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pp 1-47 | Cite as

Systems of Differential Algebraic Equations in Computational Electromagnetics

  • Idoia Cortes Garcia
  • Sebastian Schöps
  • Herbert De Gersem
  • Sascha Baumanns
Chapter
Part of the Differential-Algebraic Equations Forum book series

Abstract

Starting from space-discretisation of Maxwell’s equations, various classical formulations are proposed for the simulation of electromagnetic fields. They differ in the phenomena considered as well as in the variables chosen for discretisation. This contribution presents a literature survey of the most common approximations and formulations with a focus on their structural properties. The differential-algebraic character is discussed and quantified by the differential index concept.

Keywords

DAE index Maxwell’s equations Quasistatic approximations 

Mathematics Subject Classification (2010)

34A09 35Q61 78A25 78M12 65D30 

References

  1. 1.
    Albanese, R., Coccorese, E., Martone, R., Miano, G., Rubinacci, G.: On the numerical solution of the nonlinear three-dimensional eddy current problem. IEEE Trans. Magn. 27(5), 3990–3995 (1991). https://doi.org/10.1109/20.104976 Google Scholar
  2. 2.
    Alonso Rodríguez, A., Raffetto, M.: Unique solvability for electromagnetic boundary value problems in the presence of partly lossy inhomogeneous anisotropic media and mixed boundary conditions. Math. Models Methods Appl. Sci. 13(04), 597–611 (2003). https://doi.org/10.1142/S0218202503002672 Google Scholar
  3. 3.
    Alonso Rodríguez, A., Valli, A.: Eddy Current Approximation of Maxwell Equations. Modeling, Simulation and Applications, vol. 4. Springer, Heidelberg (2010). https://doi.org/10.1007/978-88-470-1506-7 Google Scholar
  4. 4.
    Alotto, P., De Cian, A., Molinari, G.: A time-domain 3-D full-Maxwell solver based on the cell method. IEEE Trans. Magn. 42(4), 799–802 (2006).  https://doi.org/10.1109/tmag.2006.871381 Google Scholar
  5. 5.
    Assous, F., Ciarlet, P., Labrunie, S.: Mathematical Foundations of Computational Electromagnetism. Springer, Cham (2018)Google Scholar
  6. 6.
    Außerhofer, S., Bíró, O., Preis, K.: Discontinuous Galerkin finite elements in time domain eddy-current problems. IEEE Trans. Magn. 45(3), 1300–1303 (2009)Google Scholar
  7. 7.
    Bartel, A., Baumanns, S., Schöps, S.: Structural analysis of electrical circuits including magnetoquasistatic devices. Appl. Numer. Math. 61, 1257–1270 (2011). https://doi.org/10.1016/j.apnum.2011.08.004 Google Scholar
  8. 8.
    Baumanns, S.: Coupled electromagnetic field/circuit simulation: modeling and numerical analysis. Ph.D. thesis, Universität zu Köln, Köln (2012)Google Scholar
  9. 9.
    Baumanns, S., Selva Soto, M., Tischendorf, C.: Consistent initialization for coupled circuit-device simulation. In: Roos, J., Costa, L.R.J. (eds.) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry, vol. 14, pp. 297–304. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-12294-1_38 Google Scholar
  10. 10.
    Baumanns, S., Clemens, M., Schöps, S.: Structural aspects of regularized full Maxwell electrodynamic potential formulations using FIT. In: Manara, G. (ed.) Proceedings of 2013 URSI International Symposium on Electromagnetic Theory (EMTS), pp. 1007–1010. IEEE, New York (2013)Google Scholar
  11. 11.
    Becks, T., Wolff, I.: Analysis of 3-d metallization structures by a full-wave spectral-domain technique. IEEE Trans. Microwave Theory Tech. 40(12), 2219–2227 (1992). https://doi.org/10.1109/22.179883 Google Scholar
  12. 12.
    Bedrosian, G.: A new method for coupling finite element field solutions with external circuits and kinematics. IEEE Trans. Magn. 29(2), 1664–1668 (1993). https://doi.org/10.1109/20.250726 Google Scholar
  13. 13.
    Bíró, O., Preis, K.: On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents. IEEE Trans. Magn. 25(4), 3145–3159 (1989). https://doi.org/10.1109/20.34388 Google Scholar
  14. 14.
    Bíró, O., Preis, K.: Finite element analysis of 3-d eddy currents. IEEE Trans. Magn. 26(2), 418–423 (1990). https://doi.org/10.1109/20.106343 Google Scholar
  15. 15.
    Bíró, O., Preis, K., Richter, K.R.: Various FEM formulations for the calculation of transient 3d eddy currents in nonlinear media. IEEE Trans. Magn. 31(3), 1307–1312 (1995). https://doi.org/10.1109/20.376269 Google Scholar
  16. 16.
    Boffi, D.: Finite element approximation of eigenvalue problems. Acta. Numer. 19, 1–120 (2010). https://doi.org/10.1017/S0962492910000012 Google Scholar
  17. 17.
    Bondeson, A., Rylander, T., Ingelström, P.: Computational Electromagnetics. Texts in Applied Mathematics. Springer, Berlin (2005). https://doi.org/10.1007/b136922
  18. 18.
    Bossavit, A.: Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proc. 135(8), 493–500 (1988). https://doi.org/10.1049/ip-a-1:19880077 Google Scholar
  19. 19.
    Bossavit, A.: Differential geometry for the student of numerical methods in electromagnetism. Technical Report, Électricité de France (1991)Google Scholar
  20. 20.
    Bossavit, A.: Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego (1998)Google Scholar
  21. 21.
    Bossavit, A.: On the geometry of electromagnetism. (4): ‘Maxwell’s house’. J. Jpn. Soc. Appl. Electromagn. Mech. 6(4), 318–326 (1999)Google Scholar
  22. 22.
    Bossavit, A.: Stiff problems in eddy-current theory and the regularization of Maxwell’s equations. IEEE Trans. Magn. 37(5), 3542–3545 (2001). https://doi.org/0018-9464/01<currencydollar>10.00 Google Scholar
  23. 23.
    Bossavit, A., Kettunen, L.: Yee-like schemes on a tetrahedral mesh, with diagonal lumping. Int. J. Numer. Modell. Electron. Networks Devices Fields 12(1-2), 129–142 (1999). https://doi.org/10.1002/(SICI)1099-1204(199901/04)12:1/2<129::AID-JNM327>3.0.CO;2-G Google Scholar
  24. 24.
    Bossavit, A., Kettunen, L.: Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches. IEEE Trans. Magn. 36(4), 861–867 (2000). https://doi.org/10.1109/20.877580 Google Scholar
  25. 25.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia (1995)Google Scholar
  26. 26.
    Carpenter, C.J.: Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power frequencies. IEE Proc. B Electr. Power Appl. 127(5), 332 (1980). https://doi.org/10.1049/ip-b:19800045 Google Scholar
  27. 27.
    Chen, Q., Schoenmaker, W., Chen, G., Jiang, L., Wong, N.: A numerically efficient formulation for time-domain electromagnetic-semiconductor cosimulation for fast-transient systems. IEEE Trans. Comput. Aided. Des. Integrated Circ. Syst. 32(5), 802–806 (2013).  https://doi.org/10.1109/TCAD.2012.2232709 Google Scholar
  28. 28.
    Clemens, M.: Large systems of equations in a discrete electromagnetism: formulations and numerical algorithms. IEE. Proc. Sci. Meas. Tech. 152(2), 50–72 (2005). https://doi.org/10.1049/ip-smt:20050849 Google Scholar
  29. 29.
    Clemens, M., Weiland, T.: Transient eddy-current calculation with the FI-method. IEEE Trans. Magn. 35(3), 1163–1166 (1999). https://doi.org/10.1109/20.767155 Google Scholar
  30. 30.
    Clemens, M., Weiland, T.: Regularization of eddy-current formulations using discrete grad-div operators. IEEE Trans. Magn. 38(2), 569–572 (2002). https://doi.org/10.1109/20.996149 Google Scholar
  31. 31.
    Clemens, M., Wilke, M., Weiland, T.: Linear-implicit time-integration schemes for error-controlled transient nonlinear magnetic field simulations. IEEE Trans. Magn. 39(3), 1175–1178 (2003).  https://doi.org/10.1109/TMAG.2003.810221 Google Scholar
  32. 32.
    Clemens, M., Schöps, S., De Gersem, H., Bartel, A.: Decomposition and regularization of nonlinear anisotropic curl-curl DAEs. Int. J. Comput. Math. Electr. Electron. Eng. 30(6), 1701–1714 (2011). https://doi.org/10.1108/03321641111168039 Google Scholar
  33. 33.
    Cortes Garcia, I., De Gersem, H., Schöps, S.: A structural analysis of field/circuit coupled problems based on a generalised circuit element (2018, submitted). arXiv:1801.07081Google Scholar
  34. 34.
    CST AG: CST STUDIO SUITE 2016 (2016). https://www.cst.com
  35. 35.
    De Gersem, H., Hameyer, K.: A finite element model for foil winding simulation. IEEE Trans. Magn. 37(5), 3472–3432 (2001). https://doi.org/10.1109/20.952629 Google Scholar
  36. 36.
    De Gersem, H., Weiland, T.: Field-circuit coupling for time-harmonic models discretized by the finite integration technique. IEEE Trans. Magn. 40(2), 1334–1337 (2004).  https://doi.org/10.1109/TMAG.2004.824536 Google Scholar
  37. 37.
    De Gersem, H., Hameyer, K., Weiland, T.: Field-circuit coupled models in electromagnetic simulation. J. Comput. Appl. Math. 168(1-2), 125–133 (2004). https://doi.org/10.1016/j.cam.2003.05.008 Google Scholar
  38. 38.
    Deschamps, G.A.: Electromagnetics and differential forms. Proc. IEEE 69(6), 676–696 (1981). https://doi.org/dx.doi.org/10.1109/PROC.1981.12048 Google Scholar
  39. 39.
    Dirks, H.K.: Quasi-stationary fields for microelectronic applications. Electr. Eng. 79(2), 145–155 (1996). https://doi.org/10.1007/BF01232924 Google Scholar
  40. 40.
    Dutiné, J.S., Richter, C., Jörgens, C., Schöps, S., Clemens, M.: Explicit time integration techniques for electro- and magneto-quasistatic field simulations. In: Graglia, R.D. (ed.) Proceedings of the International Conference on Electromagnetics in Advanced Applications (ICEAA) 2017. IEEE, New York (2017).  https://doi.org/10.1109/ICEAA.2017.8065562 Google Scholar
  41. 41.
    Dyck, D.N., Webb, J.P.: Solenoidal current flows for filamentary conductors. IEEE Trans. Magn. 40(2), 810–813 (2004).  https://doi.org/10.1109/TMAG.2004.824594 Google Scholar
  42. 42.
    Eller, M., Reitzinger, S., Schöps, S., Zaglmayr, S.: A symmetric low-frequency stable broadband Maxwell formulation for industrial applications. SIAM J. Sci. Comput. 39(4), B703–B731 (2017). https://doi.org/10.1137/16M1077817 Google Scholar
  43. 43.
    Estévez Schwarz, D.: Consistent initialization of differential-algebraic equations in circuit simulation. Technical Report 99-5, Humboldt Universität Berlin, Berlin (1999)Google Scholar
  44. 44.
    Gödel, N., Schomann, S., Warburton, T., Clemens, M.: GPU accelerated Adams-Bashforth multirate discontinuous Galerkin FEM simulation of high-frequency electromagnetic fields. IEEE Trans. Magn. 46(8), 2735–2738 (2010)Google Scholar
  45. 45.
    Griffiths, D.F.: Introduction to Electrodynamics. Prentice-Hall, Upper Saddle River (1999)Google Scholar
  46. 46.
    Hahne, P., Weiland, T.: 3d eddy current computation in the frequency domain regarding the displacement current. IEEE Trans. Magn. 28(2), 1801–1804 (1992). https://doi.org/10.1109/20.124056 Google Scholar
  47. 47.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, 2 edn. Springer, Berlin (2002)Google Scholar
  48. 48.
    Harrington, R.F.: Field Computation by Moment Methods. Wiley-IEEE, New York (1993)Google Scholar
  49. 49.
    Haus, H.A., Melcher, J.R.: Electromagnetic Fields and Energy. Englewood Cliffs, Prentice-Hall (1989)Google Scholar
  50. 50.
    Heaviside, O.: On the forces, stresses, and fluxes of energy in the electromagnetic field. Proc. R. Soc. Lond. Ser. I 50, 126–129 (1891)Google Scholar
  51. 51.
    Hehl, F.W., Obukhov, Y.N.: Foundations of Classical Electrodynamics – Charge, Flux, and Metric. Progress in Mathematical Physics. Birkhäuser, Basel (2003)Google Scholar
  52. 52.
    Heise, B.: Analysis of a fully discrete finite element method for a nonlinear magnetic field problem. SIAM J. Numer. Anal. 31(3), 745–759 (1994)Google Scholar
  53. 53.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Texts in Applied Mathematics. Springer, New York (2008)Google Scholar
  54. 54.
    Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1998)Google Scholar
  55. 55.
    Jochum, M.T., Farle, O., Dyczij-Edlinger, R.: A new low-frequency stable potential formulation for the finite-element simulation of electromagnetic fields. IEEE Trans. Magn. 51(3), 7402,304 (2015).  https://doi.org/10.1109/TMAG.2014.2360080 Google Scholar
  56. 56.
    Kameari, A.: Calculation of transient 3D eddy-current using edge elements. IEEE Trans. Magn. 26(5), 466–469 (1990). https://doi.org/10.1109/20.106354 Google Scholar
  57. 57.
    Kerler-Back, J., Stykel, T.: Model reduction for linear and nonlinear magneto-quasistatic equations. Int. J. Numer. Methods Eng. 111(13), 1274–1299 (2017).  https://doi.org/10.1002/nme.5507 Google Scholar
  58. 58.
    Koch, S., Weiland, T.: Time domain methods for slowly varying fields. In: URSI International Symposium on Electromagnetic Theory (EMTS 2010), pp. 291–294 (2010).  https://doi.org/10.1109/URSI-EMTS.2010.5636991
  59. 59.
    Koch, S., Weiland, T.: Different types of quasistationary formulations for time domain simulations. Radio Sci. 46(5) (2011). https://doi.org/10.1029/2010RS004637 Google Scholar
  60. 60.
    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-27555-5 Google Scholar
  61. 61.
    Larsson, J.: Electromagnetics from a quasistatic perspective. Am. J. Phys. 75(3), 230–239 (2007). https://doi.org/10.1119/1.2397095 Google Scholar
  62. 62.
    Manges, J.B., Cendes, Z.J.: Tree-cotree decompositions for first-order complete tangential vector finite elements. Int. J. Numer. Methods Eng. 40(9), 1667–1685 (1997). https://doi.org/10.1002/(SICI)1097-0207(19970515)40:9<1667::AID-NME133>3.0.CO;2-9 Google Scholar
  63. 63.
    März, R.: Differential algebraic systems with properly stated leading term and MNA equations. In: Anstreich, K., Bulirsch, R., Gilg, A., Rentrop, P. (eds.) Modelling, Simulation and Optimization of Integrated Circuits, pp. 135–151. Birkhäuser, Berlin (2003)Google Scholar
  64. 64.
    Maxwell, J.C.: A dynamical theory of the electromagnetic field. Phil. Trans. R. Soc. London CLV, 459–512 (1864)Google Scholar
  65. 65.
    Mehrmann, V.: Index Concepts for Differential-Algebraic Equations, pp. 676–681. Springer, Berlin (2015). https://doi.org/10.1007/978-3-540-70529-1_120 Google Scholar
  66. 66.
    Merkel, M., Niyonzima, I., Schöps, S.: Paraexp using leapfrog as integrator for high-frequency electromagnetic simulations. Radio Sci. 52(12), 1558–1569 (2017). https://doi.org/10.1002/2017RS006357 Google Scholar
  67. 67.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)Google Scholar
  68. 68.
    Monk, P., Süli, E.: A convergence analysis of Yee’s scheme on nonuniform grids. SIAM J. Numer. Anal. 31(2), 393–412 (1994). https://doi.org/10.1137/0731021 Google Scholar
  69. 69.
    Munteanu, I.: Tree-cotree condensation properties. ICS Newsl. (International Compumag Society) 9, 10–14 (2002). http://www.compumag.org/jsite/images/stories/newsletter/ICS-02-09-1-Munteanu.pdf
  70. 70.
    Nagel, L.W., Pederson, D.O.: Simulation program with integrated circuit emphasis. Technical Report, University of California, Berkeley, Electronics Research Laboratory, ERL-M382 (1973)Google Scholar
  71. 71.
    Nédélec, J.C.: Mixed finite elements in r 3. Numer. Math. 35(3), 315–341 (1980). https://doi.org/10.1007/BF01396415 Google Scholar
  72. 72.
    Nicolet, A., Delincé, F.: Implicit Runge-Kutta methods for transient magnetic field computation. IEEE Trans. Magn. 32(3), 1405–1408 (1996). https://doi.org/0.1109/20.497510 Google Scholar
  73. 73.
    Ostrowski, J., Hiptmair, R., Krämer, F., Smajic, J., Steinmetz, T.: Transient full Maxwell computation of slow processes. In: Michielsen, B., Poirier, J.R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Mathematics in Industry, vol. 16, pp. 87–95. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-22453-9_10 Google Scholar
  74. 74.
    Ouédraogo, Y., Gjonaj, E., Weiland, T., De Gersem, H., Steinhausen, C., Lamanna, G., Weigand, B., Preusche, A., Dreizler, A., Schremb, M.: Electrohydrodynamic simulation of electrically controlled droplet generation. Int. J. Heat Fluid Flow 64, 120–128 (2017)Google Scholar
  75. 75.
    Petzold, L.R.: Differential/algebraic equations are not ODE’s. SIAM J. Sci. Stat. Comput. 3(3), 367–384 (1982). https://doi.org/10.1137/0903023 Google Scholar
  76. 76.
    Rapetti, F., Rousseaux, G.: On quasi-static models hidden in Maxwell’s equations. Appl. Numer. Math. 79, 92–106 (2014). https://doi.org/10.1016/j.apnum.2012.11.007 Google Scholar
  77. 77.
    Rautio, J.C.: The long road to Maxwell’s equations. IEEE Spectr. 51(12), 36–56 (2014).  https://doi.org/10.1109/MSPEC.2014.6964925 Google Scholar
  78. 78.
    Raviart, P.A., Thomas, J.M.: Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comput. 31(138), 391–413 (1977)Google Scholar
  79. 79.
    Ruehli, A.E.: Equivalent circuit models for three-dimensional multiconductor systems. IEEE Trans. Microwave Theory Tech. 22(3), 216–221 (1974)Google Scholar
  80. 80.
    Ruehli, A.E., Antonini, G., Jiang, L.: The Partial Element Equivalent Circuit Method for Electro-Magnetic and Circuit Problems. Wiley, Hoboken (2015)Google Scholar
  81. 81.
    Schilders, W.H.A., Ciarlet, P., ter Maten, E.J.W. (eds.): Handbook of Numerical Analysis. Numerical Methods in Electromagnetics. Handbook of Numerical Analysis, vol. 13. North-Holland, Amsterdam (2005)Google Scholar
  82. 82.
    Schmidt, K., Sterz, O., Hiptmair, R.: Estimating the eddy-current modeling error. IEEE Trans. Magn. 44(6), 686–689 (2008).  https://doi.org/10.1109/TMAG.2008.915834 Google Scholar
  83. 83.
    Schoenmaker, W.: Computational Electrodynamics. River Publishers Series in Electronic Materials and Devices. River Publishers, Delft (2017)Google Scholar
  84. 84.
    Schöps, S.: Multiscale modeling and multirate time-integration of field/circuit coupled problems. Dissertation, Bergische Universität Wuppertal & Katholieke Universiteit Leuven, Düsseldorf (2011). VDI Verlag. Fortschritt-Berichte VDI, Reihe 21Google Scholar
  85. 85.
    Schöps, S., Bartel, A., Clemens, M.: Higher order half-explicit time integration of eddy current problems using domain substructuring. IEEE Trans. Magn. 48(2), 623–626 (2012).  https://doi.org/10.1109/TMAG.2011.2172780 Google Scholar
  86. 86.
    Schöps, S., De Gersem, H., Weiland, T.: Winding functions in transient magnetoquasistatic field-circuit coupled simulations. Int. J. Comput. Math. Electr. Electron. Eng. 32(6), 2063–2083 (2013).  https://doi.org/10.1108/COMPEL-01-2013-0004 Google Scholar
  87. 87.
    Schuhmann, R., Weiland, T.: Conservation of discrete energy and related laws in the finite integration technique. Prog. Electromagn. Res. 32, 301–316 (2001).  https://doi.org/10.2528/PIER00080112 Google Scholar
  88. 88.
    Schuhmacher, S., Klaedtke, A., Keller, C., Ackermann, W., De Gersem, H.: Optimizing the inductance cancellation behavior in an EMI filter design with the help of a sensitivity analysis. In: EMC Europe. Angers, France (2017)Google Scholar
  89. 89.
    Steinmetz, T., Kurz, S., Clemens, M.: Domains of validity of quasistatic and quasistationary field approximations. Int. J. Comput. Math. Electr. Electron. Eng. 30(4), 1237–1247 (2011). https://doi.org/10.1108/03321641111133154 Google Scholar
  90. 90.
    Taflove, A.: Computational Electrodynamics: The Finite-Difference Time-Domain-Method. Artech House, Dedham (1995)Google Scholar
  91. 91.
    Taflove, A.: A perspective on the 40-year history of FDTD computational electrodynamics. Appl. Comput. Electromagn. Soc. J. 22(1), 1–21 (2007)Google Scholar
  92. 92.
    Tischendorf, C.: Topological index calculation of DAEs in circuit simulation. Technical Report 3-4, Humboldt Universität Berlin, Berlin (1999)Google Scholar
  93. 93.
    Tonti, E.: On the formal structure of physical theories. Technical Report, Politecnico di Milano, Milano, Italy (1975)Google Scholar
  94. 94.
    Tsukerman, I.A.: Finite element differential-algebraic systems for eddy current problems. Numer. Algorithms 31(1), 319–335 (2002). https://doi.org/10.1023/A:1021112107163 Google Scholar
  95. 95.
    Webb, J.P., Forghani, B.: The low-frequency performance of h − ϕ and t − ω methods using edge elements for 3d eddy current problems. IEEE Trans. Magn. 29(6), 2461–2463 (1993). https://doi.org/10.1109/20.280983 Google Scholar
  96. 96.
    Weeks, W., Jimenez, A., Mahoney, G., Mehta, D., Qassemzadeh, H., Scott, T.: Algorithms for ASTAP – a network-analysis program. IEEE Trans. Circuit Theory 20(6), 628–634 (1973).  https://doi.org/10.1109/TCT.1973.1083755 Google Scholar
  97. 97.
    Weiland, T.: A discretization method for the solution of Maxwell’s equations for six-component fields. Int. J. Electron. Commun. (AEU) 31, 116–120 (1977)Google Scholar
  98. 98.
    Weiland, T.: On the unique numerical solution of Maxwellian eigenvalue problems in three dimensions. Part. Accel. 17(227–242) (1985)Google Scholar
  99. 99.
    Weiland, T.: Time domain electromagnetic field computation with finite difference methods. Int. J. Numer. Modell. Electron. Networks Devices Fields 9(4), 295–319 (1996). https://doi.org/10.1002/(SICI)1099-1204(199607)9:4<295::AID-JNM240>3.0.CO;2-8 Google Scholar
  100. 100.
    Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302–307 (1966).  https://doi.org/10.1109/TAP.1966.1138693 Google Scholar

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© © Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Idoia Cortes Garcia
    • 1
  • Sebastian Schöps
    • 1
  • Herbert De Gersem
    • 1
  • Sascha Baumanns
    • 2
  1. 1.Technische Universität DarmstadtGraduate School of Computational EngineeringDarmstadtGermany
  2. 2.Universität zu KölnMathematisches InstitutKölnGermany

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