Impact Forces in Injection Molding was my 2005 Applied Mathematics Honours research project and was supervised by Prof. Tim Myers.
This project uses both analytic and numerical methods to investigate the forces in an injection molding system when there is insufficient material in the mold to slow the piston before it impacts the housing. Finite element methods, usually used in engineering design, however fail to account for the behaviour of the squeeze film of lubricant that forms between the piston and flange (which tends to reduce the maximum force exerted). The most basic model assumed the lubricant was a Newtonian fluid, which was then expanded to include a non-Newtonian component in which the fluid viscosity increases exponentially with pressure. Finally, we also included a first-order model of the deformation of the piston surface due to elastohydrodynamic lubrication and found that it reduces the maximum load during impact.
The most difficult part was the first-order deformation model which suffers from numerical instability. The system is riding close to an asymptote at which pressure goes to infinity, and the gradient is so steep that even the tiniest step by the numerical integration can send the system over the asymptote (and into the complex plane).