Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations

Authors:
Qiang Du and Lili Ju

Journal:
Math. Comp. **74** (2005), 1257-1280

MSC (2000):
Primary 65N15, 65N99; Secondary 82D55

DOI:
https://doi.org/10.1090/S0025-5718-04-01719-3

Published electronically:
December 8, 2004

MathSciNet review:
2137002

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the numerical approximations of the Ginzburg- Landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal Voronoi tessellations. A reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit. We present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations. Properties of the spherical centroidal Voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field.

- Stephen L. Adler and Tsvi Piran,
*Relaxation methods for gauge field equilibrium equations*, Rev. Modern Phys.**56**(1984), no. 1, 1–40. MR**734670**, DOI https://doi.org/10.1103/RevModPhys.56.1 - Amandine Aftalion and Qiang Du,
*The bifurcation diagrams for the Ginzburg-Landau system of superconductivity*, Phys. D**163**(2002), no. 1-2, 94–105. MR**1887044**, DOI https://doi.org/10.1016/S0167-2789%2801%2900385-2 - B. Baelus and F. Peeters, Dependence of the vortex configuration on the geometry of mesoscopic flat samples,
*Phys. Rev. B*,**65**, 104515, 2002. - B. Baelus, F. Peeters and V. Schweigert, Saddle-point states and energy barriers for vortex entrance and exit in superconducting disks and rings,
*Phys. Rev. B*,**63**, 144517, 2001. - Amandine Aftalion and Qiang Du,
*The bifurcation diagrams for the Ginzburg-Landau system of superconductivity*, Phys. D**163**(2002), no. 1-2, 94–105. MR**1887044**, DOI https://doi.org/10.1016/S0167-2789%2801%2900385-2 - S. Chapman, Q. Du and M. Gunzburger, A Ginzburg-Landau model for superconducting shells, preprint.
- S. J. Chapman and D. R. Heron,
*The motion of superconducting vortices in thin films of varying thickness*, SIAM J. Appl. Math.**58**(1998), no. 6, 1808–1825. MR**1638680**, DOI https://doi.org/10.1137/S0036139996307334 - Zhiming Chen and Shibin Dai,
*Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity*, SIAM J. Numer. Anal.**38**(2001), no. 6, 1961–1985. MR**1856238**, DOI https://doi.org/10.1137/S0036142998349102 - M. Coffey, London model for the levitation force between a horizontally oriented point magnetic dipole and superconducting sphere,
*Phys. Rev. B*,**65**, 214524, 2002. - Shijin Ding and Qiang Du,
*Critical magnetic field and asymptotic behavior of superconducting thin films*, SIAM J. Math. Anal.**34**(2002), no. 1, 239–256. MR**1950834**, DOI https://doi.org/10.1137/S0036141000378619 - M. Dodgson and M.A. Moore, Vortices in a thin-film superconductor with a spherical geometry,
*Phys. Rev. B*,**55**, pp. 3816-3831, 1997. - Q. Du, Discrete gauge invariant approximations of a time-dependent Ginzburg-Landau model of superconductivity,
*Math. Comp.*,**67**, pp. 965-986, 1998. - Qiang Du, Max D. Gunzburger, and Lili Ju,
*Constrained centroidal Voronoi tessellations for surfaces*, SIAM J. Sci. Comput.**24**(2003), no. 5, 1488–1506. MR**1976306**, DOI https://doi.org/10.1137/S1064827501391576 - Qiang Du, Max D. Gunzburger, and Lili Ju,
*Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere*, Comput. Methods Appl. Mech. Engrg.**192**(2003), no. 35-36, 3933–3957. MR**1997836**, DOI https://doi.org/10.1016/S0045-7825%2803%2900394-3 - Qiang Du, Max D. Gunzburger, and Janet S. Peterson,
*Analysis and approximation of the Ginzburg-Landau model of superconductivity*, SIAM Rev.**34**(1992), no. 1, 54–81. MR**1156289**, DOI https://doi.org/10.1137/1034003 - Q. Du, M. Gunzburger and J. Peterson, Computational simulations of type-II superconductivity including pinning mechanisms,
*Phys. Rev. B*,**51**, pp. 16194-16203, 1995. - Q. Du and L. Ju, Finite volume methods on spheres and spherical centroidal Voronoi tessellations, IMA preprint, No. 1918, 2003.
- Q. Du and L. Ju, Numerical simulation of the quantized vortices on a thin superconducting hollow sphere, to appear in
*J. Computational Phys.*, 2004. - Fang-Hua Lin and Qiang Du,
*Ginzburg-Landau vortices: dynamics, pinning, and hysteresis*, SIAM J. Math. Anal.**28**(1997), no. 6, 1265–1293. MR**1474214**, DOI https://doi.org/10.1137/S0036141096298060 - Qiang Du, R. A. Nicolaides, and Xiaonan Wu,
*Analysis and convergence of a covolume approximation of the Ginzburg-Landau model of superconductivity*, SIAM J. Numer. Anal.**35**(1998), no. 3, 1049–1072. MR**1619855**, DOI https://doi.org/10.1137/S0036142996302852 - Weinan E,
*Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity*, Phys. D**77**(1994), no. 4, 383–404. MR**1297726**, DOI https://doi.org/10.1016/0167-2789%2894%2990298-4 - V. Fomin, V. Misko, J. Devreese and V. Moshchalkov, Superconducting mesoscopic square loop,
*Phys. Rev. B*,**58**, pp. 11703-11715, 1998. - W. Gropp, H. Kaper, G. Leaf, D. Levine, M. Palumbo, and V. Vinokur, Numerical simulation of vortex dynamics in type-II superconductors,
*J. Comp. Phys.*,**123**, pp. 254-266, 1996. - Emmanuel Hebey,
*Sobolev spaces on Riemannian manifolds*, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR**1481970** - B. Ianotta, Music of the Spheres,
*New Scientist*,**31**, pp.28-31, 1996. - H. Jadallah, J. Rubinstein, and P. Sternberg, Phase Transition Curves for Mesoscopic Superconducting Samples,
*Phys. Rev. Lett*,**82**, pp. 2935, 1999. - E. Polak,
*Computational methods in optimization. A unified approach*, Mathematics in Science and Engineering, Vol. 77, Academic Press, New York-London, 1971. MR**0282511** - R. Renka; Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagrams on the surface of a sphere,
*ACM Trans. Math. Soft.*,**23**, pp. 416-434, 1997. - B. Richter and R. Warburton, A new generation of superconducting gravimeters, in
*Proceedings of the 13th Intern. Sym. on Earth Tides*, Brussel, pp. 545-556, 1998, Série Géophysique, Royal Observatory of Belgium. - V. Schweigert, F. Peeters and P. Deo, Vortex Phase Diagram for Mesoscopic Superconducting Disks,
*Phys. Rev. Lett*,**81**, pp. 2783-2786, 1998. - R. Tao, X. Zhang, X. Tang, and P. Anderson, Formation of High Temperature Superconducting Balls,
*Phys. Rev. Lett*,**83**, pp. 5575-5578, 1999. - B. Tent, D. Qu, D. Shi, W. Bresser, P. Boolchand and Z. Cai, Angle dependence of magnetization in a single-domain $YBa_2Cu_3O_x$ sphere,
*Phys. Rev. B*,**58**, 11761, 1998. - M. Tinkham,
*Introduction to Superconductivity*, 2nd ed., McGraw-Hill, New York, 1994. - J. Yeo and M. Moore, Non-integer flux quanta for a spherical superconductor,
*Phys. Rev. B*,**58**, pp. 10785-10789, 1998. - Y. Xiao, S. Buchman, G Keiser, B. Muhlfelder, J. Turneaure, and C. Wu, Magnetic flux distribution on a spherical superconducting shell,
*Physica B*,**194**, pp.65-66, 1994.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65N15,
65N99,
82D55

Retrieve articles in all journals with MSC (2000): 65N15, 65N99, 82D55

Additional Information

**Qiang Du**

Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

MR Author ID:
191080

Email:
qdu@math.psu.edu

**Lili Ju**

Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455

MR Author ID:
645968

Email:
ju@ima.umn.edu

Keywords:
Ginzburg-Landau model of superconductivity,
finite volume,
gauge invariance,
convergence,
spherical centroidal Voronoi tessellations

Received by editor(s):
July 13, 2003

Received by editor(s) in revised form:
January 5, 2004

Published electronically:
December 8, 2004

Additional Notes:
This work is supported in part by NSF-DMS 0196522 and NSF-ITR 0205232

Article copyright:
© Copyright 2004
American Mathematical Society