Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-31T01:41:39.752Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomology in 3D Magneto-Quasistatics Modeling

Published online by Cambridge University Press:  03 June 2015

Paweł Dłotko  and
Ruben Specogna
Paweł Dłotko*
Affiliation:
Institute of Computer Science, Jagiellonian University, ul.. St. Łojasiewicza 6, 30-348 Kraków, Poland
Ruben Specogna*
Affiliation:
Università di Udine, Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Via delle Scienze 208, 33100 Udine, Italy
*
Email:pawel.dlotko@uj.edu.pl
Corresponding author.Email:ruben.specogna@uniud.it
Get access

Abstract

Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for defining potentials. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problemswhich involve conductive regionswith holes, cuts are needed to make the boundary value problem well defined. While an intimate connection with homology theory has been quickly recognized, heuristic definitions of cuts are surprisingly still dominant in the literature.

The aim of this paper is first to survey several definitions of cuts together with their shortcomings. Then, cuts are defined as generators of the first cohomology group over integers of a finite CW-complex. This provably general definition has also the virtue of providing an automatic, general and efficient algorithm for the computation of cuts. Some counter-examples show that heuristic definitions of cuts should be abandoned. The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem.

Keywords

55M0555N3355N9978M1078M25Algebraic topology(co)homologycomputational electromagneticscuts
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 48 - 76
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Maxwell, J.C., A Treatise on Electricity and Magnetism, Clarendon Press, Oxford, 1891.Google Scholar
[2]Carpenter, C.J., Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power frequencies, Proc. IEE, 124 (1977), pp. 1026–1034.Google Scholar
[3]Bossavit, A., Two dual formulations of the 3D eddy currents problem, COMPEL, 4 (1984), pp. 103–116.CrossRefGoogle Scholar
[4]Kron, G., Numerical solution of ordinary and partial differential equations by means of equivalent circuits, J. Appl. Phys., 126 (1945), pp. 172–186.Google Scholar
[5]Branin, F.H. Jr., The Algebraic-Topological Basis for Network Analogies and the Vector Calculus, in: Proceedings of the Symposium on Generalized Networks, Polytechnic Press, Brooklin, New York (1966), pp. 453–491.Google Scholar
[6]Tonti, E., On the formal structure of physical theories, Monograph of the Italian National Research Council, 1975 (available online).Google Scholar
[7]Weiland, T., A discretization method for the solution of Maxwell’s equations for six-component fields, Electron. Commun. (AEÜ), 31 (1977), p. 116.Google Scholar
[8]Mattiussi, C., An analysis of finite volume, finite element and finite difference methods using some concepts from algebraic topology, J. Comp. Phys., 133 (1997) pp. 289–309.CrossRefGoogle Scholar
[9]Tonti, E., Algebraic topology and computational electromagnetism, 4-th International Workshop on Electric and Magnetic Fields, Marseille, France, 12-15 May 1998, pp. 284–294.Google Scholar
[10]Bossavit, A., How weak is the Weak Solution in finite elements methods?, IEEE Trans. Magn., 34 (1998), pp. 2429–2432.CrossRefGoogle Scholar
[11]Tarhasaari, T., Kettunen, L., Bossavit, A., Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques, IEEE Trans. Magn., 35 (1999), pp. 1494–1497.Google Scholar
[12]Bossavit, A., Kettunen, L., Yee-like Schemes on Staggered Cellular Grids: A synthesis Between FIT and FEM Approaches, IEEE Trans. Magn., 36 (2000), pp. 861–867.Google Scholar
[13]Tonti, E., Finite Formulation of the Electromagnetic Field, IEEE Trans. Magn., 38 (2002), pp. 333–336.Google Scholar
[14]Bossavit, A., Computational Electromagnetism, Academic Press, 1998, ISBN-13 978-0121187101.Google Scholar
[15]Brezzi, F.,K. Lipnikov, Shashkov, M., Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal., 43 (2005), pp. 1872–1896.Google Scholar
[16]Bochev, P.V., Hyman, J.M., Principles of mimetic discretizations of differential operators, in Compatible spatial discretizations, Proceedings of IMA Hot Topics workshop on Compatible discretizations, IMA 142, eds. Arnold, D., Bochev, P., Lehoucq, R., Nicolaides, R. and Shashkov, M., Springer Verlag, (2006).Google Scholar
[17]Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E., Discrete exterior calculus, 2005, available from arXiv.org/math.DG/0508341.Google Scholar
[18]Arnold, D.N., Falk, R.S., Winther, R., Differential complexes and stability of finite element methods I: The de Rham complex, in Compatible spatial discretizations, Proceedings of IMA Hot Topics workshop on Compatible discretizations, IMA 142, eds. Arnold, D., Bochev, P., Lehoucq, R., Nicolaides, R. and Shashkov, M., Springer Verlag, (2006).Google Scholar
[19]Arnold, D.N., Falk, R.S., Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), pp. 1–155.CrossRefGoogle Scholar
[20]Arnold, D.N., Falk, R.S., Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47 (2010), pp. 281–354.CrossRefGoogle Scholar
[21]Hiptmair, R., Discrete Hodge operators, Numer. Math., 90 (2001), pp. 265–289.CrossRefGoogle Scholar
[22]Codecasa, L., Specogna, R., Trevisan, F., Base functions and discrete constitutive relations for staggered polyhedral grids, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 1117–1123.Google Scholar
[23]Codecasa, L., Specogna, R., Trevisan, F., A New Set of Basis Functions for the Discrete Geometric Approach, Journal of Computational Physics, 229 (2010), pp. 7401–7410.CrossRefGoogle Scholar
[24]Bossavit, A., Computational electromagnetism and geometry. (5): The ‘Galerkin hodge’, J. Japan Soc. Appl. Electromagn. & Mech., 8 (2000), pp. 203–209.Google Scholar
[25]Marrone, M., Properties of Constitutive Matrices for Electrostatic and Magnetostatic Problems, IEEE Trans. Magn., 40 (2004), pp. 1516–1519.Google Scholar
[26]Specogna, R., Suuriniemi, S., Trevisan, F., Geometric T-Ω approach to solve eddy-currents coupled to electric circuits, Int. J. Numer. Meth. Eng., 74 (2008), pp. 101–115.Google Scholar
[27]Dłotko, P., Specogna, R., Trevisan, F., Automatic generation of cuts suitable for the T-Ω geometric eddy-current formulation, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 3765–3781.CrossRefGoogle Scholar
[28]Specogna, R., Trevisan, F., Discrete constitutive equations in A-χ geometric eddy-currents formulation, IEEE Trans. on Magn., 41 (2005), pp. 1259–1263.Google Scholar
[29]Kotiuga, P.R., On making cuts for magnetic scalar potentials in multiply connected regions, J. Appl. Phys., 61 (1987), pp. 3916–3918.CrossRefGoogle Scholar
[30]Bott, R., Tu, L.W., Differential Forms in Algebraic Topology, Springer-Verlag (1982).Google Scholar
[31]Ren, Z., T-Ω Formulation for Eddy-Current Problems in Multiply Connected Regions, IEEE Trans. Magn., 38 (2002), pp. 557–560.Google Scholar
[32]Henrotte, F., Hameyer, K., An algorithm to construct the discrete cohomology basis functions required for magnetic scalar potential formulations without cuts, IEEE Trans. Magn., 39 (2003), pp. 1167–1170.Google Scholar
[33]Gross, P.W., Kotiuga, P.R., Electromagnetic Theory and Computation: A Topological Approach, MSRI Publication 48 (2004), Cambridge University Press, ISBN 0 521 801605.Google Scholar
[34]Dular, P., Henrotte, F., Genon, A., Legros, W., A generalized source magnetic field calculation method for inductors of any shape, IEEE Trans. Magn., 33 (1997), pp. 1398–1401.Google Scholar
[35]Dular, P., Legros, W., Nicolet, A., Coupling of local and global quantities in various finite element formulations and its application to electrostatics, magnetostatics and magnetodynamics, IEEE Trans. Magn., 34 (1998), pp. 3078–3081.Google Scholar
[36]Dular, P., Geuzaine, C., Legros, W., A natural method for coupling magnetodynamic h-formulations and circuit equations, IEEE Trans. Magn., 35 (1999), pp. 1626–1629.Google Scholar
[37]Bossavit, A., Most general ‘non-local’ boundary conditions for the Maxwell equations in a bounded region, COMPEL, 19 (2000), pp. 239–245.Google Scholar
[38]Dular, P., The benefits of nodal and edge elements coupling for discretizing global constraints in dual magnetodynamic formulations, Journal of Computational and Applied Mathematics, 168 (2004), pp. 165–178.Google Scholar
[39]Dular, P., Curl-conform source fields in finite element formulations: Automatic construction of a reduced form, COMPEL, 24 (2005), pp. 364–373.Google Scholar
[40]Hatcher, A., Algebraic topology, Cambridge University Press, 2002 (available online).Google Scholar
[41] P. Dłotko, Specogna, R., Critical analysis of the spanning tree techniques, SIAM J. Numer. Anal., 48 (2010), pp. 1601–1624.Google Scholar
[42]Massey, W. S., A Basic Course in Algebraic Topology, vol. 127 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.Google Scholar
[43]Munkres, J.R., Elements of algebraic topology, Perseus Books, Cambridge, MA, 1984.Google Scholar
[44] Computer Assisted Proofs in Dynamics library, capd.ii.uj.edu.pl.Google Scholar
[45] CHomP library, chomp.rutgers.edu.Google Scholar
[46]Dłotko, P., Specogna, R., Efficient cohomology computation for electromagnetic modeling, CMES: Computer Modeling in Engineering & Sciences, 60 (2010), pp. 247–278.Google Scholar
[47]Steinmetz, C.P., Theory and Calculation of Alternating Current Phenomena, McGraw Hill, New York, 1900.Google Scholar
[48]Dodziuk, J., Finite difference approach to the Hodge theory of harmonic forms, Amer. J. Math., 98 (1976), pp. 79–104.Google Scholar
[49]Kaczynski, T., Mischaikow, K., Mrozek, M., Computational Homology, Springer-Verlag, New York, 2004.Google Scholar
[50]Bredon, G.E., Topology and Geometry, Springer-Verlag, 1993.Google Scholar
[51]Seifert, H., Threlfall, W., A textbook of topology and Seifert: Topolgy of 3-dimensional fiberd spaces, Academic Press, 1980.Google Scholar
[52]Phung, A.-T., Labie, P., Chadebec, O., Le Floch, Y., Meunier, G., On the Use of Automatic Cuts Algorithm for T 0-T-Ω Formulation in Nondestructive Testing by Eddy Current, Intelligent Computer Techniques in Applied Electromagnetics, (2008), ISBN 978-3-540-78489-0, Springer Berlin.Google Scholar
[53]Kotiuga, P.R., Toward an Algorithm to Make Cuts for Magnetic Scalar Potentials in Finite Element Meshes, J. Appl. Phys., 63 (1988), pp. 3357–3359, erratum: 64 (1988), p. 4257.Google Scholar
[54]Kotiuga, P.R., An algorithm to make cuts for magnetic scalar potentials in tetrahedral meshes based on the finite element method, IEEE Trans. Magn., 25 (1989), pp. 4129–4131.CrossRefGoogle Scholar
[55]Harold, C.S. and Simkin, J., Cutting multiply connected domains, IEEE Trans. Magn., 21 (1985), pp. 2495–2498.Google Scholar
[56]Vourdas, A., Binns, K.J., Magnetostatics with scalar potentials in multiply connected regions, Science, Measurement and Technology, IEE Proceedings A, 136 (1989), pp. 49–54.Google Scholar
[57]Leonard, P.J., Lai, H.C., Hill-Cottingham, R.J., Rodger, D., Automatic implementation of cuts in multiply connected magnetic scalar region for 3-D eddy current models, IEEE Trans. Magn., 29 (1993), pp. 1368–1371.Google Scholar
[58]Simkin, J., Taylor, S.C., Xu, E.X., An efficient algorithm for cutting multiply connected regions, IEEE Trans. Magn., 40 (2004), pp. 707–709.Google Scholar
[59]Vourdas, A., Binns, K.J., Bossavit, A., Magnetostatics with scalar potentials in multiply connected regions (comments with reply), Science, Measurement and Technology, IEE Proceedings A, 136 (1989), pp. 260–261.Google Scholar
[60]Kotiuga, R., Magnetostatics with scalar potentials in multiply connected regions, Science, Measurement and Technology, IEE Proceedings A, 137 (1990), pp. 231–232.Google Scholar
[61]Gross, P.W., Kotiuga, P.R., A Challenge for Magnetic Scalar Potential Formulations of 3-D Eddy Current Problems: Multiply Connected Cuts in Multiply Connected Regions Which Necessarily Leave the Cut Complement Multiply Connected, Electric and Magnetic Fields, Nicolet, A., Belmans, R., eds., Plenum Press, New York, 1995, pp. 1–20.Google Scholar
[62]Crager, J.C., Kotiuga, P.R., Cuts for the Magnetic Scalar Potential in Knotted Geometries and Force-Free Magnetic Fields, IEEE Trans. Magn. 38 (2002), pp. 1309–1312.Google Scholar
[63]Kotiuga, P.R., Topology-Based Inequalities and Inverse Problems for Near Force-Free Magnetic Fields, IEEE Trans. Magn. 40 (2004), pp. 1108–1111.Google Scholar
[64]Bing, R.H., Some aspects of the topology of 3-manifolds related to the Poincaré Conjecture, Lectures on Modern Mathematics II, T.L. Saaty ed., Wiley (1964), pp. 93–128.Google Scholar
[65]Fernandes, P., Gilardi, G., Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci., 7 (1997), pp. 957–991.Google Scholar
[66]Amrouche, C., Bernardi, C., Dauge, M., Girault, V., Vector potentials in three-dimensional non-smooth domains, Mathematical Methods in the Applied Sciences, 21 (1998), pp. 823–864.Google Scholar
[67]Boffi, D., Fernandes, P., Gastaldi, L., Perugia, I., Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal., 36 (1999), pp. 1264–1290.Google Scholar
[68]Ammari, H., Buffa, A., Neédélec, J.-C., A justification of eddy currents model for the Maxwell equations, SIAM J. Appl. Math., 60 (2000), pp. 1805–1823.Google Scholar
[69]Bermuúdez, A., Rodríguez, R., Salgado, P., A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations, SIAM J. Numer. Anal., 40 (2002), pp. 1823–1849.Google Scholar
[70]Perugia, I., Schoötzau, D., The hp-local discontinuous Galerkin method fow low-frequency time-harminic Maxwell equations, Math. Comp., 72 (2002), pp. 1179–1214.Google Scholar
[71]Zheng, W., Chen, Z., Wang, L., An adaptive finite element method for the H-ψ formulation of time-dependent eddy current problems Numer. Math., 103 (2006), pp. 667–689.Google Scholar
[72]Ciarlet, P.G., Ciarlet, P., Geymonat, G., Krasucki, F., Characterization of the kernel of the operator CURL CURL, C. R. Acad. Sci. Paris, Ser. I 344 (2007), pp. 305–308.Google Scholar
[73]Bíroí, O., Valli, A., The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: Well-posedness and numerical approximation, Computer Methods in Applied Mechanics and Engineering, 196 (2007), pp. 1890–1904.Google Scholar
[74]Zheng, W., Zhang, F., Adaptive finite element frequency domain method for eddy current problems, Comput. Meth. Appl. Mech. Eng., 197 (2008), pp. 1233–1241.Google Scholar
[75]Bermuúdez, A., Rodríguez, R., Salgado, P., A finite element method for the magnetostatic problem in terms of scalar potentials, SIAM J. Numer. Anal., 46 (2008), pp. 1338–1363.Google Scholar
[76]Geymonat, G., Krasucki, F., Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains, Commun. Pure Appl. Anal, 8 (2009), pp. 295–309.Google Scholar
[77]Mrozek, M., Pilarczyk, P., Zelazna, N., Homology Algorithm Based on Acyclic Subspace, Computers and Mathematics, 55 (2008), 2395–2412.Google Scholar
[78]Kaczynski, T., Mrozek, M., Slusarek, M., Homology computation by reduction of chain complexes, Computers and Mathematics, 35 (1998), pp. 59–70.Google Scholar
[79]Dłotko, P., Kaczynski, T., Mrozek, M., Wanner, T., Coreduction Homology Algorithm for Regular CW-Complexes, Discrete Comput. Geom., 46 (2011), pp. 361–388.Google Scholar