Application of hierarchical matrices in the boundary element method for sound scattering from underwater objects

In order to determine the monostatic Target Echo Strength (TES) of underwater objects, the backscattering behavior of the structure(s) must be determined for many evaluation points, whereby these also define the respective positions of the sound source. The Boundary Element Method (BEM) is an established procedure for simulating this acoustic problem, but leads to dense complex matrices whose order corresponds to the number of surface elements. Each desired evaluation point (i.e., each sound source position) leads to a corresponding “right-hand side” for the system of equations. Depending on frequency and size, the number of required elements quickly becomes so large that the main memory of available workstations is insufficient to store the matrix and the solutions when using direct linear solution methods. In order to avoid this problem, the use of so-called hierarchical matrices was investigated as an alternative. Due to specialized compression algorithms, these matrices have significantly lower memory requirements and, when combined with iterative solvers (such as GMRES), should lead to faster solutions. This paper presents the results obtained from sample structures and compares the different solution methods in terms of memory usage, solution quality, and computational performance.