Stochastic finite element analysis of a cable-stayed bridge system with varying material properties

Stochastic seismic finite element analysis of a cable-stayed bridge whose material properties are described by random fields is presented in this paper. The stochastic perturbation technique and Monte Carlo simulation (MCS) method are used in the analyses. A summary of MCS and perturbation based stochastic finite element dynamic analysis formulation of structural system is given. The Jindo Bridge, constructed in South Korea, is chosen as a numerical example. The Kocaeli earthquake in 1999 is considered as a ground motion. During the stochastic analysis, displacements and internal forces of the considered bridge are obtained from perturbation based stochastic finite element method (SFEM) and MCS method by changing elastic modulus and mass density as random variable. The efficiency and accuracy of the proposed SFEM algorithm are evaluated by comparison with results of MCS method. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.     

Simulation of random fields on random domains

This paper focuses on the simulation of random fields on random domains. This is an important class of problems in fields such as topology optimization and multiphase material analysis. However, there is still a lack of effective methods to simulate this kind of random fields. To this end, we extend the classical Karhunen–Loève expansion (KLE) to this class of problems, and we denote this extension as stochastic Karhunen–Loève expansion (SKLE). We present three numerical algorithms for solving the stochastic integral equations arising in the SKLE. The first algorithm is an extension of the classical Monte Carlo simulation (MCS), which is used to solve the stochastic integral equation on each sampled domain. However, such approach demands remeshing each sampled domain and solving the corresponding integral equation, which can become computationally very demanding. In the second algorithm, a domain transformation is used to map the random domain into a reference domain, and only one mesh for the reference domain is required. In this way, remeshing different sample realizations of the random domain is avoided and much computational effort is thus saved. MCS is then adopted to solve the corresponding stochastic integral equation. Further, to avoid the computational effort of MCS, the third algorithm proposed in this contribution involves a reduced-order method to solve the stochastic integral equation efficiently. In this third algorithm, stochastic eigenvectors are represented as a sum of products of unknown random variables and deterministic vectors, where the deterministic vectors are efficiently computed by solving deterministic eigenvalue problems. The random variables and stochastic eigenvalues that appear in this third algorithm are calculated by a reduced-order stochastic eigenvalue problem constructed by the obtained deterministic vectors. Based on the obtained stochastic eigenvectors, the target random field is then simulated and reformulated as a classical KLE-like representation. Finally, three numerical examples are presented to demonstrate the performance of the proposed methods.     

A stochastic finite element method for fatigue reliability analysis of gear teeth subjected to bending

A stochastic finite element method (SFEM) is developed for accurate structural reliability analysis. Using the second-order three-moment reliability analytical model, this method takes into account such random factors as load, material parameters and especially geometry randomness. The calculation of the bending fatigue strength reliability of a cantilever beam is carried out as a numerical example to verify the present method. Monte-Carlo FEM and SFEM based on the first-order second-moment model are used in the example to compare with the proposed method. By incorporating the fatigue theory of gears, the present method is then used to analyze the bending fatigue strength reliability of a spur gear. The effects of random variables' coefficient of variation and skewness and the gear's correction factor (not random variable) on the gear's reliability are also investigated.     

Optimum design criteria for elastic structures subject to random dynamic loads

A new stochastic optimization method, which makes use of a constraint on structural reliability, is proposed for structures subject to dynamic random loads. A minimum weight problem is posed, in which a constraint condition imposes that the failure probability must be smaller than a given admissible level. The failure is determined by the first crossing outside the safe domain of a suitable structural response vector. The method is used to find the optimal shape of an elastic vertical column supporting a fixed mass positioned on the top, subject to a Gaussian filtered stationary stochastic horizontal acceleration process. The column, with variable annular cross-section, is described by a deterministic elastic multi-degree-of-freedom system. It is assumed that failure is reached when its lateral displacement exceeds an acceptable threshold value. Under this constraint, the structural weight is minimized and the optimal shape is determined for different structural conditions.     

A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.     

On probabilistic fatigue models for composite materials

The general purpose of this article is to review the main ideas in fatigue analysis of composites in the context of the application of probabilistic methods, both theoretical and computational. That is why most deterministic concepts of composite materials fatigue are summarized together with stochastic approaches. The application of the perturbation based Stochastic Finite Element Method (SFEM) to fatigue analysis of homogeneous and heterogeneous media is shown. Further, homogenization method in its effective modules approach is proposed below for application in fatigue processes modeling of linear elastic periodic random composites. Considering stochastic character of the analysis, the reliability tools appropriate to multicomponent materials are presented together with the specially adopted brittle and ductile fracture criteria.     

The Horseracing Simulation algorithm for evaluation of small failure probabilities

Over the past decade, the civil engineering community has ever more realized the importance and perspective of reliability-based design optimization (RBDO). Since then several advanced stochastic simulation algorithms for computing small failure probabilities encountered in reliability analysis of engineering systems have been developed: Subset Simulation (Au and Beck (2001) [2]), Line Sampling (Schuëller et al. (2004) [3]), The Auxiliary Domain Method (Katafygiotis et al. (2007) [4]), ALIS (Katafygiotis and Zuev (2007) [5]), etc. In this paper we propose a novel advanced stochastic simulation algorithm for solving high-dimensional reliability problems, called Horseracing Simulation (HRS). The key idea behind HS is as follows. Although the reliability problem itself is high-dimensional, the limit-state function maps this high-dimensional parameter space into a one-dimensional real line. This mapping transforms a high-dimensional random parameter vector, which may represent the stochastic input load as well as any uncertain structural parameters, into a random variable with unknown distribution, which represents the uncertain structural response. It turns out that the corresponding cumulative distribution function (CDF) of this random variable of interest can be accurately approximated by empirical CDFs constructed from specially designed samples. The generation of samples is governed by a process of “racing” towards the failure domain, hence the name of the algorithm. The accuracy and efficiency of the new method are demonstrated with a real-life wind engineering example.     

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poisson's ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.     

Weighted integral method in multi-dimensional stochastic finite element analysis

This paper, using the weighted integral method, proposes a new stochastic finite element method for estimating the response variability of multi-dimensional stochastic systems. Young's modules is considered to have spatial variation and is idealized as a multi-dimensional stochastic field. An essential feature of the proposed method is that the continuous stochastic field is rigorously taken care of by means of weighted integrations to construct element stiffness matrices, as the results, the issue involving the stochastic field is transformed into a problem involving only a few random variables. This may lead to substantial improvement in computational efficiency. Numerical examples show that the proposed SFEM is concluded as an efficient and accurate method.     

随机有限元-最大熵法

本文提出一种用于结构可靠性分析的随机有限元-最大熵法。它是利用随机有限元法计算结构响应量的前几阶矩,然后利用最大熵法拟会响应量的概率分布,据此算出结构的失效概率。此法具有精度较高、计算量较小的优点。     

Non-Deterministic Structural Response and Reliability Analysis Using a Hybrid Perturbation-Based Stochastic Finite Element and Quasi-Monte Carlo Method

The random interval response and probabilistic interval reliability of structures with a mixture of random and interval properties are studied in this paper. Structural stiffness matrix is a random interval matrix if some structural parameters and loads are modeled as random variables and the others are considered as interval variables. The perturbation-based stochastic finite element method and random interval moment method are employed to develop the expressions for the mean value and standard deviation of random interval structural displacement and stress responses. The lower bound and upper bound of the mean value and standard deviation of random interval structural responses are then determined by the quasi-Monte Carlo method. The structural reliability is not a deterministic value but an interval as the structural stress responses are random interval variables. Using a combination of the first order reliability method and interval approach, the lower and upper bounds of reliability for structural elements, series, parallel, parallel-series and series-parallel systems are investigated. Three numerical examples are used to demonstrate the effectiveness and efficiency of the proposed method.     

Quasi-weak and weak formulation of stochastic finite elements on static and dynamic problems—a unifying framework

To model uncertainty of spatial and/or temporal variations widely present in synthetic and natural media, a variety of displacement-based stochastic finite element methods (SFEMs) have been formulated using the standard displacement-based finite elements. In this paper, by distinguishing a quasi-weak form from a weak form in both real and random space, a unifying framework of variational formulation is presented covering both the displacement-based SFEMs and the recently proposed Green-function-based (GFB) SFEM. The study shows that Monte Carlo, perturbation, and weighted integral SFEMs correspond to the quasi-weak form, while the weak form results in spectral SFEM, pseudo-spectral SFEM, and GFB-SFEM. Within the unifying framework, dynamic problems are further addressed especially to demonstrate the unique feature of GFB-SFEM on problems with inputs characterized as random fields or random processes.     

Dimension reduction model for two-spatial dimensional stochastic wind field: Hybrid approach of spectral decomposition and wavenumber spectral representation

A renewed methodology for simulating two-spatial dimensional stochastic wind field is addressed in the present study. First, the concept of cross wavenumber spectral density (WSD) function is defined on the basis of power spectral density (PSD) function and spatial coherence function to characterize the spatial variability of the stochastic wind field in the two-spatial dimensions. Then, the hybrid approach of spectral representation and wavenumber spectral representation and that of proper orthogonal decomposition and wavenumber spectral representation are respectively derived from the Cholesky decomposition and eigen decomposition of the constructed WSD matrices. Immediately following that, the uniform hybrid expression of spectral decomposition and wavenumber spectral representation is obtained, which integrates the advantages of both the discrete and continuous methods of one-spatial dimensional stochastic field, allowing for reflecting the spatial characteristics of large-scale structures. Moreover, the dimension reduction model for two-spatial dimensional stochastic wind field is established via adopting random functions correlating the high-dimensional orthogonal random variables with merely 3 elementary random variables, such that this explicitly describes the probability information of stochastic wind field in probability density level. Finally, the numerical investigations of the two-spatial dimensional stochastic wind fields respectively acting on a long-span suspension bridge and a super high-rise building are implemented embedded in the FFT algorithm. The validity and engineering applicability of the proposed method are thus fully verified, providing a potentially effective approach for refined wind-resistance dynamic reliability analysis of large-scale complex engineering structures.     

Stochastic finite element method applied to non-linear analysis of embankments

The deterministic Finite Element Method (FEM) is a valuable tool for understanding and predicting the mechanical behaviour of earth structures. The main difficulty in the application of this technique generally arises from the large uncertainties affecting the mechanical properties of materials to be introduced in the analysis. In many instances, these parameters should actually be considered as random variables or random fields. The Stochastic Finite Element Method (SFEM) should then be used to assess the results of the analyses in probabilistic terms.In this paper, the usefulness of the SFEM approach for engineering purposes is discussed and illustrated by analyses of embankments constructed by placing successive lifts of compacted soil. Construction materials are assumed to follow a simple non-linear constitutive law (Duncan JM, Chang CY. Non-linear analysis of stress and strain in soils, Journal of the Soils Mechanics and Foundation Division, ASCE 1970;96(5):1629–1653). Stochastic finite element analyses are performed using both the First Order-Second Moment method (FOSM) and Monte Carlo simulations (MC). A simple example shows that SFEM analyses can be useful to evaluate the relative influence of each of the parameters of the constitutive model on the results. Uncertainties affecting displacements, strains and stresses predictions for a large earth dam are also presented.     

On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method

Verification of the computation of local quantities of interest, e.g. the displacements at a point, the stresses in a local area and the stress intensity factors at crack tips, plays an important role in improving the structural design for safety. In this paper, the smoothed finite element method (SFEM) is used for finding upper and lower bounds on the local quantities of interest that are outputs of the displacement field for linear elasticity problems, based on bounds on strain energy in both the primal and dual problems. One important feature of SFEM is that it bounds the strain energy of the structure from above without needing the solutions of different subproblems that are based on elements or patches but only requires the direct finite element computation. Upper and lower bounds on two linear outputs and one quadratic output related with elasticity—the local reaction, the local displacement and the J‐integral—are computed by the proposed method in two different examples. Some issues with SFEM that remain to be resolved are also discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

基于等概率近似变换的 向量型层递响应面法分析结构可靠度

基于正交变换和等概率近似变换,研究建立了随机变量为非高斯互相关的工程结构可靠度分析的向量型层递响应面法。首先利用正交变换将非高斯互相关随机变量变换为互不相关的非高斯标准随机变量,建立结构总体刚度矩阵和荷载列阵,据此定义预处理器并形成预处理随机Krylov子空间,进而利用该空间的层递基向量将结构总体节点位移向量近似展开,建立关于互不相关非高斯标准随机变量的层递响应面;然后根据等概率近似变换,将独立标准正态空间的样本点转换为层递响应面在非高斯空间中的概率配点;最后通过回归分析确定层递响应面待定系数,并利用层递响应面建立极限状态方程求解结构可靠度。分析表明:该文提出的等概率近似变换方法不仅成功地将层递响应面法拓展到非高斯互相关随机变量下的结构可靠度分析,而且方法简便、适用范围广、计算精度和效率较高,具有良好的全域性。     

Dynamic response and reliability analysis of structures with uncertain parameters

An original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.     

Reliability analysis of plane elasticity problems by stochastic spline fictitious boundary element method

In this paper a stochastic spline fictitious boundary element method (SFBEM) is proposed for reliability analysis of plane elasticity problems in conjunction with the advanced first-order-second-moment (AFOSM) method. The AFOSM method has been demonstrated to be a reliable and practical approach to the structural reliability analysis, yielding results of reasonable accuracy for the engineering applications. And as a modified method for the conventional indirect boundary element method, SFBEM can provide accurate numerical solutions at high efficiency in deterministic analyses. For the purpose of structural reliability analysis, SFBEM is introduced during the iteration process of the AFOSM method, to obtain the required values of structural responses and their derivations with respect to the random variables considered. The use of SFBEM in the formulation of the AFOSM method makes it unnecessary to construct an explicit expression to the implicit limit state function of the problem, leading to a higher efficiency and better accuracy. The present approach is validated by comparing calculated solutions with those of Monte Carlo simulation for a number of example problems and a good agreement of the results is achieved.     

First-passage reliability of high-dimensional nonlinear systems under additive excitation by the ensemble-evolving-based generalized density evolution equation

The reliability analysis of high-dimensional stochastic dynamical systems subjected to random excitations has long been one of the major challenges in civil and various engineering fields. Despite great efforts, no satisfactory method with high efficiency and accuracy has been available as yet for high-dimensional systems even when they are linear systems, not to mention generic nonlinear systems. In the present paper, a novel method by imposing appropriate absorbing boundary condition on the newly developed ensemble-evolving-based generalized density evolution equation (EV-GDEE) combined with a feasible numerical method is proposed to capture the time-variant first-passage reliability of high-dimensional systems enforced by additive white noise excitation. In the proposed method, the equivalent drift coefficients in EV-GDEE can be estimated by analytical expression or captured by some representative deterministic dynamic analyses. Further, imposing the absorbing boundary condition and then solving the EV-GDEE, a one-or two-dimensional partial differential equation (PDE), yield the remaining probability density of the response of interest. Consequently, by integrating the remaining probability density, the numerical solution of time-variant first-passage reliability can be obtained. Several numerical examples are illustrated to verify the efficiency and accuracy of the proposed method. Compared to the Monte Carlo simulation, the proposed method is of much higher efficiency. Problems to be further studied are finally discussed.     

System reliability analysis of spatial variance frames based on random field and stochastic elastic modulus reduction method

This paper presents the stochastic elastic modulus reduction method for system reliability analysis of spatial variance frames based on the perturbation stochastic finite element method (PSFEM) and the local average of a random field. The stochastic responses and reliability index of each element of a structural frame are characterized by the PSFEM and the first-order second-moment method, to properly handle the correlation structures and scale of fluctuation of random fields. A strategy of elastic modulus adjustment for the estimation of system reliability is developed to determine the range and magnitude of elastic modulus reduction, by taking the element reliability index as a governing parameter. The collapse mechanism and system reliability index of a stochastic framed structure are determined through iterative computations of the PSFEM. Compared with the failure mode approaches in traditional system reliability analysis, the proposed method avoids two major difficulties, namely the identification of significant failure modes and estimation of the joint probability of failure modes. The influences of the correlation structure and scale of fluctuation of the random field upon system reliability are investigated to demonstrate the accuracy and computational efficiency of the proposed methodology in system reliability analysis of spatial variance frames.