On the dual iterative stochastic perturbation‐based finite element method in solid mechanics with Gaussian uncertainties

The main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation‐based finite element method for both linear and nonlinear problems in solid mechanics. A general‐order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth‐order probabilistic characteristics are derived thanks to the 10th‐order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto‐plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber‐reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi‐analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd‐order terms simply vanish, it can be extended towards non‐Gaussian processes as well and completed with all the perturbation orders. Copyright © 2015 John Wiley & Sons, Ltd.     

A generalized stochastic perturbation technique for plasticity problems

The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is provided in two different ways: (i) via the straightforward differentiation of the initial incremental equation and (ii) using the modified response surface method. This methodology is illustrated with the analysis of the elasto-plastic plane truss with random Young’s modulus leading to the determination of the probabilistic moments by the hybrid stochastic symbolic-finite element method computations.     

A new probabilistic distribution matching PATC formulation using polynomial chaos expansion

In the existing probabilistic hierarchical optimization approaches, such as probabilistic analytical target cascading (PATC), all the stochastic interrelated responses are characterized only by the first two statistical moments. However, due to the high nonlinear relation between the inputs and outputs, the interrelated responses are not necessarily normally distributed. The existing approaches, therefore, may not accurately quantify the probabilistic characteristics of the interrelated responses, and would further prevent achieving the real optimal solution. To overcome this deficiency, a novel PATC approach, named PATC-PCE is developed. By employing the polynomial chaos expansion (PCE) technique, the entire distribution of interrelated response can be characterized by a PCE coefficients vector, and then matched and propagated in the hierarchy. Comparative studies show that PATC-PCE outperforms PATC in terms of yielding more accurate optimal solutions and fewer design cycles when the interrelated response are random non-normal quantities, while at a sacrifice of extra function evaluations.     

Application of the generalized perturbation-based stochastic boundary element method to the elastostatics

This paper is entirely devoted to the demonstration of a solution for some boundary value problems of isotropic linear elastostatics with random parameters using the boundary element method. The stochastic perturbation technique in its general nth-order Taylor series expansion version is used to express all the random parameters and the state functions of the problem. These expansions inserted in the classical deterministic equilibrium statement return up to the nth-order (both PDEs and matrix) equations. Contrary to the previous implementations of the stochastic perturbation technique, any order partial derivatives with respect to the random input are derived from the deterministic structural response function (SRF) at a given point. This function is approximated using polynomials by the least-squares method from the multiple solution of the initial deterministic problem solved for the expectations of random structural parameters. First two probabilistic moments have been computed symbolically here using the computational MAPLE environment, also as the polynomial expressions including perturbation parameter ε. It should be mentioned that such a generalized perturbation approach makes it possible to analyze all types of random variables (not only Gaussian) and to compute even higher probabilistic moments with a priori given accuracy. The entire methodology can be implemented after minor modifications to analyze nonlinear phenomena for both statics and dynamics of even heterogeneous domains.     

On the dynamic stochastic response of FE models

A numerical procedure to compute the mean and covariance matrix of the response of nonlinear structures modeled by large FE models is presented. Non-white, non-zero mean, non-stationary Gaussian distributed excitation is represented by the well known Karhunen–Loéve expansion, which allows to describe any type of non-white Gaussian excitation in contrast to filtered white noise which might not be easily adjusted to available statistical data. The solution procedure differs considerably from standard methodologies using a state space representation. In the proposed approach, step-by-step integration procedures developed for deterministic FE analysis are applied to compute the first two moments of the stochastic response.     

A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics

A new, generalized, multivariate dimension‐reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N‐dimensional response function into at most S‐dimensional functions, where S?N; an approximation of response moments by moments of input random variables; and a moment‐based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. 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. Eight numerical examples involving elementary mathematical functions and solid‐mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension‐reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first‐ and second‐order Taylor expansion methods, statistically equivalent solutions, quasi‐Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension‐reduction method is comparable to that of the fourth‐order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd.     

Stochastic response moments for linear systems

State space moment analysis is developed as a practical tool for investigating the response of a linear system subjected to stochastic excitation. General formulations are presented to show that the method can be used to evaluate response moments, or cumulants, of any order for both stationary and nonstationary response. The limitation is that the excitation of the linear system must be a generalized white noise called a delta-correlated process. This generalization of the Lyapunov method for finding response covariances gives a comparable matrix method for finding the higher order moments which are often important in predicting failure due to first-passage or fatigue. The technique used here involves rewriting the mth order tensor of mth order cumulants into a minimum length vector, making use of all inherent symmetry, in order to minimize the size of the resulting matrix. Easily implemented algorithms are presented for finding the terms in this matrix. General relationships are also given relating the eigenvalues and eigenvectors of this matrix for mth order cumulants to those of a much smaller matrix. This eigen solution is needed for evaluating nonstationary response cumulants, and the given relationships provide a particularly efficient method for evaluating the eigenvalues. The method is illustrated by evaluating the 35 fourth cumulants of nonstationary response for a class of two-degree-of-freedom oscillators.     

Path integral, functional method, and stochastic dynamical systems

The path-integral approach to dynamical behavior of systems subject to Gaussian white noise is presented in a straightforward manner. Starting from the Chapman-Kolmogorov equation, the transition probability density, and therefore moments and other statistics of the random response are ultimately expressed in terms of functional integrals over the sample-path space. Accordingly, various characteristic functions are replaced by a single generating functional from which moments of all orders are simply calculated through functional differentiation. This generating functional is proven to satisfy a closed system of functional differential equations. These equations are solved in the case of linear systems, their generating functional being obtained in explicit form. Also given in this paper is an integral equation satisfied by the probability densities. Three kinds of approximation method, namely perturbation expansion, Feynman's variational method, and the WKB method, are developed based on the path-integral formalism. They can be used to study the transient as well as stationary behavior of nonlinear systems.     

Filter approach to the stochastic analysis of MDOF wind-excited structures

In this paper, an approach useful for stochastic analysis of the Gaussian and non-Gaussian behavior of the response of multi-degree-of-freedom (MDOF) wind-excited structures is presented. This approach is based on a particular model of the multivariate stochastic wind field based upon a particular diagonalization of the power spectral density (PSD) matrix of the fluctuating part of wind velocity. This diagonalization is performed in the space of eigenvectors and eigenvalues that are called here wind-eigenvalues and wind-eigenvectors, respectively. From the examination of these quantities it can be recognized that the wind-eigenvectors change slowly with frequency while the first wind-eigenvalue dominates all the others in the low-frequency range. It is shown that the wind field can be modeled in a satisfactory way by taking the first wind-eigenvector as constant and by retaining only the first eigenvalue in the calculations. The described model is then used for stochastic analysis in the time domain of MDOF wind-excited structures. This is accomplished by modeling each element of the diagonalized wind-PSD matrix as the velocity PSD function of a set of second-order digital filters with viscous damping driven by white noise of selected intensity. This approach makes it easy to obtain in closed form the statistical moments of every order of the structural response, taking full advantage of the Itô calculus. Moreover, in the proposed approach, it is possible to reduce the computational effort by appropriately selecting the number of wind modes retained in the calculation.     

Small noise expansion of moment lyapunov exponents for two-dimensional systems

We construct an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. As an example, we study the moment stability of the stationary solution of nonlinear structural and mechanical systems subjected to real noise excitation. The usefulness of the moment Lyapunov exponent in predicting parameter values at which qualitative changes in the probability density function occur (stochastic bifurcation) is also illustrated.     

Probabilistic uncertainty analysis by mean-value first order Saddlepoint Approximation

Probabilistic uncertainty analysis quantifies the effect of input random variables on model outputs. It is an integral part of reliability-based design, robust design, and design for Six Sigma. The efficiency and accuracy of probabilistic uncertainty analysis is a trade-off issue in engineering applications. In this paper, an efficient and accurate mean-value first order Saddlepoint Approximation (MVFOSA) method is proposed. Similar to the mean-value first order Second Moment (MVFOSM) approach, a performance function is approximated with the first order Taylor expansion at the mean values of random input variables. Instead of simply using the first two moments of the random variables as in MVFOSM, MVFOSA estimates the probability density function and cumulative distribution function of the response by the accurate Saddlepoint Approximation. Because of the use of complete distribution information, MVFOSA is generally more accurate than MVFOSM with the same computational effort. Without the nonlinear transformation from non-normal variables to normal variables as required by the first order reliability method (FORM), MVFOSA is also more accurate than FORM in certain circumstances, especially when the transformation significantly increases the nonlinearity of a performance function. It is also more efficient than FORM because an iterative search process for the so-called Most Probable Point is not required. The features of the proposed method are demonstrated with four numerical examples.     

A spectral stochastic element free Galerkin method for the problems with random material parameter

This paper presents a spectral stochastic element free Galerkin method (SSEFGM) for the problems involving a random material property. The random material property and resulting system response quantity are represented by a probabilistic spectral expansion techniques (Karhunen–Loeve expansion and Polynomical Chaos series, respectively) and implemented into the element free Galerkin (EFG) analysis. Numerical solutions in 1D linear elastic problem with random elastic modulus are introduced, and compared with those of Monte Carlo simulation (MCS) so as to provide the validation of the proposed approach. The present SSEFGM approach can produce a probabilistic density distribution as well as a first‐ and second‐order statistical moments (mean and variance) of response quantity by a single calculation, which is distinguished from an iterative MCS. Moreover, the method is based on an element free analysis so that there is no need of nodal connectivities, which usually require more time and labourious task than main calculations. Thus the proposed SSEFGM approach can provide an alternative analysis tool for the problems contains a stochastic material property, and demands complex mesh structures. Copyright © 2004 John Wiley & Sons, Ltd.     

非高斯随机振动下包装件时变振动可靠性分析

包装件在流通过程中经常受到非高斯随机振动激励的作用,提出了一种包装件在非高斯随机振动激励条件下的时变可靠性的分析方法。结合多项式混沌扩展和Karhunen-Loeve扩展,提出了基于功率谱(或自相关函数)、均值、方差、偏斜度和峭度信息的非高斯随机振动激励的模拟方法;为减小数值分析量,应用拟蒙特卡洛法,在随机变量空间中合理控制变量的分布模拟非高斯随机振动激励,通过四阶龙格库塔法分析,用较少的随机振动模拟样本准确得到了包装件加速度响应的前四阶矩和自相关函数。基于响应的统计信息,应用该研究提出的多项式混沌扩展、Karhunen-Loeve扩展和拟蒙特卡洛分析,获得包装件加速度响应样本,计算包装件的时变可靠性,用原始蒙特卡洛法验证了计算的准确性;该方法在包装件的可靠性分析、包装系统优化等方面具有重要意义。     

Improved dynamic analysis of structures with mechanical uncertainties under deterministic input

This paper addresses the dynamic analysis of linear systems with uncertain parameters subjected to deterministic excitation. The conventional methods dealing with stochastic structures are based on series expansion of stochastic quantities with respect to uncertain parameters, by means of either Taylor expansion, perturbation technique or Neumann expansion and evaluate the first- and second-order moments of the response by solving deterministic equations. Unfortunately, these methods lead to significant error when the coefficients of variation of uncertainties are relatively large. Herein, an improved first-order perturbation approach is proposed, which considers the stochastic quantities as the sum of their mean and deviation. Comparisons with conventional second-order perturbation approach and Monte Carlo simulations illustrate the efficiency of the proposed method. Applications are discussed in order to investigate the influence of mass, damping and stiffness uncertainty on the dynamic response of the system.     

基于遗传算法的单矢量水听器多目标方位估计

矢量水听器能同时获得声场中某一点的声压标量和质点振速矢量,获得了比常规声压水听器更多的信息。矢量水听器自身是一个空间共点阵,具有一定的空间指向性,这些特点使矢量信号处理技术与声压信号处理技术具有重大差异。根据单个矢量水听器多目标分辨的数学模型,即声压和振速的偶次阶矩组成的非线性联立方程组,研究了该方程的解算方法,给出了可以使用遗传算法求解该非线性方程组的结论和计算精度。     

A classical iterative theory based on the Langevin equation for two-dimensional nonlinear terahertz spectroscopy

A classical iterative theory based on the Langevin equation is presented to obtain the nonlinear response of a system and simulate two-dimensional (2D) nonlinear terahertz (THz) spectroscopy (2DTS). Compared with the widely used method of calculating the multi-time correlation function or the Poison brackets, we start from the classical Langevin equation and use an iterative method to obtain any order of nonlinear response. The anharmonic potential (AHP) and nonlinear coordinates dependence of the dipole moment (NDM) are two types of nonlinear sources introduced here. Results are derived for general three-pulse processes with nonlinear sources, AHP or NDM, separately and with the combination of both. Only the simulative 2DTS results for the single mode case with impulsive incident THz fields are presented.     

Comparison of finite element reliability methods

The spectral stochastic finite element method (SSFEM) aims at constructing a probabilistic representation of the response of a mechanical system, whose material properties are random fields. The response quantities, e.g. the nodal displacements, are represented by a polynomial series expansion in terms of standard normal random variables. This expansion is usually post-processed to obtain the second-order statistical moments of the response quantities. However, in the literature, the SSFEM has also been suggested as a method for reliability analysis. No careful examination of this potential has been made yet. In this paper, the SSFEM is considered in conjunction with the first-order reliability method (FORM) and with importance sampling for finite element reliability analysis. This approach is compared with the direct coupling of a FORM reliability code and a finite element code. The two procedures are applied to the reliability analysis of the settlement of a foundation lying on a randomly heterogeneous soil layer. The results are used to make a comprehensive comparison of the two methods in terms of their relative accuracies and efficiencies.     

Explicit thickness integration for three-dimensional shell elements applied to non-linear analysis

The problem of multilayered degenerated 3-D shell elements for which the numerical integration is performed for each ply is that of the high generation time in non-linear analysis when the number of plies is important. But these elements give accurate results for thin and moderately thick shells, so in order to reduce the generation time explicit thickness integration is investigated. We first write an expansion of the strain-displacement matrix in power series of the thickness variable in order to obtain explicit expressions of the tangent stiffness matrix and internal force vector, appearing in the non-linear formulation. Explicit expressions of non-linear stiffness matrices are presented, using the explicit integration-first approximation. Simple expressions of several matrices, sub-matrices and vectors appearing in the formulation are given here in order to obtain an important computing-time gain. Next, some numerical validation tests comparing the classical element with numerical thickness integration and this one are discussed to prove validity of this formulation.     

Computation of higher‐order moments of generalized polynomial chaos expansions

Because of the complexity of fluid flow solvers, non‐intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher‐order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin‐type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third‐order, indeed fourth‐order moments of the polynomials are needed in this analysis. Besides, both intrusive and non‐intrusive approaches call for their computation provided that higher‐order moments of the quantities of interest need to be post‐processed. In most applications, they are evaluated by Gauss quadratures and eventually stored for use throughout the computations. In this paper, analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature‐free procedures instead. Matlab^© codes have been developed for this purpose and tested by comparisons with Gauss quadratures. Copyright © 2017 John Wiley & Sons, Ltd.