owski (Université de
Nancy I, France),
H.-J. Spies (TU Bergakademie Freiberg), S. Volkwein
(Karl-Franzens-Universität Graz, Austria)
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[Contents] | [Index] |
Cooperation with: LASERVORM Volumen- und Oberflächenbearbeitung (Mittweida),
pro-beam HÖRMANN GmbH (Neukirchen), J. Sokoowski (Université de
Nancy I, France),
H.-J. Spies (TU Bergakademie Freiberg), S. Volkwein
(Karl-Franzens-Universität Graz, Austria)
Supported by: Stiftung Industrieforschung, Köln
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.
Here, the main focus was on a model for the electron radiation flux in electron beam hardening. Compared to the laser hardening technology, the electron beam technology is much more flexible and can easily generate a wealth of different radiation profiles. Figure 1 shows two typical situations. The radiation flux profile on the left is applied for tasks where the workpiece is moved in one direction. The profile on the right-hand side corresponds to a ring-shaped exposure without workpiece movement.
Based on pdelib, a new software for electron and laser beam surface hardening, called WIAS-SHarP, has been developed. It contains additional routines to describe the phase transition kinetics during a heat treatment cycle (cf. [2]). Moreover, it allows for quite general radiation flux profiles and the implementation of two independent beam traces. To facilitate its usage, a Java-based GUI has been developed (cf. Fig. 2).
The goal of surface hardening is to achieve a desired distribution of martensite in the boundary layers of a workpiece, avoiding a melting of the surface. This goal can easily be formulated in terms of an optimal control problem. As usual, the complete control problem with PDE constraints is too big to be treated numerically. Therefore, in [3] we have used a POD ansatz for model reduction. The results (so far only in 2-d) are quite satisfactory.
However, it turns out that the special structure of our control problem with a moving heat source, which is more or less a smeared-out point source, admits the derivation of a heuristic control strategy (cf. Fig. 3).
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:
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