Cubic spline elements for modelling microstructural evolution of materials controlled by solid-state diffusion and grain-boundary migration

A new set of finite element formulations are presented in this paper to model surface diffusion, grain-boundary diffusion, grain-boundary migration and their interaction. The new formulations use the classical cubic splines both to represent material interface and to act as shape functions for the migration velocity of the interface. The smoothness of the interface is enforced such that the second order derivatives of the migration velocity is continuous anywhere on the interface. This is achieved by using the cubic spline shape functions and by introducing two new Lagrange terms in the variational principle. The work presented here is a new development to the finite element scheme which was previously developed by Pan, Cocks and their co-workers [Comput. Mater. Sci. 18 (2000) 76; Comput. Mater. Sci. 1 (1993) 95; Acta Mater. 43 (1995) 1395; Proc. Roy. Soc. London A 453 (1997) 2161] for modelling microstructural evolution of materials. The cubic spline elements provide a numerically efficient alternative to the linear elements used by Pan et al. [Proc. Roy. Soc. London A 453 (1997) 2161]. The finite element formulations are verified using a series of test cases for which analytical solutions exist in the literature. A further demonstration case of the co-sintering of two particles of different sizes is provided. The new finite element scheme has made it possible to carry out computer simulations of microstructural evolution using sophisticated and more realistic material models than ever before. The over-simplifications in various existing material models can lead to incorrect predictions. This is dramatically demonstrated in a separate paper in which the finite element scheme is used to investigate the sintering behaviour of large pores [Mech. Mater. (2003)].