ j1 j2

(xi1 , xi2

)g(

xj1i1, x

j2i2, ci1i2

)(

n∑j1=1

φ j1 (xi )g(xi1 , ci1

))( n∑j2=1

φ j2 (x2) g(xi2 , ci2

)) .

(32)

If n is the number of sample points taken along each of thevariable axis and s is the order of the component functionconsidered, starting from the zeroth-order to the l-th order,then the total number of function evaluations for interpolationpurposes is given by,

∑ls=0

(N ! (n − 1)s

)/ ((N − s)!s!) which

grows polynomially with n and s. As a few low ordercomponent functions of the HDMR or the FHDMR are used,the sample savings due to the HDMR or the FHDMR aresignificant compared to traditional sampling. Hence uncertaintyanalysis using the HDMR relies on an accurate reduced modelbeing generated with a small number of full model simulations.An arbitrarily large sample Monte Carlo analysis can beperformed on the outputs approximated by the HDMR or theFHDMR which result in the same distributions as obtained

through the Monte Carlo analysis of the full model. Thetremendous computational savings result from just having toperform interpolation instead of full model simulations foroutput determination.

6. Failure probability estimation

Eqs. (25) and (26) provides respectively, a first-orderHDMR and an FHDMR approximation g(x) of the orig-inal implicit limit state/performance function g(x) us-ing the MLS interpolation functions, constant g(c) term,first-order g(c1, . . . , ci−1, x j

i , ci+1, . . . , cN ) and second-order

g(c1, . . . , ci1−1, x j1i1, ci1+1, . . . , ci2−1, x j2

i2, ci2+1, . . . , cN )

terms. Similarly, Eqs. (31) and (32) provides respectively,the first- and second-order FHDMR approximations g(x) ofthe original implicit limit state/performance function g(x)using the MLS interpolation functions, constant g(c) term,first-order g(c1, . . . , ci−1, x j

i , ci+1, . . . , cN ) and second-order

g(c1, . . . , ci1−1, x j1i1, ci1+1, . . . , ci2−1, x j2

i2, ci2+1, . . . , cN )

terms. Therefore, the failure probability PF can be easily esti-mated by performing MCS on the first- or second-order approx-imation g(x) of the original implicit limit state/performancefunction g(x) and is given by

PF =1

NS

NS∑i=1

I[g(

xi)< 0

], (33)

where xi is i th realization of X, NS is the sampling size, I [.] isa deciding function of the fail or the safe state such that I = 1,if g

(xi)< 0 otherwise zero. A flow diagram for the response

surface generation using the HDMR and the FHDMR and thefailure probability PF estimation by MCS is shown in Figs. 1and 2, respectively. The reliability index β corresponding to thefailure probability PF can be obtained by

β = −Φ−1 (PF ) , (34)

where Φ (·) is the cumulative distribution function of a standardGaussian random variable.

Since the first- and second-order HDMR and FHDMRapproximations leads to the explicit representation of theoriginal implicit limit state/performance function, the MCScan be conducted for any sampling size. The total cost of theoriginal function evaluation entails a maximum of (n − 1) ×N+1 and (n − 1)2 (N − 1) N/2+(n − 1) N+1 by the presentmethod using the first- and second-order HDMR and FHDMRapproximations, respectively.

7. Numerical examples

Two sets of numerical examples are presented to illustratethe performance of the present methods. In Set I (Examples 1and 2), detailed comparisons are made with existing responsesurface techniques to evaluate the accuracy and computationalefficiency of the proposed methods. In Set II (Examples 3–7),comparative assessments are made between the HDMR andFHDMR based response surface approximations. An exact


Fig. 1. Flowchart of HDMR-based response surface generation.

continuous function to replace a univariate or multivariatepiece wise continuous function may not always be availablein general problems. Rather than seeking an exact continuousfunction to replace a piecewise continuous function, anequivalent continuous function can be found based on thepresent methods.

In the present study, the failure probability PF is estimatedusing the following five procedures: (1) Direct Monte Carlosimulations using the exact limit state/performance function,which may be implicitly defined. This is assumed to bethe true failure probability and is used to benchmark othermethods. (2) Response surface approximation using the first-order FHDMR. (3) Response surface approximation using thefirst-order HDMR. (4) Response surface approximation usingthe second-order FHDMR. (5) Response surface approximationusing the second-order HDMR. The coefficient of variation δof the estimated failure probability PF by direct MCS for thesampling size NS considered, is computed using

δ =

√(1− PF )

NS PF. (35)

When comparing computational efforts by various methods inevaluating the failure probability PF , the number of originallimit state/performance function evaluations is chosen as the

primary comparison tool in this paper. This is because of thefact that the number of function evaluations indirectly indicatesthe CPU time usage. For the direct MCS, the number oforiginal function evaluations is same as the sampling size.While evaluating the failure probability PF through the directMCS, the CPU time is more because it involves a number ofrepeated actual finite-element analyses. However, in the presentmethods the MCS is conducted in conjunction with the HDMRand FHDMR based response surfaces. Here, although the samesampling size as in direct MCS is considered, the number oforiginal function evaluations is very much lower. Hence, thecomputational effort expressed in terms of function evaluationsalone should be carefully interpreted for problems involvingexplicit functions. For the first-order HDMR and FHDMR, nuniformly distributed sample points are deployed along thevariable axis through the reference point. The sampling schemefor the response surface approximation of a function havingone variable (x) and two variables (x1 and x2) using the first-order HDMR and FHDMR is shown in Fig. 3(a) and (b)respectively. For the second-order HDMR and FHDMR, nuniformly distributed sample points are deployed along each ofthe variable axis to form a regular grid. The sampling schemefor the response surface approximation of a function havingtwo variables (x1 and x2) using the second-order HDMR andFHDMR is shown in Fig. 4. In all the numerical examples


Fig. 2. Flowchart of FHDMR-based response surface generation.

presented, the reference points c are taken as the mean valuesof the random variables.

7.1. Set I

7.1.1. Example 1: Hypothetical nonlinear limit state # 1 [21]An explicitly nonlinear limit state/performance function is

selected to examine the performance of the proposed methods

g(x) = exp (0.2x1 + 6.2)− exp (0.47x2 + 5.0) , (36)

where random variables x1 and x2 are assumed to beindependent and have a standard normal distribution with zeromean and unit standard deviation. This function shows highernonlinearity in the safe region away from the failure surface.Table 1 presents a detailed comparison of results obtained usingvarious existing RSMs with present methods. For evaluating thefailure probability PF , n = 5 is taken. In an effort to reducethe approximation error further, the second-order HDMR andFHDMR based response surface is adopted in evaluating thefailure probability PF using the present method. Table 1 alsoshows the failure probability PF value obtained with the presentmethod using the second-order HDMR and FHDMR basedresponse surface and the associated computational effort. Itcan be observed from Table 1 that for the same number oflimit state/performance function evaluations, when compared

Fig. 3. The sampling scheme for the first-order HDMR and FHDMR; (a) Fora function having one variable (x); and (b) For a function having two variables(x1 and x2).


Table 1Results summary for the Example 1

Method Reliability index Failure probability Number of function evaluation

Direct Monte Carlo simulation 2.351 0.009372 1000 000Response Surface with quadratic polynomial [21] 2.082 0.018660 N/Ra

Response Surface with linear polynomial (traditional sampling points) [21] 2.262 0.011850 N/Ra

Response Surface with linear polynomial (vector projected sampling points) [21] 2.349 0.009410 N/Ra

First-order HDMR 2.353 0.009312 9Second-order HDMR 2.351 0.009373 25First-order FHDMR 2.558 0.005267 9Second-order FHDMR 2.381 0.008643 25

a N/R: Not reported.

Table 2Results summary for the Example 2

Method Reliabilityindex

Failureprobability

Number of function evaluation

Direct Monte Carlo simulation 2.338 0.009690 100 000Direct Monte Carlo simulation with Importance Sampling [42] 2.340 0.009642 17 850Response Surface with quadratic polynomials without cross terms [42] 2.337 0.009731 35Response Surface with complete quadratic polynomials and Central Composite Design [42] 2.341 0.009601 44Response Surface with cubic polynomials without cross terms and Central CompositeDesign [42]

2.339 0.009676 110

Response Surface with quadratic polynomials without cross terms [42] 2.337 0.009713 35First-order HDMR 2.306 0.010578 13Second-order HDMR 2.375 0.009789 49First-order FHDMR 2.337 0.009730 13Second-order FHDMR 2.338 0.009692 49

Fig. 4. The sampling scheme for the second-order HDMR and FHDMR for afunction having two variables (x1 and x2).

with MCS, the failure probability estimation by the first- andsecond-order HDMR is better than by the first- and second-order FHDMR. This is mainly because the nonlinear limitstate/performance function in Eq. (36) is dominantly additivein nature.

7.1.2. Example 2: A cantilever beam with linear elasticbehavior [19]

The reliability evaluation of a cantilever beam with arectangular cross section, uniformly distributed load, and its

linear elastic behavior is analyzed. The limit state/performancefunction is given by the maximum vertical displacement at thefree end of the beam. The displacement must not exceed theserviceability limit, which is l/325, where l is the length of thebeam. The limit state/performance function is given explicitlyby

g(x) =l

325−wl4

8E I, (37)

where E, w, I, b and l are the Young modulus, the intensityof the uniform load per unit area, the momentum of inertia,and the width and the length of the beam, respectively. It isassumed that E and l are deterministic variables with valuesgiven by E = 2.6 × 104 MPa and l = 600 mm. Thus, thelimit state/performance function is reduced to a function of tworandom variables and is given by

g(x) = 0.01846154− 74.76923x1

x32

, (38)

where x1 is the load w (MPa) and x2 is the height h (mm).The mean values of the random variables are 1000 and 200,respectively, and the standard deviations are 200 and 30,respectively. Both load and height are non-correlated variableswith a Gaussian probability distribution.

Rajashekhar and Ellingwood [19] provided the exactsolution for this problem using the MCS with importancesampling, using 1000 simulations of the limit state/performancefunction. The reported value of the failure probability [19] wasPF = 9.607 × 10−3. Table 2 presents a detailed comparison


of results obtained using various existing RSMs with thepresent methods. For evaluating the failure probability PF ,n = 7 is taken. Table 2 also shows the failure probabilityPF value obtained with the present method using the second-order HDMR and FHDMR based response surfaces andtheir associated computational efforts. Due to the dominantlymultiplicative nature of the nonlinear limit state/performancefunction in Eq. (38), it can be observed from Table 2, for thesame number of limit state/performance function evaluationswhen compared with MCS, the failure probability estimationby the first- and second-order FHDMR is better than by thefirst- and second-order HDMR. Also, for the same level ofaccuracy in the failure probability estimation the number oflimit state/performance function evaluations using the first-order FHDMR is much less than reported in the literature [39].

7.2. Set II

7.2.1. Example 3: A cantilever beamA cantilever beam subjected to a tip load P is considered in

this example. The limit state/performance function is defined sothat the tip displacement should be less than 0.15 in

g(x) = 0.15−4Pl3

Ebh3 , (39)

where P is the tip load and b, h, l are the width, the heightand the length of the beam, which are considered as randomvariables. The mean values of the random variables are 30 in,0.8359 in and 2.5093 in, respectively and standard deviationsare 3.0 in, 0.08 in and 0.25 in, respectively for the width, theheight and the length of the beam. Young’s modulus, E of thebeam is 107 psi. Both the length and the height are consideredas log-normally distributed and the width is considered as aGaussian. For evaluating the failure probability PF , n = 7 istaken. Fig. 5 compares the variation of the failure probabilitywith the tip load obtained by different methods. A sample sizeof 105 is considered in the direct MCS to evaluate the failureprobability PF , which ranges from 0.00502 to 0.08310 andthe COV of PF corresponding to this sample size varies from0.045 to 0.011 (computed using Eq. (35)) when the tip load Pvaries from 60 lb to 100 lb. The estimated failure probabilityreported in Fig. 5 using the MCS, first-order FHDMR andHDMR are 0.0284, 0.0301 and 0.0163, respectively whenthe tip load P = 80 lb. It can be observed that,the first-order FHDMR overestimates the failure probability by 6.14%,when compared with direct MCS results, while the first-orderHDMR underpredicts by 42.45%. The accumulation of a largeamount of error using the first-order HDMR, may perhaps bedue to neglecting the higher order cooperative effects and themultiplicative nature of the limit state/performance function.However, both the first-order FHDMR and HDMR needs only19 function evaluations, while the direct MCS requires 105

number of original function evaluations respectively.Fig. 5 also presents the variation of the failure probability

PF with the tip load obtained using the second-order FHDMRand HDMR. It can be observed from Fig. 5 that thesecond-order FHDMR almost exactly estimates the failure

Fig. 5. Variation of failure probability with the tip load (Example 3).

probability (PF = 0.02831) compared with the benchmarkresult of the direct MCS (PF = 0.02836) when the tip loadP = 80 lb. The results obtained using the second-orderHDMR (PF = 0.02853) closely matches with the direct MCSestimate (PF = 0.02836) and also closer to the first-orderFHDMR (PF = 0.0301). This is attributed to consideration ofthe second-order cooperative effects in the response surfaceapproximation. For the tip load P = 80 lb, compared tothe result obtained using MCS (PF = 0.02836), the second-order HDMR produces a much closer estimate of the failureprobability (PF = 0.02853) than the result obtained usingthe first-order HDMR (PF = 0.0163). But the number offunction evaluations required using the second-order FHDMRand HDMR are 127 compared with 19 function evaluationsfor the first-order FHDMR and HDMR. Therefore, to make abalance between the computational cost in terms of functionevaluations and the accuracy, the first-order FHDMR seemsmost suitable, especially for the multiplicative nature of thelimit state/performance function.

The effect of the number of sample points used for theresponse surface generation on the reliability estimation isexamined by carrying a similar analysis while varying n from3 to 9. Fig. 6(a) and (b) present, respectively, the variationof the reliability index β and the estimated failure probabilityPF with respect to the number of sample points, for the tipload P = 80 lb. Compared with direct MCS, the error inthe estimated failure probability using different methods istabulated in Table 3. Table 3 shows that, compared with thefirst- and second-order HDMR, the first- and second-orderFHDMR resulted in a drastic reduction of the approximationerror of the estimated failure probability.

7.2.2. Example 4: Ten bar truss structureA 10-bar, linear-elastic, truss structure, shown in Fig. 7,

is considered in this example to examine the accuracyand efficiency of the proposed reliability methods. Young’smodulus of the material is 107 psi . The cross-sectional areaxi , i = 1, 2, . . . , 10 for each bar follows the normal distributionand has mean µ = 2.5 in2 and standard deviation σ =

0.5 in2. The limit state/performance function considered hereis the eigenvalue limit. The eigenvalue must be greater than


Table 3Estimation of error using different methods for Example 3

Method Number of sample points, n3 5 7 9

First-order FHDMR −58.39% +6.49% +6.14% +14.59%First-order HDMR −96.26% −55.75% −42.45% −41.89%Second-order FHDMR −1.23% −0.25% −0.18% −0.07%Second-order HDMR +1.02% +0.67% +0.59% +0.14%

Fig. 6. Variation of the reliability estimation (Example 3); (a) Reliability index,β; and (b) Probability of failure, PF .

Fig. 7. The ten bar truss structure for Example 4.

9.30 (rad/s)2. Hence, the limit state/performance function isdefined as

Fig. 8. Soil profile for Example 5.

g(x) = 1.0−Fundamental Eigenvalue

9.30≥ 0.0 (Failure) . (40)

The structure is modeled using ADINA FE [40] software.For evaluating the failure probability PF , n = 7 istaken. Table 4 compares the results obtained by the MCS,and the first-order FHDMR and HDMR. A sampling sizeNS = 105 is considered in the direct MCS to estimatethe failure probability PF . The COV of PF correspondingto this sampling size is 0.0025 (computed using Eq. (35)).Table 4 also contains the computational effort in terms ofthe number of function evaluations, associated with each ofthe methods. Compared with the failure probability obtainedusing the direct MCS (PF = 0.10627), the first-order FHDMRand HDMR underestimates the failure probability by 6.20%(PF = 0.09968) and 8.37% (PF = 0.09737), respectively.However, the first-order FHDMR and HDMR needs only 61function evaluations, while the direct MCS requires 105 numberof original function evaluations respectively.

Table 4 also shows the failure probability PF value obtainedwith the present method using the second-order FHDMR(PF = 0.10625) and second-order HDMR (PF = 0.10453)based response surface and the associated computational effort.The second-order FHDMR and HDMR resulted in a significantreduction in the error of the estimated failure probability from−6.20% to −0.02% and −8.37% to −1.64%, respectively withan increase in the number of function evaluations from 61 to1681, as compared with the first-order FHDMR and HDMR.

7.2.3. Example 5: Soil settlement problemThis is a practical engineering example studied earlier by

Ang and Tang [41]. The settlement of a point A in Fig. 8caused by the construction of a structure can be shown to beprimarily caused by the consolidation of the clay layer. Supposethe contribution of settlement due to secondary consolidation is


Table 4Estimation of the failure probability for Example 4

Method Failure probability Number of function evaluationa

Direct Monte Carlo simulation 0.10627 100 000First-order FHDMR 0.09968 61b

First-order HDMR 0.09737 61b

Second-order FHDMR 0.10625 1681c

Second-order HDMR 0.10453 1681c

a Total number of times the original performance function is calculated.b (n − 1)× N + 1 = (7− 1)× 10+ 1 = 61.c (n − 1)2 (N − 1) N/2+ (n − 1) N + 1 = (7− 1)2 (10− 1) 10/2+ (7− 1) 10+ 1 = 1681.

Table 5Statistical properties of the random variables for Example 5

Random variable Mean COV Distribution type

Cc 0.396 0.25 Gaussiane0 1.190 0.15 GaussianH 168 in 0.05 Gaussianp0 3.72 ksf 0.05 Gaussian1p 0.50 0.20 Gaussian

negligible. For normally loaded clay, the settlement S is givenby

S =Cc

1+ e0H log

p0 +1p

p0, (41)

where Cc is the compression index of the clay; e0 is the voidratio of the clay layer before loading; H is the thickness ofthe clay layer; p0 is the original effective pressure at point B(mid height of the clay layer) before loading; and 1p is theincrease in pressure at point B caused by the construction ofthe structure; “log” denotes logarithm to the base 10. Due tothe non-uniform thickness and lack of homogeneity of the claylayer, the settlement predicted by the empirical formula couldbe subject to uncertainty in predicted settlement.

Suppose satisfactory performance requires that the settle-ment be less than 2.5 in. Statistical properties of the randomvariables are presented in Table 5. Determine the probabil-ity of excessive settlement at point B in Fig. 8. The limitstate/performance function is defined as

g(x) = 2.5−Cc

1+ e0H log

p0 +1p

p0. (42)

For evaluating the failure probability PF , n = 5 is taken.Table 6 compares the results obtained by the MCS, thefirst-order FHDMR and HDMR. A sampling size NS =

106 is considered in the direct MCS to estimate thefailure probability PF . The COV of PF corresponding tothis sampling size is 0.003 (computed using Eq. (35)).Table 6 also contains the computational effort in terms ofnumber of function evaluations, associated with each of themethods. Compared with the failure probability obtainedusing direct MCS (PF = 0.08096), the first-order FHDMRand HDMR underestimates the failure probability by 0.41%(PF = 0.08063) and 18.16% (PF = 0.06625), respectively.However, the first-order FHDMR and HDMR only need 21

Fig. 9. Rotating disk for Example 6.

function evaluations respectively, while the direct MCS requires106 number of original function evaluations.

Table 6 also presents the failure probability PF valueobtained using the second-order FHDMR (PF = 0.08072)and the second-order HDMR (PF = 0.08013) based responsesurface and the associated computational effort. The second-order FHDMR and HDMR resulted in a significant reductionin the error of the estimated failure probability from −0.41%to −0.29% and −18.16% to −1.03%, respectively, with anincreasing function evaluation from 21 to 181, as comparedwith the first-order FHDMR and HDMR.

7.2.4. Example 6: Burst margin of rotating disk

Consider an annular disk of inner radius Ri , outer radiusRo, shown in Fig. 9. The disk is subject to an angular velocityω about an axis perpendicular to its plane at the center.The ultimate strength of the material is Su and the materialutilization factor is αm . The burst margin is the safety marginbefore an overstress condition occurs due to the stress onthe part being too large for the material to withstand. Thesatisfactory performance of the disk is defined when the burstmargin Mb, exceeds the threshold value of 0.37473, which canbe expressed as

Mb =

√√√√√√αm Su

/ρ(

2ωπ60

)2 (R3

o − R3i

)3 (385.82) (Ro − Ri )

. (43)









Table 7Statistical properties of the random variables of the rotating disk

Random variable αm Su (lb/in2) ρ (lb/in3) ω (rpm) Ro (in) Ri (in)

Distribution Weibulla Gaussian Uniformb Gaussian Gaussian GaussianMean 0.9378 220 000 0.29 21 000 24 8Standard deviation 0.04655 5 000 0.00577 1 000 0.50 0.30

a Scale parameter = 25.508; shape parameter = 0.958.b Uniformly distributed over (0.28–0.3).








Therefore the limit state/performance function can be definedas

g(x) = Mb (αm, Su, ρ, ω, Ro, Ri )− 0.37473. (44)

Statistical properties of the random variables are listed inTable 7. For evaluating the failure probability PF , n = 9is taken. Table 8 compares the results obtained by the MCS,and the first-order FHDMR and HDMR. A sampling sizeNS = 106 is considered in the direct MCS to estimate thefailure probability PF . The COV of PF corresponding to thissampling size is 0.031 (computed using Eq. (35)). Table 8also contains the computational effort in terms of the numberof function evaluations, associated with each of the methods.Compared with the failure probability obtained using directMCS (PF = 0.00101), the first-order FHDMR and HDMRoverestimates the failure probability by 0.99% (PF = 0.00102)and 59.41% (PF = 0.00161), respectively. However, the first-order FHDMR and HDMR need only 49 function evaluations,while the direct MCS requires 106 number of original functionevaluations respectively.

Table 8 also shows the failure probability PF value obtainedwith the present method using the second-order FHDMR(PF = 0.00101) and the second-order HDMR (PF = 0.00102)based response surface and the associated computational

effort. The second-order FHDMR estimates the exact failureprobability, while using the second-order HDMR resultedin a significant reduction in error of the estimated failureprobability from +59.41% to +0.99% with an increasingfunction evaluation from 49 to 1009, as compared with the first-order HDMR.

7.2.5. Example 7: Fatigue crack growth of the edge crackedplate

In this example, Fatigue crack growth of an edge crackedplate studied by Harkness [42], is considered. The objective ofthis example is to illustrate the effectiveness of the proposedmethod in solving fatigue reliability. Our analysis examinesthe uncertainty in fatigue life through the characterization ofthe uncertainties in the parameters governing fatigue life andmodeling crack growth.

Problem definition and inputFatigue crack growth was modeled by the Paris law [43]

in a plane strain. According to this method, the fatigue crack-initiation life NF is defined as

NF =

∫ ai

a f

da

D(1Keq)η, (45)


Table 9Statistical properties of the random variables for Example 7

Random variable Mean COV Distribution type

D 6.8× 10−12 0.05 Lognormaln 3.60 0.01 Lognormalσmax 75 0.03 Gaussianσmin 25 0.03 Gaussianai 3.1× 10−4 0.186 Uniforma

K I C 165 0.015 Lognormal

a Uniformly distributed over (2.1× 10−4–4.1× 10−4).

Fig. 10. Variation of the failure probability (Example 7).

where ai is the initial crack length and a f is the cracklength at failure; D, η are the material parameters; 1Keq isthe equivalent mode I stress intensity factor, K I , at peakloading. According to the Paris law crack growth occurs in thedirection of maximum hoop stress and crack growth direction,θ , measured from the crack tip tangent, satisfies:

sin θ1K I + (3 cos θ − 1)1K I I = 0, (46)

and 1Keq is

1Keq = cos2 (θ/2) [cos (θ/2)1K I − 3 sin (θ/2)1K I I ] . (47)

Failure is defined as the fatigue life NF , being less thanthe design threshold value, NI . The design threshold valueis based on the targets reliability and therefore, can becontrolled. It can be observed from Eq. (45) that the fatiguelife expression involves several uncertain variables, making thefatigue life uncertain. The limit state/performance function canbe expressed as

g(x) = NF (x)− NI . (48)

Since negative and very small positive values of the growthparameters D and η are unrealistic [41], shifted log-normaldistributions are assigned to both parameters. Statisticalparameters and marginal densities of the random variables arepresented in Table 9.

For evaluating the failure probability PF , n = 7 is taken.Fig. 10 compares the variation of failure probability withthe design threshold value obtained by the different methods.It can be observed that the first-order FHDMR provides a

significant accuracy to the failure probability estimation, whencompared with the direct MCS results, while using the first-order HDMR a large amount of error is accumulated in thepredicted result. Accumulation of this large amount of errorusing the first-order HDMR can be attributed to neglectingthe higher order cooperative effects in the first-order HDMRand the multiplicative nature of the limit state/performancefunction. However, both the first-order FHDMR and HDMRneed only 37 function evaluations, while the direct MCSrequires 105–109 original function evaluations respectively fordifferent design threshold values.

Fig. 10 also presents the variation of the failure probabilityPF with the design threshold value using the second-orderFHDMR and HDMR. It can be observed from Fig. 10that the second-order FHDMR almost exactly estimates thefailure probability compared to the benchmark results obtainedusing the MCS. In addition, the results obtained using thesecond-order HDMR closely match with the direct MCS.This is attributed to the consideration of the second-ordercooperative effects in the response surface approximation in thesecond-order HDMR. But the number of function evaluationsrequired using the second-order FHDMR and HDMR are577 compared with 37 function evaluations for the first-orderFHDMR and HDMR. Therefore, to make a balance betweenthe computational cost in terms of function evaluations andthe accuracy, the first-order FHDMR seems most suitable,especially for multiplicative limit state/performance functions.

8. Summary and conclusions

In the reliability assessment of real life problems, the limitstate/performance functions are most often specified implicitlythrough an FE code. Amongst existing techniques for thereliability assessment, the RSM offers the best alternative todeal with this class of problems. Difficulties in response surfaceconstruction arise due to the sampling scheme adopted and theorder of the polynomial considered. This paper addressed anassessment of the response surface based on the HDMR andthe FHDMR for predicting the failure probability of systemssubject to random loads, material properties, and geometries inan efficient manner.

Seven numerical examples are illustrated to show theperformance of the response surface approximation based onthe HDMR and the FHDMR. Comparisons were made withthe direct MCS to evaluate the accuracy and computationalefficiency of the present methods. The HDMR based responsesurface works well when the sought multivariate responsefunction has an additive nature. This is because of the structureof the right hand side of the HDMR expansion. But, formultiplicative multivariate response functions, the HDMRbased method cannot be useful. At this juncture, the factorizedform of the HDMR can be used for the response surfacegeneration of an implicit limit state/performance function. Thismethod works superior better when the response function isdominantly of a multiplicative nature. However, the inclusionof the higher order cooperative effects in the HDMR basedresponse surface provides a significant accuracy to the limit


state/performance function of a dominantly multiplicativenature, but the number of original function evaluation increasessignificantly over the first-order HDMR/FHDMR basedresponse surface approximation. Numerical implementationsalso support these observations. It can also be observedthat the HDMR/FHDMR based response surface method notonly yields a more accurate estimate of the probability offailure than the alternative approximate methods in highlynonlinear problems, but also reduces the computational effortsignificantly over direct simulation methods.

A parametric study, varying the number of sample pointsused for HDMR and FHDMR based response surfacegeneration, demonstrates that the first-order FHDMR basedresponse surface provides the desired accuracy to thepredicted failure probability with the least number of functionevaluations. In order to reduce the approximation error further,the second-order FHDMR based response surface could be usedin reliability analysis, but the number of function evaluationsincreases significantly compared to the first-order case. Thenumber of sample points n used in the response surfacegeneration was set to different values to investigate its effect onthe estimated failure probability. A very small value of n shouldbe avoided owing to loss of Uncertain information in responsesurface generation.

Acknowledgement

The authors would like to acknowledge the financial supportby the Board of Research in Nuclear Sciences, India undersanction No. 2004/36/39-BRNS/2332.

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Assessment of high dimensional model representation techniques for reliability analysis