Asymptotic and Exact Radiation Boundary Conditions for Time-dependent Scattering

Asymptotic and exact local radiation boundary conditions first derived by Hagstrom and Hariharan are reformulated as an auxiliary Cauchy problem for linear first-order systems of ordinary equations on the boundary for each harmonic on a circle or sphere in twoor three-dimensions, respectively. With this reformulation, the resulting radiation boundary condition involves first-order derivatives only and can be computed efficiently and concurrently with standard semi-discrete finite element methods for the near-field solution without changing the banded/sparse structure of the finite element equations. In 3D, with the number of equations in the Cauchy problem equal to the mode number, this reformulation is exact. If fewer equations are used, then the boundary conditions form uniform asymptotic approximations to the exact condition. Furthermore, using this approach, we formulate accurate radiation boundary conditions for the twodimensional unbounded problem on a circle. Numerical studies of time-dependent radiation and scattering are performed to assess the accuracy and convergence properties of the boundary conditions when implemented in the finite element method. The results demonstrate that the new formulation has dramatically improved accuracy and efficiency for time domain simulations compared to standard boundary treatments. INTRODUCTION When modeling radiation from structures in an acoustic medium which extends to infinity with a domain based compuCorresponding author. tational method such as the finite element method, the far-field is truncated at an artificial boundary surrounding the source of radiation. The impedance of the far-field is then represented on this boundary by either radiation boundary conditions, infinite elements, or absorbing sponge layers. Survey articles of various boundary treatments are given in (Tsynkov, 1998). If accurate boundary treatments are used, the finite computational region can be reduced so that the truncation boundary is relatively close to the radiator, and fewer elements than otherwise would be possible may be used, resulting in considerable savings in both cpu time and memory. In the frequency domain, several accurate and efficient methods for representing the impedance of the far-field are well understood, including the Dirichlet-to-Neumann (DtN) map (Keller, 1989; Grote, 1995a), and infinite elements (Burnett, 1994; Astley, 1998). However, efficient evaluation of accurate boundary treatments for the time-dependent wave equation on unbounded spatial domains has long been an obstacle for the development of reliable solvers for time domain simulations. Ideally, the artificial boundary would be placed as close as possible to the source, and the radiation boundary treatment would be capable of arbitrary accuracy at a cost and memory not exceeding that of the interior solver. A standard approach is to apply local (differential) boundary operators which annihilate leading terms in the radial multipole expansion for outgoing wave solutions. A well known sequence of boundary conditions developed for a spherical truncation boundary are the local operators derived by Bayliss and Turkel (Bayliss, 1980). However, these and other approximate local boundary conditions exhibit significant spurious reflection for high-order wave harmonics, especially as the position of the 1 Copyright  1999 by ASME truncation boundary approaches the source of radiation (Pinsky, 1991; Pinsky, 1992). In recent years, new boundary treatments have been developed which dramatically improve both the accuracy and efficiency of time domain simulations compared to approximate local radiation boundary conditions. In (Grote, 1995b; Grote, 1996), exact nonreflecting boundary conditions (NRBC) are derived involving solution of an auxiliary Cauchy problem for linear first-order systems of time-dependent differential equations on a spherical boundary for each harmonic. In (Thompson, 1999a), this NRBC is rederived based on direct application of a result given in Lamb (Lamb, 1916), with improved scaling of the first-order system of equations associated with the NRBC. Formulation of the NRBC in standard semidiscrete finite element methods with several alternative implicit and explicit timeintegrators is reported in (Thompson, 1999a; Thompson, 1999b). In (Thompson, 1999b), a modified version of the exact NRBC first derived in (Grote, 1996), is implemented in a finite element formulation. In order to obtain a symmetric system, the NRBC is reformulated with additional auxiliary variables on the truncation boundary. The modified version may be viewed as an extension of the second-order local boundary operator derived by Bayliss and Turkel (Bayliss, 1980), and gives improved accuracy when only a few harmonics are included in the spherical expansion/transformation. In (Thompson, 1999c), a method is described for calculating far field solutions concurrently with the near-field solution based on the exact NRBC. At each discrete time step, radial modes computed on a spherical artificial boundary which drive the exact NRBC for the near-field solution, are imposed concurrently as data for the radial wave equation in the far-field. The radial grid is truncated at the far-field point of interest with the modal form of the exact NRBC. The solution in the far-field is then computed from an inverse spherical harmonic transform of the radial modes. Hagstrom and Hariharan (Hagstrom, 1998) have derived a sequence of radiation boundary conditions involving first-order differential equations in time and tangential derivatives of auxiliary functions on a circular or spherical boundary. They indicate how these local conditions may be effectively implemented in a finite difference scheme using only local tangential operators, but at the cost of introducing a large number of auxiliary functions at the boundary. Numerical experiments were conducted for a model problem involving the Fourier modes of the wave equation in two-dimensions using a finite difference method. However, direct finite element implementation of this sequence in a standard Galerkin variational equation would result in a nonsymmetric system of equations. In this paper we rederive the sequence of local boundary conditions described in (Hagstrom, 1998) in terms of harmonics and reformulate the recursive equations as a Cauchy problem involving systems of first-order ordinary differential equations on the boundary, similar to that used in (Grote, 1995b; R