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Optimal control of surface heat treatments

Collaborator: D. Hömberg , W. Weiss  

Cooperation with: LASERVORM Volumen- und Oberflächenbearbeitung (Mittweida), pro-beam HÖRMANN GmbH (Neukirchen), J. Soko\l 
owski (Université de Nancy I, France), H.-J. Spies (TU Bergakademie Freiberg), S. Volkwein (Karl-Franzens-Universität Graz, Austria)

Supported by: Stiftung Industrieforschung, Köln

Description:  

In most structural components in mechanical engineering there are surface parts which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect.

Depending on the respective heat source one can distinguish between different surface hardening procedures, the most important ones being       induction hardening and radiation treatments like laser and electron beam hardening.

A. Laser and electron beam hardening.

Last year's work in our industrial project on simulation and control of laser and electron beam hardening was concerned with aspects of modeling, software engineering, and optimal control.


 
Fig. 1: Typical radiation flux profiles in electron beam hardening 
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Fig. 2: Graphical user interface of the new software WIAS-SHarP (screenshot) 
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B. Induction hardening.

In [1] a rather complete mathematical treatment of induction hardening has been given. The resulting system of model equations consists of an elliptic equation for the scalar potential, a degenerate parabolic system for the magnetic vector potential, a quasistatic momentum balance coupled with a nonlinear stress-strain relation, and a nonlinear energy balance equation. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument.

Unfortunately, in most cases the geometry of the region to be hardened does not allow to have a simple annular inductor shape. But even when the principal topology of the inductor is already fixed, the coupling distance between inductor and workpiece and the spacing of the coil turns have to be adjusted carefully in order to obtain the desired heating or hardening pattern.  

This problem is also addressed in [1]. A major issue in this connection is to find a decent mathematical formulation of the design problem. We show that induction coils can conveniently be described as tubes, constructed from space curves. To investigate the shape sensitivity with respect to perturbations of the coil, we employ the speed method for an admissible velocity field. We prove the existence of strong material derivatives for the state variables. An application of the structure theorem then allows us to conclude that the shape gradient only depends on normal variations of the velocity field. This normal velocity component, however, can be computed from a perturbation of the curve. Therefore, we are able to give a necessary optimality condition in terms of perturbations of the curve.

Probably, this new procedure of characterizing the optimal configuration of tubes will admit further applications, for instance in optimal design problems related to the flow of liquids through pipelines.

References:

  1.  D. HÖMBERG, Induction heat treatments -- Modeling, analysis and optimal design of inductors, habilitation thesis, submitted in December 2001.
  2.   D. HÖMBERG, T. STRECKENBACH, W. WEISS, Phase transitions in steel -- Mathematical models and applications, in preparation.
  3.   D. HÖMBERG, S. VOLKWEIN, Suboptimal control of laser surface hardening using proper orthogonal decomposition, WIAS Preprint no. 639, 2001.



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LaTeX typesetting by I. Bremer
9/9/2002