A phenomenon of waiting time in phase change problems driven by radiative heat transfer

In the paper (J. Food Process Eng. 2008; in press) we emphasized that during a phase change process in which the heat input is driven by a radiation transfer mechanism, a peculiar phenomenon may occur, characterized by a temporary stop of the increase of the boundary temperature due to a sudden change of the heat transfer coefficient upon phase transition. This time interval is needed to allow the thermal properties of the surface to evolve toward a state that is compatible with the heat intake rate corresponding to the new phase. The occurrence of the waiting time is motivated and studied for a general one‐dimensional Stefan problem. Then an application is presented to the much complicated problem considered in (J. Food Process Eng. 2008; in press), namely, the model for frying process. Copyright © 2008 John Wiley & Sons, Ltd.     

On a parallel multilevel solver for linear elasticity problems

In this paper an application of the additive multilevel iteration method to parallel solving of large‐scale linear elasticity problems is considered. The results are derived in the framework of the hierarchical basis finite element discretization defined on a tensor product of one‐dimensional grid and a sequence of nested triangulations . The algorithm was tested on a number of model problems, arising from bridge foundation modeling. Parallel performance of the solver is reported for Cray T3E‐600 and Sun ES/4000 computer systems. Copyright © 1999 John Wiley & Sons, Ltd.     

Weak solutions to a model with elasticity energy for electrochemically induced phase transitions in Lithium-Ion batteries

Using olivine LiFePO₄ as a model system, we study the existence of global solutions to a phase-field model with elasticity energy for Lithium-Ion batteries, which consists of a linear elasticity sub-system and nonlinear evolution equations for the order parameter and the lithium concentration. This model can be described the evolving microstructure for electrochemically induced phase transitions in electrochemical storage. Our numerical experiments are carried out to simulate the evolutions of lithium concentration and of phase interfaces for the model.     

Analysis of a system related to a model for phase transitions in dissipative isochoric thermoviscoelastic materials

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity

Abstract

In this paper we consider a semilinear heat equation (in a bounded domain Ω of ? ^N ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ? Ω, that insensitizes the L ² ? norm of the observation of the solution in another open subset 𝒪 ? Ω when ω ∩ 𝒪 ≠ ?, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L ^r -controls (r large enough) starting from insensitizing controls in L ².     

Global existence of solutions for non‐small data to non‐linear spherically symmetric thermoviscoelasticity

We consider some initial–boundary value problems for non‐linear equations of thermoviscoelasticity in the three‐dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd.     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Shape memory and phase transitions for auxetic materials

We present a mathematical model describing the auxetic‐austenitic phase transition phenomenon by a second order shape memory phase transition. The typical properties of auxetic materials, as the negative Poisson ratio ν, are described by a function of the phase ?, called order parameter, which relates the phase transition with a change of the internal order structure of the material. In our model, the auxetic phase is represented by an order parameter ? = 1, which provides a negative Poisson's ratio, while the austenitic phase will be denoted by ? = 0. Copyright © 2013 John Wiley & Sons, Ltd.     

The dynamics of transition layers in solids with discontinuous chemical potentials

We derive a two‐phase segregation model in solids under isothermal conditions where due to plastic effects the number of vacancies changes when crossing a transition layer, i.e. a reconstitutive phase transition. We show the thermodynamic correctness of the model and review the existence of weak solutions in suitable spaces. By a formal asymptotic analysis we study the dynamics of the interface and its dependence on the unsymmetric vacancy distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary stabilization for a non‐linear beam on elastic bearings

In this paper, we study the equation under non‐linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non‐linear bearing at x=L. By adding only one damping mechanism at x=L, we prove the existence of a global solution and exponential decay of the energy. Copyright © 2001 John Wiley & Sons, Ltd.     

Solvability of non‐linear parabolic boundary value problem with equivalued surface

In this paper, by means of the method of implicit discretization in time, we obtain the existence of weak solution for a class of non‐linear parabolic boundary value problem with equivalued surface. Copyright © 1999 John Wiley & Sons, Ltd.     

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.     

Existence of solutions to an anisotropic phase‐field model

We consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater. 1995; 43 (2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.     

A mathematical model for phase separation: A generalized Cahn–Hilliard equation

In this paper we present a mathematical model to describe the phenomenon of phase separation, which is modelled as space regions where an order parameter changes smoothly. The model proposed, including thermal and mixing effects, is deduced for an incompressible fluid, so the resulting differential system couples a generalized Cahn–Hilliard equation with the Navier–Stokes equation. Its consistency with the second law of thermodynamics in the classical Clausius–Duhem form is finally proved. Copyright © 2011 John Wiley & Sons, Ltd.     

The stationary Stefan problem with convection governed by a non‐linear Darcy's law

We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a non‐linear Darcy's law and the energy balance equation with second member in L¹. We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle. Copyright © 1999 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.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.