New developments in computational stochastic mechanics, part 1: theory

Nair, P.B. and Keane, A.J. (2000) New developments in computational stochastic mechanics, part 1: theory. In Proceedings of AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibition. American Institute of Aeronautics and Astronautics. pp. 1-11 .

Record type: Conference or Workshop Item (Paper)

Abstract

The focus of this paper is to develop efficient numerical schemes for analysis of systems governed by stochastic partial differential equations (PDEs). In particular, Stochastic Reduced Basis Approximation (SRBA) methods are proposed for efficient solution of large-scale linear algebraic system of equations with random coefficients. The terms of the Neumann expansion are deployed as stochastic basis vectors in the SRBA methods. The stochastic system response is expressed in terms of these basis vectors and undetermined deterministic scalars (or random functions). Variants of the Bubnov-Galerkin scheme are employed to compute the undetermined terms, which allows explicit expressions for the response quantities to be derived. This enables a complete probabilistic description of the response quantities to be obtained in a computationally efficient fashion. The application of SRBA methods in conjunction with stochastic linearization techniques to nonlinear stochastic systems is outlined. In a companion paper (Nair, P.B., and Keane, A.J., "New Developments in Computational Stochastic Mechanics, Part II: Applications, " AIAA-2000-1441), results are presented for a variety of problems to demonstrate that significant improvements over the Neumann expansion scheme can be achieved.

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