2000 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

IEEE Transactions on Antennas and Propagation
Volume 48 Number 2, February 2000

Table of Contents for this issue

Complete paper in PDF format

A Parallel Finite-Element Tearing and Interconnecting Algorithm for Solution of the Vector Wave Equation with PML Absorbing Medium

C. T. Wolfe, Member, IEEE U. Navsariwala, Member, IEEE and Stephen D. Gedney Senior Member, IEEE

Page 278.

Abstract:

A domain decomposition method based on the finite-element tearing and interconnecting (FETI) algorithm is presented for the solution of the large sparse matrices associated with the finite-element method (FEM) solution of the vector wave equation. The FETI algorithm is based on the method of Lagrange multipliers and leads to a reduced-order system, which is solved using the biconjugate gradient method (BiCGM). It is shown that this method is highly scalable and is more efficient on parallel platforms when solving large matrices than traditional iterative methods such as a preconditioned conjugate gradient algorithm. This is especially true when a perfectly matched layer (PML) absorbing medium is used to terminate the problem domain.

References

  1. I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, New York: Oxford Univ. Press, 1986.
  2. W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, New York: Springer-Verlag, 1994.
  3. Y. Saad, Iterative Methods for Sparse Linear Systems, Boston, MA: PWS-Kent, 1996.
  4. B. Nour-Omid, A. Raefsky and G. Lyzenga, "Solving finite element equations on concurrent computers", in Symposium on Parallel Computation and their Impact on Mechanics, Boston, MA, 1987.
  5. S.-H. Hsieh, G. Paulino and J. Abel, "Recursive spectral algorithms for automatic domain partitioning in parallel finite element analysis", Comput. Method. Appl. Mech. Eng., vol. 121, pp.  137-162, 1995.
  6. R. Van Driessche and D. Roose, "An improved spectral bisection algorithm and its application to dynamic load balancing", Parallel Computing, vol. 21, pp.  29-48,  Jan.  1995.
  7. G. Karypis and V. Kumar, "A fast and high quality multilevel scheme for partitioning irregular graphs", Department of Computer Science, University of Minnesota, Tech. Rep. TR 95-035, 1995.
  8. C. Farhat and M. Lesoinne, "Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics", Int. J. Numer. Methods Eng., vol. 36, pp.  745-764, 1993.
  9. C. Farhat, N. Maman and W. Brown, "Mesh partitioning for implicit computations via iterative domain decomposition: Impact and optimization of the subdomain aspect ratio", Int. J. Num. Meth. in Eng., vol. 38, pp.  989-1000,  1995.
  10. J. Patterson, T. Cwik, R. Ferraro, N. Jacobi, P. Liewer, T. Lockhart, G. Lyzenga, J. Parker and D. Simoni, "Parallel computation applied to electromagnetic scattering and radiation analysis", Electromagn., vol. 10, pp.  21-40,  Jan.-June  1990.
  11. R. D. Ferraro, "Solving partial differential equations for electromagnetic scattering problems on coarse-grained concurrent computers,"in Computational Electromagnetics and Supercomputer Architecture, T. Cwik, and J. Patterson, Eds. Cambridge, MA: EMW, 1993,vol. 7, pp.  1111-154. 
  12. R. Lee and V. Chupongstimun, "A partitioning technique for the finite element solution of electromagnetic scattering from electrically large dielectric cylinders", IEEE Trans. Antennas Propagat., vol. 42, pp.  737-741, May  1994.
  13. Y. S. Choi-Grogan, K. Eswar, P. Sadayappan and R. Lee, "Sequential and parallel implementations of the partitioning finite-element method", IEEE Trans. Antennas Propagat., vol. 44, pp.  1609-1616, Dec.  1996.
  14. B. Després, "Domain decomposition method and boundaries the Helmholtz problem", in Proc. Int. Symp. Math. Numer. Aspects Wave Propagat. Phenomena, Strasbourg, France, 1992, pp.  44-52. 
  15. B. Després, "Domain decomposition method and the Helmholtz problem", in Proc. 2nd Int. Conf. Math. Aspects of Wave Propagat. , Dover, DE, 1993, pp.  197-206. 
  16. B. Stupfel, "A fast_domain decomposition method for the solution of electromagnetic scattering by large objects", IEEE Trans. Antennas Propagat., vol. 44, pp.  1375-1385, Oct.  1996.
  17. B. Stupfel, "Absorbing boundary conditions on arbitrary boundaries for the scalar and vector wave equations", IEEE Trans. Antennas Propagat., vol. 42, pp.  773-780, 1994.
  18. C. Farhart and F. X. Roux, "A method of finite element tearing and interconnecting and its parallel solution algorithm", Int. J. Numer. Method Eng., vol. 32, pp.  1205-1227, 1991.
  19. J. P. Berenger, "A PML for the absorption of electromagnetic waves", J. Computat. Phys., vol. 114, no. 2, pp.  185-200, Oct.  1994.
  20. W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium form modified Maxwell's equations with stretched coordinates", IEEE Microwave Guided Wave Lett., vol. 7, pp.  599-604, Sept.  1994.
  21. C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz and A. Taflove, "Ultrawideband absorbing boundary condition for termination of waveguiding structures in FD-TD simulations", IEEE Microwave Guided Wave Lett., vol. 4, no. 10, pp.  344-346, Apr.  1994.
  22. D. S. Katz, E. T. Thiele and A. Taflove, "Validation and extension to three-dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes", IEEE Microwave Guided Wave Lett., vol. 4, pp.  268-270, Aug.  1994.
  23. Z. S. Sacks, D. M. Kingsland, R. Lee and J. F. Lee, "A perfectly matched anistropic absorber for use as an absorbing boundary condition", IEEE Trans. Antennas Propagat., vol. 43, pp.  1460-1463, Dec.  1995.
  24. S. D. Gedney, "An anisotropic PML absorbing media for the truncation of FDTD Lattices", IEEE Trans. Antennas Propagat., vol. 44, pp.  1630-1639, Dec.  1996.
  25. S. D. Gedney, "An anisotropic PML absorbing media for FDTD simulation for fields in lossy dispersive media", Electromagn. , vol. 16, pp.  399-416, July/Aug.  1996.
  26. S. D. Gedney, "The PML absorbing medium,"in Advances in Computational Electrodynamics: The Finite Difference Time Domain, A. Taflove, Ed. Boston, MA: Artech House, 1998.
  27. C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space", IEEE Microwave Guided Wave Lett., vol.  5, pp.  90-92, Mar.  1995.
  28. J. Gong and J. L. Volakis, "Optimal selection of uniaxial artificial absorber layer for truncating finite element meshes", Electron. Lett. , vol. 31, no. 18, pp.  1559-1561, Aug.  31, 1995 .
  29. M. R. Lyons, A. C. Polycarpou and C. A. Balanis, "On the accuracy of PMLs using a finite element formulation", in IEEE Symp. Microwave Theory Tech., San Francisco, CA, June 1996, pp.  205- 208. 
  30. D. M. Kingsland, J. Gong, J. L. Volakis and J. F. Lee, "Performance of an anisotropic artificial absorber for truncating finite-element meshes", IEEE Trans. Antennas Propagat., vol. 44, pp.  975-982, July  1996.
  31. W. C. Chew and J. M. Jin, "PML's in the discretized space: An analysis and optimization", Electromagn., vol. 16, pp.  325 -340, July/Aug.  1996.
  32. Y. Y. Botros and J. L. Volakis, "Preconditioned generalized minimal residual (GMRES) solver for domains truncated by PML absorbers", in 1998 ACES Conf., Monterey, CA, Mar. 1998, pp.  639-646. 
  33. P. C. Hammer, O. J. Marlowe and A. H. Stroud, "Numerical integration over simplexes and cones", Math. Tables  Aids Comp., vol. 10, pp.  130-137, 1956.
  34. C. Farhat and F. X. Roux, "Implicit parallel processing in structural mechanics,"in Computat. Mech. Adv., 1994,vol. 2, pp.  1-124. 
  35. D. Vanderstraeten and R. Keunings, "Optimized partitioning of unstructured finite element meshes", Int. J. Numer. Methods in Eng., vol. 38, pp.  433-450,  1995.