Estudo e implementação das equações integrais de contorno para problemas tridimensionais de elasticidade

Abstract

In this work a study about the boundary element applied to three-dimensional elastostatic problems is developed. Some approaches used to evaluate the integrals involved in the method are reviewed. Triangular isoparametric boundary elements are used, with linear or quadratic shape functions. The strongly singular integrals are evaluated indirectly using the rigid body motion method, whil the weakly singular integrals are performed numerically and analytically. Two schemes are considered in the numerical evaluation. The first one makes use of the Hammer's technique, developed to integrate triangular domains. Such technique is associated with the technique of element subdivision, leading to the use of more integration points near the singulary, in order to enhance the quadrature accuracy. In the second scheme a coordinate transformation is use to map the triangular element onto a square domain of unit side-length where the standard Gauss-Legendre quadrature is applied. Both schemes are used in the evaluation of weakly singular and non-singular integrals. The expressions developed for analytical integration of weakly singular integrals over flat linear elements are also reviewed. In addition to the classical BEM algorithm, in order to avoid any singularities, an algorithm using the collocation points outside the problem domain is presented. The displacements and stresses at internal points are evaluated in a post-process stage, using the standard Somigliana displacement identity (SDI) and the Somigliana's identity for stresses. Furthermore, a self-regularized formulation of SDI has been used in the evaluation of the displacements, providing more accurate results when the point is place close to the boundary. A program has been developed using these approaches, whose efficiency was evaluated by means of some numerical examples.