A Variational Inequality Approach to free Boundary Problems with Applications in Mould Filling

Since the early 1960s, the mathematical theory of variational inequalities has been under rapid development, based on complex analysis and strongly influenced by 'real-life' application. Many, but of course not all, moving free (Le. , a priori un­ known) boundary problems originating from engineering and economic applica­ tions can directly, or after a transformation, be formulated as variational inequal­ ities. In this work we investigate an evolutionary variational inequality with a memory term which is, as a fixed domain formulation, the result of the application of such a transformation to a degenerate moving free boundary problem. This study includes mathematical modelling, existence, uniqueness and regularity results, numerical analysis of finite element and finite volume approximations, as well as numerical simulation results for applications in polymer processing. Essential parts of these research notes were developed during my work at the Chair of Applied Mathematics (LAM) of the Technical University Munich. I would like to express my sincerest gratitude to K. -H. Hoffmann, the head of this chair and the present scientific director of the Center of Advanced European Studies and Research (caesar), for his encouragement and support. With this work I am fol­ lowing a general concept of Applied Mathematics to which he directed my interest and which, based on application problems, comprises mathematical modelling, mathematical and numerical analysis, computational aspects and visualization of simulation results.

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.- 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.- 3.3.1 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.- 4 Finite Volume Approximations for Elliptic Inequalities.- 4.1 Finite element and volume approximations for the 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 element approximations.- 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 elliptic obstacle 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.- 5 Numerical Analysis of the Evolutionary Inequalities.- 5.1 Finite element and volume approximations for the 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 finite volume solutions.- 5.2.1 Comparison of the finite element and finite 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 and the 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.- 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.- 6.2.1 Injection and compression moulding — Technical background and requirements on simulation.- 6.2.1.1 Technical background.- 6.2.1.2 Short comparison of injection/compression moulding and metal casting.- 6.2.1.3 Some aims of the numerical simulation.- 6.2.2 Balance and state equations.- 6.2.3 Rheological behaviour of polymer melts.- 6.2.4 Temperature-dependent Hele-Shaw flow in the injection and compression moulding process.- 6.2.4.1 The generalized Hele-Shaw flow.- 6.2.4.2 Viscosity models and non-isothermal effects.- 6.2.4.3 The numerical core problems.- 6.2.5 The distance concept — a geometrical approach for injection moulding.- 6.2.6 Recent three-dimensional simulation developments.- 6.3 Simulation results.- 6.3.1 Variation of gate location and thickness, non-isothermal effects, narrow flow region.- 6.3.2 Comparison with the distance model.- 6.3.3 Comparison with three-dimensional simulations.- 7 Concluding Remarks.- List of Figures.- List of Tables.- List of Symbols.