The Coupling of Monte Carlo Integration With the Boundary Integral Equation Technique to Solve Poisson Type Equations.

Abstract

The primary topic of this work is a method for dealing with boundary element formulations of nonhomogeneous terms in Poisson type equations. A concise presentation of boundary integral techniques and the corresponding fundamental solutions plays an important role in this development. Using Monte Carlo quadrature theory, an algorithm for the construction of a two-dimensional BEM Poisson equation analyzer is derived. A FORTRAN program based upon this algorithm is presented as a novel device that solves the general Poisson equation in two-dimensional and axisymmetric geometries without domain discretization. In the axisymmetric case, Monte Carlo integration is also used to effectively compute the integrals of the singular functions corresponding to some diagonal terms of the assembly matrix. Sample analyses of several engineering problems are performed with the computer program and the results are compared with solutions obtained by other means. The fine quality of the results implies that the program is generally viable for obtaining solutions to the Poisson equation. It is concluded that while the theory is applicable to transient analyses, the technique is not practical in such cases because of the large amounts of computer time needed to assemble the matrices. It is also concluded that extensions of the theory to general three-dimensional geometries pose no special problems; these are possible by drawing a simple analogy with the two-dimensional algorithm.