An FVE-FAC approach for the weldpool problem

Abstract

A block of metal is subjected to a concentrated heat source resulting in a pool of molten metal surrounded by a portion of the unmelted metal. The governing system of equations is known as the weldpool problem. The weldpool problem is discretized using finite differences to discretize time derivatives and the Finite Volume Element method (FVE) to discretize spatial derivatives. Multigrid methods, known to be effective on uniform grids, make use of overlapping uniform grids of different scales to better approximate solutions to the weldpool problem. However, the solid-liquid interface requires extremely fine grid resolution and accuracy to resolve the physical behavior of the weldpool problem at this interface. Being too costly to apply a global fine domain, a non-uniform domain is developed to utilize finer resolution along the interface while still maintaining a coarser resolution on the rest of the domain. The fast adaptive composite grid method (FAC) is introduced, incorporating the concepts of multigrid to solve the weldpool problem on this non-uniform discrete domain. FVE is then applied to the conduction equation in the solid and the convection-diffusion equation in the liquid metal to develop stencil equations for use in FAC.