Hybrid High Dimensional Model Representation for reliability analysis

Comput. Methods Appl. Mech. Engrg. 198 (2009) 753–765

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Comput. Methods Appl. Mech. Engrg.

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Hybrid High Dimensional Model Representation for reliability analysis

Rajib Chowdhury, B.N. Rao *

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

a r t i c l e i n f o

Article history:Received 21 February 2008Received in revised form 24 September2008Accepted 8 October 2008Available online 21 October 2008

Keywords:Structural reliabilityHigh Dimensional Model RepresentationFactorized High Dimensional ModelRepresentationHybrid High Dimensional ModelRepresentationMoving least squaresFailure probability

0045-7825/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.cma.2008.10.006

* Corresponding author. Tel.: +91 44 2257 4285; faE-mail address: [email protected] (B.N. Rao).

a b s t r a c t

This paper presents a new probabilistic method based on Hybrid High Dimensional Model Representation(HHDMR) for predicting the failure probability of structural and mechanical systems subject to randomloads, material properties, and geometry. High Dimensional Model Representation (HDMR) is a generalset of quantitative model assessment and analysis tools for capturing the high dimensional relationshipsbetween sets of input and output model variables. It is a very efficient formulation of the systemresponse, if higher order variable cooperative effects are weak and if the response function is dominantlyof additive nature, allowing the physical model to be captured by the first few lower order terms. But, ifmultiplicative nature of the response function is dominant then all right hand side components of HDMRmust be used to be able to obtain the best result. However, if HDMR requires all components, whichmeans 2N number of components, to get a desired accuracy, making the method very expensive in prac-tice, then Factorized HDMR (FHDMR) can be used. But in most cases the limit state/performance functionhas neither additive nor multiplicative nature, rather it has an intermediate nature. This paper presents anew HHDMR-based approximation for the limit state/performance functions having an intermediate nat-ure. The proposed approximation of an implicit limit state/performance function includes both HDMRand FHDMR expansions through a hybridity parameter. As an alternative to the conventional methodsfor reliability analysis which are very computationally demanding, when applied in conjunction withcomplex finite element models, this study aims to assess how accurately and efficiently HHDMR tech-nique can capture complex model output uncertainty. Once the approximate form of the original implicitlimit state/performance function is defined, the failure probability can be obtained by statistical simula-tion. Results of six numerical examples involving mathematical functions and structural/solid-mechanicsproblems indicate that the failure probability obtained using HHDMR-based approximation of an implicitlimit state/performance function provides significant accuracy when compared with the conventionalMonte Carlo method, while requiring fewer original model simulations.

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1. Introduction

One of the very important factors in designing of a structure isreliability. An efficient and increasingly popular method of evaluat-ing reliability is based on determining the probability that a struc-ture exceeds a threshold limit, defined by a limit state/performance function. Symbolically, the reliability problem canbe stated as [1–4]

PF � PðgðxÞ 6 0Þ ¼Z

gðxÞ60pxðxÞdx; ð1Þ

where x ¼ fx1; x2; . . . ; xNg, represent the N-dimensional randomvariables of the model under consideration; gðxÞ is the limit state/performance function, such that gðxÞ 6 0 represents the failure do-

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main; and pXðxÞ is the joint probability density function of the inputrandom variables.

Two common approaches are available for estimating the fail-ure probability PF , which can be labeled as gradient-based and sim-ulation-based methods. In the gradient-based approach, there is aneed of estimating the gradients of the limit state/performancefunction at a relevant point where probability density is largest.These methods are popularly termed as first- or second-order reli-ability method (FORM/SORM) [1–5]. In reality, the limit state/per-formance functions are implicit nature and highly non-linear.Therefore, a detailed finite element (FE) modeling of the structureis necessary in combination with reliability analysis tools. FE meth-ods for linear and non-linear structures in conjunction with FORM/SORM have been successfully applied for structural reliability com-putations [6]. But, such methods are effective in evaluating verysmall probabilities of failure for small-scale problems. In regardto the large-scale problems, merging of FORM/SORM, with com-mercial FE programs is not straightforward [7] especially whenthe non-linear problems are addressed.

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754 R. Chowdhury, B.N. Rao / Comput. Methods Appl. Mech. Engrg. 198 (2009) 753–765

In contrast to the gradient-based methods, simulation-basedmethods [8–11] hinge upon the creation of generating syntheticset of basic random variables samples and simulating the actuallimit state/performance function repeated times. But the simula-tion methods seem impractical when each simulation involvesexpensive finite element or meshfree calculations. Several issuesrelated to the applicability of FORM/SORM and the efficiency ofsimulation methods for reliability analysis have lead manyresearchers to assess and improve the viability of alternate meth-ods [12–16] in the field of reliability and system safety.

Recently authors adopted High Dimensional Model Representa-tion (HDMR) concepts to find an equivalent continuous function toreplace a univariate or multivariate piece wise continuous func-tion, rather than seeking an exact continuous function [17] andfor predicting the failure probability of structural or mechanicalsystems subjected to random loads and material properties[18,19]. Major advantages of HDMR/FHDMR approximation aretwo-folded, namely; (a) it presents higher order approximationof the limit state/performance function, and (b) it requires feweroriginal model simulations (in terms of FE analysis), as comparedto full scale simulation methods.

This paper presents Hybrid High Dimensional Model Repre-sentation (HHDMR)-based method for predicting the failure prob-ability of structural/mechanical systems subjected to randomloads, geometry and material properties. The method involvesHHDMR technique in conjunction with moving least squares(MLS) technique to approximate the original implicit limitstate/performance function with an explicit function. The paperis organized as follows. Section 2 presents a brief overview ofHDMR and its applicability to reliability analysis. Section 3 pre-sents the formulation of FHDMR. Section 4 portrays the formula-tion of HHDMR. Section 5 presents construction of limit state/performance function using HHDMR approximation. Section 6demonstrates the simulation method for evaluation of reliabilityusing HHDMR approximation. Section 7 presents six numericalexamples to illustrate the performance of the present method.Comparisons have been made with conventional approximatemethods (FORM/SORM) and direct Monte Carlo simulation(MCS) method to evaluate the accuracy and the computationalefficiency of the present methods.

2. HDMR and its application to reliability analysis

In recent years there have been efforts to develop efficientmethods to approximate multivariate functions in such a way thatthe component functions of the approximation are ordered startingfrom a constant and gradually approaching to multivariance as weproceed along the terms like first-order, second-order and so on.One such method is HDMR [20–27], which is a general set of quan-titative model assessment and analysis tools for capturing the highdimensional relationships between sets of input and output modelvariables. It is a very efficient formulation of the system response,if higher order variable correlations are weak, allowing the physi-cal model to be captured by the first few lower order terms. Prac-tically for most well-defined physical systems, only relatively loworder correlations of the input variables are expected to have a sig-nificant effect on the overall response [20–23]. HDMR expansionutilizes this property to present an accurate hierarchical represen-tation of the physical system.

Let the N-dimensional vector x ¼ fx1; x2; . . . ; xNg, represent theinput variables of the model under consideration, and gðxÞ as theresponse variable. Since the influence of the input variables onthe response variable can be independent and/or cooperative,HDMR expresses the response gðxÞ as a hierarchical correlatedfunction expansion in terms of the input variables as

gðxÞ ¼ g0 þXN

i¼1

giðxiÞ þX

16i1<i26N

gi1 i2 ðxi1 ; xi2 Þ þ � � �

þX

16i1<��<il6N

gi1 i2 ��il ðxi1 ; xi2 ; . . . ; xil Þ þ � � �

þ g12...Nðx1; x2; . . . ; xNÞ ð2Þwhere g0 is a constant term representing the zeroth-order compo-nent function or the mean response of gðxÞ. The function giðxiÞ isa first-order term expressing the effect of variable xi acting alone,although generally non-linearly, upon the output gðxÞ. The functiongi1 i2 ðxi1 ; xi2 Þ is a second-order term which describes the cooperativeeffects of the variables xi1 and xi2 upon the output gðxÞ. The higherorder terms give the cooperative effects of increasing numbers ofinput variables acting together to influence the output gðxÞ. The lastterm g12...Nðx1; x2; . . . ; xNÞ contains any residual dependence of all theinput variables locked together in a cooperative way to influencethe output gðxÞ. Once all the relevant component functions in Eq.(2) are determined and suitably represented, then the componentfunctions constitute HDMR, thereby replacing the original compu-tationally expensive method of calculating gðxÞ by the computation-ally efficient model. Usually the higher order terms in Eq. (2) arenegligible [21] such that HDMR with only few low order correla-tions, amongst the input variables are typically adequate in describ-ing the output behavior [22] resulting in rapid convergence ofHDMR expansion. However, importance of higher order terms inHDMR is problem dependent and an independent analysis such asfor example, the Sobol sensitivity indices method [28] should bemade beforehand to establish which order terms in the HDMRexpansion are not important.

Depending on the method adopted to determine the componentfunctions in Eq. (2) there are two particular HDMR expansions: AN-OVA-HDMR and cut-HDMR. ANOVA-HDMR is useful for measuringthe contributions of the variance of individual component functionsto the overall variance of the output. On the other hand, cut-HDMRexpansion is an exact representation of the output gðxÞ in the hyper-plane passing through a reference point in the variable space.

With cut-HDMR method, first a reference point c ¼fc1; c2; . . . ; cNg is defined in the variable space. In the convergencelimit that all correlated functions in Eq. (2) are considered, cut-HDMR is invariant to the choice of reference point c. However inpractice the choice of reference point c is very important for cut-HDMR especially if only terms up to first- and second-order inEq. (2) are considered, and it is shown that it is optimal to choosereference point c as mean values of the input variables [27]. Theexpansion functions are determined by evaluating the input–out-put responses of the system relative to the defined reference pointc along associated lines, surfaces, subvolumes, etc. (i.e. cuts) in theinput variable space. This process reduces to the following rela-tionship for the component functions in Eq. (2)

g0 ¼ gðcÞ; ð3ÞgiðxiÞ ¼ gðxi; ciÞ � g0; ð4Þgi1 i2 ðxi1 ; xi2 Þ ¼ gðxi1 ; xi2 ; c

i1 i2 Þ � gi1 ðxi1 Þ � gi2 ðxi2 Þ � g0; ð5Þ

where the notation gðxi; ciÞ ¼ gðc1; c2; . . . ; ci�1; xi; ciþ1; . . . ; cNÞ denotesthat all the input variables are at their reference point values exceptxi. The g0 term is the output response of the system evaluated at thereference point c. The higher order terms are evaluated as cuts inthe input variable space through the reference point. Therefore,each first-order term giðxiÞ is evaluated along its variable axisthrough the reference point. Each second-order term gi1 i2 ðxi1 ; xi2 Þ isevaluated in a plane defined by the binary set of input variablesxi1 ; xi2 through the reference point, etc. The process of subtractingoff the lower order expansion functions removes their dependenceto assure a unique contribution from the new expansion function.

R. Chowdhury, B.N. Rao / Comput. Methods Appl. Mech. Engrg. 198 (2009) 753–765 755

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

gðxÞ ¼ g0 þXN

i¼1

giðxiÞ þR2: ð6Þ

Substituting Eqs. (3) and (4) into Eq. (6) leads to

gðxÞ ¼XN

i¼1

gðc1; . . . ; ci�1; xi; ciþ1; . . . ; cNÞ � ðN � 1ÞgðcÞ þR2: ð7Þ

Now consider first-order HDMR approximation ~gðxÞ of the originalimplicit response function gðxÞ, denoted by

~gðxÞ ¼XN

i¼1

gðc1; . . . ; ci�1; xi; ciþ1; . . . ; cNÞ � ðN � 1ÞgðcÞ: ð8Þ

Comparison of Eqs. (7) and (8) indicates that the first-order approx-imation leads to the residual error gðxÞ � ~gðxÞ ¼ R2, which includescontributions from terms of two and higher order componentfunctions.

The notion of zeroth-, first-, second-order, etc. in HDMR expan-sion should not be confused with the terminology used either inthe Taylor series or in the conventional least squares-based regres-sion model. It can be shown that, the first-order component functiongiðxiÞ is the sum of all the Taylor series terms which contain and onlycontain variable xi. Hence first-order HDMR approximations shouldnot be viewed as first-order Taylor series expansions nor do they lim-it the non-linearity of gðxÞ. Furthermore, the approximations containcontributions from all input variables. Thus, the infinite number ofterms in the Taylor series are partitioned into finite different groupsand each group corresponds to one cut-HDMR component function.Therefore, any truncated cut-HDMR expansion provides a betterapproximation and convergent solution of gðxÞ than any truncatedTaylor series because the latter only contains a finite number ofterms of Taylor series. Furthermore, the coefficients associated withhigher dimensional terms are usually much smaller than that withone-dimensional terms. As such, the impact of higher dimensionalterms on the function is less, and therefore, can be neglected. Com-pared with the FORM and SORM which retains only linear and qua-dratic terms, respectively, first-order HDMR provides more accurateapproximation ~gðxÞ of the original implicit limit state/performancefunction gðxÞ.

3. Factorized HDMR

In the previous section, the response function gðxÞ is repre-sented as few low order component functions of HDMR in anadditive form. However, when the response function gðxÞ is dom-inantly of multiplicative nature, HDMR approximation may not besufficient to accurately estimate the probabilistic characteristicsof the system. Multiplicative form of HDMR for a given multivar-iate limit state/performance function gðxÞ can be represented as[29,30]

gðxÞ ¼ r0

YNi¼1

ð1þ riðxiÞÞ" # YN

i1 ;i2¼1i1<i2

ð1þ ri1 i2 ðxi1 ; xi2 ÞÞ

2664

3775� � � �

� 1þ r12...Nðx1; x2; . . . ; xNÞ½ �; ð9Þ

where r0 is a constant term, riðxiÞ is a first-order term expressing theeffect of variable xi acting alone, although generally non-linearly,upon the output gðxÞ. The function ri1 i2 ðxi1 ; xi2 Þ is a second-orderterm which describes the cooperative effects of the variables xi1

and xi2 upon the output gðxÞ and so on. The constant term, first-or-der, and higher order terms can be found by comparing Eq. (9) withEq. (2) [29]. This process reduces to the following relationship forthe component functions in Eq. (9),

r0 ¼ g0; ð10Þ

riðxiÞ ¼giðxiÞ

g0; ð11Þ

ri1 i2 ðxi1 ; xi2 Þ ¼g0gi1 i2

ðxi1 ; xi2 Þ � gi1ðxi1 Þgi2

ðxi2 Þðg0 þ gi1 ðxi1 ÞÞðg0 þ gi2 ðxi2 ÞÞ

; ð12Þ

where, g0, giðxiÞ and gi1 i2 ðxi1 ; xi2 Þ are defined in Eqs. (3)–(5) of Section2. The component functions defined in Eqs. (10)–(12) can be furthersimplified as follows:

r0 ¼ gðcÞ; ð13Þ

riðxiÞ ¼gðxi; ciÞ

gðcÞ � 1; ð14Þ

ri1 i2 ðxi1 ; xi2 Þ ¼gðcÞgðxi1 ; xi2 ; c

i1 i2 Þgðxi1 ; ci1 Þ gðxi2 ; c

i2 Þ � 1: ð15Þ

Once all the relevant component functions in Eq. (9) are determinedand suitably represented, then the component functions constitutefactorized form of HDMR called FHDMR [30]. Therefore, first-orderFHDMR approximation ~gðxÞ of the original implicit response func-tion gðxÞ can be represented as

~gðxÞ ¼ gðcÞYNi¼1

gðxi; ciÞgðcÞ

" #: ð16Þ

4. Hybrid HDMR

The methodology presented in Sections 2 and 3 is well suitedfor the limit state/performance function has dominantly of additivenature and of multiplicative nature, respectively. But in most casesthe limit state/performance function has neither additive nor mul-tiplicative nature, rather it has an intermediate nature. In this sec-tion, HHDMR methodology is presented for approximation of animplicit limit state/performance function. HHDMR expansion [31]includes HDMR and FHDMR expansions in its structure through ahybridity parameter and can be represented as

gðxÞ ¼ c g0 þXN

i¼1

giðxiÞ þX

16i1<i26N

gi1 i2 ðxi1 ; xi2 Þ þ � � � !

þ ð1� cÞ r0

YN

i¼1

ð1þ riðxiÞÞ" # YN

i1 ;i2¼1i1<i2

ð1þ ri1 i2 ðxi1 ; xi2 ÞÞ

2664

3775� � � �

0BB@

1CCA;ð17Þ

where c is the hybridity parameter, which controls the generalstructure of the implicit multivariate limit state/performance func-tion. If c ¼ 1, the function expansion in Eq. (17) merges to HDMRpresented in Eq. (2). Similarly, if c ¼ 0, the function expansion inEq. (17) merges to FHDMR presented in Eq. (9). Using Eq. (17) the(kl)th-order HHDMR approximation ~gðxÞ of the original implicit re-sponse function gðxÞ can be represented as

~gðxÞ ¼ cSkðx1; x2; . . . ; xNÞ þ ð1� cÞPlðx1; x2; . . . ; xNÞ; ð18Þwhere

S0ðx1; x2; . . . ; xNÞ ¼ g0;

S1ðx1; x2; . . . ; xNÞ ¼ S0ðx1; x2; . . . ; xNÞ þXN

i¼1

giðxiÞ;

..

.

Skðx1; x2; . . . ; xNÞ ¼ Sk�1ðx1; x2; . . . ; xNÞþ

X16i1<...<ik6N

gi1 i2 ...ikðxi1 ; xi2 ; . . . ; xikÞ; 0 6 k 6 N

ð19Þ


and

P0ðx1; x2; . . . ; xNÞ ¼ r0;

P1ðx1; x2; . . . ; xNÞ ¼ P0ðx1; x2; . . . ; xNÞYNi¼1

ð1þ riðxiÞÞ;

..

.

Plðx1; x2; . . . ; xNÞ ¼ Pl�1ðx1; x2; . . . ; xNÞY

16i1<...<ik6N

ð1þ ri1 i2 ...ik ðxi1 ; xi2 ; . . . ; xikÞÞ; 0 6 l 6 N:

ð20Þ

The most important step in this expansion is to determine thehybridity parameter c. For this purpose, a functional of the originalimplicit response function gðxÞ and the (kl)th-order HHDMRapproximation ~gðxÞ is defined as

Fðx; cÞ � gðxÞ � ~gðxÞk k2: ð21Þ

The norm of functional given in Eq. (21) can be obtained as

Fðx; cÞ �Xm1

j1¼1

. . .XmN

jN¼1

YNi¼1

jiji

!gðxj1

1 ; xj22 ; . . . ; xjN

N Þ�

�cSkðxj11 ; x

j22 ; . . . ; xjN

N Þ � ð1� cÞPlðxj11 ; x

j22 ; . . . ; xjN

N Þ�2;

0 6 k; l 6 N; ð22Þ

where m1; . . . ;mN represents the number of sample points alongeach input variable axis. ji

jithe weight function for each sample

points and defined as the reciprocal of mi such thatPmiji¼1j

iji¼ 1; 1 6 i 6 N. For example, consider a two-dimensional

function with two input variables ðx1; x2Þ. Let variable x1 be sampledalong x1-axis with five sample points ðm1 ¼ 5Þ keeping other vari-able at its reference point c2. Then j1

j1¼ ð0:2;0:2;0:2;0:2;0:2Þ,

1 6 j1 6 5. Similarly, if variable x2 is sampled along x2-axis with fivesample points ðm2 ¼ 5Þ keeping other variable at its reference pointc1. Then j2

j2¼ ð0:2;0:2;0:2;0:2;0:2Þ, 1 6 j2 6 5.

Minimizing the norm of functional estimates the value of thehybridity parameter c. This minimization criterion can be writtenas

oFðx; cÞoc

¼ 0: ð23Þ

Applying the minimization criterion given in Eq. (23), the optimumvalue for c is obtained as

c ¼ A2 þ A3 � A4 � A5

A1 þ A2 � 2A5; ð24Þ

where

A1 ¼Xm1

j1¼1

. . .XmN

jN¼1

YN

i¼1

jiji

!Skðxj1

1 ; xj22 ; . . . ; xjN

N Þ2;

A2 ¼Xm1

j1¼1

. . .XmN

jN¼1

YN

i¼1

jiji

!Plðxj1

1 ; xj22 ; . . . ; xjN

N Þ2;

A3 ¼Xm1

j1¼1

. . .XmN

jN¼1

YN

i¼1

jiji

!gðxj1

1 ; xj22 ; . . . ; xjN

N ÞSkðxj11 ; x

j22 ; . . . ; xjN

N Þ;

0 6 k; l 6 N;

A4 ¼Xm1

j1¼1

. . .XmN

jN¼1

YN

i¼1

jiji

!gðxj1

1 ; xj22 ; . . . ; xjN

N ÞPlðxj11 ; x

j22 ; . . . ; xjN

N Þ;

A5 ¼Xm1

j1¼1

. . .XmN

jN¼1

YN

i¼1

jiji

!Skðxj1

1 ; xj22 ; . . . ; xjN

N ÞPlðxj11 ; x

j22 ; . . . ; xjN

N Þ:

ð25Þ

Using the obtained c value, HHDMR approximation can be con-structed from HDMR and FHDMR. The final HHDMR expansion for

the sought multivariate limit state/performance function is deter-mined within truncation approximation.

Considering terms up to first-order in Eq. (17) yields,

gðxÞ ¼ c g0 þXN

i¼1

giðxiÞ !

þ ð1� cÞ r0

YN

i¼1

ð1þ riðxiÞÞ" # !

þR2:

ð26Þ

Therefore, using Eq. (26) first-order HHDMR approximation ~gðxÞ ofthe original implicit response function gðxÞ can be represented as

~gðxÞ ¼ cXN

i¼1

gðc1; . . . ; ci�1; xi; ciþ1; . . . ; cNÞ � ðN � 1ÞgðcÞ !

þ ð1� cÞ gðcÞYN

i¼1

gðc1; . . . ; ci�1; xi; ciþ1; . . . ; cNÞgðcÞ

" # !; ð27Þ

where the hybridity parameter c can be obtained using Eq. (24),with A1, A2, A3, A4, A5 for first-order HHDMR approximation beinggiven by

A1 ¼YNi¼1

Xmi

ji¼1

jijiS1ðc1; . . . ; ci�1; x

jii ; ciþ1; . . . ; cNÞ2;

A2 ¼YNi¼1

Xmi

ji¼1

jijiP1ðc1; . . . ; ci�1; x

jii ; ciþ1; . . . ; cNÞ2;

A3 ¼YNi¼1

Xmi

ji¼1

jijigðc1; . . . ; ci�1; x

jii ; ciþ1; . . . ; cNÞ

� S1ðc1; . . . ; ci�1; xjii ; ciþ1; . . . ; cNÞ;

A4 ¼YNi¼1

Xmi

ji¼1

jijigðc1; . . . ; ci�1; x

jii ; ciþ1; . . . ; cNÞ

� P1ðc1; . . . ; ci�1; xjii ; ciþ1; . . . ; cNÞ;

A5 ¼YNi¼1

Xmi

ji¼1

jijiS1ðc1; . . . ; ci�1; x

jii ; ciþ1; . . . ; cNÞ

� P1ðc1; . . . ; ci�1; xjii ; ciþ1; . . . ; cNÞ:

ð28Þ

It should be noted that, the hybridity parameter c obtained using A1,A2, A3, A4, A5 in Eq. (28), even for the response function of com-pletely additive or multiplicative in nature (with non-negligiblecooperative effects of input variables) may not be exactly equal to1 or 0, as A1, A2, A3, A4, A5 in Eq. (28) are approximated based onfirst-order component functions S1ðc1; . . . ; ci�1; x

jii ; ciþ1; . . . ; cNÞ and

P1ðc1; . . . ; ci�1; xjii ; ciþ1; . . . ; cNÞ where the cooperative effects of input

variables is ignored.

5. Construction of limit state/performance function

HHDMR in Eq. (17) is exact along any of the cuts, and the outputresponse gðxÞ at a point x off of the cuts can be obtained by follow-ing the procedure in steps 1–4 as follows:

Step 1: Interpolate the first-order component functiongðxi; ciÞ ¼ gðc1; c2; . . . ; ci�1; xi; ciþ1; . . . ; cNÞ with respect to the inputvalues of the point x. If for xi ¼ xj

i; n function values

gðxji; c

iÞ ¼ gðc1; . . . ; ci�1; xji; ciþ1; . . . ; cNÞ; j ¼ 1;2; . . . ; n; ð29Þ

are given at nð¼ 3;5;7 or 9Þ equally spaced sample pointsli � ðn� 1Þri=2, li � ðn� 3Þri=2, . . ., li, . . ., li þ ðn� 3Þri=2,li þ ðn� 1Þri=2 along the variable axis xi with mean li and stan-dard deviation ri, the function value for arbitrary xi can be obtainedby the MLS interpolation [32] as

gðxi; ciÞ ¼Xn

j¼1

/jðxiÞg0ðc1; . . . ; ci�1; xji; ciþ1; . . . ; cNÞ; ð30Þ


where

g0ðx1i ; c

iÞ

..

.

..

.

g0ðxni ; c

iÞ

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;¼

/1ðx1i Þ /2ðx1

i Þ � � � /nðx1i Þ

..

. ... ..

. ...

..

. ... ..

. ...

/1ðxni Þ /2ðxn

i Þ � � � /nðxni Þ

266666664

377777775

�1gðx1

i ; ciÞ

..

.

..

.

gðxni ; c

iÞ

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;:

ð31Þ

The interpolation functions /jðxiÞ can be obtained using the MLSinterpolation scheme [32].

By using Eq. (30), giðxiÞ can be generated if n function values aregiven at corresponding sample points. The same procedure shall berepeated for all the first-order component functions, i.e.,giðxiÞ; i ¼ 1;2; . . . ;N.

Step 2: Sum the interpolated values of HDMR expansion terms.This leads to first-order HDMR approximation of the function gðxÞas

~gðxÞ ¼XN

i¼1

Xn

j¼1

/jðxiÞg0ðc1; . . . ; ci�1; xji; ciþ1; . . . ; cNÞ � ðN � 1Þg0:

ð32Þ

Step 3: Multiply the interpolated values of HDMR expansionterms. This leads to first-order FHDMR approximation of the func-tion gðxÞ as

~gðxÞ ¼ g0

YN

i¼1

Pnj¼1/jðxiÞg0ðc1; . . . ; ci�1; x

ji; ciþ1; . . . ; cNÞ

g0

" #: ð33Þ

Step 4: Combine HDMR and FHDMR approximation using thehybridity parameter c. This leads to first-order HHDMR approxi-mation of the function gðxÞ as

~gðxÞ ¼ cXN

i¼1

Xn

j¼1

/jðxiÞg0ðc1; . . . ; ci�1; xji; ciþ1; . . . ; cNÞ � ðN � 1Þg0

!

þ ð1� cÞ g0

YNi¼1

Pnj¼1/jðxiÞg0ðc1; . . . ; ci�1; x

ji; ciþ1; . . . ; cNÞ

g0

" # !

ð34Þ

If n is the number of sample points taken along each of the variableaxis and s is the order of the component function considered, start-ing from zeroth-order to lth-order, then total number of functionevaluation for interpolation purpose is given by

Pls¼0ðN!ðn� 1ÞsÞ=

ððN � sÞ!s!Þ, which grows polynomially with n and s. As a few low or-der component functions of HHDMR are used, the sample savingsdue to HHDMR are significant compared to traditional sampling.Hence reliability analysis using HHDMR relies on an accurate re-duced model being generated with a small number of FE modelanalysis. An arbitrarily large sample Monte Carlo analysis can beperformed on the outputs approximated by HHDMR which resultin the same distributions as obtained through the Monte Carlo anal-ysis of the full model. The tremendous computational savings resultfrom just having to perform interpolation of reduced model insteadof full model simulations for output determination.

6. Failure probability estimation

Eq. (34) provides HHDMR approximation ~gðxÞ of the original im-plicit limit state/performance function gðxÞ using the MLS interpo-lation functions, constant gðcÞ term, first-order HDMR termgðc1; . . . ; ci�1; x

ji; ciþ1; . . . ; cNÞ. Therefore the failure probability PF

can be easily estimated by performing MCS on first-order HHDMR

approximation ~gðxÞ of the original implicit limit state/performancefunction gðxÞ and is given by

PF ¼1

NS

XNS

i¼1

I ~gðxiÞ < 0� �

; ð35Þ

where xi is ith realization of X, NS is the sampling size, I½�� is an indi-cator function of fail or safe state such that I ¼ 1, if ~gðxiÞ < 0 other-wise zero. A flow diagram of HHDMR approximation and the failureprobability PF estimation by MCS is shown in Fig. 1.

Since HHDMR approximation leads to explicit representation ofthe original implicit limit state/performance function, the MCS canbe conducted for any sampling size. The total cost of original func-tion evaluation entails a maximum of ðn� 1Þ � N þ 1 by the pres-ent method using HHDMR.

7. Numerical examples

Six numerical examples involving explicit functions (Examples1 and 2) and implicit functions from structural/solid-mechanicsproblems (Examples 3–6) are presented to illustrate the perfor-mance of the present method in conjunction with HHDMR approx-imation. An exact continuous function to replace a univariate ormultivariate piece wise continuous function may not always beavailable in general problems. Rather than seeking an exact contin-uous function to replace a piece wise continuous function, anequivalent continuous function can be found based on HDMR con-cept. To evaluate the accuracy and the efficiency of the presentmethod in conjunction with HHDMR approximation, comparisonsof the estimated failure probability PF have been made withFORM/SORM and direct MCS. The coefficient of variation (COV) dof the estimated failure probability PF by direct MCS for the sam-pling size NS considered, is computed using

d ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� PFÞ

NSPF

s: ð36Þ

When comparing computational efforts by various methods in eval-uating the failure probability PF , the number of original limit state/performance function evaluations (in terms of FE analysis) is chosenas the primary comparison tool in this paper. This is because of thefact that, number of function evaluations indirectly indicates theCPU time usage. For direct MCS, number of original function evalu-ations (in terms of FE analysis) is same as the sampling size. Whileevaluating the failure probability PF through direct MCS, CPU time ismore because it involves number of repeated actual FE analysis.However, in the present method MCS is conducted in conjunctionwith HHDMR approximation. Here, although the same samplingsize as in direct MCS is considered, the number of original functionevaluations is much less. Hence, the computational effort expressedin terms of function evaluations alone should be carefully inter-preted for problems involving explicit functions. For generation ofsurrogate model using HHDMR, equally spaced sample points aredeployed along the variable axis through the reference point. Sam-pling scheme for HDMR/FHDMR/HHDMR approximation of a func-tion having one variable (x) and two variables (x1 and x2Þ isshown in Fig. 2a and b, respectively. The number of original func-tion evaluations required for HDMR/FHDMR/HHDMR approxima-tion is problem dependent and is provided in each example toillustrate computational efficiency.

7.1. Example 1: cubic function with two variables

Consider a cubic limit state/performance function originallystudied by Grandhi and Wang [4], of the following form:

Fig. 1. Flowchart of failure probability, PF estimation using HHDMR approximation.


gðxÞ ¼ 2:2257� 0:025ffiffiffi2p

27ðx1 þ x2 � 20Þ3 þ 33

140ðx1 � x2Þ ð37Þ

with two independent normal variables. Mean and standard devia-tion of the random variables is 10 and 3, respectively. For evaluatingthe failure probability PF , HDMR/FHDMR/HHDMR approximation isconstructed by deploying five equally spaced sample points (n = 5)along each of the variable axis. The reference point is taken as themean values of the random variables. Table 1 compares the resultsobtained by the present method using HDMR, FHDMR and HHDMRapproximation with FORM, SORM, and direct MCS. A sampling sizeNS ¼ 106 is considered in direct MCS to evaluate the failure proba-bility PF and the COV of PF corresponding to this sampling size is

0.007 (computed using Eq. (36)). Table 1 also contains the compu-tational effort in terms of number of function evaluations, associ-ated with each of the methods. Compared with the failureprobability obtained using direct MCS ðPF ¼ 0:01907Þ, FORM, SORM,the present method using HDMR, FHDMR and HHDMR approxima-tion underestimates the failure probability by 31.73%ðPF ¼ 0:01302Þ, 31.73% ðPF ¼ 0:01302Þ, 21.81% ðPF ¼ 0:01491Þ,74.51% ðPF ¼ 0:00486Þ and 2.83% ðPF ¼ 0:01853Þ, respectively. Itcan be observed that, the limit state/performance function in Eq.(37) has neither dominantly additive nor multiplicative naturerather it has an intermediate nature. As the cubic additive termsdominantly regulate the function characteristics over the crossedterm, therefore, HDMR approximation provides superior result over

x

c

x1

x2

c

a

bFig. 2. Sampling scheme for first-order HDMR/FHDMR/HHDMR; (a) for a functionhaving one variable (x); and (b) for a function having two variables (x1 and x2Þ.

Table 1Estimation of failure probability for Example 1.

Method Failureprobability

Number of functionevaluationa

FORMb 0.01302 21SORMb 0.01302 204Direct Monte Carlo

simulation0.01907 1 � 106

First-order HDMR 0.01491 9c

First-order FHDMR 0.00486 9c

HHDMR 0.01853 9c

a Total number of times the original performance function is calculated.b Ref. [4].c ðn� 1Þ � N þ 1 ¼ ð5� 1Þ � 2þ 1 ¼ 9:

Table 2Estimation of error using different methods for Example 1.

Method Number of sample points, n

3 5 7 9

First-orderHDMR

�40.75%(0.01130)

�21.81%(0.01491)

�15.63%(0.01609)

�16.05%(0.01601)

First-orderFHDMR

�62.29%(0.00719)

�74.51%(0.00486)

�56.21%(0.00835)

�56.32%(0.00833)

HHDMR �30.26%(0.01330)

�2.83%(0.01853)

+3.46%(0.01973)

+5.98%(0.02021)

Note that the number inside parentheses indicates corresponding failureprobability.

Table 3Statistical properties of random variables for Example 2.

Random variable Distribution Mean COV

Nc Lognormal 5490 0.20Nf Lognormal 17100 0.20nc Lognormal 549 0.20nf Lognormal 4000 0.20h1 Normal 0.42 0.20h2 Normal 6.0 0.20


FHDMR approximation. However, as HHDMR approximation con-siders both additive and multiplicative nature of the limit state/per-formance function using optimum c, therefore, the approximationerror in failure probability is least compared with other methods.The present method using HDMR, FHDMR and HHDMR needs onlynine original function evaluations, while FORM, SORM and directMCS requires 21, 204 and 106 number of original function evalua-tions, respectively. It can also be observed that, the computationaleffort in HHDMR remains same as of HDMR and FHDMR, but theaccuracy is improved significantly.

The effect of number of sample points used for HDMR, FHDMRand HHDMR approximation on the reliability estimation is exam-ined by carrying a similar analysis varying n from 3 to 9. Comparedwith direct MCS, error in the estimated failure probability usingdifferent methods is tabulated in Table 2 along with the failureprobability values in parenthesis. The computational effort interms of number of function evaluations for all variant of approx-imation methods are 5 for n = 3, 9 for n = 5, 13 for n = 7, and 17 forn = 9. It is observed that n = 5 provides the optimum number offunction calls with maximum accuracy in evaluating the failure

probability PF with the present method using HHDMRapproximation.

7.2. Example 2: creep–fatigue limit state function

This example involves a probabilistic model for reliability anal-ysis of creep and fatigue of materials. Consider a non-linear creep–fatigue limit state/performance function studied by Lu et al. [33],having the following form:

gðNc;Nf ;nc;nf ; h1; h2Þ ¼ 2� eh1Dc þ eh1 � 2e�h2 � 1

ðe�h2Dc � 1Þ � Df ; ð38Þ

where Dc and Df correspond to the creep damage and the fatiguedamage respectively, and Dc ¼ nc=Nc , Df ¼ nf =Nf , Nc and Nf repre-sent to the creep life and the fatigue life, respectively, nc and nf arethe numbers of the creep and the fatigue load cycles. h1 and h2 arethe parameters obtained from the experimental results. Statisticalproperties of the random variables involved in the non-linearcreep–fatigue limit state/performance function are listed in Table3. For evaluating the failure probability PF , seven equally spacedsample points (n = 7) are deployed along each of the variable axisto form HDMR, FHDMR and HHDMR approximation. The referencepoint c is taken as mean values of the random variables. Table 4compares the results obtained by FORM, SORM, all variants ofHDMR approximation, line sampling [33], Wu [34] and directMCS. A sampling size NS ¼ 108 is considered in direct MCS to esti-mate the failure probability PF . The COV of PF corresponding to thissampling size is 0.0001 (computed using Eq. (36)). Table 4 also con-tains the computational effort in terms of number of function eval-uations, associated with each of the methods. Compared with thefailure probability obtained using direct MCS ðPF ¼ 1:417� 10�4Þ,FORM, HDMR, FHDMR and HHDMR underestimates the failureprobability 7.97% ðPF ¼ 1:304� 10�4Þ, 30.48% ðPF ¼ 9:850� 10�5Þ,43.44% ðPF ¼ 8:014� 10�5Þ and 0.92% ðPF ¼ 1:404� 10�4Þ, respec-tively, while SORM overestimates the failure probability by 1.55%ðPF ¼ 1:439� 10�4Þ. It can be observed that, as HHDMR approxima-tion considers both additive and multiplicative nature of the limitstate/performance function using optimum c, therefore, the approx-imation error in failure probability is least compared with othermethods. However, the present method using HDMR, FHDMR andHHDMR approximation needs only 37 function evaluations, while




FORM 1:304� 10�4 118SORM 1:439� 10�4 405Direct Monte Carlo

simulation1:417� 10�4 1� 108

Wu, 1994b 1:437� 10�4 5� 107

Line samplingc 1:462� 10�4 2000First-order HDMR 9:850� 10�5 37d

First-order FHDMR 8:014� 10�5 37d

HHDMR 1:404� 10�4 37d

a Total number of times the original performance function is calculated.b Ref. [34].c Ref. [33].d ðn� 1Þ � N þ 1 ¼ ð6� 1Þ � 6þ 1 ¼ 37.


FORM, SORM and direct MCS requires 118, 405 and 108 number oforiginal function evaluations respectively. In this aspect, presentmethod shows the accuracy and the efficiency (in terms of originalfunction evaluation), over FORM, SORM and direct MCS.

Similar to Example 1, the effect of number of sample pointsused for HDMR, FHDMR and HHDMR approximation on the reli-ability estimation is examined by carrying a similar analysis vary-ing n form 3 to 9. Compared with direct MCS, error in the estimatedfailure probability using different methods is tabulated in Table 5along with the failure probability values in parentheses.

7.3. Example 3: cantilever truss structure

A cantilever steel beam of 1.0 m, shown in Fig. 3, is consideredin this example to examine the accuracy and efficiency of the pro-posed method for the failure probability estimation. The cross-sec-tional area of the beam is (0.1 m � 0.01 m). The beam is subjectedto an in-plane moment at the free end and a concentrated load at0.4 m from the free end. The structure is modeled using ANSYS FE[35] software. The beam is discretized using ten four noded planestress elements (PLANE 182). The structure is assumed to havefailed if the square of the Von Mises stress at the support (at A inFig. 3) exceeds specified threshold Vmax. Therefore, the limitstate/performance function is defined as

gðxÞ ¼ Vmax � VðxÞ; ð39Þ



3 5

First-order HDMR �96.23% ð5:340� 10�6Þ �93.27% ð9:540First-order FHDMR �100.00% ð6:000 � 10�9Þ �99.48% ð7:400HHDMR �19.55% ð1:140� 10�4Þ +0.92% ð1:430�

Note that the number inside parentheses indicates corresponding failure probability.

1 2 3 4 5 6

A

Fig. 3. Cantilever steel b

where VðxÞ is the square of the Von Mises stress, expressed as aquadratic operator on the stress vector. In this example, loads x1

and x2, modulus of elasticity of the beam E and threshold quantityVmax are taken as random variables. The variations of E and Vmax areexpressed as E ¼ E0ð1þ ex3Þ and Vmax ¼ Vmax0ð1þ ex4Þ. Here, e issmall deterministic quantity representing the coefficient of varia-tion of the random variables and are taken to equal to 0.05,E0 ¼ 2� 105 N=m2 denotes the deterministic component of modu-lus of elasticity and Vmax0 ¼ 6:15� 109 N=m2 denotes the determin-istic component of threshold quantity. All variables are assumed tobe independent. The statistical properties of the random variablesare listed in Table 6. For HDMR/FHDMR/HHDMR approximation, se-ven equally spaced sample points (n = 7) along each of the variableaxis are deployed. Comparison of failure probability estimation ofthe function by different methods and associated computational ef-forts are listed in Table 7. A sampling size NS ¼ 104 is considered indirect MCS to evaluate the failure probability PF and the COV of PF

corresponding to this sampling size is 0.12 (computed using Eq.(36)). Compared with the benchmark solution ðPF ¼ 0:00590Þ,HHDMR approximation overestimates the failure probability byabout 0.68% ðPF ¼ 0:00594Þ. Similarly, compared with direct MCSresult, HDMR and FHDMR approximation overestimates the failureprobability by about 0.68% ðPF ¼ 0:00594Þ and 1.18�103%ðPF ¼ 0:07584Þ, respectively. However, all variant of HDMR approx-imation require 31 original FE analyses, while direct MCS requires104 number of original function evaluations, respectively. Thisshows the accuracy and the efficiency (in terms of original FE anal-ysis) of the present method, over direct MCS.

The effect of number of sample points used for HDMR/FHDMR/HHDMR approximation on the reliability estimation is examinedby carrying a similar analysis varying n form 3 to 9. Compared withdirect MCS, error in the estimated failure probability using differ-ent methods is tabulated in Table 8. It can be noticed that, HDMRand HHDMR approximation provides fairly accurate estimate offailure probability, but FHDMR method is far reaching from thebenchmark solution. Therefore, HHDMR approximation seemsmore versatile, as the nature of function approximation is capturedby the hybridity parameter c.

7.4. Example 4: dynamic problem of six degrees of freedom

This example considers a four-storey building excited by a sin-gle period sinusoidal pulse of ground motion, studied by Gavin andYau [36]. Fig. 4a shows the four-storey building with isolation sys-

7 9

� 10�6Þ �30.49% ð9:850� 10�5Þ �65.53% ð4:884� 10�5Þ� 10�7Þ �43.44% ð8:014� 10�5Þ �74.88% ð3:560� 10�5Þ10�4Þ �0.92% ð1:404� 10�4Þ �2.61% ð1:380� 10�4Þ

7 8 9 10

x2

x2

x1

eam for Example 3.

Table 6Statistical properties of random variables for Example 3.

Random variables Distribution Mean Standard deviation

x1 Gaussian 1.00 1.00x2 Gaussian 0.00 1.00x3 Lognormal 2.00 1.00x4 Lognormal 0.10 0.05




Direct Monte Carlosimulation

0.00590 1�104

First-order HDMR 0.00594 25b

First-order FHDMR 0.07584 25b

HHDMR 0.00594 25b

a Total number of times the original performance function is calculated.b ðn� 1Þ � N þ 1 ¼ ð7� 1Þ � 4þ 1 ¼ 25.

a

b

Fig. 4. Problem statement (Example 4): (a) base isolated structure with anequipment isolation system on the second floor and (b) acceleration history.


tems and Fig. 4b presents the acceleration history. The buildingcontains isolated equipment resting on the second floor. The mo-tion of the ground floor is resisted mainly by base isolation bear-ings [37] and if its displacement exceeds Dc (=0.50 m) then anadditional stiffness force contributes to the resistance. Mass, stiff-ness and damping coefficient mf , kf and cf , respectively at eachfloor are assumed to be same. There are two isolated masses, rep-resenting isolated, shock-sensitive equipment resting on the sec-ond floor. The larger mass m1 (=500 kg) is connected to the floorby a relatively flexible spring, k1 (=2500 N/m), and a damper, c1

(=350 N/m/s), representing the isolation system. The smaller massðm2 ¼ 100 kgÞ is connected to the larger mass by a relatively stiffspring k2 (=105 N/m), and a damper c2 (=200 N/m/s), representingthe equipment itself. All variables are assumed to be lognormaland statistically independent. The statistical properties of the ran-dom variables are listed in Table 9. The limit state/performancefunction is defined by the combination of three failure modes lead-ing to system failure and is the following form:

gðxÞ ¼ 12:50ð0:04�maxtjxfiðtÞ � xfi�1

ðtÞjÞi¼2;3;4 þ ð0:50

�maxtj€ugðtÞ þ €xm2 ðtÞjÞ þ 2:0ð0:25�max

tjxf2 ðtÞ

� xm1 ðtÞjÞ; ð40Þ

where xfi ðtÞ refers to the displacement of ith floor andðxfi ðtÞ � xfi�1

ðtÞÞ is the inter storey drift. €ugðtÞ is the ground accelera-tion and €xm2 ðtÞ is the acceleration smaller mass block. The displace-ment xm1 ðtÞ is of the larger mass block, and represents thedisplacement of the equipment isolation system. The limit state/performance function in Eq. (40) is the overall representation ofthree failure modes. The first term describes the damage to thestructural system due to excessive deformation. The second termrepresents the damage to equipment caused by excessive accelera-



3 5

First-order HDMR �99.86% ð8:32� 10�6Þ �27.12% (0.00750)First-order FHDMR +322.72% (0.02940) þ1:13� 103% (0.07HHDMR �9.83% (0.00532) �10.00% (0.00531)


tion. The last term represents the damage of the isolation system.The weighing factors, multiplied with each term in Eq. (40), aremainly to emphasize the equal contribution of the individual failuremodes to the overall failure of system. It is desirable that (a) interstorey drift is limited to 0.04 m, (b) the peak acceleration of theequipment is less than 0.5 m/s2, and (c) the displacement acrossthe equipment isolation system is less than 0.25 m. Eq. (40) signifiesoverall system failure, which does not necessarily occur whenabove mentioned one or two failure criteria satisfies. For evaluatingthe failure probability PF , HDMR/FHDMR/HHDMR approximation isconstructed by deploying five equally spaced sample points (n = 5)along each of the variable axis. The reference point is taken as mean

7 9

+0.68% (0.00594) +1.02% (0.00596)2840) þ1:18� 103% (0.075840) þ1:19� 103% (0.076140)

+0.68% (0.00594) +1.02% (0.00596)

Table 9Properties of the random variables for Example 4.

Random variable Units Description Mean COV

mf kg Floor mass 6� 103 0.10kf N/m Floor stiffness 3� 107 0.10cf N/m/s Floor damping coefficient 6� 104 0.20fy N Isolation yield force 2� 104 0.20dy m Isolation yield displacement 0.05 0.20kc N/m Isolation contact stiffness 3� 107 0.30T s Force period 1.0 0.20A m/m/s Force amplitude 1.0 0.50

2 3

6 5 4

1

360 in

360 in 360 in

1×105 lb 1×105 lb

Fig. 5. Ten bar truss structure for Example 5.


values of the random variables. Table 10 compares the results ob-tained by the present method using HDMR, FHDMR and HHDMRapproximation with FORM, SORM, and direct MCS and also presentsthe computational effort in terms of number of function evalua-tions, associated with each of the methods. The benchmark solutionof the failure probability is obtained by direct MCS with NS ¼ 105.The COV of PF corresponding to this sampling size is 0.0064 (com-puted using Eq. (36)). Compared with the benchmark solutionðPF ¼ 0:19599Þ, FORM and SORM overestimates the failure proba-bility by around 15.05% ðPF ¼ 0:22549Þ and 8.46% ðPF ¼ 0:21410Þ,respectively. HDMR and HHDMR overestimates the failure probabil-ity by about 1.88% ðPF ¼ 0:19968Þ and 0.11% ðPF ¼ 0:19621Þ, respec-tively, while FHDMR underestimates the failure probability by59.86% ðPF ¼ 0:31330Þ. However, HHDMR-based method requiresonly 33 function evaluations, while FORM, SORM and direct MCS re-quires 86, 356 and 106 number of original function evaluations,respectively. This shows the efficiency (in terms of original functioncalculations) and the accuracy of the present method using HHDMR,over FORM, SORM, direct MCS and also HDMR/FHDMR.

To examine the effect of number of sample points used forHDMR/FHDMR/HHDMR approximation, similar analyses are car-ried out by varying n form 3 to 9. Compared with direct MCS, errorin the estimated failure probability using different methods is tab-ulated in Table 11. The computational cost in terms of number offunction evaluations is increased from 17 to 65 for n = 3–9. It is ob-served that n = 5 provides the optimum number of function callswith acceptable accuracy in evaluating the failure probability PF

with the present method using HHDMR.

Table 10Estimation of failure probability for 6-DOF system.

Method Failure prob

FORMb 0.22549SORMb 0.21410Direct Monte Carlo simulation 0.19599First-order HDMR 0.19968First-order FHDMR 0.31330HHDMR 0.19620

a Total number of times the original performance function is calculated.b Ref. [18].c ðn� 1Þ � N þ 1 ¼ ð5� 1Þ � 8þ 1 ¼ 33.



3 5

First-order HDMR �4.78% (0.18662) +1.88% (0.19First-order FHDMR �71.56% (0.33625) �59.86% (0.HHDMR �2.95% (0.19021) +0.11% (0.19


7.5. Example 5: ten-bar truss structure

A 10-bar, linear-elastic, truss structure, shown in Fig. 5, consid-ered in this example to examine the accuracy and efficiency of theproposed reliability method. Young’s modulus of the material is107 psi. The cross-sectional area xi; i ¼ 1;2; . . . ;10 for each bar fol-lows normal distribution and has mean l ¼ 2:5 in:2 and standarddeviation r ¼ 0:5 in:2. Two concentrated forces of 105 lb are ap-plied at nodes 2 and 3, as shown in Fig. 5. Two limit state/perfor-mance functions are considered here to illustrate theperformance the proposed method. First limit state/performancefunction considers the displacement limit and second one consid-ers eigenvalue limit of the structure. According to the loading con-dition, the maximum displacement u3ðx1; x2; . . . ; x10Þ occurs at node3. The permissible displacement is limited to umax ¼ 18 in: and theeigenvalue must be greater than 9.30 (rad/sec)2. Hence, the limitstate/performance functions are defined as

Deflection limit state/performance function

gðxÞ ¼ umax � u3ðx1; x2; . . . ; x10Þ: ð41Þ

ability Number of function evaluationa

863561�105

33c

33c

33c

7 9

969) +3.19% (0.20226) +3.27% (0.20240)31330) �57.66% (0.30899) �52.23% (0.29835)620) �0.66% (0.19470) �0.10% (0.19579)


Eigenvalue limit state/performance function

gðxÞ ¼ 1:0� Fundamental eigenvalue9:30

P 0:0 ðFailureÞ: ð42Þ

7.5.1. Deflection limit state/performance functionThe structure is modeled using ADINA FE [38] software. For

evaluating the failure probability PF , HDMR/FHDMR/HHDMRapproximation is constructed by deploying five equally spacedsample points (n = 5) along each of the variable axis. The referencepoint c is taken as mean values of the random variables. Table 12compares the results obtained by the present method using HDMR,FHDMR and HHDMR with FORM, SORM, and direct MCS. A sam-pling size NS ¼ 106 is considered in direct MCS to estimate the fail-ure probability PF . The COV of PF corresponding to this samplingsize is 0.0025 (computed using Eq. (36)). Table 12 also containsthe computational effort in terms of number of function evalua-tions, associated with each of the methods. Compared with the fail-ure probability obtained using direct MCS ðPF ¼ 0:1394Þ, FORMunderestimates by 35.86% ðPF ¼ 0:0894Þ, while SORM, HDMR,FHDMR, and HHDMR overestimates the failure probability byabout 12.69% ðPF ¼ 0:1571Þ, 5.44% ðPF ¼ 0:1473Þ, 8.24%ðPF ¼ 0:1508Þ and 6.38% ðPF ¼ 0:1483Þ, respectively. All variantsof HDMR approximation includes all the higher order univariateterms in Taylor series expansion of the original function in Eq.(41), and hence the failure probability estimate by the HDMRapproximations, is more accurate than that obtained usingFORM/SORM. However, the present method using HDMR/FHDMR/HHDMR approximation needs only 41 function evalua-tions to achieve around 95% accuracy in the failure probability esti-

Table 12Estimation of failure probability for deflection limit state (Example 5).

Method Failure prob

FORMb 0.0894SORMb 0.1571Direct Monte Carlo simulation 0.1394First-order HDMR 0.1471First-order FHDMR 0.1509HHDMR 0.1483

a Total number of times the original performance function is calculated.b Ref. [18]c ðn� 1Þ � N þ 1 ¼ ð5� 1Þ � 10þ 1 ¼ 41.

Table 13Estimation of error using different methods for deflection limit state (Example 5).


3 5

First-order HDMR +3.95% (0.14491) +5.52% (0.1First-order FHDMR +4.71% (0.14596) +8.24% (0.1HHDMR +0.22% (0.13970) +6.38% (0.1


Table 14Estimation of failure probability for eigenvalue limit state (Example 5).

Method Failure prob

Direct Monte Carlo simulation 0.10627First-order HDMR 0.10010First-order FHDMR 0.10217HHDMR 0.10500

a Total number of times the original performance function is calculated.b ðn� 1Þ � N þ 1 ¼ ð5� 1Þ � 10þ 1 ¼ 41.

mate, while FORM, SORM and direct MCS requires 190, 577 and106 number of original function evaluations, respectively.

To illustrate the effect of number of sample points used forHDMR/FHDMR/HHDMR approximation on the reliability estima-tion, a similar analysis is carried out by varying n form 3 to 9. Com-pared with direct MCS, error in the estimated failure probabilityusing different methods is tabulated in Table 13.

7.5.2. Eigenvalue limit state/performance functionIn this case, for evaluating the failure probability PF , five equally

spaced sample points (n = 5) are deployed along each of the vari-able axis to form approximation using HDMR, FHDMR andHHDMR. The reference point c is taken as mean values of the ran-dom variables. Table 14 compares the results obtained by HDMR,FHDMR, HHDMR, and direct MCS. A sampling size NS ¼ 105 is con-sidered in direct MCS to estimate the failure probability PF . TheCOV of PF corresponding to this sampling size is 0.0025 (computedusing Eq. (36)). Table 14 also contains the computational effort interms of number of function evaluations, associated with each ofthe methods. Compared with the failure probability obtained usingdirect MCS ðPF ¼ 0:10627Þ, HHDMR approximation underestimatesthe failure probability by 4.58% ðPF ¼ 0:10500Þ, while HDMR andFHDMR-based approximation underestimates the failure probabil-ity by 5.81% ðPF ¼ 0:10010Þ and 3.56% ðPF ¼ 0:10217Þ, respectively.However, all variants of approximation method needs only 41function evaluations, while direct MCS requires 105 number of ori-ginal function evaluations, respectively.

The effect of number of sample points used for HDMR, FHDMRand HHDMR approximation on the reliability estimation is exam-


865771� 106

41c

41c

41c

7 9

4710) +8.06% (0.15064) +8.29% (0.15095)5088) +11.15% (0.15495) +11.41% (0.15530)4830) +7.03% (0.14920) +7.03% (0.14920)


1� 105

41b

41b

41b

Table 15Estimation of error using different methods for eigenvalue limit state (Example 5).


3 5 7 9

First-order HDMR �40.88% (0.14971) �5.81% (0.10010) �8.37% (0.09737) �8.54% (0.09719)First-order FHDMR +37.91% (0.14656) �3.56% (0.10217) �6.21% (0.09968) �6.33% (0.09954)HHDMR �41.17% (0.15002) �1.95% (0.10500) �4.58% (0.10140) �4.77% (0.10120)


Table 16Statistical properties of random variable for an edge-cracked plate.

Random variable Mean Standard deviation Distribution type

a=W 0.5 0.2309 Uniforma

s1 Variableb 0.1 NormalKIC 200 0.1 Lognormal

a Uniformly distributed over (0.3–0.7).b Varies from 2.6 to 3.1.


ined by carrying a similar analysis varying n form 3 to 9. Comparedwith direct MCS, error in the estimated failure probability usingdifferent methods is tabulated in Table 15. However the computa-tional cost in terms of number of function evaluations is increasedfrom 21 to 81 for n = 3 to 9.

7.6. Example 6: fracture-mechanics problem

This example involves an isotropic, homogeneous, edge-crackedplate, presented to illustrate the performance of proposed methodon mixed-mode probabilistic fracture-mechanics problem. Asshown in Fig. 6a, a plate of length L ¼ 16 units, width W ¼ 7 unitsis fixed at the bottom and subjected to a far-field and a shear stresss1 applied at the top. The elastic modulus and Poisson’s ratio are 1unit and 0.25, respectively. A plane strain condition is assumed.The statistical property of the random input x ¼ fa=W; s1;KICgT

is defined in Table 16. All the random variables are statisticallyuncorrelated.

Due to the far-field shear stress s1, the plate is subjected tomixed-mode deformation involving fracture modes I and II [39].The mixed-mode stress-intensity factors KIðxÞ and KIIðxÞ were cal-culated using an interaction integral method [40]. The plate wasanalyzed using the FE method involving involved 2711 nodes,832 8-noded quadrilateral elements, and 48 focused quarter-point6-noded triangular elements at the crack-tip, as shown in Fig. 6b.

The failure criterion is based on a mixed-mode fracture initia-tion using the maximum tangential stress theory [40], which leadsto the performance function

W

τ

a

L/2

L /2

Crack

a b

Fig. 6. An edge-cracked plate subject to mixed-mode deformation (Example 6): (a)geometry and loads and (b) finite element discretization.

gðxÞ ¼ KIC � KIðxÞ cos2 HðxÞ2� 3

2KIIðxÞ sin HðxÞ

� �cos

HðxÞ2

; ð43Þ

where KIC is statistically distributed fracture toughness, typicallymeasured from small-scale fracture experiments under mode Iand plane strain conditions, and HcðxÞ is the direction of crack prop-agation, given by

HðxÞ ¼2 tan�1 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ8 KIIðxÞ=KIðxÞ½ �2p

4KIIðxÞ=KIðxÞ

; if KIIðxÞ > 0;

2 tan�1 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ8 KIIðxÞ=KIðxÞ½ �2p

4KIIðxÞ=KIðxÞ

; if KIIðxÞ < 0:

0BBB@ ð44Þ

For evaluating the failure probability PF , HDMR/FHDMR/HHDMR approximation is constructed by deploying five equallyspaced sample points (n = 5) along each of the variable axis. Thereference point c is taken as mean values of the random variables.Estimation of failure probability, obtained using HDMR, FHDMR,HHDMR-based method and direct MCS, are compared in Fig. 7and plotted as a function of hs1i, where h�i is the expectation oper-ator. A sampling size NS ¼ 5� 104 is considered in direct MCS toevaluate the failure probability PF .

8. Summary and conclusions

In reliability assessment of real life problems, the limit state/per-formance functions are most often specified implicitly through a FE

2.6 2.7 2.8 2.9 3.0 3.110 -4

10 -3

10 -2

10 -1

Prob

abili

ty o

f fa

ilure

, PF

τ ∞⟨ ⟩

MCS

First-order HDMR

HHDMR

First-order FHDMR

Fig. 7. Variation of probability of failure, PF .


code. This paper presents a novel limit state/performance functionapproximation based on HHDMR concept for subsequent reliabilityanalysis of systems subject to random loads, material properties,and geometry in an efficient manner. HDMR approximation workswell when the sought multivariate response function has an addi-tive nature. But, for multiplicative nature of the multivariateresponse function, HDMR-based method becomes expensive to ob-tain accurate result. At this juncture, the factorized form of HDMRcan be used for approximating the implicit multivariate limitstate/performance function. As an alternative, this paper presentsa new HHDMR-based approximation to handle a multivariate limitstate/performance function that has neither additive nor multipli-cative nature but rather an intermediate nature. This method usesboth HDMR and FHDMR expansions. Numerical examples clearlydemonstrated the accuracy and computational efficiency of theproposed HHDMR-based approximation.

A parametric study is conducted with respect to the number ofsample points n used in the proposed approximation and its effecton the estimated failure probability is investigated. An optimumnumber of sample points n must be chosen in HHDMR approxima-tion. Very small number of sample points n should be avoided asHHDMR approximation may not capture the non-linearity outsidethe domain of sample points and it affects the estimated failureprobability significantly. However as the number of sample pointsis increased even though HHDMR approximation is able to capturethe sudden peaks and falls more accurately in the domain, it mayresult in oscillations in the tail regions which affects the estimatedfailure probability. Similar observation on HDMR approximation isreported in authors previous work [18]. It can be observed from thereported results in this paper, that n = 5 or 7 works well for allproblems.

Acknowledgement

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

References

[1] O. Ditlevsen, H.O. Madsen, Structural Reliability Methods, Wiley, Chichester,1996.

[2] H.O. Madsen, S. Krenk, N.C. Lind, Methods of Structural Safety, Prentice-Hall,Englewood Cliffs, NJ, 1986.

[3] R. Rackwitz, Reliability analysis – a review and some perspectives, Struct.Safety 23 (4) (2001) 365–395.

[4] R.V. Grandhi, L. Wang, Higher-order failure probability calculation usingnonlinear approximations, Comput. Methods Appl. Mech. Engrg. 168 (1999)185–206.

[5] K. Breitung, Asymptotic approximations for multinormal integrals, J. Engrg.Mech., ASCE 110 (3) (1984) 357–366.

[6] P.L. Liu, A. Der Kiureghian, Finite element reliability of geometrically nonlinearuncertain structures, J. Engrg. Mech., ASCE 117 (8) (1991) 1806–1825.

[7] N. Impollonia, A. Sofi, A response surface approach for the static analysis ofstochastic structures with geometrical nonlinearities, Comput. Methods Appl.Mech. Engrg. 192 (37–38) (2003) 4109–4129.

[8] R.Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley, New York,1981.

[9] R.E. Melchers, Importance sampling in structural systems, Struct. Safety 6 (1)(1989) 3–10.

[10] S.K. Au, J.L. Beck, Estimation of small failure probabilities in high dimensionsby subset simulation, Probabilistic Engrg. Mech. 16 (4) (2001) 263–277.

[11] G.I. Schuëller, H.W. Pradlwarter, P.S. Koutsourelakis, A critical appraisal ofreliability estimation procedures for high dimensions, Probabilistic Engrg.Mech. 19 (4) (2004) 463–474.

[12] J.E. Hurtado, An examination of methods for approximating implicit limit statefunctions from the viewpoint of statistical learning theory, Struct. Safety 26 (3)(2004) 271–293.

[13] J.E. Hurtado, Filtered importance sampling with support vector margin: Apowerful method for structural reliability analysis, Struct. Safety 29 (1) (2007)2–15.

[14] A. Basudhar, S. Missoum, A.H. Sanchez, Limit state function identification usingsupport vector machines for discontinuous responses and disjoint failuredomains, Probabilistic Engrg. Mech. 23 (1) (2008) 1–11.

[15] H.M. Gomes, A.M. Awruch, Comparison of response surface and neuralnetwork with other methods for structural reliability analysis, Struct. Safety26 (1) (2004) 49–67.

[16] J. Deng, Structural reliability analysis for implicit performance function usingradial basis function network, Int. J. Solids Struct. 43 (11–12) (2006) 3255–3291.

[17] R. Chowdhury, B.N. Rao, A.M. Prasad, High dimensional model representationfor piece wise continuous function approximation, Commun. Numer. MethodsEngrg. (2007), doi:10.1002/cnm.1053.

[18] R. Chowdhury, B.N. Rao, A.M. Prasad, High dimensional model representationfor structural reliability analysis, Commun. Numer. Methods Engrg. (2008),doi:10.1002/cnm.1118.

[19] R. Chowdhury, B.N. Rao, Assessment of high dimensional model representationtechniques for reliability analysis, Probabilistic Engrg. Mech. (2008),doi:10.1016/j.probengmech.2008.02.001.

[20] H. Rabitz, O.F. Alis, J. Shorter, K. Shim, Efficient input–output modelrepresentations, Comput. Phys. Commun. (1–2) (1999) 11–20.

[21] H. Rabitz, O.F. Alis, General foundations of high dimensional modelrepresentations, J. Math. Chem. 25 (2–3) (1999) 197–233.

[22] O.F. Alis, H. Rabitz, Efficient implementation of high dimensional modelrepresentations, J. Math. Chem. (2) (2001) 127–142.

[23] G. Li, C. Rosenthal, H. Rabitz, High dimensional model representations, J. Phys.Chem. A 105 (2001) 7765–7777.

[24] G. Li, S.W. Wang, H. Rabitz, High dimensional model representations generatedfrom low dimensional data samples-I. mp-Cut-HDMR, J. Math. Chem. 30 (1)(2001) 1–30.

[25] S.W. Wang, H. Levy II, G. Li, H. Rabitz, Fully equivalent operational models foratmospheric chemical kinetics within global chemistry-transport models, J.Geophys. Res. 104 (D23) (1999) 30417–30426.

[26] G. Li, S.W. Wang, H. Rabitz, S. Wang, P. Jaffé, Global uncertainty assessments byhigh dimensional model representations (HDMR), Chem. Engrg. Sci. 57 (21)(2002) 4445–4460.

[27] I.M. Sobol, Theorems and examples on high dimensional modelrepresentations, Reliab. Engrg. Syst. Safety 79 (2) (2003) 187–193.

[28] I.M. Sobol, Sensitivity estimates for nonlinear mathematical models, Math.Modeling Comput. Exp. 1 (1993) 407–414.

[29] M.A. Tunga, M. Demiralp, A factorized high dimensional model representationon the partitioned random discrete data, Appl. Numer. Anal. Comput. Math. 1(1) (2004) 231–241.

[30] M.A. Tunga, M. Demiralp, A factorized high dimensional model representationon the nods of a finite hyperprismatic regular grid, Appl. Math. Comput. 164(2005) 865–883.

[31] M.A. Tunga, M. Demiralp, Hybrid high dimensional model representation(HHDMR) on the partitioned data, J. Comput. Appl. Math. 185 (1) (2006) 107–132.

[32] P. Lancaster, K. Salkauskas, Curve and Surface Fitting: An Introduction,Academic Press, London, 1986.

[33] Z. Lu, S. Song, Z. Yue, J. Wang, Reliability sensitivity method by line sampling,Struct. Safety (2007), doi:10.1016/j.strusafe.2007.10.001.

[34] Y.T. Wu, Computational methods for efficient structural reliability andreliability sensitivity analysis, AIAA J. 32 (8) (1994) 1717–1723.

[35] ANSYS Inc., ANSYS Release 9.0 Documentation, 2004.[36] H.P. Gavin, S.C. Yau, High-order limit state functions in the response surface

method for structural reliability analysis, Struct. Safety 30 (2) (2007) 162–179.

[37] Y.K. Wen, Method for random vibration of hysteretic systems, J. Engrg. Mech.,ASCE 102 (EM2) (1976) 249–263.

[38] ADINA R&D Inc., ADINA theory and modeling guide. Report ARD 05-6, 2005.[39] T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, CRC Press

Inc., Boca Raton, Florida, 1995.[40] J.F. Yau, S.S. Wang, H.T. Corten, A mixed-mode crack analysis of isotropic

solids using conservation laws of elasticity, J. Appl. Mech. 47 (1980) 335–341.

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Hybrid High Dimensional Model Representation for reliability analysis