A Variational Inequality Approach to free Boundary Problems with Applications in Mould Filling (International Series of Numerical Mathematics, 136)

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. 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 unƯ known) boundary problems originating from engineering and economic applicaƯ tions can directly, or after a transformation, be formulated as variational inequalƯ 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 folƯ 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 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. Front Matter....Pages ii-x Introduction....Pages 1-6 Derivation of the Evolutionary Variational Inequality Approach....Pages 7-29 Properties of the Variational Inequality Solution....Pages 31-72 Finite Volume Approximations for Elliptic Variational Inequalities....Pages 73-141 Numerical Analysis of the Evolutionary Variational Inequalities....Pages 143-201 Injection and Compression Moulding as Application Problems....Pages 203-255 Concluding Remarks....Pages 257-261 Back Matter....Pages 263-294