PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 709-729.

This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired temperature can be optimally achieved. As there are finitely many control parameters but the state constraint has to be satisfied in an infinite number of points, the problem belongs to a class of semi-infinite programming problems. We present a rigorous analysis of the optimization problem and a numerical strategy based on our theoretical result.

DOI : 10.1051/m2an/2011052
Classification : 49J20, 78A25, 78A30, 35K40, 90C48
Mots clés : PDE-constrained optimization, electromagnetic induction heating, 3D time-variant heat equations, time-harmonic eddy current equations, pointwise state constraints, optimality conditions
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     title = {PDE-constrained optimization of time-dependent {3D} electromagnetic induction heating by alternating voltages},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {709--729},
     publisher = {EDP-Sciences},
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Tröltzsch, Fredi; Yousept, Irwin. PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 709-729. doi : 10.1051/m2an/2011052. http://archive.numdam.org/articles/10.1051/m2an/2011052/

[1] A. Alonso and A. Valli, Eddy Current Approximation of Maxwell Equations : Theory, Algorithms and Applications. Springer (2010). | MR | Zbl

[2] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823-864. | MR | Zbl

[3] O. Bodart, A.V. Boureau and R. Touzani, Numerical investigation of optimal control of induction heating processes. Appl. Math. Modelling 25 (2001) 697-712. | Zbl

[4] F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer-Verlag, New York (2000). | MR | Zbl

[5] A. Bossavit and J.-F. Rodrigues, On the electromagnetic induction heating problem in bounded domains. Adv. Math. Sci. Appl. 4 (1994) 79-92. | MR | Zbl

[6] E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297-1327. | MR | Zbl

[7] S. Clain and R. Touzani, A two-dimensional stationary induction heating problem. Math. Methods Appl. Sci. 20 (1997) 759-766. | MR | Zbl

[8] S. Clain, J. Rappaz, M. Swierkosz and R. Touzani, Numerical modelling of induction heating for two-dimensional geometries. Math. Models Methods Appl. Sci. 3 (1993) 805-7822. | MR | Zbl

[9] 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. | MR | Zbl

[10] J.A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Adv. Differ. Equ. 12 (2007) 1031-1078. | MR | Zbl

[11] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

[12] D. Hömberg, Induction hardening of steel - modeling, analysis and optimal design of inductors. Habilitation thesis, TU Berlin (2001).

[13] D. Hömberg, A mathematical model for induction hardening including mechanical effects. Nonlin. Anal. Real World Appl. 5 (2004) 55-90. | MR

[14] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod, Paris 1-3 (1968). | Zbl

[15] A.C. Metaxas, Foundations of Electroheat : A Unified Approach. Wiley (1996). | MR

[16] P. Monk, Finite element methods for Maxwell's equations. Clarendon press, Oxford (2003). | MR | Zbl

[17] J.C. Nédélec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315-341. | Zbl

[18] J. Nocedal and S.J. Wright, Numerical Optimization. Springer-Verlag, New York (1999). | MR | Zbl

[19] C. Parietti and J. Rappaz, A quasi-static two-dimensional induction heating problem. I : Modelling and analysis. Math. Models Methods Appl. Sci. 8 (1998) 1003-1021. | MR | Zbl

[20] C. Parietti and J. Rappaz, A quasi-static two-dimensional induction heating problem II. numerical analysis. Math. Models Methods Appl. Sci. 9 (1999) 1333-1350. | MR | Zbl

[21] J. Rappaz and M. Swierkosz, Mathematical modelling and numerical simulation of induction heating processes. Appl. Math. Comput. Sci. 6 (1996) 207-221. | MR | Zbl

[22] F. Tröltzsch, Optimal control of partial differential equations, Graduate Studies in Mathematics. American Mathematical Society, Providence, RI 112 (2010). | Zbl

[23] D. Wachsmuth and A. Rösch, How to check numerically the sufficient optimality conditions for infinite-dimensional optimization problems, in Optimal control of coupled systems of partial differential equations, Internat. Ser. Numer. Math. Birkhäuser Verlag, Basel 158 (2009) 297-317. | MR | Zbl

[24] I. Yousept, Optimal control of a nonlinear coupled electromagnetic induction heating system with pointwise state constraints. Mathematics and its Applications/Annals of AOSR 2 (2010) 45-77. | MR | Zbl

[25] I. Yousept, Optimal control of Maxwell's equations with regularized state constraints. Comput. Optim. Appl. (2011) DOI: 10.1007/s10589-011-9422-2. | MR | Zbl

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