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Retrieving Topological Information of Implicitly Represented Diffuse Interfaces with Adaptive Finite Element Discretization

Published online by Cambridge University Press:  03 June 2015

Jian Zhang  and
Qiang Du
Jian Zhang*
Affiliation:
Supercomputing center, Chinese Academy of Sciences, Beijing, P.R. China State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing, P.R. China
Qiang Du*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
*
Corresponding author.Email:zhangjian@sccas.cn
Email:qdu@math.psu.edu
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Abstract

We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework. Utilizing an adaptive finite element implementation with effective gradient recovery techniques, we discuss how the Euler number can be accurately computed directly from the nu-merically solved phase field functions or order parameters. Numerical examples and applications to the topological analysis of point clouds are also presented.

Keywords

65N3057M5074A5092C05
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013 , pp. 1209 - 1226
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Anderson, D., McFadden, G. and Wheeler, A., Diffuse-interface methods in fluid mechanics, Annual Review of Fluid Mechanics, 30: 139–165 (1998).CrossRefGoogle Scholar
[2]Babuska, I. and Strouboulis, T., The Finite Element Method and its Reliability, Oxford University Press, London (2001).Google Scholar
[3]Barrett, J. W., Blowey, J. F., and Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal. 37 (1999), pp. 286–318.Google Scholar
[4]Braun, R. J. and Murray, B. T., Adaptive phase-field computations of dendritic crystal growth, Journal of Crystal Growth, 174 (1997), pp. 41–53.Google Scholar
[5]Caginalp, G. and Chen, X. F., Phase field equations in the singular limit of sharp interface problems, In On the evolution of phase boundaries (Minneapolis, MN, 1990-91), pp. 1–27. Springer, New York, 1992.Google Scholar
[6]Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258–267.Google Scholar
[7]Chen, L.Q., Phase-field models for microstructure evolution, Annual Review of Materials Science, 32 (2002), pp. 113–140.CrossRefGoogle Scholar
[8]Chen, L.Q. and Shen, J., Applications of semi-implicit Fourier-spectral method to phase field equations, Computer Physics Communications, 108 (1998), pp. 147–158.Google Scholar
[9]Chen, Z., Nochetto, R. and Schmidt, A., Error control and adaptivity for a phase relaxation model, M2AN, 34 (2000), 775–797.CrossRefGoogle Scholar
[10]Collins, A., Zomorodian, A., Carlsson, G. and Gulbas, L., A barcode shape descriptor for curve point cloud data, Computers and Graphics, 28 (2004), 881–894.Google Scholar
[11]Döbereiner, H., Evans, E., Kraus, M., Seifert, U., and Wortis, M., Mapping vesicle shapes into the phase diagram: a comparison of experiment and theory, Phys. Rev. E, 55, pp. 4458–4474, 1997.Google Scholar
[12]Du, Q., Phase field calculus, curvature-dependent energies, and vesicle membranes, Philosophical Magazine, 91 (2011), pp. 165–181.Google Scholar
[13]Du, Q., Liu, C. and Wang, X., A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198, pp. 450468, 2004.CrossRefGoogle Scholar
[14]Du, Q., Liu, C. and Wang, X., Retrieving topological information for phase field model, SIAM J. Appl. Math., 65, pp. 1913–1932,2005.CrossRefGoogle Scholar
[15]Du, Q., Liu, C., Ryham, R. and Wang, X., Diffuse Interface Energies Capturing the Euler Number: Relaxation and Renomalization, Comm. Math. Sci., 5, pp. 233–242,2007.CrossRefGoogle Scholar
[16]Du, Q. and Nicolaides, R., Numerical analysis of a continuum model of phase transition, SIAM J. Num. Anal. 28 (1991), pp. 1310–1322.Google Scholar
[17]Du, Q. and Zhang, J., Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations, SIAM J. Sci. Comp., 30, pp. 1634–1657,2008.Google Scholar
[18]Du, Q. and Zhu, L., Analysis of a Mixed Finite Element Method for a Phase Field Elastic Bending Energy Model of Vesicle Membrane Deformation, J. Computational Mathematics, 24, pp. 265–280, 2006.Google Scholar
[19]Elliott, C. and French, D., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA Journal of Applied Mathematics, 38, 97–128,1987.CrossRefGoogle Scholar
[20]Evans, L.C., Soner, H. M., and Souganidis, P. E., Phase transitions and generalized motion by mean curvature Comm. Pure Appl. Math., 45, pp. 1097–1123,1992.Google Scholar
[21]Feng, X and Prohl, A, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numerische Mathematik, 99, pp. 47–84, 2004.CrossRefGoogle Scholar
[22]Feng, W.M., Yu, P., Hu, S.Y., Liu, Z.K., Du, Q. and Chen, L.Q., Spectral Implementation of An Adaptive Moving Mesh Method for Phase-field Equations, 220 (2006), 498–510.Google Scholar
[23]Ghrist, R., Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., 45 (2008), 61–75.Google Scholar
[24]Heimsund, B., Tai, X.-C. and Wang, J., Superconvergence for the gradient of finite element approximations by L2 projections, SIAM J. Numer. Anal., 40, pp. 1263–1280,2003.Google Scholar
[25]Hu, W., Li, R. and Tang, T., A multi-mesh adaptive finite element approximation to phase field models, Communications in Computational Physics, 5, pp. 1012–1029,2009.Google Scholar
[26]Kunert, G. and Nicaise, S., Zienkiewicz - Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 1013–1043.Google Scholar
[27]Kwon, Y., Thornton, K. and Voorhees, P., The topology and morphology of bicontinuous interfaces during coarsening, Euro. Phys. Lett., 86, 46005,2009.Google Scholar
[28]Lakhany, A. M., Marek, I., and Whiteman, J. R., Superconvergence results on mildly structured triangulations, Comput. Methods Appl. Mech. Engrg., 189 (2000), pp. 1–75.Google Scholar
[29]Lowengrub, J. and Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), pp. 2617–2654.Google Scholar
[30]Lipowsky, R., The conformation of membranes, Nature, 349,475–481,1991.Google Scholar
[31]Mouritsen, O., Life - As a Matter of Fat: The Emerging Science of Lipidomics, Springer, Berlin, 2005.Google Scholar
[32]Ou-Yang, Z., Liu, J., and Xie, Y., Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, World Scientific, Singapore, 1999.Google Scholar
[33]Provatas, N., Goldenfeld, N. and Dantzig, J., Efficient Computation of Dendritic Microstruc-tures Using Adaptive Mesh Refinement, Phys. Rev. Lett., 80 (1998), 3308–3311.Google Scholar
[34]Seifert, U., Berndl, K. and Lipowsky, R., Configurations of fluid membranes and Vesicles, Phys. Rev. A, 44, pp. 1182–1202,1991.Google Scholar
[35]Steinbach, I., Phase-field models in materials science, Modelling Simul. Mater. Sci. Eng., 17, pp.073001, 2009.Google Scholar
[36]Taylor, J., Some Mathematical Challenges in Materials Science, Bull. of AMS, 40, 69–87,2003.Google Scholar
[37]Verfürth, R., A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner Skripten zur Numerik, B.G. Teubner, Stuttgart (1995).Google Scholar
[38]Zhang, Z. and Naga, A., A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput., 26, pp. 1192–1213,2005.Google Scholar
[39]Zienkiewicz, O.C. and Zhu, J.Z., The superconvergence patch recovery and a posteriori error estimates. part 1: the recovery technique, Int. J. Numer. Methods Engrg., 33, pp. 1331–1364, 1992.Google Scholar
[40]Zienkiewicz, O. C. and Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Numer. Methods Engrg., 33 (1992), pp. 1365–1382.Google Scholar