Implementation of Non-Reflecting Boundaries in a Space-Time Finite Element Method for Structural Acoustics

Abstract

Abstract This paper examines the development and implementation of second-order accurate non-reflecting boundary conditions in a time-discontinuous Galerkin finite element method for structural acoustics in unbounded domains. The formulation is based on a multi-field space-time variational equation for both the acoustic fluid and elastic solid together with their interaction. This approach to the modeling of the temporal variables allows for the consistent use of high-order accurate adaptive solution strategies for unstructured finite elements in both time and space. An important feature of the method is the incorporation of temporal jump operators which allow for discretizations that are discontinuous in time. Two alternative approaches are examined for implementing non-reflecting boundaries within a time-discontinuous Galerkin finite element method; direct implementation of the exterior acoustic impedance through a weighted variational equation in time and space, and indirectly through a decomposition into two equations involving an auxiliary variable defined on the non-reflecting boundary. The idea for the indirect approach was originally developed in (Kallivokas, 1991) in the context of a standard semi-discrete formulation. Extensions to general convex boundaries are also discussed — numerical results are presented for acoustic scattering from an elongated structure using a first-order accurate boundary condition applied to an elliptical absorbing boundary.