A space-time finite element method for structural acoustics in infinite domains part 2: Exact time-dependent non-reflecting boundary conditions

Abstract

In Part 1, a new space-time finite element method for transient structural acoustics in exterior domains was given. The formulation employs a finite computational fluid domain surrounding the structure and incorporates local time-dependent non-reflecting boundary conditions on the fluid truncation boundary. In this paper, new exact time-dependent non-reflecting boundary conditions are developed for solutions of the scalar wave equation in three space dimensions. These high-order accurate absorbing boundary conditions are based on the exact impedance relation for the acoustic fluid through the Dirichlet-to-Neumann (DtN) map in the frequency domain and are exact for solutions consisting of the first N spherical wave harmonics. Time-dependent boundary conditions are obtained through an inverse Fourier transform procedure. Two alternative sequences of boundary conditions are derived; the first involves both temporal and spatial derivatives (local in time and local in space version), and the second involves temporal derivatives and a spatial integral (local in time and non-local in space version). These non-reflecting boundary conditions are incorporated as ‘natural’ boundary conditions in the space-time variational equation, i.e. they are enforced weakly in both space and time. Several numerical examples involving transient radiation are presented to illustrate the high-order accuracy and efficiency achieved by the new space-time finite element formulation for transient structural acoustics with non-reflecting boundaries.