Commun. Comput. Phys.,
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Volume 3.


A General Moving Mesh Framework in 3D and its Application for Simulating the Mixture of Multi-Phase Flows

Yana Di 1, Ruo Li 2*, Tao Tang 3

1 P.O. Box 2719, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China.
2 LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China.
3 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong; and Faculty of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China.

Received 20 December 2006; Accepted (in revised version) 10 June 2007
Available online 30 October 2007

Abstract

In this paper, we present an adaptive moving mesh algorithm for meshes of unstructured polyhedra in three space dimensions. The algorithm automatically adjusts 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 based on 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 will cost as little as possible comparing with the solution evolution part. Since the mesh redistribution procedure normally requires to solve large size matrix equations, we will 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-grid method. 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 describes the motion of mixtures of two incompressible fluids. Numerical results on two- and three-dimensional simulations will be presented.

AMS subject classifications: 65M20, 65M50, 65M60

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Key words: Moving mesh methods, multi-phase flows, unstructured tetrahedra, phase field model, Navier-Stokes equations, finite element method.

*Corresponding author.
Email: yndi@lsec.cc.ac.cn (Y.-N. Di), rli@math.pku.edu.cn (R. Li), ttang@math.hkbu.edu.hk (T. Tang)
 

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