Level set methods to compute minimal surfaces in a medium with exclusions (voids)

Interfaces and Free Boundaries

Volumn 7, Issue 2, 2005, Pages 161-185

  Torres, Monica ^a   Chopp, David ^b   Walsh, Timothy ^c  

^a NORTHWESTERN UNIVERSITY   (United States)

^b Northwestern University   (United States)

^c SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22544475904     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/119     Document Type: Article

References (59)

1. ADALSTEINSSON, D. & SETHIAN, J. A. A fast level set method for propagating interfaces. J. Comput. Phys. 118 (1995), 269-277. Zbl 0823.65137 MR 1329634
2. ADALSTEINSSON, D. & SETHIAN, J. A. A level set approach to a unified model for etching, deposition, and lithography. III. Redeposition, reemission, surface diffusion, and complex simulations. J. Comput. Phys. 138 (1997), 193-223. Zbl 0952.65113 MR 1484289
3. ADALSTEINSSON, D. & SETHIAN, J. A. The fast construction of extension velocities in level set methods. J. Comput. Phys. 148 (1999), 2-22. Zbl 0919.65074 MR 1665209
4. BARLES, G. & GEORGELIN, C. A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995), 484-500. Zbl 0831.65138 MR 1324298
5. BARLES, G. & SOUGANIDIS, P. E. A new approach to front propagation problems: theory and applications. Arch. Rat. Mech. Anal. 141 (1998), 237-296. Zbl 0904.35034 MR 1617291
6. BRAKKE, K. A. The Motion of a Surface by its Mean Curvature. Princeton Univ. Press, Princeton, NJ (1978). Zbl 0386.53047 MR 0485012
7. BRAKKE, K. A. The surface evolver. Experiment. Math. 1 (1992), 141-165. Zbl 0769.49033 MR 1203871
8. BRAKKE, K. A. The surface evolver and the stability of liquid surfaces. Philos. Trans. Roy. Soc. London Ser. A 354 (1996), 2143-2157. Zbl 0870.76065 MR 1421042
9. BRAKKE, K. A. & SULLIVAN, J. M. Using symmetry features of the surface evolver to study foams. Visualization and Mathematics (Berlin-Dahlem, 1995), Springer, Berlin (1997), 95-117 MR 1607360
10. CAFFARELLI, L. A. & DE LA LLAVE, R. Planelike minimizers in periodic media. Comm. Pure Appl. Math. 54 (2001), 1403-1441. Zbl 1036.49040 MR 1852978
11. CHOPP, D. L. Computing minimal surfaces via level set curvature flow. J. Comput. Phys. 106 (1993), 77-91. Zbl 0786.65015 MR 1214016
12. CHOPP, D. L. Computation of self-similar solutions for mean curvature flow. Experiment. Math. 3 (1994), 1-15. Zbl 0811.53011 MR 1302814
13. CHOPP, D. L. Some improvements of the fast marching method. SIAM J. Sci. Comput. 23 (2001), 230-244. Zbl 0991.65105 MR 1860913
14. CHOPP, D. L. & SETHIAN, J. A. Flow under curvature: Singularity formation, minimal surfaces, and geodesics. Experiment. Math. 2 (1993), 235-255. Zbl 0806.53004 MR 1281473
15. CONCUS, P., FINN, R., & HOFFMAN, D. E. (eds.) Geometric Analysis and Computer Graphics (Berkeley, 1988). Math. Sci. Res. Inst. Publ. 17, Springer, New York (1991). Zbl 0776.68009 MR 1081324
16. DEMKOWICZ, L. 2d hp-adaptive finite element package (2dhp90) version 2.0. TICAM Report 02-06, The University of Texas at Austin, 2002.
17. DEMKOWICZ, L., GERDES, K., C., S., BAJER, A., & WALSH, T. Hp90: A general and flexible fortran 90 hp-fe code. Computing and Visualization in Science 1 (1998), 145-163. Zbl 0912.68014
18. DUBOIS-VIOLETTE, E. & PANSU, B. E. (eds.) International Workshop on Geometry and Interfaces (Aussois, 1990). Les Éditions de Physique, Les Ulis (1990). MR 1090143
19. EVANS, L. C. Convergence of an algorithm for mean curvature motion. Indiana Univ. Math. J. 42 (1993), 533-557. Zbl 0802.65098 MR 1237058
20. EVANS, L. C. & GARIEPY, R. F. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL (1992). Zbl 0804.28001 MR 1158660
21. EVANS, L. & SPRUCK, J. Motion of level sets by mean curvature. I. J. Differential Geom. 33 (1991), 635-681. Zbl 0726.53029 MR 1100206
22. EVANS, L. & SPRUCK, J. Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992), 321-332. Zbl 0776.53005 MR 1068927
23. EVANS, L. & SPRUCK, J. Motion of level sets by mean curvature. III. J. Geom. Anal. 2 (1992), 121-150. Zbl 0768.53003 MR 1151756
24. EVANS, L. & SPRUCK, J. Motion of level sets by mean curvature. IV. J. Geom. Anal. 5 (1995), 77-114. Zbl 0829.53040 MR 1315658
25. FEDERER, H. Geometric Measure Theory. Grundlehren Math. Wiss. 153, Springer, New York (1969). Zbl 0176.00801 MR 0257325
26. FINN, R. Equilibrium Capillary Surfaces. Springer (1986). Zbl pre00583.35002 MR 0816345
27. GIUSTI, E. Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984). Zbl pre00545.49018 MR 0775682
28. GOMES, D. A. Stability of rotating liquid films. Quart. J. Mech. Appl. Math. 55 (2002), 327-343. Zbl 1036.76018 MR 1919971
29. GRAYSON, M. A. The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 (1987), 285-314. Zbl 0667.53001 MR 0906392
30. GRAYSON, M. A. A short note on the evolution of a surface by its mean curvature. Duke Math. J. 58 (1989), 555-558. Zbl 0677.53059 MR 1016434
31. GRÜTER, M. Optimal regularity for codimension one minimal surfaces with a free boundary. Manuscripta Math. 58 (1987), 295-343. Zbl 0609.35012 MR 0893158
32. HSU, L., KUSNER, R. & SULLIVAN, J. Minimizing the squared mean curvature integral for surfaces in space forms. Experiment. Math. 1 (1991), 191-207. Zbl 0778.53001 MR 1203874
33. HUISKEN, G. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20 (1984), 237-266. Zbl pre00556.53001 MR 0772132
34. KRAYNIK, A. & REINELT, D. Simple shearing flow of a dry kelvin soap foam. J. Fluid Mech. 311 (1996), 327-342. Zbl 0864.76008.
35. LEONI, F. Convergence of and aproximation scheme for curvature-dependent motions of sets. SIAM J. Numer. Anal. 39 (2001), 1115-1131. Zbl 1008.65067 MR 1870835
36. LI, Z. L., LIN, X. B., TORRES, M., & ZHAO, H. K. Generalized Snell's law for weighted minimal surfaces in heterogeneous media. Methods Appl. Anal. 10 (2003), 199-214. Zbl pre02110033 MR 2074748
37. MALLADI, R. & SETHIAN, J. A. Image processing via level set curvature flow. Proc. Nat. Acad. Sci. U.S.A. 92 (1995), 7046-7050. Zbl 1023.68689 MR 1343443
38. MALLADI, R. & SETHIAN, J. A. Level set methods for curvature flow, image enhancement, and shape recovery in medical images. Visualization and Mathematics (Berlin-Dahlem, 1995), Springer, Berlin (1997), 329-345. MR 1607328
39. MERRIMAN, B., BENCE, J. K., & OSHER, S. J. Motion of multiple functions: a level set approach. J. Comput. Phys. 112 (1994), 334-363. Zbl 0805.65090 MR 127728
40. MICHALET, X., FOURCADE, B., & BENSIMON, D. Fluctuating vesicles of nonspherical topology. Phys. Rev. Lett. 72 (1994), 168-171.
41. MITTELMANN, H. Symmetric capillary surfaces in a cube. Math. Comput. Simulation 35 (1993), 139-152 MR 1290499
42. NITSCHE, J. C. C. Lectures on Minimal Surfaces. Vol. 1. Cambridge Univ. Press, Cambridge (1989). Zbl 0688.53001 MR 1015936
43. OSHER, S. & SETHIAN, J. A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988), 12-49. Zbl 0659.65132 MR 0965860
44. PHELAN, R., WEAIRE, D., & BRAKKE, K. Computation of equilibrium foam structures using the surface evolver. Experiment. Math. 4 (1995), 181-192. Zbl 0861.76014 MR 1387476
45. RACZ, L., SZEKELY, J., & BRAKKE, K. A general statement of the problem and description of a proposed method of calculation for some meniscus problems in materials processing. ISIJ Int. 33 (1993), 328-335.
46. SETHIAN, J. Fast marching methods. SIAM Rev. 41 (1999), 199-235. Zbl 0926.65106 MR 1684542
47. SETHIAN, J. A. A fast marching level set method for monotonically advancing fronts. Proc. Nat. Acad. Sci. 93 (1996), 1591-1595. Zbl 0852.65055 MR 1374010
48. SETHIAN, J. A. Level Set Methods. Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge Monogr. Appl. Comput. Math. 3, Cambridge Univ. Press, Cambridge (1996). Zbl 0859.76004 MR 1409367
49. SETHIAN, J. A. Level Set Methods and Fast Marching Methods. Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science. 2nd ed., Cambridge Monogr. Appl. Comput. Math. 3, Cambridge Univ. Press, Cambridge (1999). Zbl 0973.76003 MR 1700751
50. SETHIAN, J. A. Advancing interfaces: level set and fast marching methods. ICIAM 99 (Edinburgh), Oxford Univ. Press, Oxford (2000), 222-233. Zbl 0994.65090 MR 1824446
51. SETHIAN, J. A. Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169 (2001), 503-555. Zbl 0988.65095 MR 1836524
52. SETHIAN, J. A. A review of level set and fast marching methods for image processing. Modern Methods in Scientific Computing and Applications (Montreal, QC, 2001), NATO Sci. Ser. II Math. Phys. Chem. 75, Kluwer, Dordrecht (2002), 365-396. Zbl pre02110608 MR 2004360
53. SETHIAN, J. A. & CHOPP, D. L. Physics and geometry of flow under curvature: singularity formation, minimal surfaces, geodesics, and surface tension. Motion by Mean Curvature and Related Topics (Trento, 1992), de Gruyter, Berlin (1994), 168-185. Zbl 0815.49027 MR 1277398
54. SETHIAN, J. A. & SMEREKA, P. Level set methods for fluid interfaces. Annual Review of Fluid Mechanics, Vol. 35, Annual Reviews, Palo Alto, CA (2003), 341-372. Zbl 1041.76057 MR 1967017
55. SHAPIRO, G. The Mathematics of Soap Films: Explorations with Maple. Amer. Math. Soc. (2000).
56. SHAPIRO, G. Geometric Partial Differential Equations and Image Analysis. Cambridge Univ. Press (2001).
57. SIMON, L. Lectures on Geometric Measure Theory. Austral. Nat. Univ. Centre for Math. Analysis, Canberra (1983). Zbl 0546.49019 MR 0795227
58. TEGART, J. Three-dimensional fluid interfaces in cylindrical containers. AIAA paper AIAA-912174, 27th Joint Propulsion Conference, Sacramento, CA (1991).
59. TORRES, M. Plane-like minimal surfaces in periodic media with exclusions. SIAM J. Math. Anal. 36 (2004), 523-551. Zbl pre02139025 MR 2111789