Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering

Abstract

We study the coupled surface and grain boundary motion in bi-and tricrystals in three space dimensions, building on previous work by the authors on the simplified two dimensional case. The motion of the interfaces, which in this paper are presented by two-dimensional hypersurfaces, is described by two types of normal velocities: motion by mean curvature and motion by surface diffusion. Three hypersurfaces meet at triple junction lines, where junction conditions need to hold. Similarly, boundary conditions are prescribed where an interface meets an external boundary, and these conditions naturally give rise to contact angles. We present a variational formulation of the flows, which leads to a fully practical finite element approximation that exhibits excellent mesh properties, with no mesh smoothing or remeshing required in practice. For the introduced parametric finite element approximation we show well-posedness and, in general, unconditional stability, i.e. there is no restriction on the chosen time step size. Moreover, the induced discrete equations are linear and easy to solve. A generalization to anisotropic surface energies is straightforward. Several numerical results in two and three space dimensions are presented, including simulations for thermal grooving and sintering. Three dimensional simulations featuring quadruple junction points, nonstandard boundary contact angles and fully anisotropic surface energies are also presented.

Finite element methods for fourth order axisymmetric geometric evolution equations

“…In most cases, the contact energies are assumed to be the same, so that (0) ∂S = (1) ∂S = 0, which leads to 90 • contact angle conditions in (2.9), and means that (2.8) collapses to (2.7). See [6] for more details on contact angles and contact energies. We note that a necessary condition to admit a solution to (2.9a) or to (2.9b) is that | (p) ∂S | ≤ 1, but we do allow for more general values in (2.8).…”

A parametric finite element method for solid-state dewetting problems with anisotropic surface energies

“…where the curve X h 2 , τ 4 (ρ, t) belongs to the piecewise linear finite element vector spaces and at the interval nodes ρ = ρ j , its values are equal to the values of numerical solutions X h 2 , τ 4 (ρ j , t). We remark here that the above L ∞ -norm in (2.39) has been adapted in [38][39][40][41] for studying numerically convergence rate of the PFEM for time evolution of a closed curve under motion by mean curvature or the Willmore flow.…”

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

“…in thermal grooving ( [57]), in interface motion in polycrystalline two-phase materials ( [21]), in sintering processes ( [60]), and in the evolution of boundaries in the electromigration of intergranular voids (see [8]). A parametric finite element approximation of such flows for curve networks in the plane has been considered in [6], while its extension to surface clusters is the subject of the forthcoming article [13]. However, in this paper we will only consider the purely second and fourth order flows (2.2) and (2.3), respectively, as well as their anisotropic counterparts.…”