Skip to main content
SpringerLink
Log in

Physical Framework and Models

  • Chapter
  • First Online:
Mathematical Foundations of Computational Electromagnetism

Part of the book series: Applied Mathematical Sciences ((AMS,volume 198))

  • 1780 Accesses

Abstract

The aim of this first chapter is to present the physics framework of electromagnetism, in relation to the main sets of equations, that is, Maxwell’s equations and some related approximations. In that sense, it is neither a purely physical nor a purely mathematical point of view. The term model might be more appropriate: sometimes, it will be necessary to refer to specific applications in order to clarify our purpose, presented in a selective and biased way, as it leans on the authors’ personal view. This being stated, this chapter remains a fairly general introduction, including the foremost models in electromagnetics. Although the choice of such applications is guided by our own experience, the presentation follows a natural structure.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Unless otherwise specified, in this chapter, a domain is an open region of space. Another meaning is given for the mathematical studies, starting in Chap. 2.

  2. 2.

    H is sometimes called the magnetizing field.

  3. 3.

    The standard differential operators , div, \( \operatorname {\mathrm {\mathbf {grad}}}\), and Δ are mathematically defined in Sect. 1.5.1.

  4. 4.

    By definition, δ Σ is the surface Dirac mass on Σ, so one has \(\int \varrho v = \int _\varSigma \sigma _\varSigma v_{|\varSigma } \,dS\) for ad hoc functions v.

  5. 5.

    See the end of the section.

  6. 6.

    By definition, \(\delta _{{\boldsymbol {x}}_0}\) is the Dirac mass in x 0, so one has \(\int \varrho _0 v = q_0 v({\boldsymbol {x}}_0)\) for ad hoc functions v.

  7. 7.

    Or: electrostatically.

  8. 8.

    To deserve the label fixed frequency problem, one assumes a non-vanishing value of the pulsation. Otherwise, one solves a static problem, cf. Sect. 1.4.1.

  9. 9.

    The fields and are respectively called electric and magnetic polarizations .

  10. 10.

    Other conditions on lead to the same conclusion. For instance, if is a real-valued, even function of ω that can be expressed as a rational fraction, with decaying condition for large |ω|.

  11. 11.

    In particular, this is the case for the Lorentz force (1.75). As a matter of fact, div v F(t, x, v) = q (div v E + div v (v ×B)) = 0, since the electromagnetic fields are independent of v in the phase space.

  12. 12.

    Note, however, that in the more general case of a kinetic description given by Eq. (1.81) for several species, one can still prove that ϱ and J defined by Eqs. (1.84–1.85) satisfy the differential charge conservation equation (1.10). This is a straightforward consequence of Eq. (1.86).

  13. 13.

    It can happen that, in Maxwell’s equations, parts of ϱ and J are due to external charge and current sources. In this case, E and B depend in an affine way on f.

  14. 14.

    Id est, consider \(f_\alpha ({\boldsymbol {v}})\approx A_\alpha \exp (-B_\alpha |{\boldsymbol {v}}-\boldsymbol {u}_\alpha |{ }^2)\), with A α , B α  > 0.

  15. 15.

    See the upcoming Sect. 1.5 for a precise definition.

  16. 16.

    More precisely, ω is a pulsation and the corresponding frequency is ω∕(2π).

  17. 17.

    One may also use the interface conditions to describe electromagnetic fields globally in \(\mathbb {R}^3\): this is an integral representation. More precisely [167, §5.5], consider that \(\mathbb {R}^3\) is split into two media M + and M −, one of them being bounded, and let Σ be the interface between the two media. If one is interested in the electromagnetic fields that are governed by the homogeneous time-harmonic equations in M + and M −, then, assuming that the jump j Σ  = −[H ×n Σ ] Σ (condition (1.12 right)) is known, one can use integral representation formulas for the values of E(x) and H(x), for all \({\boldsymbol {x}}\in \mathbb {R}^3\setminus \varSigma \). The integrals are taken over Σ and depend only on j Σ . In the same spirit, one can represent the (different) values of E ±(x Σ ) and H ±(x Σ ) for all x Σ  ∈ Σ. Within this framework, one may generalize these results in the presence of magnetic polarization by assuming that the magnetic current on Σ, m Σ  = [E ×n Σ ] Σ , is also different from 0. In this case, one ends up with integral representation formulas of E and H, with integrals over Σ that depend on j Σ and m Σ . In the same manner, one may use the jump relation σ Σ  = [D ⋅n Σ ] Σ (1.11 left) to solve a diffraction problem expressed as a scalar Helmholtz equation, assuming σ Σ is known, where the unknown is the scalar electric potential.

  18. 18.

    Instead of B(O, R), one can choose any reasonable volume in which the computations ought to be performed: a cube, as in Fig. 1.3 (right, rightmost Γ A ), etc.

  19. 19.

    For instance (see [187]), if the artificial boundary Γ A is a cylinder of radius R and axis Oz, one gets

    with E = E r e r  + E θ e θ  + E z e z in cylindrical coordinates.

  20. 20.

    Manipulating Maxwell’s equations thusly is certainly admissible, since one is dealing with artificial media, in which the electromagnetic fields are artifacts…

  21. 21.

    Indeed, the unit outward normal vector to ∂B(O, R) is n = e r . Moreover, since x = r e r on ∂B(O, R), for an outgoing plane wave that propagates normally to ∂B(O, R) (k out  = k e r ), one finds k out  ⋅x = kr. Respectively, for an incoming plane wave that propagates normally to ∂B(O, R) (k in  = −k e r ), k in  ⋅x = −kr.

References

  1. S. Abarbanel, D. Gottlieb, J.S. Hestaven, Non-linear PML equations for time-dependent electromagnetics in three dimensions. J. Sci. Comput. 28, 125–137 (2006)

    Article  MathSciNet  Google Scholar 

  2. J.J. Ambrosiano, S.T. Brandon, E. Sonnendrücker, A finite element formulation of the Darwin PIC model for use on unstructured grids. J. Comput. Phys. 121, 281–297 (1995)

    Google Scholar 

  3. X. Antoine, H. Barucq, A. Bendali, Bayliss-Turkell-like radiation conditions on surfaces of arbitrary shape. J. Math. Anal. Appl. 229, 184–211 (1999)

    Article  MathSciNet  Google Scholar 

  4. A.A. Arseneev, Local uniqueness and existence of classical solution of Vlasov’s system of equations. Sov. Math. Dokl. 15, 1223–1225 (1974)

    Google Scholar 

  5. A.A. Arseneev, Global existence of a weak solution of Vlasov’s system of equations. U.S.S.R. Comput. Math. Math. Phys. 15, 131–143 (1975)

    Google Scholar 

  6. F. Assous, P. Degond, E. Heintzé, P.-A. Raviart, J. Segré, On a finite element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109, 222–237 (1993)

    Article  MathSciNet  Google Scholar 

  7. F. Assous, F. Tsipis, A PIC method for solving a paraxial model of highly relativistic beams. J. Comput. Appl. Math. 227, 136–146 (2008)

    Article  MathSciNet  Google Scholar 

  8. F.V. Atkinson, On Sommerfeld’s radiation condition. Philos. Mag. 40, 645–651 (1949)

    Google Scholar 

  9. A. Bachelot, V. Lange, Time dependent integral method for Maxwell’s equations with impedance boundary condition, in Boundary Element Technology, vol. 10, ed. by C.A. Brebbia, M.H. Aliabadah (Wessex Institute of Technology, Southampton, 1995), pp. 137–144

    Google Scholar 

  10. E. Bécache, S. Fauqueux, P. Joly, Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 399–433 (2003)

    Article  MathSciNet  Google Scholar 

  11. E. Bécache, P. Joly, On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations. Math. Model. Numer. Anal. 36, 87–119 (2002)

    Article  Google Scholar 

  12. S. Benachour, F. Filbet, P. Laurençot, E. Sonnendrücker, Global existence for the Vlasov-Darwin system in \(\mathbb {R}^3\) for small initial data. Math. Methods Appl. Sci. 26, 297–319 (2003)

    Google Scholar 

  13. A. Bendali, P. Guillaume, Non-reflecting boundary conditions for waveguides. Math. Comput. 68, 123–144 (1999)

    Article  MathSciNet  Google Scholar 

  14. A. Bendali, L. Halpern, Conditions aux limites absorbantes pour le système de Maxwell dans le vide en dimension 3. C. R. Acad. Sci. Paris Ser. I 307, 1011–1013 (1988)

    Google Scholar 

  15. J.-P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MathSciNet  Google Scholar 

  16. J.-P. Bérenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127, 363–379 (1996)

    Article  MathSciNet  Google Scholar 

  17. A.J. Berkhout, Seismic Migration (Elsevier, Amsterdam, Oxford, New York, Tokyo, 1984)

    Google Scholar 

  18. C.K. Birdsall, A.B. Langdon, Plasmas Physics via Computer Cimulation (McGraw-Hill, New York, 1985)

    Google Scholar 

  19. V.A. Bokil, M.W. Buksas, Comparison of finite difference and mixed finite element methods for perfectly matched layer models. Commun. Comput. Phys. 2, 806–826 (2007)

    Google Scholar 

  20. F. Bouchut, F. Golse, M. Pulvirenti, Kinetic Equations and Asymptotic Theory. Series in Applied Mathematics (Gauthier-Villars, 2000)

    Google Scholar 

  21. M. Cessenat, Mathematical Methods in Electromagnetism. Linear Theory and Applications. Advances in Mathematics for Applied Sciences, vol. n41 (World Scientific, Singapore, 1996)

    Google Scholar 

  22. F.F. Chen, Introduction to Plasma Physics (Plenum Press, New York, London, 1974)

    Google Scholar 

  23. W.C. Chew, W.H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microw. Opt. Technol. Lett. 7, 599–604 (1994)

    Article  Google Scholar 

  24. P. Ciarlet, Jr., E. Sonnendrücker, A decomposition of the electric field. Application to the Darwin model. Math. Methods Appl. Sci. 7(8), 1085–1120 (1997)

    Google Scholar 

  25. J.F. Claerbout, Imaging the Earth’s Interior (Blackwell Scientific Publications, Oxford, 1985)

    Google Scholar 

  26. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Theory (Springer, Berlin, 1998)

    Google Scholar 

  27. C.G. Darwin, The dynamical motion of particles. Philos. Mag. 39, 537–551 (1920)

    Google Scholar 

  28. R. Dautray, J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques (Masson, Paris, 1987)

    Google Scholar 

  29. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer, Berlin, 1990)

    MATH  Google Scholar 

  30. P.A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  31. P. Degond, Local existence of the solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity. Math. Methods Appl. Sci. 8, 533–558 (1986)

    Article  MathSciNet  Google Scholar 

  32. P. Degond, P.-A. Raviart, An analysis of the Darwin model of approximation to Maxwell’s equations. Forum Math. 4, 13–44 (1992)

    Google Scholar 

  33. L. Desvillettes, Quelques outils d’analyse pour les équations de Vlasov et de Landau, in Ecole CEA-EDF-INRIA, Equations cinétiques et applications à la physique (2005)

    Google Scholar 

  34. R.J. DiPerna, P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42, 729–757 (1989)

    Article  MathSciNet  Google Scholar 

  35. E. Durand, Electrostatique (Masson, Paris, 1964)

    Google Scholar 

  36. D.M. Eidus, The principle of limiting absorption. Russ. Math. Surv. 24, 97–167 (1969)

    Article  Google Scholar 

  37. B. Engquist, A. Majda, Absorbing boundary conditions for acoustic and elastic wave equations. Math. Comput. 31, 629–651 (1977)

    Google Scholar 

  38. X. Feng, Absorbing boundary conditions for electromagnetic wave propagation. Math. Comput. 68, 145–168 (1999)

    Article  MathSciNet  Google Scholar 

  39. R. Feynman, Leighton, Sands, The Feynman Lectures on Physics: Electromagnetism (Addison-Wesley, Reading, 1964)

    Google Scholar 

  40. S.D. Gedney, An anisotropic perfectly-matched layer absorbing medium for the truncation of FT-TD lattices. IEEE Trans. Antennas Propag. 44, 1630–1639 (1996)

    Article  Google Scholar 

  41. S.D. Gedney, The perfectly matched layer absorbing medium, in Advances in Computational Electrodynamics, ed. by A. Taflove (Artech House, 1998), pp. 263–344

    Google Scholar 

  42. D. Givoli, A spatially exact non-reflecting boundary condition for time dependent problems. Comput. Methods Appl. Mech. Eng. 95, 97–113 (1992)

    Article  MathSciNet  Google Scholar 

  43. R.T. Glassey, W. Strauss, Singularity formation in a colisionless plasma could only occur at high velocities. Arch. Ration. Mech. Anal. 92, 56–90 (1986)

    Google Scholar 

  44. P.W. Gross, P.R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach. MSRI Publications Series (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  45. M.J. Grote, J.B. Keller, On nonreflecting boundary conditions. J. Comput. Phys. 122, 231–243 (1995)

    Article  MathSciNet  Google Scholar 

  46. Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions. Commun. Math. Phys. 154, 245–263 (1993)

    Article  MathSciNet  Google Scholar 

  47. C. Hazard, M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell’s equations. SIAM J. Math. Anal. 27, 1597–1630 (1996)

    Article  MathSciNet  Google Scholar 

  48. A.D. Ioannidis, I.G. Stratis, A. Yannacopoulos, Electromagnetic wave propagation in dispersive bianisotropic media, in Proc. of the 6th Int. Workshop on Advances in Scattering and Biomedical Engineering, ed. by D. Fotiadis et al. (World Scientific, 2003), pp. 295–304

    Google Scholar 

  49. J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999)

    Google Scholar 

  50. B. Jensen, The quantum extension of the Drude-Zener theory in polar semiconductors, in Handbook of Optical Constants of Solids, ed. by E.D. Palik (Academic Press, New York, 1998), pp. 169–188

    Chapter  Google Scholar 

  51. P. Joly, An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation. SeMA J. 57, 5–48 (2012)

    Article  MathSciNet  Google Scholar 

  52. P. Joly, B. Mercier, Une nouvelle condition transparente d’ordre 2 pour les équations de Maxwell en dimension 3. Technical Report INRIA, vol. 1047 (1989)

    Google Scholar 

  53. D.S. Jones, Methods in Electromagnetic Wave Propagation (Clarendon Press, Oxford, 1998)

    Google Scholar 

  54. A. Karlsson, G. Kristensson, Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain. Technical Report (Revision No. 2: August 1999) Lund Institute of Technology, CODEN:LUTEDX/(TEAT-7005)/1-36/(1989), Lund (1999)

    Google Scholar 

  55. N.A. Krall, A.W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, New York, 1973)

    Article  Google Scholar 

  56. A. Lakhtakia, V.K. Varadan, V.V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media. Lecture Notes in Physics (Springer, Berlin, 1989)

    Google Scholar 

  57. G. Laval, Matière et rayonnement (Ecole Polytechnique, Palaiseau, 1986)

    Google Scholar 

  58. A.E. Lifschitz, Magnetohydrodynamics and Spectral Theory (Kluwer Academic, 1989)

    Book  Google Scholar 

  59. I. Lindell, A. Shivola, S. Tertyakov, A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, Boston, 1994)

    Google Scholar 

  60. S.A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007)

    Google Scholar 

  61. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, Berlin, 1969)

    Chapter  Google Scholar 

  62. J.-C. Nédélec, Acoustic and Electromagnetic Equations. Applied Mathematical Sciences, vol. 144 (Springer, New York, 2001)

    Book  Google Scholar 

  63. H.M. Nussenzveig, Causality and Dispersion Relations (Academic Press, New York, 1972)

    Google Scholar 

  64. C. Rappaport, Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space. IEEE Microw. Guid. Wave Lett. 5, 90–92 (1995)

    Article  MathSciNet  Google Scholar 

  65. P.-A. Raviart, E. Sonnendrücker, A hierarchy of approximate models for the Maxwell equations. Numer. Math. 73, 329–372 (1996)

    Article  MathSciNet  Google Scholar 

  66. Z. Sacks, D. Kinsland, R. Lee, J.F. Lee, A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propag. 43, 1460–1463 (1995)

    Article  Google Scholar 

  67. J. Sanchez Hubert, E. Sanchez Palencia, Vibration and Coupling of Continuous Systems (Springer, 1989)

    Google Scholar 

  68. M. Sesques, Conditions aux limites artificielles pour le système de Maxwell, PhD Thesis, Bordeaux I University, France, 1990 (in French)

    Google Scholar 

  69. S. Silver, Microwave Antenna Theory and Design. M.I.T. Radiation Laboratory Series, vol. 12 (McGraw-Hill, New York, 1949)

    Google Scholar 

  70. F. Simpson, K. Bahr, Practical Magnetotellurics (Cambridge University Press, Cambridge, 2005)

    Book  Google Scholar 

  71. A. Sommerfeld, Die Greensche Funktion der Schwingungsgleichung. Jber. Deutsch. Math. Verein. 21, 309–353 (1912)

    MATH  Google Scholar 

  72. T.H. Stix, Waves in Plasmas (American Institute of Physics, New York, 1992)

    Google Scholar 

  73. S. Ukai, T. Okabe, On classical solution in the large in time of two dimensional Vlasov’s equation. Osaka J. Math. 15, 245–261 (1978)

    MathSciNet  MATH  Google Scholar 

  74. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1985)

    Google Scholar 

  75. C.H. Wilcox, An expansion theorem for electromagnetic fields. Commun. Pure Appl. Math. 9, 115–134 (1956)

    Article  MathSciNet  Google Scholar 

  76. S. Wollman, An existence and uniqueness theorem for the Vlasov-Maxwell system. Commun. Pure Appl. Math. 37, 457–462 (1984)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ariel University, Ariel, Israel

    Franck Assous

  2. ParisTech, ENSTA, Palaiseau, France

    Patrick Ciarlet

  3. Université de Lorraine, Institut Élie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

    Simon Labrunie

Authors
  1. Franck Assous
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrick Ciarlet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Simon Labrunie
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Assous, F., Ciarlet, P., Labrunie, S. (2018). Physical Framework and Models. In: Mathematical Foundations of Computational Electromagnetism. Applied Mathematical Sciences, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-70842-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation