Discontinuous Galerkin methods with plane waves for the displacement-based acoustic equation

Abstract

Several special finite element methods have been proposed to solve Helmholtz problems in the mid-frequency regime, such as the Partition of Unity Method, the Ultra Weak Variational Formulation and the Discontinuous Enrichment Method. The first main purpose of this paper is to present a discontinuous Galerkin method with plane waves (which is a variant of the Discontinuous Enrichment Method) to solve the displacement-based acoustic equation. The use of the displacement variable is often necessary in the context of fluid–structure interactions. A well-known issue with this model is the presence of spurious vortical modes when one uses standard finite elements such as Lagrange elements. This problem, also known as the locking phenomenon, is observed with several other vector based equations such as incompressible elasticity and electromagnetism. So this paper also aims at assessing if the special finite element methods suffer from the locking phenomenon in the context of the displacement acoustic equation. The discontinuous Galerkin method presented in this paper is shown to be very accurate and stable, i.e. no spurious modes are observed. The optimal choice of the various parameters are discussed with regards to numerical accuracy and conditioning. Some interesting properties of the mixed displacement–pressure formulation are also presented. Furthermore, the use of the Partition of Unity Method is also presented, but it is found that spurious vortical modes may appear with this method.

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