Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration

This paper presents a new module towards the development of efficient computational stochastic mechanics. Specifically, the possibility of an adaptive polynomial chaos expansion is investigated. Adaptivity in this context refers to retaining, through an iterative procedure, only those terms in a representation of the solution process that are significant to the numerical evaluation of the solution. The technique can be applied to the calculation of statistics of extremes for nongaussian processes. The only assumption involved is that these processes be the response of a nonlinear oscillator excited by a general stochastic process. The proposed technique is an extension of a technique developed by the second author for the solution of general nonlinear random vibration problems. Accordingly, the response process is represented using its Karhunen-Loeve expansion. This expansion allows for the optimal encapsulation of the information contained in the stochastic process into a set of discrete random variables. The response process is then expanded using the polynomial chaos basis, which is a complete orthogonal set in the space of second-order random variables. The time dependent coefficients in this expansion are then computed by using a Galerkin projection scheme which minimizes the approximation error involved in using a finite-dimensional subspace. These coefficients completely characterize the solution process, and the accuracy of the approximation can be assessed by comparing the contribution of successive coefficients. A significant contribution of this paper is the development and implimentation of adaptive schemes for the polynomial chaos expansion. These schemes permit the inclusion of only those terms in the expansion that have a significant contribution.     

Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced‐dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

A polynomial chaos method for arbitrary random inputs using B-splines

Isogeometric analysis which extends the finite element method through the usage of B-splines has become well established in engineering analysis and design procedures. In this paper, this concept is considered in context with the methodology of polynomial chaos as applied to computational stochastic mechanics. In this regard it is noted that many random processes used in several applications can be approximated by the chaos representation by truncating the associated series expansion. Ordinarily, the basis of these series are orthogonal Hermite polynomials which are replaced by B-spline basis functions. Further, the convergence of the B-spline chaos is presented and substantiated by numerical results. Furthermore, it is pointed out, that the B-spline expansion is a generalization of the Legendre multi-element generalized polynomial chaos expansion, which is proven by solving several stochastic differential equations.     

An iterative polynomial chaos approach toward stochastic elastostatic structural analysis with non-Gaussian randomness

Stochastic analysis of structure with non-Gaussian material property and loading in the framework of polynomial chaos (PC) is considered. A new approach for the solution of stochastic mechanics problem with random coefficient is presented. The major focus of the method is to consider reduced size of expansion in an iterative manner to overcome the problem of large system matrix in conventional PC expansion. The iterative method is based on orthogonal expansion of stochastic responses and generation of an iterative PC based on the responses of the previous iteration. The polynomials are evaluated using Gram-Schmidt orthogonalization process. The numbers of random variables in PC expansion are reduced by considering only the dominant components of the response characteristics, which is evaluated using Karhunen-Loève (KL) expansion. In case of random material field problem, the KL expansion is used to discretize and simulate the non-Gaussian random field. Independent component analysis (ICA) is carried out on the non-Gaussian KL random variables to minimize statistical dependence. The usefulness of the proposed method in terms of accuracy and computational efficiency is examined. From the numerical analysis of three different types of structural mechanics problems, the proposed iterative method is observed to be computationally more efficient and accurate than conventional PC method for solution of linear elastostatic structural mechanics problems.     

基于同伦分析方法的随机结构静力响应求解

该文提出了一种基于同伦分析方法的求解含随机参数结构的静力响应的新方法。该方法将随机静力平衡方程重新进行同伦构造,利用含随机变量和趋近函数的同伦级数展式来表示结构的随机静力位移响应,该同伦级数的各阶确定性系数和趋近函数可通过对一系列的变形方程求解得到。由于趋近函数的引入,该同伦级数解相较于传统的摄动法有更大的收敛范围,对于含较大变异性随机参数的结构也能获得不错的求解精度。同时,该文提出了一种降维策略来提高该方法的计算效率。数值算例表明,与目前广泛应用的广义正交多项式展开法（GPC）相比,从计算精度上看,该文方法的3阶展开与GPC 2阶展开相当,该文方法的6阶展开与GPC 4阶展开相当,而计算时间上前者均明显少于后者。此外,该文方法也可以方便地应用到随机结构的几何非线性分析当中,并具有较好的计算精度和计算效率。     

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

A methodology is introduced for rapid reduced‐order solution of stochastic partial differential equations. On the random domain, a generalized polynomial chaos expansion (GPCE) is used to generate a reduced subspace. GPCE involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the Askey series. A proper orthogonal decomposition (POD) approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain. POD methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas. They have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom. This concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations (PDEs) in natural convection processes involving randomness in the porosity of the medium and the Rayleigh number. The results indicate that owing to the multiplicative nature of the concurrent model reduction, extremely large computational gains are realized without significant loss of accuracy. Copyright © 2005 John Wiley & Sons, Ltd.     

Global sensitivity analysis by polynomial dimensional decomposition

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.     

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.     

Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions

Given their mathematical structure, methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory. This is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition. This enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non‐continuous dependence of the solution on the stochastic parameters. In particular, given the polynomial structure of these approximations, the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated. An enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non‐smooth functions by integrating a priori knowledge about their behaviour. The proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues. It is observed that through judicious selection of enrichment functions, the spectrum of such a random system can be more efficiently characterized, even for systems with closely spaced eigenvalues. Copyright © 2007 John Wiley & Sons, Ltd.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic fracture mechanics using polynomial chaos

Crack propagation in metals has long been recognized as a stochastic process. As a consequence, crack propagation rates have been modeled as random variables or as random processes of the continuous. On the other hand, polynomial chaos is a known powerful tool to represent general second order random variables or processes. Hence, it is natural to use polynomial chaos to represent random crack propagation data: nevertheless, no such application has been found in the published literature. In the present article, the large replicate experimental results of Virkler et al. and Ghonem and Dore are used to illustrate how polynomial chaos can be used to obtain accurate representations of random crack propagation data. Hermite polynomials indexed in stationary Gaussian stochastic processes are used to represent the logarithm of crack propagation rates as a function of the logarithm of stress intensity factor ranges. As a result, crack propagation rates become log-normally distributed, as observed from experimental data. The Karhunen–Loève expansion is used to represent the Gaussian process in the polynomial chaos basis. The analytical polynomial chaos representations derived herein are shown to be very accurate, and can be employed in predicting the reliability of structural components subject to fatigue.     

Identification of composite uncertain material parameters from experimental modal data

Stochastic analysis of structures using probability methods requires the statistical knowledge of uncertain material parameters. This is often quite easier to identify these statistics indirectly from structure response by solving an inverse stochastic problem. In this paper, a robust and efficient inverse stochastic method based on the non-sampling generalized polynomial chaos method is presented for identifying uncertain elastic parameters from experimental modal data. A data set on natural frequencies is collected from experimental modal analysis for sample orthotropic plates. The Pearson model is used to identify the distribution functions of the measured natural frequencies. This realization is then employed to construct the random orthogonal basis for each vibration mode. The uncertain parameters are represented by polynomial chaos expansions with unknown coefficients and the same random orthogonal basis as the vibration modes. The coefficients are identified via a stochastic inverse problem. The results show good agreement with experimental data.     

Dynamic response of tunnels in stochastic soils by the boundary element method

A stochastic boundary element method (SBEM) is developed in this work for evaluating the dynamic response of underground openings excited by seismically induced, horizontally polarized shear waves under steady-state conditions. The surrounding geological medium is viewed as an elastic continuum exhibiting large randomness in its mechanical properties, which implies that the wave number of the propagating signal is a function of a random variable. Suitable Green's functions are proposed and used within the context of the SBEM formulation. More specifically, a series expansion for the Green's functions is employed, where the basis functions are orthogonal polynomials of a random argument (polynomial chaos). These are subsequently incorporated in the SBEM formulation, which employs the usual quadratic, isoparametric line elements for modeling the surfaces of the problem in question. Finally, this formulation is used for the solution of a few problems of engineering interest involving buried cavities (tunnels). We note that the present approach departs from earlier boundary element derivations based on perturbations, which are valid for ‘small’ amounts of randomness in the elastic continuum.     

Robust topology optimization of phononic crystals with random field uncertainty

The uncertain spatial variation of material properties can remarkably affect the band gap characteristics of phononic crystals (PnCs). It is necessary to consider this issue when designing and manufacturing PnC materials/structures. This paper investigates a robust topology optimization method for designing the microstructures of PnCs by considering random‐field material properties. Herein, the spatial distribution of the material properties is first represented by a random field and then discretized into uncorrelated stochastic variables with the expansion optimal linear estimation method; stochastic band gap analysis is then conducted with polynomial chaos expansion. Furthermore, a robust topology optimization formulation of PnCs is proposed on the basis of the relative elemental density, where a weighted objective function handles the compromise of the mean value and standard deviation of the PnC band gap. The band gap response is analyzed, employing the finite element method for each sample of polynomial chaos expansion. In this context, the sensitivities of the stochastic band gap behaviors to the design variables are also derived. Numerical examples demonstrate that the proposed method can generate meaningful optimal topologies of PnCs with a relatively large width and less sensitive band gap. Additionally, the effects of the weight factors in the objective function and the variation coefficient of material properties are discussed.     

Hybridization of stochastic reduced basis methods with polynomial chaos expansions

We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost.     

A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty

The present paper proposes a method for stochastic problems that have uncertainty in the boundary geometry. The method is developed by applying the spectral stochastic approach to the boundary element method and is called the spectral stochastic boundary element method (SSBEM). In the SSBEM, the uncertainty in the boundary geometry is represented by the Karhunen–Loève expansion. It is shown that, by utilizing material derivative, variation of boundary element matrices associated with the geometrical fluctuation of the boundary can be approximated by the Taylor expansion. The solution is represented by a stochastic process expressed in the form of polynomial chaos expansion. The stochastic equation is then projected on a homogeneous chaos space. This procedure reduces a stochastic equation to an ordinary linear matrix equation that can be solved by conventional schemes. The SSBEM can estimate not only mean values and variances of the solutions but also their probability density functions. In order to examine the performance, the SSBEM is applied to two-dimensional elastostatic and elastodynamic problems with geometrical boundary uncertainty. Computation results of the SSBEM exhibit good agreement with those obtained by Monte Carlo simulation. The efficiency of the SSBEM is verified by comparison of their computation times.     

An adaptive spectral Galerkin stochastic finite element method using variability response functions

A methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

Fundamental solutions for SH-waves in a continuum with large randomness

Fundamental solutions for horizontally polarized shear (SH) waves propagating in continuous medium with an arbitrarily large random wave number are derived herein. These fundamental solutions, or Green's functions, besides being useful in their own right, also serve as kernel functions for integral equation formulations that can be used in the numerical solution of elastic wave scattering problems of practical importance. Thus, the present work serves as an extension of earlier derivations of boundary integral equation statements based on the perturbation approach by removing the assumption of small fluctuations of key medium properties about their mean values. The methodology developed here is based on a series expansion of the fundamental solutions of the SH wave equation under time harmonic conditions using an orthogonal polynomial basis (polynomial chaos) for the randomness. The position-dependent coefficients of this expansion are subsequently found from the resulting vector wave equation, which is uncoupled through use of the eigensolution of its system matrix. Finally, some representative cases are solved and the results are contrasted with those obtained by the perturbation method. At the same time, the accuracy of the solution to the number of terms used in the polynomial expansion is investigated.     

Collocation-based stochastic finite element analysis for random field problems

A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM.