Nonreflecting boundary conditions for finite element methods based upon off-surface boundary integral equations.

Abstract

In this work, an off-surface boundary integral method (OSBI) is presented as a mesh termination scheme for finite element formulations of acoustics and elasto-dynamics problems. By using finite element approximations for both the unknown Dirichlet boundary condition and the unknown Neumann boundary condition on the artificial boundary, the discretized boundary integral equation is used to solve for the discrete Neumann boundary condition term on the artificial boundary in terms of the unknown Dirichlet boundary condition term; this expression is then substituted into finite element formulations. As the observation points are placed off-surface, the difficulties are avoided in evaluating the singular boundary integral kernels. Of particular emphasis in this work is to show that new integral operators avoid singularity, the artificial boundary can be arbitrarily shaped, and no special operators need to be derived. It is easy to embed this method into a standard finite element code by supplying a library of fundamental solutions. Nonuniqueness can be suppressed either by choosing the observation points accordingly or by resorting to a Burton-Miller approach, which does not result in hyper-singular kernels owing to off-surface placement of observation points. For acoustics and elastodynamics equations, comparison is made of the new OSBI technique with the DtN method and several popular local nonreflecting boundary conditions (NRBC's). The OSBI approach is accurate, robust to parameter changes and has a uniform convergence, whereas local NRBC's are shown to be sensitive to parameter changes and may not have a uniform convergence. Notably, the OSBI method is competitive with the DtN method, and is not restricted to circular artificial boundaries. Even though acoustics and elastodynamics equations are considered, the approach developed here is general and can be extended to other time-dependent and time-harmonic problems.