Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
interface evolution; anisotropic energy; weighted mean curvature; obstacle problem; thresholding method; convolution kernels; topology change; numerical analysis
Summary:
We extend thresholding methods for numerical realization of mean curvature flow on obstacles to the anisotropic setting where interfacial energy depends on the orientation of the interface. This type of schemes treats the interface implicitly, which supports natural implementation of topology changes, such as merging and splitting, and makes the approach attractive for applications in material science. The main tool in the new scheme are convolution kernels developed in previous studies that approximate the given anisotropy in a nonlocal way. We provide a detailed report on the numerical properties of the proposed algorithm.
References:
[1] Bao, W., Jiang, W., Srolovitz, D. J., Wang, Y.: Stable equilibria of anisotropic particles on substrates: A generalized Winterbottom construction. SIAM J. Appl. Math. 77 (2017), 2093-2118. DOI 10.1137/16M1091599 | MR 3730550 | Zbl 1386.74059
[2] Bertagnolli, M., Marchese, M., Jacucci, G.: Modeling of particles impacting on a rigid substrate under plasma spraying conditions. J. Thermal Spray Technology 4 (1995), 41-49. DOI 10.1007/BF02648527
[3] Bonnetier, E., Bretin, E., Chambolle, A.: Consistency result for a non monotone scheme for anisotropic mean curvature flow. Interface Free Bound. 14 (2012), 1-35. DOI 10.4171/IFB/272 | MR 2929124 | Zbl 1254.35128
[4] Campinho, P., Behrndt, M., Ranft, J., Risler, T., Minc, N., Heisenberg, C.-P.: Tensionoriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebrafish epiboly. Nature Cell Biology 15 (2013), 1405-1414. DOI 10.1038/ncb2869
[5] Elsey, M., Esedo\={g}lu, S.: Threshold dynamics for anisotropic surface energies. Math. Comput. 87 (2018), 1721-1756 \99999DOI99999 10.1090/mcom/3268 . DOI 10.1090/mcom/3268 | MR 3787390 | Zbl 1397.65156
[6] Esedo\={g}lu, S., Jacobs, M.: Convolution kernels and stability of threshold dynamics methods. SIAM J. Numer. Anal. 55 (2017), 2123-2150. DOI 10.1137/16M1087552 | MR 3693605 | Zbl 1372.65253
[7] Esedo\={g}lu, S., Jacobs, M., Zhang, P.: Kernels with prescribed surface tension & mobility for threshold dynamics schemes. J. Comput. Phys. 337 (2017), 62-83. DOI 10.1016/j.jcp.2017.02.023 | MR 3623147 | Zbl 1415.65278
[8] Esedo\={g}lu, S., Otto, F.: Threshold dynamics for networks with arbitrary surface tensions. Commun. Pure Appl. Math. 68 (2015), 808-864. DOI 10.1002/cpa.21527 | MR 3333842 | Zbl 1334.82072
[9] Huang, J., Kim, F., Tao, A. R., Connor, S., Yang, P.: Spontaneous formation of nanoparticle stripe patterns through dewetting. Nature Materials 4 (2005), 896-900. DOI 10.1038/nmat1517
[10] Ishii, H., Pires, G. E., Souganidis, P. E.: Threshold dynamics type approximation schemes for propagating fronts. J. Math. Soc. Japan 51 (1999), 267-308. DOI 10.2969/jmsj/05120267 | MR 1674750 | Zbl 0935.53006
[11] Jiang, W., Bao, W., Thompson, C. V., Srolovitz, D. J.: Phase field approach for simulating solid-state dewetting problems. Acta Mater. 60 (2012), 5578-5592. DOI 10.1016/j.actamat.2012.07.002
[12] Laux, T., Otto, F.: Convergence of the thresholding scheme for multi-phase mean-curvature flow. Calc. Var. Partial Differ. Equ. 55 (2016), Article ID 129, 74 pages. DOI 10.1007/s00526-016-1053-0 | MR 3556529 | Zbl 1388.35121
[13] Merriman, B., Bence, J. K., Osher, S. J.: Motion of multiple junctions: A level set approach. J. Comput. Phys. 112 (1994), 334-363. DOI 10.1006/jcph.1994.1105 | MR 1277282 | Zbl 0805.65090
[14] Misiats, O., Yip, N. Kwan: Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete Contin. Dyn. Syst. 36 (2016), 6379-6411. DOI 10.3934/dcds.2016076 | MR 3543592 | Zbl 1353.65097
[15] Ruuth, S. J., Merriman, B.: Convolution-generated motion and generalized Huygens' principles for interface motion. SIAM J. Appl. Math. 60 (2000), 868-890. DOI 10.1137/S003613999833397X | MR 1740854 | Zbl 0958.65021
[16] Ševčovič, D., Yazaki, S.: Evolution of plane curves with a curvature adjusted tangential velocity. Japan J. Ind. Appl. Math. 28 (2011), 413-442. DOI 10.1007/s13160-011-0046-9 | MR 2846183 | Zbl 1291.35109
[17] Thompson, C. V.: Solid-state dewetting of thin films. Annual Review Materials Research 42 (2012), 399-434. DOI 10.1146/annurev-matsci-070511-155048
[18] Wang, Y., Jiang, W., Bao, W., Srolovitz, D. J.: Sharp interface model for solid-state dewetting problems with weakly anisotropic surface energies. Phys. Rev. B 91 (2015), Article ID 045303. DOI 10.1103/PhysRevB.91.045303
[19] Winterbottom, W. L.: Equilibrium shape of a small particle in contact with a foreign substrate. Acta Metallurgica 15 (1967), 303-310. DOI 10.1016/0001-6160(67)90206-4
[20] Xu, X., Wang, D., Wang, X.-P.: An efficient threshold dynamics method for wetting on rough surfaces. J. Comput. Phys. 330 (2017), 510-528. DOI 10.1016/j.jcp.2016.11.008 | MR 3581477 | Zbl 1378.76087
[21] Yagisita, H.: Non-uniqueness of self-similar shrinking curves for an anisotropic curvature flow. Calc. Var. Partial Differ. Equ. 26 (2006), 49-55. DOI 10.1007/s00526-005-0357-2 | MR 2217482 | Zbl 1116.53041
Partner of
EuDML logo