This monograph studies an evolutionary variational inequality approach to a degenerate moving free boundary problem. 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. 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.
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.
