Generalised Adaptive Cross Approximation for Convolution Quadrature based Boundary Element Formulation

Abstract

The acoustic wave equation is solved in time domain with a boundary element formulation. The time discretisation is performed with the generalised convolution quadrature method and for the spatial approximation standard lowest order elements are used. Collocation and Galerkin methods are applied. In the interest of increasing the efficiency of the boundary element method, a low-rank approximation such as the adaptive cross approximation (ACA) is carried out. We discuss about a generalisation of the ACA to approximate a three-dimensional array of data, i.e., usual boundary element matrices at several complex frequencies. This method is used within the generalised convolution quadrature (gCQ) method to obtain a real time domain formulation. The behaviour of the proposed method is studied with three examples, a unit cube, a unit cube with a reentrant corner, and a unit ball. The properties of the method are preserved in the data sparse representation and a significant reduction in storage is obtained.

Figures (15)

• Figure 1: Unit cube: Geometry and discretisation parameters
• Figure 2: Cube: $L_{max}$-error versus refinement in space and time
• Figure 3: Cube: Number of used frequencies for the Dirichlet problem
• Figure 4: Cube: Number of used frequencies for the Neumann problem
• Figure 5: Cube (level 4): Used complex frequencies with a colour code for the number of matrix blocks, where this frequency is active