A Novel Conversion Technique from Nodal to Edge Finite Element Data Structure for Electromagnetic Analysis
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 analysisReferences
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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.1063/1.338584
6. D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19: 1–120, 2010, https://doi.org/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, https://doi.org/10.1016/S0045-7949%2801%2900003-7
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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.1002/%28SICI%291099-1204%28200003/06%2913%3A2/3<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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.9734/JAMCS/2018/39632
27. J.C. Nedelec, Mixed finite elements in R3, Numerische Mathematik, 35: 315–341, 1980, https://doi.org/10.1007/BF01396415
28. R. Otin, Regularized Maxwell equations and nodal finite elements for electromagnetic field computations, Electromagnetics, 30(1–2): 190–204, 2010, https://doi.org/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, https://doi.org/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, https://doi.org/10.1109/75.401074
31. F. Rapetti, A. Bossavit, Whitney forms of higher degree, SIAM Journal on Numerical Analysis, 47(3): 2369–2386, 2009, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.1109/20.497506
2. A. Ahagon, T. Kashimoto, Three-dimensional electromagnetic wave analysis using high order edge elements, IEEE Transactions on Magnetics, 31(3): 1753–1756, 1995, https://doi.org/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, https://doi.org/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, https://doi.org/10.1063/1.338584
6. D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19: 1–120, 2010, https://doi.org/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, https://doi.org/10.1016/S0045-7949%2801%2900003-7
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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.1002/%28SICI%291099-1204%28200003/06%2913%3A2/3<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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.9734/JAMCS/2018/39632
27. J.C. Nedelec, Mixed finite elements in R3, Numerische Mathematik, 35: 315–341, 1980, https://doi.org/10.1007/BF01396415
28. R. Otin, Regularized Maxwell equations and nodal finite elements for electromagnetic field computations, Electromagnetics, 30(1–2): 190–204, 2010, https://doi.org/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, https://doi.org/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, https://doi.org/10.1109/75.401074
31. F. Rapetti, A. Bossavit, Whitney forms of higher degree, SIAM Journal on Numerical Analysis, 47(3): 2369–2386, 2009, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/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, https://doi.org/10.1109/20.497506
