International Journal of Computational Methods in Engineering Science and Mechanics, 6:41–58, 2005 Copyright c  Taylor & Francis Inc. ISSN: 15502287 print / 15502295 online DOI: 10.1080/15502280590888649 A Meshless Method for Computational Stochastic Mechanics S. Rahman and H. Xu Department of Mechanical Engineering, The University of Iowa, Iowa City, IA, USA This paper presents a stochastic meshless method for proba- bilistic analysis of linear-elastic structures with spatially varying random material properties. Using Karhunen-Lo` eve (K-L) expan- sion, the homogeneous random field representing material prop- erties was discretized by a set of orthonormal eigenfunctions and uncorrelated random variables. Two numerical methods were de- veloped for solving the integral eigenvalue problem associated with K-L expansion. In the first method, the eigenfunctions were ap- proximated 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 con- verted 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 charac- teristics 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 simu- lation. 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 alter- native to the finite element method for solving stochastic-mechanics problems. Keywords Element-Free Galerkin Method, Karhunen-Lo` eve Expan- sion, Meshless Method, Neumann Expansion, Random Field, Stochastic Finite Element Method, Wavelets Received 5 November 2002; accepted 10 May 2003. 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. Address correspondence to S. Rahman, Department of Mechani- cal Engineering, The University of Iowa, Iowa City, IA 52242, USA. E-mail: rahman@engineering.uiowa.edu 1. INTRODUCTION In recent years, much attention has been focused on colloca- tion [1, 2] or Galerkin-based [3–8] meshfree methods to solve computational mechanics problems without using a structured grid. Among these methods, the element-free Galerkin method (EFGM) [4] is particularly appealing, due to its simplicity and its use of a formulation that corresponds to the well-established fi- nite element method (FEM). Similar to other meshless methods, EFGM employs moving least-squares approximation [9] that permits the resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. Since no element connectiv- ity data are needed, burdensome meshing or remeshing required by FEM is avoided. This issue is particularly important for crack propagation in solids for which FEM may become ineffective in addressing substantial remeshing [10, 11]. Hence, EFGM and other meshless methods provide an attractive alternative to FEM in solving computational-mechanics problems. However, most developments in meshless methods have fo- cused on deterministic problems. Research in probabilistic mod- eling using EFGM or other meshless methods has not been widespread and is only now gaining attention [12, 13]. For exam- ple, using perturbation expansions of response, Rahman and Rao [12] recently developed a stochastic meshless method to pre- dict the second-moment characteristics of response for one- and two-dimensional structures. A good agreement was obtained be- tween the results of the perturbation method and the Monte Carlo simulation when random fluctuations were small. Later, Rahman and Rao [13] incorporated the first-order reliability method in conjunction with meshless equations to predict accurate proba- bilistic characteristics of response and reliability. However, both of these methods involved spatial discretization of the structural domain to achieve parametric representation of the random field. This requires a large number of random variables for multi- dimensional domain discretization, and consequently, the com- putational effort for probabilistic meshless analysis can become very large. An alternative approach involves spectral represen- tation of random field, such as the Karhunen-Lo` eve expansion, which, in general, permits decomposition of random field into 41