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June  2015, 8(3): 619-647. doi: 10.3934/dcdss.2015.8.619

Asymptotic boundary element methods for thin conducting sheets

Kersten Schmidt 1, and  Ralf Hiptmair 2,

1. 

Research Center MATHEON, Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
2. 

Seminar for Applied Mathematics, ETH Zurich, 8092 Zürich, Switzerland

Received  November 2013 Revised  June 2014 Published  October 2014

Various asymptotic models for thin conducting sheets in computational electromagnetics describe them as closed hyper-surfaces equipped with linear local transmission conditions for the traces of electric and magnetic fields. The transmission conditions turn out to be singularly perturbed with respect to limit values of parameters depending on sheet thickness and conductivity. We consider the reformulation of the resulting transmission problems into boundary integral equations (BIE) and their Galerkin discretization by means of low-order boundary elements. We establish stability of the BIE and provide a priori $h$-convergence estimates.
Keywords: Boundary element method, transmission condition, asymptotic expansions, thin conducting sheets..
Mathematics Subject Classification: 65N38, 35C20, 35J25, 41A60, 35B40, 78M30, 78M3.
Citation: Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New-York, London, 1975.

[2]

A. Bendali, Boundary element solution of scattering problems relative to a generalized impedance boundary condition, in Partial differential equations: Theory and numerical solution., 406 of Chapman & Hall/CRC Res. Notes Math., 2000, 10-24.

[3]

A. Bendali and K. Lemrabet, Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell, Asymptotic Analysis, 57 (2008), 199-227.

[4]

A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56 (1996), 1664-1693. doi: 10.1137/S0036139995281822.

[5]

D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, 2007. doi: 10.1088/0957-0233/13/9/704.

[6]

Concepts Development Team, Webpage of Numerical C++ Library Concepts 2, URL http://www.concepts.math.ethz.ch, 2014.

[7]

B. Engquist and J.-C. Nédélec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers, Technical report, Ecole Polytechnique Paris, 1993, Rapport interne du C.M.A.P.

[8]

P. Frauenfelder and C. Lage, Concepts - An Object-Oriented Software Package for Partial Differential Equations, Math. Model. Numer. Anal., 36 (2002), 937-951. doi: 10.1051/m2an:2002036.

[9]

H. Haddar, P. Joly and H. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci, 15 (2005), 1273-1300. doi: 10.1142/S021820250500073X.

[10]

H. Haddar, P. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations, Math. Models Meth. Appl. Sci., 18 (2008), 1787-1827. doi: 10.1142/S0218202508003194.

[11]

T. Levi-Civita, La teoria elettrodinamica di Hertz di fronte ai fenomeni di induzione, Rend. Lincei (5), 11 (1902), 75-81.

[12]

I. Mayergoyz and G. Bedrosian, On calculation of 3-D eddy currents in conducting and magnetic shells, Magnetics, IEEE Transactions on, 31 (1995), 1319-1324. doi: 10.1109/20.376271.

[13]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.

[14]

T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, 3D magnetic field analysis using special elements, Magnetics, IEEE Transactions on, 26 (1990), 2379-2381. doi: 10.1109/20.104737.

[15]

C. Poignard, Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Meth. Appl. Sci., 31 (2008), 443-479. doi: 10.1002/mma.923.

[16]

C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Meth. Appl. Sci., 32 (2009), 435-453. doi: 10.1002/mma.1045.

[17]

S. Sauter and C. Schwab, Boundary Element Methods, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-540-68093-2.

[18]

K. Schmidt and A. Chernov, A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model, SIAM J. Appl. Math, 73 (2013), 1980-2003. doi: 10.1137/120901398.

[19]

K. Schmidt and R. Hiptmair, Asymptotic boundary element methods for thin conducting sheets, Preprint 2013-15, Inst. f. Mathematik, TU Berlin, 2013.

[20]

K. Schmidt and A. Chernov, Robust Families of Transmission Conditions of High Order for Thin Conducting Sheets, INS Report 1102, Institut for Numerical Simulation, University of Bonn, 2011.

[21]

K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets, Z. Angew. Math. Phys., 61 (2010), 603-626. doi: 10.1007/s00033-009-0043-x.

[22]

K. Schmidt and S. Tordeux, High order transmission conditions for thin conductive sheets in magneto-quasistatics, ESAIM: M2AN, 45 (2011), 1115-1140. doi: 10.1051/m2an/2011009.

[23]

O. Steinbach, Numerische Näherungsverfahren für elliptische Randwertprobleme. Finite Elemente und Randelemente, B.G. Teubner-Verlag, 2003.

[24]

O. Tozoni and I. Mayergoyz, Calculation of three-dimensional electromagnetic fields, Technika, Kiev, 1974, (in Russian).

[25]

L. Vernhet, generalized impedance boundary condition, Math. Meth. Appl. Sci., 22 (1999), 587-603.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New-York, London, 1975.

[2]

A. Bendali, Boundary element solution of scattering problems relative to a generalized impedance boundary condition, in Partial differential equations: Theory and numerical solution., 406 of Chapman & Hall/CRC Res. Notes Math., 2000, 10-24.

[3]

A. Bendali and K. Lemrabet, Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell, Asymptotic Analysis, 57 (2008), 199-227.

[4]

A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56 (1996), 1664-1693. doi: 10.1137/S0036139995281822.

[5]

D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, 2007. doi: 10.1088/0957-0233/13/9/704.

[6]

Concepts Development Team, Webpage of Numerical C++ Library Concepts 2, URL http://www.concepts.math.ethz.ch, 2014.

[7]

B. Engquist and J.-C. Nédélec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers, Technical report, Ecole Polytechnique Paris, 1993, Rapport interne du C.M.A.P.

[8]

P. Frauenfelder and C. Lage, Concepts - An Object-Oriented Software Package for Partial Differential Equations, Math. Model. Numer. Anal., 36 (2002), 937-951. doi: 10.1051/m2an:2002036.

[9]

H. Haddar, P. Joly and H. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci, 15 (2005), 1273-1300. doi: 10.1142/S021820250500073X.

[10]

H. Haddar, P. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations, Math. Models Meth. Appl. Sci., 18 (2008), 1787-1827. doi: 10.1142/S0218202508003194.

[11]

T. Levi-Civita, La teoria elettrodinamica di Hertz di fronte ai fenomeni di induzione, Rend. Lincei (5), 11 (1902), 75-81.

[12]

I. Mayergoyz and G. Bedrosian, On calculation of 3-D eddy currents in conducting and magnetic shells, Magnetics, IEEE Transactions on, 31 (1995), 1319-1324. doi: 10.1109/20.376271.

[13]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.

[14]

T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, 3D magnetic field analysis using special elements, Magnetics, IEEE Transactions on, 26 (1990), 2379-2381. doi: 10.1109/20.104737.

[15]

C. Poignard, Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Meth. Appl. Sci., 31 (2008), 443-479. doi: 10.1002/mma.923.

[16]

C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Meth. Appl. Sci., 32 (2009), 435-453. doi: 10.1002/mma.1045.

[17]

S. Sauter and C. Schwab, Boundary Element Methods, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-540-68093-2.

[18]

K. Schmidt and A. Chernov, A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model, SIAM J. Appl. Math, 73 (2013), 1980-2003. doi: 10.1137/120901398.

[19]

K. Schmidt and R. Hiptmair, Asymptotic boundary element methods for thin conducting sheets, Preprint 2013-15, Inst. f. Mathematik, TU Berlin, 2013.

[20]

K. Schmidt and A. Chernov, Robust Families of Transmission Conditions of High Order for Thin Conducting Sheets, INS Report 1102, Institut for Numerical Simulation, University of Bonn, 2011.

[21]

K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets, Z. Angew. Math. Phys., 61 (2010), 603-626. doi: 10.1007/s00033-009-0043-x.

[22]

K. Schmidt and S. Tordeux, High order transmission conditions for thin conductive sheets in magneto-quasistatics, ESAIM: M2AN, 45 (2011), 1115-1140. doi: 10.1051/m2an/2011009.

[23]

O. Steinbach, Numerische Näherungsverfahren für elliptische Randwertprobleme. Finite Elemente und Randelemente, B.G. Teubner-Verlag, 2003.

[24]

O. Tozoni and I. Mayergoyz, Calculation of three-dimensional electromagnetic fields, Technika, Kiev, 1974, (in Russian).

[25]

L. Vernhet, generalized impedance boundary condition, Math. Meth. Appl. Sci., 22 (1999), 587-603.

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