scholarly journals Theoretical Framework of a Variational Formulation for Nonlinear Heat Transfer with Phase Changes

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Eric Feulvarch ◽  
Jean-Christophe Roux ◽  
Jean-Michel Bergheau
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Theoretical Framework ◽  
Phase Changes ◽  
Time Step ◽  
Numerical Resolution ◽  
Nonlinear Heat Transfer ◽  
Uniqueness Of The Solution

This paper discusses mathematical results for a variational formulation dedicated to heat transfer with phase changes. Practical finite element experiences show that the studied formulation can lead to difficulties for the numerical resolution at each time step. The aim of the paper is to show that such numerical pathologies do not come from the basic variational formulation by showing existence and uniqueness of the solution.

A nonlinear system for irreversible phase changes

1990 ◽  
Vol 1 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Dominique Blanchard ◽  
Hamid Ghidouche
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Phase Changes ◽  
The Other ◽  
System Modelling ◽  
Mathematical Study ◽  
Uniqueness Of The Solution ◽  
Change Problem

This paper is concerned with the mathematical study of a nonlinear system modelling an irreversible phase change problem. Uniqueness of the solution is proved using the accretivity of the system in (L1)2. Expressing one of the two unknowns as an explicit functional of the other reduces the system to a single nonlinear evolution equation and ultimately leads to an existence theorem.In this paper the existence and uniqueness of the solution of a nonlinear system modelling some irreversible phase changes is established.

scholarly journals CONVECTIVE AND RADIATIVE THERMAL TRANSFER WITH MULTIPLE REFLECTIONS: ANALYSIS AND APPROXIMATION BY A FINITE ELEMENT METHOD

2001 ◽  
Vol 11 (02) ◽  
pp. 229-262 ◽  
Author(s):  
J. MONNIER ◽  
J. P. VILA
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Existence And Uniqueness ◽  
Multiple Reflections ◽  
Convection Diffusion ◽  
Thermal Transfer ◽  
Uniqueness Of The Solution ◽  
Nonlinear Integrodifferential System ◽  
Account Heat ◽  
Element Method

We study a 3D steady-state thermal model taking into account heat transfer by convection, diffusion and radiation with multiple reflections (grey bodies). This model is a nonlinear integrodifferential system which we solve numerically by a finite element method. Some results of existence and uniqueness of the solution are proved, the numerical analysis is detailed, error estimates are given and two-dimensional numerical results of thermal exchanges under a car bonnet are presented.

Adaptive Multigrid Methods for Extended Fluid-Structure Interaction (eXFSI) Problem: Part I — Mathematical Modelling

2015 ◽  
Author(s):  
Bhuiyan Shameem Mahmood Ebna Hai ◽  
Markus Bause
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Variational Formulation ◽  
Fluid Structure Interaction ◽  
Multigrid Methods ◽  
Time Step ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Time Stepping ◽  
Crank Nicolson

This contribution is the first part of three papers on Adaptive Multigrid Methods for eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art of recent developments in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research will focus on the newly developed mathematical model of a new FSI problem which is called eXtended Fluid-Structure Interaction (eXFSI) problem in ALE framework. This model is used to design an on-live Structural Health Monitoring (SHM) system in order to determine the wave propagation in moving domains and optimum locations for SHM sensors. eXFSI is strongly coupled problem of typical FSI with a wave propagation problem on the fluid-structure interface, where wave propagation problems automatically adopted the boundary conditions from of the typical FSI problem at each time step. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as an one step-θ scheme; from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with Newton’s method whereas the spatial discretization is done with a Galerkin finite element scheme. To control computational costs we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaptation during the computation. The implementation is accomplished via the software library package DOpElib and deal.II for the computation of different eXFSI configurations.

The Characteristic Radial Basis Meshless Method for Convection-Dominated Diffusion Equations

2012 ◽  
Vol 182-183 ◽  
pp. 1756-1760 ◽  
Author(s):  
Xin Qiang Qin ◽  
Xian Bao Duan ◽  
Wei Guo
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Diffusion Equations ◽  
Characteristic Finite Element ◽  
Radial Basis ◽  
Uniqueness Of The Solution ◽  
Convection Dominated ◽  
Diffusion Problems

A characteristic radial basis meshless method (CRBM) is developed for numerically solving convection-dominated diffusion equations. This method is a truly meshless technique without mesh discretization, and it is numerically stable and more efficient than the characteristic finite element method (CFEM) as demonstrated by the provided numerical results for convection-dominated diffusion problems. Moreover, the existence and uniqueness of the solution to the method are proved.

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