About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 215109, 15 pages
http://dx.doi.org/10.1155/2012/215109
Research Article

Multiresolution Analysis for Stochastic Finite Element Problems with Wavelet-Based Karhunen-Loève Expansion

Karlsruhe Institute of Technology (KIT), Institute of Engineering Mechanics, Kaiserstraße 12, Building 10.23, 76131 Karlsruhe, Germany

Received 28 October 2011; Revised 12 March 2012; Accepted 25 March 2012

Academic Editor: M. D. S. Aliyu

Copyright © 2012 Carsten Proppe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Multiresolution analysis for problems involving random parameter fields is considered. The random field is discretized by a Karhunen-Loève expansion. The eigenfunctions involved in this representation are computed by a wavelet expansion. The wavelet expansion allows to control the spatial resolution of the problem. Fine and coarse scales are defined, and the fine scales are taken into account by projection operators. The influence of the truncation level for the wavelet expansion on the computed reliability is documented.

1. Introduction

Reliability analysis for problems involving random parameter fields is concerned with the solution of stochastic elliptic boundary value problems. For the solution of stochastic boundary value problems, the random parameter field has to be discretized. This can be accomplished either by a Karhunen-Loève expansion or a projection on a polynomial basis [1, 2]. As a result of this discretization, the random parameter field can be approximated by an expression containing only a finite number of random variables. The reliability problem reduces then to the computation of failure probabilities with respect to a finite number of random variables.

After introduction of the discretization, the stochastic boundary value problem becomes equivalent to a deterministic one, for which the approximation on the physical domain, and on the stochastic domain can be treated differently. The deterministic boundary value problem can be solved by standard approximation methods. On the physical domain, finite element (FE) approximations are prevalent. Galerkin projection methods or collocation schemes on the stochastic domain give rise to stochastic Galerkin methods [1] and stochastic collocation schemes [35]. Their theoretical foundation has been laid in [69], where local and global polynomial chaos expansions were investigated and where a priori error estimates have been proved for a fixed number of terms of the Karhunen-Loève expansion. Approximate solutions of the stochastic boundary value problem can be viewed as local stochastic response surfaces [10] that depend on three parameters: a discretization parameter for the physical domain, a discretization parameter for the stochastic domain, and the discretization level of the input random field.

Although the focus of many investigations concerning the approximate solution of stochastic boundary value problems lies on the computation of the first- and second-order moments of the solution, already, in [1], reliability computation techniques were described, that are based on series representations of the response distribution, on the reliability index, or on Monte Carlo simulation methods. In most of the papers that followed, for example, [11, 12] and references therein, either the reliability index, or simulation methods has been employed. Alternatively, once the random field is discretized, sampling-based response surfaces [13, 14] can be applied as well. Except for the stochastic Galerkin method, all methods provide nonintrusive algorithms, which allow combining the solution procedure with repetitive runs of an FE solver for deterministic problems.

In this paper, the focus is on the discretization of the random field itself. The discretization of the random field may enforce a rather fine FE mesh and thus increase the computational effort. The Karhunen-Loève expansion of the random field requires the solution of an eigenvalue problem for the determination of the expansion functions. Recently, a wavelet expansion of the eigenfunctions has been introduced [15]. The advantage of wavelet bases resides in the fact that they are of localized compact support, which lead to sparse representations of functions and integral operators. Moreover, the discrete wavelet transform provides an efficient means to solve the integral equation related to the determination of the eigenfunctions. Wavelet expansions have also been studied in the context of simulation of random fields [16, 17] and for the solution of stochastic dynamic systems by polynomial chaos expansion [18].

Here, the wavelet representation of the eigenfunctions is introduced into the stochastic FE approximation procedure in order to control the spatial resolution of the eigenfunctions. This leads to a multiresolution approximation scheme, where finer scales are taken into account by projection operators. The truncation level of the wavelet expansion has been identified as an additional parameter of the metamodel. As the multiresolution analysis is based on the eigenfunctions involved in the Karhunen-Loève expansion of the random field, it can be combined with stochastic Galerkin, stochastic collocation, or sampling-based methods. In this paper, only the latter are considered in the examples.

The paper is organized as follows: the next section gives an introduction to multiresolution analysis. Section 3 discusses random field discretization by Karhunen-Loève expansion with emphasis on wavelet approximations of the eigenfunctions. Section 4 provides basic information on the approximate solution of the stochastic boundary value problem. Section 5 explains the proposed multiresolution approximation, and Section 6 gives some details on reliability assessment. In Section 7, the proposed procedure is validated by examples and relative errors for the failure probability are given. Finally, conclusions are drawn in Section 8.

2. Multiresolution Analysis

A multiresolution analysis is a sequence of subspaces 𝑉𝑗𝐿2(), {0}𝑉2𝑉1𝑉0𝑉1𝑉2𝐿2(), with𝑗𝑉𝑗=𝐿2(),𝑗𝑉𝑗={0},𝑓()𝑉𝑗2𝑓𝑗𝑉0,(2.1) generated by a scaling function 𝜑𝐿2() via 𝜑𝑗,𝑘(𝑥)=2𝑗/2𝜑(2𝑗𝑥𝑘) and 𝑉𝑗=span{𝜑𝑗,𝑘𝑘}. Especially, from 𝜑𝑉0𝑉1, there is a sequence {𝑘}𝑘 of real numbers, such that𝜑(𝑥)=2𝑘𝑘𝜑(2𝑥𝑘).(2.2) For every 𝑗, denote with 𝑊𝑗 the orthogonal complement of 𝑉𝑗 in 𝑉𝑗1: 𝑉𝑗1=𝑉𝑗𝑊𝑗. It follows that 𝑉𝑚=𝑗𝑚+1𝑊𝑗, 𝐿2()=𝑗𝑊𝑗 and that 𝑓()𝑊𝑗𝑓(2𝑗)𝑊0.

A wavelet is a function 𝜓𝐿2() with0<||||𝜓(𝜔)2|𝜔|d𝜔<,(2.3) where𝜓(𝜔)=lim𝑛(2𝜋)1/2𝑛𝑛𝜓(𝑥)exp(𝑖𝑥𝜔)d𝑥.(2.4) It can be shown [19] that there exists a wavelet 𝜓(𝑥), such that 𝜓𝑗,𝑘(𝑥)=2𝑗/2𝜓(2𝑗𝑥𝑘) is an orthonormal basis for 𝑊𝑗. Thus, any function 𝑓𝐿2() can be decomposed into𝑓=𝑃𝑚𝑓+𝑚𝑗=𝑘𝑓,𝜓𝑗,𝑘𝜓𝑗,𝑘,(2.5) where 𝑃𝑚 is the projection operator on 𝑉𝑚. As we are interested in functions with compact support, the second sum in (2.5) is finite. For 𝑚=0, we have after truncation at level 𝜈𝑓(𝑥)=𝑘𝑓,𝜑0,𝑘𝜑0,𝑘(𝑥)+0𝑗=𝜈𝑘𝑓,𝜓𝑗,𝑘𝜓𝑗,𝑘+𝑒(𝑥),(2.6) where the error 𝑒(𝑥) is an element of 𝑊(𝜈+1). Note that all sums are finite, if 𝑓(𝑥) has compact support.

The same representation can be obtained for functions 𝐿2(𝑑) with 𝑑. For example, for 𝑑=2, the functions𝜙𝑗,𝑘1,𝑘2(𝑥,𝑦)=2𝑗𝜑2𝑗𝑥𝑘1𝜑2𝑗𝑦𝑘2,𝑘1,𝑘2,(2.7) constitute an orthonormal basis of 𝑉𝑗 and the functions𝜓𝑗,𝑘1,𝑘2(𝑥,𝑦)=2𝑗𝜓2𝑗𝑥𝑘1𝜑2𝑗𝑦𝑘2,𝑘1,𝑘2𝜓,𝜐𝑗,𝑘1,𝑘2(𝑥,𝑦)=2𝑗𝜑2𝑗𝑥𝑘1𝜓2𝑗𝑦𝑘2,𝑘1,𝑘2𝜓,𝑑𝑗,𝑘1,𝑘2(𝑥,𝑦)=2𝑗𝜓2𝑗𝑥𝑘1𝜓2𝑗𝑦𝑘2,𝑘1,𝑘2,(2.8) where superscript stands for horizontal, 𝜐 for vertical, and 𝑑 for diagonal translation of the unidimensional wavelet 𝜓(𝑥), constitute an orthonormal basis of 𝑊𝑗.

3. Karhunen-Loève Discretization of Random Fields

Let 𝐷 be a convex bounded open set in 𝑛 and (Ω,,𝑃) a complete probability space, where Ω is the set of outcomes, the 𝜎-field of events, and 𝑃[01] a probability measure.

We consider a random field 𝛼𝐷×Ω that has a continuous and square-integrable covariance function𝐶(𝑥,𝑦)=Ω([𝛼]([𝛼](𝛼(𝑥,𝜔)𝐸𝑥))(𝛼(𝑦,𝜔)𝐸𝑦))d𝑃(𝜔),(3.1) where the expectation operator 𝐸[𝛼](𝑥)=Ω𝛼(𝑥,𝜔)d𝑃(𝜔) denotes the mean value of the random field. It is assumed that 𝛼(𝑥,𝜔) is bounded and coercive, that is, there exist positive constants 𝑎min, 𝑎max, such that𝑃𝜔Ω𝑎min<𝛼(𝑥,𝜔)<𝑎max𝑥𝐷=1.(3.2) Due to the properties of the covariance function, the operator 𝑇𝐿2(𝐷)𝐿2(𝐷),𝑇𝑤=𝐷𝐶(𝑥,𝑦)𝑤(𝑥)d𝑥,(3.3) is compact and self-adjoint and thus admits a spectrum of decreasing nonnegative eigenvalues {𝜆𝑖}𝑖=1,𝐷𝐶(𝑥,𝑦)𝑤𝑖(𝑥)d𝑥=𝜆𝑖𝑤𝑖(𝑦).(3.4) The corresponding eigenfunctions {𝑤𝑖(𝑥)}𝑖=1 are orthonormal in 𝐿2(𝐷). The random variables given by𝜉𝑖(1𝜔)=𝜆𝑖𝐷([𝛼](𝛼(𝑥,𝜔)𝐸𝑥))𝑤𝑖(𝑥)d𝑥(3.5) are uncorrelated (but in general not independent), have zero mean and unit variance, and allow to represent the random field by the Karhunen-Loève expansion[𝛼](𝛼(𝑥,𝜔)=𝐸𝑥)+𝑖=1𝜆𝑖𝜉𝑖(𝜔)𝑤𝑖(𝑥),(3.6) that converges in 𝐿2(𝐷×Ω) [20]. Conditions for stronger convergence properties are given in [8]. The Karhunen-Loève expansion is usually truncated by retaining only the first 𝑀 terms. In order to keep the computational effort small, a fast decay of the spectrum of (3.3) is important. It is shown in [21] that fast eigenvalue decay corresponds to smoothness of the covariance function.

The solution of the Fredholm integral equation (3.4) can be computed by means of the wavelet basis introduced before (cf., e.g., [15]). For the 𝑘th eigenfunction 𝑤𝑘(𝑥), an approximation with a finite number of basis functions Ψ𝑖(𝑥), 𝑖=1,2,,𝑛, of 𝑉0 and 𝑊𝑗, 0𝑗𝜈, is given by𝑤𝑘(𝑥)=𝑛𝑖=1𝑑𝑖(𝑘)Ψ𝑖(𝑥).(3.7) A Galerkin technique can be applied in order to compute the eigenvalue 𝜆𝑘 and the coefficients of the normalized (𝐷𝑤2𝑘(𝑥)𝑑𝑥=1) approximate eigenfunctions. To this end, the representation (3.7) is inserted into the Fredholm integral equation (3.4) and the equation is multiplied with a basis function Ψ𝑗(𝑦). After integration over the domain 𝐷, one obtains the algebraic eigenvalue problem𝑛𝑖=1𝐷𝐶(𝑥,𝑦)Ψ𝑖(𝑥)Ψ𝑗(𝑦)d𝑥d𝑦𝑑𝑖(𝑘)=𝜆𝑘𝑑𝑗(𝑘),𝑗=1,2,𝑛,(3.8) where the coefficients on the left hand side can be obtained from discrete wavelet transforms of the covariance function 𝐶(𝑥,𝑦).

More details and convergence studies for this approximation of the eigenfunctions may be found in [15]. Here, the decomposition is introduced in order to obtain a coarse approximation (by taking scales until a certain level 𝜇>𝜈 into account) and a fine approximation (scales from 𝜇1 to 𝜈).

4. Stochastic Linear Elliptic Boundary Value Problems

Consider the following model problem with stochastic operator and deterministic input function on 𝐷×Ω: find 𝑢𝐷×Ω, such that 𝑃 almost surely:𝛼(𝑥,𝜔)𝑢(𝑥,𝜔)=𝑔(𝑥)on𝐷,𝑢(𝑥,𝜔)=0on𝜕𝐷.(4.1) It is assumed that the deterministic input function 𝑔(𝑥) is square-integrable.

We are interested in the probability that a functional 𝐹(𝑢) of the solution 𝑢(𝑥,𝜔) exceeds a threshold 𝐹0, that is, we want to evaluate the integral𝑃𝐹=Ω𝜒(𝐹0,)(𝐹(𝑢(𝑥,𝜔)))d𝑃(𝜔),(4.2) where 𝜒𝐼(), the indicator function, assumes the value 1 in the interval 𝐼 and vanishes elsewhere.

The variational formulation of the stochastic boundary value problem necessitates the introduction of the Sobolev space 𝐻10(𝐷) of functions having generalized derivatives in 𝐿2(𝐷) and vanishing on the boundary 𝜕𝐷 with norm 𝑢𝐻10(𝐷)=(𝐷|𝑢|2d𝑥)1/2, the space 𝐿2𝑃(Ω) of square integrable random variables and the tensor product space 𝐻10(𝐷)𝐿2𝑃(Ω) of 𝐻10(𝐷)-valued random fields with finite second-order moments, equipped with the inner product(𝑢,𝜐)𝐻10(𝐷)𝐿2(Ω)=Ω𝐷𝑢(𝑥,𝜔)𝑣(𝑥,𝜔)d𝑥d𝑃(𝜔).(4.3) The variational formulation of the stochastic linear elliptic boundary value problem (4.1) then reads find 𝑢𝐻10(𝐷)𝐿2𝑃(Ω), such that for all 𝜐𝐻10(𝐷)𝐿2𝑃(Ω):Ω𝐷𝛼(𝑥,𝜔)𝑢𝜐d𝑥d𝑃(𝜔)=Ω𝐷𝑔(𝑥)𝜐(𝑥,𝜔)d𝑥d𝑃(𝜔).(4.4) The assumptions on the random field 𝛼(𝑥,𝜔) guarantee the continuity and coercivity of the bilinear form in (4.4). Existence and uniqueness of a solution to (4.4) follow from the Lax-Milgram lemma.

For a prescribed, uniformly bounded random field 𝛼(𝑥,𝜔), the random variables 𝜉𝑖(𝜔) in (3.6) would be dependent non-Gaussian random variables whose joint distribution function is very difficult to identify. If, on the other hand, independent but bounded distributions are prescribed for 𝜉𝑖(𝜔), 𝑖=1,2,,𝑀, the random field 𝛼(𝑥,𝜔) is not necessarily bounded for 𝑀. Thus, one is left with Gaussian distributions for 𝜉𝑖(𝜔) and 𝛼(𝑥,𝜔), with transformations of Gaussian random fields [22] or with some situations, where nonnegative distributions for 𝜉𝑖(𝜔) lead to meaningful (e.g., Erlang-) distributions for 𝛼(𝑥,𝜔).

Some authors [11, 23] consider models that contain a finite number of random variables as parameters of a boundary value problem. For this kind of problems, Babuška et al. [8] investigate convergence properties of several approximation schemes. The examples studied in this paper belong to this class of problems. It is henceforth assumed that the random field 𝛼(𝑥,𝜔) is given by a truncated sum𝛼𝑀[𝛼](𝑥,𝜔)=𝐸(𝑥)+𝑀𝑖=1𝜆𝑖𝜉𝑖(𝜔)𝑤𝑖(𝑥),(4.5) where 𝜉𝑖(𝜔) are continuous and independent random variables with zero mean and unit variance, where Γ𝑖=𝜉𝑖(Ω) are bounded intervals in and the parameters 𝜆𝑖 and functions 𝑤𝑖(𝑥) are the first 𝑀 eigenvalues and corresponding eigenfunctions of the operator (3.3). In the examples, 𝜉𝑖(𝜔) are Gaussian random variables truncated at ±3𝜎, 𝜎=1.

The representation of 𝛼𝑀(𝑥,𝜔) by a finite number of random variables allows to consider a deterministic auxiliary problem:𝐸[𝛼](𝑥)+𝑀𝑖=1𝜆𝑖𝑦𝑖𝑤𝑖(𝑥)𝑢(𝑥,𝑦)=𝑔(𝑥),for(𝑥,𝑦)D×Γ,(4.6) where Γ=𝑀𝑖=1Γ𝑖𝑀. From the expression for 𝑢(𝑥,𝑦), one obtains 𝑢(𝑥,𝜔) by replacing the vector 𝑦 with the vector of random variables 𝜉𝑖(𝜔), 𝑖=1,2,,𝑀.

This equation is discretized on finite dimensional approximation spaces for the physical domain. For 𝐻10(𝐷), a family of standard FE approximation spaces 𝑋𝐻10(𝐷) of continuous piecewise linear functions in a regular triangulation 𝒯 of 𝐷 with mesh parameter is considered.

Denote with 𝑁𝑖(𝑥), 𝑖=1,2,,𝑁, a basis of 𝑋𝐻10(𝐷). The solution 𝑢(𝑥,𝑦) is approximated by𝑢(𝑥,𝑦)=𝑁𝑖=1𝑢𝑖(𝑦)𝑁𝑖(𝑥).(4.7) For a fixed value 𝑦, the unknown coefficients 𝑢𝑖(𝑦), 𝑖=1,2,,𝑁, can be computed from the solution of the FE problem𝐊(0)+𝑀𝑠=1𝐊(𝑠)𝑦𝑠𝐮(𝑦)=𝐠,(4.8) where the matrices 𝐊(0), 𝐊(𝑠), 𝑠=1,2,,𝑀 and the vector 𝐠 are given by 𝐾(0)𝑖𝑗=𝐷𝐸[𝛼](𝑥)𝑁𝑖(𝑥)𝑁𝑗(𝐾𝑥)d𝑥,(4.9)(𝑠)𝑖𝑗=𝜆𝑠𝐷𝑤𝑠(𝑥)𝑁𝑖(𝑥)𝑁𝑗𝑔(𝑥)d𝑥,(4.10)𝑖=𝐷𝑔(𝑥)N𝑖(𝑥)d𝑥,𝑖,𝑗=1,2,,𝑁,(4.11) and 𝐮(𝑦) is the vector containing the nodal displacements 𝑢𝑖(𝑦), 𝑖=1,2,,𝑁, for the fixed value 𝑦. The matrices 𝐊(𝑠), 𝑠=1,2,,𝑀, can be interpreted, for example, as FE stiffness matrices for a spatial variation of elastic properties.

5. Multiresolution Approximation

By means of multiresolution analysis, each of the matrices 𝐊(𝑠) in (4.10) can be decomposed as 𝐊(𝑠)=𝐊𝑐(𝑠)+𝐊𝑓(𝑠), where 𝐊𝑐(𝑠) contains the low-frequency content, expressed by the multiresolution scheme until the level 𝜇, while 𝐊𝑓(𝑠) takes the high frequency content of the levels 𝜇1 until 𝜈 into account.

Given a fixed value for 𝑦, the decomposition of 𝐊(𝑠) leads to a decomposition of the global stiffness matrix:𝐊𝑐+𝐊𝑓𝐮=𝐠,(5.1) where 𝐊𝑐=𝐊0+𝑀𝑠=1𝐊𝑐(𝑠) and 𝐊𝑓=𝑀𝑠=1𝐊𝑓(𝑠). Also the solution 𝐮 is decomposed into 𝐮𝑐 and 𝐮𝑓, where 𝐮𝑐 is represented on a coarse mesh and 𝐮𝑓 on a fine mesh. With the help of an interpolation matrix 𝐏, the solution 𝐮 is written as 𝐮=𝐏𝐮𝑐+𝐮𝑓. Inserting these decompositions into equation (5.1) leads to𝐊𝑐𝐏𝐮𝑐+𝐊𝑐𝐮𝑓+𝐊𝑓𝐏𝐮𝑐+𝐊𝑓𝐮𝑓=𝐠.(5.2) This equation is split into 𝐊𝑐𝐏𝐮𝑐𝐊=𝐠,(5.3)𝑐𝐮𝑓+𝐊𝑓𝐏𝐮𝑐+𝐊𝑓𝐮𝑓=𝟎.(5.4) Multiplication of the first equation with 𝐏𝑇 projects this equation onto the coarse mesh:𝐏𝑇𝐊𝑐𝐏𝐮𝑐=𝐏𝑇𝐠.(5.5) For 𝐮𝑓, a coarse mesh approximation is computed in the following manner:(i)the term 𝐊𝑓𝐮𝑓 in (5.4) is neglected;(ii)𝐮𝑓 is represented on the coarse mesh by ̃𝐮𝑓, thus 𝐮𝑓̃𝐮=𝐏𝑓.

After multiplication of (5.4) with 𝐏𝑇, one then obtains𝐏𝑇𝐊𝑐𝐏̃𝐮𝑓=𝐏𝑇𝐊𝑓𝐏𝐮𝑐(5.6) for the coarse mesh approximation ̃𝐮𝑓 of 𝐮𝑓. Thus, the linear system of equations in (5.5) and (5.6) has to be resolved only on the reduced set of degrees of freedom given by the coarse mesh, which saves a considerable amount of CPU time. The interpolation matrix 𝐏 can be generated from FE interpolation functions.

6. Reliability Assessment

Once the algebraic problems are solved and the correction to the coarse scale solution is computed, an approximation for 𝑢(𝑥,𝜔) has been obtained. The approximation quality depends on the following parameters:(i)partition of the physical domain and the stochastic domain,(ii)truncation of the Karhunen-Loève expansion (𝑀),(iii)wavelet scales (𝜇,𝜈).

The wavelet scales 𝜇,𝜈 appear as an additional parameter that influences the approximation quality.

For solving reliability problems, (4.8) yields a functional relationship between the input random variables and 𝑢(𝑥,𝑦). It is then possible to compute the most probable point of failure (MPP), that is, the point 𝜉Γ with 𝐹(𝑢(𝑥,𝜉))=𝐹0 with lowest Euclidean norm. The norm of the MPP may serve as a control variable for the approximation quality and for the adaptation of the parameters mentioned above.

The MPP may also be useful for the evaluation of the integral in (4.2) by means of variance reduced Monte Carlo simulation (importance sampling). To this end, a sampling density ̃𝑝(𝑦) is introduced by shifting the original probability density function 𝑝(𝑦) of the random variables 𝜉𝑖(𝜔), 𝑖=1,,𝑀, to the previously obtained MPP, and (4.2) is approximated by𝑃𝐹𝑁𝑠𝑗=1𝜒(𝐹0,)𝐹𝑢𝑥,𝑦𝑗𝑝𝑦𝑗̃𝑝(𝑦𝑗)𝑦̃𝑝𝑗,(6.1) where the sampling points 𝑦𝑗, 𝑗=1,2,,𝑁𝑠, are generated according to ̃𝑝(𝑦).

7. Examples

7.1. Example 1: Clamped Square Plate

The first example deals with a standard problem for stochastic FE techniques, a clamped thin square plate under uniform in-plane tension 𝑞 (cf. [1]). The problem is depicted in Figure 1. The plate has unit length 𝑙. The product of Young’s modulus and the thickness of the plate are assumed to be an isotropic normal random field with covariance function𝐶𝑥1,𝑦1;𝑥2,𝑦2=𝜎2||𝑥exp1𝑥2||𝑙𝑥||𝑦1𝑦2||𝑙𝑦,(7.1) with standard deviation 𝜎=0.2 and unit mean value. Poisson’s ratio is set to 0.3.

215109.fig.001
Figure 1: Thin square plate under uniform in-plane tension.

A first investigation is concerned with the efficiency of the multiresolution analysis. To this end, relative errors for the maximum longitudinal displacement have been computed by comparing the coarse mesh—coarse level (𝜇=1) solution and the coarse solution corrected by the fine level solution (𝜈=4) to the fine mesh—fine level reference solution (i.e., 𝜇=0, 𝜈=4). Dividing the relative error of the coarse scale solution by the relative error of the corrected solution yields an improvement factor that indicates the efficiency of the multiresolution correction. For a coarse mesh of 2×2 quadrilateral elements only and a fine mesh of 4×4 quadrilateral elements, Figures 2 and 3 display the improvement factors for a varying number of Karhunen-Loève expansion terms and for a variation of the correlation length 𝑙𝑐=𝑙𝑥=𝑙𝑦, respectively. Here and in the following, the fine mesh is generated from the coarse mesh by halving the edge length of the elements. For Figure 2, the correlation length was 𝑙𝑐=0.5, and, for Figure 3, 12 Karhunen-Loève expansion terms have been retained. The figures reveal that the multiresolution scheme is more efficient if fluctuations become more important, that is for a large number of Karhunen-Loève expansion terms and low correlation lengths.

215109.fig.002
Figure 2: Efficiency of the multiresolution scheme with respect to the number of Karhunen-Loève expansion terms.
215109.fig.003
Figure 3: Efficiency of the multiresolution scheme with respect to the correlation length.

Next, the influence of the multiresolution correction on the prediction of failure has been investigated. A threshold value of 1.5 is assumed for the maximum longitudinal displacement, and failure occurs if the maximum longitudinal displacement exceeds this threshold value.

The random field has been discretized by a Karhunen-Loève expansion with 𝑀=4 terms for a correlation length 𝑙𝑐=1. The eigenfunctions were computed either on a coarse (8×8 elements) or a fine (16×16 elements) mesh. A reference solution MPP has been computed on the fine mesh including up to seven scales for the wavelet expansion of the eigenfunctions, and a reference result for the failure probability has been obtained on the fine mesh from importance sampling with 30000 samples. These results allow to evaluate the error due to the multiresolution scheme. To this end, relative errors for the Euclidean norm of the MPP and the failure probability with respect to the reference results have been computed from the multiresolution scheme with one coarse level (𝜇=1) and up to seven fine levels. Figure 4 presents the development of the relative errors for the norm of the design point.

215109.fig.004
Figure 4: Relative error of the norm for the MPP.

Figure 5 displays the relative error of the failure probability. From the figure, an exponential decrease of the error with respect to the number of fine scales involved can be deduced. The coarse mesh solution needed less than half of the CPU time of the fine scale solution.

215109.fig.005
Figure 5: Relative error of the failure probability.
7.2. Example 2: Soil-Structure Interaction

The second example, previously introduced in [24], considers a soil structure interaction problem. In contrast to the first example, it deals with an anisotropic autocorrelation function of the random field and an anisotropic FE mesh. The settlement of a foundation (width 2𝐵=10m), represented by a uniform pressure of 0.2 MPa, on an elastic soil layer of thickness 𝑡=30m lying on a rigid substratum, is investigated. A plane strain deformation of the soil is assumed with linear elastic material properties (cf. Figure 6). The Young’s modulus is assumed to be a random field with mean value 50 MPa, 20% coefficient of variation, and an exponential correlation function as in (7.1) but with correlation length 𝑙𝑥=250m, 𝑙𝑦=100m. Poisson’s ratio is set to 0.3. Six truncated Gaussian random variables (𝑀=6) for the representation of the random field have been considered.

215109.fig.006
Figure 6: Soil structure interaction problem.

Starting from the center of the foundation, the right half of the soil layer is meshed by quadrilateral finite elements until a length of 𝐿=60m, as indicated in Figure 6. The coarse mesh consists of 24 elements, while the fine mesh is divided into 96 elements. The reliability problem deals with the displacement of point A in Figure 6, situated at the center of the foundation and on the surface of the soil, where the vertical displacements of the corresponding deterministic problem (i.e., considering only the mean value of Young’s modulus) attain their maximum. Failure is defined as exceedance of a limit value of 0.1 m for the vertical displacement at point A. Figures 7 and 8 summarize the behavior of the relative errors for the Euclidean norm of the MPP and for the failure probability, respectively. As in the previous example, 𝜇 has been set to −1 and 𝜈 has been varied. It can be seen that, while the error in the norm is relatively small, rather large errors for the probability of failure may occur. As in the previous example, the relative error for the failure probability decreases exponentially with the wavelet level 𝜈; however, this decrease starts only after taking a certain number of fine scales (4 and more) into account.

215109.fig.007
Figure 7: Relative error of the norm for the MPP.
215109.fig.008
Figure 8: Relative error of the failure probability.

8. Conclusions

In this paper, wavelet-based Karhunen-Loève expansion has been extended to multiresolution analysis of stochastic FE problems. The proposed procedure allows to include fine scale corrections of the FE solution by means of coarse mesh computations. Improvement due to these corrections has been highlighted by examples, and it could be seen that the relative error of the failure probability decreases exponentially with the wavelet level.

Although the procedure has been combined in this paper to importance sampling-based computation of failure probabilities, it applies to stochastic Galerkin and stochastic collocation methods as well, because it is solely based on the decomposition of the Karhunen-Loève eigenfunctions. However, it relies on linear problems.

Typical applications for multiresolution analysis in the context of stochastic FE methods arise in multiscale analysis of material properties, especially if there is scale coupling and length scales that correspond to the length scales of the representative volume element have to be taken into account. For this kind of application, the limitation of wavelets to rather simple geometries, as those in the examples, will not be a drawback, since simple geometries are frequently used on the micro- and mesoscale and more complex geometries could be embedded into simple ones for the solution of the Fredholm integral equation.

References

  1. R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, NY, USA, 1991. View at Zentralblatt MATH
  2. D. Xiu and G. E. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential equations,” SIAM Journal on Scientific Computing, vol. 24, no. 2, pp. 619–644, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. Acharjee and N. Zabaras, “A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes,” Computers and Structures, vol. 85, no. 5-6, pp. 244–254, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. J. Baroth, Ph. Bressolette, C. Chauvière, and M. Fogli, “An efficient SFE method using Lagrange polynomials: application to nonsingular mechanical problems with uncertain parameters,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 45–48, pp. 4419–4429, 2007. View at Publisher · View at Google Scholar
  5. S. Huang, S. Mahadevan, and R. Rebba, “Collocation-based stochastic finite element analysis for random field problems,” Probabilistic Engineering Mechanics, vol. 22, no. 2, pp. 194–205, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. M. K. Deb, I. M. Babuška, and J. T. Oden, “Solution of stochastic partial differential equations using Galerkin finite element techniques,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 48, pp. 6359–6372, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. H. G. Matthies and A. Keese, “Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 12–16, pp. 1295–1331, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. I. Babuška, R. Tempone, and G. E. Zouraris, “Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 12–16, pp. 1251–1294, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. I. Babuška, F. Nobile, and R. Tempone, “A stochastic collocation method for elliptic partial differential equations with random input data,” SIAM Journal on Numerical Analysis, vol. 45, no. 3, pp. 1005–1034, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. C. Proppe, “Reliability computation with local polynomial chaos approximations,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 89, no. 1, pp. 28–37, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. K. Choi, R. V. Grandhi, and R. A. Canfield, “Structural reliability under non-Gaussian stochastic behavior,” Computers and Structures, vol. 82, no. 13-14, pp. 1113–1121, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. B. Sudret and A. Der Kiureghian, “Comparison of finite element reliability methods,” Probabilistic Engineering Mechanics, vol. 17, no. 4, pp. 337–348, 2002. View at Publisher · View at Google Scholar · View at Scopus
  13. J. E. Hurtado, “An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory,” Structural Safety, vol. 26, no. 3, pp. 271–293, 2004. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Proppe, “Estimation of failure probabilities by local approximation of the limit state function,” Structural Safety, vol. 30, no. 4, pp. 277–290, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. K. K. Phoon, S. P. Huang, and S. T. Quek, “Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme,” Probabilistic Engineering Mechanics, vol. 17, no. 3, pp. 293–303, 2002. View at Publisher · View at Google Scholar · View at Scopus
  16. K. Gurley and A. Kareem, “Applications of wavelet transforms in earthquake, wind and ocean engineering,” Engineering Structures, vol. 21, no. 2, pp. 149–167, 1999. View at Scopus
  17. B. A. Zeldin and P. D. Spanos, “Random field representation and synthesis using wavelet bases,” Journal of Applied Mechanics, vol. 63, no. 4, pp. 946–952, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, “Uncertainty propagation using Wiener-Haar expansions,” Journal of Computational Physics, vol. 197, no. 1, pp. 28–57, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1992.
  20. M. Loève, Probability Theory, Springer, Berlin, Germany, 1977.
  21. R. A. Todor and C. Schwab, “Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients,” Research Report number 2006-05, Seminar für Angwandte Mathematik, ETH Zürich, Switzerland, 2006.
  22. C. Proppe, “A stochastic finite-element method for transformed normal random parameter fields,” International Journal for Uncertainty Quantification, vol. 1, no. 3, pp. 189–201, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. J. Baroth, L. Bodé, P. Bressolette, and M. Fogli, “SFE method using Hermite polynomials: an approach for solving nonlinear mechanical problems with uncertain parameters,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 44–47, pp. 6479–6501, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. B. Sudret and A. Der Kiureghian, “Stochastic finite element methods and reliability—state of the art,” Tech. Rep. UCB/SEMM-2000/08, Department of Civil & Environmental Engineering, University of California, Berkeley, Calif, USA, 2000.