Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects

Druet, Pierre-Etienne; Klein, Olaf; Sprekels, Jürgen; Tröltzsch, Fredi; Yousept, Irwin

doi:10.1137/090760544

Cited by 28 publications

(15 citation statements)

References 14 publications

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“…Optimal control of Maxwell's equations and coupled systems involving these have been subject to intensive research in the recent past. We only mention the work of Tröltzsch et al [16,[33][34][35]44] and Yousept [46][47][48][49][50]. However, most of these contributions deal with stationary or time harmonic Maxwell's equation.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control of the Inhomogeneous Relativistic Maxwell--Newton--Lorentz Equations

Meyer¹,

Schnepp²,

Thoma³

2016

SIAM J. Control Optim.

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Abstract. This note is concerned with an optimal control problem governed by the relativistic Maxwell-Newton-Lorentz equations, which describes the motion of charges particles in electro-magnetic fields and consists of a hyperbolic PDE system coupled with a nonlinear ODE. An external magnetic field acts as control variable. Additional control constraints are incorporated by introducing a scalar magnetic potential which leads to an additional state equation in form of a very weak elliptic PDE. Existence and uniqueness for the state equation is shown and the existence of a global optimal control is established. Moreover, first-order necessary optimality conditions in form of Karush-KuhnTucker conditions are derived. A numerical test illustrates the theoretical findings.

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Section: Introductionmentioning

confidence: 99%

Optimal Control of the Inhomogeneous Relativistic Maxwell--Newton--Lorentz Equations

Meyer¹,

Schnepp²,

Thoma³

2016

SIAM J. Control Optim.

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“…In two recent papers the simpler frequency domain situation of Joule heating has been studied. In [8] the Boccardo-Galluet approach has been used to prove existence of a weak solution, while in [4] new regularity results established in [3] have been used to prove existence and stability in the frequency domain setting. In [11,12] the eddy current model has been considered in the time domain.…”

Section: Introductionmentioning

confidence: 99%

Analysis and simulations of multifrequency induction hardening

Hömberg

Petzold

Rocca

2015

Nonlinear Analysis: Real World Applications

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We study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwell's equations coupled with an internal energy balance and an ODE for the volume fraction of austenite, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system.

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“…We mention Bodart et al [3] concerning a numerical study in a two-dimensional setting without pointwise state constraints. We also refer to [9,24] for the analysis of optimal control of 3D stationary induction heating problems (see also [25] for the numerical analysis of optimal control problems of Maxwell's equations). To our best knowledge, the mathematical analysis and the numerical treatment for optimal control of 3D induction heating problems with time-dependent temperature and pointwise state constraints have not been investigated in literature.…”

Section: Introductionmentioning

confidence: 99%

PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages

Tröltzsch

Yousept

2012

ESAIM: M2AN

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Abstract. 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. Mathematics Subject Classification.

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Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects

Cited by 28 publications

References 14 publications

Optimal Control of the Inhomogeneous Relativistic Maxwell--Newton--Lorentz Equations

Optimal Control of the Inhomogeneous Relativistic Maxwell--Newton--Lorentz Equations

Analysis and simulations of multifrequency induction hardening

PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages

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