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End date: 2019 × Start date: 2016 × Start date: 1990 × Type: report × Subject: phase transition ×

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Now showing 1 - 10 of 16
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    A large-deviations approach to gelation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Andreis, Luisa; König, Wolfgang; Patterson, Robert
    A @large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t = 1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdos-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdos-Rényi graphs are connected.
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    On unwanted nucleation phenomena at the wall of a VGF chamber
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Dreyer, Wolfgang; Duderstadt, Frank; Eichler, Stefan; Naldzhieva, Margarita
    This is preliminary study on a phenomenon that happens during crystal growth of GaAs in a vertical gradient freeze (VGF) device. Here unwanted polycrystals nucleate at the chamber wall and move into the interior of the crystal. This happens within an undercooled region in the vicinity of the triple point, where the liquid-solid interface meets the chamber wall. The size and shape of that region is modelled by the Gibbs-Thomson law, which will be rederived in this paper. Hereafter we identify the crucial parameter, whose proper adjustment may minimize the undercooled region.
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    Asymptotic analysis for Korteweg models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christiane; Rohde, Christiane
    This paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.
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    On the clustering property of the random intersection graphs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Yao, Xin; Chen, Jinwen; Zhang, Changshui; Li, Yanda
    A random intersection graph mtlmcalG_V,W,p is induced from a random bipartite graph mtlmcalG^*_V,W,p with vertices classes mtlV, mtlW and the edges incident between mtlv in V and mtlw in W with probability mtlp. Two vertices in mtlV are considered to be connected with each other if both of them connect with some common vertices in mtlW. The clustering properties of the random intersection graph are investigated completely in this article. Suppose that the vertices number be mtlN = mabsV and mtlM=mabsW and mtlM = N^alpha, p=N^-beta, where mtlalpha > 0,, beta > 0, we derive the exact expressions of the clustering coefficient mtlC_v of vertex mtlv in mtlmcalG_V,W,p. The results show that if mtlalpha < 2beta and mtlalpha neq beta, mtlC_v decreases with the increasing of the graph size; if mtlalpha = beta or mtlalpha geq 2beta, the graph has the constant clustering coefficients, in addition, if mtlalpha > 2beta, the graph connecChangshui Zhangts almost completely. Therefore, we illustrate the phase transition for the clustering property in the random intersection graphs and give the condition that mtlriG being high clustering graph.
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    Analysis and simulation of multifrequency induction hardening
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hömberg, Dietmar; Petzold, Thomas; Rocca, Elisabetta
    We study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwells 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 a-priori 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.
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    Low Mach asymptotic preserving scheme for the Euler-Korteweg model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Giesselmann, Jan
    We present an all speed scheme for the Euler-Korteweg model.We study a semi-implicit time-discretisation which treats the terms, which are stiff for low Mach numbers, implicitly and thereby avoids a dependence of the timestep restriction on the Mach number. Based on this we present a fully discrete finite difference scheme. In particular, the scheme is asymptotic preserving, i.e., it converges to a stable discretisation of the incompressible limit of the Euler-Korteweg model when the Mach number tends to zero.
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    A quasi-incompressible diffuse interface model with phase transition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Aki, Gonca; Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christine
    This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.
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    Optimal control for a phase field system with a possibly singular potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Colli, Pierluigi; Gilardi, Gianni; Marinoschi, Gabriela; Rocca, Elisabetta
    In this paper we study a distributed control problem for a phase-field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions.
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    Sliding modes for a phase-field system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Barbu, Viorel; Colli, Pierluigi; Gilardi, Gianni; Marinoschi, Gabriela; Rocca, Elisabetta
    In the present contribution the sliding mode control (SMC) problem for a phasefield model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the statefeedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target phi*. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.
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    Shape optimization for a sharp interface model of distortion compensation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Sturm, Kevin; Hintermüller, Michael; Hömberg, Dietmar
    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.