Stochastic boundary elements for two-dimensional potential flow in homogeneous domains

Abstract

A stochastic boundary element formulation is presented for the analysis of two-dimensional steady state potential flow through homogeneous domains. The operator of the governing differential equation is assumed to be random and is described by a set of correlated random variables. The perturbation method, in conjunction with the boundary element method, is employed to derive the systems of equations for the unknown boundary variables and their respective first and second order derivatives with respect to the random variables. These quantities are then used to calculate the desired response statistics. A general procedure is developed which is next applied for the specific cases of random geometric configuration and random material parameter. The random geometric configuration is modeled using a finite set of correlated random variables. The random material parameter is modeled as a homogeneous random field which allows the use of deterministic fundamental solutions and integral representations for homogeneous domains. The random field is first discretized into a set of correlated random variables and then the general procedure is applied. A transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations. The results for the boundary variables are used to calculate the response statistics of internal potentials. These calculations require the modeling of the interior of the domain under consideration. Several models for representing the interior of the domain are presented for both random configuration and random material parameter and their influence on the response statistics is analysed. Distributed sources are considered in the present study using the particular integral approach. A number of numerical examples are presented to demonstrate the validity of the present formulations. The results obtained from the present analyses are compared with those obtained from Monte Carlo simulations with 5000 samples and a good agreement of results is observed.