25th Annual GAMM-Seminar Leipzig on

FEM and BEM for time-dependent wave problems

Geometrical Singularities of Retarded Potentials

Elke Ostermann (Leibniz Universität Hannover)

We will present our latest results concerning the modeling of transient wave propagation using the boundary element method. The special structure of the fundamental solution of the wave equation leads to a close interaction of space and time variables in a so-called retarded time-argument. The corresponding retarded potentials lead naturally to sparse matrices in contrast to the dense matrices usually associated with the BEM, but in each time step the matrix has to be stored, creating a history of matrices. The sparsity of these matrices is a result of the intersection of the support of the discrete retarded potential of the trial functions (light cones) with the support of the test functions. Thus the actual number of interacting elements is rather small.

In this talk we will focus on the analysis of retarded potentials on triangles. Here we pay special attention to the smoothness of higher order derivatives. We were able to show, that there exist additional singularities compared to time-independent potentials. We discuss these non-nearfield-like singularities of the retarded potentials and their consequence for the numerical evaluation of the Galerkin integrals.

Moreover, we present stable numerical simulations for the transient sound radiation in three dimensions using the developed quadrature scheme.

This is joint work with E. P. Stephan and M. Maischak

[1] E. P. Stephan, M. Maischak, E. Ostermann. Transient Boundary Element Method and Numerical Evaluation of Retarded Potentials, Computational Science - ICCS 2008, LNCS, no. 5102, pp. 321--330

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