Numerical Methods for Solidification Processes in Materials Science

摘要

Our work on moving interface problems in materials science combines fast PDE solvers such as boundary integral methods with fast geometric algorithms and semi-Lagrangian implicit representations to build effective numerical methods. We developed an implicit boundary integral method for computing periodic dendrite formation in the symmetric model of unstable solidification 7 and fast algorithms for evaluating heat. potentials 2 which speeded up out method by several orders of magnitude. In 6, we combined the boundary integral method of 7 with fast algorithms from 2,3,8 and the level set method of 4: the level set method handled topological changes effectively while fast boundary integral techniques ensured accuracy and efficiency in the velocity evaluation. We developed and analyzed efficient and accurate new vortex methods for modeling convection in the melt 4,10,11, together with new error analyses 12 and quadrature rules 9 for general integral equations. Since 1999, we have focused on the development and implementation of highly effective new numerical methods for general moving interface problems and widely applicable subsidiary computations. We summarize three projects below: the fast modular semi-Lagrangian method for general moving interfaces (described in Publications P112 and references 13-16). accurate contouring methods in two and three dimensions, and fast solution of two-point boundary value problems P3,P4.