Un esquema completamente discreto basado en elementos finitos para el problema de corrientes inducidas
Main Article Content
Keywords
Modelo evolutivo de corrientes inducidas, formulación en términos de potenciales, esquema completamente discreto, elementos finitos, estimaciones de error.
Resumen
El modelo de corrientes inducidas se obtiene a partir de las ecuaciones de Maxwell, despreciando las corrientes de desplazamiento de la Ley de AmpèreMaxwell. Bíró & Valli realizaron recientemente el análisis de existencia y unicidad de solución y el análisis teórico de convergencia para una de las formulaciones más populares del problema de corrientes inducidas en regimen armónico, conocida como “formulación en potenciales A; V A”. En el presente artículo se extiende el análisis realizado por Bíró & Valli al modelo evolutivo general de corrientes inducidas. Presentamos un esquema completamente discreto para la formulación, basado en una aproximación temporal usando un método de Euler implícito y una aproximación espacial a través del método de elementos finitos. Además, demostramos que el problema discreto resultante es un problema bien planteado y obtenemos estimaciones del error que muestran convergencia óptima.
MSC: 78M10, 65N30
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Referencias
[1] A. B. J. Reece and T. W. Preston, Finite Elements Methods in Electrical Power Engineering. New York: Oxford University Press, 2000. 112
[2] A. Bossavit, Computational electromagnetism, ser. Electromagnetism. San Diego, CA: Academic Press Inc., 1998. 112
[3] H. Ammari, A. Buffa, and J.-C. Nédélec, “A justification of eddy currents model for the Maxwell equations,” SIAM Journal on Applied Mathematics, vol. 60, no. 5, pp. 1805–1823, 2000. 113
[4] A. Alonso Rodríguez and A. Valli, Eddy current approximation of Maxwell equations, ser. MS&A. Modeling, Simulation and Applications. Springer-Verlag Italia, Milan, 2010, vol. 4. [Online]. Available:
http://dx.doi.org/10.1007/978-88-470-1506-7 113, 124
[5] S. Meddahi and V. Selgas, “An H-based FEM-BEM formulation for a time dependent eddy current problem,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1061–1083, 2008. 113, 124
[6] C. Ma, “The finite element analysis of a decoupled T-Φ scheme for solving eddycurrent problems,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 352–361, 2008. 113
[7] R. Acevedo, S. Meddahi, and R. Rodríguez, “An E-based mixed formulation for a time-dependent eddy current problem,” 2009. 113, 124
[8] T. Kang and K. Kim, “Fully discrete potential-based finite element methods for a transient eddy current problem,” Computing, vol. 85, no. 4, pp. 339–362, 2009. 113
[9] R. A. Prato Torres, E. P. Stephan, and F. Leydecker, “A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates,” Computing, vol. 88, no. 3-4, pp. 131–154, 2010. [Online]. Available: http://dx.doi.org/10.1007/s00607-010-0089-9 113
[10] R. Acevedo and S. Meddahi, “An E-based mixed FEM and BEM coupling for a time-dependent eddy current problem,” IMA Journal of Numerical Analysis, vol. 31, no. 2, pp. 667–697, 2011. 113, 124
[11] A. Bermúdez, R. López, R. Rodríguez, and P. Salgado, “Numerical solution of transient eddy current problems with input current intensities as boundary data,” IMA Journal of Numerical Analysis, vol. 32, pp. 1001–1029, 2012. 113
[12] J. Camaño and R. Rodríguez, “Analysis of a FEM-BEM model posed on the conducting domain for the time-dependent eddy current problem,” J. Comput. Appl. Math., vol. 236, no. 13, pp. 3084–3100, 2012. [Online]. Available:http://dx.doi.org/10.1016/j.cam.2012.01.030 113
[13] O. Bíró and K. Preis, “On the use of the magnetic vector potential in the finiteelement analysis of three-dimensional eddy currents,” Magnetics, IEEE Transactions on, vol. 25, no. 4, pp. 3145–3159, 1989. 113, 120, 121, 124
[14] O. Bíró and A. Valli, “The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: wellposedness and numerical approximation,” Comput. Methods Appl. Mech.
Engrg., vol. 196, no. 13-16, pp. 1890–1904, 2007. [Online]. Available: http://dx.doi.org/10.1016/j.cma.2006.10.008 113, 114, 121, 124, 127
[15] A. Bermúdez, R. Rodríguez, and P. Salgado, “FEM for 3D eddy current problems in bounded domains subject to realistic boundary conditions. An application to metallurgical electrodes,” Computer Methods in Applied Mechanics and Engineering, vol. 12, no. 1, pp. 67–114, 2005. 113, 124
[16] C. Yongbin, Y. Junyou, Y. Hainian, and T. Renyuan, “Study on eddy current losses and shielding measures in large power transformers,” Magnetics, IEEE Transactions on, vol. 30, no. 5, pp. 3068–3071, 1994. 113
[17] R. Acevedo and R. Rodríguez, “Analysis of the A, V-A- potential formulation for the eddy current problem in a bounded domain,” Electron. Trans. Numer. Anal., vol. 26, pp. 270–284, 2007. 114, 140
[18] P. Monk, Finite element methods for Maxwell’s equations, ser. Numerical Mathematics and Scientific Computation. New York: Oxford University Press, 2003. [Online]. Available: http://dx.doi.org/10.1093/acprof:oso/9780198508885.001.0001 115, 116, 118
[19] A. Buffa and P. Ciarlet Jr., “On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra,” Mathematical Methods in the Applied Sciences, vol. 24, no. 1, pp. 9–30, 2001. 116, 117
[20] ——, “On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications,” Mathematical Methods in the Applied Sciences, vol. 24, no. 1, pp. 31–48, 2001. 116
[21] A. Buffa, M. Costabel, and D. Sheen, “On traces for H(curl,Ω) in Lipschitz domains,” J. Math. Anal. Appl., vol. 276, no. 2, pp. 845–867, 2002. [Online]. Available: http://dx.doi.org/10.1016/S0022-47X(02)00455-9 116, 117
[22] E. Zeidler, Nonlinear functional analysis and its applications. II/A. New York: Springer-Verlag, 1990. [Online]. Available: http://dx.doi.org/10.1007/978-1-4612-0985-0 119
[23] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, ser. Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 1986, vol. 5. [Online]. Available:
http://dx.doi.org/10.1007/978-3-642-61623-5 121, 124, 129
[24] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, “Vector potentials in three-dimensional non-smooth domains,” Mathematical Methods in the Applied Sciences, vol. 21, no. 9, pp. 823–864, 1998. 124, 129
[25] D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed finite elements, compatibility conditions, and applications, ser. Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2008, vol. 1939. [Online]. Available: http://dx.doi.org/10.1007/978-3-540-78319-0 124
[26] A. Bermúdez, R. Rodríguez, and P. Salgado, “A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations,” SIAM J. Numer. Anal., vol. 40, no. 5, pp. 1823–1849, 2002. [Online]. Available: http://dx.doi.org/10.1137/S0036142901390780 124
[27] M. Zlámal, “Finite Element Solution of Quasistationary Nonlinear Magnetic Field,” Rairo-Analyse numérique, vol. 16, no. 2, pp. 161–191, 1982. 126, 127
[28] ——, “Addendum to the paper ’Finite element solution of quasistationary nonlinear magnetic field’,” RAIRO Anal. Numér., vol. 17, no. 4, pp. 407–415, 1983. 126, 127
[29] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, ser. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 1997, vol. 49. 127
[30] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. New York: Wiley, 2002. 127
[31] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, ser. Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 1994, vol. 23. 133
[32] P. G. Ciarlet, The finite element method for elliptic problems, ser. Classics in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2002, vol. 40. [Online]. Available:
http://dx.doi.org/10.1137/1.9780898719208 140
[33] M. Costabel and M. Dauge, “Singularities of electromagnetic fields in polyhedral domains,” Arch. Ration. Mech. Anal., vol. 151, no. 3, pp. 221–276, 2000. [Online]. Available: http://dx.doi.org/10.1007/s002050050197 140
[34] M. Costabel, M. Dauge, and S. Nicaise, “Singularities of eddy current problems,” M2AN Math. Model. Numer. Anal., vol. 37, no. 5, pp. 807–831, 2003. [Online]. Available: http://dx.doi.org/10.1051/m2an:2003056 140
[35] M. Costabel and M. Dauge, “Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements,” Numer. Math., vol. 93, no. 2, pp. 239–277, 2002. [Online]. Available:
http://dx.doi.org/10.1007/s002110100388 140
[36] O. Bíró, “Edge element formulations of eddy current problems,” Comput. Methods Appl. Mech. Engrg., vol. 169, no. 3-4, pp. 391–405, 1999. [Online]. Available: http://dx.doi.org/10.1016/S0045-7825(98)00165-0 140
[2] A. Bossavit, Computational electromagnetism, ser. Electromagnetism. San Diego, CA: Academic Press Inc., 1998. 112
[3] H. Ammari, A. Buffa, and J.-C. Nédélec, “A justification of eddy currents model for the Maxwell equations,” SIAM Journal on Applied Mathematics, vol. 60, no. 5, pp. 1805–1823, 2000. 113
[4] A. Alonso Rodríguez and A. Valli, Eddy current approximation of Maxwell equations, ser. MS&A. Modeling, Simulation and Applications. Springer-Verlag Italia, Milan, 2010, vol. 4. [Online]. Available:
http://dx.doi.org/10.1007/978-88-470-1506-7 113, 124
[5] S. Meddahi and V. Selgas, “An H-based FEM-BEM formulation for a time dependent eddy current problem,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1061–1083, 2008. 113, 124
[6] C. Ma, “The finite element analysis of a decoupled T-Φ scheme for solving eddycurrent problems,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 352–361, 2008. 113
[7] R. Acevedo, S. Meddahi, and R. Rodríguez, “An E-based mixed formulation for a time-dependent eddy current problem,” 2009. 113, 124
[8] T. Kang and K. Kim, “Fully discrete potential-based finite element methods for a transient eddy current problem,” Computing, vol. 85, no. 4, pp. 339–362, 2009. 113
[9] R. A. Prato Torres, E. P. Stephan, and F. Leydecker, “A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates,” Computing, vol. 88, no. 3-4, pp. 131–154, 2010. [Online]. Available: http://dx.doi.org/10.1007/s00607-010-0089-9 113
[10] R. Acevedo and S. Meddahi, “An E-based mixed FEM and BEM coupling for a time-dependent eddy current problem,” IMA Journal of Numerical Analysis, vol. 31, no. 2, pp. 667–697, 2011. 113, 124
[11] A. Bermúdez, R. López, R. Rodríguez, and P. Salgado, “Numerical solution of transient eddy current problems with input current intensities as boundary data,” IMA Journal of Numerical Analysis, vol. 32, pp. 1001–1029, 2012. 113
[12] J. Camaño and R. Rodríguez, “Analysis of a FEM-BEM model posed on the conducting domain for the time-dependent eddy current problem,” J. Comput. Appl. Math., vol. 236, no. 13, pp. 3084–3100, 2012. [Online]. Available:http://dx.doi.org/10.1016/j.cam.2012.01.030 113
[13] O. Bíró and K. Preis, “On the use of the magnetic vector potential in the finiteelement analysis of three-dimensional eddy currents,” Magnetics, IEEE Transactions on, vol. 25, no. 4, pp. 3145–3159, 1989. 113, 120, 121, 124
[14] O. Bíró and A. Valli, “The Coulomb gauged vector potential formulation for the eddy-current problem in general geometry: wellposedness and numerical approximation,” Comput. Methods Appl. Mech.
Engrg., vol. 196, no. 13-16, pp. 1890–1904, 2007. [Online]. Available: http://dx.doi.org/10.1016/j.cma.2006.10.008 113, 114, 121, 124, 127
[15] A. Bermúdez, R. Rodríguez, and P. Salgado, “FEM for 3D eddy current problems in bounded domains subject to realistic boundary conditions. An application to metallurgical electrodes,” Computer Methods in Applied Mechanics and Engineering, vol. 12, no. 1, pp. 67–114, 2005. 113, 124
[16] C. Yongbin, Y. Junyou, Y. Hainian, and T. Renyuan, “Study on eddy current losses and shielding measures in large power transformers,” Magnetics, IEEE Transactions on, vol. 30, no. 5, pp. 3068–3071, 1994. 113
[17] R. Acevedo and R. Rodríguez, “Analysis of the A, V-A- potential formulation for the eddy current problem in a bounded domain,” Electron. Trans. Numer. Anal., vol. 26, pp. 270–284, 2007. 114, 140
[18] P. Monk, Finite element methods for Maxwell’s equations, ser. Numerical Mathematics and Scientific Computation. New York: Oxford University Press, 2003. [Online]. Available: http://dx.doi.org/10.1093/acprof:oso/9780198508885.001.0001 115, 116, 118
[19] A. Buffa and P. Ciarlet Jr., “On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra,” Mathematical Methods in the Applied Sciences, vol. 24, no. 1, pp. 9–30, 2001. 116, 117
[20] ——, “On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications,” Mathematical Methods in the Applied Sciences, vol. 24, no. 1, pp. 31–48, 2001. 116
[21] A. Buffa, M. Costabel, and D. Sheen, “On traces for H(curl,Ω) in Lipschitz domains,” J. Math. Anal. Appl., vol. 276, no. 2, pp. 845–867, 2002. [Online]. Available: http://dx.doi.org/10.1016/S0022-47X(02)00455-9 116, 117
[22] E. Zeidler, Nonlinear functional analysis and its applications. II/A. New York: Springer-Verlag, 1990. [Online]. Available: http://dx.doi.org/10.1007/978-1-4612-0985-0 119
[23] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, ser. Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 1986, vol. 5. [Online]. Available:
http://dx.doi.org/10.1007/978-3-642-61623-5 121, 124, 129
[24] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, “Vector potentials in three-dimensional non-smooth domains,” Mathematical Methods in the Applied Sciences, vol. 21, no. 9, pp. 823–864, 1998. 124, 129
[25] D. Boffi, F. Brezzi, L. F. Demkowicz, R. G. Durán, R. S. Falk, and M. Fortin, Mixed finite elements, compatibility conditions, and applications, ser. Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2008, vol. 1939. [Online]. Available: http://dx.doi.org/10.1007/978-3-540-78319-0 124
[26] A. Bermúdez, R. Rodríguez, and P. Salgado, “A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations,” SIAM J. Numer. Anal., vol. 40, no. 5, pp. 1823–1849, 2002. [Online]. Available: http://dx.doi.org/10.1137/S0036142901390780 124
[27] M. Zlámal, “Finite Element Solution of Quasistationary Nonlinear Magnetic Field,” Rairo-Analyse numérique, vol. 16, no. 2, pp. 161–191, 1982. 126, 127
[28] ——, “Addendum to the paper ’Finite element solution of quasistationary nonlinear magnetic field’,” RAIRO Anal. Numér., vol. 17, no. 4, pp. 407–415, 1983. 126, 127
[29] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, ser. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 1997, vol. 49. 127
[30] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. New York: Wiley, 2002. 127
[31] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, ser. Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 1994, vol. 23. 133
[32] P. G. Ciarlet, The finite element method for elliptic problems, ser. Classics in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2002, vol. 40. [Online]. Available:
http://dx.doi.org/10.1137/1.9780898719208 140
[33] M. Costabel and M. Dauge, “Singularities of electromagnetic fields in polyhedral domains,” Arch. Ration. Mech. Anal., vol. 151, no. 3, pp. 221–276, 2000. [Online]. Available: http://dx.doi.org/10.1007/s002050050197 140
[34] M. Costabel, M. Dauge, and S. Nicaise, “Singularities of eddy current problems,” M2AN Math. Model. Numer. Anal., vol. 37, no. 5, pp. 807–831, 2003. [Online]. Available: http://dx.doi.org/10.1051/m2an:2003056 140
[35] M. Costabel and M. Dauge, “Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements,” Numer. Math., vol. 93, no. 2, pp. 239–277, 2002. [Online]. Available:
http://dx.doi.org/10.1007/s002110100388 140
[36] O. Bíró, “Edge element formulations of eddy current problems,” Comput. Methods Appl. Mech. Engrg., vol. 169, no. 3-4, pp. 391–405, 1999. [Online]. Available: http://dx.doi.org/10.1016/S0045-7825(98)00165-0 140
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mar 21, 2013
Article Details
Cómo citar
Acevedo, R., & Loaiza, G. (2013). Un esquema completamente discreto basado en elementos finitos para el problema de corrientes inducidas. Ingeniería Y Ciencia, 9(17), 111–145. https://doi.org/10.17230/ingciecia.9.17.6
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Universidad del Cauca
Biografía del autor/a
Ramiro Acevedo, Universidad del Cauca
Doctor en Ingeniería MatemáticaGerardo Loaiza, Universidad del Cauca
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