The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods

TLDR: In this article, the authors compared three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries.

Abstract: The treatment of domain integrals has been a topic of interest almost since the inception of the boundary element method (BEM). Proponents of meshless methods such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM) have typically pointed out that these meshless methods obviate the need for an interior discretization. Hence, the DRM and MRM maintain one of the biggest advantages of the BEM, namely, the boundary-only discretization. On the other hand, other researchers maintain that classical domain integration with an interior discretization is more robust. However, the discretization of the domain in complex multiply-connected geometries remains problematic. In this research, three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries are compared. The methods include the DRM, classical cell-based domain integration, and a novel auxiliary domain method. The auxiliary domain method allows the evaluation of the domain integral by constructing an approximately C 1 extension of the domain integrand into the complement of the multiply-connected domain. This approach combines the robustness and accuracy of direct domain integral evaluation while, at the same time, allowing for a relatively simple interior discretization. Comparisons are made between these three methods of domain integral evaluation in terms of speed and accuracy.