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

Published online by Cambridge University Press:  20 August 2015

Bo Li  and
John Shopple
Bo Li*
Affiliation:
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
John Shopple*
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112. La Jolla, CA 92093-0112, USA
*
Corresponding author.Email:bli@math.ucsd.edu
Email:jshopple@math.ucsd.edu
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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.

Keywords

65MLevel set methodfinite elementinterface-fitted meshcurvature approximationvelocity extensionreinitializationsolidificationdendritesmolecular solvationvariational implicit-solvent models
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 32 - 56
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Che, J., Dzubiella, J., Li, B., and McCammon, J. A., Electrostatic free energy and its variations in implicit solvent models, J. Phys. Chem. B., 112 (2008), 3058–3069.CrossRefGoogle ScholarPubMed
[2]Chen, S., Merriman, B., Osher, S., and Smereka, P., A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), 8–29.Google Scholar
[3]Cheng, L.-T., Dzubiella, J., McCammon, J. A., and Li, B., Application of the level-set method to the implicit solvation of nonpolar molecules, J. Chem. Phys., 127 (2007), 084503.Google Scholar
[4]Cheng, L.-T., Xie, Y., Dzubiella, J., McCammon, J. A., Che, J., and Li, B., Coupling the level-set method with molecular mechanics for variational implicit solvation of nonpolar molecules, J. Chem. Theory. Comput., 5 (2009), 257–266.CrossRefGoogle ScholarPubMed
[5]Deckelnick, K. P., Dziuk, G., and Elliott, C. M., Computation of geometric PDEs and mean curvature flow, Acta. Numer., 14 (2005), 139–232.Google Scholar
[6]Di Pietro, D. A., Lo Forte, S., and Parolini, N., Mass preserving finite element implementations of the level set method, Appl. Numer. Math., 56 (2006), 1179–1195.CrossRefGoogle Scholar
[7]Dzubiella, J., Swanson, J. M. J., and McCammon, J. A., Coupling hydrophobicity, dispersion, and electrostatics in continuum solvent models, Phys. Rev. Lett., 96 (2006), 087802.Google Scholar
[8]Dzubiella, J., Swanson, J. M. J., and McCammon, J. A., Coupling nonpolar and polar solvation free energies in implicit solvent models, J. Chem. Phys., 124 (2006), 084905.CrossRefGoogle ScholarPubMed
[9]Fried, M., A level set based finite element algorithm for the simulation of dendritic growth, Comput. isual. Sci., 7 (2004), 97–110.Google Scholar
[10]Gong, Y., Li, B., and Li, Z., Immersed-interface finite-element method for elliptic interface problems with non-homogeneous jump conditions, SIAM J. Numer. Anal., 46 (2008), 472–495.CrossRefGoogle Scholar
[11]Groß, S., Reichelt, V., and Reusken, A., A finite element based level set method for two-phase incompressible flows, Comput. Visual. Sci., 9 (2004), 239–257.Google Scholar
[12]Osher, S., and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, 2003.Google Scholar
[13]Osher, S., and Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1998), 12–49.Google Scholar
[14]Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge, 1999.Google Scholar
[15]Tornberg, A.-K., Interface Tracking Methods with Application to Multiphase Flows, PhD thesis, KTH, Stockholm, Sweden, (2000).Google Scholar
[16]Tornberg, A.-K., and Engquist, B., A finite element based level set method for multiphase flow applications, Comput. Visual. Sci., 3 (2000), 93–101.Google Scholar
[17]Xie, H., Ito, K., Li, Z., and Toivanen, J., A finite element method for interface problems with locally modified triangulations, Contemp. Math., 466 (2008), 179–190.Google Scholar
[18]Xie, X., and Li, Z., A finite element method for elasticity interface problems with locally modified triangulations, Int. J. Numer. Methods., 2009, in press.Google Scholar