A partition of unity boundary element method for transient wave propagation
David Stark  1@  , Heiko Gimperlein  1, 2@  
1 : Heriot-Watt University
Department of Mathematics Heriot-Watt University Edinburgh, EH14 4AS -  United Kingdom
2 : University of Paderborn
Institute for Mathematics University of Paderborn Warburger Str. 100 33098 Paderborn -  Germany

This article proposes a time-domain partition of unity boundary element method for wave propagation problems at high frequency. It is based on travelling plane waves as enrichment functions in a time-domain boundary element solver.

Boundary element methods reduce a boundary problem for the wave equation outside an emitter to a time-domain integral equation of the first kind on the boundary. The resulting equation is solved by a conforming space-time Galerkin method with piecewise polynomial shape functions multiplied by plane-wave enrichments. The scheme is proven to be stable as a variational method for a coercive problem.

The practical stability and convergence are first investigated for a model Dirichlet problem, in which high-frequency waves are emitted from a ball. The numerical results study the convergence as the number of plane-wave enrichments is increased. The method is stable also for large time-steps, thereby allowing coarse meshes both in space and in time. The results are extended to more complex geometries with a view towards benchmarks for the sound radiation of tires and applications to traffic noise. We thereby show that the proposed method is suitable for real-world high-frequency transient sound emission problems and prove its relevance for large-scale engineering computations of traffic noise.

As for the well-studied time-independent partition of unity methods, we observe the deterioration of the method for large numbers of enrichment functions, due to conditioning problems.

This is the first work on enriched methods for time-dependent integral equations. Our approach extends the extensive work on partition of unity methods for time-harmonic wave propagation to truly transient problems in space and time. It is particularly relevant for applications which involve many frequencies, as well as for nonlinear problems that cannot be reformulated as a PDE in frequency domain, e.g. dynamic contact beyond the quasistatic approximation.

A main challenge is the accurate assembly of the Galerkin matrix, which is based on a geometrically graded hp-quadrature method. We consider this and further algorithmic issues.

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