Professor Gunnar Aronsson has since 1989 developed a research group concerned with mathematical modeling of some non-Newtonian flow, in particular polymer shaping technologies using asymptotic methods and e.g. convex analysis to obtain a simple and intuitive model for injection moulding. The benefit of the theoretical analysis is deeper understanding and the possibility to extend the model to obtain better accuracy. The work started with a Master's Thesis by Lennart Pettersson 1989, followed by Master's Theses by Torbjörn Andersson 1990 and Mats Aigner 1992, continued by a Licentiate Thesis by Klas Samuelsson and a Doctoral Thesis by Ulf Janfalk 1993, and recently Master's Theses by Mikael Lundquist 1995 and Fredrik Berntsson 1996. Janfalk's thesis is written essentially in pure mathematics but the results obtained are of great value for our modeling of injection moulding and compression moulding (work in progress by Andréas Bergwall). This Licentiate Thesis finished in 1997 concludes the current state of the research on polymer shaping using injection moulding in Gunnar Aronsson's group.
Instead of using the standard approach of solving the flow equations using some finite element method we use the so-called distance model to obtain a simpler algorithm. Then, instead of being forced to write complex software and use powerful computers, we are able to present a simple algorithm which runs fast on an ordinary PC. The weaknesses of the simple method we use are easier to analyze and understand than e.g. the complex interaction of mesh generation and flow-front tracking that renders problems hard to understand and analyze in the standard approach. We can not claim better accuracy than the standard approach, but have a simpler program that performs the computations without user interaction and still is not depending on extensive heuristic selections of parameters critical for the numerical method. In particular, no mesh needs to be created.
The purpose of this thesis is to study the concept of the distance model in the case of non-uniform thickness of the mould and non-planar objects with thin walls. Currently we have restricted us to a domain decomposable into polygons with constant thickness to obtain an effective and simple algorithm but the method as such does not impose such restrictions. As in the case of uniform thickness we are using Dijkstra's algorithm to compute the shortest path in a graph (that is modified to cope with the case of non-uniform thickness) and this turns out to give an algorithm that is fast and simple to implement.
In Section 2 we describe the ideas and assumptions leading to the distance model; a summary of the physical background is given, as well as the most important equations leading to the pseudo-circle principle.
In Section 3 we summarize known results in the case of uniform thickness as developed by Gunnar Aronsson and programmed and investigated by Mats Aigner and Mikael Lundquist, as well as results by others in constructing and investigating the complexity of algorithms for solving the problems that appear.
In Section 4 we state and analyze the main problem of the investigation in this thesis, namely The Weighted Region Problem (WRP); we also summarize results by others. We give a proof of the existence of a shortest path in the WRP, point out some minor mistakes in published papers, investigate properties of a shortest path and give a uniqueness result under certain natural restrictions. By analyzing a specific case we illustrate some phenomena that can occur in the WRP.
In Section 5 we formulate our algorithm for solving the WRP stated in Section 4 as well as some investigations of the complexity of the algorithm.
In Section 6 we investigate an idea for improving the method, still using the formulation as a WRP, by including influence from the boundary of the mould.
In Section 7 we give the results of our numerical experiments, including execution time on different computers and some remarks regarding an error estimate given in Section 5.
In Section 8 we explain our visualization of results from our computations, specifically how to detect one interesting phenomenon described in Section 1.3, namely weld lines. The problem of detecting air traps from the results of our computations has been solved by Fredrik Berntsson.
In Section 9 we make some final remarks on the method and make suggestions for future research.
Peter Johansson (pejoh@mai.liu.se)