A continuous approach to discrete ordering on S2

Multiscale Modeling and Simulation

Volumn 9, Issue 1, 2011, Pages 314-334

  Backofen, Rainer ^a   Gräf, Manuel ^b   Potts, Daniel ^b   Praetorius, Simon ^a   Voigt, Axel ^a   Witkowski, Thomas ^a  

^a DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

^b CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Dynamic density functional theory; Ordering on curved surfaces; PDE on surfaces; Phase field crystal; Thomson problem

Indexed keywords

CURVED SURFACES; DYNAMIC DENSITY FUNCTIONAL THEORY; PDE ON SURFACES; PHASE FIELDS; THOMSON PROBLEM;

ASYMPTOTIC ANALYSIS; FAST FOURIER TRANSFORMS; FINITE ELEMENT METHOD; IMAGE SEGMENTATION; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SPECTROSCOPY; SPHERES;

DENSITY FUNCTIONAL THEORY;

EID: 79955920215     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/100787532     Document Type: Article

References (51)

1. S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), pp. 7-15. (Pubitemid 128353810)
2. M. Shub and S. Smale, Complexity of Bezout's theorem, iii: Condition number and packing, J. Complexity, 9 (1993), pp. 4-14.
3. J. J. Thomson, On the structure of the atom: An investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results on the theory of atomic structure, Philos. Mag., 7 (1904), pp. 237-265.
4. T. Erber and G. M. Hockney, Equilibrium configurations of n equal charges on a sphere, J. Phys. A, 24 (1991), pp. L1369-L1377.
5. A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G. Nikolaides, A. Travesset, and D. A. Weitz, Grain boundary scars and spherical crystallography, Science, 299 (2003), pp. 1716-1718. (Pubitemid 36337194)
6. L. Ramos, T. C. Lubensky, N. Dan, P. Nelson, and D. A. Weitz, Surfactant-mediated two-dimensional crystallization of colloidal crystals, Science, 286 (1999), pp. 2325-2328. (Pubitemid 30016890)
7. A. D. Dinsmore, M. F. Hsu, M. G. Nikolaides, M. Marquez, A. R. Bausch, and D. A. Weitz, Colloidosomes: Selectively permeable capsules composed of colloidal particles, Science, 298 (2002), pp. 1006-1009. (Pubitemid 35247303)
8. K. Stratford, R. Adhikari, I. Pagonabarraga, J. C. Desplat, and M. E. Cates, Colloidal jamming at interfaces: A route to fluid-bicontinuous gels, Science, 309 (2005), pp. 2198- 2201. (Pubitemid 41398167)
9. Y. G. Sun, W. M. Choi, H. Q. Jiang, Y. Y. Huang, and J. A. Rogers, Controlled buckling of semiconductor nanoribbons for strechable electronics, Nat. Nanotechnology, 1 (2007), pp. 201-207.
10. J. Lindmar, L. Mirny, and D. R. Nelson, Virus shapes and buckling transitions in spherical shells, Phys. ReV. E, 68 (2003), 051910.
11. R. Zandi, D. Reguera, R. F. Bruisma, W. M. Gelbart, and J. Rudnick, Origin of icosahedral symmetry in viruses, Proc. Natl. Acad. Sci. USA, 101 (2004), pp. 15556-15560. (Pubitemid 39473526)
12. R. Zandi and D. Reguera, Mechnical properties of viral capsids, Phys. ReV. E, 72 (2005), 021917.
13. M. J. Bowick and L. Giomi, Two-dimensional matter: Order, curvature and defects, AdV. Phys., 58 (2009), pp. 449-563.
14. K. R. Elder, M. Katakowski, M. Haataja, and M. Grant, Modeling elasticity in crystal growth, Phys. ReV. Lett., 88 (2002), 245701.
15. N. Provatas and K. Elder, Phase-field Methods in Materials Science and Engineering, 1st ed., Wiley-VCH, Berlin, Germany, 2010.
16. R. Backofen, A. Voigt, and T. Witkowski, Particles on curved surfaces: A dynamic approach by a phase-field-crystal model, Phys. ReV. E, 81 (2010), 025701(R).
17. T. Erber and G. M. Hockney, Complex systems: Equilibrium configurations of n equal charges on a sphere (2 ≤ n ≤ 112), Adv. Chem. Phys., 98 (1997), pp. 495-594. (Pubitemid 127030750)
18. S. van Teefelen, R. Backofen, A. Voigt, and H. Löwen, Derivation of the phase-fieldcrystal model for colloidal solidification, Phys. Rev. E, 79 (2009), 051404.
19. U. M. B. Marconi and P. Tarazona, Dynamic density functional theory of fluids, J. Chem. Phys., 110 (1999), pp. 8032-8044.
20. A. J. Archer and R. Evans, Dynamic density functional theory and its application to spinodal decomposition, J. Chem. Phys., 121 (2004), pp. 4246-4254.
21. T. V. Ramakrishnan and M. Yussouff, 1st-principles order-parameter theory of freezing, Phys. ReV. B, 19 (1979), pp. 2775-2794.
22. K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing, Phys. ReV. B, 75 (2007), 064107.
23. K. A. Wu and A. Karma, Phase-field crystal modeling of equilibrium bcc-liquid interfaces, Phys. ReV. B, 88 (2007), 245701.
24. A. Jaatinen, C. V. Achim, K. R. Elder, and T. Ala-Nissila, Thermodynamics of bcc metals in phase-field-crystal models, Phys. ReV. E, 80 (2009), 031602.
25. C. M. Elliott and G. Dziuk, Finite elements on evolving surfaces, IMA J. Numer. Anal., 27 (2007), pp. 262-292.
26. S. Vey and A. Voigt, Amdis - adaptive multidimensional simulations, Comput. Vis. Sci., 10 (2007), pp. 57-66.
27. D. A. Calthoun, C. Helzel, and R. J. LeVeque, Logically rectangular grids and finite volume methods for PDEs in circular and spherical domains, SIAM ReV., 50 (2008), pp. 723-752.
28. M. Bertalmio, L. T. Cheng, S. Osher, and G. Sapiro, Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys., 174 (2001), pp. 759-780.
29. A. Rätz and A. Voigt, PDE's on surfaces - a diffuse interface approach, Commun. Math. Sci., 4 (2006), pp. 575-590.
30. A. Rätz and A. Voigt, A diffuse-interface approximation for surface diffusion including adatoms, Nonlinearity, 20 (2007), pp. 177-192.
31. J. Lowengrub, A. Rätz, and A. Voigt, Phase-field approximation of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission, Phys. ReV. E, 79 (2009), 031926.
32. K. E. Teigen, X. Li, J. Lowengrub, F. Wang, and A. Voigt, A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 7 (2009), pp. 1009-1037.
33. P. Schwartz, D. Adalsteinsson, P. Coletta, A. P. Arkin, and M. Onsum, Numerical computation of diffusion on a sphere, Proc. Natl. Acad. Sci. USA, 102 (2005), pp. 11151-11156.
34. S. J. Ruuth and B. Merriman, A simple embedding method for solving partial differential equations on surfaces, J. Comput. Phys., 227 (2008), pp. 2118-2129.
35. C. M. Elliott and B. Stinner, Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface, Math. Models Methods Appl. Sci., 19 (2009), pp. 787-802.
36. C. Landsberg and A. Voigt, A multigrid finite element method for reaction-diffusion systems on surfaces, Comput. Vis. Sci., 13 (2010), pp. 177-185.
37. A. Voigt and T. Witkowski, Hybrid parallelization of an adaptive finite element code, Kybernetika, 46 (2010), pp. 316-327.
38. R. Backofen, A. Rätz, and A. Voigt, Nucleation and growth by a phase-field-crystal (pfc) model, Phil. Mag. Lett., 87 (2007), pp. 813-820. (Pubitemid 47621500)
39. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Press, New York, 2000.
40. J. S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, 2007.
41. C. Müller, Spherical Harmonics, Springer, Aachen, 1966.
42. L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Comm., 108 (1998), pp. 147-158. (Pubitemid 128392371)
43. S. Kunis and D. Potts, Fast spherical Fourier algorithms, J. Comput. Appl. Math., 161 (2003), pp. 75-98. (Pubitemid 37379463)
44. J. Keiner, S. Kunis, and D. Potts, Using NFFT3 - a software library for various nonequispaced fast Fourier transforms, ACM Trans. Math. Software, 36 (2009), pp. 1-30.
45. J. Keiner and D. Potts, Fast evaluation of quadrature formulae on the sphere, Math. Comp., 77 (2008), pp. 397-419.
46. S. Kunis, M. Gräf, and D. Potts, On the computation of nonnegative quadrature weights on the sphere, Appl. Comput. Harmon. Anal., 27 (2009), pp. 124-132.
47. W. Freeden, T. Gervens, and M. Schreiner, Constructive Approximation on the Sphere, Oxford University Press, Oxford, 1998.
48. A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc., 350 (1998), pp. 523-538.
49. M. J. Bowick, D. R. Nelson, and A. Travesset, Interacting topological defects on frozen topographies, Phys. ReV. B, 62 (2000), pp. 8738-8751. (Pubitemid 32367337)
50. K. Deckelnick, G. Dziuk, and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14 (2005), pp. 139-232. (Pubitemid 40763883)
51. B. Li, J. Lowengrub, A. Rätz, and A. Voigt, Geometric evolution laws for thin crystalline films: Modelling and numerics, Commun. Comput. Phys., 6 (2009), pp. 433-482.