volume | PIER Journals

Abstract

To solve the low-frequency breakdown inherent from the electric field integral equation (EFIE), an alternative new form of the EFIE is proposed by using the Coulomb-gauge Green's function of quasi-static approximation. Different from the commonly adopted Lorentz-gauge EFIE, the Coulomb-gauge EFIE separates the solenoidal and irrotational surface currents explicitly, which captures inductive and capacitive responses through electrodynamic and electrostatic Green's functions, respectively. By applying existing techniques such as the loop-tree decomposition, frequency normalization, and basis rearrangement, the Coulomb-gauge EFIE also can remedy the low-frequency breakdown problem. Through comparative studies between the Lorentz-gauge and Coulomb-gauge EFIE approaches from mathematical, physical and numerical aspects, the Coulomb-gauge EFIE approach shows the capability of solving low-frequency problems and achieves almost the same accuracy and computational costs compared to the Lorentz-gauge counterpart.

1. Chew, W. C., M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool, London, UK, 2009.
doi:10.2200/S00102ED1V01Y200807CEM012

2. Araujo, M. G., J. M. Taboada, J. Rivero, and F. Obelleiro, "Comparison of surface integral equations for left-handed materials," Progress In Electromagnetics Research, Vol. 118, 425-440, 2011.
doi:10.2528/PIER11031110 Google Scholar

3. Rao, S. M., D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag., Vol. 30, 409-418, 1982.
doi:10.1109/TAP.1982.1142818 Google Scholar

4. Wilton, D. R., J. S. Lin, and S. M. Rao, "A novel technique to calculate the electromagnetic scattering by surfaces of arbitrary shape," URSI Radio Science Meeting Dig., 24, Los Angeles, CA, Jun. 1981. Google Scholar

5. Eibert, T. F., "Iterative-solver convergence for loop-star and loop-tree decompositions in method-of-moments solutions of the electric-field integral equation," IEEE Antennas Propag. Mag., Vol. 46, 80-85, Jun. 2004.
doi:10.1109/MAP.2004.1374101 Google Scholar

6. Vecchi, G., "Loop-star decomposition of basis functions in the discretization of the EFIE," IEEE Trans. Antennas Propag., Vol. 47, 339-346, Feb. 1999.
doi:10.1109/8.761074 Google Scholar

7. Zhao, J. S. and W. C. Chew, "Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies," IEEE Trans. Antennas Propag., Vol. 48, 1635-1645, Oct. 2000.
doi:10.1109/8.899680 Google Scholar

8. Yeom, J.-H., H. Chin, H.-T. Kim, and K.-T. Kim, "Block matrix preconditioner method for the electric field integral equation (EFIE) formulation based on loop-star basis functions," Progress In Electromagnetics Research, Vol. 134, 543-558, 2013. Google Scholar

9. Lee, J. F., R. Lee, and R. J. Burkholder, "Loop star basis functions and a robust preconditioner for EFIE scattering problems," IEEE Trans. Antennas Propag., Vol. 51, 1855-1863, Aug. 2003.
doi:10.1109/TAP.2003.814736 Google Scholar

10. Christiansen, S. H. and J. C. Nedelec, "A preconditioner for the electric field integral equation based on Calderón formulas," SIAM J. Numer. Anal., Vol. 40, 1100-1135, Sep. 2002.
doi:10.1137/S0036142901388731 Google Scholar

11. Yan, S., J. M. Jin, and Z. Nie, "EFIE analysis of low-frequency problems with loop-star decomposition and Calderón multiplicative preconditioner," IEEE Trans. Antennas Propag., Vol. 58, 857-867, Mar. 2010.
doi:10.1109/TAP.2009.2039336 Google Scholar

12. Andriulli, F. P., K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, "A multiplicative Calderón preconditioner for the electric field integral equation," IEEE Trans. Antennas Propag., Vol. 56, 2398-2412, Aug. 2008.
doi:10.1109/TAP.2008.926788 Google Scholar

13. Buffa, A. and S. H. Christiansen, "A dual finite element complex on the barycentric refinement," Math. Comput., Vol. 76, 1743-1769, 2007.
doi:10.1090/S0025-5718-07-01965-5 Google Scholar

14. Valdes, F., F. P. Andriulli, K. Cools, and E. Michielssen, "High-order div- and quasi curl-conforming basis functions for Calderón multiplicative preconditioning of the EFIE," IEEE Trans. Antennas Propag., Vol. 59, 1321-1337, Apr. 2011.
doi:10.1109/TAP.2011.2109692 Google Scholar

15. Andriulli, F. P., A. Tabacco, and G. Vecchi, "Solving the EFIE at low frequencies with a conditioning that grows only logarithmically with the number of unknowns," IEEE Trans. Antennas Propag., Vol. 58, 1614-1624, May 2010.
doi:10.1109/TAP.2010.2044325 Google Scholar

16. Qian, Z. G. and W. C. Chew, "An augmented electric field integral equation for high-speed interconnect analysis," Microw. Opt. Technol. Lett., Vol. 50, 2658-2662, Oct. 2008.
doi:10.1002/mop.23736 Google Scholar

17. Qian, Z. G. and W. C. Chew, "Enhanced A-EFIE with perturbation method," IEEE Trans. Antennas Propag., Vol. 58, 3256-3264, Oct. 2010.
doi:10.1109/TAP.2010.2055795 Google Scholar

18. Chen, Y. P., L. J. Jiang, Z. G. Qian, and W. C. Chew, "An augmented electric field integral equation for layered medium Green's function," IEEE Trans. Antennas Propag., Vol. 59, 960-968, Mar. 2011.
doi:10.1109/TAP.2010.2103042 Google Scholar

19. Yan, S., J. M. Jin, and Z. P. Nie, "Analysis of electrically large problems using the augmented EFIE with a Calderón preconditioner," IEEE Trans. Antennas Propag., Vol. 59, 2303-2314, Jun. 2011.
doi:10.1109/TAP.2011.2143672 Google Scholar

20. Pan, Y. C. and W. C. Chew, "A fast multipole method for embedded structure in a stratified medium," Progress In Electromagnetics Research, Vol. 44, 1-38, 2004.
doi:10.2528/PIER03050602 Google Scholar

21. Ergul, O. and L. Gurel, "Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm," Progress In Electromagnetics Research, Vol. 108, 81-99, 2010.
doi:10.2528/PIER10071104 Google Scholar

22. Bogaert, I., J. Peeters, and D. De Zutter, "Error control of the vectorial nondirective stable plane wave multilevel fast multipole algorithm," Progress In Electromagnetics Research, Vol. 111, 271-290, 2011.
doi:10.2528/PIER10090604 Google Scholar

23. Pan, X.-M., L. Cai, and X.-Q. Sheng, "An efficient high order multilevel fast multipole algorithm for electromagnetic scattering analysis," Progress In Electromagnetics Research, Vol. 126, 85-100, 2012.
doi:10.2528/PIER12020203 Google Scholar

24. Wang, W. and N. Nishimura, "Calculation of shape derivatives with periodic fast multipole method with application to shape optimization of metamaterials," Progress In Electromagnetic Research, Vol. 127, 46-64, 2012. Google Scholar

25. Gope, D., A. Ruehli, and V. Jandhyala, "Solving low-frequency EM-CKT problems using the PEEC method," IEEE Transactions on Advanced Packaging, Vol. 30, No. 2, May 2007.
doi:10.1109/TADVP.2007.896000 Google Scholar

26. Jiang, L. J. and A. Ruehli, "On the frequency barrier of surface integral equations from a circuit point of view," Progress In Electromagnetics Research Symposium Abstracts, 46, Cambridge, USA, Jul. 5-8, 2010. Google Scholar

27. Song, Z., D. Su, F. Duval, and A. Louis, "Model order reduction for PEEC modeling based on moment matching," Progress In Electromagnetics Research, Vol. 114, 285-299, 2011. Google Scholar

28. Song, Z., F. Dai, D. Su, S. Xie, and F. Duval, "Reduced PEEC modeling of wire-ground structures using a selective mesh approach," Progress In Electromagnetics Research, Vol. 123, 355-370, 2012.
doi:10.2528/PIER11112109 Google Scholar

29. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley India Pvt Ltd., 2007.

30. Nevels, R. D. and K. J. Crowell, "A Coulomb gauge analysis of a wire scatterer," IEE Proc., Pt. H, Vol. 137, 384-388, Dec. 1990. Google Scholar

31. Bladel, J., Electromagnetic Field, 3rd Ed., Wiley-Interscience, 2007.

32. Chew, W. C., E. Michielssen, J. M. Song, and J. M. Jin, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Inc., Norwood, MA, 2001.

33. Saad, Y. and M. H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 7, 856-869, Jul. 1986. Google Scholar

34. Chu, Y. H. and W. C. Chew, "Large-scale computation for electrically small structures using surface-integral equation method," Microw. Opt. Technol. Lett., Vol. 47, No. 6, 525-530, Dec. 20, 2005.
doi:10.1002/mop.21219 Google Scholar

35. http://www.ansys.com/Products/Simulation.

Ms. Xiaoyan Y. Z. Xiong

Dr. Li Jun Jiang

Dr. Wei E. I. Sha

Dr. Yat-Hei Lo