volume | PIER Journals

Abstract

This paper concerns the implementation of the perfectly matched layer (PML) absorbing boundary condition in the framework of a mimetic differencing scheme for Maxwell's Equations. We use mimetic versions of the discrete curl operator on irregular logically rectangular grids to implement anisotropic tensor formulation of the PML. The form of the tensor we use is fixed with respect to the grid and is known to be perfectly matched in the continuous limit for orthogonal coordinate systems in which the metric is constant, i.e. Cartesian coordinates, and quasi-perfectly matched (quasi-PML) for curvilinear coordinates. Examples illustrating the effectiveness and long-term stability of the methods are presented. These examples demonstrate that the grid-based coordinate implementation of the PML is effective on Cartesian grids, but generates systematic reflections on grids which are orthogonal but non-Cartesian (quasi-PML). On non-orthogonal grids progressively worse performance of the PML is demonstrated. The paper begins with a summary derivation of the anisotropic formulation of the perfectly matched layer and mimetic differencing schemes for irregular logically rectangular grids.

1. Balanis, C. A., Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989.

2. Berenger, J.-P., "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, Vol. 114, 185-200, October 1994.
doi:10.1006/jcph.1994.1159 Google Scholar

3. Gedney, S. D., "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Transactions on Antennas and Propagation, Vol. 44, No. 12, 1630-1639, December 1996.
doi:10.1109/8.546249 Google Scholar

4. Gedney, S. D., "An anisotropic PML absorbing media for the FDTD simulation of fields in lossy and dispersive media," Electromagnetics, Vol. 16, 399-415, 1996.
doi:10.1080/02726349608908487 Google Scholar

5. He, J.-Q. and Q. H. Liu, "A nonuniform cylindrical FDTD algorithm with improved PML and quasi-PML absorbing boundary conditions," IEEE Transactions on Geoscience and Remote Sensing, Vol. 37, No. 2, 1066-1072, March 1999.
doi:10.1109/36.752224 Google Scholar

6. Hyman, J., M. Shashkov, and S. Stienberg, "The numerical solution of diffusion problems in strongly heterogeneous nonisotropic materials," Journal of Compuational Physics, Vol. 132, 130-148, 1997.
doi:10.1006/jcph.1996.5633 Google Scholar

7. Hyman, J. M. and M. Shashkov, "Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids," Applied Numerical Mathematics, Vol. 25, 413-442, 1997.
doi:10.1016/S0168-9274(97)00097-4 Google Scholar

8. Hyman, J. M. and M. Shashkov, "Natural discretizations for the divergence, gradient and curl on logically rectangular grids," Applied Numerical Mathematics, Vol. 33, No. 4, 81-104, 1997. Google Scholar

9. Hyman, J. M. and M. Shashkov, "Mimetic discretizations for Maxwell’s equations," Journal of Compuational Physics, Vol. 151, 881-909, 1999.
doi:10.1006/jcph.1999.6225 Google Scholar

10. Jackson, J. D., Classical Electrodynamics, 2nd edition, John Wiley & Sons, New York, 1975.

11. Mittra, R. and U. Pekel, "A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves," IEEE Microwave and Guided Wave Letters, Vol. 5, No. 3, 84-86, March 1995.
doi:10.1109/75.366461 Google Scholar

12. Sacks, Z. S., D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Transactions on Antennas and Propagation, Vol. 43, No. 12, 1460-1463, December 1995.
doi:10.1109/8.477075 Google Scholar

13. Taflove, A. (ed.), Advances in Computational Electrodynamics. The finite-difference time-domain method, Artech House, Boston, Mass., 1998.

14. Teixeira, F. L. and W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microwave and Guided Wave Letters, Vol. 7, No. 11, 371-373, November 1997.
doi:10.1109/75.641424 Google Scholar

15. Teixeira, F. L. and W. C. Chew, "A general approach to extend Berenger’s absorbing boundary condition to anisotropic and dispersive media," IEEE Transactions on Antennas and Propagation, Vol. 46, No. 9, 1386-1387, September 1998.
doi:10.1109/8.719984 Google Scholar

16. Teixeira, F. L. and W. C. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microwave and Guided Wave Letters, Vol. 7, No. 9, 285-287, September 1997.
doi:10.1109/75.622542 Google Scholar

17. Tong, M.-S., Y. Chen, M. Kuzuoglu, and R. Mittra, "A new anisotropic perfectly matched layer medium for mesh truncation in finite difference time domain analysis," Int. J. Electronics, Vol. 86, No. 9, 1085-1091, 1999.
doi:10.1080/002072199132860 Google Scholar

18. Yang, B. and P. G. Petropoulos, "Plane-wave analysis and comparison of split-field, biaxial and uniaxial PML methods as ABC’s for pseudospectral electromagnetic wave simulations in curvilinear coordinates," Journal of Computational Physics, Vol. 146, 747-774, 1998.
doi:10.1006/jcph.1998.6082 Google Scholar

19. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966. Google Scholar

Dr. M. W. Buksas