Computation of transient radiation in semi-infinite regions based on exact nonreflecting boundary conditions and mixed time integration

Abstract

Transient radiation in a semi-infinite region, bounded by a planar infinite baffle with a local acoustic source is considered. The numerical simulation of the transient radiation problem requires an artificial boundary Γ, here chosen to be a hemisphere, which separates the computational region from the surrounding unbounded acoustic medium. Inside the computational region we use a semidiscrete finite element method. On Γ, we apply the exact nonreflecting boundary condition (NRBC) first derived by Grote and Keller for the free-space problem. Since the problem is symmetric about the infinite planar surface, in order to satisfy the rigid baffle condition it is sufficient to restrict the indices in the spherical harmonic expansion which defines the NRBC and scale the radial harmonics which drive auxiliary equations on the boundary. The Fourier expansion in the circumferential angle appearing in the NRBC may be used to efficiently model axisymmetric problems in two dimensions. A new mixed explicit-implicit time integration method which retains the efficiency of explicit pressure field updates without the need for diagonal matrices in the auxiliary equations on Γ is presented. Here, the interior finite element equations are integrated explicitly in time while the auxiliary equations are integrated implicitly. The result is a very natural and highly efficient algorithm for large-scale wave propagation analysis. Numerical examples of fully transient radiation resulting from a piston transducer mounted in an infinite planar baffle are compared to analytical solutions to demonstrate the accuracy of the mixed time integration method with the NRBC for the half-space problem.