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- ItemOptimal control for the thermistor problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hömberg, Dietmar; Meyer, Christian; Rehberg, Joachim; Ring, WolfgangThis paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.
- ItemSimulationsbasierte Regelung der Laserhärtung von Stahl(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Alder, Holger; Hömberg, Dietmar; Weiss, WolfBei der Oberflaechenhärtung mit Hilfe von Laserstrahlen ist eine konstante Einhärtetiefe erwünscht, wobei gleichzeitig Anschmelzungen vermieden werden sollen. Um Anschmelzungen zu verhindern, kann die Temperatur im Auftreffpunkt des Lasers gemessen werden und die Laserleistung entsprechend geregelt werden. Eine konstante Temperatur fährt bei geometrisch komplizierten Bauteilen jedoch nicht zu einer konstanten Einhärtetiefe. In dieser Arbeit wird ein Verfahren aufgezeigt, wobei durch numerische Simulationen eine nichtkonstante Oberflächentemperatur berechnet wird, die eine konstante Einhärtetiefe liefert. Die berechnete Oberflächentemperatur kann als Solltemperatur im realen Prozess benutzt werden.
- ItemOptimal control of robot guided laser material treatment(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Hömberg, Dietmar; Steinbrecher, Andreas; Stykel, TatjanaIn this article we will consider the optimal control of robot guided laser material treatments, where the discrete multibody system model of a robot is coupled with a PDE model of the laser treatment. We will present and discuss several optimization approaches of such optimal control problems and its properties in view of a robust and suitable numerical solution. We will illustrate the approaches in an application to the surface hardening of steel.
- ItemA comparison of analytical cutting force models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Rott, Oliver; Hömberg, Dietmar; Mense, CarstenThe modeling of dynamic processes in milling and the determination of stable cutting conditions have become increasingly important for the optimization of manufacturing processes. Analytic approaches and time domain simulations based on simplified dynamic systems are used to identify chatter-free machining conditions. Stresses applied to the system are generally estimated by cutting force models. The goal of this paper is to compare the influence of the cutting force models on the stability limits. Numerical simulations of a simplified, generic milling machine model are therefore performed, while varying the cutting force approach. In order to distinguish stable from unstable cutting conditions a numerical stability criterion is used. The resulting stability charts are then investigated and analyzed to show the effect of the different cutting force models.
- ItemA milling model with thermal effects including the dynamics of machine and work piece(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Rott, Oliver; Rasper, Patrick; Hömberg, Dietmar; Uhlmann, EckartThis paper deals with the development of a new mathematical model that characterizes the structure-process 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 multi-body system, the work piece is described as a linear thermo-elastic 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 thermo-mechanical system are presented at the end.
- ItemPhase transformation modeling and parameter identification from dilatometric investigations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Suwanpinij, Piyada; Togobytska, Nataliya; Keul, Christoph; Weiss, Wolf; Prahl, Ulrich; Hömberg, Dietmar; Bleck, WolfgangThe 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.
- ItemThe heat treatment of steel - a mathematical control problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Hömberg, Dietmar; Kern, DanielaThe 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 machine-based control.
- ItemOn a thermomechanical model of phase transitions in steel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Chełminski, Krzysztof; Hömberg, Dietmar; Kern, DanielaWe investigate a thermomechanical model of phase transitions in steel. The strain is assumed to be additively decomposed into an elastic and a thermal part as well as a contribution from transformation induced plasticity. The resulting model can be viewed as an extension of quasistatic linear thermoelasticity. We prove existence of a unique solution and conclude with some numerical simulations.
- ItemOn a mathematical model for laser-induced thermotherapy(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Fasano, Antonio; Hömberg, Dietmar; Naumov, DmitriWe study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat 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.
- ItemA mathematical model for case hardening of steel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Fasano, Antonio; Hömberg, Dietmar; Panizzi, LuciaA mathematical model for the gas carburizing of steel is presented. Carbon is dissolved in the surface layer of a low-carbon 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 phase fractions. We prove existence and uniqueness of a solution and finally present some numerical simulations.