A LAGRANGIAN-EULERIAN FINITE-ELEMENT FORMULATION FOR STEADY-STATE SOLIDIFICATION PROBLEMS

An accurate finite-element methodology is developed to solve a steady-state solidification problem for pure materials. Generally, this type of problem is governed by a set of conduction-advection differential equations. We consider a solidifying body moving at a constant velocity. At a steady state, the solid/liquid interface is fixed to an observer in a Eulerian frame, and the movement of the interface to an observer in a Lagrangian frame is determined by an energy balance equation at the interface. To determine the interface position in the Eulerian frame, we use the composition of the steady-state velocity of the moving body and the solid/liquid interface velocity in the Lagrangian frame through an iterative process. Meanwhile, the finite-element mesh is updated with a transfinite mapping scheme. In this new methodology, a weak formulation is applied to the interface energy balance equation to calculate the interface velocity in the Lagrangian frame. Numerical results are compared with analytical solutions for one-dimensional steady-state solidification problems, and excellent agreement is achieved. Several two-dimensional examples are provided to demonstrate the capability of the new methodology.