A sample-based stochastic finite element method for structural reliability analysis
AA sample-based stochastic finite element method forstructural reliability analysis
Zhibao Zheng a,b, ∗ , Hongzhe Dai a,b , Yuyin Wang a,b , Wei Wang a,b a Key Lab of Structures Dynamic Behavior and Control, Harbin Institute of Technology,Ministry of Education, Harbin 150090, China b School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China
Abstract
This paper presents a new methodology for structural reliability analysis viastochastic finite element method (SFEM). A novel sample-based SFEM isfirstly used to compute structural stochastic responses of all spatial pointsat the same time, which decouples the stochastic response into a combina-tion of a series of deterministic responses with random variable coefficients,and solves corresponding stochastic finite element equation through an iter-ative algorithm. Based on the stochastic response obtained by the SFEM,the limit state function described by the stochastic response and the mul-tidimensional integral encountered in reliability analysis can be computedwithout any difficulties, and failure probabilities of all spatial points are cal-culated once time. The proposed method can be applied to high-dimensionalstochastic problems, and one of the most challenging issues encountered inhigh-dimensional reliability analysis, known as Curse of Dimensionality, canbe circumvented without expensive computational costs. Three practical ex- ∗ Corresponding author
Email address: [email protected] (Zhibao Zheng)
Preprint submitted to Journal August 19, 2020 a r X i v : . [ phy s i c s . c o m p - ph ] A ug mples, including large-scale and high-dimensional reliability analysis, aregiven to demonstrate the accuracy and efficiency of the proposed method incomparison to the Monte Carlo simulation. Keywords:
Reliability analysis; Stochastic finite element method; Highdimensions; Stochastic responses
1. Introduction
As a powerful tool to quantify the uncertainty in practical problems,reliability analysis nowadays has become an indispensable cornerstone forsolving complex problems in many fields [1, 2], such as structural design andoptimization, decision management, etc. Despite the progress in existingmethods of modeling and analysis, the estimation of the failure probabilityin reliability analysis is challenging to achieve [3, 4]. On one hand, the mul-tidimensional integral encountered in reliability analysis for computing thefailure probability often lies in high-dimensional stochastic spaces (hundredsto more), which is prohibited because of expensive computational costs. Onanother hand, the failure region, that is, the region of unacceptable systemperformance, is usually complicated and irregular, which leads that the fail-ure surface is rarely known explicitly and only can be evaluated by numericalsolutions.Many methods are proposed in order to evaluate multidimensional inte-grals arising in reliability analysis. The most straightforward method is theMonte Carlo simulation (MCS), which is based on the law of large numbersand almost converges to the exact value when the number of samples is largeenough [5]. The MCS doesn’t depend on the dimension of stochastic spaces,2hus it doesn’t encounter the Curse of Dimensionality. However, the com-putational cost for estimating a small failure probability is expensive, whichis prohibited in practical complex problems. As a very robust technique, itis usually used to check the effectiveness of other methods. Some variationshave been proposed to improve the MCS, such as importance sampling, sub-set simulation, etc [4, 6, 7]. A typically kind of non-sampling methods for reli-ability analysis are First/Second Order Reliability Method (FORM/SORM)[8, 9]. These method are based on first/second order series expansion ap-proximation of the failure surface at the so-called design point, then theresulting approximate integral is calculated by asymptotic method. Thsesmethods generally have good accuracies and efficiencies for low-dimensionaland weakly nonlinear problems, however considerable errors arise in high-dimensional stochastic spaces and nonlinear failure surfaces [10]. Severalmehtods have been proposed to improve the performance of these methods[11]. In order to decrease the computational cost for reliability analysis, sur-rogate model methods are receiving particular attention and continuouslygaining in significance, which are based on a functional surrogate represen-tation as an approximation of the limit state function. The surrogate modelis constructed in an explicit expression via a set of observed points, then thefailure probability can be estimated with cheap computational costs. Theconstruction of the surrogate model is crucial, and available surrogate mod-els include response surface method [12, 13], Kriging method [14], supportvector machine [15], high dimensional model representation [16], polynomialchaos expansion [17, 18, 19], etc.In most practical cases, the limit state function in reliability analysis3uilds a relationship between stochastic spaces of input parameters and thefailure probability via the stochastic response of the system [17, 20, 21], thusthe determination of the stochastic response of the system is crucial. Fordecades, the stochastic finite element method (SFEM), especially the spec-tral stochastic finite element method and its extensions [22, 23, 24, 25, 26],has received particular attentions. As an extension of the classical deter-ministic finite element method to the stochastic framework, SFEM has beenproven efficient both numerically and analytically on numerous problems inengineering and science [27]. In this method, the unknown stochastic re-sponse is projected onto a stochastic space spanned by (generalized) polyno-mial chaos basis, and stochastic Galerkin method is then adopted to trans-form the original finite element equation into a deterministic finite elementequation, whose size can be up to orders of magnitude larger than that ofthe corresponding stochastic problems [22, 23]. Extrme computational costsarise as the number of stochastic dimensions and the number of polynomialchaos expansion terms increase, thus the high resolution solution of SFEM isstill challenging due to the increased memory and computational resourcesrequired, especially for high-dimensional and large-scale stochastic problems[25, 26].In this paper, in order to overcome the difficulties encountered in exist-ing SFEM, we adopt a novel sample-based stochastic finite element method[28] to compute stochastic responses of target systems. In this method, theunknown stochastic response is expanded into a combination of a series ofdeterministic responses with random variable coefficients described by sam-ples. More importantly, it can be applied to high-dimensional and large-scale4tochastic problems with a high accuracy and efficiency. Based on the ob-tained stochastic response, the limit state function and the multidimensionalintegrals in reliability analysis can be computed without any difficulties, andfailure probabilities of all spatial points are calculated once time withoutexpensive computational costs.The paper is organized as follows: Section 2 presents a novel sample-based stochastic finite element method for determining structural stochasticresponses. Reliability analysis based on stochastic responses is described inSection 3. Following this, the algorithm implementation of the proposedmethod is elaborated in Section 4. Three practical problems are used todemonstrate good performances of the proposed method in Section 5. Someconclusions and prospects are discussed in Section 6.
2. Stochastic responses determination using SFEM
As an extension of deterministic finite element method (FEM), SFEMhas become a common tool for computing structural stochastic responses[22, 27]. Modeling random system parameters and environmental sources byuse of random fields [29, 30], it becomes available to integrate discretizationmethods for structural responses and random fields to arrive at a system ofstochastic finite element equations as K ( θ ) u ( θ ) = F ( θ ) (1)where K ( θ ) is the stochastic global stiffness matrix representing propertiesof the physical model under investigation, u ( θ ) is the unknown stochasticresponse and F ( θ ) is the load vector associated with the source terms.5s one of the most important problems of SFEM, it’s a great challengeto compute the high-precision solution of Eq.(1). Spectral stochastic finiteelement method (SSFEM) is a popular method in the past few decades,which represents the stochastic response through polynomial chaos expansion(PCE) and transform Eq.(1) into a deterministic finite element equationby stochastic Galerkin projection [25, 27]. The size of the deterministicfinite element equation is much larger than that of the original problem, thusexpensive computational costs limit SSFEM to low-dimensional stochasticproblems. In order to overcome these difficulties, a novel sample-based SFEMis developed to solve Eq.(1) in [28], which represents the unknown stochasticresponse u ( θ ) as u ( θ ) = k (cid:88) i =1 λ i ( θ ) d i (2)where { λ i ( θ ) } ki =1 and { d i } ki =1 are unknown random variables and unknown de-terministic vectors, respectivey. Solution u ( θ ) is approximated after k termstruncated, and the more terms k retains, the more accurate approximationcan be obtained. In order to compute the couple { λ k ( θ ) , d k } , supposing thatthe k − { λ i ( θ ) , d i } k − i =1 have been obtained and substituting Eq.2 intoEq.(1) yields, K ( θ ) (cid:34) k − (cid:88) i =1 λ i ( θ ) d i + λ k ( θ ) d k (cid:35) = F ( θ ) (3)It’s not easy to determine λ k ( θ ) and d k at the same time. In order to avoidthis difficulty, λ k ( θ ) and d k are computed one after another. For determinedrandom variable λ k ( θ ) (or given as an initial value), d k can be computed by6sing stochastic Galerkin method, which corresponds to E (cid:40) λ k ( θ ) K ( θ ) (cid:34) k − (cid:88) i =1 λ i ( θ ) d i + λ k ( θ ) d k (cid:35)(cid:41) = E { λ k ( θ ) F ( θ ) } (4)where E {·} is the expectation operator. Once d k has been determined inEq.(4), the random variable λ k ( θ ) can be subsequently computed via multi-plying d k on both sides of Eq.(3). It yields d Tk K ( θ ) (cid:34) k − (cid:88) i =1 λ i ( θ ) d i + λ k ( θ ) d k (cid:35) = d Tk F ( θ ) (5)The couple { λ k ( θ ) , d k } can be computed by repeating Eq.(4) and Eq.(5) untilit converges to the required accuracy. For the practical implementation, d k is unitized as d Tk d k = 1, and the convergence error of the couple { λ k ( θ ) , d k } is defined as, ε local = E (cid:8) ( λ k,j ( θ ) d k,j ) − ( λ k,j − ( θ ) d k,j − ) (cid:9) E (cid:8) ( λ k,j ( θ ) d k,j ) (cid:9) = 1 − E (cid:8) λ k,j − ( θ ) (cid:9) E (cid:8) λ k,j ( θ ) (cid:9) (6)which measures the difference between λ k,j ( θ ) and λ k,j − ( θ ) and the calcu-lation is stopped when λ k,j ( θ ) is almost the same as λ k,j − ( θ ). Further, thestop criterion of the number k that are retained of the stochastic solution u ( θ ) is defined as ε global = E (cid:8) u k ( θ ) − u k − ( θ ) (cid:9) E { u k ( θ ) } = 1 − k − (cid:80) i,j =1 E { λ i ( θ ) λ j ( θ ) } d Ti d jk (cid:80) i,j =1 E { λ i ( θ ) λ j ( θ ) } d Ti d j (7)In a practical way, the stochastic global stiffness matrix K ( θ ) and stochas-tic global load vector F ( θ ) in stochastic finite element equation Eq.(1) are7btained by assembling stochastic element stiffness matrices and stochasticelement load vector as K ( θ ) = M (cid:88) l =0 ξ l ( θ ) K l , F ( θ ) = Q (cid:88) m =0 η m ( θ ) F m (8)Based on Eq.(8), Eq.(4) can be simplified and written as˜ K kk d k = Q (cid:88) m =0 h mk F m − k − (cid:88) i =1 ˜ K ik d i (9)where deterministic matrices ˜ K ij are given by˜ K ij = M (cid:88) l =0 c lij K l (10)and coefficients c ijk and h ij are computed by c ijk = E { ξ i ( θ ) λ j ( θ ) λ k ( θ ) } , h ij = E { η i ( θ ) λ j ( θ ) } (11)The size of ˜ K ij in Eq.(10) is the same as the original stochastic finiteelement equation Eq.(1), which can be solved by existing deterministic FEMtechniques [31, 32], thus it is readily applied to large-scale stochastic prob-lems. Similarly, Eq.(5) can be simplified and written as a k ( θ ) λ k ( θ ) = b k ( θ ) (12)where random variables a k ( θ ) and b k ( θ ) are given by a k ( θ ) = M (cid:88) l =0 g klk ξ l ( θ ) , b k ( θ ) = Q (cid:88) m =0 f km η m ( θ ) − k − (cid:88) i =1 M (cid:88) l =0 g kli ξ l ( θ ) λ i ( θ ) (13)and coefficients g ijk and f ij are computed by g ijk = d Ti K j d k , f ij = d Ti F j (14)8he common methods solving Eq.(12) by representing the random vari-able λ k ( θ ) in terms of a set of polynomial chaos have expensive computa-tional costs [25, 27]. In order to avoid this difficulty, a sample-based methodis adopted to determine λ k ( θ ). For sample realizations { θ ( n ) } Nn =1 of all con-sidered random parameters θ , sample matrices of random variables a k ( θ ) and b k ( θ ) are written as ˜ a k ( θ ) = ˜ ξ ( θ ) g k, · ,k , ˜ b k ( θ ) = ˜ η ( θ ) f k − (cid:16) ˜ ξ ( θ ) g k, · , k − (cid:12) ˜ λ ( k − ( θ ) (cid:17) [ ] ( k − × (15)where ˜ a k ( θ ) , ˜ b k ( θ ) ∈ R N × , (cid:12) reprents element-by-element multiplicationof ˜ ξ ( θ ) g k, · , k − and ˜ λ ( k − ( θ ), and the sample matrices of random variables { ξ i ( θ ) } Mi =0 , { η i ( θ ) } Qi =0 , { λ i ( θ ) } k − i =1 are given by ˜ ξ ( θ ) = ξ (cid:0) θ (1) (cid:1) · · · ξ M (cid:0) θ (1) (cid:1) ... ... . . . ...1 ξ (cid:0) θ ( N ) (cid:1) · · · ξ M (cid:0) θ ( N ) (cid:1) ∈ R N × ( M +1) (16) ˜ η ( θ ) = η (cid:0) θ (1) (cid:1) · · · η Q (cid:0) θ (1) (cid:1) ... ... . . . ...1 η (cid:0) θ ( N ) (cid:1) · · · η Q (cid:0) θ ( N ) (cid:1) ∈ R N × ( Q +1) (17) ˜ λ ( k − ( θ ) = λ (cid:0) θ (1) (cid:1) · · · λ k − (cid:0) θ (1) (cid:1) ... . . . ... λ (cid:0) θ ( N ) (cid:1) · · · λ k − (cid:0) θ ( N ) (cid:1) ∈ R N × ( k − (18)and the coefficient matrices are obtained by g k = [ g kij ] ∈ R ( M +1) × k , f k = [ f km ] ∈ R ( Q +1) × (19)where g k, · ,k ∈ R ( M +1) × represents the k -th column of the matrix g k and g k, · , k − ∈ R ( M +1) × k − represents the 1-st column to ( k − g k . By use of the sample realizations ˜ a k ( θ ) and ˜ b k ( θ ), samplerealizations ˜ λ k ( θ ) of the random variable λ k ( θ ) can be obtained by ˜ λ k ( θ ) = ˜ a k ( θ ) ˜ b k ( θ ) (20)Statistics methods are readily introduced to obtain probability character-istics of the random variable λ k ( θ ) from samples ˜ λ k ( θ ). The computationalcost for solving Eq.(20) mainly comes from computing the sample vectors ˜ a k ( θ ) and ˜ b k ( θ ) in Eq.(15). It is very low even for high-dimensional stochas-tic problems, that is, Eq.(15) is insensitive to the dimensions of ˜ ξ ( θ ) and ˜ η ( θ ), which avoid the Curse of Dimensionality to great extent. Hence, theproposed method is particularly appropriate for high-dimensional stochasticproblems in practice.
3. Reliability analysis
Reliability analysis is typically described by a scalar limit state function g ( θ ) and corresponding failure probability P F , which requires the evaluationof the following multidimensional integral [3, 4] P F = (cid:90) g ( θ ) ≤ f ( θ ) dθ (21)where g ( θ ) ≤ f ( θ ) represents the joint probabilitydensity function (PDF) of random variables associated with system param-eters and environmental sources. The integral Eq.(21) for determining thefailure probability is usually difficult to evaluate since the limit state surfaceor failure surface g ( θ ) = 0 has a very complicated geometry and f ( θ ) isdefined in high-dimensional stochastic spaces. In most cases the form of the10imit state function g ( θ ) is not known explicitly and numerical methods areemployed for the evaluation of Eq.(21). Existing reliability analysis meth-ods generally evaluate the failure probability of a single point. For generalpurpose, the spatial limit state function g ( x , θ ) is considered as a vector func-tion of spatial positions x . Similarly, the spatial failure probability function P F ( x ) is defined as P F ( x ) = (cid:90) g ( x ,θ ) ≤ f ( x , θ ) dθ (22)Due to the introduction of spatial positions x , the failure probabilityfunction P F ( x ) in Eq.(22) is more difficult to compute than that in Eq.(21).In fact, g ( x , θ ) is typically represents a complicated relation between theinputs and the failure modes via the solution of a potential highly complexstochastic system. Here, we compute the solution of the complex stochasticsystem by use of SFEM mentioned in Section 2. Considering the stochasticresponse u ( θ ) = k (cid:80) i =1 λ i ( θ ) d i and substituting it into g ( x , θ ) yield g ( x , θ ) = g (cid:32) k (cid:88) i =1 λ i ( θ ) d i (cid:33) (23)where the random parameters of the system are integrated in the randomvariables { λ i ( θ ) } ki =1 and the spatial parameter x is discretized and embeddedin the deterministic vectors { d i } ki =1 . Thus, the failure probability function P F ( x ) in Eq.(22) can be rewritten as P F ( x ) = Pr (cid:34) g (cid:32) k (cid:88) i =1 λ i ( θ ) d i (cid:33) ≤ (cid:35) (24)The most straightforward and efficient way to compute Eq.(24) is MonteCarlo simulation and its variations, where random samples are generated11ccording to the distribution of θ . Numbers of the points land in the failuredomain are counted to estimate the failure probability. Similar to the processof Monte Carlo simulation, we utilize a sample-based method to estimateEq.(24). Random samples (cid:8) λ i (cid:0) θ ( n ) (cid:1)(cid:9) Nn =1 ( N is the number of samples) inEq.(24) have been generated by use of Eq.(20), thus the failure probabilityfunction P F ( x ) can be evaluated in the following form P F ( x ) = 1 N N (cid:88) n =1 I (cid:34) g (cid:32) k (cid:88) i =1 λ i (cid:0) θ ( n ) (cid:1) d i (cid:33)(cid:35) (25)where I ( · ) is the indicator function satisfying I ( s ) = , s ≤ , s > P F ( x i ) for each spatial point x i once time,which provides a simple but effective way to identify multiple failure modesof complex structures. Hence, the proposed method provides an efficient andunified framework for reliability analysis, and is particularly appropriate forhigh-dimensional and complex stochastic problems in practice.12 lgorithm 1 Initialize samples of the random variables (cid:8) ξ l (cid:0) θ ( n ) (cid:1)(cid:9) Nn =1 , l = 1 , · · · , M and (cid:8) η m (cid:0) θ ( n ) (cid:1)(cid:9) Nn =1 , m = 1 , · · · , Q ; while ε global > ε do Initialize samples of the random variable (cid:8) λ k, (cid:0) θ ( n ) (cid:1)(cid:9) Nn =1 ; repeat Compute the response component d k,j by solving Eq.(9); Compute the random variable λ k,j ( θ ) via Eq.(20); until ε local < ε u k ( θ ) = k − (cid:80) i =1 λ i ( θ ) d i + λ k ( θ ) d k , k ≥ end while Compute the spatial limit state function g ( x , θ ) via Eq.(23); Compute the spatial failure probability function P F ( x ) via Eq.(25);
4. Algorithm implementation
The resulting procedures for solving the stochastic finite element equationEq.(1) and computing the the failure probability function P F ( x ) via Eq.(22)are summarized in Algorithm 1, which includes two parts in turn. The firstpart is from step 2 to step 9, which is to compute the stochastic response u ( θ ) and includes a double-loop iteration procedure. The inner loop, whichis from step 4 to step 7, is used to determine the couple of { λ k ( θ ) , d k } , whilethe outer loop, which is from step 2 to step 9, corresponds to recursivelybuilding the set of couples such that the approximate solution u k ( θ ) satisfiesEq.(1). In step 2 and step 7, iteration errors ε global and ε local are calculatedvia Eq.(7) and Eq.(6), and corresponding convergence errors ε and ε are13equired precisions. The second part includes step 10 and step 11, where thelimit state function g ( x , θ ) is generated in step 10 based on the stochasticresponse u ( θ ) obtained in step 8 and the failure probability function P F ( x )is computed in step 11.
5. Numerical examples
In this section, we present three examples, including the reliability analy-sis of a beam-bar frame, the reliability analysis of a roof truss defined in100-dimensional stochastic spaces and the global reliability analysis of aplate, to illustrate the accuracy and efficiency of the proposed method incomparison to 1 × times Monte Carlo simulations. For all consideredexamples, 1 × initial samples for each random variable (cid:8) ξ l (cid:0) θ ( n ) (cid:1)(cid:9) × n =1 , Figure 1: Model of the two-layer frame. η m (cid:0) θ ( n ) (cid:1)(cid:9) × n =1 and (cid:8) λ k, (cid:0) θ ( n ) (cid:1)(cid:9) × n =1 are generated, and the convergenceerrors in step 2 and step 7 of Algorithm 1 are set as ε = 1 × − and ε = 1 × − , respectively. A two-layer frame is shown in Fig.1, which consists of horizontal and ver-tical beams, and is stabilized with diagonal bars. Probability distributionsof independent random variables associated with material properties, geom-etry properties and loads are listed in Table 1. In this example, we considerthe failure probability of a single point and the limit state function g ( θ ) isdefined by the maximum joint displacement of the frame as g ( θ ) = max i (cid:113) u x i + u y i − c · u mean (27)where u mean = mean (cid:16) max i (cid:113) u x i ( θ ) + u y i ( θ ) (cid:17) is the mean value of the max-imum joint displacement, and the scalar c is related to different failure prob- Table 1: Probability distributions of random variables in the Example 5.1. variable description distribution mean variance E beam Youngs modulus of beam normal 210 MPa 0.2 A beam cross-sectional area of beam lognormal 100 mm I beam moment of inertia of beam lognormal 800 mm E bar Youngs modulus of bar normal 210 MPa 0.2 A bar cross-sectional area of bar lognormal 100 mm F load 1 normal 10 kN 0.2 F load 2 normal 10 kN 0.215bilities, that is, the failure probability decreases as the scalar c increases. Inthis paper, the maximum joint displacement of the frame can be identifiedautomatically by the proposed method instead of selecting manually, sincethe proposed method can calculate the stochastic response of all nodes oncetime.In order to compute the failure probability P f , we firstly compute thestochastic response of the frame by using the first part of Algorithm 1. Itis seen from Fig.2 that only 4 iterations can achieve the required precision ε = 1 × − , which demonstrates the fast convergence rate of the proposedmethod. Correspondingly, the number of couples { λ k ( θ ) , d k } that consti-tute the stochastic response is adopted as k = 4, as shown in Fig.3. Withthe increasing of the number of couples, the ranges of corresponding randomvariables are more closely approaching to zero, which indicates that the con- k -6 -4 -2 E rr o r Figure 2: Iteration errors of different retained items. P D F λ d -2 0 2 × -3 λ d -1 0 1 × -3 λ d -5 0 5 × -4 λ Figure 3: Solutions of the couples { λ i ( θ ) , d i } i =1 . c -6 -4 -2 P f Monte CarloSFEMAbsolute Error
Figure 4: Failure probabilities of different scalar c . c are shown in Fig.4, here the scalar c is set from 1.01 to1.16. The failure probability P f computed from the proposed method rangesfrom 10 to 10 − , which is fairly close to that obtained from the MonteCarlo simulation even for a very small failure probability. The absolute errorbetween the proposed method and MC simulation demonstrate the accuracyand efficiency of the proposed method. In this example, we consider that a stochastic wind load acts verticallydownward on a roof truss. As shown in Fig.5, the roof truss from [28] in-cludes 185 spatial nodes and 664 elements, where material properties of allmembers are set as Young’s modulus E = 209GPa and cross-sectional areas × × × . Figure 5: Model of the roof truss. = 16cm . The stochastic wind load f ( x, y, θ ) is a random field with thecovariance function C ff ( x , y ; x , y ) = σ f e −| x − x | / l x −| y − y | / l y , where thevariance σ f = 1 .
2, the correlation lengths l x = l y = 24, and it can be ex-panded into a series form by use of Karhunen-Lo`eve expansion [29, 30, 33]with M -term truncated as f ( x, y, θ ) = M (cid:88) i =0 ξ i ( θ ) √ ν i f i ( x, y ) (28)where { ξ i ( θ ) } Mi =1 are uncorrelated standard Gaussian random variables, ν i and f i ( x, y ) are eigenvalues and eigenfunctions of the covariance function C ff ( x , y ; x , y ), which can be obtained by solving a eigen equation [34], ν = ξ ( θ ) ≡ f ( x, y ) = 10kN.Similar to Example 5.1, we consider the failure probability at the max-imum displacement of the roof truss and the limit state function g ( θ ) isdefined by the maximum displacement as g ( θ ) = max i u i ( θ ) − c · u mean (29)where u mean = mean (cid:16) max i u i ( θ ) (cid:17) is the mean value of the maximum dis-placement, u i ( θ ) are vertical displacements of all spatial nodes, and the scalar c is related to different failure probabilities.A stochastic finite element equation of the stochastic response u ( θ ) isobtained based on the expansion Eq.(28) of the stochastic wind load. Inorder to show the effectiveness of the proposed method for high-dimensionalreliability analysis, we adopt the stochastic dimension M = 100 in Eq.(28).It is seen from Fig.6 that seven iterations can achieve the required preci-sion ε = 1 × − , which indicates the fast convergence rate of the pro-posed method even for very high stochastic dimensions. The deterministic19 k -6 -4 -2 E rr o r Figure 6: Iteration errors of different retained items. d P D F λ d -5 0 5 × -3 λ d -2 0 2 × -3 λ d -2 0 2 × -3 λ d -1 0 1 × -3 λ d -5 0 5 × -4 λ d -4 -2 0 2 4 × -4 λ Figure 7: Solutions of the couples { λ i ( θ ) , d i } i =1 . response components { d i } i =1 and corresponding random variables { λ i ( θ ) } i =1 are shown in Fig.7. The computational time for solving couples { λ i ( θ ) , d i } i =1 in this example is less than half a minute by use of a personal laptop (dual-core, Intel i7, 2.40GHz), which indicates that Algorithm 1 is still less com-putational costs for high-dimensional stochastic problems.20he resulted approximate probability density function (PDF) of the max-imum stochastic displacement of the whole roof truss compared with thatobtained from the Monte Carlo simulation is seen in Fig.8, which indicatesthat the result of seven-term approximation is in very good accordance withthat from the Monte Carlo simulation. Further increasing the number ofcouples won’t significantly improve the accuracy since the series in Eq.(2)has converged. It is noted that, the tail of the probability distribution iscrucial for reliability analysis. The proposed method is sample-based, whichprovides the possibility for high-precision reliability analysis. P D F Monte CarloSFEM
Figure 8: PDF of the maximum displacement
In this example, the scalar parameter c in Eq.29 is set from 1.10 to 1.31,and failure probabilities of different scalar c are shown in Fig.9. The accuracyand efficiency of the proposed method is verified again in comparison to theMonte Carlo simulation. The accuracy is reduced when the failure proba-21ility P f is close to 10 − , but it is still very close to that from the MonteCarlo simulation. In this sense, the Curse of Dimensionality encounteredin high-dimensional reliability analysis, can be overcome successfully withcheap computational costs. c -6 -5 -3 -1 P f Monte CarloSFEMAbsolute Error
Figure 9: Failure probabilities of different scalar c . In this example, we consider a Kirchhoff-Love thin plate subjected to adeterministic distributed load q = − and simply supported on fouredges, which is modified from [35]. As shown in Fig.10, parameters of thisproblem are set as length L = 4m, width D = 2m, thickness t = 0 .
05m andPoisson’s ratio ν = 0 .
3. For the sake of simplicity, we neglect self-weightof the plate and assume Young’s modulus E ( x, y, θ ) as the realization ofa Gaussian random field with mean function µ E = 210GPa and covariance22unction C EE ( x , y ; x , y ) = σ E e −| x − x | / l x −| y − y | / l y with correlation lengths l x = 2m, l y = 4m, σ E = 22GPa, l x = 2m, l y = 4m. Similar to Eq.28, Young’smodulus E ( x, y, θ ) is represented by Karhunen-Lo`eve expansion with 10-term truncated as E ( x, y, θ ) = µ E + (cid:88) i =1 ξ i ( θ ) E i ( x, y ) (30) without a considerable increment in the computational cost, since all computations using the SSFEM withthe projection on the homogeneous chaos approach were completed in terms of minutes. A thin plate with length m, width m and thickness t = 0 . m, simply supported on its four edges, issubjected to a static distributed load, q = − kN/m , as shown in Figure 4.17. The Poisson ratio is set to be ν = 0 . and the self-weight of the plate is neglected.The Young modulus is assumed to be uncertain, so that it is described by a 2D random field withknown mean value µ E = 210 GPa and standard deviation σ E = 22 GPa. The field is described by a 2Dexponential covariance kernel C ( x , x ) = exp ( − | x − y | /l − | x − y | /l ) , with correlation lengthsin x e y , l = 1 and l = 2 . The plate is divided into MZC (Melosh-Zienkiewicz-Cheung) plate finiteelements, then, a total of nodes and degrees of freedom are defined.Figure 4.17 – Love-Kirchhoff plate: Model definition (units in m).A complete presentation of the finite element theory of plates (either Love-Kirchhoff or Reissner-Mindlin)is presented in Oñate (2009b). In this case, the vector of movements is given by u = [ w, θ x , θ y ] T , where w isthe vertical displacement of the plate (deflection), and θ x , θ y are the rotations in the x and y axes, respectively.For instance, using the four-noded MZC plate element, the stiffness matrix and force vector of each elementare given by: K ( e ) = ˆ − ˆ − B T b ( ξ, η ) ˆ D b B b ( ξ, η ) (cid:12)(cid:12)(cid:12) J ( e ) ( ξ, η ) (cid:12)(cid:12)(cid:12) d ξ d η = ωab b a − ν + a b +
710 2 ν + b a +
110 2 ν + a b + · · · a b − ν + ν + b a +
110 4 b a − ν + ν · · · ν + a b + ν a b − ν + · · · ν + a b − ... ... ... ... ... a b − ν + ν + a b − · · · a b − ν + × q= -
10 kN/m Figure 10: Model of the plate.
In this example, we consider the failure probabilities of all spatial points,which can be considered as a global reliability analysis. The global limitstate function g ( x , θ ) is defined by the stochastic displacement of the plateexceeding a critical threshold as, g ( x , θ ) = u ω ( x , θ ) − c · u ω,mean ( x ) (31)where u ω ( x , θ ) is the vertical stochastic displacement field of all spatial nodes, u ω,mean ( x ) = mean ( u ω ( x , θ )) is corresponding mean displacement field of u ω ( x , θ ), and the scalar parameter is adopted as c = 1 . k -6 -4 -2 E rr o r Figure 11: Iteration errors of different retained items.Figure 12: Solutions of the couples { λ i ( θ ) , d i } i =1 . u ( θ ) is intro-duced as u ( θ ) = [ u ω ( θ ) , u x ( θ ) , u y ( θ )] T , which are the vertical displacement,rotations in x and y axes, respectively, then 2583 degrees of freedom are de-fined and corresponding stochastic finite element equation can be obtained.As shown in Fig.11, the required precision ε = 1 × − can be achieved aftersix iterations, which demonstrate that the proposed method can be appliedto large-scale stochastic problems. Fig.12 shows the vertical displacementcomponents { d i } i =1 and corresponding random variables { λ i ( θ ) } i =1 , whichagain indicates that the first few couples dominate the solution even for verycomplex stochastic problems. Figure 13: Failure probability nephogram
Based on the vertical stochastic displacement u ω ( x , θ ) obtained by theproposed method, the global failure probability P f ( x ) of the plate can be25alculated by use of Eq.(25), that is, the step 11 in Algorithm 1. The fail-ure probability nephogram in a discrete form shown in Fig.13 has a goodaccordance with that from the Monte Carlo simulation, which demonstratesthe effectiveness and accuracy of the proposed method for global reliabilityanalysis. It is noted that, failure probabilities P f ( x i ) of all spatial nodes con-stitute the global failure probability P f ( x ) in Fig.13, thus some difficultiesencountered in existing approaches can be circumvented, such as computingthe design point (a point lying on the failure surface which has the highestprobability density among other points on the failure surface). In this way,the proposed method presents a new strategy for reliability analysis.
6. Conclusion
This paper proposes an efficient and unified methodology for structuralreliability analysis and illustrates its accuracy and efficiency using three prac-tical examples. The proposed method firstly compute structural stochasticresponses by using a novel stochastic finite element method and the fail-ure probability is subsequently calculated based on the obtained stochasticresponses. As shown in three considered examples, the proposed methodhas the same implementation procedure for different problems and allowsto deal with high-dimensional and large-scale stochastic problems with verylow computational costs. The Curse of Dimensionality encountered in high-dimensional reliability analysis can thus be circumvented with great success.In addition, the proposed method gives a high-precision solution of globalreliability analysis, which overcomes some difficulties encountered in existingapproaches and provides a new strategy for reliability analysis of complex26roblems. In these senses, the methodology proposed in this paper is partic-ularly appropriate for large-scale and high-dimensional reliability analysis ofpractical interests and has great potential in the reliability analysis in scienceand engineering. In the follow-up research, we hopefully further improve thetheoretical analysis of the proposed method and apply it to a wider range ofproblems, such as reliability analysis of time-dependent problems and non-linear problems [36, 37, 38].
Acknowledgments
This research was supported by the Research Foundation of Harbin In-stitute of Technology and the National Natural Science Foundation of China(Project 11972009). These supports are gratefully acknowledged.