A combined finite element and finite difference scheme for computer simulation of microstructure evolution and its application to pore–boundary separation during sintering

Abstract

This paper presents two new developments to the numerical technique previously developed by Pan and Cocks [cf. J. Pan, A.C.F. Cocks, A numerical technique for the analysis of coupled surface and grain-boundary diffusion, Acta Metall. 43 (1995) 1395–1406; J. Pan, A.C.F. Cocks, S. Kucherenko, Finite element analysis of coupled grain-boundary and surface diffusion with grain-boundary migration, Proc. R. Soc. London A 453 (1998) 2161–2184] for the computer simulation of microstructure evolution of materials. Coupled grain-boundary diffusion, surface diffusion and grain-boundary migration are considered as the underlying mechanisms for the evolution. The first development is that a set of “link elements” are developed to link the finite difference scheme of Pan and Cocks (1995) with the finite element scheme of Pan et al. (1998) for the surface diffusion and grain-boundary migration parts of the problem respectively. Unlike the method used in Pan and Cocks (1995), these link elements are designed to link the two discretisation schemes away from the interface junction so that the dihedral angle can be maintained in the variational sense at the junction while the finite difference scheme can still be used for most of the interface network. Such a combined scheme is more efficient than the full finite element scheme because most of the degrees of freedom for surface diffusion and grain-boundary migration no longer contribute to the global linear simultaneous equations. The second development is that an implicit time integration method is implemented. In general the implicit time integration method allows much larger timesteps to be used than that allowed by an explicit method. The two new developments together significantly improved the efficiency of the numerical scheme. Several test cases are provided to verify the numerical scheme. As an example of application, the effect of pore shrinkage on pore-boundary separation is investigated using the numerical scheme. It is shown that the existing separation criteria significantly over-predict separation.