High-order adaptive arbitrary-Lagrangian-Eulerian (ALE) simulations of solidification

A high-order-accurate method for simulation of solidification is presented. The solidification front is tracked using a triangular, Arbitrary-Lagrangian-Eulerian moving mesh, and a mesh adaption algorithm is used to allow simulations of unsteady problems with large interfacial movement. An improved mesh coarsening algorithm is presented that maintains high quality deforming meshes while reducing the amount of interpolation needed to transfer solutions between meshes. An hp-finite element method is used to resolve the thermal and flow fields. This is combined with an A-stable diagonally-implicit Runge-Kutta temporal scheme. The method was demonstrated to give a temporal order of accuracy near 3 by comparing to a 1D analytic solution of melting. The spatial accuracy was calculated to be nearly 5th order for an approximation degree, p, equal to 4. Even for this simple case, the mesh adaption algorithm improved the accuracy over a simulation where the mesh only deformed. For a practical demonstration, the algorithm was used to simulate horizontal ribbon growth of single-crystal silicon and was able to resolve solutions where the solid layer thickness decreased by a factor of 20 over the course of the simulation.