Finite Element Simulations of Solidification with a Fixed Mesh Front Tracking Method

Abstract

Moving boundary problems in which the location of an interface must be detennined as part of the solution arise in many scientific and engineering applications one of utmost
importance of which is crystal growth. There are three basic ways to address these problems numerically: (1) Fixed grid methods based on a finite different or finite element
discretization in which the basic computational mesh is fixed and the interface is tracked using either a field variable or a discrete set of points that define it. (2) Adaptive grid methods also based on finite differences or finite elements in which the interface is described by grid points in the computational mesh. (3) Boundary integral methods in which the problem is cast in the form of an integral representation of the interface. Here, we present a fixed grid method based on finite elements where the interface is described by a set of tracer points. We solve the conservation equations of energy and solute transport and pay special attention to the accurate calculation of the interface velocity.
We also show that the method is second-order accurate. Example applications including the solidification of a binary alloy under a temperature gradient and the growth of dendrites into an under-cooled liquid are presented.