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Engineering ComputationsStructural reliability and stochastic finite element methods: State-of-the-artreview and evidence-based comparisonMuhannad Aldosary, Jinsheng Wang, Chenfeng Li,
Article information:To cite this document:Muhannad Aldosary, Jinsheng Wang, Chenfeng Li, (2018) "Structural reliability and stochasticfinite element methods: State-of-the-art review and evidence-based comparison", EngineeringComputations, Vol. 35 Issue: 6, pp.2165-2214, https://doi.org/10.1108/EC-04-2018-0157Permanent link to this document:https://doi.org/10.1108/EC-04-2018-0157
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Structural reliability andstochastic finite element methods
State-of-the-art review andevidence-based comparison
Muhannad Aldosary, Jinsheng Wang and Chenfeng LiZienkiewicz Centre for Computational Engineering and Energy Safety Research
Institute, College of Engineering, Swansea University, Swansea, UK
AbstractPurpose – This paper aims to provide a comprehensive review of uncertainty quantification methodssupported by evidence-based comparison studies. Uncertainties are widely encountered in engineeringpractice, arising from such diverse sources as heterogeneity of materials, variability in measurement, lack ofdata and ambiguity in knowledge. Academia and industries have long been researching for uncertaintyquantification (UQ) methods to quantitatively account for the effects of various input uncertainties on thesystem response. Despite the rich literature of relevant research, UQ is not an easy subject for noviceresearchers/practitioners, where many different methods and techniques coexist with inconsistent input/output requirements and analysis schemes.
Design/methodology/approach – This confusing status significantly hampers the researchprogress and practical application of UQ methods in engineering. In the context of engineering analysis,the research efforts of UQ are most focused in two largely separate research fields: structural reliabilityanalysis (SRA) and stochastic finite element method (SFEM). This paper provides a state-of-the-artreview of SRA and SFEM, covering both technology and application aspects. Moreover, unlike standardsurvey papers that focus primarily on description and explanation, a thorough and rigorouscomparative study is performed to test all UQ methods reviewed in the paper on a common set ofreprehensive examples.
Findings – Over 20 uncertainty quantification methods in the fields of structural reliability analysisand stochastic finite element methods are reviewed and rigorously tested on carefully designednumerical examples. They include FORM/SORM, importance sampling, subset simulation, responsesurface method, surrogate methods, polynomial chaos expansion, perturbation method, stochasticcollocation method, etc. The review and comparison tests comment and conclude not only on accuracyand efficiency of each method but also their applicability in different types of uncertainty propagationproblems.
Originality/value – The research fields of structural reliability analysis and stochastic finite elementmethods have largely been developed separately, although both tackle uncertainty quantification inengineering problems. For the first time, all major uncertainty quantification methods in both fields arereviewed and rigorously tested on a common set of examples. Critical opinions and concluding remarks aredrawn from the rigorous comparative study, providing objective evidence-based information for furtherresearch and practical applications.
Keyword Structural reliability, Stochastic finite element method, Uncertainty quantification,Uncertainty propagation
Paper type Literature review
The authors would like to thank the support from the College of Engineering in University of Anbar,the Sêr Cymru National Research Network in Advanced Engineering and Materials, the ChinaScholarship Council and the Royal Academy of Engineering.
Structuralreliability
2165
Received 2 April 2018Revised 5May 2018
Accepted 5May 2018
Engineering ComputationsVol. 35 No. 6, 2018
pp. 2165-2214© EmeraldPublishingLimited
0264-4401DOI 10.1108/EC-04-2018-0157
The current issue and full text archive of this journal is available on Emerald Insight at:www.emeraldinsight.com/0264-4401.htm
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1. IntroductionThe analysis and design of complex structures or engineering systems rely heavily onpredictions from numerical models (e.g. finite element analysis), while the accuracy of thenumerical results depends on the proximity between the digital representation and the real-world system. The feasibility, applicability and confidence level of numerical models arefrequently challenged by the presence of various evitable uncertainties in engineeringstructures and systems. This has motivated the research of uncertainty quantification (UQ) inengineering analysis, which has historically been pursued in two largely separate researchfields: structural reliability analysis (SRA) and stochastic finite elementmethod (SFEM).
SRA and the associated risk assessment have long been established in the civilengineering community, where the earliest research work can be dated back to over half acentury ago. Over the years, SRA has gradually grown from an academic research topic to amore applied field with emphasizing applications in such critical structures as dams,tunnels, nuclear stations and offshore structures. It aims to provide a rational framework toaddress uncertainties in structural analysis such that design can be more objective and lessdependent on ideal assumptions. Despite the continuous progress in SRA, the estimation ofstructural reliability remains a challenging problem for structural engineers. The SRAtheory is formulated around a core concept, namely, the probability of failure:
Pf ¼ Prob g xð Þ# 0½ � ¼ð
g xð Þ# 0
fX xð Þ dx (1)
where fX(x) is the joint probability density function (PDF) of random vector X and g(x) is thelimit state function (also known as the performance function) with g(x)# 0 denoting the failuredomain and g(x) > 0 the safe domain. The definition of Pf is simple, but its exact evaluationthrough direct integration is often intractable for practical problems, where the dimension ofthe integral is usually high and the limit state surface has complicated shape and topology.Moreover, direct integration is completely unfeasible when the joint PDF fX(x) is unknown.Difficulties in computing the failure probability Pf has led to the development of various SRAmethods, such as first- and second-order reliability methods, Monte Carlo simulation (MCS) andimportant sampling.
SFEM is another major UQ research field in engineering analysis. The development ofthe standard finite element method (FEM) started from the 1950s, and in the past fewdecades, it has become the dominant engineering simulation tool for analyzing materials,structures and subsystems in almost all engineering sectors, especially in automotive,aerospace, manufacturing and civil engineering industries. As a deterministic analysis tool,all input variables in the standard FEMmust be uniquely specified and the output solutionsare also uniquely resolved in the form of constant values. To account for the variousuncertainties encountered in engineering practice, researchers have been trying to extendthe standard FEM into SFEM by incorporating random variables into the mathematical andcomputational formulations. The simulation capacity of SFEM has been growing steadilyover the past decade as a result of continuous algorithm development and growth incomputing power. In the wider context, SFEM aims to provide numerical solutions tostochastic partial differential equations, and it has been applied in very diverse engineeringtopics including solid materials, structures, fluid flow, acoustics and heat transfer problems.
The aim of this study is to present a comprehensive and critical review of all existingSRA and SFEM approaches, with a particular emphasis on the potential of practicalapplications. The advantages and drawbacks of individual methods are elaborated through
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a rigorous comparative study, which provides objective evidence-based information for thefeasibility and performance of different methods with respect to specific applications. Therest of the paper is organized as follows: the methods for transformation and discretizationof random variables and fields are introduced in Section 2; technical summaries of variousSRA and SFEM approaches are presented in Section 3 and Section 4, respectively; based ona set of carefully designed representative examples, a rigorous comparative study ispresented in Section 5, covering all SRA and SFEM approaches reviewed in the paper; andSection 6 draws the concluding remarks from the unbiased comparative study.
2. Transformation and discretization techniques for random variables andrandom fieldsUncertainties in engineering analysis are often associated with material properties (e.g. massdensity, elasticity, permeability and damping factors), geometry and boundary conditions of theconcerned structure, and they are commonlymodeled as discrete random variables or continuousrandom fields. For both SRA and SFEM, it is often essential and beneficial to performtransformation between different types of random variables and establish discretization ofrandom fields. A random field f(x, u ) can be defined as a curve in the probability space (H,X, P),containing a collection of random variables indexed by a parameter x [X, whereX is a subset ofRd defined by the system geometry. For a given pointx0, f(x0, u ) is a randomvariable, while for agiven event u 0, f(x, u 0) is a realization of the random field (Sudret et al., 2003; Sudret and DerKiureghian, 2000). The random field f(x, u ) can be univariate or multivariate depending onwhether the quantity f(x) attached to the point x is a random variable or a random vector. Also, f(x, u ) can be one- or multi-dimensional depending on the dimensionality of X. Differentmathematical and computational methods have been developed for the transformation anddiscretization task, and a brief summary is provided below.
2.1 Radom variable transformationRandom variables quantified directly from practical observations can be of variousprobabilistic distributions, whose representation and statistical computation can be verydifferent. To ease the task in SRA and SFEM, it is often beneficial to transform the originalrandom variables from the physical space to the standard normal space for mathematical andcomputational processing. A number of transformation techniques have been developed for thetask, among which Rosenblatt transformation and Nataf transformation are two most widelyused approaches (Hohenbichler and Rackwitz, 1981; Kiureghian and Liu, 1986). LetX denote arandom vector defined in the physical space; the transformation ofX is defined as:
U ¼ T Xð Þ (2)
where U is a standard Gaussian random vector with zero mean and unit covariance matrix.If the joint cumulative distribution function (CDF) ofX is known, Rosenblatt transformationcan be applied (Hohenbichler and Rackwitz, 1981):
T1 ¼ U�1 FX1 x1ð Þ� �T2 ¼ U�1½FX2ðx2jx1Þ�
� � �TM ¼ U�1½FXM ðxM jx1; . . . ; xM�1Þ�
(3)
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where FXiðxijx1; . . . ; xi�1Þ is the conditional distribution function and U is the standardnormal CDF. However, in practical problems, the probabilistic information of randomvariables is usually limited to the marginal distribution and the correlation coefficients. Forthese cases, Rosenblatt transformation is unfeasible and Nataf transformation is suggestedby Kiureghian and Liu (1986). Specifically, the random vector Z is obtained by using themarginal PDF ofXi:
Zi ¼ U�1 FXi xið Þ� �; i ¼ 1; . . . ;M (4)
where Z is a Gaussian random vector with the correlation matrix R0 and FXi xið Þ is themarginal distribution function. The joint PDF of X is determined by using the inversetransformation of equation (4) as:
fX xð Þ ¼ fX1 x1ð Þ . . . fXn xnð Þ wn z;R0ð Þw z1ð Þ . . . w znð Þ (5)
The unknown correlation matrix R0 can be obtained by solving the following implicitequation:
r ij ¼ððþ1
�1
xi � m i
s i
� � xj � m j
s j
� �w 2 zi; zj; r 0;ijð Þdzidzj (6)
for which semi-empirical formulations have been adopted to simplify the calculation(Kiureghian and Liu, 1986). After defining the correlation matrix R0 of the vector Z, thetransformation to the standard space can be expressed as:
T Xð Þ ¼ L�10 :Z ¼ L�1
0 : f U�1 FX1 x1ð Þ� �; . . . ; U�1 FXn xnð Þ� �gT (7)
where the transformation matrix L0 is determined by the Cholesky decomposition of R0, i.e.R0 ¼ L0 :L
T0 .
2.2 Radom field discretizationIntroduced by Ghanem and Spanos (1991), Spanos and Ghanem (1989) and Ghanem(1998), the Karhunen–Loève (K-L) expansion is arguably the most widely used tool forrandom field discretization, especially for uncertainties associated with inputparameters. The K-L expansion represents the random field as a linear combination oforthogonal basis terms, which are determined by solving the Fredholm integralequation: ð
X
R x; x0� �w i x
0ð Þ dXx0 ¼ l i w i xð Þ 8 i ¼ 1; . . . : (8)
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where the kernel autocovariance function R(x, x0) is bounded, symmetric and positivedefinite. Thus, all eigenvalues l i are real and positive, and the deterministic functions w i(x)form a complete orthogonal basis of Hilbert spaces q2(X). The K-L expansion of the randomfield f(x, u ) is:
f x; uð Þ ¼ m xð Þ þX1i¼1
ffiffiffiffiffil i
pj i uð Þ f i xð Þ (9)
where {j i(u ), i = 1,. . .} are a set of uncorrelated random variables, and they becomeindependent Gaussian random variables if f(x, u ) is a Gaussian field. In practice, only afinite number of terms in the K-L expansion are used, where the truncation is performedafter sorting the eigenvalues l i in a descending order.
Analytical solutions to the K-L expansion are available for simple geometries and specialforms of the autocovariance function. However, for more general cases, numerical solutionsto equation (8) are required to obtain the corresponding K-L expansion. These numericalsolutions usually have a high computational cost, while the obtained approximations arerarely optimal. The effectiveness of the K-L expansion is affected by the accuracy of theeigenpair l i and w i(x) (Stefanou, 2009), for which several effective solution methods to theFredholm integral equation (8) have been reported in the literature (Phoon et al., 2002;Schwab and Todor, 2006).
To overcome the computational difficulties in the K-L expansion of random fields Li et al.(2008, 2006) and Feng et al. (2014, 2016) proposed the Fourier–Karhunen–Loève (F-K-L)expansion, which is based on the spectral representation theory of wide-sense stationarystochastic fields and the standard dimensionality reduction technology of principalcomponent analysis. The F-K-L expansion is independent of the detailed shape of therandom structure and is a completely meshfree scheme, thus avoiding the meshconvergence and mesh sensitivity problems often encountered in mesh-based discretizationschemes. In all our experiments where the discretization of a continuous random field isrequired, the F-K-L expansion has exhibited a much higher accuracy and efficiency overmesh-based discretization schemes. The superior accuracy and efficiency of the F-K-Lexpansion are due to the harmonic essence of wide-sense stationary stochastic fields.
3. Structural reliability analysisAs outlined in Section 1, the SRA theory is centered on the concept of failure probability Pf,which is essentially a multi-dimensional integral over the failure domain determined by thelimit state function. The computational cost used to calculate the probability of failure canbe prohibitive, especially for evaluating small Pf values, which is almost always the case inthe practical world. Various approximation methods have been developed to evaluate thefailure probability, forming a rich literature of SRA. These SRA methods can be broadlyclassified into three categories: Taylor series-based approaches such as the first-orderreliability analysis method (FORM) and second-order reliability analysis method (SORM),simulation-based methods such as MCS and its variants and surrogate methods such as theresponse surface method (RSM) and the Kriging meta-model. A brief summary of variousSRA approaches are presented in the following subsections.
3.1 First-order reliability analysis method and second-order reliability analysis methodAs one of the oldest SRA method, FORM applies the first-order Taylor expansion tolinearize the limit state surface in the standard normal space at the so-called most probable
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point (MPP) U*, which has the highest likelihood among all points in the failure region(Hohenbichler and Rackwitz, 1981; Hasofer and Lind, 1974; Rackwitz and Fiessler, 1978):
g Xð Þ � g Uð Þ ¼ g U *ð Þ þ rg U *ð ÞT U � U *ð Þ (10)
where $g(U*) is the gradient vector at U*. The reliability index b is defined as the shortestdistance from the origin to the failure surface, and the failure probability is approximated byPf = U(–b ). Following the definition of the reliability index, MPP can be obtained from aconstrained optimization problem in the standard normal space as:
U * ¼ argmin Q Uð Þ ¼ 12kU k2
g Uð Þ# 0
( )(11)
A number of optimization algorithms are available to solve this problem, and interestedreaders can refer to Rackwitz and Fiessler (1978); Lemaire (2009); and Zhang and DerKiureghian (1995) for more information.
To further improve the accuracy of FORM, a second-order Taylor expansion at MPP canbe adopted to approximate the limit state surface, and this leads to the formulation of SORM(Breitung, 1984; Kiureghian and Destefano, 1991; Tvedt, 1988; Koyluoglu and Nielsen, 1994;Zhao and Ono, 1999). Specifically, the limit state surface is approximated by a quadraticsurface in the standard normal space:
g Uð Þ ¼ aTU * � aTU þ 12
U � U *ð ÞTB U � U *ð Þ (12)
where a ¼ rg U *ð Þjrg U *ð Þj, B ¼ r2g U *ð Þ
jrg U *ð Þj and $2g(U*) is the Hessian evaluated at the MPP. Several
equations are available to obtain the failure probability in the context of SORM, amongwhich an asymptotic equation proposed by Breitung (1984) is widely used:
Pf ¼ U �bFORMð ÞYN�1
j¼1
1þ bFORM kj� ��1=2 (13)
where kj, j = 1, . . ., N – 1 are principal curvatures at the MPP. Attempts to improve theefficiency and accuracy of SORM are still going on (Adhikari, 2004; Zhang and Du, 2010; Leeet al., 2012; Mansour and Olsson, 2014; Lu et al., 2017).
3.2 Simulation based methodsIf the limit state function is highly nonlinear, large errors may be introduced into the failureprobability calculation when using FORM or SORM, as the limit state surface isapproximated only with lower-order Taylor expansions. To achieve better accuracy in thesecases, an alternative approach is the simulation-based SRA methods, which generatesamples of the limit state function with finite element simulation and directly estimate thefailure probability through numerical integration.
3.2.1 Monte Carlo simulation. The MCS method is arguably the most robust andversatile SRA approach, albeit at a high computational cost (Feng et al., 2010). The failureprobability in MCS is defined as the ratio of the number of samples in the failure domain tothe total number of samples:
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P f ¼Nf
N¼ 1
N
XNi¼1
I gi xð Þ# 0½ � (14)
where the sampling points x are generated according the PDF fX(x), Nf is the number ofsampling points such that g(x)# 0, N is the total number of sampling points, gi(x) is the ithrealization of the limit state surface and I[g(x)] is an indicator function taking values of unityif g(x)# 0 and zero otherwise.
With a convergence rate of O 1ffiffiffiN
p �
, the computational cost of MCS can be prohibitivelyhigh for complex problems, especially when time-consuming numerical codes such as finiteelement analysis are involved in sample generation. To address this problem, manyvariance reduction techniques such as importance sampling (IS), directional sampling (DS),Latin hypercube sampling (LHS), line sampling (LS) and subset simulation (SS) have beenproposed to conduct failure probability estimation with a reduced computational cost. Thesemethods are briefly recapped in the following subsections, while more details are referred tothe original papers and summary textbooks where appropriate.
3.2.2 Importance sampling. The key idea of IS to distribute the sampling points in theregion of the greatest importance such that the failure probability evaluation can beaccelerated. Specifically, equation (1) is reformulated as:
Pf ¼ð
g xð Þ# 0
fX xð ÞhX xð Þ hX xð Þdx (15)
where hX(x) is the IS density function. Hence, the probability of failure can be approximatedas:
Pf ¼ 1N
XNj¼1
I xjð ÞfX xjð ÞhX xjð Þ (16)
where the sampling points xj, j = 1, . . ., N are generated according to the distribution hXinstead of fX. The effectiveness of IS depends on the selection of an appropriate hX(x) suchthat the probabilistic sampling in equation (16) can be prioritized for the region of thegreatest importance, therefore achieving a better convergence rate. Although there is nogeneral conclusion on the choice of hX(x) (Schueller and Stix, 1987; Engelund and Rackwitz,1993; Bucher and Macke, 2002), it has been argued that the MPP and its neighborhood canbe a good option for the region of the greatest importance unless additional information onthe limit state function and the failure probability are available. The MPP can be identifiedby FORM. However, unlike FORM or SORM, the IS estimation is not sensitive to the exactposition of MPP; therefore, it does not need to be determined up to a high accuracy.
It is well known that the MPP and its neighborhood do not always describe the most“important” region of the failure domain, especially in high dimensional space. Analternative approach is to place the sampling points inside the failure domain to create theoptimal IS density function. In earlier attempts, a rejection sampling scheme following theoriginal PDF fX(x) was adopted (Ang et al., 1992), but this is extremely inefficient in caseswhere the failure probability is small. To improve the efficiency, a Markov chain metropolisalgorithm was introduced, and points xi with a distribution of hopt can be obtained asintermediate states of an irreducible Markov chain (Kass et al., 1998; Au and Beck, 1999).
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The initial point x0 can be selected either by rejection sampling or using engineeringassessment. Subsequently, a kernel sampling density estimator is created usingNv points xiobtained by theMarkov chain:
k xð Þ ¼ 1Nv
XNv
i¼1
1
wkið ÞMK
x � xi
wki
� �(17)
wherew is the windowwidth, ki is the local bandwidth factor,M is dimensional space andKis the kernel PDF, often set as the multivariate normal. The density k(x) is used as the ISdensity to estimate Pf based on equation (16). Several methods have been developed toadjust w and ki so that the estimate obtained equation (17) is optimal (Au and Beck, 1999;Nn, 1995). There are many other further improvements following the IS approach, and someof these variants rely on the concept of adaptive sampling (Wang and Ang, 1994; Bucher,1988), which has shown promising results in evaluating the failure probability.
3.2.3 Directional sampling. The DS method was originally proposed to evaluate themultivariate distribution function and has been adopted to calculate the probability offailure in the U-space for general structural reliability problems (Bjerager, 1988; Ditlevsenet al., 1990; Melchers, 1994; Moarefzadeh and Melchers, 1999). The DS method generatesuniformly distributed direction vectors, and along each direction, a one-dimensionalintegration is performed. The reference (Katsuki and Frangopol, 1994) used a series ofhyperspherical segments, whose radii follow a x 2 distribution to investigate the actual limitstate surface in the standard normal space. Practically, a sequence of N random directionvectors a jð Þ ¼ u jð Þ
ku jð Þk ; j ¼ 1; . . . ;N are generated first, then rj = {r|g(ra(j)) = 0} are founditeratively. Finally, the sum of the failure probabilities associated with those segments givesthe approximated probability of failure:
Pf ¼ 1N
XNj¼1
1� x 2M r2j �h i
(18)
where x 2M is the x 2 CDF withM d.o.f. The DS method is relatively more efficient compared
to other MCS approaches, but its performance drops dramatically when the limit statesurface is highly nonlinear. As the prior knowledge of the limit state is rarely available,search-based IS has been proposed, which increases the cost of computation. Moreover,the randomly generated samples in DS may not be optimal, and several new approacheshave been proposed to better identify integration directions (e.g. spherical t-design, spiralpoints and Fekete points) (Nie and Ellingwood, 2000). Once determined, the DS directionscan be readily reused for calculations of other probabilistic integrations. The efficiency ofDS was significantly improved recently (Nie and Ellingwood, 2005) by utilizingdeterministic point sets to preserve the underlying joint probability distribution and byusing neural networks to focus the simulation effort in the significant regions.
3.2.4 Subset simulation. SS (Au and Beck, 2001; Au and Wang, 2014) expresses thefailure probability as a product of a series of conditional probabilities for some chosenintermediate failure events, whose estimations are cheaper than evaluating directly theoverall failure probability. The conditional probabilities are obtained using the MarkovChains Monte Carlo (MCMC) simulation based on a modified Metropolis–Hastingsalgorithm. The efficiency and accuracy of SS depend on the ability of the MCMC algorithmto accurately estimate the conditional probabilities with a minimum number of samples. The
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idea of SS has attracted wider attention: it was suggested in Ching et al. (2005a, 2005b) tosplit the trajectory to increase the acceptance rate of a candidate sample; a spherical SS forhigh-dimensional problems was proposed in Katafygiotis and Cheung (2007); and anoptimal scaling technique was developed in Zuev and Katafygiotis (2011) for the modifiedMetropolis–Hasting algorithm, with a theoretical analysis for the optimal value of theconditional failure probability. SS methods have been widely applied in various UQproblems, including structures subjected to uncertain earthquake ground motions (Au andBeck, 2001; Au and Wang, 2014; Ching et al., 2005a, 2005b; Katafygiotis and Cheung, 2007;Au and Beck, 2003; Tee et al., 2014), aerospace engineering (Song et al., 2009), geotechnicalengineering (Wang et al., 2011) and nuclear engineering (Wang et al., 2015).
The failure of a practical structure F = {g(x)# 0} is usually a rare event, correspondingto a small failure region in the random parameter space. Let F1 � F2 � . . . � Fm = F denotea decreasing sequence of failure events, which are defined as Fi = {g(x) # yi} withdecreasing yi values for the limit state surface and ym = 0. Following the definition of failureprobability, it can be calculated as (Au and Beck, 2001):
Pf ¼ P Fð Þ ¼ P Fmð Þ ¼ P Fm jFm�1ð ÞP Fm�1ð Þ ¼ � � � ¼ P F1ð ÞYmi¼2
P Fi jFi�1ð Þ (19)
Although the original failure probability Pf may be small, by choosing the appropriateintermediate failure events {Fi, i = 1,. . ., m}, it is possible to evaluate more efficiently theassociated conditional probabilities in equation (19). To calculate the probability of failurefrom equation (19), one needs to compute the probabilities P(F1) and {P(FijFi – 1):i = 2,. . .,m}. The first threshold y1 is obtained by a crude MCS, such that P(F1) = p0, where p0 istarget probability for each subset step. For further thresholds, new sampling pointscorresponding to the conditional events (FijFi�1) are obtained from MCMC using on amodified Metropolis–Hastings algorithm (Au and Beck, 2003; Zuev and Katafygiotis, 2011),and the conditional probability P(FjjFj–1) can be estimated as:
Pj ¼ P Fj jFj�1� � � 1
N
XNi¼1
IFj g xið Þð Þ (20)
Finally, the failure probability of the target event is calculated as Pf ¼Ym
j¼1Pj.
3.2.5 Latin hypercube sampling. The LHS method is a popular tool to improve theefficiency of crude Monte Carlo sampling. The LHS was originally proposed in Mckay et al.(1979) and has been further developed for different purposes by many researchers (Iman andConover, 1982; Stein, 1987; Owen, 1994; Ziha, 1995; Olsson and Sandberg, 2002). The LHS isvery efficient for estimating mean values and standard deviations (Olsson and Sandberg,2002), but it is only slightly more efficient than the crude MCS for estimating smallprobabilities (Pebesma and Heuvelink, 1999). Recently, some studies demonstrated therobustness of the LHS and the accuracy of the estimated probability of failure for bothregular and irregular configurations (Monteiro, 2016). LHS aims to spread the sample pointsmore evenly across the domain; therefore, with the same sample size, it is more stable andmore accurate than the estimation produced by Monte Carlo sampling (Olsson et al., 2003).The main idea is a stratification of the probability distribution by dividing the CDF curveinto equal intervals and then choosing one sample randomly inside each stratification.Taking the two-dimensional sample space as an example, a square grid is a Latin square ifand only if there is only one sample in each row and each column. LHS extends this concept
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to arbitrary dimensions, whereby each sample is the only one in each dimension-alignedhyperplane including it. Unlike MCS, where the generation of later samples are completelyindependent from the early samples, the LHS requires to remember the history of samplegeneration such that its later samples do not overlap in the hyperplane of the early samples.It is worth to mention that even though the marginal distribution of each variable isefficiently represented, there is a risk that some spurious correlation will appear (Olssonet al., 2003). The correlation between random variables can be introduced during the LHSprocess that rearranges the samples to form pairs with the desired correlation level, which isknown as rank correlation. LHS with rank correlation technique is effective to generatesampling matrix with correlation structure rather close to the target correlation matrix(Helton and Davis, 2003). Yu et al. (2009) applies LHS with Cholesky decomposition tominimize the correlation between samples of random variables in probabilistic space.Several authors have improved LHS by determining optimal pairings that either enhancespace-filling or reduce spurious correlation (increasing orthogonality) (Monteiro, 2016).
It is worth to note that other types of space-filling random sampling strategies such asSobol series and Halton sequences (Morokoff and Caflisch, 1995) can also be used for SRA.
3.3 Surrogate methodsIn the aforementioned SRAmethods, finite element structural analysis is performed in everyiteration of FORM/SORM or is required for every sample solution using the simulation-based approach, which can be extremely time-consuming for complex structures. To reducethe computational cost associated with the finite element simulation, an alternativeapproach is to approximate the actual limit state function with a surrogate model (alsocalled meta-model) that is of a simpler form, after which the probability of failure can beefficiently estimated from the surrogate without resolving the actual limit function usingfinite element simulation.
3.3.1 Response surface method. The RSM (Faravelli, 1989; Zheng and Das, 2000;Roussouly et al., 2013) fits and identifies an approximate response surface model from inputand output data collected from experimental/numerical studies, such that the actual limitstate function g(x) can be replaced with a simple function (often polynomial) g xð Þ for fastevaluation of the failure probability. In practice, quadratic functions with or without crossterms are often used to approximate the response surface:
g xð Þ � g xð Þ ¼ a0 þXMi¼1
aixi þXMi¼1
aiixi2 þXMi¼1
XMj¼1;j 6¼i
aijxi : xj (21)
where a = {a0,} a = {a0, ai, aii, aij} are unknown coefficients andM denotes the number ofrandom variables. The unknown polynomial coefficients {a} are determined by the leastsquare regression technique by using a sufficient number of experimental points. Afterconstructing the response surface g xð Þ, the reliability analysis can be performed on g xð Þinstead of the actual limit state function g(x).
A number of design schemes are available to select the experimental points forconstruction of the approximate response surface, among which the central compositedesign is a well-known approach. However, this scheme requires N = 2n þ 2n þ 1evaluations of the exact limit state function, where the computational cost will betremendous for large-scale structures with a large number of random variables. To reducethe number of fitting points, an adaptive interpolation scheme was proposed in Bucher andBourgund (1990), where a new center point xM for interpolation is chosen on a straight line
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from the mean vector x to the design point x* obtained from the first constructed responsesurface:
xM ¼ lX þ x* � lX
� g lXð Þg lXð Þ � g x*ð Þ (22)
This adaptive interpolation scheme ensures that the new center point is closer to the exactlimit surface g(x) = 0. The RSM has been continuously improved over the years, and theimprovements include the adaptive iteration procedure (Rajashekhar and Ellingwood, 1993),the gradient projection method to select sampling points (Kim and Na, 1997), the coupledRSM and moment method (Lee and Kwak, 2006), the adoption of the moving least-squaresmethod for better response surface fitting (Goswami et al., 2016), the use of exponentialresponse surface (Hadidi et al., 2017), the artificial neural network based RSM and thesupport vector machine-based RSM (Gomes and Awruch, 2004; Hurtado, 2004; Dai et al.,2015; Jiang et al., 2017), etc.
3.3.2 Kriging. Kriging (also known as Gaussian process modeling) is another popularmeta-modeling technique that is widely reported in the structural reliability and SFEMliterature. Kriging was originally developed for geostatistics in the 1950s and 1960s byKrige and then by Matheron (1973), and the method gained attention in the field of computerexperiments in the 1980s. Kriging interpolates exactly the experimental design points andprovides estimations of the local variance of the predictions, which provide an indication ofuncertainty associated with the Kriging model. In the 1990s, Kriging had been intensivelyused in optimization problems with active learning methods such as efficient globaloptimization (Jones et al., 1998).
The application of the Kriging method in the context of structural reliability is relativelyrecent, introduced in Romero et al. (2004), where Kriging with polynomial regression wascompared with finite element interpolation on progressive lattice samplings with analyticalfunctions. In Kaymaz (2005), a Kriging model implemented using the MATLAB toolboxDACE (Lophaven et al., 2002a) was combined with FORM to compute the structural failureprobability, after which the result was further compared with RSM. The sampling strategyto build the Kriging meta-model was improved in Bichon et al. (2008) by using an activelearning approach to iteratively add new samples to the experimental design, which reducesthe number of calls to the time-demanding performance function.
Technically, the Kriging model is a stochastic interpolation algorithm representing theoutput of a computer model M(x) as a random process, which is assumed to be a Gaussianrandom process indexed by x [ DX � RM. The first step of Kriging is to define thisstochastic field with its parameters according to a design of experiments. Then, the BestLinear Unbiased Predictor is used to estimate the value in a given point. A Kriging modelconsists of two parts (Echard et al., 2011), as shown in the equation below:
MK xð Þ ¼ bTf xð Þ þ Z xð Þ (23)
The first term in the above equation is the linear regression part and the second term is thenonparametric part. The linear regression part is similar to the polynomial model in a RSM,and it consists of the basis functions f(x) = {f1(x), f2(x),. . ., fP(x)}
T and the regressioncoefficients b = {b 1, b 2,. . .,b P}
T, which need to be determined. The second part inequation (23) is used to model the deviation from the first term and consists of the randomprocess Z(x), which is assumed to be a Gaussian stationary process with zero mean. Thecovariance of Z(x) can be defined as:
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Cov Z xið Þ; Z xjð Þ� � ¼ s 2R xi; xj; hð Þ; i; j ¼ 1; . . . ; N (24)
where N is the number of experimental points, s 2 is the progress variance and R(xi, xj; h) isthe spatial correlation function, which controls the smoothness of the model, the influence ofother nearby points and differentiability of the response surface. The correlation functionR = R(xi, xj; h) describes the correlation between two samples of the input space, e.g. xi andxj, and depends on the hyper-parameter h. In the context of meta-modeling, it is of interest toprovide a prediction Kriging model for a new point x. LetX = {x1,. . ., xN} denote the set ofknown points of the computer model whose response is Y = {y1 =M(x1),. . ., yN =M(xN)}
T.For a new point x, the expected value m Y xð Þ and variance s Y 2 xð Þ of the Kriging modelprediction Y xð Þ can be calculated as (Santner et al., 2003):
m Y xð Þ ¼ bTf xð Þ þ r xð ÞTR�1 Y � Fbð Þ (25)
s Y 2 xð Þ ¼ s 2 � f xð ÞT r xð ÞTh i 0 FT
F R
" #�1f xð Þr xð Þ
" #(26)
where b = (FTR�1F)�1FTR�1y, F is the information matrix of generation terms, r(x) is thecorrelation vector between an unknown point x and all known experimental points x andRis the correlation matrix defined by Rij= (xi, xj; h), i, j= 1,. . .,N.
To construct a Kriging meta-model, the functional basis of Kriging trend f(x) needs to beselected properly. Then, an appropriate correlation function R(xi, xj; h) is needed to estimatethe unknown parameters h by solving the optimization problem. Using the optimal value ofh, the rest of the unknown Kriging parameters (s 2, b) can be calculated. Finally, predictionsfor new points can be made in terms of the mean and variance of Y xð Þ, according toequations (25) and (26). Before providing the Kriging meta-model in equation (23), theunknown hyper-parameters need to be estimated by solving an optimization problem. Themaximum likelihood estimation and the cross-validation methods are among the mostpopular approaches to estimate the best parameters of the spatial correlation functions(Bachoc, 2013).
The Kriging meta-models predict the value of the limit state surface most accurately inthe vicinity of the experimental design samples x, but these samples are generally notoptimal to estimate failure probability. Thus, an adaptive Kriging MCS (AK-MCS) wasintroduced in Echard et al. (2011) and Schobi et al. (2015) to improve the accuracy of thesurrogate model in the neighborhood of the limit state function. Using an actively learningmethod, AK-MCS adaptively establishes the Kriging meta-model, which is then combinedwith MCS to evaluate the probability of failure. By using the Kriging meta-model toapproximate the limit state function, AK-MCS saves the costly evaluation of the actual limitstate function. There have been different learning functions reported in the literature to usewith Kriging (Bichon et al., 2008; Echard et al., 2011; Dani et al., 2008; Srinivas et al., 2012).
3.3.3 The moment method. The moment method (Lee and Kwak, 2006; Evans, 1972; Seoand Kwak, 2002) computes the statistical moments of the limit state function and fits thesemoments with an empirical distribution system such as the Johnson system, the Pearsonsystem and Gram-Charier series. The fitted distribution is then used to calculate theprobability of failure. Unlike RSM and Kriging methods, which build surrogate models forthe limit state function, the moment method builds the surrogate models for the probabilitydistribution of the limit state function. Different methods have been proposed to evaluate the
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statistical moments required, including the generalized method of moments (Hall, 2005), thehigh-dimensional model representation (Rabitz and Alis, 1999) and the dimension reductionmethod (Rahman and Xu, 2004). A nonlinear system of equations for point estimate ofprobability was combined with the Johnson distribution system and Gram–Charlier series inHong (1996). Five-moment method formulas were investigated in Zhao and Ono (2001),where a point estimate using Rosenblatt transformation and quadrature points for eachrandom variable were adopted. A design of experiment technique was proposed in Taguchi(1978) to use three-level experiments for each random variable to calculate the first twomoments of the limit state function. This approach was further improved in Derrico andZaino (1988) where levels and weighs were set equivalent to the nodes and weights in theGuass–Herminte quadrature formula. The approach was further extended by Seo and Kwak(2002) from normally distributed random variables to non-normal cases by deriving anexplicit formulation of three levels and weights for general distributions. The fourth-moment method was suggested in Zhao and Lu (2007).
The cost of the moment method can be high, as it increases exponentially with theincrease in random variables. Among others, one way to reduce the associatedcomputational cost is the univariate dimension reduction method (Rahman and Xu, 2004;Xu and Rahman, 2004), which decomposes the multi-dimensional performance function intomultiple univariate functions, therefore accelerating the calculation of the N-dimensionstatistical moment integration. Specifically, the performance function g(x) is approximatedby a sum of univariate functions, depending on only one random variable with the othervariables fixed to the reference point.
4. Stochastic finite element methodsUnlike the deterministic FEM, which has a standard Galerkin formulation, the SFEMs donot have a unique and generally agreed framework. Instead, there are various SFEMapproaches, such as the perturbation method, the polynomial chaos expansion (PCE)method, the stochastic collocation method and joint diagonalization (JD) method (Li et al.,2006). These different SFEM approaches have their own advantages and drawbacks, andthey have been formulated following various assumptions and have different mathematicalsetups. At the highest level, SFEM schemes can be roughly classified into two groups:intrusive methods and non-intrusive methods. The intrusive methods rely on dedicatedformulations of a stochastic version of the original model. As such, their solution schemeshave to be derived from scratch for each new class of problems. The non-intrusive methods,however, do not require modification to the deterministic FEM codes, which is a majoradvantage for analyzing reliability problems involving complex finite element analysis.These methods typically build surrogate models to approximate the system response basedon sample solutions. It is worth to note that the SFEM is still in its early stage and furtherdevelopment is needed to advance the SFEM to a level where both reasonable mathematicalaccuracy and sufficient computational efficiency are achieved. All major SFEM approachesare briefly recapped in the following subsections, while their technical details are referred tothe original papers or the summary papers where appropriate.
4.1 The perturbation methodThe perturbation SFEM can estimate the mean and covariance of the system response (Liuet al., 1986a, 1986b; Van den Nieuwenhof and Coyette, 2003; Kami�nski, 2013), with very wideapplications in different engineering fields, including geotechnical problems (Baecher andIngra, 1981; Phoon et al., 1990) and nonlinear dynamic problems (Liu et al., 1986a, 1986b).The main idea of the perturbation method is to expand all input random variables with
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respect to their respective mean values via Taylor expansion, which can then be used toderive the analytical expression for the variation of desired system response due to a smallvariation of those random variables. The unknown coefficients in the expansion areobtained by grouping like polynomials and canceling the corresponding coefficients. Themain limitation of the perturbation method is on its accuracy, which holds only for low-levelinput uncertainties (Wang et al., 2015; Wang et al., 2013). The standard perturbation schemedoes not provide higher-order statistical estimates, and there is no easy way to extend theformulation to the higher-order situation without invoking significantly complicatedimplementation and greatly increasing computational cost.
4.2 The Neumann expansion methodProposed in Shinozuka and Deodatis (1988) and Yamazaki et al. (1988), the Neumannexpansion method uses a truncated series expansion to approximate the solution of adiscretized linear equation system, avoiding the cost of a direct matrix inversion. Themethod was further applied to non-linear static and dynamic problems (Araujo and Awruch,1994) and also to cover geometric uncertainties (Chakraborty and Dey, 1998). The first stepof the Neumann expansion method is to split the stochastic stiffness matrix and load vectorinto their corresponding mean and random deviatoric parts. Then, expanding the stiffnessmatrix by Neumann expansion yields:
u ¼Xar¼0
�Qð Þru0 þXar¼0
�Qð ÞrK 0�1 DF (27)
The above Neumann expansion converges if all eigenvalues ofQ =K0�1DK are always less
than unity (Wang et al., 2013). The computational cost of the Neumann expansion methodincreases with the number of terms required in equation (27). Therefore, for problems withlarge random fluctuations, the Neumann expansion series may lose its advantage, and itcould become even more expensive than the direct Monte Carlo method (Wang et al., 2013).To overcome this computational difficulty and enhance the computational efficiency ofmatrix inversions, improvements have been made by combining the Neumann expansionmethod and the preconditioned conjugate gradient method (Papadrakakis andPapadopoulos, 1996). Another way to improve the efficiency and the accuracy is byintroducing the convergence parameter l , which can be determined as a solution to adistance minimization problem for an approximation of the inverse of the system matrix(Avila da and Beck, 2015).
4.3 Joint diagonalization strategyAs a general strategy to solve stochastic linear systems, the JD method (Li et al., 2006, 2010;Li, 2006) is applicable to any real symmetric matrices. The main idea of JD is tosimultaneously diagonalize all matrices in the system to obtain an average eigen-structureusing a sequence of orthogonal similarity transformation, which gradually decreases the off-diagonal elements of the matrices. The stochastic linear algebraic system is then decoupledby JD, and its approximate solution is explicitly obtained by inverting the resulting diagonalstochastic matrix and performing the corresponding similarity transformation. For JD, theclassical Jacobi method is modified to solve the resulting average eigenvalue problem. Thestrategy is simply an approximation, except if all the matrices have exactly the same eigen-structure. In this approach, the linear static equationKu=F is first reorganized as follows:
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K 0 þ j 1K 1 þ j 2K 2 þ . . .þþj mKmð Þu ¼ F (28)
where Kj, V are real symmetric deterministic matrices and j j, V are random variables.Assume that there exists an invertible Q to simultaneously diagonalize all matrices Kj, Vsuch that:
Q�1K jQ ¼ Kj ¼ diag l j1; l j2; . . . ; l jnð Þ; j ¼ 1; . . . ;mð Þ (29)
where l ji, (i = 1,. . ., n) are eigenvalues of the n n real symmetric matrixKj, V. The productof the Givens rotation matrices forms the transformmatrix:
Q ¼ GT1 G
T2 . . .GT
k ; (30)
where Gj = Gj(p, q, u ) is the Givens rotation matrices and k is the total number of Givenstransformations. After determining the transformation matrix and eigenvalues, the randomequation system can be readily calculated:
u � Q K0 þ j 1K1 þ . . .þ j mKmð Þ�1Q�1F (31)
The major computational cost of the proposed approach is the Jacobi-like JD procedure,which is proportional to the total number of matricesm. This implies that the algorithm canbe easily parallelized and the total computational cost is proportional to the total number ofrandom variables in the system.
4.4 Polynomial chaos expansionIntroduced by Ghanem and Spanos (1991), the PCE approach uses Hermit polynomials ofGaussian random variables (known as PCE or Wiener chaos expansion) to represent thesolution of the stochastic equations. By using a Galerkin scheme to project the randomsolution onto the Hermit polynomials, the solution can be resolved in the form of PCEcoefficients by solving a deterministic equation system. The classical PCE based onHermit polynomials have been extended to the so-called generalized PCE (gPC) by usingdifferent orthogonal polynomial basis functions corresponding to the probabilitydistributions of other non-normal variables (Wan and Karniadakis, 2006; Xiu andKarniadakis, 2002, 2003). The PCE approach provides a very flexible framework toinvestigate numerical solutions for stochastic partial differential equations, andpractically, it has marked the foundation of SFEM as an independent and fast-growingresearch field in computational mechanics and computational engineering. The PCEscheme has been researched in a very wide context of engineering problems, rangingfrom structural dynamics and random vibrations to computational fluid dynamics andthermodynamics problems (Rupert and Miller, 2007; Lin and Karniadakis, 2006;Sepahvand et al., 2007). The basic theory of PCE is outlined below, while the various PCE-based SFEM approaches are summarized in the following subsections. For simplicity, allSFEM techniques directly or indirectly related to PCE are grouped together in thissection. It is noted that for different emphasis, the methods reviewed here may well beconsidered as independent techniques in other studies.
In the PCE approach, any second-order random variable or stochastic process, i.e. thequantities with finite variance, may be expanded as follows:
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Y ¼Xa2NM
yac a nð Þ (32)
where c a(n) are a set of basis functions, ya refers to the deterministic coefficients to be solved, nrepresents standard normal random variables vector, a = {a1,. . .,aM} is the multi-index andMis the number of input random variables. These basis functions form the approximation spaceand can be represented by some orthogonal polynomials. For M-dimensional of PCE up toorder p, the total number of terms P required in the expansion is:
P ¼ M þ pp
� �¼ M þ pð Þ!
M !p!(33)
The number of polynomials P in the expansion grows very fast for small increases in p or inthe number of the random variablesM (Sudret and Kiureghian, 2002).
The PCE method can be classified into intrusive and non-intrusive approaches. TheGalerkin method is the best example for intrusive approach (Ghiocel and Ghanem, 2002;Sudret et al., 2003, 2006). The non-intrusive approaches include the probabilisticcollocation method (Foo et al., 2008) and the sparse quadrature (Nobile et al., 2008), whichdepend on repeated running of the computational model for selected realizations ofrandom variablesX.
4.4.1 The Galerkin solution schemes. In this approach, the PCE coefficients aredetermined by solving a system of linear equations derived by making the residual to beorthogonal to the approximation space. Considering the linear static elastic analysisequationKu = F with n degrees of freedom, the Galerkin-based PCE formulation leads to alinear system of size (n . P n . P), which reflects the high computational cost associatedwith the standard PCE approach. A number of improvements have been reported to increasethe computational efficiency while keeping the memory usage to a minimum (Ghanem andKruger, 1996; Elman et al., 2005; Verhoosel et al., 2006). As the block diagonal of the systemare fully determined by a single component matrix K0, whereas Ki represents the randomfluctuations and has the same sparsity structure, the block Gauss–Jacobi preconditioningwas proposed for cost reduction (Chung et al., 2005). An iterative solution was suggested inPellissetti and Ghanem (2000) to avoid the assembly of the (n. P n. P) matrix system and tosolve the systems [kjk]{uk} = {Fj}, which require only the storage of the deterministic sizeKi matrices and the corresponding hj ic jc ki coefficients. The preconditioned conjugategradient solvers have also been adapted for determining the solution of a linear SFEMsystem (Saad and van der Vorst, 2000; Tootkaboni et al., 2009).
The accuracy of the Galerkin-based PCE was checked in terms of representing the PDFof the strain values for the composite panel structure (Ngah and Young, 2007). It was foundthat low-order (such as 1-2) PCE provided reasonable accuracy, but higher-order expansionwas required to achieve good accuracy in the tails of the distribution, which are essential inreliability analysis. Furthermore, it is noted that the results become unstable and highlyinaccurate, where the stiffness has been modeled using normally distributed randomparameters and the variation of these parameters exceed 30 per cent. This is because theassumption of Gaussian random fields allows zero or negative values for the stiffnessmatrix, which is non-physical.
4.4.2 Collocation method. Stochastic collocation method estimates the PCE coefficientsby calculating the stochastic response at selected points in the multidimensional space of theinput random variables, which are referred to as the collocation points. Due to the
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orthogonality of the PCE basis, the deterministic coefficients (yi) for the PCE approximationof a random parameter (Y) are estimated as:
yi ¼ 1c 2
i
ðY : c i : f nð Þ : dn (34)
where f(n) represents the joint PDF of the random vector n. Thus, each coefficient yi isnothing but the orthogonal projection of the random response Y onto the correspondingbasis function c i.
The primary computational effort arises from the evaluation of the above integral, whichis performed usually by either deterministic techniques, i.e. the tensor product method(Baroth et al., 2007; Bressolette et al., 2010) and sparse grid method (Smolyak, 1963; Gerstnerand Griebel, 1998; Barthelmann et al., 2000), or probabilistic approaches, i.e. MCS. Recently,quasi-random numbers (Sudret et al., 2007) have also been investigated as an alternativeoption to replace MCS (Niederreiter, 1992). The simplest general technique forapproximating the multidimensional integral is to use a tensor product of one-dimensionalquadrature rules. In this case, a so-called product grid is constructed, where the total numberof points is the product of the number of integration points in each dimension. However, thetensor products would require an unacceptably high number of function evaluations forhigh-dimensional integrals. To address this deficiency, Smolyak’s quadrature (Smolyak,1963) is often used, which reduces dramatically the number of collocation points whileconserving a high level of accuracy. It makes use of sparse and nested grids, where thefunction evaluations are performed only at important points, and furthermore, the pointsused at one level can be re-used in the next one.
Instead of directly estimating the PCE coefficients from equation (34), the least-squareregression was adopted in Sudret et al. (2006) to fit the PCE model on a set of speciallyselected collocation points. A critical aspect of the linear regression approach is the choice ofdata points, for which the LHS has been commonly adopted. Moreover, quasi-randomsequences such as the Sobol’ or Halton sequence have also been adopted in the experimentaldesign (Blatman and Sudret, 2008). Isukapalli (1999) selected the points corresponding to theroots of the orthogonal polynomial of one degree higher than the maximum order of thecurrent PCE. However, different selections of experimental design points can result indifferent coefficients, which raise questions about the stability of the regression techniques.Furthermore, it was shown in Berveiller (2005) that the selection number of 2P (Isukapalli,1999) does not yield accurate estimations in most applications, and sequentially, anempirical rule of (M – 1)P regression points was suggested. The minimal size of theexperimental design points required for an accurate solution of the regression problem(Sudret, 2008) may evenmake the computation intractable in high dimensions.
It is desirable to predict in advance the truncation error to guide the selection of themaximal polynomial degree of PCE. The standard procedure is to test several truncationschemes of increasing polynomial degrees and check the convergence for the interestquantities. Also, a posteriori error estimate was suggested in Blatman and Sudret (2010) toevaluate the accuracy of any truncated PCE. To obtain a fair error estimation withinreasonable computational cost, the leave-one-out (LOO) cross validation (Allen, 1971) can beadopted. The main idea of LOO is to build a surrogate model and compute the error by usingdifferent sets of experimental design points.
The main drawbacks of PCE-based SFEM approaches is the high computational costs.Even a low-order PCE can lead to a large number of unknown coefficients, and therefore, theexperimental design may become unaffordable. Most PCE terms correspond to polynomials
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representing interactions between input variables, but in many practical applications, thehigh-order interactions terms are insignificant and even negligible. Some sparserepresentations of the truncation have been studied in Blatman and Sudret (2010), Blatman(2009) and Blatman and Sudret (2011a, 2011b) to reduce the computational cost, where theleast-angle regression (LARS) algorithm was proposed to keep only the most influentialpolynomials.
4.4.3 Combining polynomial chaos expansion and Kriging. The PCE approximation maybecome inefficient for high nonlinear performance function with a large number of randomvariables (Yadav and Rahman, 2014; Lee and Chen, 2009). To overcome this problem, thePCE with LARS and the universal Kriging model were combined to obtain a new family ofthe optimized meta-model (Kersaudy et al., 2015). The PCE treats the global behavior of themodel, whereas Kriging interpolates the local variations as a function of the nearbyexperimental design points. The new technique is defined as a universal Kriging model:
MPCK xð Þ ¼Xa2NM
yac a nð Þ þ Z xð Þ (35)
where the first term is an orthogonal PCE describing the trend of the model, whereas thesecond term is used to model the deviation from the main trend, and it consists of a Gaussianstationary process with zero mean and the covariance as introduced in equation (24). Thepurpose of this technique is to choose polynomials that bring the most relevant informationto the Kriging meta-model. Note that the new surrogate model can be interpreted as auniversal Kriging model, the trend of which consists of a set of orthogonal polynomials.There are two ways to combine the PCE and Kriging: sequential PC-Kriging and optimalPC-Kriging.
In sequential PC-Kriging (SPCK), the set of polynomials and the Kriging meta-models aredetermined sequentially. First, the optimal set of the polynomials is determined by theLARS algorithm. Then these results can be used as a suitable trend for the universal Krigingmodel. At the end of the algorithm, the accuracy of the calibrated meta-model can beestimated by determining the LOO error. In optimal PC-Kriging (OPCK), the PC-Krigingmeta-model is obtained iteratively. As in SPCK, the optimal set of polynomials is evaluatedby LARS. The LARS algorithm results in a sparse set of polynomials which are rankedaccording to their correlation to the current residual at each LARS iterations (in decreasingorder). Each polynomial is then added individually to the trend of a PC-Kriging model. Ineach iteration, a new PC-Krigingmodel is calibrated.
5. Comparative studiesIn this section, four examples are tested to compare the performance of the representativeSRA and SFEM approaches reviewed above and to extract general guidelines for selectingthe most appropriate UQ approach for specific application needs. The first example consistsof a group of explicitly defined limit functions under different nonlinearities and differentnumber and type of random inputs. The rest examples are based on different structuralproblems in the presence of both continuous and discontinuous uncertainties.
5.1 Examples for explicit limit state functionsAs listed in Table I, several commonly used limit state functions are tested in this example.The reference values are obtained using the crude MCS. The failure probability togetherwith the time and the number of functional calls is evaluated for each case. The results for
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Case
Limit-statefunctio
nsRandom
variables
Descriptio
nRef.
1g¼
x 1þ2x
2þ2x
3þx 4�
5x5�5x
6þ0:001X6 i¼
1
sin100x
ið
Þx 1
�4:LN
120;12
ðÞ
x 5:LN
50;15
ðÞ
x 6:LN
40;12
ðÞ
Linear
limitstatefunctio
nwith
noiseterm
(Eng
elun
dandRackw
itz,1993)
2g=x 1x 2
–146.14
x 1:N
78064:4;11709:7
ðÞ
x 2:N
0:0104;0:00156
ðÞ
Multip
leMPP
s(Eng
elun
dandRackw
itz,1993)
3g¼
2þ0:015X9 i¼
1
x2 i�x 1
0
x 1–10:N
(0,1)
Quadraticlim
itstatefunctio
nwith
tenterm
s(Kiuregh
ianetal.,1987)
4g¼
�0:5
x 1�x 2
ðÞ2�
x 1þx 2
ðÞ ffiffiffi 2pþ3
x 1–2:N(0,1)
Concavelim
itstatefunctio
n(BorriandSp
eranzini,1997)
Table I.Limit-state functions
description
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Cases 1, 2, 3 and 4 are shown in Tables II-V, respectively. These cases are investigated todemonstrate the accuracy, efficiency and robustness of different reliability methods.
Case 1 is a linear limit state function with noise terms, where the FORM gives acceptableresults for the reliability index but the SORM gives very poor results. This is because thenoise terms in the performance function causes the total principal curvatures of the limitstate surface to exceed the applicable range of SORM. If the noise term is removed from theperformance function, the probability of failure will convergent to the exact solution(PSORM
f ¼ 1:264 10�2). The limit state function in u space and FORM iteration for Case 2are shown in Figure 1(a) and (b). In this case, FORM has significant errors for theperformance function with multiple MPP problems, as shown in Table III. This means thatFORM is not suitable for performance function with multiple MPP. The same reasonexplains the bad result by SORM. The results given in Table IV show that FORM andSORM provide quite good results for slightly non-linear problem as given in Case 3compared with those of MCS, i.e. with a relative error equal to 5.99 and 0 per cent for FORMand SORM, respectively. However, this is not the case for concave limit state function givenin Case 4 where both FORM and SORM fail to converge.
The IS has been proposed to reduce the number of samples in the conventional MCS.This technique can shorten the computational time to a certain extent as shown in allresults, except for Case 4 (concave limit state function). The SS approach is applied with aconditional failure probability at each level equal to p0 = 0.1 and with the number of samplesset to N = 500 at each conditional level. The number of conditional levels is chosen to coverthe required response level whose failure probability is estimated. From Tables II-V, it isseen that the average values of failure probability produced by SS/MCMCmostly agree withthe results of MCS. Figure 1(c) plots the number of subsets for Case 2; however, it uses sevensubsets until the results converge. The value obtained for the probability of failure with thecombined DS and IS method gives good accuracy with less than 1 per cent relative error.
The RSM approach can provide results with sufficient accuracy for certain cases, butbecomes computationally impractical for problems including a large number of nonlinearrandom variables or with multiple MPPs. Besides, it is hard to build the appropriateresponse surface without knowing the location of MPP, and there is no guarantee that thesurrogate surface is a sufficiently close fit in all regions of interest.
Table II.Comparison ofreliabilityapproximations forcase 1: covPf ¼ 0:1
Method Pf 10–2 bFunctioncalls Time (s)
Relativeerror «b%
FORM 0.943 2.3481 1,165 1 4.2SORM (curvature fitting method) 0.00055 4.3950 1,245 1.02 95SORM (curvature fitting method) 0.00076 4.3255 1,245 1.02 95MCS 1.212 2.2533 8,20,000 1.5 ReferenceIS 1.200 2.2572 6565 1.2 0.17DSþ IS 1.190 2.2511 2,680 0.75 0.1SS 1.220 2.2511 18,994 0.11 0.1RSM (second-order without cross term)þIS 1.240 2.2455 650 3.45 0.35PCE–Quadrature (Smolyak) 1.160 2.2708 455 0.49 0.78PCE–OLS 1.182 2.2629 120 0.34 0.42PCE–LARS 1.227 2.2485 40 0.25 0.21AK-MCS 1.213 2.2530 40 35 0.01Moment method – Zhao and Ono 1.219 2.2511 42 0.07 0.098Moment method – FFMM 1.212 2.2533 729 1.2 0
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Figure 1.Case 2 multiple MPPs
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The random variables in Case 1 are independent lognormal variables. Let fZig6i¼1 beGaussian random variables N(0,1). By using Hermite polynomials to approximate fXig6i¼1,we get:
Xi � XN :i Zið Þ ¼XNk¼0
ci;kHk Zið Þ; ci;k ¼ exp m i þs 2
i
2
� �s k
i
k!(36)
Then, the surrogate model can be constructed as follows:
gN Zð Þ ¼ XN :1 þ 2XN :2 þ 2XN :3 þ XN :4 � 5XN :5 � 5XN :6 þ 0:001X6i¼1
sin 100XN :ið Þ
(37)
Adaptive LARS is now compared to the ordinary least squares (OLS) algorithm with thedegree of polynomial p = 2. From the results of Case 1, it can be observed that adaptive
Table III.Comparison ofreliabilityapproximations forCase 2: (multipleMPPs) covPf ¼ 0:1
Method Pf = 10–7 bFunctioncalls Time (s)
Relativeerror «b%
FORM 0.285 5.428 20 0.03 5.83SORM Found curvatures 1MCS 1.460 5.1285 > 109
IS 1.1791 5.1686 8220 0.2 0.78DSþ IS 1.610 5.1101 57 0.34SS 1.418 5.134 57134 0.72 0.11RSM (second-order without cross term)þ IS Not convergedNon-intrusive PCE–Quadrature (Smolyak) 1.604 5.110 30 0.7 0.36Non-intrusive PCE–OLS (full PC) 0.803 5.240 80 0.13 2.17Non-intrusive PCE–LARS (sparse PC) 1.5054 5.1220 75 80 0.12AK-MCS 1.501 5.1233 30 30 0.10Moment method – FFMM 1.3255 5.1467 9 0.04 0.35
Table IV.Comparison ofreliabilityapproximations forCase 3: (quadratic LSten terms)
Method Pf 10–2 bFunctioncalls Time (s)
Relativeerror «b%
FORM 2.28 2.00 24 0.02 5.97SORM 1.67 2.127 236 0.03 0SORM (Breitung) 1.75 2.108 236 0.03 0.89MCS 1.67 2.127 5,90,000 1.07 ReferenceIS 1.62 2.138 27124 0.96 0.52SS 1.65 2.131 18999 0.1 0.19RSM (second-order without cross term)þ IS 1.64 2.135 1050 2.9 0.38PCE–Quadrature (Smolyak) 1.75 2.108 1771 2.45 0.89PCE–OLS 1.71 2.117 500 0.33 0.47PCE–LARS 1.68 2.126 250 0.27 0.05AK-MCS 1.53 2.1628 40 40 1.68Moment method – FFMM 1.4871 2.1735 59,049 0.04 2.19
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LARS appears to be more efficient than OLS. In particular, it yields a relative error errLOO =5 10�4 using only N = 40 model evaluations, whereas N = 120 simulations are used forOLS algorithm to get errLOO = 0.0023. However, with a sufficient number of samples ofperformance function, an accurate failure probability estimation can always be obtained.
The Kriging meta-model-based MCS procedure is also applied to these cases. First, aninitial Kriging predictor is built for the limit state function using 100 uniformly generatedpoints within a hypersphere. Based on this initial prediction, the refinement procedureintroduced in AK-MCS is then used to add new points at each refinement iteration. As can beseen from Tables II-V, the results provided by the Kriging meta-model method are quiteclose to those given byMCS, which means that the Kriging predictor is rather accurate in allcases.
The full factorial moment method (FFMM) gives accurate results for Cases 1,2,3 and 4but with high computational cost for Cases 1 and 3. This is due to the exponential increase incomputation time with input dimension. Thus, it is only suitable for low dimensions.
5.2 Tunnel analysisA plane strain problem of a simplified tunnel model is considered in this example, and thegeometry and load parameters are shown in Figure 2(a). The deterministic parametersconsidered in this example are the unite weight of the rock mass g = 25 kN/m2, Poison’sratio v = 0.3 and the surcharge load p = 1,900 kPa. Young’s modulus of elasticity of the rockis considered spatially varying with mean(E(x, v )) = 15 108 Pa and
Cov E x1;vð Þ; E x2;vð Þ� � ¼ 5:0625 1016exp � x2 � x1ð Þ2 þ y2 � y1ð Þ24
�Pa2. A plane-strain
finite element mesh consisting of 267 nodes and 472 triangular elements is shown in Figure2(b). Subject to surcharge load, the maximum settlement of the tunnel is taken as the keyfactor for measuring the serviceability of the tunnel; thus, the performance function isexpressed as:
g E x;vð Þ� � ¼ d max � ju E x;vð Þ� �j (38)
where dmax is the maximum allowable settlement at the top of the tunnel and u(E(x, v )) isthe crown settlement dependent on the random field E(x, v ). For illustrative purposes, themaximum allowable settlement is assumed to be 20 mm. As no exact K-L expansion solution
Table V.Comparison of
reliabilityapproximations forCase 4: (concave LS)
Method Pf 10�1 b Function calls Time (s)Relative
error «b%
FORM No convergenceSORM No convergenceMCS 1.0537 1.2516 90,000 0.137 –IS 1.0411 1.2584 2,191,012 190 0.54DSþ IS 1.02 1.2702 32 75 1.49SS 1.0680 1.2437 10,000 0.1 0.63RSM (second-order without cross term)þ IS No convergencePCE–Quadrature (Smolyak) 0.994 1.2849 30,012 0.6 2.7PCE–OLS 1.0469 1.2553 20 0.20 0.29PCE–LARS 1.0408 1.2586 15 0.17 0.55AK-MCS 1.0458 1.2559 28 3.5 0.34Moment method-FFMM 1.137 1.2071 9 0.04 3.56
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Figure 2.(a) Geometry of theunderground tunnel;(b) finite elementmesh and boundaryconditions of thetunnel; and (c)Young’s modulus ofelasticity for the rockreconstructed fromF-K-L expansion
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is available for representing this random field, only the F-K-L discretization is used for thisstochastic field. To determine the number of random variables needed in the F-K-Ldiscretization, the approximation error due to truncation is controlled within 10 per cent interms of the difference of variance, which leads to 13 random variables. A specificrealization of the random Young’s modulus field of the rock is shown in Figure 2(c), wherethe random Young’s modulus varies significantly by �30 and 20 per cent from the meanvalue. The stochastic system of linear algebraic equations is composed of 14 534 534 realsymmetric matrices, including the mean matrix. The failure of probability, reliability index,number of evaluation model and computational time are investigated using different SRAand SFEMmethods, and the results are summarized in Table VI.
The reference result is obtained as 2.6345 for the reliability index and 4.2130 10�3
for the probability of failure from MCS with 1,000,000 simulations, where COV = 1.5 percent as shown in Figure 3(b). From Table VI and Figure 3(a), the following observationsare made:
� The FORM is inaccurate in dealing with such large number of random variableswith a nonlinear performance function.
� SORM and RSM have similar results and are reasonably accurate compared withthe MCS reference with relative errors of 0.67 and 0.83 per cent in reliability index,respectively. According to the number of model evaluations, RSM is more efficientthan SORM in this test.
Table VI.Comparison of SRAand SFEMmethods
for the tunnelexample
Method Pf 10–3 bFunctioncalls Time (s) Notes
Relativeerror «b%
FORM 1.9827 2.8809 236 15 – 9.35SORM (CurvatureFitting method)
4.8704 2.5849 589 30 – 1.88
SORM (CurvatureFitting method)
4.4370 2.6169 589 30 ImprovedBreitung
0.67
MCS 4.2130 2.6345 1,000,000 40,766 covPf ¼ 0:015IS 4.2270 2.6334 45,736 1,666 covPf ¼ 0:01 0.04SS 4.590 2.6053 2,800 115 covPf ¼ 0:197 1.11RSM (second-orderwithout cross term)þ IS
4.4921 2.6127 116 14 covPf ¼ 0:01 0.83
Neumann expansion 4.2270 2.6334 45,736 2,134 covPf ¼ 0:01 0.04MCS with Neumann–Lambda-first
4.0213 2.6503 37,141 1,692 covPf ¼ 0:01 0.60
Intrusive PCE (Galerkin) – – – – – –Non-intrusive PCE–Quadrature (Smolyak)
4.4905 2.6128 3,654 255 Error = 6.766e-4 0.82
Non-intrusive PCE–OLS(full PC)
4.6108 2.6037 700 45 LOO = 4.924e-05p = 3
Size of full basis:118
1.17
Non-intrusive PCE –LARS (sparse PC)
4.1797 2.6372 350 33 LOO 8.8977e-04p = 3
Size of sparsebasis: 115
0.1
AK-MCS 4.2850 2.6287 140 427 covPf ¼ 0:015 0.22JDþ IS 4.2181 2.6341 – 323 0.02
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Figure 3.(a) Convergence of theprobability of failurefor differentreliability methods;(b) Monte Carloconvergences; and (c)AK-MCSconvergence
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� The JD method obtains a good path-wise solution (strong solution) to theprobability of failure with the best accuracy and low computational cost, with theratio of the off-diagonal entries to the Frobenius norm being 6.024 10�3.Nevertheless, when the magnitude of the uncertainties becomes large (more than20 per cent), the JD scheme yields an oscillating solution, which gives poor results[Figure 5(a) and (b)]. Moreover, for problems with the probability of failure in anorder higher than 10�3, the JD leads to less accurate results compared with otherreliability methods [Figure 4(a) and (b)].
� The instructive PCE (based on a third-order expansion, 13 random variables andtotal degree of freedom at 534) requires to solve a large system of 299,040 linearequations to determine the PCE coefficients. To reduce the cost, the non-intrusivemethods OLS and LARS are tested in this example, where the adaptive LARSprocedure converges at the cost of only N = 700 runs and gives an accurate resultfor the probability of failure. The obtained PCE provides a normalized LOO error of8.8977e-04.
� The results obtained from the AK-MCS compare very well to the reference as soonas N = 140 points are used in the experimental design [Figure 3(c)]. The error is lessthan 0.2 per cent in the generalized reliability index for probabilities of failure in theorder of 10�3.
Table VI reveals that in relative terms, the JD, AK-MCS and PCE-LARS schemes have thebest balance of accuracy and efficiency. The classical Neumann method converges muchslower than all other methods. The intrusive PCE is not particularly suitable for complexstructures with many uncertain variables (e.g. 10).
A parametric study is carried out by varying the threshold u from 18 to 22 mm toexamine the SRA and SFEM methods at different levels of the failure probability and thereliability index. The results are summarized in Table IX and Figure 4(a) and (b). The errorof the JD method increases with the growth of the failure probability, while the error of othermethods increases as the failure probability decreases.
The PDF and CDF of the performance function considered and the number of theexperimental design points selected in various SRA methods are plotted in Figure 6(a) and (b).Figure 6(a) shows that the PDF curve obtained from JD differs significantly from the other PDFcurves. The PDF curves for other methods are similar and agree well with the MCS result witha sample size of 5,000. The CDF curve obtained from the JD is also different from the referencecurve, except in their tails. This observation confirms that JD is only suitable for evaluatinglower level of failure probability [Figure 6(b)]. In any case, the accuracy in representing the tailsof the CDF of the performance function should be carefully evaluated, as they are the singlemost critical part for failure probability evaluation.
Figure 5(a) and (b) show the effect of the coefficient of variance of the input random fieldcovE on the probability of failure and the reliability index of the tunnel. It is noted that covEhas a significant influence on Pf. The increase in covE will cause an increase in Pf in thisexample.
The mean, standard deviation, skewness and kurtosis of the distribution of the crownsettlement are also considered. The MCS provides the reference value with a two-digitaccuracy (100,000 model evaluations are performed), and the reference mean, standarddeviation, skewness and kurtosis are 17.79 mm, 0.75 mm,�0.3221 and 3.1476, respectively.The statistical moments are also estimated using different SRA and SFEM schemes, and theresults are compared and are shown in Table VII. Good accurate estimates of the fourth-order moments are obtained by these methods. Also, the results of the fourth statistical
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Figure 4.(a) Influence of thethresholds on theprobability of failureand (b) influence ofthe thresholds on theBeta index
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moments of the performance function are calculated using various methods and are recodedin Table VIII. Good results are obtained using these methods, except RSM.
Figure 7(a) and (b) show the effect of (covE) on the statistical moments. It is found that themean and the standard deviation of the settlement are very sensitive to covE. For example,when covE increases from 0.15 to 0.4, the mean value increases from 17.79 to 18.63 mm andthe standard deviation increases from 0.54 to 2.6 mm. Also, it is observed that whenincreasing covE, the perturbation and JD methods give poor results for the statisticalmoments (Figures 3-7 and Tables VI-IX).
5.3 Sheet pile problemA sheet pile is studied in this example where the settlement of an elastic soil layer displaysspatial variability in its material properties. The uncertainty in the soil material properties ismodeled by Gaussian random fields. The site consists of a 5.0 m deep excavation withcantilever sheet piles, without anchors or bottom support [Figure 8(a)], in a homogeneous
Table VII.Comparison of
moment analysis forthe crown settlement
of the tunnel
Method mU cm sU cm Function calls
MCS 17.79 0.75 100,000RSM (second-order without cross term) 17.78 0.721 116Intrusive PC Galerkin – – –Non-intrusive PCE–OLS (full PC) 17.79 0.75 1,000Non-intrusive PCE–LARS (sparse PC) 17.79 0.75 700Perturbation (second-order) 17.68 0.76 –MCS-Neumann 17.79 0.75 50Kriging 17.66 0.75 80JD 17.61 0.76 50
Figure 5.Estimate of (a)
probability of failureand (b) reliabilityindex in terms of
coefficient of varianceof the rock evaluatedby different reliability
methods
Table VIII.Comparison of
statistical momentfor performance
function g(E(x,v )) =dmax – |u(E(x,v ))|
Statistical moments MCS JD Neumann RSM Kriging PCE-OLS PCE-LARS
Mean mm 2.21 2.39 2.21 2.22 2.21 2.21 2.21Standard deviation mm 0.751 0.818 0.751 0.722 0.751 0.7591 0.751Skewness �0.3221 �0.3074 �0.3221 �0.1693 �0.3221 �0.3213 �0.3062Kurtosis 3.1476 3.1305 3.1476 2.9464 3.1476 3.1411 3.0787
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Figure 6.(a) PDFs and (b)CDFs of theperformance function
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soil layer of dense cohesionless sand with uncertain spatially varying mechanical properties.The soil is modeled in two dimensions with plane-strain finite elements and external loadingequal to 10 kPa. It is assumed that the Poisson’s ratio of the soil takes a constant value andthe only random material property is Young’s modulus. The deterministic and probabilisticproperties of the sheet pile and soil are shown in Figure 8(a) and Table X, respectively. Thespatial variability of the soil is modeled by a homogeneous random field, with the followingauto-correlation function:
r sð Þ ¼ exp � t xlx� t z
lz
� �(39)
where s is the vector of absolute distances in the horizontal and vertical directions, and lx =20 m, lz = 5 m are the correlation length in x and z directions for the Young’s modulus E ofthe soil. The first four terms in the F-K-L representation are considered, resulting in fourGaussian random variables j i. Figure 9(b) shows a realization of the Gaussian random fieldrepresenting the uncertainty of Young’s modulus.
A refined mesh was first used to get the exact maximum deflection of the sheet pile.Then, different meshes were tried to design an optimal mesh that meets the 1 per cent errorrequirement in the maximum deflection. The selected mesh is shown in Figure 8(b), whichcontains 1,542 nodes and 1,460 elements. The pile is modeled using beam elements
Figure 7.(a) Mean value and (b)
standard deviationvalue of the
settlement in terms ofcoefficient of varianceof the rock evaluated
by differentstochastic methods
Table IX.Estimates of the
generalizedreliability index for
various values of thethreshold settlement
of the tunnel
Threshold mmReference IS FORM SORM Subset RSM
Beta Beta Error (%) Beta Error (%) Beta Error (%) Beta Error (%)
18 0.317 0.561 76.9 0.444 40.0 0.332 4.7 0.296 6.519 1.528 1.793 17.3 1.571 2.8 1.501 1.8 1.537 0.620 2.633 2.881 9.4 2.617 0.6 2.605 1.1 2.613 0.821 3.539 3.842 8.6 3.550 0.3 3.602 1.8 3.553 0.422 4.370 4.686 7.2 4.373 0.1 4.452 1.9 4.394 0.6Threshold mm PCE-OLS PCE-LARS AKMCS JD Neumann
Beta Error (%) Beta Error (%) Beta Error (%) Beta Error (%) Beta Error (%)18 0.339 7.0 0.322 1.7 0.306 3.5 0.502 58.4 0.328 3.519 1.546 1.2 1.526 0.1 1.516 0.8 1.632 6.8 1.553 1.720 2.598 1.3 2.652 0.7 2.629 0.2 2.634 0.0 2.597 1.421 3.570 0.9 3.649 3.1 3.662 3.5 3.553 0.4 3.594 1.622 4.431 1.4 4.586 5.0 4.753 8.8 4.394 0.6 4.416 1.1
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corresponding to a Larssen 24 profile that behaves equally to the sheet pile in bending andaxial resistance. The interaction between the sheet pile and the surrounding soil is modeledusing spring elements. The maximum horizontal ux displacement occurs at the top of thesheet pile. The failure occurs when the value of ux exceeding a threshold of umax = 2.5 cm, asexpressed by the following limit state function:
g Xð Þ ¼ umax � ux (40)
Given the random variablesX and by using the FEM, we can solve the value of ux.The mean and standard deviation of the random displacement at the top of the sheet pile
is now considered. Figure 9(a) depicts the deformed shape of the sheet pile and soilcomputed with the mean value of the random field. The reference value is obtained by LHSwith 100,000 model evaluations. The mean and standard deviation for the responsereference are 9.86 and 3.87 mm, respectively. The results obtained using different SRA andSFEM schemes are listed in Table XI and Figures 10(a) and (b). It appears that the first-orderperturbation method gives acceptable estimation only when the variance of the inputrandom field is less than 0.25, while the second-order perturbation gives better estimates forthe full range of random parameters considered here. The results from the Neumann andPCE methods give better accuracy than the perturbation methods. As for the efficiencyaspect, the intrusive PCE method takes more computational effort than all other methods,except MCS. In contrast, the non-intrusive PCE by using the OLS or LARS and Krigingmeta-modeling required less model evaluations.
The probability of failure is investigated by different SRA and SFEM approaches, andthe results are summarized in Table XII. The accuracy of each method has been investigatedfor different variance values of the input random field and plotted in Figure 11. The AK-MCS approach is seen to have the optimal balance between the accuracy in the reliabilityindex and the computational cost, with a relative error of 1 per cent and only 152 finiteelement runs. FORM runs fastest with the minimum limit state function evaluations, but it
Figure 8.(a) Sheet pile wall(without anchors) and(b) finite elementdiscretization of thesoil
Table X.Material properties ofthe soil in the sheetpile example
Variable Dist. Mean kPa Std kPa COV
Esoil Normal 50,000 15,000 0.25
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gives the worst result in terms of accuracy. The accuracy of the simulation methods (e.g. SS)are better than that of the FORM and PCE methods, but required a large number of limitstate function evaluations to converge with an acceptable level of accuracy. The non-intrusive PCE, including both OLS and LARS, is also tested here, with the polynomial orderset to 4. The random field is discretized with four random variables, leading to a totaldimensionality of the computational model of 70 (i.e. the number of basis). The experimentaldesign using LHS consists of N = 157,123 samples for OLS and LARS. The accuracy of thePCE for this problem is low compared with the reference value. This is because the
Figure 9.(a) Deformed shape ofthe structure and (b)
realization of ahomogeneous
anisotropic normalrandom field
representing theYoung’s modulus E
of the soil
Table XI.Comparison of
moment analysis forsheet pile example
Method mU cm sU cm Function calls Time (s) Notes
MCS 0.986 0.387 100,000 105,830 LHSIntrusive PC–Galerkin 1.44 0.93 – 10,772 F-K-L with 4 Rv. p = 4Non-intrusive PCE–OLS 0.985 0.39 157 514 p = 4Non-intrusive PCE–LARS 0.984 0.39 123 150.7 p = 4Perturbation I 0.973 0.82 – 37 F-K-L with 4 RvMCS with Neumann–Lambda 0.983 0.38 – 90 –
Figure 10.Estimate of (a)
expected value and(b) variance value ofthe displacement ofsheet pile in terms of
coefficient of varianceof the soil evaluated
by the IS,perturbation, PC and
Neumannmethod
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probability of failure for this problem is very small and, as alluded in the previous examples,the PCE can be used in one-shot for not too small probabilities of failures. Moreover, thecomputation time may, however, blow up when more than a couple of random variables areused to discretize the random field.
After collecting all the results, it clarifies that the AK-MCS method is the most attractivefor problems involving random field because it is accurate with reasonable computationalcost even for large coefficients of variation of the input. SORM is the second good method forthe reliability index evaluation. The PCE approach also gives accurate results when appliedat third and higher orders. The computation time may, however, blow up when more thantwo or three random variables are used to discretize the random field. In contrast, usingintrusive polynomial chaos in this context requires increasing the order of expansion,leading rapidly to intractable calculations.
Table XII.Comparison ofreliabilityapproximations forsheet pile example(continue)
Method Pf 10�3 bFunctioncalls Time (s) Notes
Relativeerror «b%
FORM 1.1805 3.0406 118 255 – 4.11SORM (curvature fittingmethod) 1.7936 2.9123 180 502 Improved Breitung 0.29SORM (curvature fittingmethod) 1.7004 2.9289 180 502 Breitung 0.28IS 1.7464 2.9207 67,518 50,145 covPf ¼ 0:01 –SS 1.9400 2.8878 2,778 20,596 covPf ¼ 0:22 1.23Neumann expansion Not convergedMCS with Neumann–Lambda 1.400 3.0141 – 7,811 covPf ¼ 0:01 3.19Intrusive PCE (Galerkin) Required high capacityPCE – Quadrature (Smolyak) array exceeds maximum array sizePCE – OLS 1.9207 2.8909 157 593 p = 4 1.02PCE – LARS 1.9012 2.8941 123 397 p = 4 0.91AK-MCS 1.8900 2.8959 152 238 covPf ¼ 0:07 0.85
Figure 11.Estimate ofprobability of failurein terms of coefficientof variance of the soilevaluated by differentreliability methods(sheet pile example)
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5.4 Frame structureA three-span, five-story frame structure subjected to horizontal loads is considered in thisexample, as shown in Figure 12. Three applied loads, two Young’s modulus, eight momentsof inertia and eight cross-section areas of the frame components are modeled by randomvariables. There are in total M = 21 random variables, denoted by X = {P1, P2, P3, E4, . . .,I6,. . ., A14,. . ., A21}
T. The applied loads and the material properties are assumed to follow alognormal distribution and truncated Gaussian distribution over [0,þ1). It is noted that theGaussian random variables have been truncated to avoid non-physical negative realizations.Moreover, the various input random variables are correlated [refer to Blatman and Sudret(2010) for the complete description]. The response of interest is the horizontal displacement uat the top right corner of the top floor, with the horizontal loads acting at individual floorlevels. The model response can be eventually recast as a function of independent standardGaussian random variables j i by using Nataf transformation so that it may be expandedonto a PCE made of normalized Hermite polynomials. Of interest are the mean and thestandard deviation of the random maximal displacement U =M(Z(X)). The reference valuefor this example is obtained with the quadrature scheme with a two-digit accuracy (15,179model valuations are performed). The reference mean and standard deviation value for thedeflection are 0.069 and 0.021 ft, respectively. The statistical moments estimated withdifferent methods are listed in Table XIII.
The maximum degree pmax of the spars polynomial chaos (PC) is set equal to 7, its targetapproximation error to 10�2, its maximum interaction order jmax to 2 and the cut-off values« 1, « 2 to 5 10�5. The LHS design is used to generate collocation points to calculate PCEcoefficients and LARS coefficients, and the experimental design points are enriched until theregression problem is well-posed. An initial sampling of the size N = M þ 1 = 22 is used.The LARS approximation leads to gain factor of 2.75 (120 finite element runs instead of 330).The expectation and variance value of the deflection for the frame evaluated by differentvalues for the coefficient of variance of material cov(E4) are presented in Figure 13(a) and (b).
Table XIII.Comparison of
moment analysis forframe structure
example
Method mU ft sU ft Function calls Time (s) Notes
MCS 0.0652 0.0202 100,000 LHSMCS with Neumann–Lambda 0.0662 0.0202 1,000 3.1 LHSNon-intrusive PCE – OLS (full PC) 0.0653 0.0200 330 3.5 p = 2: Jmax = 2Non-intrusive PCE – LARS (sparse PC) 0.0649 0.0200 120 0.48 LOO = 0.011
Figure 12.Three-span, five-
story frame structuresubjected to lateral
loads
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The perturbation method gives poor results especially when increasing cov(E4), while allother methods give accurate results compared with MCS. Figure 14 presents the PDF for theresponse, which shows a good agreement with theMCS.
A reliability analysis is carried out with respect to the limit state function g(X) =umax – u1(X), where u1(X) is the horizontal component of the top displacement and umax isa given threshold set to 0.2 ft. The reference value of the failure probability Pf is obtainedby MCS by using (1,000,000) model frame evaluations to get a coefficient of variation lessthan 0.05 on Pf. Estimates of the probability of failure and reliability index are calculatedby different methods, and the results are summarized in Table XIV. It can be observedthat FORM requires 209 function evaluations (frame analysis) with the relative error of bequal to 11 per cent, while SORM requires 680 function evaluations and give moreaccurate result than FORM. The results of failure probability analyzed using differentmethods are plotted in Figures 15 and 16(a) and (b). It can be observed that all methods,except for FORM, give reasonable results. Furthermore, using N = 24,159 samples to
Figure 14.PDFs for the response(frame structure)
Figure 13.(a) Expected valueand (b) variance valueof the responseevaluated withdifferent value of thecoefficient of varianceof the Young’smodulus of thematerialE4
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construct the kernel sampling density, the IS simulation results give an estimate of theprobability of failure with the approximated error on b equal to 5 per cent. Theprobability of failure Pf = 2.0575 10�4, which was obtained by DS using 10,000directions, gives about 4 per cent approximation error. Nevertheless, other commonly
Table XIV.Comparison of
reliabilityapproximations for
frame structureexample
Method Pf = 10�4 bFunctioncalls Time (s) Notes
Relativeerror «b%
FORM 0.7846 3.7800 209 10 – 11.62SORM (curvature fittingmethod)
1.3930 3.6344 252 11.4 ImprovedBreitung
7.32
SORM (curvature fittingmethod)
1.3396 3.6445 252 11.4 Breitung 7.61
SORM (point fittingmethod)
1.3823 3.6364 680 31 ImprovedBreitung
7.38
SORM (point fittingmethod)
1.3396 3.6445 680 31 Breitung 7.62
MCS 3.5400 3.3865 1,000,000 43936 covPf ¼ 0:05 –IS 1.7698 3.5722 24,159 31 covPf ¼ 0:05 5.48DS 2.0575 3.5326 51,309 2323 covPf ¼ 0:27 4.31SS 3.2720 3.4080 212,960 9548 covPf ¼ 0:04 0.63RSM (second-orderwithout cross term)
2.2344 3.5107 731 – – 3.67
Neumann expansion – – – – Not convergedMCS with Neumann–Lambda
1.3762 3.6375 9,782 28 covPf ¼ 0:025 7.41
Non-intrusive PCE–OLS 1.4555 3.6231 4,048 39 p = 3 6.98Non-intrusive PCE–LARS 1.1059 3.6935 700 23 p = 3 9.06AK-MCS 3.1400 3.4192 142 18.4 covPf ¼ 0:056 0.97Moment method (FFMM) 3.7340 3.3718 600 26.4 – 0.43UDR 3.6280 3.3797 600 26.4 – 0.2
Figure 15.Influence of thethreshold on thereliability index(frame structure)
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used reliability methods, such as SS, over predict the failure probability by 0.6 per centcompared with MCS by using five subsets but with high computation effort. Due to thevery slow convergence of the Neumann series, we used only the MCS–Neumann withLambda parameter, which leads to a significantly more accurate solution thanconventional Neumann with smaller effort. Also, estimates of the failure probability arecomputed by post-processing a full third-order PC (OLS) and sparse PC (LARS)approximation. The latter is built up by setting the target approximation error equal to10�3 (this is an overall mean square error that drives the convergence of the adaptivePCE). From the result, it can be note that both PCE approximations yield relative errorson b less than 9 per cent. Additionally, we can observe LARS to be efficient as it makesuse of only N = 700 runs of the finite element model. In comparison, using a full third-
order PCE would require performance more thanM þ p
p
� �¼ 21þ 3
3
� �¼ 2024 and
would thus require about 2P� 4,048 finite element runs to get accurate results. Hence, thecomputational cost of a full PC requires 5.78 runs of the LARS method. It is also seen thatPCE in the least square sense provides an approximation whose accuracy is controlledglobally. In the tail of the distribution of the performance function, it does not guarantee aperfect control of the accuracy. This frame structure has been modelled with a sparsepolynomial chaos expansion, and a maximum degree of six was necessary to obtainaccurate results for the performance function. However, this approach should not be usedfor very small probabilities less than Pf = 10�4. In such a case, it should be better to useother kinds of meta-models to ensure the accuracy. The AK-MCS can approximate theprobability of failure with accuracy similar to an MCS with high efficiency. This exampleshows that AK-MCS can be applied on moderate dimension problems with greateffectiveness. For the MPP-based univariate method with numerical integration and theunivariate method with simulation (Rahman and Wei, 2006), the linear approximation ofthe limit state function in rotated Gaussian space gN(vN) was used. Both versions ofthe univariate methods provide more accurate results than FORM and SORM withslightly more expensive computational effort, but significantly more efficient thansimulation methods. The probability of failure is also evaluated by the FFMM. Theresults show that it leads to accurate results but with large number of computationalmodel runs.
Figure 16.(a) Influence of thethreshold on theprobability of failureand (b) distributiontail for probability offailure
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6. ConclusionA comprehensive review and rigorous study of SRA and SFEMs have been completed.The accuracy, efficiency and robustness of each method are analyzed in detail againstthe common set of representative examples, including explicit limit-state functions andgeotechnical and structural problems. The results from the comparative study providecomprehensive evidence-based references for the performance of different SRA andSFEM approaches. The main concluding remarks are summarized below:
� As the most ancient SRA methods, the theory and computational framework ofFORM and SORM are well established in their own context. However, both schemeshave severe drawbacks when dealing with practical structures whose limit-statefunctions can be implicitly defined, non-smooth, of high randomness dimensionalityand with large degrees of freedoms.
� The simulation methods have become increasingly attractive in recent years due tothe availability of cheaper and fast computers. The main drawback is its highcomputational cost, especially when dealing with rare events. A number ofnumerical techniques have been developed to reduce the cost, including IS, LHS, DSand SS methods.
� The RSM offers a vehicle to efficiently combine the powerful FEMs with reliability-related problems. Methods have also been developed to rationally reduce the size ofthe probabilistic models by using importance measures.
� The SFEM enables the treatment of spatially varying stochastic structuralheterogeneity. This is of essential importance, for instance, in the treatment ofgeotechnical problems, structural problems involving high-frequency vibrationsand in problems of elastic stability.
� In the context of intrusive SFEM, a number of algorithms have been developed,including the perturbation, Neumann expansion, JD and PCE. The advantages anddrawbacks of each method are presented in Table XV.
� The surrogate model-based SRA has become quite popular in the literature. Itis most suitable for cases where the performance function has implicit formand needs to be evaluated point-wise by numerical methods such as FEM.However, most surrogate models suffer from the number of unknowncoefficients increasing quickly with the increase in the number of variables (thecurse of dimensionality) and selection of the experimental points to constructthe surrogate model.
� The meta-models that are used in this paper are PCE (OLS and LARS algorithms),Kriging meta-model and the combined PCK technique. PCE is a regressiontechnique that models well the global behavior of the computational model. On theother side, Kriging is known to model well local variations because of itsinterpolation properties. In fact, PCK is a special case of universal Kriging modelswhere the trend is modeled by orthogonal polynomials. From the comparativestudy, it is observed that accuracy increases by using PCK instead of the traditionalmeta-modeling techniques.
� The meta-models are used to compute rare events, such as small failureprobabilities by a popular strategy such as PCE-IS and AK-MCS and APCK-MCS. In the shown applications, the results of AK-MCS are almost better thanthat from PCE-IS. But at the same time, these results have large relative errors(Figures 17 and 18).
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Method
Adv
antages
Disadvantages
FORM
Efficiency,effectiv
enessandsimplicity
whenthelim
itstatefunctio
nisexplicitly
defined
The
accuracy
depend
son
theshapeof
thelim
itstatesurfaceandcanonly
beassessed
throug
hcomparisonwith
MCS
Conv
ergenceissues
whentheLS
Fisnonlineara
nd/orinthepresence
ofnon-
Gaussianrand
omvariablesalso
forL
SFwith
multip
leMPP
Itisnotsuitableforthe
LSFwhich
isim
plicitlydefined
Requiresthesolutio
nof
anoptim
izationproblem
tofind
thesm
allestdistance
tothelim
itstate
Itrequ
ires
search
MPP
.Nogu
aranteeto
determ
inethecorrectM
PPSO
RM
Moreaccuratethan
FORM
Moreefficientcom
paredwith
simulationmethods
Compu
tatio
nalefficiency
isno
greaterthanthat
ofMCS
whenthenu
mbero
frand
omvariablesislarge(app
roximately100)(Skagg
sandBarry,1996)
The
accuracy
depend
son
theshapeof
thefailu
resurfaceandcanonly
beassessed
incomparisonwith
MCS
Itisnotp
articularly
suita
bleforthe
implicitlydefinedlim
itstatefunctio
nItdepend
son
FORM’sresult
MCS
MCS
isvery
robu
stinthesensethat
itcanhand
lecomplex
limitstates
MCS
aresimple,flexible,friendlyto
parallelcom
putin
g,andnotrestrictedby
the
numbero
frandom
variables
Highcompu
tatio
nalcostespecially
fors
mallprobabilityvalues
ISItisdesign
edtoredu
cethevariance
oftheMonteCa
rloestim
atorsfora
given
samplesize
ISalgorithm
ismoreefficientthanMCS
anddirectionalsim
ulation
Itisabletosimulatemorerare
rand
omevents
Itrequ
ires
inform
ationaboutthe
locatio
nof
thelim
itstate(s),especially
thepart
closesttotheorigin
inU-space,tobe
useful
Itfacesdifficulties
whenappliedto
high
-dim
ension
problemsandwhenthelim
itstatefunctio
nhasmultip
leMPP
sDS
Itisrelativ
elyefficientcom
paredtoMCS
The
accuracy
dropsdrastically
whentheperformance
functio
nishigh
lynonlinearu
nlessthenu
mbero
fsam
plingdirections
islarge
SSIthasbeen
capableto
solvevery
complex
problems(com
plex
non-lin
earlim
itstatefunctio
ninhigh
dimensions)(Pradlwartera
ndSchu
eller,2011)
Itisrelativ
elyefficientcom
paredtoothers
imulationmethods
Itiscapableto
redu
cethecompu
tatio
naleffortsneeded
byMCS
byoneto
two
orders
ofmagnitudes;theneeded
samplesize
isstill
essentialthatisthe
draw
back
ofits
generality
RSM
Itgivescomparableresults
from
othercom
putatio
nally
moredemanding
approach
Identifying
thesensitive
parametersinvolved
incontrolling
thesystem
response
andhelpsin
quality
controlm
easures
Reduces
thenu
mbero
frun
sSimplified
relatio
nshipthatcanbe
used
forp
racticalengineeringpu
rposes
Itcansometim
eslead
tofalseMPP
,and
thereisno
guaranteethat
thefitted
surfaceisin
factasufficiently
closefitinallregions
ofinterest
The
training
pointsforthe
constructio
nofRSM
modelareem
pirically
selected
Neumann
Itisasimpleform
ulation
The
compu
tatio
nalefficiency
ofthemethoddepend
son
therang
eof
rand
omfluctuations.T
hefluctuationrang
eallowed
bytheNeumannexpansionmethod
isgenerally
high
erthan
theperturbatio
nmethods
Notsuita
bleforp
roblem
swith
largerand
omvariations
The
conv
ergenceisvery
slow
Perturbatio
nOnlythefirsttwomom
entsof
rand
omvariablesneed
tobe
know
nItisrestricted
tosm
allvariability,andtheaccuracy
decreaseswhentherand
omfluctuationincreases
(contin
ued)
Table XV.Advantages anddisadvantages of thereliability methods
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Method
Adv
antages
Disadvantages
JDThismethodisnotsensitiv
eto
thenu
mbero
frandomvariablesinthesystem
Itisbetter
andmoreaccuratethan
Neumannexpansionandperturbatio
nmethod
There
hasnotb
eenamathematicallyrigorous
prooffor
itsaccuracy
Itisused
with
therand
omfieldproblem
andinapplicablewith
therand
omvariableproblems
IntrusivePC
ECo
mparedwith
theMCS
,the
compu
tatio
nalcosto
fgPC
issign
ificantly
lower
The
compu
tatio
nisefficienta
ndrobu
stThe
useof
samplingisnotrequired
Intrusivemethods
have
theinconv
eniencyof
requ
iringfullre-program
mingof
conv
entio
nalfi
niteelem
entsoftw
are
Theirlowefficiency,because
thesimulationruns
increase
exponentially
with
the
numbero
frandom
variablesandtheordero
fthe
polynomialsused
The
amount
ofcompu
tatio
nrequ
ired
fora
givenproblemismuchgreaterthan
thatof
theequivalent
determ
inistic
problem
Collocatio
nmethod(non-
intrusive)
The
lownu
mbero
fsim
ulationruns,w
henfewrand
omparametersareinvolved.
Easyto
implem
entw
ithanalysiscodestreatedas
blackboxes
Sparse
gridsredu
cethe“curse
ofdimensionality
”drastically.
Whenthesamerand
omvariableoccurs
inmanydifferentp
roblem
s,itis
sufficienttocalculatetheintegrationpointsetonce
andto
save
itforfurtheruse
Less
rigorous
andtheira
ccuracyispoorer
than
thefullprojectio
nmethods
using
theGalerkinapproach
AK-M
CSItisindepend
entfromthetype
oftheresponse
surfacefunctio
nto
befitted.A
lso
itisnotg
reatly
affected
bythechoice
ofthespace
Kriging
meta-modelismoreefficientthantheotherm
eta-modelsfoun
din
literature
The
parametersof
theKriging
methodsign
ificantly
affectthereliabilityresults.
How
ever,there
isno
guidance
howto
selectappropriatevalues
forthe
parameters
Someparametersused
inKriging
aredeterm
ined
byoptim
izationapproach.
These
causes
nogu
aranteeagood
resultforthe
structuralreliabilityproblems
Classical
mom
ent
methods
The
probability
offailu
recanbe
compu
ted,even
whentheCD
Fsor
PDFs
ofrand
omvariablesareun
know
nItissimpleandefficiency,w
hileitrequ
ires
neith
eriteratio
nnorthe
compu
tatio
nof
derivativ
es
The
expression
ofthefourth
mom
entreliabilityindexistoocomplicated
forthe
engineeringuse.
Itismoresuita
blefora
negativ
eskew
ness
than
fora
positiv
evalue
Itisoftenconsidered
moresuita
bleforh
ighprobability
problems.How
ever,at
probability
levelsless
than
10–5 ,P f
dedu
cedby
thePearsonsystem
might
notb
ereliable
Ifused
thethirdpolynomial(cubic)norm
aldistribu
tionto
predictthe
probability,the
mom
entm
ethodbecomes
generally
inapplicabletoalim
it-state
functio
nwith
morethan
thirdpower
rand
omvariables
Twoerrors
ources
exist:estim
atingfour
statisticalmom
entsandapproxim
ating
theresponse
PDFor
CDFbasedon
thefour
actualmom
ents
Improved
mom
ent
method-
FFMM
Itshow
svery
robu
stperformance
interm
sof
accuracy
againstthe
non-
norm
ality
ofinpu
tsItsuffersfrom
theexponentially
increasing
numberofnodeswith
rand
ominpu
tdimensions,anditseem
ssuita
bleonly
forp
roblem
swith
very
small
dimensionality
becauseof
the“curse
ofdimensionality
”
Improved
mom
ent
method-UDR
Easyto
implem
entand
use
Itisalso
notedthat
theaccuracy
ofUDRdeteriorates
astheinteractioneffect
increasesandthehigh
-order
mom
ents(skewness
andku
rtosis)aremore
vulnerableto
thiserrorthanlow-order
mom
ents
Itmay
benotb
eadequatefora
system
with
alargenu
mbero
frandom
variables
orstrong
nonlinearity
duetothehigh
-dim
ensionalintegralsretained
inthe
residu
eerroro
fthismethod
Table XV.
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ReferencesAdhikari, S. (2004), “Reliability analysis using parabolic failure surface approximation”, Journal of
EngineeringMechanics-ASCE, Vol. 130 No. 12, pp. 1407-1427.Allen, C.D. (1971), The Prediction Sum of Squares as a Criterion for Selecting Prediction Variables, T.R.
23, Editor, Dept. of Statistics, University of Kentucky, Lexington, KY.Ang, G.L., Ang, A.H.S. and Tang, W.H. (1992), “Optimal importance-sampling density estimator”,
Journal of EngineeringMechanics-ASCE, Vol. 118 No. 6, pp. 1146-1163.Araujo, J.M. and Awruch, A.M. (1994), “On stochastic finite-elements for structural-analysis”,
Computers & Structures, Vol. 52 No. 3, pp. 461-469.Au, S.K. and Beck, J.L. (1999), “A new adaptive importance sampling scheme for reliability
calculations”, Structural Safety, Vol. 21 No. 2, pp. 135-158.Au, S.K. and Beck, J.L. (2001), “Estimation of small failure probabilities in high dimensions by subset
simulation”, Probabilistic EngineeringMechanics, Vol. 16 No. 4, pp. 263-277.Au, S.K. and Beck, J.L. (2003), “Subset simulation and its application to seismic risk based on dynamic
analysis”, Journal of EngineeringMechanics, Vol. 129 No. 8, pp. 901-917.
Figure 17.The accuracy via theefficiency for thestructural reliabilitymethods
Figure 18.Bar chart of (a) theaccuracy and (b) theefficiency for thestructural reliabilitymethods
EC35,6
2206
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Au, S.K. andWang, Y. (2014), Engineering Risk Assessment with Subset Simulation, JohnWiley & Sons,Hoboken, NJ, p. 315.
Avila da, S.C.R. and Beck, A.T. (2015), “New method for efficient Monte Carlo-Neumann solution oflinear stochastic systems”, Probabilistic EngineeringMechanics, Vol. 40, pp. 90-96.
Bachoc, F. (2013), “Cross validation and maximum likelihood estimations of hyper-parameters ofGaussian processes with model misspecification”, Computational Statistics & Data Analysis,Vol. 66, pp. 55-69.
Baecher, G.B. and Ingra, T.S. (1981), “Stochastic fem in settlement predictions”, Journal of theGeotechnical Engineering Division-ASCE, Vol. 107 No. 4, pp. 449-463.
Baroth, J., Bressolette, P., Chauvière, C. and Fogli, M. (2007), “An efficient SFE method usingLagrarge polynomials: application to nonlinear mechanical problems with uncertainparameters”, Computer Methods in Applied Mechanics and Engineering, Vol. 196 Nos 45/48,pp. 4419-4429.
Barthelmann, V., Novak, E. and Ritter, K. (2000), “High dimensional polynomial interpolation on sparsegrids”,Advances in Computational Mathematics, Vol. 12 No. 4, pp. 273-288.
Berveiller, M. (2005), Stochastic Finite Elements: Intrusive and Non Intrusive Methods for ReliabilityAnalysis, Université Blaise Pascal, Clermont-Ferrand.
Bichon, B.J., Eldred, M.S., Swiler, L.P., Mahadevan, S. and McFarland, J.M. (2008), “Efficient globalreliability analysis for nonlinear implicit performance functions”, AIAA Journal, Vol. 46 No. 10,pp. 2459-2468.
Bjerager, P. (1988), “Probability integration by directional simulation”, Journal of EngineeringMechanics-ASCE, Vol. 114 No. 8, pp. 1285-1302.
Blatman, G. (2009), Adaptive Sparse Polynomial Chaos Expansions for Uncertainty Propagation andSensitivity Analysis, Université Blaise Pascal, Clermont-Ferrand.
Blatman, G. and Sudret, B. (2008), “Sparse polynomial chaos expansions and adaptive stochastic finiteelements using a regression approach”, Comptes RendusMecanique, Vol. 336 No. 6, pp. 518-523.
Blatman, G. and Sudret, B. (2010), “An adaptive algorithm to build up sparse polynomial chaosexpansions for stochastic finite element analysis”, Probabilistic Engineering Mechanics, Vol. 25No. 2, pp. 183-197.
Blatman, G. and Sudret, B. (2011a), “Adaptive sparse polynomial chaos expansion based on least angleregression”, Journal of Computational Physics, Vol. 230 No. 6, pp. 2345-2367.
Blatman, G. and Sudret, B. (2011b), “Principal component analysis and least angle regression inspectral stochastic finite element analysis”, Applications of Statistics and Probability in CivilEngineering, pp. 669-676.
Borri, A. and Speranzini, E. (1997), “Structural reliability analysis using a standard deterministic finiteelement code”, Structural Safety, Vol. 19 No. 4, pp. 361-382.
Breitung, K. (1984), “Asymptotic approximations for multinormal integrals”, Journal of EngineeringMechanics-Asce, Vol. 110 No. 3, pp. 357-366.
Bressolette, P., Fogli, M. and Chauviere, C. (2010), “A stochastic collocation method for large classes ofmechanical problems with uncertain parameters”, Probabilistic Engineering Mechanics, Vol. 25No. 2, pp. 255-270.
Bucher, C.G. (1988), “Adaptive sampling – an iterative fast Monte-Carlo procedure”, Structural Safety,Vol. 5 No. 2, pp. 119-126.
Bucher, C.G. and Bourgund, U. (1990), “A fast and efficient response-surface approach for structuralreliability problems”, Structural Safety, Vol. 7 No. 1, pp. 57-66.
Bucher, C. and Macke, M. (2002), “Time-variant reliability – computational approaches based onFORM and importance sampling”, AMAS Course on Reliability-Based Optimization RBO’02,pp. 9-34.
Structuralreliability
2207
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Chakraborty, S. and Dey, S.S. (1998), “A stochastic finite element dynamic analysis of structures withuncertain parameters”, International Journal ofMechanical Sciences, Vol. 40 No. 11, pp. 1071-1087.
Ching, J., Au, S.K. and Beck, J.L. (2005a), “Reliability estimation for dynamical systems subject tostochastic excitation using subset simulation with splitting”, Computer Methods in AppliedMechanics and Engineering, Vol. 194 Nos 12/16, pp. 1557-1579.
Ching, J., Beck, J.L. and Au, S.K. (2005b), “Hybrid subset simulation method for reliability estimation ofdynamical systems subject to stochastic excitation”, Probabilistic EngineeringMechanics, Vol. 20No. 3, pp. 199-214.
Chung, D.B., Gutiérrez, M.A., Graham-Brady, L.L. and Lingen, F.J. (2005), “Efficient numericalstrategies for spectral stochastic finite element models”, International Journal for NumericalMethods in Engineering, Vol. 64 No. 10, pp. 1334-1349.
Dai, H.Z., Zhang, H. and Wang, W. (2015), “A multiwavelet neural network-Based Response surfacemethod for structural reliability analysis”, Computer-Aided Civil and Infrastructure Engineering,Vol. 30 No. 2, pp. 151-162.
Dani, V., Hayes, T.P. and Kakade, S.M. (2008), “Stochastic linear optimization under bandit feedback”,The 21st Annual Conference on Learning Theory (COLT 2008).
Derrico, J.R. and Zaino, N.A. (1988), “Statistical tolerancing using a modification of Taguchi method”,Technometrics, Vol. 30 No. 4, pp. 397-405.
Ditlevsen, O., Melchers, R.E. and Gluver, H. (1990), “General multidimensional probability integrationby directional simulation”, Computers & Structures, Vol. 36 No. 2, pp. 355-368.
Echard, B., Gayton, N. and Lemaire, M. (2011), “AK-MCS: an active learning reliability methodcombining kriging andMonte Carlo simulation”, Structural Safety, Vol. 33 No. 2, pp. 145-154.
Elman, H.C., Ernst, O.G., O’Leary, D.P. and Stewart, M. (2005), “Efficient iterative algorithms for thestochastic finite element method with application to acoustic scattering”, Computer Methods inAppliedMechanics and Engineering, Vol. 194 Nos 9/11, pp. 1037-1055.
Engelund, S. and Rackwitz, R. (1993), “A benchmark study on importance sampling techniques instructural reliability”, Structural Safety, Vol. 12 No. 4, pp. 255-276.
Evans, D.H. (1972), “An application of numerical integration techniques to statistical tolerancing, III –general distributions”,Technometrics, Vol. 14 No. 1, pp. 23-35.
Faravelli, L. (1989), “Response-surface approach for reliability-Analysis”, Journal of EngineeringMechanics-Asce, Vol. 115 No. 12, pp. 2763-2781.
Feng, J.W., Li, C.F., Cen, S. and Owen, D.R.J. (2014), “Statistical reconstruction of two-phase randommedia”, Computers & Structures, Vol. 137, pp. 78-92.
Feng, J.W., Li, C.F., Cen, S. and Owen, D.R.J. (2016), “Statistical reconstruction and Karhunen-LoeveExpansion for multiphase random media”, International Journal for Numerical Methods inEngineering, Vol. 105 No. 1, pp. 3-32.
Feng, Y.T., Li, C.F. and Owen, D.R.J. (2010), “A directed Monte Carlo solution of linear stochasticalgebraic system of equations”, Finite Elements in Analysis and Design, Vol. 46 No. 6,pp. 462-473.
Foo, J., Wan, X.L. and Karniadakis, G.E. (2008), “The multi-element probabilistic collocation method(ME-PCM), error analysis and applications”, Journal of Computational Physics, Vol. 227 No. 22,pp. 9572-9595.
Gerstner, T. and Griebel, M. (1998), “Numerical integration using sparse grids”, Numerical Algorithms,Vol. 18 Nos 3/4, pp. 209-232.
Ghanem, R. (1998), “Probabilistic characterization of transport in heterogeneous media”, ComputerMethods in AppliedMechanics and Engineering, Vol. 158 Nos 3/4, pp. 199-220.
Ghanem, R.G. and Kruger, R.M. (1996), “Numerical solution of spectral stochastic finite elementsystems”, ComputerMethods in AppliedMechanics and Engineering, Vol. 129 No. 3, pp. 289-303.
EC35,6
2208
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Ghanem, R.G. and Spanos, P.D. (1991), “Spectral stochastic finite-element formulation for reliability-analysis”, Journal of EngineeringMechanics-ASCE, Vol. 117 No. 10, pp. 2351-2372.
Ghiocel, D.M. and Ghanem, R.G. (2002), “Stochastic finite-element analysis of seismic soil-structureinteraction”, Journal of EngineeringMechanics-ASCE, Vol. 128 No. 1, pp. 66-77.
Gomes, H.M. andAwruch, A.M. (2004), “Comparison of response surface and neural network with othermethods for structural reliability analysis”, Structural Safety, Vol. 26 No. 1, pp. 49-67.
Goswami, S., Ghosh, S. and Chakraborty, S. (2016), “Reliability analysis of structures by iterativeimproved response surface method”, Structural Safety, Vol. 60, pp. 56-66.
Hadidi, A., Azar, B.F. and Rafiee, A. (2017), “Efficient response surface method for high-dimensionalstructural reliability analysis”, Structural Safety, Vol. 68, pp. 15-27.
Hall, A.R. (2005), GeneralizedMethod ofMoments, Oxford University Press, New York, NY.Hasofer, A.M. and Lind, N.C. (1974), “Exact and invariant second-moment code format”, Journal of the
EngineeringMechanics Division-Asce, Vol. 100 No. Nem1, pp. 111-121.
Helton, J.C. and Davis, F.J. (2003), “Latin hypercube sampling and the propagation of uncertainty inanalyses of complex systems”, Reliability Engineering & System Safety, Vol. 81 No. 1, pp. 23-69.
Hohenbichler, M. and Rackwitz, R. (1981), “Non-normal dependent vectors in structural safety”, Journalof the EngineeringMechanics Division-ASCE, Vol. 107 No. 6, pp. 1227-1238.
Hong, H.P. (1996), “Point-estimate moment-based reliability analysis”, Civil Engineering Systems,Vol. 13 No. 4, pp. 281-294.
Hurtado, J.E. (2004), “An examination of methods for approximating implicit limit state functions fromthe viewpoint of statistical learning theory”, Structural Safety, Vol. 26 No. 3, pp. 271-293.
Iman, R.L. and Conover, W.J. (1982), “A distribution-free approach to inducing rank correlation amonginput variables”, Communications in Statistics Part B-Simulation and Computation, Vol. 11No. 3, pp. 311-334.
Isukapalli, S.S. (1999),Uncertainty Analysis of Transport-Transformation Models, The State Universityof New Jersey, New Brunswick, Newark.
Jiang, Y., Zhao, L., Beer, M., Patelli, E., Broggi, M., Luo, J. and Zhang, J. (2017), “Multiple responsesurfaces method with advanced classification of samples for structural failure function fitting”,Structural Safety, Vol. 64, pp. 87-97.
Jones, D.R., Schonlau, M. and Welch, W.J. (1998), “Efficient global optimization of expensive black-boxfunctions”, Journal of Global Optimization, Vol. 13 No. 4, pp. 455-492.
Kami�nski, M.M. (2013), The Stochastic Perturbation Method for Computational Mechanics, JohnWiley & Sons, Hoboken, NJ, p. 19, 330.
Kass, R.E., Carlin, B.P., Gelman, A. and Neal, R.M. (1998), “Markov chain Monte Carlo in practice: aroundtable discussion”,American Statistician, Vol. 52 No. 2, pp. 93-100.
Katafygiotis, L.S. and Cheung, S.H. (2007), “Application of spherical subset simulation method andauxiliary domain method on a benchmark reliability study”, Structural Safety, Vol. 29 No. 3,pp. 194-207.
Katsuki, S. and Frangopol, D.M. (1994), “Hyperspace division method for structural reliability”, Journalof EngineeringMechanics-ASCE, Vol. 120 No. 11, pp. 2405-2427.
Kaymaz, I. (2005), “Application of kriging method to structural reliability problems”, Structural Safety,Vol. 27 No. 2, pp. 133-151.
Kersaudy, P., Sudret, B., Varsier, N., Picon, O. and Wiart, J. (2015), “A new surrogate modelingtechnique combining kriging and polynomial chaos expansions – application to uncertaintyanalysis in computational dosimetry”, Journal of Computational Physics, Vol. 286, pp. 103-117.
Kim, S.H. and Na, S.W. (1997), “Response surface method using vector projected sampling points”,Structural Safety, Vol. 19 No. 1, pp. 3-19.
Structuralreliability
2209
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Kiureghian, A.D. and Destefano, M. (1991), “Efficient algorithm for 2nd-Order Reliability-Analysis”,Journal of EngineeringMechanics-ASCE, Vol. 117 No. 12, pp. 2904-2923.
Kiureghian, A.D. and Liu, P.L. (1986), “Structural reliability under incomplete probability information”,Journal of EngineeringMechanics-ASCE, Vol. 112 No. 1, pp. 85-104.
Kiureghian, A.D., Lin, H.Z. and Hwang, S.J. (1987), “2nd-order reliability approximations”, Journal ofEngineeringMechanics-Asce, Vol. 113 No. 8, pp. 1208-1225.
Koyluoglu, H.U. and Nielsen, S.R.K. (1994), “New approximations for sorm integrals”, Structural Safety,Vol. 13 No. 4, pp. 235-246.
Lee, I., Noh, Y. and Yoo, D. (2012), “A novel second-order reliability method (Sorm) using noncentral orgeneralized chi-squared distributions”, Journal of Mechanical Design, Vol. 134 No. 10.
Lee, S.H. and Chen,W. (2009), “A comparative study of uncertainty propagation methods for black-box-type problems”, Structural andMultidisciplinary Optimization, Vol. 37 No. 3, pp. 239-253.
Lee, S.H. and Kwak, B.M. (2006), “Response surface augmented moment method for efficient reliabilityanalysis”, Structural Safety, Vol. 28 No. 3, pp. 261-272.
Lemaire, M. (2009), Structural Reliability2009, ISTE;Wiley, London, Hoboken, NJ, p. 488.Li, C.F., Adhikari, S., Cen, S., Feng, Y.T. and Owen, D.R.J. (2010), “A joint diagonalisation approach for
linear stochastic systems”, Computers & Structures, Vol. 88 Nos 19/20, pp. 1137-1148.Li, C.F., Feng, Y.T., Owen, D.R.J. and Davies, I.M. (2006), “Fourier representation of random media
fields in stochastic finite element modelling”, Engineering Computations, Vol. 23 No. 7,pp. 794-817.
Li, C.F., Feng, Y.T. and Owen, D.R.J. (2006), “Explicit solution to the stochastic system of linearalgebraic equations (alpha(1)A(1)þ alpha(2)A(2)þ center dot center dot center dotþ alpha(m)A(m)) x = b”, Computer Methods in Applied Mechanics and Engineering, Vol. 195 Nos 44/47,pp. 6560-6576.
Li, C.F., Feng, Y.T., Owen, D.R.J., Li, D.F. and Davis, I.M. (2008), “A fourier-Karhunen-LoeveDiscretization scheme for stationary random material properties in SFEM”, InternationalJournal for Numerical Methods in Engineering, Vol. 73 No. 13, pp. 1942-1965.
Li, C.F. (2006), Stochastic Finite Element Modelling of Elementary Random Media, University of WalesSwansea, Swansea.
Lin, G. and Karniadakis, G.E. (2006), “A discontinuous Galerkin method for two-temperature plasmas”,ComputerMethods in AppliedMechanics and Engineering, Vol. 195 Nos 25/28, pp. 3504-3527.
Liu, W.K., Belytschko, T. and Mani, A. (1986a), “Probabilistic finite-elements for nonlinear structuraldynamics”, ComputerMethods in AppliedMechanics and Engineering, Vol. 56 No. 1, pp. 61-81.
Liu, W.K., Belytschko, T. and Mani, A. (1986b), “Random field finite-elements”, International Journalfor Numerical Methods in Engineering, Vol. 23 No. 10, pp. 1831-1845.
Lophaven, S.N., Nielsen, H.B. and Søndergaard, J. (2002a), DACE, a Matlab Kriging Toolbox, Version2.0, IMM-TR-2002-12: Technical University of Denmark, Lyngby.
Lu, Z.H., Hu, D.Z. and Zhao, Y.G. (2017), “Second-Order Fourth-Moment Method for structuralreliability”, Journal of EngineeringMechanics, Vol. 143 No. 4.
Mansour, R. and Olsson, M. (2014), “A closed-form second-order reliability method using noncentralchi-squared distributions”, Journal of Mechanical Design, Vol. 136 No. 10.
Matheron, G. (1973), “The intrinsic random functions and their applications”, Advances in AppliedProbability, Vol. 5 No. 3, pp. 439-468.
Mckay, M.D., Beckman, R.J. and Conover, W.J. (1979), “A comparison of three methods for selectingvalues of input variables in the analysis of output from a computer code”,Technometrics, Vol. 21No. 2, pp. 239-245.
Melchers, R.E. (1994), “Structural system reliability assessment using directional simulation”,Structural Safety, Vol. 16 Nos 1/2, pp. 23-37.
EC35,6
2210
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Moarefzadeh, M.R. and Melchers, R.E. (1999), “Directional importance sampling for ill-proportionedspaces”, Structural Safety, Vol. 21 No. 1, pp. 1-22.
Monteiro, R. (2016), “Sampling based numerical seismic assessment of continuous span RC bridges”,Engineering Structures, Vol. 118, pp. 407-420.
Morokoff, W.J. and Caflisch, R.E. (1995), “Quasi-Monte Carlo integration”, Journal of ComputationalPhysics, Vol. 122 No. 2, pp. 218-230.
Ngah, M.F. and Young, A. (2007), “Application of the spectral stochastic finite element method forperformance prediction of composite structures”, Composite Structures, Vol. 78 No. 3,pp. 447-456.
Nie, J. and Ellingwood, B.R. (2000), “Directional methods for structural reliability analysis”, StructuralSafety, Vol. 22 No. 3, pp. 233-249.
Nie, J.S. and Ellingwood, B.R. (2005), “Finite element-based structural reliability assessment usingefficient directional simulation”, Journal of EngineeringMechanics, Vol. 131 No. 3, pp. 259-267.
Niederreiter, H. (1992), “Random number generation and quasi-Monte Carlo methods”, CBMS-NSFRegional Conference Series in Applied Mathematics1992, Society for Industrial and AppliedMathematics, Philadelphia, PA, Vol. 6, p. 241.
Nn, S. (1995),Optimization of Entropy Under Uncertainty, University of California, San Diego.Nobile, F., Tempone, R. and Webster, C.G. (2008), “An anisotropic sparse grid stochastic collocation
method for partial differential equations with random input data”, Siam Journal on NumericalAnalysis, Vol. 46 No. 5, pp. 2411-2442.
Olsson, A.M.J. and Sandberg, G.E. (2002), “Latin hypercube sampling for stochastic finite elementanalysis”, Journal of EngineeringMechanics-Asce, Vol. 128 No. 1, pp. 121-125.
Olsson, A., Sandberg, G. and Dahlblom, O. (2003), “On latin hypercube sampling for structuralreliability analysis”, Structural Safety, Vol. 25 No. 1, pp. 47-68.
Owen, A.B. (1994), “Controlling correlations in latin hypercube samples”, Journal of the AmericanStatistical Association, Vol. 89 No. 428, pp. 1517-1522.
Papadrakakis, M. and Papadopoulos, V. (1996), “Robust and efficient methods for stochastic finiteelement analysis using Monte Carlo simulation”, Computer Methods in Applied Mechanics andEngineering, Vol. 134 Nos 3/4, pp. 325-340.
Pebesma, E.J. and Heuvelink, G.B.M. (1999), “Latin hypercube sampling of Gaussian random fields”,Technometrics, Vol. 41 No. 4, pp. 303-312.
Pellissetti, M.F. and Ghanem, R.G. (2000), “Iterative solution of systems of linear equations arising inthe context of stochastic finite elements”, Advances in Engineering Software, Vol. 31 Nos 8/9,pp. 607-616.
Phoon, K.K., Quek, S.T., Chow, Y.K. and Lee, S.L. (1990), “Reliability-analysis of pile settlement”,Journal of Geotechnical Engineering-Asce, Vol. 116 No. 11, pp. 1717-1735.
Phoon, K.K., Huang, S.P. and Quek, S.T. (2002), “Implementation of Karhunen-Loeve expansion forsimulation using a Wavelet-Galerkin scheme”, Probabilistic Engineering Mechanics, Vol. 17No. 3, pp. 293-303.
Pradlwarter, H.J. and Schueller, G.I. (2011), “Reliability of deterministic non-linear systems subjected tostochastic dynamic excitation”, International Journal for Numerical Methods in Engineering,Vol. 85 No. 9, pp. 1160-1176.
Rabitz, H. and Alis, O.F. (1999), “General foundations of high-dimensional model representations”,Journal of Mathematical Chemistry, Vol. 25 Nos 2/3, pp. 197-233.
Rackwitz, R. and Fiessler, B. (1978), “Structural reliability under combined random load sequences”,Computers & Structures, Vol. 9 No. 5, pp. 489-494.
Rahman, S. and Wei, D. (2006), “A univariate approximation at most probable point for higher-orderreliability analysis”, International Journal of Solids and Structures, Vol. 43 No. 9, pp. 2820-2839.
Structuralreliability
2211
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Rahman, S. and Xu, H. (2004), “A univariate dimension-reduction method for multi-dimensionalintegration in stochastic mechanics”, Probabilistic Engineering Mechanics, Vol. 19 No. 4,pp. 393-408.
Rajashekhar, M.R. and Ellingwood, B.R. (1993), “A new look at the response-surface approach forreliability-analysis”, Structural Safety, Vol. 12 No. 3, pp. 205-220.
Romero, V.J., Swiler, L.P. and Giunta, A.A. (2004), “Construction of response surfaces. based onprogressive-lattice-sampling experimental designs with application to uncertainty propagation”,Structural Safety, Vol. 26 No. 2, pp. 201-219.
Roussouly, N., Petitjean, F. and Salaun, M. (2013), “A new adaptive response surface method forreliability analysis”, Probabilistic EngineeringMechanics, Vol. 32, pp. 103-115.
Rupert, C.P. and Miller, C.T. (2007), “An analysis of polynomial chaos approximations for modelingsingle-fluid-phase flow in porous medium systems”, Journal of Computational Physics, Vol. 226No. 2, pp. 2175-2205.
Saad, Y. and van der Vorst, H.A. (2000), “Iterative solution of linear systems in the 20th century”,Journal of Computational and AppliedMathematics, Vol. 123 Nos 1/2, pp. 1-33.
Santner, T., Williams, B. and Notz, W. (2003), The Design and Analysis of Computer Experiments,Springer series in Statistics, Springer, Berlin.
Schobi, R., Sudret, B. and Marelli, S. (2015), “Rare event estimation using polynomial-Chaos-Kriging”,ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering,Vol. 3 No. 2.
Schueller, G.I. and Stix, R. (1987), “A critical-appraisal of methods to determine failure probabilities”,Structural Safety, Vol. 4 No. 4, pp. 293-309.
Schwab, C. and Todor, R.A. (2006), “Karhunen-Loeve approximation of random fields by generalizedfast multipole method”, Journal of Computational Physics, Vol. 217 No. 1, pp. 100-122.
Seo, H.S. and Kwak, B.M. (2002), “Efficient statistical tolerance analysis for general distributions usingthree-point information”, International Journal of Production Research, Vol. 40 No. 4, pp. 931-944.
Sepahvand, K., Marburg, S. and Hardtke, H.J. (2007), “Numerical solution of one-dimensional waveequation with stochastic parameters using generalized polynomial chaos expansion”, Journal ofComputational Acoustics, Vol. 15 No. 4, pp. 579-593.
Shinozuka, M. and Deodatis, G. (1988), “Response variability of stochastic finite-element systems”,Journal of EngineeringMechanics-Asce, Vol. 114 No. 3, pp. 499-519.
Skaggs, T.H. and Barry, D.A. (1996), “Assessing uncertainty in subsurface solute transport: efficientfirst-order reliability methods”, Environmental Software, Vol. 11 No. 1-3, pp. 179-184.
Smolyak, S.A. (1963), “Quadrature and interpolation formulas for tensor products of certain classes offunctions”, Soviet Math Dokl, Vol. 4, pp. 240-243.
Song, S.F., Lu, Z.Z. and Qiao, H.W. (2009), “Subset simulation for structural reliability sensitivityanalysis”, Reliability Engineering & System Safety, Vol. 94 No. 2, pp. 658-665.
Spanos, P.D. and Ghanem, R. (1989), “Stochastic finite-element expansion for random-media”, Journal ofEngineeringMechanics-Asce, Vol. 115 No. 5, pp. 1035-1053.
Srinivas, N., Krause, A., Kakade, S.M. and Seeger, M.W. (2012), “Information-theoretic regret bounds forgaussian process optimization in the bandit setting”, Ieee Transactions on Information Theory,Vol. 58 No. 5, pp. 3250-3265.
Stefanou, G. (2009), “The stochastic finite element method: past, present and future”, ComputerMethods in AppliedMechanics and Engineering, Vol. 198 Nos 9/12, pp. 1031-1051.
Stein, M. (1987), “Large sample properties of simulations using Latin hypercube sampling”,Technometrics, Vol. 29 No. 2, pp. 143-151.
Sudret, B. (2008), “Global sensitivity analysis using polynomial chaos expansions”, ReliabilityEngineering & System Safety, Vol. 93 No. 7, pp. 964-979.
EC35,6
2212
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Sudret, B. and Kiureghian, A.D. (2002), “Comparison of finite element reliability methods”, ProbabilisticEngineeringMechanics, Vol. 17 No. 4, pp. 337-348.
Sudret, B. and Der Kiureghian, A. (2000), “Stochastic finite element methods and reliability-a state-of-the-art report”.
Sudret, B., Berveiller, M. and Lemaire, M. (2003), “Application of a stochastic finite element procedure toreliability analysis”, in Maes, M. and Huyse, L. (Eds), Proceeding 11th IFIP WG7.5 Conferenceon Reliability and Optimization of Structural Systems, Banff, Balkema, Rotterdam, pp. 319-327.
Sudret, B., Berveiller, M. and Lemaire, M. (2006), “A stochastic finite element procedure for momentand reliability analysis”, European Journal of Computational Mechanics, Vol. 15 Nos 7-8,pp. 825-866.
Sudret, B., Blatman, G. and Berveiller, M. (2007), “Quasi random numbers in stochastic finite elementanalysis – application to global sensitivity analysis”, Applications of Statisics and Probability inCivil Engineering, Vol. 8 No. 3, pp. 257-258.
Taguchi, G. (1978), “Performance analysis design”, International Journal of Production Research,Vol. 16 No. 6, pp. 521-530.
Tee, K.F., Khan, L.R. and Li, H.S. (2014), “Application of subset simulation in reliability estimation ofunderground pipelines”, Reliability Engineering & System Safety, Vol. 130, pp. 125-131.
Tootkaboni, M., Graham-Brady, L. and Schafer, B.W. (2009), “Geometrically non-linear behavior ofstructural systems with random material property: an asymptotic spectral stochastic approach”,ComputerMethods in AppliedMechanics and Engineering, Vol. 198 Nos 37/40, pp. 3173-3185.
Tvedt, L. (1988), “Second-order reliability by an exact integral”, Proceeding of 2nd IFIP WorkingConference on Reliability and Optimization on Structural, System, Thoft-Christensen, P., EditorSpringer,NewYork, NY, pp. 377-384.
Van den Nieuwenhof, B. and Coyette, J.P. (2003), “Modal approaches for the stochastic finite elementanalysis of structures with material and geometric uncertainties”, Computer Methods in AppliedMechanics and Engineering, Vol. 192 Nos 33/34, pp. 3705-3729.
Verhoosel, C.V., Gutierrez, M.A. and Hulshoff, S.J. (2006), “Iterative solution of the random eigenvalueproblem with application to spectral stochastic finite element systems”, International Journal forNumerical Methods in Engineering, Vol. 68 No. 4, pp. 401-424.
Wan, X.L. and Karniadakis, G.E.M. (2006), “Multi-element generalized polynomial chaos for arbitraryprobability measures”, Siam Journal on Scientific Computing, Vol. 28 No. 3, pp. 901-928.
Wang, G.S. and Ang, A.H.S. (1994), “Adaptive kernel-method for evaluating structural systemreliability”, Structural Safety & Reliability, Vol. 5 No. 2, pp. 1495-1500.
Wang, B., Wang, D., Jiang, J., Zhang, J. and Sun, P. (2015), “Efficient functional reliability estimation fora passive residual heat removal system with subset simulation based on importance sampling”,Progress in Nuclear Energy, Vol. 78, pp. 36-46.
Wang, X.Y., Cen, S., Li, C.F. and Owen, D.R.J. (2015), “A priori error estimation for the stochasticperturbation method”, Computer Methods in Applied Mechanics and Engineering, Vol. 286,pp. 1-21.
Wang, X., Cen, S. and Li, C. (2013), “Generalized neumann expansion and its application in stochasticfinite element methods”,Mathematical Problems in Engineering, Vol. 2013, p. 13.
Wang, Y., Cao, Z.J. and Au, S.K. (2011), “Practical reliability analysis of slope stability by advancedMonte Carlo simulations in a spreadsheet”, Canadian Geotechnical Journal, Vol. 48 No. 1,pp. 162-172.
Xiu, D.B. and Karniadakis, G.E. (2002), “The wiener-askey polynomial chaos for stochastic differentialequations”, Siam Journal on Scientific Computing, Vol. 24 No. 2, pp. 619-644.
Xiu, D.B. and Karniadakis, G.E. (2003), “Modeling uncertainty in flow simulations via generalizedpolynomial chaos”, Journal of Computational Physics, Vol. 187 No. 1, pp. 137-167.
Structuralreliability
2213
Dow
nloa
ded
by S
wan
sea
Uni
vers
ity, P
rofe
ssor
Che
nfen
g L
i At 1
4:03
11
Febr
uary
201
9 (P
T)
Xu, H. and Rahman, S. (2004), “A generalized dimension-reduction method for multidimensionalintegration in stochastic mechanics”, International Journal for Numerical Methods inEngineering, Vol. 61 No. 12, pp. 1992-2019.
Yadav, V. and Rahman, S. (2014), “Adaptive-sparse polynomial dimensional decomposition methodsfor high-dimensional stochastic computing”, Computer Methods in Applied Mechanics andEngineering, Vol. 274, pp. 56-83.
Yamazaki, F., Shinozuka, M. and Dasgupta, G. (1988), “Neumann expansion for stochastic finite-element analysis”, Journal of EngineeringMechanics-Asce, Vol. 114 No. 8, pp. 1335-1354.
Yu, H., Chung, C.Y., Wong, K.P., Lee, H.W. and Zhang, J.H. (2009), “Probabilistic load flow evaluationwith hybrid Latin hypercube-sampling and cholesky decomposition”, Ieee Transactions onPower Systems, Vol. 24 No. 2, pp. 661-667.
Zhang, J.F. and Du, X.P. (2010), “A second-order reliability method with first-order efficiency”, Journalof Mechanical Design, Vol. 132 No. 10.
Zhang, Y. and Der Kiureghian, A. (1995), Two Improved Algorithms for Reliability Analysis, inRackwitz, R., Augusti, G. and Borr, A., (Eds), Springer, Boston, MA.
Zhao, Y.G. and Lu, Z.H. (2007), “Applicable range of the fourth-moment method for structuralreliability”, Journal of Asian Architecture and Building Engineering, Vol. 6 No. 1, pp. 151-158.
Zhao, Y.G. and Ono, T. (1999), “New approximations for SORM: Part 2”, Journal of EngineeringMechanics-Asce, Vol. 125 No. 1, pp. 86-93.
Zhao, Y.G. and Ono, T. (2001), “Moment methods for structural reliability”, Structural Safety, Vol. 23No. 1, pp. 47-75.
Zheng, Y. and Das, P.K. (2000), “Improved response surface method and its application to stiffenedplate reliability analysis”, Engineering Structures, Vol. 22 No. 5, pp. 544-551.
Ziha, K. (1995), “Descriptive sampling in structural safety”, Structural Safety, Vol. 17 No. 1, pp. 33-41.Zuev, K.M. and Katafygiotis, L.S. (2011), “Modified metropolis-hastings algorithm with delayed
rejection”, Probabilistic EngineeringMechanics, Vol. 26 No. 3, pp. 405-412.
Corresponding authorChenfeng Li can be contacted at: [email protected]
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