Comparação de desempenho entre as formulações singular e hipersingular do método dos elementos de contorno

Abstract

In this work, the Singular and Hypersingular Formulations of the Boundary Element Method are presented and their results are compared on two dimensional stationary scalar field problems , governed by the Laplace Equation. In the conventional formulations of the Boundary Element Method, a special attention must be given to the singular integrations that appear in the boundary integrals kernel, typical of the method. This special attention is for more important in the case of Hypersingular Formulations. With the adequate treatment of these integrals it is possible to demonstrate that they are convergent in the sense of the Cauchy Principal Value and that they can have their treatment simplified according to some discretization functional nodes positioning and procedures which are typical of the Boundary Element Method Although both formulations characteristics have been presented already in specialized literature, many interesting particularities of the Hypersingular Formulation are not well known and better evaluation, specially through comparative results with the classic singular formulation. The objective of this work is to present these integral formulations of the Laplace Equation and their numerical treatment in the simplest form, considering the kernel properties and the order of the boundary elements used in the boundary discretization. By performing a comparison between the numerical results percentage error, made in the potential and its derivative calculations in examples with known analytical solutions, the performance of each formulation is discussed. This analysis finishes with a comparison with a referring example, using another numerical method.