On numerical methods for time-dependent eddy current problems for the Maxwell equations in inhomogeneous media
A.A. Arbuzova*, R.Z. Dautovb**, E.M. Karchevskiib***, M.M. Karchevskiib****, D.V. Chistyakov
aTGT Oilfield Services Company, Kazan, 420108 Russia
bKazan Federal University, Kazan, 420008 Russia
E-mail: *info@tgtoil.com, **Rafail.Dautov@kpfu.ru, ***Evgenii.Karchevskii@kpfu.ru, ****Mikhail.Karchevsky@kpfu.ru
Received March 15, 2018
Abstract
E-based and H-based formulations for time-dependent eddy current problems for the Maxwell equations in inhomogeneous media have been derived. The concept of generalized solutions for boundary-value problems in bounded regions for obtained systems of equations has been formulated. Conditions for the existence and uniqueness of the generalized solutions have been established. Axisymmetric problems have been thoroughly considered, and a class of test problems has been proposed. Their exact solutions have the same key features as the solutions of the original problems. Numerical methods based on finite element approximations of the three-dimensional operators have been constructed. Particular attention has been paid to the methods on tetrahedral elements. Linear Lagrange elements, zero-order and first-order Nédélec elements have been used. The computational efficiency of the proposed finite element approximations has been analyzed by solving the constructed test problems. For small gaps in the coefficients of equations and regular finite element meshes, the first-order Nédélec elements have certain advantages in terms of accuracy and computational costs.
Keywords: Maxwell equations, eddy current approximation, finite element method, test problems
Acknowledgments. The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project no. 1.12878.2018/12.1.
References
1. Landau L.D., Pitaevskii L.P., Lifshitz E.M. Electrodynamics of Continuous Media. Butterworth-Heinemann, 1984. 460 p.
2. Duvant G., Lions J.L. Inequalities in Mechanics and Physics. Berlin, Heidelberg, Springer-Verlag, 1976. xvi, 400 p. doi: 10.1007/978-3-642-66165-5.
3. Tamm I.E. Osnovy teroii elektrichestva [Fundamentals of The Theory of Electricity]. Moscow, Mir, 1979. 684 p. (In Russian)
4. Galanin M.P., Popov Yu.P. Kvazistatsionarnye elektromagnitnye polya v neodnorodnykh sredakh [Quasi-Stationary Electromagnetic Fields in Inhomogeneous Media]. Moscow, Nauka, Fizmatlit, 1995. 315 p. (In Russian)
5. Acevedo R., Meddahi S., Rodriguez R. An E-based mixed formulation for a time-dependent eddy current problem. Math. Comput., 2009, vol. 78, no. 268, pp. 1929–1949. doi: 10.1090/S0025-5718-09-02254-6.
6. Kalinin A.V., Morozov S.F. The Maxwell equations system in the quasi-stationary magnetic approximation. Vestn. Nizhegorod. Gos. Univ. Ser. Mat. Model. Optim. Upr., 2001, no. 1, pp. 97–106. (In Russian)
7. Rapettia R., Rousseaux B. On quasi-static models hidden in Maxwell's equations. Appl. Numer. Math., 2014, vol. 79, pp. 92–106. doi: 10.1016/j.apnum.2012.11.007.
8. Ladyzhenskaya O.A. The Boundary Value Problems of Mathematical Physics. New York, Springer-Verlag, 1985. xxx, 322 p. doi: 10.1007/978-1-4757-4317-3.
9. Mikhlin S.G. Lineinye uravneniya v chastnykh proizvodnykh [Linear Partial Differential Equations]. Moscow, Vyssh. Shk., 1977. 423 p. (In Russian)
10. Karchevskii M.M., Pavlova M.F. Uravneniya matematicheskoi fiziki. Dopolnitel'nye glavy [Equations of Mathematical Physics. Additional Chapters]. St. Petersburg, Lan', 2016. 276 p. (In Russian)
11. Kalinin A.V. Matematicheskie zadachi fizicheskoi diagnostiki. Korrektnost' zadach elektromagnitnoi teorii v statsionarnom i kvazistatsionarnom priblizhenii [Mathematical Problems of Physical Diagnostics. The Correctness of the Problems of the Electromagnetic Theory in the Stationary and Quasi-Stationary Approximations]. Nizhny Novgorod, Izd. Nizhegorod. Univ., 2007. 121 p. (In Russian)
12. Gaevsky H., Greger K., Zakharias K. Nelineinye operatornye uravneniya i operatornye differentsial'nye uravneniya [Nonlinear Operator Equations and Operator Differential Equations]. Moscow, Mir, 1978. 336 p. (In Russian)
13. Zeidler E. Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators. New York, Springer-Verlag, 1990. xviii, 467 p. doi: 10.1007/978-1-4612-0985-0.
14. Ciarlet P.G. The Finite Element Method for Elliptic Problems. SIAM, 2002. xxiii, 529 p. doi: 10.1137/1.9780898719208.
15. Nédélec J.C. A new family of mixed finite elements in R3. Numer. Math., 1986, vol. 50, no. 1, pp. 57–81. doi: 10.1007/BF01389668.
16. Rodriguez A.A., Valli A. Eddy Current Approximation of Maxwell Equations. Theory, Algorithms and Applications. Milan, Springer-Verlag, 2010. xiii, 347 p. doi: 10.1007/978-88-470-1506-7.
For citation: Arbuzov A.A., Dautov R.Z., Karchevskii E.M., Karchevskii M.M., Chistyakov D.V. On numerical methods for time-dependent eddy current problems for the Maxwell equations in inhomogeneous media. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 3, pp. 477–494. (In Russian)
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