Investigation of nonplanar perfectly matched absorbers for finite-element mesh truncation

We present a detailed theoretical and numerical investigation of the perfectly matched layer (PML) concept as applied to the problem of mesh truncation in the finite-element method (FEM). We show that it is possible to extend the Cartesian PML concepts involving half-spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems. By using the method of separation of variables, it is possible to solve the boundary value problems in these geometries. The analytical solutions demonstrate that under certain conditions, outgoing waves are absorbed with negligible reflection, and the transmitted wave is attenuated within the PML. To reduce the white space in radiation or scattering problems, conformal PMLs are constructed via parametric mappings. It is also verified that the PML concept, which was originally introduced for problems governed by Maxwell's equations, can be extended to cases governed by the scalar Helmholtz equation. Finally, numerical results are presented to demonstrate the use of the PML in FEM mesh truncation.