Goal-oriented adaptive refinement for phase field modeling with finite elements

Phase field modeling is very often performed with the finite-difference method for equally spaced grids. Typically its solutions are highly non-homogenous; and, therefore, non-equally spaced grids with dense meshes at interfaces between different phases and coarse meshes in homogenous regions would be more advantageous with respect to both, efficiency and reliability of the numerical solutions. To this end, in the present work, an adaptive strategy with finite elements for phase field modeling is adopted, where the time step and the grid size are selected on the basis of goal-oriented error estimation. In order to account for nonlinearity of the variational equations, we introduce a secant form for the dual problem, which for practical purposes is approximated by a tangent form. In a numerical example, we investigate transformation and retransformation for a two-phase system in a square region subjected to thermal loading. Copyright (c) 2013 John Wiley & Sons, Ltd.