Optimal Control of Inductive Heating of Ferromagnetic Materials

  • Inductive heating is a technological process where a steel workpiece is surrounded by an electromagnetic coil to which currents at various frequencies and time-varying amplitudes are applied. The amplitudes are considered as the controls and the objective is to heat the workpiece up to a desired temperature profile at the final time of the heating process. The workpiece is then quenched which due to a phase transition in the crystallographic structure of the steel leads to a hardening of the surface of the workpiece. For the inductive heating process, the state equations represent a coupled system of nonlinear partial differential equations consisting of the eddy currents equations in the coil, the workpiece, and the surrounding air, and a heat equation in the workpiece. The nonlinearity stems from the temperature dependent nonlinear material laws for steel both with regard to its electromagnetic and thermal behavior. Following the principle 'Discretize first, then optimize', weInductive heating is a technological process where a steel workpiece is surrounded by an electromagnetic coil to which currents at various frequencies and time-varying amplitudes are applied. The amplitudes are considered as the controls and the objective is to heat the workpiece up to a desired temperature profile at the final time of the heating process. The workpiece is then quenched which due to a phase transition in the crystallographic structure of the steel leads to a hardening of the surface of the workpiece. For the inductive heating process, the state equations represent a coupled system of nonlinear partial differential equations consisting of the eddy currents equations in the coil, the workpiece, and the surrounding air, and a heat equation in the workpiece. The nonlinearity stems from the temperature dependent nonlinear material laws for steel both with regard to its electromagnetic and thermal behavior. Following the principle 'Discretize first, then optimize', we consider a semi-discretization in time by the implicit Euler scheme which leads to a discrete-time optimal control problem. We prove the existence of a minimizer for the discrete-time optimal control problem and derive the first order necessary optimality conditions.show moreshow less

Download full text files

Export metadata

Statistics

Number of document requests

Additional Services

Share in Twitter Search Google Scholar
Metadaten
Author:Oleg Boyarkin, Ronald H. W. HoppeGND
URN:urn:nbn:de:bvb:384-opus4-23389
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/2338
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2013-10)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Release Date:2013/05/06
Tag:optimal control; inductive heating; eddy currents equations; heat equation
GND-Keyword:Induktive Erwärmung; Wärmeleitungsgleichung; Wirbelstrom; Mathematische Modellierung; Optimale Kontrolle; Implizites Euler-Verfahren
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht mit Print on Demand