Stochastic boundary elements for two-dimensional potential flow in non-homogeneous media

Abstract

A stochastic boundary element formulation is presented for the treatment of two-dimensional problems of steady-state potential flow in non-homogeneous media that involve a random operator in the governing differential equation. The randomness is introduced through the material parameter of the domain which is described as a non-homogeneous random field. The random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem. This leads to differential equations for the unknown potential and its first- and second-order derivatives, respectively, evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field. An approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order. These solutions are introduced into the differential equations and upon equating similar order terms, a sequence of Poisson's equations is obtained for each order. A transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables. The resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first- and second-order derivatives which are then used to compute the desired response statistics. Quadratic, conforming boundary elements are used in the boundary integration and four-node quadrilateral cells are used in the domain integration. Strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object. The singular domain integrals are calculated analytically. Direct solution techniques are used to calculate the response variables and their derivatives, respectively. A number of example problems are presented and the results are compared with those obtained from Monte Carlo simulations. A good agreement of the results is observed.