Modeling Microstructure Evolution Using Gradient-Weighted Moving Finite Elements

Abstract

Microstructure evolution, where grain boundaries evolve by mean curvature motion, is modeled in three dimensions (3-D) using gradient-weighted moving finite elements (GWMFE). To do this, we modify and extend an existing 2-D GWMFE code to create a new code GRAIN3D which makes the 3-D microstructure modeling possible. The right-hand side term which drives the GWMFE motion can be viewed as surface tension forces, that is, as the negative gradient of the surface integral of a constant energy density $\mu$ on the triangular interfacial grid. Extensions to the method include equations for the motion of tetrahedra that are conformally attached to the moving piecewise linear triangular facets which represent the GWMFE discretization of the evolving grain boundaries. We present some new regularization terms which control element quality, as well as preventing element collapse in the simulation. New capabilities for changing the mesh topology are used to keep the grid edge lengths below a maximum allowable length hmax and to mimic actual changes in the physical topology, such as collapse and disappearance of individual grains. Validating runs are performed on some test cases that can be analytically solved, including collapse of a spherical grain and the case of columnar microstructure. In the spherical collapse case, the GWMFE method appears to have an error in the surface area collapse rate $-{dA\over dt}$ which is ${\cal O}((\Delta\theta)^2)$, where $\Delta\theta$ is a measure of the angular resolution of the mesh. Finally, a run is presented where a true 3-D microstructure (possessing triple lines and quadruple points in the interior and triple points on the exterior boundaries) is evolved to a "2-D" columnar microstructure and finally evolved down to a single grain.