On a Reformulated Convolution Quadrature Based Boundary Element Method

Boundary Element formulations in time domain suffer from two problems. First, for hyperbolic problems not too much fundamental solutions are available and, second, the time stepping procedure is expensive in storage and has stability problems for badly chosen time step sizes. The first problem can be overcome by using the Convolution Quadrature Method (CQM) for time discretisation. This as well improves the stability. However, still the storage requirements are large. A recently published reformulation of the CQM by Banjai and Sauter [Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47, 227-249] reduces the time stepping procedure to the solution of decoupled problems in Laplace domain. This new version of the CQM is applied here to elastodynamics. The storage is reduced to nearly the amount necessary for one calculation in Laplace domain. The properties of the original method concerning stability in time are preserved. Further, the only parameter to be adjusted is still the time step size. The drawback is that the time history of the given boundary data has to be known in advance. These conclusions are validated by the examples of an elastodynamic column and a poroelastodynamic half space.