A new set of finite element formulations was presented in order to model surface diffusion, grain-boundary diffusion, grain-boundary migration and their interaction. The new formulations used classic cubic splines to represent material interfaces and to act as shape functions for the migration velocity of the interface. The smoothness of the interface was controlled so that the second-order derivatives of the migration velocity were continuous anywhere on the interface. This was achieved by using cubic spline shape functions and by introducing 2 new Lagrange terms into the variational principle. This was a new development of the finite element scheme of Pan, Cocks et al., for modelling the microstructural evolution of materials. The cubic spline elements were a numerically efficient alternative to the linear elements used by the latter. The finite element formulations were verified by using a series of test cases for which analytical solutions existed.

Cubic Spline Elements for Modelling Microstructural Evolution of Materials Controlled by Solid-State Diffusion and Grain-Boundary Migration. H.N.Chng, J.Pan: Journal of Computational Physics, 2004, 196[2], 724-50