Mesh Movement Governed by Entropy Production

The aim of this paper is to study the feasibility of using (irreversible) entropy production as driving force for a moving mesh. Such a method should be able to capture or track physical phenomena such as viscous and thermal boundary layers, shock waves and regions with chemical reactions. A brief outline is given for the FEM-ALE approach, which forms the basis of the moving mesh method. Mesh velocity is a degree of freedom when the Arbitrary-Lagrangian-Eulerian (ALE) formulation for governing PDEs is used. A Finite Element Method (FEM, viz. Galerkin’s method) is applied as discretization of the spatial domain. Two methods will be suggested for determination of mesh motion: mesh displacement method and mesh velocity method. Both methods have an analogy with respectively solid mechanics and fluid mechanics. The general PDE for mesh motion is based on the equation of motion used in continuum mechanics. Body force for mesh motion is determined by temperature and the gradient of entropy production. This implementation introduces a coupling between mesh motion and the governing physics. The suggested method is implemented in the commercial software package Femlab (release 3.0a). A heat conduction problem in a 1-dimensional geometry is selected as physical problem. Numerical solutions for both mesh methods are shown. For the mesh velocity method the geometry is also extended with a prescribed moving boundary. Finally, there are some remarks about stabilization methods for convection-dominated problems.