DFG-Forschungszentrum Matheon Weierstrass Institute for Applied Analysis and Stochastics Technische Universität Berlin

Matheon Project C11: Modeling and optimization of phase transitions in steel

Factual information: duration, research team, collaboration, software, funding

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Duration April 2003 -- May 2014
Project leaders Michael Hintermüller (since May 2010)
Humboldt-University of Berlin
Department of Mathematics
Unter den Linden 6
10099 Berlin
phone: +49 (0)30 - 2093 2668 (office)
+49 (0)30 - 2093 5844 (secretary)

email: hint[at]math.hu-berlin.de

Dietmar Hömberg
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39
10117 Berlin
phone: +49 (0)30 - 20372 491 (office)
+49 (0)30 - 20372 555 (secretary)

email: dietmar.hoemberg[at]wias-berlin.de
Technical University of Berlin
Department of Mathematics
Strasse des 17. Juni 136
10623 Berlin
phone: +49 (0)30 - 31428 034 (office)
+49 (0)30 - 31479 687 (secretary)

Responsible Kevin Sturm
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39
10117 Berlin
phone: +49 (0)30 - 20372 413
email: kevin.sturm[at]wias-berlin.de
internal There are connections to C9 and C28 on numerical optimal control and to C18 and C26 on thermomechanical modeling of phase transitions in steel.
external W. Bleck, RWTH Aachen
C. Chelminski, Warschau
A. Fasano, Florenz
A. Khludnev, Novosibirsk
Th. Lübben, Bremen
H.-J. Pesch, Bayreuth
J. Sokolowski, Nancy
H.-J. Spies, Freiberg
S. Volkwein, Graz
M. Wolff, Bremen
M. Yamamoto, Tokyo
industrial BÄHR-Thermoanalyse
Endress + Hauser Flowtec AG, Reinach (Schweiz)
Photon Laser Engineering, Berlin
pro-beam Anlagenbau GmbH, München
Rücker AG
Software WIAS-SharP (Surface Hardening Program)
Support Project C11 is funded by DFG Research Center MATHEON "Mathematics for Key Technologies"


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Steel is a very important material that provides a lot of desirable properties. On the one hand, it is economically and ecologically advantageous due to its recycability. Moreover, although used as a mass product for more than 150 years, steel is an innovative material. For instance, more than 50% of the steel grades produced by ThyssenKrupp have been developed in the last 10 years.

One of the most important features of steel is the fact that one can change its mechanical properties by thermal manipulations. The reason for this behavior lies in occurring solid--solid phase transitions, i.e. changes in the crystal structure. This feature is utilized in the heat treatment of steel, which is a process of controlled heating and cooling to achieve a desired distribution of metallurgical phases corresponding to locally varying, heterogeneous physical properties.

During the last years, after a long period of steel being mainly subject to mere experimental investigations, the thermomechanical modeling of phase transitions in steel became an active research topic of physical metallurgy (cf., e.g., [1], [2], [3], [4]). Pioneering analytical results, both with respect to analysis of the resulting multi-field problems and related to optimal control problems can be found in [5], [6].

The results of the mathematical investigations are fruitful to industry as companies show a growing interest in simulation and control of heat treatments. First, the production of modern multiphase steels usually happens in a very narrow process window, which in turn necessitates a precise process control which can be achieved by simulation based knowledge. Secondly, to increase the degree of automation there is a strong tendency to include heat treatments in the process chain. Then the dimensional accuracy of workpieces is of paramount importance, there are a lot of tasks concerning distortion, i.e. undesired alterations in size and shape caused by heat treatments.

Further Information:


[1] Denis, S., Sjöström, S., Simon, A., Coupled temperature, stress, phase transformation calculation model numerical illustration of the internal stresses evolution during cooling of an eutectoid carbon steel cylinder Met. Trans., 18A, 1987, pp. 1203--1212.

[2] Fischer, F. D., Reisner, G., Werner, E., Tanaka, K., Cailletaud, G., Antretter, T., A new view on transformation induced plasticity (TRIP) Int. J. Plasticity, 16, 2000, pp. 723--748.

[3] Wolff, M., Böhm, M., Löwisch, G., Schmidt, A., Modelling and testing of transformation-induced plasticity and stress-dependent phase transformations in steel via simple experiments Comput. Materials Sci., 32, 2005, pp. 604--610.

[4] Wolff, M., Böhm, M., Helm, D., Material behavior of steel -- modeling of complex phenomena and thermodynamic consistency Int. J. Plasticity, 24, 2008, pp. 746--774.

[5] Hömberg, D., A mathematical model for induction hardening including mechanical effects Nonlinear Anal. Real World Appl., 5, 2004, pp. 55--90.

[6] Hömberg, D., Volkwein, S., Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition Math. Comput. Modelling, 38, 2003, pp. 1003--1028.

Achievements of the project

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    goto Derivation of a thermomechanical standard model for the phase transitions in steel that serves as the basis for numerical analysis as well as optimal control

    goto Mathematical analysis of the state system, existence of weak solutions, special cases in which uniqueness can be obtained

    goto Investigation of optimal control problems for thermomechanical problems

    goto Application to concrete heat treatments

    goto Investigation of laser welding and remelting

Current research

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The problem under consideration is given by the fact that internal stresses are accumulated during steel process chains (i.e. production of roller bearing rings) and released during the subsequent heat treatment. This may lead to undesired alterations in size and shape of the workpiece, so-called distortion. Heat treatment, as it often reveals the internal stresses, was first mistaken to be the origin of the distortion, but in fact, it is a possible remedy. We use optimal control theory to identify cooling strategies that compensate the unwanted shape alterations, either by controlling phase growth rates, using the so-called TrIP effect (Transformation induced plasticity), or by controlling the product phase mixture, using the phases' different expansion properties.

Cooperation with the institute for material science IWT Bremen


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  1. Anthonissen, M., Hömberg, D., Weiss, W., Real-time control of surface remelting. Progress in Industrial Mathematics at ECMI 2004, A. Di Bucchianico, R.M.M. Mattheij, M.A.Peletier, editors, volume 8 of Mathematics in Industry, Springer, 2005, pp. 356--360.

  2. Dreyer, W., Duderstadt, F., Hömberg, D., Modelling and simulation of capacitor impulse welding. W. Jäger and H.-J. Krebs, editors, Mathematics - Key Technology for the Future. Joint Projects Between Universities and Industry, Statusseminar zum BMBF-Fördergebiet "Ausgewählte Gebiete der Mathematik", Frankfurt am Main, 11.-12.12.2002, pages 233--242, Berlin/Heidelberg, 2003. Springer.

  3. Fuhrmann, J., Hömberg, D., Sokolowski, J., Modeling, simulation and control of laser heat treatments. K.-H. Hofmann, I. Lasiecka, G. Leugering, and J. Sprekels, editors, Optimal control of complex structures (Oberwolfach, 2000), volume 139 of Internat. Ser. Numer. Math., pages 71--82. Birkhäuser, Basel, 2002.

  4. Hömberg, D., Volkwein, S., Weiss, W., Optimal control strategies for the surface hardening of steel. Proceedings of the 2nd International Conference on Thermal Process Modelling and Computer Simulation, S. Denis, P. Archambault, J.-M. Bergheau, R. Fortunier, editors, volume 120 of Journal de Physique IV, EDP Sciences, 2004, pp. 325--335.

  5. Panizzi, L., Fasano, A., Hömberg, D., Modeling, analysis and simulations of case hardening of steel Progress in Industrial Mathematics at ECMI 2008, A. Fitt, J. Norbury, H. Ockendon, E. Wilson, eds., 15 of Mathematics in Industry, Springer, Berlin et al., 2010, pp. 965--970.

  6. Hömberg, D., Rott, O., Modeling, analysis and stability of milling processes including workpiece effects Progress in Industrial Mathematics at ECMI 2008, A. Fitt, J. Norbury, H. Ockendon, E. Wilson, eds., 15 of Mathematics in Industry, Springer, Berlin et al., 2010, pp. 493---498.

  7. Rasper, P., Rott, O., Hömberg, D., Uhlmann, E., Analysis of uncertainties in the stability prediction for milling processes Conference Proceedings, CIRP 2nd International Conference ``Process Machine Interactions'', June 10--11, 2010, Vancouver (CD), Y. Altintas, B. Denkena, Ch. Brecher, eds., 2010, pp. M18/1--M18/12.

Refereed publications:
  1. Alder, H., Hömberg, D., Weiss, W., Simulationsbasierte Regelung der Laserhärtung von Stahl. WIAS Preprint No. 1085 (2005); HTM Z. Werkst. Wärmebeh. Fertigung 61 (2006), pp. 103--108.

  2. Chelminski, K., Hömberg, D., Kern, D., On a thermomechanical model of phase transitions in steel. WIAS Preprint No. 1225 (2007); Adv. Math. Sci. Appl. 18 (2008), pp. 119--140.

  3. Fasano, A., Hömberg, D., Panizzi, L. A mathematical model for case hardening of steel WIAS Preprint No. 1283 (2007); Math. Models Methods Appl. Sci., 19 (2009), pp. 2101--2126.

  4. Hömberg, D., Khludnev, A., A thermoelastic contact problem with a phase transition. WIAS Preprint No. 914 (2004); IMA J. Appl. Math., 71 (2006), pp. 479--495.

  5. Hömberg, D., Sokolowski, J., Optimal shape design of inductor coils for induction hardening. SIAM Journal on Control and Optimization 42 (2003).

  6. Hömberg, D., A mathematical model for induction hardening including mechanical effects. Nonlinear Analysis: Real World Problems, 5 (2004), pp. 55--90.

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

  8. Hömberg, D., Weiss, W., PID-control of laser surface hardening of steel. WIAS Preprint No. 876 (2003); IEEE Trans. Control Syst. Technol., 14 (2006), pp. 896--904.

  9. Hömberg, D., Togobytska, N., Yamamoto, M., On the evaluation of dilatometer experiments. WIAS Preprint No. 1298 (2008); Appl. Anal., 88 (2009), pp. 669--681.

  10. Hömberg, D., Kern, D., The heat treatment of steel - a mathematical control problem. WIAS Preprint No. 1402 (2009); Materialwiss. Werkstofftech., 40 (2009), pp 438--442.

  11. Suwanpinij, P., Togobytska, N., Prahl, U., Weiss, W., Hömberg, D., Bleck, W., Numerical cooling strategy design for hot rolled dual phase steel WIAS Preprint No. 1507 (2010); Steel Res. Int., 81, 2010, pp. 1001---1009.

  12. Hömberg, D., Meyer, Ch., Rehberg, J. Ring, W., Optimal control for the thermistor problem WIAS Preprint No. 1363 (2008); SIAM J. Control Optim. 48, 2010, pp. 3449---3481.

  13. Fasano, A., Hömberg, D., Naumov, D., On a mathematical model for laser-induced thermotherapy, WIAS Preprint No. 1368 (2008); Appl. Math. Modelling, 34, 2010, pp. 3831---3840.

  14. Chelminski, K., Hömberg, Rott, O., Mathematical analysis of a thermomechanical milling process GAMM-Mitt., 34, 2011, pp. 59---63.

  15. Chelminski, K., Hömberg, Rott, O.: On a thermomechanical milling model WIAS Preprint No. 1361 (2008); Nonlinear Anal. Real World Appl., 12} (2011) pp. 615--632.

  16. Hömberg, D., Rocca, E., A model for resistance welding including phase transitions and Joule heating WIAS Preprint No. 1496 (2008); Math. Methods Appl. Sci., 34, 2011, pp. 2077---2088.

  17. Hömberg, D., Liu, J., Togobytska, N., Identification of the thermal growth characteristics of coagulated tumor tissue in laser-induced thermotherapy WIAS Preprint No. 1600 (2011); Math. Methods Appl. Sci., to appear.

  1. Seedorf, R., Regelungsalgorithmen für die Laserhärtung von Stahl. Diploma thesis, Technische Fachhochschule Berlin (2003).

  2. Seibold, M., Optimale Steuerung thermomechanischer Phasenübergänge in Stahl. Diploma thesis, TU Berlin (2004).

  3. Panizzi, L., A mathematical model for the case hardening of steel PhD thesis, TU Berlin and Scuola Normale Superiore, Pisa (2010).

  4. Kern, D., Analysis and numerics for a thermomechanical phase transition model in steel PhD thesis, TU Berlin (2011).

  5. Käbisch, S., Anwendung eines halbglatten Newtonverfahrens zur Auswertung von Dilatometerdaten Bachelor thesis, TU Berlin (2010).

  6. Sturm, K., On a visco-elastic milling model: existence, uniqueness and numerical analysis Diploma thesis, TU Berlin (2011).

last modified January 17, 2012