| Abstract: | In this paper, we present an adaptive moving mesh algorithm for meshesof unstructured polyhedra in three space dimensions. The algorithm automaticallyadjusts the size of the elements with time and position in the physical domain to resolve the relevant scales in multiscale physical systems while minimizing computational costs. The algorithm is a generalization of the moving mesh methods basedon harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To make 3D moving mesh simulations possible,the key is to develop an efficient mesh redistribution procedure so that this part willcost as little as possible comparing with the solution evolution part. Since the meshredistribution procedure normally requires to solve large size matrix equations, wewill describe a procedure to decouple the matrix equation to a much simpler block-tridiagonal type which can be efficiently solved by a particularly designed multi-gridmethod. To demonstrate the performance of the proposed 3D moving mesh strategy,the algorithm is implemented in finite element simulations of fluid-fluid interface interactions in multiphase flows. To demonstrate the main ideas, we consider the formation of drops by using an energetic variational phase field model which describesthe motion of mixtures of two incompressible fluids. Numerical results on two- andthree-dimensional simulations will be presented. |
