A variational inequality approach to free boundary problems with applications in mould filling自由边界问题变分不等式法及在模填充中的应用

作者： Jörg Steinbach著

出 版 社：

出版时间： 2002-5-1字数：版次： 1页数： 294印刷时间： 2002/05/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783764365820包装： 精装内容简介

This monograph is devoted to the study of an evolutionary variational inequality approach to a degenerate moving free boundary problem. The inequality approach of obstacle type results from the application of an integral transformation. It takes an intermediate position between elliptic and parabolic inequalities and comprises an elliptic differential operator, a memory term and time-dependent convex constraint sets. The study of such inequality problems is motivated by applications to injection and compression moulding, to electro-chemical machining and other quasi-stationary Stefan type problems.

The mathematical analysis of the problem covers existence, uniqueness, regularity and time evolution of the solution. This is carried out in the framework of the variational inequality theory. The numerical solution in two and three space dimensions is discussed using both finite element and finite volume approximations. Finally, a description of injection and compression moulding is presented in terms of different mathematical models, a generalized Hele-Shaw flow, a distance concept and Navier-Stokes flow.

This volume is primarily addressed to applied mathematicians working in the field of nonlinear partial differential equations and their applications, especially those concerned with numerical aspects. However, the book will also be useful for scientists from the application areas, in particular, applied scientists from engineering and physics.

目录

Preface

1 Introduction

2 Evolutionary Variational Inequality Approach

2.1 The degenerate free boundary problem

2.2 Some application problems

2.3 Different fixed domain formulations

2.3.1 Front tracking and fixing methods versus fixed domain formulations exemplified by injection and compression moulding

2.3.2 Weak formulation

2.3.3 The evolutionary variational inequality approach

3Properties of the Variational Inequality Solution

3.1 Problem setting and general notations

3.2 Existence and uniqueness result

3.3 Monotonicity properties and regularity with respect to time

3.3.1 Time-independent convex sets

3.3.2 Time-dependent convex sets

3.4 Regularity with respect to space variables

3.4.1 Dirichlet boundary conditions

3.4.2 Boundary conditions of Neumann/Newton type

3.5 Some remarks on further regularity results

4Finite Volume Approximations for Elliptic Inequalities

4.1 Finite element and volume approximations forthe obstacle problem

4.1.1 The elliptic obstacle problem

4.1.2 Finite element approximations for the obstacle problem

4.1.3 Basics of finite volume approximations

4.1.4 Finite volume approximations for the obstacle problem

4.2 Comparison of finite volume and finite elementapproximations

4.3 Error estimates for the finite volume solution

4.4 Penalization methods for the finite volume obstacle problem .

4.4.1 Discrete maximum principle

4.4.2 Discussion of penalization techniques

4.4.3 Iterative solution of the penalization problems

4.5 The Signorini problem as a boundary obstacle problem

4.6 Results from numerical experiments for ellipticobstacle problems

4.6.1 Examples with known exact solution

4.6.2 Numerical results for the error between the finite element and the finite volume solution

4.6.3 Error behaviour of the finite volume and the penalization solutions

5Numerical Analysis of the Evolutionary Inequalities

5.1 Finite element and volume approximations forthe evolutionary problems

5.1.1 Formulation of the finite element and finite volume approximations

5.1.2 Properties of the discrete inequality problems

5.1.3 Time evolution of the finite volume solution

5.2 Error estimates for the finite element and finitevolume solutions

5 2 1 Cnmp~rison of the finite clemcnt and finitc volume approximations

5.2.2 A priori estimates for the finite element and finite volume solutions

5.2.3 Convergence rate for the finite element and finite volume solutions

5.3 Penalization methods for the evolutionary finite volume inequalities

5.3.1 Discussion of penalization techniques

5.3.2 Iterative solution of the penalization problems

5.4 Numerical experiments for evolutionary variational inequalities

5.4.1 Two evolutionary variational inequalities andthe related free boundary problems

5.4.2 Numerical results for the errors between exact,finite element and finite volume solution

5.4.3 Error behaviour of the penalization solutions

……

6Injection and Compression Mouding as Application problems

7 Concluding remarks

Bibliography

List of Figures

List of Tables

List of Symbols

Index