A space-time finite element method for structural acoustics in infinite domains part 1: Formulation, stability and convergence

Abstract

A space-time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated. In particular, time-discontinuous Galerkin and Galerkin Least-Squares (GLS) variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence. The formulation employs a finite computational fluid domain surrounding the structure and incorporates time-dependent non-reflecting boundary conditions on the fluid truncation boundary. Energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid-structure problem with absorbing boundaries The methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space-time meshes are used to track waves propagating along space-time characteristics. An important feature of the space-time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time. For additional stability, least-squares operators based on local residuals of the structural acoustics equations including the non-reflecting boundary conditions are incorporated. The energy decay estimates and high-order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example.