Discretization of Electromagnetic Problems: The "Generalized Finite Differences" Approach

Handbook of Numerical Analysis

Volumn 13, Issue , 2005, Pages 105-197

  Bossavit, Alain ^a  

^a UNIV PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29544450619     PISSN: 15708659     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1570-8659(04)13002-0     Document Type: Review

References (102)

1. Albanese R., and Rubinacci G. Integral formulations for 3-D eddy-currents computation using edge-elements. IEE Proc. A 135 (1988) 457-462
2. Armstrong M.A. Basic Topology (1979), McGraw-Hill, London
3. 2AN 19 (1985) 7-32
4. Babuška I., and Aziz A.K. On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976) 214-226
5. Baez J., and Muniain J.P. Gauge Fields, Knots and Gravity (1994), World Scientific, Singapore
6. Baldomir D., and Hammond P. Geometry of Electromagnetic Systems (1996), Oxford Univ. Press, Oxford
7. Bank R.E., and Rose D.J. Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787
8. Bänsch E. Local mesh refinement in 2 and 3 dimensions. Impact Comput. Sci. Engrg. 3 (1991) 181-191
9. Bey J. Tetrahedral grid refinement. Computing 55 (1995) 355-378
10. Bossavit A. Eddy-currents and forces in deformable conductors. In: Hsieh R.K.T. (Ed). Mechanical Modellings of New Electromagnetic Materials: Proc. IUTAM Symp., Stockholm, April 1990 (1990), Elsevier, Amsterdam 235-242
11. Bossavit A. Solving Maxwell's equations in a closed cavity, and the question of spurious modes. IEEE Trans. Magn. 26 (1990) 702-705
12. Bossavit A. A puzzle. ICS Newsletter 3 2 (1996) 7 3 (3), 14; 4 (1) (1997) 17-18
13. Bossavit A. Computational Electromagnetism (1998), Academic Press, Boston
14. Bossavit A. Computational electromagnetism and geometry. J. Japan Soc. Appl. Electromagn. Mech. 6 (1998) 17-28 114-123, 233-240, 318-326; 7 (1999) 150-159, 249-301, 401-408; 8 (2000) 102-109, 203-209, 372-377
15. Bossavit A. On axial and polar vectors. ICS Newsletter 6 (1999) 12-14
16. Bossavit A. Most general 'non-local' boundary conditions for the Maxwell equations in a bounded region. COMPEL 19 (2000) 239-245
17. Bossavit A. 'Stiff' problems in eddy-current theory and the regularization of Maxwell's equations. IEEE Trans. Magn. 37 (2001) 3542-3545
18. Bossavit A. On the notion of anisotropy of constitutive laws: some implications of the 'Hodge implies metric' result. COMPEL 20 (2001) 233-239
19. Bossavit A. On the representation of differential forms by potentials in dimension 3. In: van Rienen U., Günther M., and Hecht D. (Eds). Scientific Computing in Electrical Engineering (2001), Springer-Verlag, Berlin 97-104
20. Bossavit A. Mixed-hybrid methods in magnetostatics: complementarity in one stroke. IEEE Trans. Magn. 39 (2003) 1099-1102
21. Bossavit A., and Kettunen L. Yee-like schemes on a tetrahedral mesh, with diagonal lumping. Int. J. Numer. Modelling 12 (1999) 129-142
22. Branin Jr. F.H. An abstract mathematical basis for network analogies and its significance in physics and engineering. Matrix and Tensor Quarterly 12 (1961) 31-49
23. Burke W.L. Applied Differential Geometry (1985), Cambridge Univ. Press, Cambridge
24. Di Carlo A., and Tiero A. The geometry of linear heat conduction. In: Schneider W., Troger H., and Ziegler F. (Eds). Trends in Applications of Mathematics to Mechanics (1991), Longman, Harlow 281-287
25. Carpenter C.J. Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power frequencies. Proc. IEE 124 (1977) 1026-1034
26. Chavent G., and Roberts J.E. A unified presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Resources 14 (1991) 329-348
27. Clemens M., and Weiland T. Transient eddy-current calculation with the FI-method. IEEE Trans. Magn. 35 (1999) 1163-1166
28. Cohen G., Joly P., and Tordjman N. Construction and analysis of higher order elements with mass-lumping for the wave equation. Mathematical Aspects of Wave Propagation Phenomena (1993), SIAM, Philadelphia 152-160
29. Costabel M., and Dauge M. Singularités des équations de Maxwell dans un polyèdre. C. R. Acad. Sci. Paris I 324 (1997) 1005-1010
30. de Cougny H.L., and Shephard M.S. Parallel refinement and coarsening of tetrahedral meshes. Int. J. Numer. Meth. Engng. 46 (1999) 1101-1125
31. Coulomb J.L., Zgainski F.X., and Maréchal Y. A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements. IEEE Trans. Magn. 33 (1997) 1362-1365
32. 2AN 32 (1998) 631-649
33. Van Dantzig D. The fundamental equations of electromagnetism, independent of metrical geometry. Proc. Cambridge Phil. Soc. 30 (1934) 421-427
34. Van Dantzig D. On the geometrical representation of elementary physical objects and the relations between geometry and physics. Nieuw. Archief vor Wiskunde 3 (1954) 73-89
35. Dirichlet G.L. Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angew. Math. 40 (1850) 209
36. Dular P., Hody J.-Y., Nicolet A., Genon A., and Legros W. Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms. IEEE Trans. Magn. 30 (1994) 2980-2983
37. Ebeling F., Klatt R., Krawczyk F., Lawinsky E., Weiland T., Wipf S.G., Steffen B., Barts T., Browman M.J., Cooper R.K., Deaven H., and Rodenz G. The 3-D MAFIA group of electromagnetic codes. IEEE Trans. Magn. 25 (1989) 2962-2964
38. Eckmann B. Topology, algebra, analysis - relations and missing links. Notices AMS 46 (1999) 520-527
39. Elmkies A., and Joly P. Éléments finis d'arête et condensation de masse pour les équations de Maxwell: le cas de dimension 3. C. R. Acad. Sci. Paris Sér. I 325 (1997) 1217-1222
40. Ergatoudis J.G., Irons B.M., and Zienkiewicz O.C. Curved, isoparametric, 'quadrilateral' elements for finite element analysis. Int. J. Solids Struct. 4 (1968) 31-42
41. Firestone F.A. A new analogy between mechanical and electrical systems. J. Acoust. Soc. Am. (1933) 249-267
42. Gallouet T., and Vila J.P. Finite volume element scheme for conservation laws of mixed type. SIAM J. Numer. Anal. 28 (1991) 1548-1573
43. Gelbaum B.R., and Olmsted J.M.H. Counterexamples in Analysis (1964), Holden-Day, San Francisco
44. Goldhaber A.S., and Trower W.P. Resource letter MM-1: Magnetic monopoles. Am. J. Phys. 58 (1990) 429-439
45. Gradinaru V., and Hiptmair R. Whitney elements on pyramids. ETNA 8 (1999) 154-168
46. Halmos P.R. Measure Theory (1950), Van Nostrand, Princeton
47. Hamouda L., Bandelier B., and Rioux-Damidau F. Mixed formulation for magnetostatics. Proc. Compumag (2001) (paper PE4-11)
48. Harrison J. Continuity of the integral as a function of the domain. J. Geometric Anal. 8 (1998) 769-795
49. Haugazeau Y., and Lacoste P. Condensation de la matrice masse pour les éléments finis mixtes de H(rot). C. R. Acad. Sci. Paris I 316 (1993) 509-512
50. Heinrich B. Finite Difference Methods on Irregular Networks (1987), Akademie-Verlag, Berlin
51. Henle A. A Combinatorial Introduction to Topology (1994), Dover, New York
52. Hilton P.J., and Wylie S. Homology Theory, An Introduction to Algebraic Topology (1965), Cambridge Univ. Press, Cambridge
53. Huang J., and Xi S. On the finite volume element method for general self-adjoint elliptic problems. SIAM J. Numer. Anal. 35 (1998) 1762-1774
54. Hyman J.M., and Shashkov M. Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl. 33 (1997) 81-104
55. Jänich K. Vector Analysis (2001), Springer, New York
56. Kaasschieter E.F., and Huijben A.J.M. Mixed-hybrid finite elements and streamline computation for the potential flow problem. Numer. Meth. PDE 8 (1992) 221-266
57. Kameari A. Symmetric second order edge elements for triangles and tetrahedra. IEEE Trans. Magn. 35 (1999) 1394-1397
58. Kheyfets A., and Wheeler J.A. Boundary of a boundary and geometric structure of field theories. Int. J. Theor. Phys. 25 (1986) 573-580
59. Koenig H.E., and Blackwell W.A. Linear graph theory: a fundamental engineering discipline. IRE Trans. Edu. 3 (1960) 42-49
60. Kottler F. Maxwell'sche Gleichungen und Metrik. Sitzungber. Akad. Wien IIa 131 (1922) 119-146
61. Lax P.D., and Richtmyer R.D. Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9 (1956) 267-293
62. Lee J.-F., and Sacks Z. Whitney elements time domain (WETD) methods. IEEE Trans. Magn. 31 (1995) 1325-1329
63. Leis R. Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien. Math. Z. 106 (1968) 213-224
64. Madsen I., and Tornehave J. From Calculus to Cohomology (1997), Cambridge Univ. Press, Cambridge
65. Mattiussi C. The finite volume, finite element, and finite difference methods as numerical methods for physical field problems. Adv. Imag. Electron Phys. 113 (2000) 1-146
66. Maubach J.M. Local bisection refinement for N-simplicial grids generated by reflection. SIAM J. Sci. Stat. 16 (1995) 210-227
67. Mittra R., Ramahi O., Khebir A., Gordon R., and Kouki A. A review of absorbing boundary conditions for two and three-dimensional electromagnetic scattering problems. IEEE Trans. Magn. 25 (1989) 3034-3038
68. Monk P., and Süli E. A convergence analysis of Yee's scheme on nonuniform grids. SIAM J. Numer. Anal. 31 (1994) 393-412
69. Mosé R., Siegel P., Ackerer P., and Chavent G. Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity?. Water Resources Res. 30 (1994) 3001-3012
70. Munteanu I. Tree-cotree condensation properties. ICS Newsletter 9 (2002) 10-14
71. Nicolaides R., and Wang D.-Q. Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions. Math. Comp. 67 (1998) 947-963
72. Post E.J. The constitutive map and some of its ramifications. Ann. Phys. 71 (1972) 497-518
73. Rapetti F., Dubois F., and Bossavit A. Integer matrix factorization for mesh defect detection. C. R. Acad. Sci. Paris 334 (2002) 717-720
74. Ren Z. Autogauging of vector potential by iterative solver - numerical evidence. 3d Int. Workshop on Electric and Magnetic Fields, A.I.M. (31 Rue St-Gilles, Liège) (1996) 119-124
75. de Rham G. Relations entre la topologie et la théorie des intégrales multiples. L'Enseignement Math. 35 (1936) 213-228
76. de Rham G. Variétés différentiables (1960), Hermann, Paris
77. Rosen J. Transformation properties of electromagnetic quantities under space inversion, time reversal, and charge conjugation. Am. J. Phys. 41 (1973) 586-588
78. Rudin W. Functional Analysis (1973), McGraw-Hill, New York
79. Schatz A.H., Sloan I.H., and Wahlbin L.B. Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point. SIAM J. Numer. Anal. 33 (1996) 505-521
80. Schouten J.A. Tensor Analysis for Physicists (1989), Dover, New York
81. Schutz B. Geometrical Methods of Mathematical Physics (1980), Cambridge Univ. Press, Cambridge
82. Seifert H., and Threlfall W. A Textbook of Topology (1980), Academic Press, Orlando (first German ed., 1934)
83. Shaw R., and Yeadon F.J. On (a×b)×c. Am. Math. Monthly 96 (1989) 623-629
84. Smyth J.B., and Smyth D.C. Critique of the paper 'The electromagnetic radiation from a finite antenna'. Am. J. Phys. 45 (1977) 581-582
85. Sorkin R. The electromagnetic field on a simplicial net. J. Math. Phys. 16 (1975) 2432-2440
86. Süli E. Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419-1430
87. Taylor E.F., and Wheeler J.A. Spacetime Physics (1992), Freeman, New York
88. Teixeira F.L., and Chew W.C. Lattice electromagnetic theory from a topological viewpoint. J. Math. Phys. 40 (1999) 169-187
89. Tonti E. On the geometrical structure of electromagnetism. In: Ferrarese G. (Ed). Gravitation, Electromagnetism and Geometrical Structures (1996), Pitagora, Bologna 281-308
90. Tonti E. A direct formulation of field laws: the cell method. CMES 2 (2001) 237-258
91. Trapp B., Munteanu I., Schuhmann R., Weiland T., and Ioan D. Eigenvalue computation by means of a tree-cotree filtering technique. IEEE Trans. Magn. 38 (2002) 445-448
92. Uman M.A. Reply to Smyth and Smyth. Am. J. Phys. 45 (1977) 582
93. Veblen O., and Whitehead J.H.C. The Foundations of Differential Geometry (1932), Cambridge Univ. Press, Cambridge
94. Weiland T. Maxwell's grid equations. Proc. URSI Int. Symp. Electromagnetic Theory, Sydney (1992) 37-39
95. Weiland T. Time domain electromagnetic field computation with finite difference methods. Int. J. Numer. Modelling 9 (1996) 295-319
96. Weiland T. Three dimensional resonator mode computation by finite difference methods. IEEE Trans. Magn. 21 (1985) 2340-2343
97. Weiser A., and Wheeler M.F. On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375
98. Van Welij J.S. Calculation of eddy currents in terms of H on hexahedra. IEEE Trans. Magn. 21 (1985) 2239-2241
99. White, D.A., Koning, J.M. (2000). A novel approach for computing solenoidal eigenmodes of the vector Helmholtz equation. CEFC'00 (p. 328 of the "digest of abstracts")
100. Whitney H. Geometric Integration Theory (1957), Princeton Univ. Press, Princeton
101. Yee K.S. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. AP 14 (1966) 302-307
102. Yosida K. Functional Analysis (1980), Springer-Verlag, Berlin (first ed., 1965)