Transforming Domain Integrals Into BoundaryIntegrals Using Global Shape Functions And LeastSquares Approximations

Abstract

In different types of problems, the Boundary Integral Equation (BIE) formulation involves domain integrals in which neither the field variables nor their derivatives are present. In those eases, it is desirable to find a convenient way to compute the domain integrals in such a fashion that is not necessary to introduce integration cells inside the domain. Several methods have been proposed to accomplish this goal and among the most relevant ones are the following: Analytical integration, the use of Fourier expansions, the Galerkin vector technique, the Multiple Reciprocity Method (MRM) and the Dual Reciprocity Method (DRM). In all those instances, the approach that has been followed is either an analytical one or relies on the use of interpolation techniques. In the present paper a different alternative to transform domain integrals into boundary integrals is presented. Here the density function that appears in the domain integrals is approximated in terms of global shape functions which are polynomials, trigonometric functions or other kind of functions using a least squares regression. Those functions must be such that a particular solution for each one of the them can be found analytically. Once the approximation is obtained, it is used instead of the density function and each one of the resulting domain integrals is transformed into boundary integrals using the original BIE formulation for the problem. This approach is applied for the solution of Poisson's equation and two numerical examples are presented to show its accuracy and convenience.