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 fur Angewandte Mathematik und Physik (IAMP), Technikumstrasse 9, Postfach, CH-8401 Winterthur, Switzerland

3. 

Technische Universität Berlin, Institut für 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 & 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, 216 (2001), 317. Google Scholar

[2]

K. Altmann, Numerische Verfahren der Optimalsteuerung von Magnetfeldern,, Phd thesis, (2013). Google Scholar

[3]

K. Altmann, S. Stingelin and F. Tröltzsch, On some optimal control problems for electric circuits,, International Journal of Circuit theory, 42 (2014), 808. doi: 10.1002/cta.1889. Google Scholar

[4]

F. Bachinger, U. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems,, Numer. Math., 100 (2005), 593. doi: 10.1007/s00211-005-0597-2. Google Scholar

[5]

G. Bärwolff and M. Hinze, Optimization of semiconductor melts,, ZAMM Z. Angew. Math. Mech., 86 (2006), 423. doi: 10.1002/zamm.200410247. Google Scholar

[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. doi: 10.1137/090760544. Google Scholar

[7]

R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation,, SIAM J. Control Optim., 45 (2006), 1822. doi: 10.1137/050624236. Google Scholar

[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. doi: 10.1016/j.jmaa.2004.11.022. Google Scholar

[9]

M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces,, GAMM-Mitt., 30 (2007), 149. doi: 10.1002/gamm.200790004. Google Scholar

[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, 45 (2005), 261. doi: 10.1007/3-540-27909-1_10. Google Scholar

[11]

D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening,, Numer. Funct. Anal. Optim., 42 (2003), 1087. doi: 10.1137/S0363012900375822. Google Scholar

[12]

L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows,, Appl. Math. Optim., 32 (1995), 143. doi: 10.1007/BF01185228. Google Scholar

[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. doi: 10.1006/jcph.1996.0213. Google Scholar

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications,, Academic Press, (1980). Google Scholar

[15]

M. Kolmbauer, The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems,, Phd thesis, (2012). Google Scholar

[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). doi: 10.1137/110842533. Google Scholar

[17]

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems,, Numer. Math., 90 (2001), 117. doi: 10.1007/s002110100282. Google Scholar

[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. doi: 10.1137/S0036142900382612. Google Scholar

[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. doi: 10.1137/040617753. Google Scholar

[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. Google Scholar

[21]

S. Nicaise and F. Tröltzsch, A coupled Maxwell integrodifferential model for magnetization processes., Mathematische Nachrichten, 287 (2014), 432. doi: 10.1002/mana.201200206. Google Scholar

[22]

S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system,, SIAM J. Sci. Comput., 26 (2005), 1369. doi: 10.1137/S1064827502400534. Google Scholar

[23]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications,, 112, (2010). Google Scholar

[24]

S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition,, Lecture notes, (2007). Google Scholar

[25]

I. Yousept, Optimal control of Maxwell's equations with regularized state constraints,, Comput. Optim. Appl., 52 (2012), 559. doi: 10.1007/s10589-011-9422-2. Google Scholar

[26]

I. Yousept and F. Tröltzsch., PDE-constrained optimization of time-dependent 3d electromagnetic induction heating by alternating voltages., ESAIM M2AN, 46 (2012), 709. doi: 10.1051/m2an/2011052. Google Scholar

show all references

References:
[1]

K. Afanasiev and M. Hinze, Adaptive control of a wake flow using proper orthogonal decomposition,, in Shape Optimization & Optimal Design, 216 (2001), 317. Google Scholar

[2]

K. Altmann, Numerische Verfahren der Optimalsteuerung von Magnetfeldern,, Phd thesis, (2013). Google Scholar

[3]

K. Altmann, S. Stingelin and F. Tröltzsch, On some optimal control problems for electric circuits,, International Journal of Circuit theory, 42 (2014), 808. doi: 10.1002/cta.1889. Google Scholar

[4]

F. Bachinger, U. Langer and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems,, Numer. Math., 100 (2005), 593. doi: 10.1007/s00211-005-0597-2. Google Scholar

[5]

G. Bärwolff and M. Hinze, Optimization of semiconductor melts,, ZAMM Z. Angew. Math. Mech., 86 (2006), 423. doi: 10.1002/zamm.200410247. Google Scholar

[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. doi: 10.1137/090760544. Google Scholar

[7]

R. Griesse and K. Kunisch, Optimal control for a stationary MHD system in velocity-current formulation,, SIAM J. Control Optim., 45 (2006), 1822. doi: 10.1137/050624236. Google Scholar

[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. doi: 10.1016/j.jmaa.2004.11.022. Google Scholar

[9]

M. Hinze, Control of weakly conductive fluids by near wall Lorentz forces,, GAMM-Mitt., 30 (2007), 149. doi: 10.1002/gamm.200790004. Google Scholar

[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, 45 (2005), 261. doi: 10.1007/3-540-27909-1_10. Google Scholar

[11]

D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening,, Numer. Funct. Anal. Optim., 42 (2003), 1087. doi: 10.1137/S0363012900375822. Google Scholar

[12]

L. S. Hou and A. J. Meir, Boundary optimal control of MHD flows,, Appl. Math. Optim., 32 (1995), 143. doi: 10.1007/BF01185228. Google Scholar

[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. doi: 10.1006/jcph.1996.0213. Google Scholar

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications,, Academic Press, (1980). Google Scholar

[15]

M. Kolmbauer, The Multiharmonic Finite Element and Boundary Element Method for Simulation and Control of Eddy Current Problems,, Phd thesis, (2012). Google Scholar

[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). doi: 10.1137/110842533. Google Scholar

[17]

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems,, Numer. Math., 90 (2001), 117. doi: 10.1007/s002110100282. Google Scholar

[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. doi: 10.1137/S0036142900382612. Google Scholar

[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. doi: 10.1137/040617753. Google Scholar

[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. Google Scholar

[21]

S. Nicaise and F. Tröltzsch, A coupled Maxwell integrodifferential model for magnetization processes., Mathematische Nachrichten, 287 (2014), 432. doi: 10.1002/mana.201200206. Google Scholar

[22]

S. S. Ravindran, Real-time computational algorithm for optimal control of an MHD flow system,, SIAM J. Sci. Comput., 26 (2005), 1369. doi: 10.1137/S1064827502400534. Google Scholar

[23]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications,, 112, (2010). Google Scholar

[24]

S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition,, Lecture notes, (2007). Google Scholar

[25]

I. Yousept, Optimal control of Maxwell's equations with regularized state constraints,, Comput. Optim. Appl., 52 (2012), 559. doi: 10.1007/s10589-011-9422-2. Google Scholar

[26]

I. Yousept and F. Tröltzsch., PDE-constrained optimization of time-dependent 3d electromagnetic induction heating by alternating voltages., ESAIM M2AN, 46 (2012), 709. doi: 10.1051/m2an/2011052. Google Scholar

[1]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[2]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217

[3]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537

[4]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

[5]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57

[6]

Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261

[7]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[8]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[9]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[10]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885

[11]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022

[12]

Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119

[13]

Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019042

[14]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

[15]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[16]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517

[17]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069

[18]

Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17

[19]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

[20]

Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (7)

[Back to Top]