STOCHASTIC BOUNDARY ELEMENTS FOR 2-DIMENSIONAL POTENTIAL FLOW IN NONHOMOGENEOUS MEDIA

A stochastic boundary element formulation is presented for the treatme nt of two-dimensional problems of steady-state potential flow in non-h omogeneous media that involve a random operator in the governing diffe rential equation. The randomness is introduced through the material pa rameter of the domain which is described as a non-homogeneous random f ield. The random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation o f the problem. This leads to differential equations for the unknown po tential and its first- and second-order derivatives, respectively, eva luated at the mathematical expectations of the random variables result ing from the discretization of the random field. An approximate method is applied for the solution of these equations by expressing the pote ntial and its derivatives as a sum of functions of descending order. T hese solutions are introduced into the differential equations and upon equating similar order terms, a sequence of Poisson's equations is ob tained for each order. A transformation of the correlated random varia bles into an uncorrelated set is performed to reduce the number of num erical operations by retaining a small number of transformed random va riables. The resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and th eir respective first- and second-order derivatives which are then used to compute the desired response statistics. Quadratic, conforming bou ndary elements are used in the boundary integration and four-node quad rilateral cells are used in the domain integration. Strongly singular terms of the boundary element matrices are obtained indirectly by appl ying a state of uniform unit potential over the entire contour of the object. The singular domain integrals are calculated analytically. Dir ect solution techniques are used to calculate the response variables a nd their derivatives, respectively. A number of example problems are p resented and the results are compared with those obtained from Monte C arlo simulations. A good agreement of the results is observed.