Y. Ruan, A LAGRANGIAN-EULERIAN FINITE-ELEMENT FORMULATION FOR STEADY-STATE SOLIDIFICATION PROBLEMS, Numerical heat transfer. Part B, Fundamentals, 26(3), 1994, pp. 335-351
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 e
quations. We consider a solidifying body moving at a constant velocity
. At a steady state, the solid/liquid interface is fixed to an observe
r in a Eulerian frame, and the movement of the interface to an observe
r in a Lagrangian frame is determined by an energy balance equation at
the interface. To determine the interface position in the Eulerian fr
ame, 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 up
dated with a transfinite mapping scheme. In this new methodology, a we
ak formulation is applied to the interface energy balance equation to
calculate the interface velocity in the Lagrangian frame. Numerical re
sults are compared with analytical solutions for one-dimensional stead
y-state solidification problems, and excellent agreement is achieved.
Several two-dimensional examples are provided to demonstrate the capab
ility of the new methodology.