IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 5, MAY 2002 1297 Improvement to the PML Boundary Condition in the FEM Using Mesh Compression Arnan Mitchell, Member, IEEE, David M. Kokotoff, Member, IEEE, and Michael W. Austin, Member, IEEE Abstract—Numerical errors encountered when using the per- fectly matched layer (PML) absorbing boundary condition with the finite-element method are investigated to discover more effi- cient implementation schemes. Closed-form expressions for the nu- merical reflection at an interface between two general biaxial ma- terials are applied to the special case of a PML boundary. Expres- sions for an anisotropically compressed mesh are then derived, re- vealing that reflections can be greatly reduced through increasing mesh density only where it is required. Significant improvements over previously reported PML boundaries are demonstrated. Index Terms—Finite-element methods, numerical errors, PML. I. INTRODUCTION T HE finite-element method (FEM) has become an invalu- able tool for the analysis of electromagnetic problems with complex geometries. In particular, the three-dimensional (3-D) FEM allows for the rigorous analysis of a broad range of prac- tical structures. The routine use of the FEM in design problems can, however, become cumbersome due to the vast computa- tional resources often required. It is interesting to note that a significant proportion of the un- knowns in typical problems are utilized in modeling the free space separating the geometry of interest from the boundary condition that terminates the solution space. Small reflections from these terminating boundaries, due to their imperfect imple- mentation, can significantly affect the finite-element solution. Increasing the distance from the geometry of interest to the ter- minating boundary can minimize the effects of these reflections. The increase in distance to the terminating boundary comes at the cost of an increased number of elements, an effect that can become crippling when considering 3-D models. Thus, much attention has been paid to the reduction of reflection errors from these boundaries. The reflectionless perfectly matched layer (PML) boundary [1], [2] offers great promise for the reduction of these reflections, however, as discussed in [3], the PML boundary condition, when applied to the FEM, is not, in fact, reflectionless. It would seem that in order to implement a PML boundary that sufficiently reduces these reflections, an excessive number of unknowns is required, defeating the purpose of including such a boundary. Manuscript received May 3, 2000. A. Mitchell and M. W. Austin are with the Department of Communication and Electronic Engineering, Royal Melbourne Institute of Technology University, Melbourne, Vic. 3000, Australia. D. M. Kokotoff was with the Royal Melbourne Institute of Technology Uni- versity, Melbourne, Vic. 3000, Australia. He is now with Gabriel Electronics, Scarborough, ME 04074 USA. Publisher Item Identifier S 0018-9480(02)04066-8. An investigation [4] has developed expressions for the re- flection coefficient from a PML interface in a finite-difference time-domain (FDTD) model on a rectangular grid for interfaces between isotropic media and a PML truncation. We have previ- ously investigated the nature of reflection errors caused by trian- gular finite-element discretization at boundaries in biaxial ma- terials [5], deriving closed-form expressions that have proven to be a very good model of the numerical errors observed in prac- tical finite-element simulations. A PML interface is a special case of such a boundary and, thus, those closed-form expres- sions should be equally well suited to the analysis of the asso- ciated numerical errors. Section II of this investigation applies the closed-form ex- pressions derived in [5] to the special case of a PML boundary interface. The resulting expressions are compared to practical simulations to ensure their validity and then their form is ex- amined to better understand the behavior of the PML and how the associated numerical errors depend on the parameters of the problem. It is found that the numerical reflection error depends only on discretization parameters normal to the PML interface, suggesting that uniform mesh refinement may be an inefficient approach to suppressing numerical reflections. Thus, Section III re-derives the expressions for an anisotropically distorted mesh and the resulting expressions are again applied to the special case of the PML. A significant reduction in numerical reflec- tions from PML boundaries is predicted and finite-element sim- ulations verify that such gains are also obtained in practice. Comparison with the previous investigation of Polycarpou et al. [3] verify that an anisotropic mesh compression can offer a very efficient means of implementing a PML boundary. II. NUMERICAL REFLECTION FROM A PML In [5], expressions for the numerical dispersion and reflec- tion expected from an interface between two general biaxial materials were derived. This was done for two-dimensional tri- angular elements for both edge and node-basis functions. Al- though uniform equilateral triangles were assumed, these ex- pressions were demonstrated to provide a reasonable approxi- mation to a practical mesh. It was concluded that these expressions for numerical inaccu- racies could provide a useful tool for the investigation of PML performance in the FEM. An investigation of numerical disper- sion and reflection of the special case of a PML boundary is thus conducted in Section II-A. A. Closed-Form Reflection From the PML For brevity, the closed-form expressions for numerical dis- persion and reflection from an arbitrary interface between two 0018-9480/02$17.00 © 2002 IEEE