HALFSPACE FORMULATIONS FOR THE BOUNDARY ELEMENT METHOD IN 3D-ACOUSTICS USING THE FAST MULTIPOLE METHOD

摘要

The Boundary Element Method (BEM) is a well introduced technique for solving acoustic radiation problems in the frequency domain. In contrast to the Finite Element Method a three dimensional problem is not discretized by volume elements but by surface elements, which leads to a considerable decrease of the number of elements. The resulting system matrices of the standard BEM, however, are fully populated. Solving the problem with these dense matrices by iterative solvers leads to a quadratic complexity with regard to the degrees of freedom. Using the Fast Multipole Method (FMM) a reduction to a quasi linear complexity can be achieved. In particular, the necessary matrix vector multiplications are approximated causing a considerable reduction of the computation time. In acoustics often a halfspace boundary condition needs to be analyzed and the BEM allows different approaches to handle this. One option is a modified fundamental solution to include a rigid plane a priori. Another possibility is to mirror the discretization at this plane leading to a set of elements in the real and the mirrored domain. With respect to the FMM, special algorithms exist to realize these two approaches. By using a modified fundamental solution, elements are located only in the real domain, i.e. the hierarchical structure needs to be setup only for this domain. In the case of the mirror technique, the hierarchical structure includes all elements in the real as well as in the mirrored one. Depending on the problem, these two approaches for acoustic halfspace problems have different characteristics. In the current paper the different algorithms will be described and compared to each other looking at different representative acoustic problems.