A Meshless Method for Computational Stochastic Mechanics

Abstract

Abstract This paper presents a stochastic meshless method for probabilistic analysis of linear-elastic structures with spatially varying random material properties. Using Karhunen-Loève (K-L) expansion, the homogeneous random field representing material properties was discretized by a set of orthonormal eigenfunctions and uncorrelated random variables. Two numerical methods were developed for solving the integral eigenvalue problem associated with K-L expansion. In the first method, the eigenfunctions were approximated as linear sums of wavelets and the integral eigenvalue problem was converted to a finite-dimensional matrix eigenvalue problem that can be easily solved. In the second method, a Galerkin-based approach in conjunction with meshless discretization was developed in which the integral eigenvalue problem was also converted to a matrix eigenvalue problem. The second method is more general than the first, and can solve problems involving a multi-dimensional random field with arbitrary covariance functions. In conjunction with meshless discretization, the classical Neumann expansion method was applied to predict second-moment characteristics of the structural response. Several numerical examples are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and the Monte Carlo simulation. Since mesh generation of complex structures can be far more time-consuming and costly than the solution of a discrete set of equations, the meshless method provides an attractive alternative to the finite element method for solving stochastic-mechanics problems. Keywords: Element-Free Galerkin MethodKarhunen-Loève ExpansionMeshless MethodNeumann ExpansionRandom FieldStochastic Finite Element MethodWavelets The authors would like to acknowledge the financial support of the U.S. National Science Foundation (Grant No. CMS-9900196). Dr. Ken Chong was the Program Director. Notes a u 1 and u 2 represent horizontal and vertical displacements, respectively. ϵ11 and ϵ22 represent normal strains in x 1 and x 2 directions, respectively; and ϵ 12 represents shear strain. a u 1 and u 2 represent horizontal and vertical displacements, respectively. ϵ11 and ϵ22 represent normal strains in x 1 and x 2 directions, respectively; and ϵ 12 represents shear strain. a u 1 and u 2 represent horizontal and vertical displacements, respectively. ϵ11 and ϵ22 represent normal strains in x 1 and x 2 directions, respectively; and ϵ 12 represents shear strain. a u 1 and u 2 represent horizontal and vertical displacements, respectively. ϵ11 and ϵ22 represent normal strains in x 1 and x 2 directions, respectively; and ϵ 12 represents shear strain.