Numéro
J. Phys. IV France
Volume 120, December 2004
Page(s) 325 - 335
DOI http://dx.doi.org/10.1051/jp4:2004120037


J. Phys. IV France 120 (2004) 325-335

DOI: 10.1051/jp4:2004120037

Optimal control strategies for the surface hardening of steel

D. Hömberg1, S. Volkwein2 and W. Weiss1

1  Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
2  Institut für Mathematik Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria


Abstract
We discuss control strategies for the surface hardening of steel with laser or electron beam. The goal is to acchieve a prescribed hardening depth avoiding surface melting. Our mathematical model consists of a system of ODEs for the phase volume fractions coupled with the heat equation. The system is solved semi-implicitely using the finite element method. For the optimal control we discuss two approaches: model reduction using POD (Proper Orthogonal Decomposition) and a feedback control of temperature. The numerical results prove that it is not sufficient to control the surface temperature in order to obtain a uniform hardening depth. Instead the best strategy should be to compute the optimal temperature in the hot spot of the beam by solving the control problem and use this temperature as the set-point for the pyrometer control of the real process.



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