Equivalent transmission conditions for the time-harmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet. (English) Zbl 1373.35299
The authors consider the time-harmonic Maxwell equations in a 3-dimensional domain with a thin conducting layer of thickness of order \(\varepsilon\) inside the domain. By using a multiscale expansion, the authors derive formally the asymptotic limits of order 1 and 2. The transition conditions on the midsurface of the thin layer involve generalized Poincaré-Steklov maps between tangential components of the magnetic field and the electric field, and involve second order surface differential operators. It would be interesting to study further the limiting problems involving the transmission conditions derived in this paper and examine rigorously the convergence of the solutions of the original problem near the thin layer as \(\varepsilon\to 0\).
Reviewer: Xingbin Pan (Shanghai)
MSC:
| 35Q61 | Maxwell equations |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
time-harmonic Maxwell’s equations; thin conducting sheets; asymptotic expansions; multiscale expansions; impedance transmission conditionsReferences:
| [1] | H. Bateman, {\it The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell’s Equations}, Cambridge University Press, Cambridge, UK, 1915. · JFM 45.1324.05 |
| [2] | O. Biro, K. Preis, K. Richter, R. Heller, P. Komarek, and W. Maurer, {\it FEM calculation of eddy current losses and forces in thin conducting sheets of test facilities for fusion reactor components}, IEEE Trans. Magn., 28 (1992), pp. 1509-1512. |
| [3] | D. Braess, {\it Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics}, 3rd ed., Cambridge University Press, Cambridge, UK, 2007. · Zbl 1118.65117 |
| [4] | G. Caloz, M. Dauge, E. Faou, and V. Péron, {\it On the influence of the geometry on skin effect in electromagnetism}, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 1053-1068, http://dx.doi.org/10.1016/j.cma.2010.11.011, http://hal.archives-ouvertes.fr/hal-00503170/en/. · Zbl 1225.78003 |
| [5] | G. Caloz, M. Dauge, and V. Péron, {\it Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism}, J. Math. Anal. Appl., 370 (2010), pp. 555-572, http://dx.doi.org/DOI: 10.1016/j.jmaa.2010.04.060. · Zbl 1194.35452 |
| [6] | G. Cohen and M. Duruflé, {\it Non spurious spectral-like element methods for Maxwell’s equations}, J. Comput. Math, 25 (2007), pp. 282-304. |
| [7] | B. Delourme, H. Haddar, and P. Joly, {\it On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems}, Math. Models Methods Appl. Sci., 23 (2013), pp. 2433-2464, http://hal.inria.fr/hal-00682357. · Zbl 1288.78016 |
| [8] | M. Duruflé, V. Péron, and C. Poignard, {\it Time-harmonic Maxwell equations in biological cells–the differential form formalism to treat the thin layer}, Confluentes Math., 3 (2011), pp. 325-357, http://dx.doi.org/10.1142/S1793744211000345. · Zbl 1235.78035 |
| [9] | M. Duruflé, V. Péron, and C. Poignard, {\it Thin layer models for electromagnetism}, Commun. Comput. Phys., 16 (2014), pp. 213-238, http://hal.archives-ouvertes.fr/hal-00918634. |
| [10] | C. Geuzaine, P. Dular, and W. Legros, {\it Dual formulations for the modeling of thin electromagnetic shells using edge elements}, IEEE Trans. Magn., 36 (2000), pp. 799-803. |
| [11] | H. Haddar and Z. Jiang, {\it Axisymmetric eddy current inspection of highly conducting thin layers via asymptotic models}, Inverse Problems, 31 (2015), 115005, http://stacks.iop.org/0266-5611/31/i=11/a=115005. · Zbl 1334.35333 |
| [12] | R. Hiptmair, {\it Symmetric coupling for eddy current problems}, SIAM J. Numer. Anal., 40 (2002), pp. 41-65, http://dx.doi.org/10.1137/S0036142900380467. · Zbl 1010.78011 |
| [13] | H. Igarashi, A. Kost, and T. Honma, {\it Impedance boundary condition for vector potentials on thin layers and its application to integral equations}, European Phys. J. Appl. Phys., 1 (1998), pp. 103-109, http://dx.doi.org/10.1051/epjap:1998123, http://www.epjap.org/article_S1286004298001232. |
| [14] | J.-M. Jin, J. L. Volakis, C. Yu, and A. Woo, {\it Modeling of resistive sheets in finite element solutions}, IEEE Trans. Antennas Propagation, 40 (1992), pp. 727-731. |
| [15] | E. Knott and T. Senior, {\it Non-Specular Radar Cross Section Study}, Tech. report 011764-1-T, University of Michigan Radiation Lab., Ann Arbor, MI, 1974. |
| [16] | L. Krähenbühl and D. Muller, {\it Thin layers in electrical engineering. Example of shell models in analysing eddy-currents by boundary and finite element methods}, IEEE Trans. Magn., 29 (1993), pp. 1450-1455. |
| [17] | T. Levi-Civita, {\it La teoria elettrodinamica di Hertz di fronte ai fenomeni di induzione}, Rend. Lincei (5), 11 (1902), pp. 75-81. \newblock · JFM 34.0934.01 |
| [18] | I. Mayergoyz and G. Bedrosian, {\it On calculation of 3-D eddy currents in conducting and magnetic shells}, IEEE Trans. Magn., 31 (1995), pp. 1319-1324. |
| [19] | W. McLean, {\it Strongly Elliptic Systems and Boundary Integral Equations}, Cambridge University Press, Cambridge, UK, 2000. · Zbl 0948.35001 |
| [20] | J. McWhirter, {\it Computation of three-dimensional eddy currents in thin conductors}, IEEE Trans. Magn., 18 (1982), pp. 456-460. |
| [21] | P. Monk, {\it Finite Element Methods for Maxwell’s Equations}, Numer. Math. Sci. Comput., Oxford University Press, New York, 2003, http://dx.doi.org/10.1093/acprof:oso/9780198508885.001.0001. · Zbl 1024.78009 |
| [22] | T. Nakata, N. Takahashi, K. Fujiwara, and Y. Shiraki, {\it 3D magnetic field analysis using special elements}, IEEE Trans. Magn., 26 (1990), pp. 2379-2381. |
| [23] | J.-C. Nédélec, {\it Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems}, Appl. Math. Sci. 144, Springer, New York, 2001. · Zbl 0981.35002 |
| [24] | V. Péron, {\it Modélisation Mathématique de Phénomènes Électromagnétiques dans des Matériaux à Fort Contraste}, Ph.D. thesis, Université Rennes 1, Rennes, France, 2009, http://tel.archives-ouvertes.fr/tel-00421736/fr/. |
| [25] | J. Poltz and K. Romanowski, {\it Solution of quasi-stationary field problems by means of magnetic scalar potential}, IEEE Trans. Magn., 19 (1983), pp. 2425-2428. |
| [26] | D. Rodger and N. Atkinson, {\it Finite element method for 3D eddy current flow in thin conducting sheets}, IEE Proceedings A, 135 (1988), pp. 369-374. |
| [27] | K. Schmidt and A. Chernov, {\it Robust Families of Transmission Conditions of High Order for Thin Conducting Sheets}, Tech. report 1102, Institute for Numerical Simulation, University of Bonn, Bonn, Germany, 2011, http://page.math.tu-berlin.de/7Ekschmidt/pub/SchmidtChernovPreprint2011.pdf. |
| [28] | K. Schmidt and A. Chernov, {\it A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model}, SIAM J. Appl. Math., 73 (2013), pp. 1980-2003, http://dx.doi.org/10.1137/120901398. · Zbl 1290.78020 |
| [29] | K. Schmidt and A. Chernov, {\it Robust transmission conditions of high order for thin conducting sheets in two dimensions}, IEEE Trans. Magn., 50 (2014), pp. 41-44, http://dx.doi.org/10.1109/TMAG.2013.2285437. |
| [30] | K. Schmidt and R. Hiptmair, {\it Asymptotic boundary element methods for thin conducting sheets}, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), pp. 619-647. · Zbl 1302.78027 |
| [31] | K. Schmidt and S. Tordeux, {\it Asymptotic modelling of conductive thin sheets}, Z. Angew. Math. Phys., 61 (2010), pp. 603-626, http://dx.doi.org/10.1007/s00033-009-0043-x, http://page.math.tu-berlin.de/7Ekschmidt/pub/SchmidtTordeux2010.pdf. · Zbl 1235.78042 |
| [32] | K. Schmidt and S. Tordeux, {\it High order transmission conditions for thin conductive sheets in magneto-quasistatics}, ESAIM Math. Model. Numer. Anal., 45 (2011), pp. 1115-1140, http://dx.doi.org/10.1051/m2an/2011009. · Zbl 1273.78029 |
| [33] | T. Senior, {\it Approximate boundary conditions}, IEEE Trans. Antennas Propagation, 29 (1981), pp. 826-829. |
| [34] | O. Tozoni and I. Mayergoyz, {\it Analysis of Three-Dimensional Electromagnetic Fields}, Technika, Kiev, 1974 (in Russian). |
| [35] | B. I. Wohlmuth, {\it A mortar finite element method using dual spaces for the Lagrange multiplier}, SIAM J. Numer. Anal., 38 (2000), pp. 989-1012, http://dx.doi.org/10.1137/S0036142999350929. · Zbl 0974.65105 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
