# American Institute of Mathematical Sciences

June  2015, 8(3): 579-605. doi: 10.3934/dcdss.2015.8.579

## Optimal control of magnetic fields in flow measurement

 1 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 2 ZHAW School of Engineering, Institut fur Angewandte Mathematik und Physik (IAMP), Technikumstrasse 9, Postfach, CH-8401 Winterthur, Switzerland 3 Technische Universität Berlin, Institut für Mathematik, Str. des 17. Juni 136, Sekr. MA 4-5, D-10623 Berlin, Germany

Received  November 2013 Revised  April 2014 Published  October 2014

Optimal control problems are considered for transient magnetization processes arising from electromagnetic flow measurement. The magnetic fields are generated by an induction coil and are defined in 3D spatial domains that include electrically conducting and nonconducting regions. Taking the electrical voltage in the coil as control, the state equation for the magnetic field and the electrical current generated in the induction coil is a system of integro-differential evolution Maxwell equations. The aim of the control is a fast transition of the magnetic field in the conduction region from an initial polarization to the opposite one. First-order necessary optimality condition and numerical methods of projected gradient type are discussed for associated optimal control problems. To deal with the extremely long computing times for this problem, model reduction by standard proper orthogonal decomposition is applied. Numerical tests are shown for a simplified geometry and for a 3D industrial application.
Citation: Serge Nicaise, Simon Stingelin, Fredi Tröltzsch. Optimal control of magnetic fields in flow measurement. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 579-605. doi: 10.3934/dcdss.2015.8.579
##### References:
 [1] K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition, in Shape Optimization & Optimal Design, Lect. Notes in Pure and Appl. Math., 216 (2001), 317-332. [2] K. Altmann, Numerische Verfahren der Optimalsteuerung von Magnetfeldern, Phd thesis, Technical University of Berlin, 2013. [3] K. Altmann, S. Stingelin and F. Tröltzsch, On some optimal control problems for electric circuits, International Journal of Circuit theory, 42 (2014), 808-830. doi: 10.1002/cta.1889. [4] F. Bachinger, U. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), 593-616. doi: 10.1007/s00211-005-0597-2. [5] G. Bärwolff and M. Hinze, Optimization of semiconductor melts, ZAMM Z. Angew. Math. Mech., 86 (2006), 423-437. doi: 10.1002/zamm.200410247. [6] P. E. Druet, O. Klein, J. Sprekels, F. Tröltzsch and I. Yousept, Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects, SIAM J. Control Optim., 49 (2011), 1707-1736. doi: 10.1137/090760544. [7] R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845. doi: 10.1137/050624236. [8] M. Gunzburger and C. Trenchea, Analysis and discretization of an optimal control problem for the time-periodic MHD equations, J. Math. Anal. Appl., 308 (2005), 440-466. doi: 10.1016/j.jmaa.2004.11.022. [9] M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158. doi: 10.1002/gamm.200790004. [10] M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, in Dimension reduction of large-scale systems, Lect. Notes Comput. Sci. Eng.Berlin, Springer, 45 (2005), 261-306. doi: 10.1007/3-540-27909-1_10. [11] D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117. doi: 10.1137/S0363012900375822. [12] L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows, Appl. Math. Optim., 32 (1995), 143-162. doi: 10.1007/BF01185228. [13] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, J. Comput. Phys., 128 (1996), 319-330. doi: 10.1006/jcph.1996.0213. [14] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. [15] M. Kolmbauer, The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems, Phd thesis, 2012. [16] M. Kolmbauer and U. Langer, A Robust Preconditioned MinRes Solver for Distributed Time-Periodic Eddy Current Optimal Control Problems, SIAM J. Sci. Comput., 34 (2012), B785-B809. doi: 10.1137/110842533. [17] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117-148. doi: 10.1007/s002110100282. [18] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numerical Analysis, 40 (2002), 492-515. doi: 10.1137/S0036142900382612. [19] C. Meyer, P. Philip and F. Tröltzsch, Optimal control of a semilinear {PDE} with nonlocal radiation interface conditions, SIAM J. Control Optimization, 45 (2006), 699-721. doi: 10.1137/040617753. [20] S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, To appear in Computational Methods in Applied Mathematics, (2014). doi: 10.1515/cmam-2014-0022. [21] S. Nicaise and F. Tröltzsch, A coupled Maxwell integrodifferential model for magnetization processes. Mathematische Nachrichten, 287 (2014), 432-452. doi: 10.1002/mana.201200206. [22] S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), 1369-1388 (electronic). doi: 10.1137/S1064827502400534. [23] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, 112, American Math. Society, Providence, 2010. [24] S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition, Lecture notes, Institute of Mathematics and Scientific Computing, University of Graz, 2007. [25] I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581. doi: 10.1007/s10589-011-9422-2. [26] I. Yousept and F. Tröltzsch., PDE-constrained optimization of time-dependent 3d electromagnetic induction heating by alternating voltages. ESAIM M2AN, 46 (2012), 709-729. doi: 10.1051/m2an/2011052.

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##### References:
 [1] K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition, in Shape Optimization & Optimal Design, Lect. Notes in Pure and Appl. Math., 216 (2001), 317-332. [2] K. Altmann, Numerische Verfahren der Optimalsteuerung von Magnetfeldern, Phd thesis, Technical University of Berlin, 2013. [3] K. Altmann, S. Stingelin and F. Tröltzsch, On some optimal control problems for electric circuits, International Journal of Circuit theory, 42 (2014), 808-830. doi: 10.1002/cta.1889. [4] F. Bachinger, U. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), 593-616. doi: 10.1007/s00211-005-0597-2. [5] G. Bärwolff and M. Hinze, Optimization of semiconductor melts, ZAMM Z. Angew. Math. Mech., 86 (2006), 423-437. doi: 10.1002/zamm.200410247. [6] P. E. Druet, O. Klein, J. Sprekels, F. Tröltzsch and I. Yousept, Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects, SIAM J. Control Optim., 49 (2011), 1707-1736. doi: 10.1137/090760544. [7] R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation, SIAM J. Control Optim., 45 (2006), 1822-1845. doi: 10.1137/050624236. [8] M. Gunzburger and C. Trenchea, Analysis and discretization of an optimal control problem for the time-periodic MHD equations, J. Math. Anal. Appl., 308 (2005), 440-466. doi: 10.1016/j.jmaa.2004.11.022. [9] M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces, GAMM-Mitt., 30 (2007), 149-158. doi: 10.1002/gamm.200790004. [10] M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, in Dimension reduction of large-scale systems, Lect. Notes Comput. Sci. Eng.Berlin, Springer, 45 (2005), 261-306. doi: 10.1007/3-540-27909-1_10. [11] D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening, Numer. Funct. Anal. Optim., 42 (2003), 1087-1117. doi: 10.1137/S0363012900375822. [12] L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows, Appl. Math. Optim., 32 (1995), 143-162. doi: 10.1007/BF01185228. [13] L. S. Hou and S. S. Ravindran, Computations of boundary optimal control problems for an electrically conducting fluid, J. Comput. Phys., 128 (1996), 319-330. doi: 10.1006/jcph.1996.0213. [14] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980. [15] M. Kolmbauer, The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems, Phd thesis, 2012. [16] M. Kolmbauer and U. Langer, A Robust Preconditioned MinRes Solver for Distributed Time-Periodic Eddy Current Optimal Control Problems, SIAM J. Sci. Comput., 34 (2012), B785-B809. doi: 10.1137/110842533. [17] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117-148. doi: 10.1007/s002110100282. [18] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numerical Analysis, 40 (2002), 492-515. doi: 10.1137/S0036142900382612. [19] C. Meyer, P. Philip and F. Tröltzsch, Optimal control of a semilinear {PDE} with nonlocal radiation interface conditions, SIAM J. Control Optimization, 45 (2006), 699-721. doi: 10.1137/040617753. [20] S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, To appear in Computational Methods in Applied Mathematics, (2014). doi: 10.1515/cmam-2014-0022. [21] S. Nicaise and F. Tröltzsch, A coupled Maxwell integrodifferential model for magnetization processes. Mathematische Nachrichten, 287 (2014), 432-452. doi: 10.1002/mana.201200206. [22] S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system, SIAM J. Sci. Comput., 26 (2005), 1369-1388 (electronic). doi: 10.1137/S1064827502400534. [23] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, 112, American Math. Society, Providence, 2010. [24] S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition, Lecture notes, Institute of Mathematics and Scientific Computing, University of Graz, 2007. [25] I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581. doi: 10.1007/s10589-011-9422-2. [26] I. Yousept and F. Tröltzsch., PDE-constrained optimization of time-dependent 3d electromagnetic induction heating by alternating voltages. ESAIM M2AN, 46 (2012), 709-729. doi: 10.1051/m2an/2011052.
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