GALERKIN METHOD FOR STOCHASTIC ALGEBRAIC EQUATIONS. AND PLATES ON RANDOM ELASTIC FOUNDATION

Mircea Grigoriu

Cornell University, Ithaca, NY 14853, USA
E-mail: mdg12@cornell. edu

Abstract

A new perspective is presented on the Galerkin solution for linear stochastic algebraic equations, that is, linear algebraic equations with random coefficients. It is shown that (1) a stochastic algebraic equation has an optimal Galerkin solution, that is, a Galerkin solution that is best in the mean square sense, and (2) the optimal Galerkin solution is equal to the conditional expectation of the exact solution with respect to a a -field coarser than the a – field relative to which this solution is measurable. Galerkin solutions that are not optimal are called sub-optimal. Both optimal and sub­optimal Galerkin solutions are defined and constructed. Optimal and sub-optimal Galerkin solutions are used to calculate statistics of the displacement of a simply supported plate sitting on a random elastic foundation. The accuracy of these Galerkin solutions is assessed by Monte Carlo simulation.

1. Introduction

Ordinary or partial differential equations with random coefficients, input and/or boundary conditions, referred to as stochastic differential equations, are used to formulate a broad range of mechanics problems. It is common to approximate the solution of stochastic differential equations by that of algebraic equations with random coefficients, called stochastic algebraic equations. There are no general and efficient methods for finding the probability law of the solution of a stochastic algebraic equation. Taylor series, perturbation, Neumann series, decomposition, equivalent linearization, it­eration, and other approximate techniques can be used to solve these equations (Deb et al. 2001; Ghanem and Spanos, 1991; Grigoriu, 2002: section 8.3.1). Monte Carlo simulation is the only available method capable of providing estimates for the probability law of the solution of general stochastic equations, but can be numerically prohibitive if applied to solve realistic problems.

Our objectives are to (1) present an alternative interpretation of the Galerkin method for solving stochastic algebraic equations and (2) apply this method to calculate statistics for the displacement of a plate supported by a random elastic foundation. It is shown that there is an optimal Galerkin solution for a stochastic algebraic equation, that is, a Galerkin solution minimizing the mean square error. The optimal Galerkin solution is equal to the conditional expectation of the exact solution of a stochastic algebraic equation with respect to a a-field coarser than the a-field with respect to which this solution is measurable. Galerkin solutions that are not optimal are said to be sub-optimal. Second-moment properties, distributions, and other statistics are developed for both optimal and sub­optimal Galerkin solutions. Statistics are calculated for optimal and sub-optimal Galerkin solutions for a simply supported plate sitting on a random elastic foundation. The accuracy of the Galerkin solutions is evaluated by Monte Carlo simulation.