A Novel Conversion Technique from Nodal to Edge Finite Element Data Structure for Electromagnetic Analysis

  • Durgarao Kamireddy Indian Institute of Technology Guwahati
  • Arup Nandy Indian Institute of Technology Guwahati

Abstract

Standard nodal finite elements in the electromagnetic analysis have a well-known limitation of the occurrence of a spurious solution. In order to circumvent the problem, a penalty function method or a regularization method is used with the potential formulation. These methods solve the problem partially by pushing the spurious mode to the higher end of the spectrum. But it fails to capture singular eigenvalues in the case of the problem domains with sharp edges and corners. To circumvent this limitation, edge elements have been developed for the electromagnetic analysis where degree of freedom is along the edges. But most of the preprocessors develop complex meshes in the nodal framework. In this work, we have developed a novel technique to convert nodal data structure to edge data structure for electromagnetic analysis. We have explained the conversion algorithm in details, mentioning associated complexities with relevant examples. The performance of the developed algorithm has been demonstrated extensively with several examples.

Keywords

FEM, Electromagnetics, Edge finite elements, Eigenvalue analysis,

References

1. M. Agrawal, C.S. Jog, Monolithic formulation of electromechanical systems within the context of hybrid finite elements, Computational Mechanics, 59 (3): 443–457, 2017, doi: 10.1007/s00466-016-1356-1.
2. A. Ahagon, T. Kashimoto, Three-dimensional electromagnetic wave analysis using high order edge elements, IEEE Transactions on Magnetics, 31(3): 1753–1756, 1995, doi: 10.1109/20.376375.
3. M. Ainsworth, J.F. Coyle, P.D. Ledger, K. Morgan, Computing Maxwell eigenvalues by using higher order edge elements in three dimensions, IEEE Transactions on Magnetics, 39(5): 2149–2153, 2003, doi: 10.1109/TMAG.2003.817097.
4. Nandy Arup Kumar, Robust Finite Element Strategies for Structures, Acoustics, Electromagnetics and Magneto-hydrodynamics, Ph.D. thesis, Indian Institute of Science Bangalore, Department of Mechanical Engineering, IISc, Bangalore, India, https://etd.iisc.ac.in/handle/2005/2913.
5. M.L. Barton, Z.J. Cendes, New vector finite elements for three-dimensional magnetic field computation, Journal of Applied Physics, 61(8): 3919–3921, 1987, doi: 10.1063/1.338584.
6. D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19: 1–120, 2010, doi: 10.1017/S0962492910000012.
7. D. Boffi, M. Farina, L. Gastaldi, On the approximation of Maxwell’s eigenproblem in general 2D domains, Computers & Structures, 79: 1089–1096, 2001, doi: 10.1016/S0045-7949(01)00003-7.
8. D. Boffi, P. Fernandes, L. Gastaldi, I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM Journal on Numerical Analysis, 36(4): 1264–1290, 1999, doi: 10.1137/S003614299731853X.
9. A. Bossavit, A rationale for ‘edge-elements’ in 3-D fields computations, IEEE Transactions on Magnetics, 24(1): 74–79, 1988, doi: 10.1109/20.43860.
10. A. Bossavit, J.-C. Vérité, A mixed FEM-BIEM method to solve 3-D eddy-current problems, IEEE Transactions on Magnetics, 18(2): 431–435, 1982, doi: 10.1109/TMAG.1982.1061847.
11. J.H. Bramble, T. Kolev, J.E. Pasciak, The approximation of the Maxwell eigenvalue problem using a least-squares method, Mathematics of Computation, 74(252): 1575–1598, 2005, doi: 10.1090/S0025-5718-05-01759-X.
12. Z.J. Cendes, Vector finite elements for electromagnetic field computation, IEEE Transactions on Magnetics, 27(5): 3958–3966, 1991, doi: 10.1109/20.104970.
13. M. Dauge, Benchmark computations for Maxwell equations for the approximation of highly singular solutions, 2004, http://perso.univ-rennes1.fr/monique.dauge/core/index.html.
14. A. Elsherbeni, D. Kajfez, S. Zeng, Circular sectoral waveguides, IEEE Antennas and Propagation Magazine, 33(6): 20–27, 1991, doi: 10.1109/74.107352.
15. L.E. Garcia-Castillo, M. Salazar-Palma, Second-order Nédélec tetrahedral element for computational electromagnetics, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 13, 261–287, 2000, doi: 10.1002/(SICI)1099-1204(200003/06)13:2/3<261::AID-JNM360>3.0.CO;2-L.
16. R.D. Graglia, D.R. Wilton, A.F. Peterson, Higher order interpolatory vector bases for computational electromagnetics, IEEE Transactions on Antennas and Propagation, 45(3): 329–342, 1997, doi: 10.1109/8.558649.
17. R.F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, NY, 1961, https://cds.cern.ch/record/230916.
18. J.-M. Jin, The Finite Element Method in Electromagnetics, Third ed., John Wiley & Sons, New Jersey, 2014.
19. C.S. Jog, A. Nandy, Mixed finite elements for electromagnetic analysis, Computers and Mathematics with Applications, 68(8): 887–902, 2014, doi: 10.1016/j.camwa.2014.08.006.
20. D. Kamireddy, A. Nandy, Combination of triangular and quadrilateral edge element for the eigenvalue analysis of electromagnetic wave propagation, European Journal of Molecular & Clinical Medicine, 7(11): 1656–1663, 2020, https://ejmcm.com/article_5696.html.
21. D. Kamireddy, A. Nandy, Creating edge element from four node quadrilateral element, IOP Conference Series: Materials Science and Engineering, 1080, 2021, doi: 10.1088/1757-899x/1080/1/012015.
22. M. Koshiba, K. Hayata, M. Suzuki, Finite-element formulation in terms of the electricfield vector for electromagnetic waveguide problems, IEEE Transactions on Microwave Theory and Techniques, 33(10): 900–905, 1985, doi: 10.1109/TMTT.1985.1133148.
23. J.F. Lee, D.K. Sun, Z.J. Cendes, Tangential vector finite elements for electromagnetic field computation, IEEE Transactions on Magnetics, 27(5): 4032–4035, 1991, doi: 10.1109/20.104986.
24. A. Nandy, C.S. Jog, An amplitude finite element formulation for electromagnetic radiation and scattering, Computers & Mathematics with Applications, 71(7): 1364–1391, 2016, doi: 10.1016/j.camwa.2016.02.013.
25. A. Nandy, C.S. Jog, A monolithic finite-element formulation for magnetohydrodynamics, Sadhana – Academy Proceedings in Engineering Sciences, 43: 1–18, 2018, doi: 10.1007/s12046-018-0905-z.
26. A. Nandy, C.S. Jog, Conservation properties of the trapezoidal rule for linear transient electromagnetics, Journal of Advances in Mathematics and Computer Science, 26(4): 1–26, 2018, doi: 10.9734/JAMCS/2018/39632.
27. J.C. Nedelec, Mixed finite elements in R3, Numerische Mathematik, 35: 315–341, 1980, doi: 10.1007/BF01396415.
28. R. Otin, Regularized Maxwell equations and nodal finite elements for electromagnetic field computations, Electromagnetics, 30(1–2): 190–204, 2010, doi: 10.1080/02726340903485489.
29. K.D. Paulsen, D.R. Lynch, Elimination of vector parasites in finite element Maxwell solutions, IEEE Transactions on Microwave Theory and Techniques, 39(3): 395–404, 1991, doi: 10.1109/22.75280.
30. U. Pekel, R. Mittra, An application of the perfectly matched layer (PML) concept to the finite element method frequency domain analysis of scattering problems, IEEE Microwave and Guided Wave Letters, 59(8): 258–260, 1995, doi: 10.1109/75.401074.
31. F. Rapetti, A. Bossavit, Whitney forms of higher degree, SIAM Journal on Numerical Analysis, 47(3): 2369–2386, 2009, doi: 10.1137/070705489.
32. C.J. Reddy, M.D. Deshpande, C.R. Cockrell, F.B. Beck, Finite element method for eigenvalue problems in electromagnetics, NASA Technical Paper, 3485, http://ecee.colorado.edu/~ecen5004/PDFs/FiniteElementCJReddy.pdf.
33. Seung-Cheol Lee, Jin-Fa Lee, R. Lee, Hierarchical vector finite elements for analyzing waveguiding structures, IEEE Transactions on Microwave Theory and Techniques, 51(8): 1897–1905, 2003, doi: 10.1109/TMTT.2003.815263.
34. X.Q. Sheng, J.M. Jin, C.C. Lu, W.C. Chew, On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering, IEEE Transactions on Antennas and Propagation, 46(3): 303–311, 1998, doi: 10.1109/8.662648.
35. J.P.Webb, Edge elements and what they can do for you, IEEE Transactions on Magnetics, 29: 1460–1465, 1993, doi: 10.1109/CEFC.1992.720787.
36. J.P. Webb, Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements, IEEE Transactions on Antennas and Propagation, 47: 1244–1253, 1999, doi: 10.1109/8.791939.
37. J.P. Webb, V.N. Kanellopoulos, Absorbing boundary conditions for the finite element solution of the vector wave equation, Microwave and Optical Technology Letters, 2: 370–372, 1989, doi: 10.1002/mop.4650021010.
38. Wolfram Research, Inc., Mathematica 10.4.1, Champaign, IL (2020), https://www.wolfram.com.
39. T.V. Yioultsis, T. Tsiboukis, Development and implementation of second and third order vector finite elements in various 3-D electromagnetic field problems, IEEE Transactions on Magnetics, 33(2): 1812–1815, 1997, doi: 10.1109/20.582630.
40. T.V. Yioultsis, Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions, IEEE Transactions on Magnetics, 32(3): 1389–1392, 1996, doi: 10.1109/20.497506.
Published
Apr 14, 2022
How to Cite
KAMIREDDY, Durgarao; NANDY, Arup. A Novel Conversion Technique from Nodal to Edge Finite Element Data Structure for Electromagnetic Analysis. Computer Assisted Methods in Engineering and Science, [S.l.], v. 28, n. 4, p. 291–319, apr. 2022. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/384>. Date accessed: 28 june 2022. doi: http://dx.doi.org/10.24423/cames.384.
Section
Articles