Thermomechanical models are the basis for the description of numerous technological processes. The consideration of phase transitions and of inelastic constitutive laws leads to coupled new systems of partial differential equations. Up to now a qualitative solution theory for these systems is available only in special cases.
A further important question is the identification of unknown materials parameters. Typically, quantities like heat conductivity or the latent heat released or consumed during phase transitions do not only depend on temperature but also on the respective phase. In stock removal processes modelling and identification of toolinduced heating of the workpiece come along.
These tasks are dealt with in collaboration with partners from engineering sciences and from industry. An important challenge is the further interdisciplinary development of measurement techniques based on the mathematical investigations.
Publications
Monographs

S. Rodt, P.I. Schneider, L. Zschiedrich, T. Heindel, S. Bounouar, M. Kantner, Th. Koprucki, U. Bandelow, S. Burger, S. Reitzenstein, Deterministic quantum devices for optical quantum communication, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 285359, (Chapter Published), DOI 10.1007/9783030356569_8 .
Abstract
Photonic quantum technologies are based on the exchange of information via single photons. The information is typically encoded in the polarization of the photons and security is ensured intrinsically via principles of quantum mechanics such as the nocloning theorem. Thus, all optical quantum communication networks rely crucially on the availability of suitable quantumlight sources. Such light sources with close to ideal optical and quantum optical properties can be realized by selfassembled semiconductor quantum dots. These highquality nanocrystals are predestined singlephoton emitters due to their quasi zerodimensional carrier confinement. Still, the development of practical quantumdotbased sources of single photons and entangledphoton pairs for applications in photonic quantum technology and especially for the quantumrepeater scheme is very demanding and requires highly advanced device concepts and deterministic fabrication technologies. This is mainly explained by their random position and emission energy as well as by the low photonextraction efficiency in simple planar device configurations. 
M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Multidimensional modeling and simulation of semiconductor nanophotonic devices, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 241283, (Chapter Published), DOI 10.1007/9783030356569_7 .
Abstract
Selfconsistent modeling and multidimensional simulation of semiconductor nanophotonicdevices is an important tool in the development of future integrated light sources and quantumdevices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically exploretheir optimization potential. The efficient implementation of multidimensional numerical simulationsfor computeraided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in devicescale modeling of quantum dot based singlephotonsources and laser diodes by selfconsistently coupling the optical Maxwell equations with semiclassical carrier transport models using semiclassical and fully quantum mechanical descriptionsof the optically active region, respectively. For the simulation of realistic devices with complex,multidimensional geometries, we have developed a novel hpadaptive finite element approachfor the optical Maxwell equations, using mixed meshes adapted to the multiscale properties ofthe photonic structures. For electrically driven devices, we introduced novel discretization andparameterembedding techniques to solve the driftdiffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on variousapplications, including verticalcavity surfaceemitting lasers, grating couplers and singlephotonsources. 
L. Ghezzi, D. Hömberg, Ch. Landry, eds., Math for the Digital Factory, 27 of Mathematics in Industry / The European Consortium for Mathematics in Industry, Springer International Publishing AG, Cham, 2017, x+348 pages, (Collection Published), DOI 10.1007/9783319639574 .

P. Colli, G. Gilardi, D. Hömberg, E. Rocca, eds., Special Issue dedicated to Jürgen Sprekels on the Occasion of his 65th Birthday, 35, no. 6 of Discrete Contin. Dyn. Syst. Ser. A, American Institute of Mathematical Sciences, Springfield, 2015, 472 pages, (Collection Published).

P. Deuflhard, M. Grötschel, D. Hömberg, U. Horst, J. Kramer, V. Mehrmann, K. Polthier, F. Schmidt, Ch. Schütte, M. Skutella, J. Sprekels, eds., MATHEON  Mathematics for Key Technologies, 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, 453 pages, (Collection Published).

B. Denkena, D. Hömberg, E. Uhlmann, Mathematik für Werkzeugmaschinen und Fabrikautomatisierung, in: Produktionsfaktor Mathematik. Wie Mathematik Technik und Wirtschaft bewegt, M. Grötschel, K. Lucas, V. Mehrmann, eds., acatech diskutiert, acatech, Springer, Berlin, Heidelberg, 2008, pp. 279299, (Chapter Published).
Articles in Refereed Journals

D. Hömberg, K. Krumbiegel, N. Togobytska, Optimal control of multiphase steel production, Journal of Mathematics in Industry, 9 (2019), pp. 132, DOI 10.1186/s133620190063x .
Abstract
An optimal control problem for the production of multiphase steel is investigated, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the ferrite phase fraction. The optimal control problem is analyzed and the firstorder necessary and secondorder sufficient optimality conditions are derived. For the numerical solution of the control problem reduced sequential quadratic programming (rSQP) method with a primaldual active set strategy (PDAS) was applied. The numerical results were presented for the optimal control of a cooling line for production of hot rolled MoMn dual phase steel. 
D. Hömberg, S. Lu, M. Yamamoto, Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by onepoint Dirichlet data, Journal of Differential Equations, 266 (2019), pp. 75257544, DOI 10.1016/j.jde.2018.12.004 .
Abstract
We consider an inverse problem arising in laserinduced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a onepoint measurement. 
G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rateindependent damage in thermoviscoelastic materials with inertia, Journal of Dynamics and Differential Equations, 30 (2018), pp. 13111364, DOI 10.1007/s108840189666y .
Abstract
We present a model for rateindependent, unidirectional, partial damage in viscoelastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rateindependent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the AmbrosioTortorelli phasefield model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled timediscrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rateindependent limit model for displacements and damage, which is independent of temperature. 
CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperaturedependent phasefield model for phase separation and damage, Archive for Rational Mechanics and Analysis, 225 (2017), pp. 177247.
Abstract
In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for CahnHilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461490] in the framework of FourierNavierStokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 13451369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 25192586] for the study of PDE systems for phase transition and damage. Our globalintime existence result is obtained by passing to the limit in a carefully devised timediscretization scheme. 
D. Hömberg, F.S. Patacchini, K. Sakamoto, J. Zimmer, A revisited JohnsonMehlAvramiKolmogorov model and the evolution of grainsize distributions in steel, IMA Journal of Applied Mathematics, 82 (2017), pp. 763780, DOI 10.1093/imamat/hxx012 .
Abstract
The classical JohnsonMehlAvramiKolmogorov approach for nucleation and growth models of diffusive phase transitions is revisited and applied to model the growth of ferrite in multiphase steels. For the prediction of mechanical properties of such steels, a deeper knowledge of the grain structure is essential. To this end, a FokkerPlanck evolution law for the volume distribution of ferrite grains is developed and shown to exhibit a lognormally distributed solution. Numerical parameter studies are given and confirm expected properties qualitatively. As a preparation for future work on parameter identification, a strategy is presented for the comparison of volume distributions with area distributions experimentally gained from polished micrograph sections. 
Y. Guo, D. Hömberg, G. Hu, J. Li, H. Liu, A time domain sampling method for inverse acoustic scattering problems, Journal of Computational Physics, 314 (2016), pp. 647660.
Abstract
This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic nearfield measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of timedependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and illposed, the proposed imaging algorithms are totally “direct” involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving timedependent wave equations. 
K. Sturm, M. Hintermüller, D. Hömberg, Distortion compensation as a shape optimisation problem for a sharp interface model, Computational Optimization and Applications. An International Journal, 64 (2016), pp. 557588.
Abstract
We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface. 
D. Hömberg, Q. Liu, J. MontalvoUrquizo, D. Nadolski, Th. Petzold, A. Schmidt, A. Schulz, Simulation of multifrequencyinductionhardening including phase transitions and mechanical effects, Finite Elements in Analysis and Design, 121 (2016), pp. 86100.
Abstract
Induction hardening is a well known method for the heat treatment of steel components. With the concept of multifrequency hardening, where currents with two different frequency components are provided on a single inductor coil, it is possible to optimize the hardening zone to follow a given contour, e.g. of a gear. In this article, we consider the simulation of multifrequency induction hardening in 3D. The equations to solve are the vector potential formulation of Maxwell's equations describing the electromagnetic fields, the balance of momentum to determine internal stresses and deformations arising from thermoelasticity and transformation induced plasticity, a rate law to determine the distribution of different phases and the heat equation to determine the temperature distribution in the workpiece. The equations are solved using adaptive finite element methods. The simulation results are compared to experiments for discs and for gears. A very good agreement for the hardening profile and the temperature is observed. It is also possible to predict the distribution of residual stresses after the heat treatment. 
G. Hu, M. Yamamoto, Hölder stability estimate of the Robin coefficient in corrosion detection problems with a single boundary measurement, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 31 (2015), pp. 115009/1115009/20.

TH. Petzold, D. Hömberg, D. Nadolski, A. Schulz, H. Stiele, Adaptive FiniteElementeSimulation des MehrfrequenzInduktionshärtens, HTM Journal for Heat Treatment and Materials, 70 (2015), pp. 3339.

R. Rossi, M. Thomas, From an adhesive to a brittle delamination model in thermoviscoelasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015), pp. 159.
Abstract
We address the analysis of a model for brittle delamination of two viscoelastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques. 
CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 45874612.
Abstract
In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damagedependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging. 
D. Hömberg, Th. Petzold, E. Rocca, Analysis and simulations of multifrequency induction hardening, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 8497.
Abstract
We study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwell's equations coupled with an internal energy balance and an ODE for the volume fraction of austenite, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable apriori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system. 
W. Bleck, D. Hömberg, U. Prahl, P. Suwanpinij, N. Togobytska, Optimal control of a cooling line for production of hot rolled dual phase steel, Steel Research International, 85 (2014), pp. 13281333.
Abstract
In this article, the optimal control of a cooling line for production of dual phase steel in a hot rolling process is discussed. In order to achieve a desired dual phase steel microstructure an optimal cooling strategy has to be found. The cooling strategy should be such that a desired final distribution of ferrite in the steel slab is reached most accurately. This problem has been solved by means of mathematical control theory. The results of the optimal control of the cooling line have been verified in hot rolling experiments at the pilot hot rolling mill at the Institute for Metal Forming (IMF), TU Bergakademie Freiberg. 
D. Hömberg, S. Lu, K. Sakamoto, M. Yamamoto, Parameter identification in nonisothermal nucleation and growth processes, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 30 (2014), pp. 035003/1035003/24.
Abstract
We study nonisothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an energy balance to account for recalescence effects. The main novelty of our work is the identification of temperature dependent nucleation rates. We prove that such rates can be uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation. 
K. Chełminski, D. Hömberg, O. Rott, On a thermomechanical milling model, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 12 (2011), pp. 615632.
Abstract
This paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions. 
A. Fasano, D. Hömberg, D. Naumov, On a mathematical model for laserinduced thermotherapy, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems. Elsevier Science Inc., New York, NY. English, English abstracts., 34 (2010), pp. 38313840.
Abstract
We study a mathematical model for laserinduced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bioheat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well. 
A. Fasano, D. Hömberg, L. Panizzi, A mathematical model for case hardening of steel, Mathematical Models & Methods in Applied Sciences, 19 (2009), pp. 21012126.
Abstract
A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a lowcarbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions. We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations. 
D. Hömberg, D. Kern, The heat treatment of steel  A mathematical control problem, Materialwissenschaft und Werkstofftechnik, 40 (2009), pp. 438442.
Abstract
The goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with machinebased control. 
D. Hömberg, N. Togobytska, M. Yamamoto, On the evaluation of dilatometer experiments, Applicable Analysis. An International Journal, 88 (2009), pp. 669681.
Abstract
The goal of this paper is a mathematical investigation of dilatometer experiments to measure the kinetics of solidsolid phase transitions in steel upon cooling from the high temperature phase. Usually, the data are only used for measuring the start and end temperature of the phase transition. In the case of several coexisting product phases, lavish microscopic investigations have to be performed to obtain the resulting fractions of the different phases. In contrast, we show that the complete phase transition kinetics including the final phase fractions are uniquely determined by the dilatometer data and present some numerical identification results. 
K. Chełminski, D. Hömberg, D. Kern, On a thermomechanical model of phase transitions in steel, Advances in Mathematical Sciences and Applications, 18 (2008), pp. 119140.

P. Suwanpinij, N. Togobytska, Ch. Keul, W. Weiss, U. Prahl, D. Hömberg, W. Bleck, Phase transformation modeling and parameter identification from dilatometric investigations, Steel Research International, 79 (2008), pp. 793799.
Abstract
The goal of this paper is to propose a new approach towards the evaluation of dilatometric results, which are often employed to analyse the phase transformation kinetics in steel, especially in terms of continuous cooling transformation (CCT) diagram. A simple task of dilatometry is deriving the start and end temperatures of the phase transformation. It can yield phase transformation kinetics provided that plenty metallographic investigations are performed, whose analysis is complicated especially in case of several coexisting product phases. The new method is based on the numerical solution of a thermomechanical identification problem. It is expected that the phase transformation kinetics can be derived by this approach with less metallographic tasks. The first results are remarkably promising although further investigations are required for the numerical simulations. 
H. Alder, D. Hömberg, W. Weiss, Simulationsbasierte Regelung der Laserhärtung von Stahl, HTM Zeitschrift für Werkstoffe. Wärmebehandlung. Fertigung, 61 (2006), pp. 103108.

D. Hömberg, W. Weiss, PID control of laser surface hardening of steel, IEEE Transactions on Control Systems Technology, 14 (2006), pp. 896904.

D. Hömberg, A. Khludnev, A thermoelastic contact problem with a phase transition, IMA Journal of Applied Mathematics, 71 (2006), pp. 479495.

D. Hömberg, M. Yamamoto, On an inverse problem related to laser material treatments, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 22 (2006), pp. 18551867.

D. Hömberg, A mathematical model for induction hardening including mechanical effects, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 5 (2004), pp. 5590.

D. Hömberg, S. Volkwein, Control of laser surface hardening by a reducedorder approach using proper orthogonal decomposition, Mathematical and Computer Modelling, 38 (2003), pp. 10031028.
Contributions to Collected Editions

L. Capone, Th. Petzold, D. Hömberg, D. Ivanov, A novel integrated tool for parametric geometry generation and simulation for induction hardening of gears, in: Heat Treat 2017: Proceedings of the 29th AMS Heat Treating Society Conference, October 2426, 2017, Columbus, Ohio, USA, ASM International, Materials Park, Ohio, 2017, pp. 234242.

P. Das, J.I. Asperheim, B. Grande, Th. Petzold, D. Hömberg, Simulation of temperature profile in longitudinal welded tubes during highfrequency induction (HFI) welding, in: Heat Treat 2017: Proceedings of the 29th AMS Heat Treating Society Conference, October 2426, 2017, Columbus, Ohio, USA, ASM International, Materials Park, Ohio, 2017, pp. 534538.

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rateindependent damage in thermoviscoelastodynamics, in: MURPHYSHSFS2014: 7th International Workshop on MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1012009/20.
Abstract
This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the ratedependent heat equation, and with the momentum balance featuring inertia and viscosity according to KelvinVoigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a timedependent Dirichlet loading. 
Q. Liu, Th. Petzold, D. Nadolski, R. Pulch, Simulation of thermomechanical behavior subjected to induction hardening, in: Scientific Computing in Electrical Engineering, SCEE 2014, Wuppertal, Germany, July 2014, A. Bartel, M. Clemens, M. Günther, E.J.W. TER Maten, eds., 23 of Mathematics in Industry, Springer International Publishing Switzerland, Cham, 2016, pp. 133142.

C. Carstensen, M. Hintermüller, D. Hömberg, F. Tröltzsch, C  Production, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 151153.

M. Hintermüller, D. Hömberg, O. Klein, J. Sprekels, F. Tröltzsch, C4  PDEconstrained optimization with industrial applications, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 207222.

D. Hömberg, E. Rocca, Th. Petzold, Multifrequency induction hardening  A challenge for industrial mathematics, in: The Impact of Applications on Mathematics  Proceedings of the Forum of Mathematics for Industry 2013, M. Wakayama, ed., 1 of Mathematics for Industry, Springer, Tokyo et al., 2014, pp. 257264.

D. Hömberg, S. Lu, K. Sakamoto, M. Yamamoto, Nucleation rate identification in binary phase transition, in: The Impact of Applications on Mathematics  Proceedings of the Forum of Mathematics for Industry 2013, M. Wakayama, ed., 1 of Mathematics for Industry, Springer, Tokyo et al., 2014, pp. 227243.

P. Krejčí, E. Rocca, J. Sprekels, Liquidsolid phase transitions in a deformable container, in: Continuous Media with Microstructure, B. Albers, ed., Springer, Berlin/Heidelberg, 2010, pp. 285300.

L. Panizzi, A. Fasano, D. Hömberg, Modeling, analysis and simulations of case hardening of steel, in: 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. 965970.

D. Hömberg, O. Rott, Modeling, analysis and stability of milling processes including workpiece effects, in: 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. 493498.

O. Rott, P. Rasper, D. Hömberg, E. Uhlmann, A milling model with thermal effects including the dynamics of machine and work piece, B. Denkena, ed., Proceedings, 1st International Conference on Process Machine Interactions, Hannover, September 34, 2008, PZH Produktionstechnisches Zentrum GmbH, Garbsen, 2008, pp. 369378.
Abstract
This paper deals with the development of a new mathematical model that characterizes the structureprocess interaction for a complex milling system. The structure is divided into a work piece and a machine part, which are represented by different models. While the machine dynamics is characterized by a standard multibody system, the work piece is described as a linear thermoelastic continuum. The coupling of both parts is carried out by an empirical process model permitting an estimate of heat and coupling forces occurring during milling. This work reports the derivation of the governing equations emphasizing the coupling and summarizes the numerical algorithms being applied to solve the coupled equation system. The results of numerical simulations that show the dynamics of the complex thermomechanical system are presented at the end. 
D. Hömberg, D. Kern, The heat treatment of steel  A mathematical control problem, in: Proceedings of the 2nd International Conference on Distortion Engineering  IDE 2008, 1719 September 2008, Bremen, Germany, H.W. Zoch, Th. Lübben, eds., IWT, Bremen, 2008, pp. 201209.

D. Hömberg, D. Kern, W. Weiss, Die Wärmebehandlung von Stahl  ein Optimierungsproblem, in: Distortion Engineering  Verzugsbeherrschung in der Fertigung III , 3 of Sonderforschungsbereich 570, Universität Bremen, Kolloquium, 2006, pp. 3955.

M. Anthonissen, D. Hömberg, W. Weiss, Realtime control of surface remelting, in: Progress in Industrial Mathematics at ECMI 2004, A. Di Bucchianico, R.M.M. Mattheij, M.A. Peletier, eds., 8 of Mathematics in Industry, Springer, 2005, pp. 356360.

D. Hömberg, S. Volkwein, W. Weiss, Optimal control strategies for the surface hardening of steel, in: Proceedings of the 2nd International Conference on Thermal Process Modelling and Computer Simulation, S. Denis, P. Archambault, J.M. Bergheau, R. Fortunier, eds., 120 of J. Physique IV, EDP Sciences, 2004, pp. 325335.
Talks, Poster

K. Hopf, Global existence analysis of energyreactiondiffusion systems, Variational Methods for Evolution, September 13  19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 15, 2020.

M.J. Arenas Jaén, Modelling, simulation and optimization of inductive preheating for thermal cutting of steel plates, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium 27 ``MSO for Steel Production and Manufacturing'', June 18  22, 2018, Budapest, Hungary, June 19, 2018.

TH. Petzold, The MIMESIS project  An example for an interdisciplinary research project, LeibnizKolleg for Young Researchers: Chances and Challenges of Interdisciplinary Research, Thematic Workshop ``Models and Modelling'', November 9  11, 2016, LeibnizGemeinschaft, Berlin, November 9, 2016.

D. Hömberg, Analysis and simulation of Joule heating problems, Mathematisches Kolloquium, Bergische Universität Wuppertal, Fachgruppe Mathematik und Informatik, June 21, 2016.

D. Hömberg, Math for steel production and manufacturing, MACSI10  Empowering Industrial Mathematical and Statistical Modelling for the Future, December 8  9, 2016, University of Limerick, Ireland, December 9, 2016.

M. Heida, Modeling of fluid interfaces, Jahrestagung der Deutschen MathematikerVereinigung, Minisymposium ``Mathematics of Fluid Interfaces'', September 21  25, 2015, Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Hamburg, September 23, 2015.

CH. Heinemann, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, 3rd Workshop of the GAMM Activity Group Analysis of Partial differential Equations, September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, October 1, 2015.

D. Hömberg, Nucleation, growth, and grain size evolution in multiphase materials, INdAM Workshop ``Special Materials in Complex Systems  SMaCS 2015'', May 18  22, 2015, Rome, Italy, May 21, 2015.

D. Hömberg, Optimal coefficient control for semilinear parabolic equations, Fudan University, School of Mathematical Sciences, Shanghai, China, March 10, 2015.

TH. Petzold, Adaptive FiniteElementeSimulation des MehrfrequenzInduktionshärtens in 3D, HärtereiKongress, October 22  24, 2014, Köln, October 23, 2014.

TH. Petzold, Finite element simulations and experiments for multifrequency induction hardening, IUTAM Symposium on ThermomechanicalElectromagnetic Coupling in Solids: Microstructural and Stability Aspect, June 16  18, 2014, Paris, France, June 18, 2014.

TH. Petzold, Modelling and simulation of multifrequency induction hardening of steel parts, 7th International Scientific Colloquium ``Modelling for Electromagnetic Processing'' (MEP 2014), September 16  19, 2014, Leibniz Universität Hannover, September 19, 2014.

TH. Petzold, Modelling and simulation of multifrequency induction hardening for gear components, The 18th European Conference on Mathematics for Industry 2014 (ECMI 2014), Minisymposium ``Recent Trends in Modelling, Analysis, and Simulation of Induction Heat Treatments'', June 9  13, 2014, Taormina, Italy, June 13, 2014.

M. Thomas, Rateindependent systems with viscosity and inertia: Existence and evolutionary Gammaconvergence, Workshop ``Variational Methods for Evolution'', December 14  20, 2014, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials with inertia, International Workshop ``Variational Modeling in Solid Mechanics'', September 22  24, 2014, University of Udine, Department of Mathematics and Informatics, Italy, September 23, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials with inertia, Oberseminar ``Analysis und Angewandte Mathematik'', Universität Kassel, Institut für Mathematik, December 1, 2014.

D. Hömberg, Modelling, simulation and control of surface heat treatments, Norwegian University of Science and Technology, Department of Physics, Trondheim, October 31, 2014.

D. Hömberg, Nucleation, growth, and grain size evolution in dual phase steels, Workshop ``Recent Developments and Challenges in Interface and Free Boundary Problems'', March 25  28, 2014, University of Warwick, UK, March 26, 2014.

D. Hömberg, Nucleation, growth, and grain size evolution in dual phase steels, The 18th European Conference on Mathematics for Industry 2014 (ECMI 2014), Minisymposium 37: Simulation and Control of Hotrolling, June 9  13, 2014, Taormina, Italy, June 9, 2014.

D. Hömberg, Nucleation, growth, and grain size evolution in dual phase steels, Wrocław University of Technology, Institute of Mathematics and Computer Science, Poland, July 1, 2014.

D. Hömberg, Oberflächenbearbeitung mit Mathematik, Opel Innovation Day, Rüsselsheim, November 7, 2014.

D. Hömberg, Optimal control and shape design problems in thermomechanics, BMSWIAS Summer School ``Applied Analysis for Materials'', August 25  September 5, 2014, Berlin Mathematical School, Technische Universität Berlin, September 1, 2014.

J. Sprekels, PrandtlIshlinskii operators and elastoplasticity, Spring School on ``Rateindependent Evolutions and Hysteresis Modelling'', May 27  31, 2013, Politecnico di Milano, Università degli Studi di Milano, Italy.

D. Kern, Analysis and simulation of a thermomechanical model of phase transitions in steel, Universität Bremen, SFB 570 ``Distortion Engineering'', February 2, 2011.

N. Togobytska, Simulation, Optimierung und Regelung von Gefügebildung und mechanischen Eigenschaften beim Warmwalzen von Mehrphasenstählen, 9. Kolloquium zum DFG SPP 1204 ``Algorithmen zur schnellen, werkstoffgerechten Prozesskettengestaltung und analyse in der Umformtechnik'', Siegen, April 6, 2011.

N. Togobytska, Simulation, optimisation and control of microstructure evolution and mechanical properties during hot rolling of multiphase steels, 10. Kolloquium zum DFG SPP 1204 ``Algorithmen zur schnellen, werkstoffgerechten Prozesskettengestaltung und analyse in der Umformtechnik'', Freiberg, November 8, 2011.

D. Hömberg, Optimal control problems in thermomechanics, Schwerpunktskolloquium ``Analysis und Numerik'', Universität Konstanz, Fachbereich Mathematik und Statistik, January 20, 2011.

N. Togobytska, An inverse problem for laserinduced thermotherapy arising in tumor tissue imaging, Chemnitz Symposium on Inverse Problems 2010, September 23  24, 2010, September 24, 2010.

N. Togobytska, Simulation, Optimierung und Regelung von Gefügebildung und mechanischen Eigenschaften beim Warmwalzen von Mehrphasenstählen, 8. Kolloquium zum DFG SPP 1204 ``Algorithmen zur schnellen, werkstoffgerechten Prozesskettengestaltung und analyse in der Umformtechnik'', Aachen, November 10, 2010.

D. Hömberg, Modelling, simulation and control of phase transformations in steel, Nippon Steel Corporation, Chiba, Japan, September 10, 2010.

D. Hömberg, Multiphase steels, heat treatment and distortion  Mathematical challenges in steel production and manufacturing, Summer School ``High Performance Computing'' (organizer: TU Ilmenau), September 29  October 2, 2010, Upstalsboom Hotel Friedrichshain, Berlin, September 30, 2010.

D. Kern, Optimal control of a thermomechanical model including transformation induced plasticity, SIAM Conference on Control and its Applications, July 6  8, 2009, Denver, USA, July 8, 2009.

N. Togobytska, Parameter identification for the phase transformations in steel, Conference on Applied Inverse Problems 2009, July 20  24, 2009, University of Vienna, Austria, July 21, 2009.

N. Togobytska, Parameter identification from dilatometric investigations, SIAM Conference on Control and its Applications, July 6  8, 2009, Denver, USA, July 8, 2009.

N. Togobytska, Simulation, Optimierung und Regelung von Gefügebildung und mechanischen Eigenschaften beim Warmwalzen von Mehrphasenstählen, 8. Kolloquium zum DFG SPP 1204 ``Algorithmen zur schnellen, werkstoffgerechten Prozesskettengestaltung und analyse in der Umformtechnik'', WIAS Berlin, June 23, 2009.

D. Hömberg, Coupling of process, machine, and workpiece in production processes  A challenge for industrial mathematics, Workshop ``Industrial Mathematics and its Practice'', February 23  24, 2009, University of Tokyo, Japan, February 23, 2009.

D. Hömberg, Die Wärmebehandlung von Stahl  Thermomechanische Modellierung, Simulation und Optimierung, Technische Universität Dortmund, Fakultät Maschinenbau, January 22, 2009.

D. Hömberg, Direct and inverse problems related to phase transitions and distortion in modern multiphase steels, Workshop ``Mathematical Models and Analytical Problems for Special Materials'', July 9  11, 2009, Università degli Studi di Brescia, Italy, July 9, 2009.

D. Hömberg, Distortion compensation  An optimal control approach, 24th IFIP TC 7 Conference on System Modelling and Optimization, July 27  31, 2009, Buenos Aires, Argentina, July 27, 2009.

D. Hömberg, Interactions between machine, workpiece, and process dynamics in milling machines, SIAM Conference on Mathematics for Industry: Challenges and Frontiers (MI09), October 9  10, 2009, San Francisco, USA, October 9, 2009.

D. Hömberg, Optimal control of heat treatments and stability of milling processes  Two case studies from industrial mathematics, Worcester Polytechnic Institute, Mechanical Engineering Department, USA, October 7, 2009.

D. Hömberg, The mathematics of distortion, University of Delaware, Department of Mathematical Sciences, Newark, USA, October 6, 2009.

O. Rott, A milling model with thermal effects including the dynamics of machine and workpiece, 1st International Conference on Process Machine Interactions (PMI 2008), September 3  4, 2008, Leibniz Universität Hannover, September 3, 2008.

O. Rott, Modeling, analysis and stability of milling processes including workpiece effects, The European Consortium for Mathematics in Industry (ECMI 2008), June 30  July 4, 2008, University College, London, UK, July 3, 2008.

O. Rott, Numerical solution of a milling model including thermoelastic workpiece effects, Universität Dortmund, Fachbereich für Mathematik, June 23, 2008.

O. Rott, Parameter für ein MKSModell einer Fräsmaschine, DFG SPP 1180 Workshop ``Parameteridentifikation bei Werkzeugmaschinen'', February 21  22, 2008, WIAS Berlin, February 22, 2008.

O. Rott, Semiimplizite Zeitintegration eines gekoppelten Systems aus partiellen und gewöhnlichen Differentialgleichungen mit Retardierung, DFG SPP 1180 Workshop ``Modellierungstechnologien'', May 29  30, 2008, Dresden, May 30, 2008.

N. Togobytska, On the evaluation of the dilatometer experiments, Chemnitz Symposium on Inverse Problems 2008, September 25  26, 2008, September 25, 2008.

N. Togobytska, Simulation, Optimierung und Regelung von Gefügebildung und mechanischen Eigenschaften beim Warmwalzen von Mehrphasenstählen, Antragskolloquium zum DFG Schwerpunktprogramm 1204/2, Bonn, May 20, 2008.

W. Bleck, D. Hömberg, Ch. Keul, U. Prahl, P. Suwanpinij, N. Togobytska, Simulation, Optimierung und Regelung von Gefügebildung und mechanischen Eigenschaften beim Warmwalzen von Mehrphasenstählen, Workshop ``MEFORM 2008: Simulation von Umformprozessen'', Freiberg, March 26  28, 2008.

D. Hömberg, Modellierung und Optimierung der Gefügeumwandlung in niedrig legierten Stählen und Anwendungen, Salzgitter Mannesmann Forschung GmbH, February 19, 2008.

D. Hömberg, On a mathematical model for highspeed milling including the dynamics of machine and workpiece, Conference ``Direct, Inverse and Control Problems for PDE's'' (DICOP 08), September 22  26, 2008, Cortona, Italy, September 26, 2008.

D. Hömberg, Prozesskette Stahl, Workshop of scshape Matheon with Siemens AG (Industry Sector) in cooperation with Center of Knowledge Interchange (CKI) of Technische Universität (TU) Berlin and Siemens AG, TU Berlin, September 29, 2008.

D. Hömberg, Solidsolid phase transitions  Analysis, optimal control and industrial application, Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland, February 14, 2008.

D. Hömberg, The heat treatment of steel  A mathematical control problem, 2nd International Conference on Distortion Engineering 2008, September 17  19, 2008, Bremen, September 19, 2008.

D. Kern, Optimal control of a thermomechanical model of phase transitions in steel, 13th CzechFrenchGerman Conference on Optimization, September 17  21, 2007, Heidelberg, September 20, 2007.

W. Weiss, Control of laser surface hardening, 6th International Congress on Industrial and Applied Mathematics (ICIAM 2007), July 16  20, 2007, ETH Zürich, Switzerland, July 19, 2007.

W. Weiss, Simulationsbasierte Regelung der Laserhärtung von Stahl, RuhrUniversität Bochum, Institut für Werkstoffe, May 30, 2007.

D. Hömberg, D. Kern, Optimal control of a thermomechanical model of phase transitions in steel, 6th International Congress on Industrial and Applied Mathematics (ICIAM 2007), July 16  20, 2007, ETH Zürich, Switzerland, July 19, 2007.

D. Hömberg, Mathematics for steel production and manufacturing, Nippon Steel, Kimitsu, Japan, March 1, 2007.

D. Hömberg, On a thermomechanical phase transition model for the heat treatment of steel, Universidad de Cádiz, Departamento de Matemáticas, Puerto Real, Spain, January 15, 2007.

D. Hömberg, On a thermomechanical phase transition model for the heat treatment of steel, Fudan University, Department of Mathematics, Shanghai, China, March 5, 2007.

D. Hömberg, Phase transition models for multiphase steels, Industrial and Interdisciplinary Workshop ``Problems Related to the Manufacture of Multiphase Steels'', University of Oxford, Oxford Centre for Industrial and Applied Mathematics, UK, November 2, 2007.

D. Hömberg, Solidsolid phase transitions in steel  modeling, simulation and optimal control, Universitá di Pavia, Dipartimento di Matematica, Italy, March 27, 2007.

D. Hömberg, Thermomechanical phase transition models  analysis, optimal control and industrial applications, University of Oxford, Oxford Centre for Industrial and Applied Mathematics, UK, October 11, 2007.

D. Hömberg, Die Wärmebehandlung von Stahl  ein Optimierungsproblem, Universität Bremen, SFB 570 ``Distortion Engineering'', March 2, 2006.

D. Hömberg, Modellierung, Simulation und Optimierung der Wärmebehandlung von Stahl, Endress+Hauser Flowtec AG, Reinach, Switzerland, May 15, 2006.

D. Hömberg, Optimal control of a thermomechanical phase transition model, 12th IEEE International Conference on Methods and Models in Automation and Robotics, August 28  31, 2006, Miedzyzdroje, Poland, August 29, 2006.

D. Hömberg, Optimal control of thermomechanical phase transitions, Workshop ``Inverse and Control Problems for PDE's'', March 13  17, 2006, Rome, Italy, March 13, 2006.

D. Hömberg, Phasenübergänge in Stahl, Summer School ``Simulation und Anwendungen von Mikrostrukturen'', August 14  18, 2006, Föhr.

D. Hömberg, Thermomechanical models of phase transitions  modelling, control and industrial applications, Escuela Politécnica Nacional, Departamento de Matématica, Quito, Ecuador, November 13, 2006.

D. Hömberg, A thermomechanical phase transition model for the surface hardening of steel, International Conference ``Free Boundary Problems: Theory and Applications'', June 7  12, 2005, Coimbra, Portugal, June 11, 2005.

D. Hömberg, Control of laser material treatments, SIAM Conference on Mathematics for Industry, October 24  26, 2005, Detroit Marriott Renaissance Center, USA, October 25, 2005.

D. Hömberg, Die Laserhärtung von Stahl  Modellierung, Analysis und optimale Steuerung, Universität Bayreuth, Mathematisches Institut, June 30, 2005.

D. Hömberg, Entwicklung eines Prognosetools zur Identifizierung von stabilen Fräsprozessen, DFG SPP 1180, Garbsen, May 19, 2005.

D. Hömberg, Laser material treatments  modeling, simulation, and optimal control, Michigan State University, Department of Mathematics, East Lansing, USA, October 27, 2005.

D. Hömberg, Modelling, simulation and control of laser material treatments, Scuola Normale Superiore, Pisa, Italy, November 22, 2005.

D. Hömberg, On a thermomechanical model of surface heat treatments, EQUADIFF 11 International conference on differential equations, July 25  29, 2005, Comenius University, Bratislava, Slovakia, July 28, 2005.

D. Hömberg, Optimal control of solidsolid phase transitions including mechanical effects, Workshop ``Optimal Control of Coupled Systems of PDE'', April 17  23, 2005, Mathematisches Forschungsinstitut Oberwolfach, April 22, 2005.

D. Hömberg, Von der Stahlhärtung bis zur Krebstherapie  Simulations und Optimierungsaufgaben in Lehre und Forschung, FEMLAB Konferenz 2005, November 3  4, 2005, Frankfurt am Main, November 3, 2005.

D. Hömberg, Modellierung, Analysis und optimale Steuerung der Lasermaterialbearbeitung, Kolloquium der Angewandten Mathematik, Universität Münster, December 3, 2004.

D. Hömberg, Modelling, simulation and control issues in laserinduced thermotherapy, 13th European Conference on Mathematics for Industry (ECMI 2004), June 21  25, 2004, Eindhoven University of Technology, Eindhoven, Netherlands, June 22, 2004.

D. Hömberg, On a thermomechanical model of phase transitions in steel, INdAM Workshop ``Dissipative Models in Phase Transitions'', September 5  11, 2004, Cortona, Italy, September 10, 2004.

D. Hömberg, On a thermomechanical model of surface heat treatments, University of Tokyo, Department of Mathematical Sciences, Japan, October 14, 2004.

D. Hömberg, Optimal control of laser surface hardening, University of Chiba, Department of Mathematics and Informatics, Japan, October 19, 2004.

D. Hömberg, Simulation und Optimierung der Lasermaterialbearbeitung, Seminar des Forschungsschwerpunktes Photonik, Technische Universität Berlin, Optisches Institut, June 18, 2004.

D. Hömberg, Thermoelastische Phasenübergänge in Stahl, HumboldtUniversität zu Berlin, Institut für Mathematik, May 13, 2004.

D. Hömberg, Widerstandsschweißen und Oberflächenhärtung von Stahl  Modellierung, Analysis und optimale Steuerung, Colloquium of Sfb 393, Technische Universität Chemnitz, Institut für Mathematik, February 13, 2004.

D. Hömberg, A mathematical model for capacitor resistance welding, 5th International Congress on Industrial and Applied Mathematics (ICIAM 2003), July 7  11, 2003, Sydney, Australia, July 10, 2003.
External Preprints

J. Fischer , K. Hopf , M. Kniely, A. Mielke, Global existence analysis of energyreactiondiffusion systems, Preprint no. arXiv:2012.03792, Cornell University, 2020.
Abstract
We establish globalintime existence results for thermodynamically consistent reaction(cross)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model speciesdependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the nonisothermal case lies in the intrinsic presence of crossdiffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where nonintegrable diffusion fluxes or reaction terms appear.