volume | PIER Journals

Abstract

A broadband preconditioner based on a modified version of the sparsified nested dissection ordering (m-spaNDO) technique is proposed for the full wave discrete exterior calculus (DEC) A-Φformulation solver in electromagnetics. The matrix equation discretized by the DEC A-Φ solver is in general complex symmetric and indefinite. When conductive media and disparate mesh are involved, the DEC A-Φ matrix equation is ill-conditioned, and proper preconditioner must be utilized to accelerate iterative solver convergence. In this letter, an introduction to the DEC A-Φ solver is provided, followed by the implementation details of the m-spaNDO preconditioner. Numerical examples in this paper show that the proposed m-spaNDO preconditioner can effectively accelerate the convergence of iterative solvers in solving ill-conditioned problems. The m-spaNDO preconditioned DEC A-Φ solver has O(N logN) computational complexity and the efficiency of the preconditioner is independent of change in parameters such as frequency and conductivity in the problem, which indicates the broadband stable nature of the m-spaNDO preconditioner.

1. Chew, Weng Cho, "Vector potential electromagnetics with generalized gauge for inhomogeneous media: Formulation," Progress In Electromagnetics Research, Vol. 149, 69-84, 2014.

2. Zhang, Boyuan, Dong-Yeop Na, Dan Jiao, and Weng Cho Chew, "An A-Φ formulation solver in electromagnetics based on discrete exterior calculus," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 8, 11-21, 2022.

3. Zhao, Yanpu and W. N. Fu, "A new stable full-wave Maxwell solver for all frequencies," IEEE Transactions on Magnetics, Vol. 53, No. 6, 1-4, 2017.

4. Hestenes, Magnus R. and Eduard Stiefel, "Methods of conjugate gradients for solving linear systems," Journal of Research of the National Bureau of Standards, Vol. 49, No. 6, 409-436, 1952.
doi:10.6028/jres.049.044

5. Fletcher, Roger, "Conjugate gradient methods for indefinite systems," Numerical Analysis: Proceedings of the Dundee Conference on Numerical Analysis, 73-89, Berlin, Heidelberg, 1976.

6. Saad, Youcef and Martin H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 3, 856-869, 1986.

7. Chew, W. C., "Computational electromagnetics: The physics of smooth versus oscillatory fields," Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 362, No. 1816, 579-602, 2004.

8. Chai, Wenwen and Dan Jiao, "Theoretical study on the rank of integral operators for broadband electromagnetic modeling from static to electrodynamic frequencies," IEEE Transactions on Components, Packaging and Manufacturing Technology, Vol. 3, No. 12, 2113-2126, 2013.

9. Cambier, Léopold, Chao Chen, Erik G. Boman, Sivasankaran Rajamanickam, Raymond S. Tuminaro, and Eric Darve, "An algebraic sparsified nested dissection algorithm using low-rank approximations," SIAM Journal on Matrix Analysis and Applications, Vol. 41, No. 2, 715-746, 2020.
doi:10.1137/19M123806X

10. Ambikasaran, S., "Fast algorithms for dense numerical linear algebra and applications," Stanford University, Stanford, CA, 2013.

11. Chandrasekaran, Shiv, Patrick Dewilde, Ming Gu, T. Pals, Xiaorui Sun, Alle-Jan van der Veen, and Daniel White, "Some fast algorithms for sequentially semiseparable representations," SIAM Journal on Matrix Analysis and Applications, Vol. 27, No. 2, 341-364, 2005.

12. Ho, Kenneth L. and Lexing Ying, "Hierarchical interpolative factorization for elliptic operators: Differential equations," Communications on Pure and Applied Mathematics, Vol. 69, No. 8, 1415-1451, 2016.

13. Chew, Weng Cho and C.-C. Lu, "The use of Huygens' equivalence principle for solving the volume integral equation of scattering," IEEE Transactions on Antennas and Propagation, Vol. 41, No. 7, 897-904, 1993.

14. Taflove, Allen, Susan C. Hagness, and Melinda Piket-May, "Computational electromagnetics: The finite-difference time-domain method," The Electrical Engineering Handbook, Vol. 3, 629-670, 2005.

15. Jin, Jian-Ming, The Finite Element Method in Electromagnetics, John Wiley & Sons, 2015.

16. George, Alan, "Nested dissection of a regular finite element mesh," SIAM Journal on Numerical Analysis, Vol. 10, No. 2, 345-363, 1973.

17. Deschamps, Georges A., "Electromagnetics and differential forms," Proceedings of the IEEE, Vol. 69, No. 6, 676-696, 1981.
doi:10.1109/PROC.1981.12048

18. Desbrun, Mathieu, Anil N. Hirani, Melvin Leok, and Jerrold E. Marsden, "Discrete exterior calculus," arXiv:math.DG/0508341, 2005.

19. Klema, Virginia and Alan Laub, "The singular value decomposition: Its computation and some applications," IEEE Transactions on Automatic Control, Vol. 25, No. 2, 164-176, 1980.

20. Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp, "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions," SIAM Review, Vol. 53, No. 2, 217-288, 2011.
doi:10.1137/090771806

21. Chan, Tony F., "Rank revealing QR factorizations," Linear Algebra and Its Applications, Vol. 88, 67-82, 1987.

Dr. Boyuan Zhang

Professor Weng Cho Chew