In this thesis we present the Stefan problem with two boundary conditions, one constant and one time-dependent. This problem is a classic example of a free boundary problem in partial differential equations, with a free boundary moving in time. Some properties are being proved for the one-dimensional case and the important Stefan condition is also derived. The importance of the maximum principle, and the existence of a unique solution are being discussed. To numerically solve this problem, an analysis when the time t goes to zero is being done. The approximative solutions are shown graphically with proper error estimates.

Universiẗat, Diss., 1999--Freiburg (Breisgau).

Universiẗat, Diss., 2003--Bonn.

Bei der Kristallzüchtung insbesondere von Halbleitern hat die Form des freien Randes (dem Interface zwischen fester und flüssiger Phase) einen starken Einfluss auf die Qualität des Kristalls. Die Dissertation befasst sich mit der Optimalsteuerung der Form und des Verlaufs des freien Randes. Als Vorlage für die in der Arbeit betrachteten Modellkonfigurationen dient das VGF-Verfahren (Vertical Gradient Freeze). Der Erstarrungsprozess wird durch ein Zweiphasen-Stefan-Problem mit durch Konvektion und Lorentzkräfte getriebener Strömung beschrieben. Der freie Rand wird als Graph formuliert. Das Kontrollziel besteht in der Ansteuerung eines gewünschten Verlaufs des freien Randes. Als Kontrollgrößen dient die Temperatur auf der Wand des Schmelztiegels und/oder wandnahe oder verteilte Lorentzkräfte. Das Kontrollziel wird durch Minimierung eines geeigneten Kosten-Funktionals erreicht. Das daraus resultierende Minimierungsproblem wird mit einem Adjungierten-Ansatz gelöst. Anhand numerischer Experimente mit Aluminium und Gallium-Arsenid Schmelzen wird gezeigt, das das vorgestellte Verfahren gut funktioniert.

Bei der Kristallzüchtung insbesondere von Halbleitern hat die Form des freien Randes (dem Interface zwischen fester und flüssiger Phase) einen starken Einfluss auf die Qualität des Kristalls. Die Dissertation befasst sich mit der Optimalsteuerung der Form und des Verlaufs des freien Randes. Als Vorlage für die in der Arbeit betrachteten Modellkonfigurationen dient das VGF-Verfahren (Vertical Gradient Freeze). Der Erstarrungsprozess wird durch ein Zweiphasen-Stefan-Problem mit durch Konvektion und Lorentzkräfte getriebener Strömung beschrieben. Der freie Rand wird als Graph formuliert. Das Kontrollziel besteht in der Ansteuerung eines gewünschten Verlaufs des freien Randes. Als Kontrollgrößen dient die Temperatur auf der Wand des Schmelztiegels und/oder wandnahe oder verteilte Lorentzkräfte. Das Kontrollziel wird durch Minimierung eines geeigneten Kosten-Funktionals erreicht. Das daraus resultierende Minimierungsproblem wird mit einem Adjungierten-Ansatz gelöst. Anhand numerischer Experimente mit Aluminium und Gallium-Arsenid Schmelzen wird gezeigt, das das vorgestellte Verfahren gut funktioniert.

Die vorliegende Arbeit befasst sich mit der Verallgemeinerung von Reaktions-Diffusions-Systemen auf Subdiffusion. Die subdiffusive Dynamik auf mesoskopischer Skala wurde mittels Continuous-Time Random Walks mit breiten Wartezeitverteilungen modelliert. Die Reaktion findet auf mikroskopischer Skala, d.h. während der Wartezeiten, statt und unterliegt dem Massenwirkungsgesetz. Die resultierenden Integro-Differentialgleichungen weisen im Integralkern des Transportterms eine Abhängigkeit von der Reaktion auf. Im Falle der Degradation A->0 wurde ein allgemeiner Ausdruck für die Lösungen beliebiger Dirichlet-Randwertprobleme hergeleitet. Die Annahme, dass die Reaktion dem Massenwirkungsgesetz unterliegt, ist eine entscheidende Voraussetzung für die Existenz stationärer Profile unter Subdiffusion. Eine nichtlineare Reaktion stellt die irreversible autokatalytische Reaktion A+B->2A unter Subdiffusion dar. Es wurde ein Analogon zur Fisher-Kolmogorov-Petrovskii-Piscounov-Gleichung (FKPP) aufgestellt und die resultierenden propagierenden Fronten untersucht. Numerische Simulationen legten die Existenz zweier Regimes nahe, die sowohl mittels eines Crossover-Argumentes als auch durch analytische Berechnungen untersucht wurden. Das erste Regime ist charakterisiert durch eine Front, deren Breite und Geschwindigkeit sich mit der Zeit verringert. Das zweite, fluktuationsdominierte Regime liegt nicht im Geltungsbereich der kontinuierlichen Gleichung und weist eine stärkere Abnahme der Frontgeschwindigkeit sowie eine atomar scharf definierte Front auf. Ein anderes Szenario, bei dem eine Spezies A in ein mit immobilen B-Partikeln besetztes Medium hineindiffundiert und gemäß dem Schema A+B->(inert) reagiert, wurde ebenfalls betrachtet. Diese Anordnung wurde näherungsweise als ein Randwertproblem mit einem beweglichen Rand (Stefan-Problem) formuliert. Die analytisch gewonnenen Ergebnisse bzgl. der Position des beweglichen Randes wurden durch numerische Simulationen untermauert. / The present work studies the generalization of reaction-diffusion schemes to subdiffusion. The subdiffusive dynamics was modelled by means of continuous-time random walks on a mesoscopic scale with a heavy-tailed waiting time pdf lacking the first moment. The reaction was assumed to take place on a microscopic scale, i.e. during the waiting times, obeying the mass action law. The resultant equations are of integro-differential form, and the reaction explicitly affects the transport term. The long ranged memory of the subdiffusion kernel is modified by a factor accounting for the reaction of particles during the waiting times. The degradation A->0 was considered and a general expression for the solution to arbitrary Dirichlet Boundary Value Problems was derived. For stationary solutions to exist in reaction-subdiffusion, the assumption of reactions according to classical rate kinetics is essential. As an example for a nonlinear reaction-subdiffusion system, the irreversible autocatalytic reaction A+B->2A under subdiffusion is considered. A subdiffusive analogue of the classical Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation was derived and the resultant propagating fronts were studied. Two different regimes were detected in numerical simulations, and were discussed using both crossover arguments and analytic calculations. The first regime is characterized by a decaying front velocity and width. The fluctuation dominated regime is not within the scope of the continuous description. The velocity of the front decays faster in time than in the continuous regime, and the front is atomically sharp. Another setup where reactants A penetrate a medium initially filled with immobile reactants B and react according to the scheme A+B->(inert) was also considered. This problem was approximately described in terms of a moving boundary problem (Stefan-problem). The theoretical predictions concerning the moving boundary were corroborated by numerical simulations.

This work deals with the one-dimensional Stefan problem with a general time- dependent boundary condition at the fixed boundary. The solution will be obtained using a discrete random walk method and the results will be compared qualitatively with analytical- and finite difference method solutions. A critical part has been to model the moving boundary with the random walk method. The results show that the random walk method is competitive in relation to the finite difference method and has its advantages in generality and low effort to implement. The finite difference method has, on the other hand, higher accuracy for the same computational time with the here chosen step lengths. For the random walk method to increase the accuracy, longer execution times are required, but since the method is generally easily adapted for parallel computing, it is possible to speed up. Regarding applications for the Stefan problem, there are a large range of examples such as climate models, the diffusion of lithium-ions in lithium-ion batteries and modelling steam chambers for oil extraction using steam assisted gravity drainage.

O seguinte trabalho procura estudar problemas de fronteira móvel, conhecidos por problemas de Stefan, bem como aproximar suas soluções. Aplicações de problemas de Stefan encontram-se, por exemplo, na física termal de mudança de estados, presente em diversos fenômenos físicos e químicos naturais e na indústria. Devido a não-linearidade, a maior parte destes problemas não possuem solução analítica conhecida e uma técnica comum para se aproximar soluções é o método de balanceamento integral, inicialmente estudado por Goodman (1958). Este método e suas variações propõem perfis de aproximação no domínio da solução e resolvem uma versão integral da equação diferencial. O problema se resume a resolver uma equação diferencial ordinária no tempo envolvendo a profundidade de penetração do calor e o perfil de aproximação proposto. O trabalho estuda tais métodos para problemas termais clássicos em primeiro lugar, de modo que a extensão para problemas de Stefan seja natural. Refinamentos são apresentados, bem como uma técnica de subdivisão do espaço que resulta num esquema numérico. A técnica de imobilização e fronteira é desenvolvida e aplicada em diversos momentos, a fim de simplificar a utilização dos métodos integrais. / The current work aims to study moving boundary problems, known as Stefan problems, and approximate their solutions. Applications of Stefan problems are found in situations where there is change of physical state, present in several natural and industrial physical and chemical phenomena. Due to their inherent nonlinearity, most of these problems have no known analytic solution and a common technique to approximate solutions is the heat balance integral method, originally studied by Goodman (1958). This method and its variations propose an approximating profile and solve an integral version of the differential equation. The problem is reduced to solving an ordinary differential equation in time involving the depth of heat penetration and the proposed profile. This work studies such classic methods to thermal problems first, in a way that the extension to Stefan problems is natural. Refinements are presented, as well as a technique of subdividing the space domain which results in a numerical scheme. The technique of boundary immobilization is developed and applied at different times in order to simplify the use of these methods.

Optimal control (motion planning) of the free interface in classical two-phase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The first-order optimality system is derived on a formal basis. It provides gradient information based on the adjoint temperature and adjoint level set function. Suitable discretization schemes for the forward and adjoint systems are described. Numerical examples verify the correctness and flexibility of the proposed scheme.

The one dimensional transient heat conduction equation was numerically modeled through matrix diagonalization and three time-discretization schemes. The discrete methods were first-order backward, second-order backward, and implicit finite difference schemes. All simulations used the central difference formula in the space dimension. Two relevant physical systems were considered: a uniformly conducting slab and a melting block of ice. The latter lead to a moving boundary system, or Stefan problem. The multiphysics of melting was numerically modeled through alternating updates of temperature and melt front profiles. Iterative simulations were run with regularly refined discretization meshes in both systems. In the case of the conducting slab, temperature at a fixed point in space and time was considered. For the Stefan problem, the melt front movement after a set time was the physical solution of interest. The accuracy of the convergent results was increased using Richardson acceleration and the Wynn's epsilon algorithm. Accuracy was improved for the moving boundary problem as well, but to a significantly lesser degree. The relative errors improved by five and two orders of magnitude for the conducting block and melting ice simulations, respectively. These relative errors were used to determine that matrix diagonalization is the most accurate numerical solution among the four considered. In both simulation convergence and acceleration potential, matrix diagonalization was superior to the implicit and explicit discretization solutions. However, matrix diagonalization required significantly more computational time. With the enhancement of convergence acceleration, the finite difference schemes obtained similar relative errors to the diagonalization model. This demonstrated the value of convergence acceleration in the classic dilemma for every programmer. There is always a balance struck between model sophistication, accuracy, and computational time. Convergence acceleration allows for a simpler numerical model to achieve comparable accuracy, and in less time than that required for sophisticated numerical models. The numerical models were also compared for stability through parameters that arose in each simulation. These parameters were the Courant-Friederichs-Lewy (CFL) condition and diagonalized eigenvalues. Though diagonalization was found to be the most accurate, it was determined that the backwards finite difference solutions are the easiest to evaluate for stability. In these solution methods, the CFL value allows the stability to be determined prior to running the simulation.