Note on Perfectly Matched Layers

An interesting and related direction in current research into boundary termination (and one into which we invested some considerable time and effort) involves the use of so-called perfectly matched layers (PMLs) [13,14] as boundary terminations in problems to be solved over an unbounded spatial domain. The idea, generally speaking, is to surround a numerical problem domain with a layer of a material which creates as little numerical reflection as possible, while also attenuating waves that enter from the problem interior.

Absorbing boundary conditions (ABCs) [185] were long used for this purpose in (2+1)D and (3+1)D electromagnetic problems; in terms of the (2+1)D parallel-plate problem (which is equivalent to (2+1)D TE or TM mode electromagnetics), the layer is chosen to be matched to the characteristic impedance of the plates, namely $\sqrt{l/c}$. As such, it can be thought of as an extension to (2+1)D of the reflectionless matched termination that can be applied to a (1+1)D transmission line. Unfortunately, in higher dimensions, such a termination is reflection-free only for waves at normal incidence, and there will be significant backscatter into the problem interior at oblique incidence; furthermore, the amount of reflection is frequency-dependent.

Berenger [13] solved this problem, at least in theory, by proposing a new unphysical medium as an absorbing material. For the parallel-plate problem, the dependent variables in this new medium are the current density, and two orthogonal (``split'') voltage components; if the layer is infinitely thick, then it indeed absorbs and attenuates waves of any frequency or angle of incidence. The problem here, as has been pointed out in [1,2,189] is that the proposed medium can be described by a system which, though hyperbolic, is not symmetric hyperbolic and thus not of the form of (3.1), and what is worse, is not even strongly hyperbolic [82]; strong hyperbolicity is the necessary requirement for the initial value problem to be well-posed. As a result, lower-order perturbations such as those that might result from numerical discretization, can render such a system ill-posed, and susceptible to numerical instability. (It is worth noting that the MDKC representations that we have disussed in this chapter have only been applied to symmetric hyperbolic systems of the particular form of (3.1). An MDKC representation of a (3+1)D PML medium has been proposed in [129], but in this case, the asymmetries in the system were lumped into dependent source terms, and MD-passivity does not immediately follow.) Other more physical reformulations of the PML in terms of an anisotropic frequency-dependent medium [153,216] and stretched complex coordinates [185] do not alleviate this problem significantly, and other similar aproaches, such as sponge layers [138] and the transparent absorbing boundary [137] and Lorentz materials [217] appear to have similar problems.

New PML-type media, which can be described by symmetric hyperbolic systems, were put forth in [2,189]; they are of the form of (3.1), but for these media the symmetric part of the ${\bf B}$ matrix is not positive semi-definite, so an energy estimate of the form of (3.7) is not available. In particular, though one can indeed develop MDKC representations for these systems, the non-positivity of the symmetric part of ${\bf B}$ leads to active (though purely resistive) coupling between the various circuit loops. This is somewhat curious, because it is shown in [2] that field quantities in the absorbing medium decay as a function of distance from the boundary in any direction, so it would be expected that these media are indeed passive. Several questions arise here which are related to the general issue of the when a passive MDKC representation can be derived from a physically passive system. In particular: what kind of symmetries are required of the various system matrices? Is it possible to represent systems which are not symmetric hyperbolic, but only strongly hyperbolic (in which case non-reciprocal reactive elements would be necessary)?

Finally, we mention that although these absorbing layers have been proposed for use in electromagnetic field simulation problems, they apply equally well to the associated mechanical and acoustic systems; a version of the layer intended for use in fluid dynamic problems was put forth in [93]. Applications in musical and room acoustics would seem to be manifold (calculating the sound fields radiating from the open end of a musical instrument into a large space, or in open-air architectural acoustics problems come to mind as two possible examples).