ANALYTICAL 3D BOUNDARY ELEMENT IMPLEMENTATION OF FLAT TRIANGLE AND QUADRILATERAL ELEMENTS FOR POTENTIAL AND LINEAR ELASTICITY PROBLEMS
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NEY AUGUSTO DUMONT, TATIANA GALVÃO KURZ
This paper introduces a formulation for 3D potential and linear elasticity problems that end up with the analytical handling of all regular, improper, quasi-singular, singular and hypersingular integrals of an implementation using linear triangle (T3) elements. The extension to flat Q4 and T6 elements is almost straightforward. Results at arbitrarily located internal points are also given analytically. The formulation is based on a generalized transformation to subtriangle coordinates that simplifies the problem’s description and enables the adequate interpretation of all relevant geometric features of a discretized boundary segment, so that it becomes possible to arrive at manageable analytical expressions of all integrals. The paper outlines the main concepts and computational features of the proposed formulation, based on an array with all pre-evaluated integrals required in an implementation. An example of 3D potential problems illustrates all particular cases and the most challenging topological configurations one might deal with in practical applications. The procedure may be easily implemented in a general boundary element code, as the usual numerical quadrature schemes for source points sufficiently far from the integration field remain applicable. There is a work in progress for the implementation of the procedure in the frame of a fast multipole algorithm.
collocation boundary element method, numerical integration, analytical integration, 3D problems