On motion by curvature of a network with a triple junction

Pozzi, Paola; Stinner, Björn

The SMAI Journal of computational mathematics, Tome 7 (2021), pp. 27-55.

We numerically study the planar evolution by curvature flow of three parametrised curves that are connected by a triple junction in which conditions are imposed on the angles at which the curves meet. One of the key problems in analysing motion of networks by curvature law is the choice of a tangential velocity that allows for motion of the triple junction, does not lead to mesh degeneration, and is amenable to an error analysis. Our approach consists in considering a perturbation of a classical smooth formulation. The problem we propose admits a natural variational formulation that can be discretized with finite elements. The perturbation can be made arbitrarily small when a regularisation parameter shrinks to zero. Convergence of the new semi-discrete finite element scheme including optimal error estimates are proved. These results are supported by some numerical tests. Finally, the influence of the small regularisation parameter on the properties of scheme and the accuracy of the results is numerically investigated.

Classification : 65M12, 65M15, 65M60
Mots clés : curve shortening flow, network, triod, Herring’s condition, Young’s law, semi-discrete scheme

[1] Balažovjech, M.; Mikula, K. A Higher Order Scheme for a Tangentially Stabilized Plane Curve Shortening Flow with a Driving Force, SIAM J. Sci. Comput., Volume 33 (2011) no. 5, pp. 2277-2294 | DOI | MR | Zbl

[2] Barrett, J. W.; Garcke, H.; Nürnberg, R. On the Variational Approximation of Combined Second and Fourth Order Geometric Evolution Equations, SIAM J. Sci. Comput., Volume 29 (2007) no. 3, pp. 1006-1041 | DOI | MR | Zbl

[3] Barrett, J. W.; Garcke, H.; Nürnberg, R. The Approximation of Planar Curve Evolutions by Stable Fully Implicit Finite Element Schemes That Equidistribute, Numer. Methods Partial Differ. Equations, Volume 27 (2011) no. 1, pp. 1-30 | DOI | MR | Zbl

[4] Barrett, J. W.; Garcke, H.; Nürnberg, R. Chapter 4 - Parametric Finite Element Approximations of Curvature-driven Interface Evolutions, Geometric Partial Differential Equations - Part I (Bonito, Andrea; Nochetto, Ricardo H, eds.) (Handbook of Numerical Analysis), Volume 21, Elsevier, 2020, pp. 275-423 | DOI

[5] Bretin, E.; Masnou, S. A New Phase Field Model for Inhomogeneous Minimal Partitions, and Applications to Droplets Dynamics, Interfaces Free Bound., Volume 19 (2017) no. 2, pp. 141-182 | DOI | MR | Zbl

[6] Bronsard, L.; Garcke, H.; Stoth, B. A Multi-Phase Mullins-Sekerka System: Matched Asymptotic Expansions and an Implicit Time Discretisation for the Geometric Evolution Problem, Proc. R. Soc. Edinb., Sect. A, Math., Volume 128 (1998) no. 3, pp. 481-506 | DOI | MR | Zbl

[7] Bronsard, L.; Reitich, F. On Three-Phase Boundary Motion and the Singular Limit of a Vector-Valued Ginzburg-Landau Equation, Arch. Ration. Mech. Anal., Volume 124 (1993) no. 4, pp. 355-379 | DOI | MR | Zbl

[8] Bronsard, L.; Wetton, B. A Numerical Method for Tracking Curve Networks Moving with Curvature Motion, J. Comput. Phys., Volume 120 (1993) no. 1, pp. 66-87 | DOI | MR

[9] Cahn, J. W. Critical Point Wetting, J. Chem. Phys., Volume 66 (1977) no. 8, pp. 3667-3672 | DOI

[10] Deckelnick, K.; Dziuk, G. On the Approximation of the Curve Shortening Flow, Calculus of Variations, Applications and Computations: Pont-à-Mousson, 1994 (Pitman Research Notes in Mathematics Series), Longman Scientific & Technical, 1994, pp. 100-108 | Zbl

[11] Deckelnick, K.; Dziuk, G.; Elliott, C. M. Computation of Geometric Partial Differential Equations and Mean Curvature Flow, Acta Numer., Volume 14 (2005), pp. 139-232 | DOI | MR | Zbl

[12] Deckelnick, K.; Elliott, C. M. Finite Element Error Bounds for a Curve Shrinking with Prescribed Normal Contact to a Fixed Boundary, IMA J. Numer. Anal., Volume 18 (1998) no. 4, pp. 635-654 | DOI | MR | Zbl

[13] Du, Q.; Feng, X. Chapter 5 - The Phase Field Method for Geometric Moving Interfaces and their Numerical Approximations, Geometric Partial Differential Equations - Part I (Bonito, A.; Nochetto, R. H., eds.) (Handbook of Numerical Analysis), Volume 21, Elsevier, 2020, pp. 425-508 | DOI

[14] Dziuk, G. An Algorithm for Evolutionary Surfaces, Numer. Math., Volume 58 (1991) no. 1, pp. 603-611 | DOI | MR | Zbl

[15] Dziuk, G. Convergence of a Semi-Discrete Scheme for the Curve Shortening Flow, Math. Models Methods Appl. Sci., Volume 4 (1994) no. 4, pp. 589-606 | DOI | MR | Zbl

[16] Elliott, C. M.; Fritz, H. On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow Based on the DeTurck Trick, IMA J. Numer. Anal. (2016), drw020 | DOI | Zbl

[17] Esedoḡlu, S.; Jacobs, M.; Zhang, P. Kernels with Prescribed Surface Tension & Mobility for Threshold Dynamics Schemes, J. Comput. Phys., Volume 337 (2017), pp. 62-83 | DOI | MR | Zbl

[18] Garcke, H.; Nestler, B.; Stoth, B. A Multiphase Field Concept: Numerical Simulations of Moving Phase Boundaries and Multiple Junctions, SIAM J. Appl. Math., Volume 60 (1999) no. 1, pp. 295-315 | DOI | MR | Zbl

[19] Gilbert, E. N.; Pollak, H. O. Steiner Minimal Trees, SIAM J. Appl. Math., Volume 16 (1968) no. 1, pp. 1-29 | DOI | MR | Zbl

[20] Herring, C. Surface Diffusion as a Motivation for Sintering, The Physics of Powder Metallurgy, McGraw-Hill, 1951, pp. 143-179

[21] Mackenzie, J. A.; Nolan, M.; Rowlatt, C. F.; Insall, R. H. An Adaptive Moving Mesh Method for Forced Curve Shortening Flow, SIAM J. Sci. Comput., Volume 41 (2019) no. 2, p. A1170-A1200 | DOI | MR | Zbl

[22] Mantegazza, C.; Novaga, M.; Pluda, A.; Schulze, F. Evolution of Networks with Multiple Junctions (2016) (https://arxiv.org/abs/1611.08254)

[23] Merriman, B.; Bence, J. K.; Osher, S. J. Motion of Multiple Junctions: A Level Set Approach, J. Comput. Phys., Volume 112 (1994) no. 2, pp. 334-363 | DOI | Zbl

[24] Mikula, K.; Remešíková, M.; Sarkoci, P.; Ševčovič, D. Manifold Evolution with Tangential Redistribution of Points, SIAM J. Sci. Comput., Volume 36 (2014) no. 4, p. A1384-A1414 | DOI | MR | Zbl

[25] Pan, Z.; Wetton, B. Numerical Simulation and Linear Well-posedness Analysis for a Class of Three-phase Boundary Motion Problems, J. Comput. Appl. Math., Volume 236 (2012) no. 13, pp. 3160-3173 | DOI | MR | Zbl

[26] Priester, L.; Yu, D. Triple Junctions at the Mesoscopic, Microscopic and Nanoscopic Scales, Materials Science and Engineering: A, Volume 188 (1994) no. 1-2, pp. 113-119 | DOI

[27] Saye, R. I.; Sethian, J. A. Chapter 6 - A Review of Level Set Methods to Model Interfaces Moving under Complex Physics: Recent Challenges and Advances, Geometric Partial Differential Equations - Part I (Bonito, A.; Nochetto, R. H., eds.) (Handbook of Numerical Analysis), Volume 21, Elsevier, 2020, pp. 509-554 | DOI

[28] Smith, K. A.; Solis, F. J; Chopp, D. L. A Projection Method for Motion of Triple Junctions by Level Sets, Interfaces Free Bound., Volume 4 (2002) no. 3, pp. 263-276 | DOI | MR | Zbl

[29] Solonnikov, V. A. Boundary Value Problems of Mathematical Physics. III, Proceedings of the Steklov institute of Mathematics (1965), American Mathematical Society, 1967 no. 83

[30] Taylor, J. E.; Cahn, J. W.; Handwerker, C. A. Geometric Models of Crystal Growth, Acta Metallurgica et Materialia, Volume 40 (1992) no. 7, pp. 1443-1474 | DOI

[31] Teschl, G. Ordinary Differential Equations and Dynamical Systems, 140, American Mathematical Society, 2012, 364 pages | MR | Zbl

[32] Young, T. An Essay on the Cohesion of Fluids, Philosophical Transactions of the Royal Society of London, Volume 95 (1805), pp. 65-87