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Derivation of a homogenized two-temperature model from the heat equation

Laurent Desvillettes, François Golse, Valeria Ricci (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat,...

Diffuse-interface treatment of the anisotropic mean-curvature flow

Michal Beneš (2003)

Applications of Mathematics

We investigate the motion by mean curvature in relative geometry by means of the modified Allen-Cahn equation, where the anisotropy is incorporated. We obtain the existence result for the solution as well as a result concerning the asymptotical behaviour with respect to the thickness parameter. By means of a numerical scheme, we can approximate the original law, as shown in several computational examples.

Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model

Nicolas Bouillard, Robert Eymard, Raphaele Herbin, Philippe Montarnal (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion also takes place leads to a system of two parabolic equations and one ordinary differential equation coupled with a stiff reaction term. This system is discretized by a finite volume scheme which is suitable for the approximation of the discontinuous reaction term of unknown sign. Discrete solutions are shown to exist and converge towards a weak solution of the continuous problem. Uniqueness...

Direct approach to mean-curvature flow with topological changes

Petr Pauš, Michal Beneš (2009)

Kybernetika

This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves Γ ( t ) : S → ℝ 2 , t ≧ 0 . The curves are driven by the normal velocity v which is the function of curvature κ and the position. The evolution law reads as: v = - κ + F . The motion law is treated using direct approach numerically solved by two schemes, i. e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved...

Elastoplastic reaction of a container to water freezing

Pavel Krejčí (2010)

Mathematica Bohemica

The paper deals with a model for water freezing in a deformable elastoplastic container. The mathematical problem consists of a system of one parabolic equation for temperature, one integrodifferential equation with a hysteresis operator for local volume increment, and one differential inclusion for the water content. The problem is shown to admit a unique global uniformly bounded weak solution.

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