Accurate Non-reflecting Boundary Conditions for Time-dependent Acoustic Scattering

Accurate radiation boundary conditions for the time-dependent wave equation are formulated in the finite element method as an auxiliary problem for each radial harmonic on a spherical boundary. The method is based on residuals of an asymptotic expansion for the time-dependent radial harmonics. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local/sparse character of the finite element equations. The method has the ability to vary separately, and up to any desired order, the radial and transverse modal orders of the radiation boundary condition. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, the conditions are exact. If fewer equations are used, then the boundary conditions form high-order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the radiation boundary conditions. The results demonstrate dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition.