Computational implementation for 3D problems

Abstract

The dual reciprocity method (DRM) is a boundary element technique to approach domain-dominant problems without losing the boundary-only nature of the boundary element method. The DRM converts domain integrals into boundary integrals by means of approximation functions. The DRM is general and the number of applications solved using the procedure has been increasing in the literature since the early 1990s. However, the DRM faces a serious drawback when applied to large problems: the resulting system of equations is dense and frequently ill-conditioned. A way to overcome this inconvenient feature is by using domain subdivision in the limiting case when the resulting internal mesh looks like a finite element grid. This technique is known as the dual reciprocity method multidomain (DRM-MD) approach. The DRM-MD approach produces a sparse and well-conditioned system of equations. It has been successfully applied to a variety of problems in 2D domains and has showed good performance. The extension of the procedure to 3D cases is not straightforward since factors such as continuity of the elements, DRM approximation function, scaling, number of internal DRM nodes, etc. that largely affect the performance of the code need to be selected. In this chapter, several aspects regarding the computational implementation of the DRM-MD codes are presented and reviewed. A general assembly procedure is proposed, which can be used in both 2D and 3D problems and can be easily adapted to approach problems with different governing equations. Results of numerical examples using several different schemes for 3D problems are shown to provide an insight on 3D DRM-MD implementation.