American Institute of Mathematical Sciences

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

Optimal control of magnetic fields in flow measurement

Serge Nicaise 1, , Simon Stingelin 2, and  Fredi Tröltzsch 3,

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.
Keywords: numerical solution, optimal control, induction law, Evolution Maxwell equations, vector potential formulation, adjoint equation, proper orthogonal decomposition., model reduction, degenerate parabolic equation, integro-differential equation.
Mathematics Subject Classification: Primary: 49K20, 35K565; Secondary: 49M05, 78A2.
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, Lect. Notes in Pure and Appl. Math., 216 (2001), 317-332.  Google Scholar

[2]

K. Altmann, Numerische Verfahren der Optimalsteuerung von Magnetfeldern, Phd thesis, Technical University of Berlin, 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), 808-830. 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), 593-616. 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), 423-437. 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), 1707-1736. 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), 1822-1845. 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), 440-466. 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), 149-158. 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, Lect. Notes Comput. Sci. Eng.Berlin, Springer, 45 (2005), 261-306. 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), 1087-1117. 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), 143-162. 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), 319-330. doi: 10.1006/jcph.1996.0213.  Google Scholar

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 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), B785-B809. doi: 10.1137/110842533.  Google Scholar

[17]

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117-148. 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), 492-515. 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), 699-721. 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), 432-452. 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), 1369-1388 (electronic). doi: 10.1137/S1064827502400534.  Google Scholar

[23]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, 112, American Math. Society, Providence, 2010.  Google Scholar

[24]

S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition, Lecture notes, Institute of Mathematics and Scientific Computing, University of Graz, 2007. Google Scholar

[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.  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), 709-729. 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, Lect. Notes in Pure and Appl. Math., 216 (2001), 317-332.  Google Scholar

[2]

K. Altmann, Numerische Verfahren der Optimalsteuerung von Magnetfeldern, Phd thesis, Technical University of Berlin, 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), 808-830. 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), 593-616. 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), 423-437. 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), 1707-1736. 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), 1822-1845. 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), 440-466. 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), 149-158. 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, Lect. Notes Comput. Sci. Eng.Berlin, Springer, 45 (2005), 261-306. 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), 1087-1117. 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), 143-162. 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), 319-330. doi: 10.1006/jcph.1996.0213.  Google Scholar

[14]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 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), B785-B809. doi: 10.1137/110842533.  Google Scholar

[17]

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), 117-148. 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), 492-515. 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), 699-721. 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), 432-452. 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), 1369-1388 (electronic). doi: 10.1137/S1064827502400534.  Google Scholar

[23]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, 112, American Math. Society, Providence, 2010.  Google Scholar

[24]

S. Volkwein, Model Reduction Using Proper Orthogonal Decomposition, Lecture notes, Institute of Mathematics and Scientific Computing, University of Graz, 2007. Google Scholar

[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.  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), 709-729. 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, 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, 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]

Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021, 11 (3) : 579-599. doi: 10.3934/mcrf.2021013
[8]

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
[9]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[10]

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
[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]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
[13]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885
[14]

Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021021
[15]

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
[16]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046
[17]

Victor Ginting. An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, 2021, 29 (5) : 3405-3427. doi: 10.3934/era.2021045
[18]

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, 9 (4) : 433-444. doi: 10.3934/naco.2019042
[19]

Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053
[20]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3351-3386. doi: 10.3934/dcdss.2020440

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy