A regularized mesh transformation method for the computation of crystalline microstructures

Since the finite-element approximations of microstructures are strongly mesh dependent, the use of the mesh transformation method based on the energy minimizing principle is considered a natural approach in the finite-element computation of microstructures. However, without a control on the mesh quality, the mesh can become increasingly irregular in the process of energy minimization and thus jeopardize the convergence of the algorithm. In this paper, a mesh quality control term based on the mesh regularity is introduced to regularize the mesh transformation procedure. The existence and convergence of the regularized mesh transformation method are proved. Numerical experiments show that the method does help to produce much better numerical results in the computation of microstructures.