volume | PIER Journals

Abstract

The finite element implement of the generalized Lorenz gauged A formulation has been proposed for low-frequency modeling. However, the inverse of mass matrix of intermediate scalar in the finite element implement leads to additional computation cost and dense coefficient matrix. In this paper we propose to adopt a diagonal lumping mass matrix in the finite element discretization of the generalized Lorenz gauged double-curl operator in charge-free electromagnetic problems. Consequently, a sparser discrete system with improved condition number is thus obtained which is more favourable for low-frequency modeling in frequency-domain analysis. Furthermore, we apply the diagonal lumping formulation in time-domain analysis, showing that it can remedy spurious linear growth problem. Numerical examples are used to demonstrate the validity.

1. Jin, J. M., The Finite Element Method in Electromagnetics, 3rd Ed., John Wiley & Sons, 2015.

2. Lee, S. C., J. F. Lee, and R. Lee, "Hierarchical vector finite elements for analyzing waveguiding structures," IEEE Transactions on Microwave Theory and Techniques, Vol. 51, No. 8, 1897-1905, 2003.
doi:10.1109/TMTT.2003.815263 Google Scholar

3. Lee, S. H. and J. M. Jin, "Application of the treecotree splitting for improving matrix conditioning in the full-wave finite-element analysis of high-speed circuits," Microwave and Optical Technology Letters, Vol. 50, No. 6, 1476-1481, 2008.
doi:10.1002/mop.23403 Google Scholar

4. Zhu, J. and D. Jiao, "A theoretically rigorous full-wave finite-element-based solution of Maxwell’s equations from DC to high frequencies," IEEE Transactions on Advanced Packaging, Vol. 33, No. 4, 1043-1050, 2010.
doi:10.1109/TADVP.2010.2057428 Google Scholar

5. Zhu, J. and D. Jiao, "A rigorous solution to the low-frequency breakdown in full-wave finiteelement- based analysis of general problems involving inhomogeneous lossless/lossy dielectrics and nonideal conductors," IEEE Transactions on Microwave Theory and Techniques, Vol. 59, No. 12, 3294-3306, 2011.
doi:10.1109/TMTT.2011.2171707 Google Scholar

6. Venkatarayalu, N. V., M. N. Vouvakis, Y. B. Gan, et al. "Suppressing linear time growth in edge element based finite element time domain solution using divergence free constraint equation," Antennas and Propagation Society International Symposium, 193-196, 2005. Google Scholar

7. Hwang, C. T. and R. B.Wu, "Treating late-time instability of hybrid finite-element/finite-difference time-domain method," IEEE Transactions on Antennas and Propagation, Vol. 47, No. 2, 227-232, 1999.
doi:10.1109/8.761061 Google Scholar

8. Golias, N. A. and T. D. Tsiboukis, "Magnetostatics with edge elements: A numerical investigation in the choice of the tree," IEEE Transactions on Magnetics, Vol. 30, No. 5, 2877-2880, 1994.
doi:10.1109/20.312537 Google Scholar

9. Kikuchi, F., "Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism," Computer Methods in Applied Mechanics and Engineering, Vol. 64, No. 1, 509-521, 1987. Google Scholar

10. Chen, Z., Q. Du, and J. Zou, "Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients," SIAM Journal on Numerical Analysis, Vol. 37, No. 5, 1542-1570, 2000.
doi:10.1137/S0036142998349977 Google Scholar

11. Benzi, M., G. H. Golub, and J. Liesen, "Numerical solution of saddle point problems," Actanumerica, Vol. 14, 1-137, 2005. Google Scholar

12. Bespalov, A. N., "Finite element method for the eigenmode problem of a RF cavity resonator," Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 3, No. 3, 163-178, 1988.
doi:10.1515/rnam.1988.3.3.163 Google Scholar

13. Hiptmair, R., "Finite elements in computational electromagnetism," Acta Numerica, Vol. 11, 237-339, 2002. Google Scholar

14. Chew, W. C., "Vector potential electromagnetics with generalized gauge for inhomogeneous media: Formulation," Progress In Electromagnetics Research, Vol. 149, 69-84, 2014.
doi:10.2528/PIER14060904 Google Scholar

15. Li, Y. L., S. Sun, Q. I. Dai, et al. "Finite element implementation of the generalized-Lorenz gauged A-Φ formulation for low-frequency circuit modeling," IEEE Transactions on Antennas and Propagation, Vol. 64, No. 10, 4355-4364, 2016.
doi:10.1109/TAP.2016.2593748 Google Scholar

16. Li, Y. L., S. Sun, Q. I. Dai, and W. C. Chew, "Vectorial solution to double curl equation with generalized coulomb gauge for magnetostatic problems," IEEE Transactions on Magnetics, Vol. 51, No. 8, 1-6, 2015.
doi:10.1109/TMAG.2015.2423267 Google Scholar

17. Bossavit, A. and L. Kettunen, "Yee-like schemes on a tetrahedral mesh, with diagonal lumping," International Journal of Numerical Modelling Electronic Networks Devices and Fields, Vol. 12, 129-142, 1999.
doi:10.1002/(SICI)1099-1204(199901/04)12:1/2<129::AID-JNM327>3.0.CO;2-G Google Scholar

18. Magele, C., H. Stogner, and K. Preis, "Comparison of different finite element formulations for 3D magnetostatic problems," IEEE Transactions on Magnetics, Vol. 24, No. 1, 31-34, 1988.
doi:10.1109/20.43846 Google Scholar

Dr. Peng Jiang

Dr. Guozhong Zhao

Dr. Qun Zhang

Dr. Zhenqun Guan