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Article information
2018 , Volume 23, ¹ 3, p.92-108
Shurina E.P., Dobrolyubova D.V., Shtanko E.I.
A reduced formulation for modelling time-harmonic electromagnetic field in the media with thin highly conductive inclusions
Purpose. In this paper, we propose a reduced variational formulation for the Helmholtz equation for the electric field, in which thin highly conductive objects are approximated by surfaces with the equivalent surface current density. We conduct a study aimed at defining the range of application for the reduced variational formulation, focusing on highly contrasting thin objects of various geometrical shape and arrangement in a wide frequency range. Methodology. The modelling is performed on unstructured tetrahedral meshes. Since the reduced variational formulation treats thin highly conductive objects as surfaces, no volume mesh is constructed inside of them. We compare the results obtained by the vector FEM using the proposed variational formulation with the results obtained using standard formulation. Findings. Due to the fact that the proposed variational formulation does not require volume meshing of the thin objects, its computational cost is significantly lower. However, the reduced formulation yields correct results in a restricted frequency range. It also imposes some limitations on the minimal contrast and maximal thickness of the thin highly conductive objects. Originality/value. The proposed reduced variational formulation can be applied to simulate the time-harmonic electric field in the media with thin highly conductive inclusions of either regular or chaotic arrangement, as well as thin shielding plates or casings of various geometrical forms.
[full text]
Keywords: vector finite element method, reduced models, media with thin inclusions, electromagnetic shielding
doi: 10.25743/ICT.2018.3.16017
Author(s):
Shurina Ella Petrovna
Dr. , Professor
Position: General Scientist
Office: Novosibirsk State Technical University, Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
Address: 630073, Russia, Novosibirsk, Karl Marx Ave. 20
Phone Office: (343) 223-72-95
E-mail: shurina@online.sinor.ru
Dobrolyubova Darya Vladimirovna
Office: Trofimuk Institute of Petroleum Geology and Geophysics SB RAS, Novosibirsk State Technical University, Novosibirsk State University
Address: Russia, Novosibirsk, Novosibirsk, Karl Marx Ave. 20
Shtanko Ekaterina Igorevna
Office: Trofimuk Institute of Petroleum Geology and Geophysics SB RAS, Novosibirsk State University
Address: Russia, Novosibirsk, Novosibirsk, Karl Marx Ave. 20
References:
[1] Le-Duc, T., Meunier, G., Chadebec, O., Guichon, J. M., Bastos, J. P. A. General Integral Formulation for the 3D Thin Shell Modeling. IEEE Transactions on Magnetics. 2013; 49(5):1989-1992.
[2] Schmidt, K. High-order numerical modelling of highly conductive thin sheets: Doctoral Thesis. Zurich, Eidgenössische Technische Hochschule ETH; 2008: 189.
[3] Bardi, I., Peng, G., Petersson, L. E. R. Modeling Periodic Layered Structures by Shell Elements Using the Finite-Element Method. IEEE Transactions on Magnetics. 2016; 52(3):1-4.
[4] Bensaid, S., Trichet, D., Fouladgar, J. Electromagnetic and thermal behaviors of multilayer anisotropic composite materials. IEEE transactions on magnetics. 2006; 42(4):995-998.
[5] Schmidt, K., Tordeux, S. High order transmission conditions for thin conductive sheets in magneto-quasistatics. ESAIM: Mathematical Modelling and Numerical Analysis. 2011; 45(6):1115-1140.
[6] Schmidt, K., Chernov, A. A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model. SIAM Journal on Applied Mathematics. 2013; 73(6):1980-2003.
[7] Schmidt, K., Hiptmair, R. Asymptotic boundary element methods for thin conducting sheets. Discrete and Continuous Dynamical Systems. Series S. 2015; 8(3):619-647.
[8] Erdozain, A., Péron, V. Impedance Transmission Conditions for the Electric Potential across a Highly Conductive Casing. Waves-2015, Karlsruhe, Germany; 2015: 3.
[9] Peron, V., Schmidt, K., Durufle, M. Equivalent transmission conditions for the timeharmonic Maxwell equations in 3D for a medium with a highly conductive thin sheet. SIAM Journal on Applied Mathematics. 2016; 76(3):1031-1052.
[10] Igarashi, H., Kost, A., Honma, T. Impedance boundary condition for vector potentials on thin layers and its application to integral equations. The European Physical Journal Applied Physics. 1998; (1):103–109.
[11] Geuzaine, C. High order hybrid finite element schemes for Maxwell’s equations taking thin structures and global quantities into account: Doctoral thesis. University of Liege; 2001: 174.
[12] Biro, O., Preis, K., Richter, K.R., Heller, R., Komarek, P., Maurer, W. FEM calculation of eddy current losses and forces in thin conducting sheets of test facilities for fusion reactor components. IEEE Transactions on Magnetics. 1992; 28(2):1509-1512.
[13] Nakata, T., Takahashi, N., Fujiwara, K., Shirak,Y. 3D magnetic field analysis using special elements. IEEE Transactions on Magnetics. 1990; 26(5):2379-2381.
[14] Mayergoyz, I. D, Bedrosian, G. On calculation of 3-D eddy currents in conducting and magnetic shells. IEEE Transactions on Magnetics. 1995; 31(3):1319–1324.
[15] Guerin, C., Maunier, G. 3-D magnetic scalar potential finite element formulation for conducting shells coupled with an external circuit. IEEE Transactions on Magnetics. 2012; 48(2):823-826.
[16] Krahenbuhl, L., Muller, D. Thin layers in electrial engineering. Example of shell models in analysing eddy-currents by boundary and finite element methods. IEEE Transactions on Magnetics. 1993; (29):1450–1455.
[17] Boubekeur, M., Kameni, A., Modave, A., Bernard, L., Pichon, L. Modeling of Weakly Conducting Thin Sheets in the Discontinuous Galerkin Method for Shielding Effectiveness Evaluation. Applied Computational Electromagnetics Society Journal. 2013; 28(10):931-937.
[18] Boubekeur, M., Kameni, A., Bernard, L., Modave, A., Pichon, L. 3D modeling of thin resistive sheets in the discontinuous Galerkin method for transient scattering analysis. Proceedings of the 19th Conference on the Computation of Electromagnetic Fields (COMPUMAG 2013), Budapest, Hungary; 2013: 5-7.
[19] Pebernet, L., Ferrieres, X., Pernet, S., Michielsen, B. L., Rogier, F., Degond, P. Discontinuous Galerkin method applied to electromagnetic compatibility problems: introduction of thin wire and thin resistive material models. IET Science, Measurement & Technology. 2008; 2(6):395-401.
[20]Sivukhin, D.V. Obshchiy kurs fiziki. Uchebnoe posobie: Dlya vuzov. V 5 t. T. III. Elektrichestvo. 4-e izdanie [General Course of Physics. Vol. 3: Electricity]. Moscow: Fizmatlit; 2004: 560. (In Russ.)
[21] Ren, Z. Degenerated Whitney Prism Elements - General Nodaland Egde Shell Elements For Field Computation in Thin Structures. IEEE Transactions on Magnetics. 1998; 34(5):2547-2550.
[22] Balandin, M.Y., Shurina, E.P. Vektornyy metod konechnykh elementov: Uchebnoe posobie [Vector finite-element method]. Novosibirsk: NGTU; 2001: 69. [In Russ.]
[23] Shurina, E.P., Dobrolyubova, D.V., Shtanko E.I. The modified vector finite element method formulation for the media with thin highly conductive plates. Sbornik statey VII Mezhdunarodnoy nauchno-prakticheskoy konferentsii “Mnogoyadernye protsessory, parallel'noe programmirovanie, plis, sistemy obrabotki signalov” (MPPOS-2017) [Proceedings of the International Scientific and Practical Conference "Multi-core processors, parallel programming, PLD, signal processing systems"]. Russia, Barnaul, March 10-11; 2017: pp. 64-72. [In Russ.]
[24] Shurina, E.P., Dobrolyubova, D.V., Mikhaylova, E.I. Osobennosti konechno-elementnogo modelirovaniya sred s tonkimi provodyashchimi plastinami. Kraevye zadachi i matematicheskoe modelirovanie. Tematicheskiy sbornik nauchnykh trudov [Features of finite element modeling for the media with thin conductive plates. Boundary problems and mathematical modeling. Scientific works]. Novokuznetsk; 2017: 218-225. [In Russ.]
[25] Vendik, I.B., Vendik, O.G. Metamaterials and their application in microwaves: A review. Technical Physics. The Russian Journal of Applied Physics. 2013; 58(1):1-24.
[26] Shapiro, D. N. Osnovy teorii elektromagnitnogo ekranirovaniya [Fundamentals of the theory of electromagnetic shielding]. Leningrad:Energiya; 1975: 112.
[27] Pardo, D., Torres-Verdin, C., Paszynski, M. Simulations of 3D DC borehole resistivity measurements with a goal-oriented hp finite-element method. Part II: through-casing resistivity instruments. Computational Geosciences. 2008; 12(1):83-89.
[28] Wang, L.-B., Zhang, J.-W., See, K.Y. Svimonishvili, T Ultra-Thin and Flexible Multi-Band Rejection EMI Shield. Journal of Electromagnetic Analysis and Applications. 2014; (6):163-173.
[29] Unal, E., Gokcen, A. Kutlu, Y. Effective Electromagnetic Shielding. IEEE Microwave Magazine. 2006; (7):48-54.
[30] Webb, J. P. Hierarchal Vector Basis Functions of Arbitrary Order for Triangular and Tetrahedral Finite Elements. IEEE Transactions on Antennas and Propagation. 1999; 47(8):1244-1253.
Bibliography link:
Shurina E.P., Dobrolyubova D.V., Shtanko E.I. A reduced formulation for modelling time-harmonic electromagnetic field in the media with thin highly conductive inclusions // Computational technologies. 2018. V. 23. ¹ 3. P. 92-108
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