An efficient dimension-adaptive uncertainty propagation approach

In recent years, there has been a growing interest in uncertainty propagation, and a wide variety of uncertainty propagation methods exist in literature. In this paper, an uncertainty propagation approach is developed by using high-dimensional model representation (HDMR) and dimension reduction (DR) method technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. To save the computational cost, a dimension-adaptive version of the additive decomposition is proposed to detect the important component functions to reduce the terms. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Two numerical examples show the efficiency and accuracy of the proposed method.     

Hybrid high dimensional model representation (HHDMR) on the partitioned data

A multivariate interpolation problem is generally constructed for appropriate determination of a multivariate function whose values are given at a finite number of nodes of a multivariate grid. One way to construct the solution of this problem is to partition the given multivariate data into low-variate data. High dimensional model representation (HDMR) and generalized high dimensional model representation (GHDMR) methods are used to make this partitioning. Using the components of the HDMR or the GHDMR expansions the multivariate data can be partitioned. When a cartesian product set in the space of the independent variables is given, the HDMR expansion is used. On the other hand, if the nodes are the elements of a random discrete data the GHDMR expansion is used instead of HDMR. These two expansions work well for the multivariate data that have the additive nature. If the data have multiplicative nature then factorized high dimensional model representation (FHDMR) is used. But in most cases the nature of the given multivariate data and the sought multivariate function have neither additive nor multiplicative nature. They have a hybrid nature. So, a new method is developed to obtain better results and it is called hybrid high dimensional model representation (HHDMR). This new expansion includes both the HDMR (or GHDMR) and the FHDMR expansions through a hybridity parameter. In this work, the general structure of this hybrid expansion is given. It has tried to obtain the best value for the hybridity parameter. According to this value the analytical structure of the sought multivariate function can be determined via HHDMR.     

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.     

Two-stage stochastic lot-sizing problem under cost uncertainty

For production planning problems, cost parameters can be uncertain due to marketing activities and interest rate fluctuation. In this paper, we consider a single-item two-stage stochastic lot-sizing problem under cost parameter uncertainty. Assuming cost parameters will increase or decrease after time period p each with certain probability, we minimize the total expected cost for a finite horizon problem. We develop an extended linear programming formulation in a higher dimensional space that can provide integral solutions by showing that its constraint matrix is totally unimodular. We also project this extended formulation to a lower dimensional space and obtain a corresponding extended formulation in the lower dimensional space. Final computational experiments demonstrate that the extended formulation is more efficient and performs more stable than the two-stage stochastic mixed-integer programming formulation.     

The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

A Galerkin finite element method is developed for the two dimensional/three dimensional nonlinear time-dependent three-species Lotka–Volterra competition-diffusion equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation are proved. An error estimate for the numerical solution is obtained. Numerical computations are carried out to examine the expected orders of accuracy in the error estimates.     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs

In this paper we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations. The control objective is to minimize the expectation of a tracking cost functional, and the control is of the deterministic, distributed type. The main analytical tool is the Wiener-Itô chaos or the Karhunen-Loève expansion. Mathematically, we prove the existence of an optimal solution; we establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations; we represent the input data in their Wiener-Itô chaos expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method; we also derive error estimates in terms of both types of discretizations.     

具有白噪声的随机格点系统的随机吸引子的Kolmogorov熵

基于耗散的随机格点系统解的渐近行为理论,主要运用元素分解法与有限维空间中多面体球覆盖的拓扑性质,研究了具有白噪声的随机Klein-Gordon-Schrdinger格点动力系统的随机吸引子的Kolmogorov熵,并得到它的一个上界.     

Stochastic equations and ergodicity for two-type continuous-state branching processes with immigration in Lévy random environments

This paper establishes a stochastic differential equation system with both positive and negative jumps and proves the existence and uniqueness of the strong solution and presents an equivalent condition for ergodicity of the solution. The strong solution is called two-type continuous-state branching processes with immigration in Lévy random environments. The model can be extended to any finite dimensional case.     

Parallel implementation of the p‐version of the finite element method for elliptic equations on a shared‐memory architecture

An implementation of the p‐version of the finite element method for solving two‐dimensional linear elliptic problems on a shared‐memory parallel computer is analyzed. The idea is to partition the problem among the available processors and perform computations corresponding to different elements in parallel. The parallelization is based on a domain decomposition technique using the Lagrange multipliers. The numerical experiments carried out on the Sequent system indicate very high performance of the mixed finite element algorithm in terms of attained speedups. This revised version was published online in August 2006 with corrections to the Cover Date.     

Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C⁰ finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.     

Stochastic multi-scale finite element based reliability analysis for laminated composite structures

This paper proposes a novel multi-scale approach for the reliability analysis of composite structures that accounts for both microscopic and macroscopic uncertainties, such as constituent material properties and ply angle. The stochastic structural responses, which establish the relationship between structural responses and random variables, are achieved using a stochastic multi-scale finite element method, which integrates computational homogenisation with the stochastic finite element method. This is further combined with the first- and second-order reliability methods to create a unique reliability analysis framework. To assess this approach, the deterministic computational homogenisation method is combined with the Monte Carlo method as an alternative reliability method. Numerical examples are used to demonstrate the capability of the proposed method in measuring the safety of composite structures. The paper shows that it provides estimates very close to those from Monte Carlo method, but is significantly more efficient in terms of computational time. It is advocated that this new method can be a fundamental element in the development of stochastic multi-scale design methods for composite structures.     

基于Monte-Carlo随机有限元方法的随机边界条件下自然对流换热不确定性研究

为分析边界条件不确定性对方腔内自然对流换热的影响,发展了一种求解随机边界条件下自然对流换热不确定性传播的Monte-Carlo随机有限元方法.通过对输入参数场随机边界条件进行Karhunen-Loeve展开及基于Latin(拉丁)抽样法生成边界条件随机样本,数值计算了不同边界条件随机样本下方腔内自然对流换热流场与温度场,并用采样统计方法计算了随机输出场的平均值与标准偏差.根据计算框架编写了求解随机边界条件下方腔内自然对流换热不确定性的MATLAB随机有限元程序,分析了随机边界条件相关长度与方差对自然对流不确定性的影响.结果表明:平均温度场及流场与确定性温度场及流场分布基本相同;随机边界条件下Nu数概率分布基本呈现正态分布,平均Nu数随着相关长度和方差增加而增大;方差对自然对流换热的影响强于相关长度的影响.     

频域有限元计算的扩展面向目标误差估计

研究了针对频域有限元直接动态分析的面向目标误差估计以及误差范围估计计算方法．面向目标的误差估计方法就是专门针对如何准确和经济地估算特定值误差的一种方法,利用原问题的共轭偶问题进行计算．频域有限元的直接动态分析是模拟频域扫描实验的一种计算方法,专门针对谐振激励的线性动态响应问题,利用将原自由度分解为实部和虚部描述频率的变化,从而计算变形体的动态响应．利用扩展针对有限元的面向目标误差估计的自由度,将该方法应用到直接动态分析中进行误差估计．通过建立同时包含实部和虚部自由度的能量弱形式及偶问题,并将其数值实现,估算频域直接动态分析有限元解的误差及误差范围,并通过悬臂梁的激振算例进行了验证．     

A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates

Probabilistic analysis is becoming more important in mechanical science and real-world engineering applications. In this work, a novel generalized stochastic edge-based smoothed finite element method is proposed for Reissner–Mindlin plate problems. The edge-based smoothing technique is applied in the standard FEM to soften the over-stiff behavior of Reissner–Mindlin plate system, aiming to improve the accuracy of predictions for deterministic response. Then, the generalized nth order stochastic perturbation technique is incorporated with the edge-based S-FEM to formulate a generalized probabilistic ES-FEM framework (GP_ES-FEM). Based upon a general order Taylor expansion with random variables of input, it is able to determine higher order probabilistic moments and characteristics of the response of Reissner–Mindlin plates. The significant feature of the proposed approach is that it not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also overcomes the inherent drawbacks of conventional second-order perturbation approach, which is satisfactory only for small coefficients of variation of the stochastic input field. Two numerical examples for static analysis of Reissner–Mindlin plates are presented and verified by Monte Carlo simulations to demonstrate the effectiveness of the present method.     

Stochastic analysis of dynamical systems with delayed control forces

Reduction of structural vibration in actively controlled dynamical system is usually performed by means of convenient control forces dependent of the dynamic response. In this paper the existent studies will be extended to dynamical systems subjected to non-normal delta-correlated random process with delayed control forces. Taylor series expansion of the control forces has been introduced and the statistics of the dynamical response have been obtained by means of the extended Itô differential rule. Numerical application provided shows the capabilities of the proposed method to analyze stochastic dynamic systems with delayed actions under delta-correlated process contrasting statistics of response with estimates from Monte-Carlo (MC) simulation.     

Numerical Prediction of Macroscopic Material Failure

The aim of this contribution is the numerical determination of macroscopic material properties based on constitutive relationships characterising the microscale. A macroscopic failure criterion is computed using a three dimensional finite element formulation. The proposed finite element model implements the Strong Discontinuity Approach (SDA) in order to include the localised, fully nonlinear kinematics associated with the failure on the microscale. This numerical application exploits further the Enhanced–Assumed–Strain (EAS) concept to decompose additively the deformation gradient into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. This finite element formulation is then used for the modelling of the microscale and for the discretisation of a representative volume element (RVE). The macroscopic material behaviour results from numerical computations of the RVE. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于嵌入式多项式混沌展开法的随机边界下流动与传热问题不确定性量化

提出了一种嵌入式多项式混沌展开(polynomial chaos expansion, PCE)的随机边界条件下流动与传热问题不确定性量化方法及有限元程序框架.该方法利用Karhunen-Loeve展开表达随机输入边界条件,以及嵌入式多项式混沌展开法表达输出随机场;同时利用谱分解技术将控制方程转化为一组确定性控制方程,并对每个多项式混沌进行求解得到其统计特征.与Monte-Carlo法相比,该方法能够准确高效地预测随机边界条件下流动与传热问题的不确定性特征,同时可以节省大量计算资源.