Hysteresis in phase‐field models with thermal memory

Phase‐field systems as mathematical models for phase transitions have drawn increasing attention in recent years. However, while capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming, a new approach has recently been proposed by the last two authors which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. In this paper this approach is extended to cases where the material exhibits an additional thermal memory, i.e. where the heat flux contains a time convolution of the spatial gradient of temperature. It is shown that the corresponding system of field equations admits a unique strong solution that depends continuously on the data of the system. Copyright © 2000 John Wiley & Sons, Ltd.     

Hysteresis operators in phase-field models of Penrose-fife type

Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.     

A characteristic line based method to build finite-dimensional medoels of heat exchangers

surface heat exchngers are typical simulated with simplified models obtained through segmentation of the heat exchanging fluid path in a number of consecutive lumps In order to aviod major drawbacks of this approach, which may be very misleading for control design purpose, we propose a method, based on the intergration of the PDE system by the method of characteristic lines, for the construction of numerical heat exchangers models. It can be proved that the time response of such new models is indeed rid of parasitic oscillation and suitable for the understanding of complex dynamic phenomena occurring and suitable for the understanding of complex dynamic phenomena occurring in long residence time heat exchangers, both with one- and two- phase flow. In this paper, particular attention is paid to the problem of generating finite dimensional dynamic system by application of the characteristic lines method and computing the frequency responce of such models. Actually, since the characteristic lines method is not naturally is not straightforward to define Finally, the accuracy of CL models is compared with classical models of comparable complexity, with special reference to real application cases, taken from the power generation field.     

Minimizing transient times in a coupled solid‐state laser model

We consider a system of ordinary differential equations modelling the dynamics of two coupled solid‐state lasers. Under the dynamics, this system may execute transitions between in‐phase and out‐of‐phase states. For satellite communications and high‐speed data transfer the transition times should be reduced to their shortest possible duration. In this paper, we apply optimal control theory to find the values of various laser parameters (e.g. the amplitude of the injected field, detunings, and coupling constants) which minimize the transient times between out‐of‐phase and in‐phase states. The effect of each parameter is shown to be independent of the other two, and the transient time is shown to be a strictly increasing function of detuning and a strictly decreasing function of the coupling constant and amplitude of the injected field. The effect of initial conditions on transient times is also analysed. Published in 2005 by John Wiley & Sons, Ltd.     

A model for resistance welding including phase transitions and Joule heating

In this paper, we introduce a new model for solid–liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase‐field model and the consideration of phase‐dependent physical parameters through a mixture ansatz. The PDE system resulting from our modeling approach couples a strongly nonlinear heat equation, a non‐smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with a quasistatic electric charge conservation law. We prove the existence of weak solutions in the three‐dimensional (3D) case, whereas the regularity result and the uniqueness of solution is stated only in the two‐dimensional case. Indeed, uniqueness for the 3D system is still an open problem. Copyright © 2011 John Wiley & Sons, Ltd.     

An approximate method for analysing stochastic mechanical systems

An approximate method for analysing diffusion processes in a natural mechanical system when there are perturbing forces similar to normal white noise is proposed. It is based on orthogonal expansions of the one-dimensional probability density of the state vector in a suitable Hubert space of functions which are square-integrable with respect to a certain measure in the phase space (manifold) of the system. The method consists of solving a special system of linear ordinary differential equations for the expansion coefficients, and is suitable for computer implementation. The method is rigorously proved. The motion of a two-dimensional mathematical pendulum in a random medium is investigated as an example.     

Mathematical methods for restoration of images in magnetism of nanoparticles

It is shown that the formalism of ??restoration and improvement of image qualities?? can be considered as an effective tool for mathematical correction of initial experimental conditions in various cases of magnetometry. As a concrete example, the case of identification of magnetic phase transitions, both field and temperature ones, in nanoparticle systems with log-normal distribution of their sizes has been considered.     

A mathematical model for induction hardening including mechanical effects

We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We take care of effects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions.The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L¹ data. We prove existence of a weak solution to the complete system using a truncation argument.     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.     

On the swapping algorithm

The Metropolis‐coupled Markov chain method (or “Swapping Algorithm”) is an empirically successful hybrid Monte Carlo algorithm. It alternates between standard transitions on parallel versions of the system at different parameter values, and swapping two versions. We prove rapid mixing for two bimodal examples, including the mean‐field Ising model. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 66–97, 2002     

Singular limit of a transmission problem for the parabolic phase-field model

A transmission problem describing the thermal interchange between two regions occupied by possibly different fluids, which may present phase transitions, is studied in the framework of the Caginalp-Fix phase field model. Dirichlet (or Neumann) and Cauchy conditions are required. A regular solution is obtained by means of approximation techniques for parabolic systems. Then, an asymptotic study of the problem is carried out as the time relaxation parameter for the phase field tends to 0 in one of the domains. It is also proved that the limit formulation admits a unique solution in a suitable weak sense.     

A continuum theory for first‐order phase transitions based on the balance of structure order

First‐order phase transitions are modelled by a non‐homogeneous, time‐dependent scalar‐valued order parameter or phase field. The time dependence of the order parameter is viewed as arising from a balance law of the structure order. The gross motion is disregarded and hence the body is regarded merely as a heat conductor. Compatibility of the constitutive functions with thermodynamics is exploited by expressing the second law through the classical Clausius–Duhem inequality. First, a model for conductors without memory is set up and the order parameter is shown to satisfy a maximum theorem. Next, heat conductors with memory are considered. Different evolution problems are established through a system of differential equations whose form is related to the manner in which the memory property is represented. Copyright © 2007 John Wiley & Sons, Ltd.     

On in‐out splitting of incident fields and the far‐field behaviour of Herglotz wavefunctions

It is easy to write down entire solutions of the Helmholtz equation: Examples are plane waves and Herglotz wavefunctions. We are interested in the far‐field behaviour of these solutions motivated by the following question: When is it legitimate to split the far field of such an entire solution into the sum of an incoming spherical wave and an outgoing spherical wave? We review the relevant literature (there are disjoint physical and mathematical threads), and then we answer the question for Herglotz wavefunctions, using a combination of the 2‐dimensional method of stationary phase and some explicit examples.     

A finite element method for the axisymmetric three‐field Stokes system with a discontinuous pressure

A quadrilateral based velocity‐pressure‐extrastress tensor mixed finite element method for solving the three‐field Stokes system in the axisymmetric case is studied. The method derived from Fortin's Q₂ − P₁ velocity‐pressure element is to be used in connection with the standard Galerkin formulation. This makes it particularly suitable for the numerical simulation of viscoelastic flow. It is proven to be second‐order convergent in the natural weighted Sobolev norms, for the system under consideration. The crucial result that the method is uniformly stable is proven for the case of rectangular meshes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 739–763, 1999     

Consistency of iterative operator‐splitting methods: Theory and applications

In this article, we describe a different operator‐splitting method for decoupling complex equations with multidimensional and multiphysical processes for applications for porous media and phase‐transitions. We introduce different operator‐splitting methods with respect to their usability and applicability in computer codes. The error‐analysis for the iterative operator‐splitting methods is discussed. Numerical examples are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Stability and optimal control of thermomechanical processes with non-convex free energies of Ginzburg–Landau type

In this paper we study a coupled non-linear system of partial differential equations that models the dynamics of structural phase transitions in a one-dimensional non-viscous and heat-conducting solid. The corresponding Helmholtz free energy density is assumed in Ginzburg–Landau form; to allow for phase transitions and hysteresis phenomena, it is not assumed convex in the order parameter. It is shown that the solution of the system depends continuously upon the data, and we prove an existence result for an associated optimal control problem.     

Error Estimates for the Adaptive Computation of a Scalar Three Well Problem

We investigate the numerical approximation of Young measure solutions appearing as generalised solutions in scalar non‐convex variational problems. A priori and a posteriori error estimates for a macroscopic quantity, i.e., the stress, are given. Numerical experiments for a scalar three well problem, occurring as a subproblem in the theory of phase transitions in crystalline solids, show that the computational effort can be significantly reduced using an adaptive mesh‐refinement strategy combined with an active set technique by Carstensen and Roubíček.     

Weak solutions for Falk's model of shape memory alloys

In this paper we discuss the system of two partial differential equations governing the dynamics of phase transitions in shape memory alloys. We consider the one‐dimensional model proposed by Falk, in which a term containing a fourth‐derivative appears. The main purpose is to show the uniqueness for weak solutions of the problem by using the approximate dual equations for the system without growth condition for the free energy function. Copyright © 2000 John Wiley & Sons, Ltd.     

Phase-field-based modeling and simulation of solidification and deformation behavior of technological alloys

The purpose of this work is the formulation and application of a continuum field approach to the phenomenological modeling of the behavior of technological alloys undergoing phase transitions and attendant inelastic deformation. To describe the phase transition, a phase–field approach is utilized. For the fully coupled system, an algorithmic formulation is derived based on efficient finite element techniques. Applications will be discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reaction–diffusion systems for the microscopic cellular model of the cardiac electric field

The paper deals with a mathematical model for the electric activity of the heart at microscopic level. The membrane model used to describe the ionic currents is a generalization of the phase‐I Luo–Rudy, a model widely used in 2‐D and 3‐D simulations of the action potential propagation. From the mathematical viewpoint the model is made up of a parabolic reaction diffusion system coupled with an ODE system. We derive existence and some regularity results. Copyright © 2006 John Wiley & Sons, Ltd.