Abstract
In electromagnetics, the finite element method has become the most used tool to study several applications from transformers and rotating machines in low frequencies to antennas and photonic devices in high frequencies. Unfortunately, this approach usually leads to (very) large systems of equations and is thus very computationally demanding. This contribution compares three model order reduction techniques for the solution of nonlinear low frequency electromagnetic applications (in the so-called magnetoquasistatic regime) to efficiently reduce the number of equations—leading to smaller and faster systems to solve.
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References
Amsallem, D.: Interpolation on manifolds of CFD-based fluid and finite element-based structural reduced-order models for on-line aeroelastic predictions. Ph.D. dissertation, Stanford, USA (2010)
Astrid, P., et al.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control, 53(10), 2237–2251 (2008)
Barrault, M., et al.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004)
Bossavit, A.: Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego (1998)
Bui-Thanh, T., Damodaran, M., Willcox, K.E.: Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 42, 1505–1516 (2004)
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)
Clenet, S., et al.: Model order reduction of non-linear magnetostatic problems based on POD and DEI methods. IEEE Trans. Magn. 50, 33–36 (2014)
Cohen, A., DeVore, R.: Kolmogorov widths under holomorphic mappings. IMA J. Numer. Anal. (2015)
Dular, P., et al.: A general environment for the treatment of discrete problems and its application to the finite element method. IEEE Trans. Magn. 34(5), 3395–3398 (1998)
Gyselinck, J.: Twee-Dimensionale Dynamische Eindige-Elementenmodellering van Statische en Roterende Elektromagnetische Energieomzetters. PhD thesis (1999)
Gyselinck, J., et al.: Calculation of eddy currents and associated losses in electrical steel laminations. IEEE Trans. Magn. 35(3), 1191–1194 (1999)
Hiptmair, R., Xu, J.-C.: Nodal auxiliary space preconditioning for edge elements. In: 10th International Symposium on Electric and Magnetic Fields, France (2015)
Kolmogoroff, A.: Über die beste Annäaherung von Funktionen einer gegebenen Funktionen-klasse. Ann. Math. Second Ser. 37, 107–110 (1936)
Maday, Y., Patera, A., Turinici, G.: A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17, 437–446 (2002)
Montier, L., et al: Robust model order reduction of a nonlinear electrical machine at start-up through reduction error estimation. In: 10th International Symposium on Electric and Magnetic Fields, France (2015)
Paquay, Y., Brüls, O., Geuzaine, C.: Nonlinear interpolation on manifold of reduced order models in magnetodynamic problems. IEEE Trans. Magn. 52(3), 1–4 (2016)
Ryckelynck, D.: A priori hyperreduction method: an adaptive approach. J. Comput. Phys. 202(1), 346–366 (2005)
Schilders, W., et al.: Model Order Reduction: Theory, Research Aspects and Applications, vol. 13. Springer, Berlin (2008)
Sirovich, L.: Turbulence and the dynamics of coherent structures. Part I: Coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)
Sorensen, D. Private discussions (2015)
Volkwein, S.: Proper orthogonal decomposition and singular value decomposition. Universität Graz/Technische Universität Graz. SFB F003-Optimierung und Kontrolle (1999)
Zlatko, D., Gugercin, S.: A New Selection Operator for the Discrete Empirical Interpolation Method–improved a priori error bound and extensions. SIAM J. Sci. Comput. 38(5), A631–A648 (2016)
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This work was funded in part by the Belgian Science Policy under grant IAP P7-02 (Multiscale Modelling of Electrical Energy Systems) and by the F.R.S.—FNRS (Belgium).
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Paquay, Y., Brüls, O., Geuzaine, C. (2017). Model Order Reduction of Nonlinear Eddy Current Problems Using Missing Point Estimation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_27
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DOI: https://doi.org/10.1007/978-3-319-58786-8_27
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