[Next]:  Optimization problems related to industrial resource  
 [Up]:  Projects  
 [Previous]:  Optimization of diffractive structures and reconstruction  
 [Contents]   [Index] 

Optimal control of surface heat treatments

Collaborator: E. Bänsch (FG 3), M.H. Farshbaf Shaker, D. Hömberg, W. Weiss

Cooperation with: A. Khludnev (Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia), J. Soko\lowski (IECN/INRIA Lorraine, Vandoeuvre-lès-Nancy, France), F. Tröltzsch (Technische Universität Berlin), S. Volkwein (Karl-Franzens-Universität Graz, Austria)

Supported by: DFG-Forschungszentrum ``Mathematik für Schlüsseltechnologien'' (Research Center ``Mathematics for Key Technologies''), project C11


1. Thermomechanical models of phase transitions in steel (E. Bänsch, D. Hömberg, W. Weiss).

While the interplay between temperature and phase volume fractions is well understood, the incorporation of mechanical effects is still a challenging task. The metallurgical phases have material parameters with different thermal characteristics, hence their effective values have to be computed by a mixture ansatz. The different densities of the metallurgical phases result in a different thermal expansion. This thermal and transformation strain is the major contribution to the evolution of internal stresses during heat treatments. Experiments with phase transformations under applied loading show an additional irreversible deformation even when the equivalent stress corresponding to the load is far below the normal yield stress. This effect is called transformation-induced plasticity. The irreversible deformation leads to a mechanical dissipation that acts as a source term in the energy balance.

Neglecting the influence of internal stresses on the transformation kinetics, a consistent mathematical model which takes care of all these effects has been developed and analyzed in [1]. The new model has been implemented in an existing adaptive finite element code, [9]. In a simplified situation without inelastic dissipation term in the energy balance, an optimal control problem has been investigated, [8].

In [2] a thermoelastic contact problem with phase transitions has been studied.

2. Optimal control of laser surface treatments (M.H. Farshbaf Shaker, D. Hömberg, W. Weiss).

Besides of the reduced order approach to tackle the optimal control problem of laser surface hardening, which has been considered in [4, 5], the emphasis of last year's work lay on the development of new nonlinear PID control algorithms for two- and three-dimensional situations. The results are published in [3] and [7].

Figure 1 shows the result of a simulation with constant laser energy (top) in comparison with the application of a linear PID control of subsurface temperature (bottom left) and the result of a simulation with the new nonlinear PID algorithm (bottom right).

Fig. 1: Control of laser surface hardening to achieve a constant hardening depth:
uncontrolled (top); linear PID subsurface control (bottom left);
nonlinear PID subsurface control (bottom right).

3. Laser-induced thermotherapy (M.H. Farshbaf Shaker, D. Hömberg, W. Weiss).

Fig. 2: Mechanism of laser thermotherapy

In a first step towards life sciences applications we have started to apply our laser hardening model to the case of laser-induced thermotherapy (Fig. 2). This is a cancer therapy in which laser light is guided through a transparent catheter into a tumor. The absorbed light leads to a heating of the tissue. In contrast to hyperthermia cancer treatments where the temperature does not exceed, say, 43 oC, in this process the tissue is heated up to more than 60 oC leading to a coagulation of the tumor tissue and thus a destruction of the tumor.

In [6] our software WIAS-SHarP has been used to simulate this process using an Arrhenius ansatz to model the tissue damage due to coagulation.

Future work will concern improved models for tissue damage and the investigation of optimal control problems related to therapy planning and the design of applicators.


  1. D. HÖMBERG, A mathematical model for induction hardening including mechanical effects, Nonlinear Anal., Real World Appl., 5 (2004), pp. 55-90.

  2. D. HÖMBERG, A. KHLUDNEV, A thermoelastic contact problem with a phase transition, in preparation.

  3. D. HÖMBERG, W. WEISS, PID-control of laser surface hardening of steel, WIAS Preprint no. 876, 2003.

  4. D. HÖMBERG, S. VOLKWEIN, W. WEISS, Optimal control strategies for the surface hardening of steel, to appear in: J. Physique IV.

  5. D. HÖMBERG, S. VOLKWEIN, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition, Math. Comput. Modelling, 38 (2003), pp. 1003-1028.

  6. A. JÄTZSCHMANN, Ein Simulator für die laserinduzierte Thermotherapie, diploma thesis, Technische Fachhochschule Berlin, 2003.

  7. R. SEEDORF, Regelungsalgorithmen für die Laserhärtung von Stahl, diploma thesis, Technische Fachhochschule Berlin, 2003.

  8. M. SEIBOLD, Optimale Steuerung thermomechanischer Phasenübergänge in Stahl, diploma thesis, in preparation.

  9. M. WOLF ET AL., Modellierung der Abkühlung von Stahlbrammen, Berichte aus der Technomathematik no. 00-07, Universität Bremen, 2000.

 [Next]:  Optimization problems related to industrial resource  
 [Up]:  Projects  
 [Previous]:  Optimization of diffractive structures and reconstruction  
 [Contents]   [Index] 

LaTeX typesetting by I. Bremer