Stochastic boundary elements in elastostatics

A stochastic boundary element formulation for the treatment of boundary value problems in two-dimensional elastostatics that involve a random operator is presented. A general perturbation procedure is formulated for the set of correlated random variables governing the response of the solid. This procedure is then specialized for the cases of (a) random geometry and (b) random material properties. The problems involving a random configuration are analyzed using the random variable model, and those with a random material property are analyzed using the random field model. The random field is first discretized into a set of correlated random variables, which are then transformed into an uncorrelated set to simplify the analysis. The derivatives of the boundary element matrices appearing in the systems of equations from the perturbation of the random variables are derived analytically. The direct solution methods are used to obtain the response variables and their first- and second-order derivatives, respectively. Quadratic, conforming boundary elements are employed in the boundary element discretization and the strongly singular terms of the boundary element matrices and their first- and second-order derivatives are obtained using the conditions associated with rigid body motions of the solid. The present formulation has been evaluated for a number of example problems through comparisons with the solutions obtained by Monte Carlo simulation. A good agreement of the results is observed.