Study and solution of BEM-singular integral equation method in the case of concentrated loads

The Boundary Element Method has been widely used in order to obtain numerical solutions in many problems in science and engineering. On the other hand the singular integral equation method can treat successfully the singular fields appearing in engineering problems when the right hand side of it is a regular function. In this paper we consider the case where the right hand side of a Boundary Integral Equation, singular integral equation is a distribution. This problem arises in the case of concentrated forces and/or moments. Our main scope is to propose appropriate methods for the numerical solution of singular integral equations when the loading is a concentrated force or a moment. Before that we are obliged to analyze the solvability of such equations. The main key for the proof of the solvability is the proof of existence of Cauchy integrals of distributions. As a by product we find a very interesting result that the solution behaves as 1/r1/r or 1/r2 at the point where the concentrated force or moment, respectively, is applied. Obviously the existence of essential singularities in the solution creates difficulties in the numerical treatment of integral equations. Thus two appropriate numerical methods and an interpolatory quadrature method are proposed for the numerical solution of BIE–SIE. Numerical examples illustrate the importance of the proposed methods.