A Reflection-Transmission Analysis of a Fictitious Domain Method for Wave Scattering by a Sound-Hard Obstacle

Abstract eng:
We consider the wave scattering by a sound-hard obstacle in a fluid. In previous works, we had proposed a method based on a first order in time mixed variational formulation allowing to use regular meshes for most of the computations. The boundary condition was weakly taken into account by the fictitious domain method. The resulting system was discretized with two families of finite elements compatible with mass lumping. Through some numerical examples we showed that the first family (which provides optimal rate of convergence in absence of boundary [1]; pure propagation) does not always take into account correctly the boundary condition. To overcome this problem, we had introduced a second family of finite elements for which we have obtained a convergence proof [2]. Inspired on this work we had extended this mixed finite element to the more complex case of elastodynamic equations with a free surface boundary condition [3]. Several numerical experiments as well as a numerical convergence analysis show that the numerical method computes a good approximate solution. The resulting numerical scheme has the following properties: allows rapid computations, leads to quasi-explicit schemes after time discretization, permits to control the CFL number and accounts complex geometries avoiding the staircase phenomena. In this study we present a plane wave reflection-transmission analysis of the method with both families of finite elements for the scalar case that confirms most of the previous results and provides the quantitative behavior of the numerical error with respect to the discretization parameters (in some particular situations). We consider several configurations where the geometry and the meshes present a periodicity in one direction. This allows to study the interaction of numerical plane waves with the obstacle and to compare them with the continuous case. Some ideas on the extension of this study to elastodynamics will be given.

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