Overview

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 applications 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 non-linear partial differential equations and their applications, especially those concerned with numerical aspects. However, the book should also be useful for scientists from the application areas - in particular, applied scientists from engineering and physics.

Full Product Details

Author:   Jörg Steinbach
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2002 ed.
Volume:   136
Dimensions:   Width: 15.50cm , Height: 1.90cm , Length: 23.50cm
Weight:   1.340kg
ISBN:  

9783764365820

ISBN 10:   376436582
Pages:   294
Publication Date:   01 February 2002
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
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Table of Contents

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.- 3 Properties 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.- 3.4 Regularity with respect to space variables.- 3.5 Some remarks on further regularity results.- 4 Finite Volume Approximations for Elliptic Inequalities.- 4.1 Finite element and volume approximations for the obstacle problem.- 4.2 Comparison of finite volume and finite element approximations.- 4.3 Error estimates for the finite volume solution.- 4.4 Penalization methods for the finite volume obstacle problem.- 4.5 The Signorini problem as a boundary obstacle problem.- 4.6 Results from numerical experiments for elliptic obstacle problems.- 5 Numerical Analysis of the Evolutionary Inequalities.- 5.1 Finite element and volume approximations for the evolutionary problems.- 5.2 Error estimates for the finite element and finite volume solutions.- 5.3 Penalization methods for the evolutionary finite volume inequalities.- 5.4 Numerical experiments for evolutionary variational inequalities.- 6 Injection and Compression Moulding as Application Problems.- 6.1 Classical Hele-Shaw flows and related moving boundary problems.- 6.2 Mathematical modelling of injection and compression moulding.- 6.3 Simulation results.- 7 Concluding Remarks.- List of Figures.- List of Tables.- List of Symbols.

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