Finite Element Formulation of Exact Dirichlet-to-Neumann Radiation
Conditions on Elliptic and Spheroidal Boundaries
Lonny L. Thompson, Runnong Huan and Cristian Ianculescu
Advanced Computational Mechanics Research Laboratory
Department of Mechanical Engineering and Engineering Mechanics
Clemson University
Clemson, South Carolina 29634-0921
NCA-Vol. 26, Proceedings of the ASME Noise Control and Acoustics Division - 1999, ASME 1999, pp. 497-510;
1999 International Mechanical Engineering Congress and Exposition,
Symposium on Computational Acoustics,
Nov. 14-19, 1999, Nashville, TN.
Abstract
Exact Dirichlet-to-Neumann (DtN)
radiation boundary conditions are derived in elliptic and spheroidal
coordinates and formulated in a finite element method for the Helmholtz
equation in unbounded domains.
The DtN map matches the first N wave harmonics exactly
at the artificial boundary.
The use of elliptic and spheroidal boundaries enables the efficient
solution of scattering from elongated objects in two- and three- dimensions
respectively.
Modified DtN conditions based on first and
second order local boundary operators are also derived in
elliptic and spheroidal coordinates, in a form suitable for
finite element implementation.
The modified DtN conditions are more accurate
than the DtN boundary condition, yet require no extra memory and
little extra cost.
Direct implementation involves
non-local spatial integrals leading to a dense, fully populated submatrix.
A matrix-free interpretation
of the non-local DtN map for elliptic and spheroidal boundaries,
suitable for iterative solution of the resulting complex-symmetric
system is described.
For both the DtN and modified DtN conditions,
we describe efficient and effective SSOR preconditioners
with Eisenstat's trick based on the matrix partition associated with the interior mesh and local boundary operator. Numerical examples
of scattering from elliptic and spheroidal boundaries are computed
to demonstrate the efficiency and accuracy of the boundary treatments
for elongated structures.
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