Commun. Comput. Phys., 10 (2011), pp. 32-56.


An Interface-Fitted Finite Element Level Set Method with Application to Solidification and Solvation

Bo Li 1*, John Shopple 2

1 Department of Mathematics and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112. La Jolla, CA 92093-0112, USA.
2 Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112. La Jolla, CA 92093-0112, USA.

Received 23 May 2009; Accepted (in revised version) 24 September 2009
Available online 7 February 2011
doi:10.4208/cicp.230510.240910a

Abstract

A new finite element level set method is developed to simulate the interface motion. The normal velocity of the moving interface can depend on both the local geometry, such as the curvature, and the external force such as that due to the flux from both sides of the interface of a material whose concentration is governed by a diffusion equation. The key idea of the method is to use an interface-fitted finite element mesh. Such an approximation of the interface allows an accurate calculation of the solution to the diffusion equation. The interface-fitted mesh is constructed from a base mesh, a uniform finite element mesh, at each time step to explicitly locate the interface and separate regions defined by the interface. Several new level set techniques are developed in the framework of finite element methods. These include a simple finite element method for approximating the curvature, a new method for the extension of normal velocity, and a finite element least-squares method for the reinitialization of level set functions. Application of the method to the classical solidification problem captures the dendrites. The method is also applied to the molecular solvation to determine optimal solute-solvent interfaces of solvation systems.

AMS subject classifications: 65M

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Key words: Level set method, finite element, interface-fitted mesh, curvature approximation, velocity extension, reinitialization, solidification, dendrites, molecular solvation, variational implicit-solvent models.

*Corresponding author.
Email: bli@math.ucsd.edu (B. Li), jshopple@math.ucsd.edu (J. Shopple)
 

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