In the context of a project on additive manufacturing, molten aluminum is fed vertically through a pipe and deposited at the exit onto a cold substrate moving in the horizontal direction, thus forming a free surface in air. Upon deposition, the liquid aluminum is cooled by the substrate and ultimately solidifies. Both the solidification front and free surface of the aluminum constitute unknowns of our manufacturing process and may be predicted only with the help of advanced simulation techniques that solve free boundary problems (FBPs).
FBPs are a class of partial differential equations (PDE) where the boundary of the domain is unknown a priori and is an output of the PDE solution. A classical example is the Stefan problem where a semi-infinite material, initially in liquid state, is brought into contact at one end with a cold source (below freezing temperature). A resulting solidification front advances through the material in time and thus forms the free boundary. Another example is the die swell problem where a liquid exits a pipe into air. In this case, the shape of the liquid jet is the free boundary and depends on the flow rate of the fluid. In FBPs, traditional fixed-domain techniques for PDEs like the finite element method (FEM), the finite difference method, etc. cannot be implemented in standard form as they do not accommodate unknown domains; however, they may be applied when adequately coupled with iteration.
In this Master’s project, the student will be tasked with the development of numerical schemes for the solution of FBPs found in the context of solidification. An iterative approach predicated on the Newton-Raphson method and FEM will be employed to solve these FBPs with the help of the Fenics finite element library. Upon the development of robust algorithms for solidification modeling, the student will couple fluid flow to their simulation and finally, if time permits, they will apply their algorithm to a simplified version of the aforementioned additive manufacturing process. Knowledge of numerical methods for PDEs, as well as an ability to program will be crucial in this project. Interest in physics, namely those involved in the additive manufacturing process at hand is also welcome
Contact: Tyler Benkley