From the Collocation Boundary Element Method to a Meshless Formulation

Abstract

The present developments rely on a consistent formulation of the conventional, collocation boundary element method with the aim to establish a computationally less intensive procedure, although not necessarily less accurate, for large-scale, two-dimensional and three-dimensional problems of potential and elasticity. One shows that both the double-layer and the single-layer potential matrices, H and G, respectively, whose evaluation requires dealing with singular and improper integrals, may be obtained in an expedite way that circumvents almost any numerical integration – except for a few regular integrals. Although both H and G are full populated, special solution schemes (not developed in the paper) may be conceived to dramatically decrease the storage allocation required in the iterative solution of the matrix system. The evaluation of results at internal points is also straightforward, as the fundamental solutions of the boundary element method may be assumed as the domain trial functions. The evaluation of results takes into account boundary-layer effects, although special domain functions should be required to adequately simulate stress gradients related to notches and cracks. The paper focuses on the mathematical fundamentals of the formulation. A few examples illustrate the applicability of the method and some convergence issues. Application to large scale problems shall be dealt with opportunely.