Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes a rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior.
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Keywords: Free boundary problem, Willmore flow, phase-field model, diffuse interface, finite elements
@article{COCV_2021__27_1_A16_0,
author = {R\"atz, Andreas and R\"oger, Matthias},
title = {A new diffuse-interface approximation of the {Willmore} flow},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021013},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021013/}
}
TY - JOUR AU - Rätz, Andreas AU - Röger, Matthias TI - A new diffuse-interface approximation of the Willmore flow JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021013/ DO - 10.1051/cocv/2021013 LA - en ID - COCV_2021__27_1_A16_0 ER -
%0 Journal Article %A Rätz, Andreas %A Röger, Matthias %T A new diffuse-interface approximation of the Willmore flow %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021013/ %R 10.1051/cocv/2021013 %G en %F COCV_2021__27_1_A16_0
Rätz, Andreas; Röger, Matthias. A new diffuse-interface approximation of the Willmore flow. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 14. doi: 10.1051/cocv/2021013
[1] , and , Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differ. Equ. 12 (2007) 361–380.
[2] , and , A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222 (2007) 441–462.
[3] , and , Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31 (2008) 225–253.
[4] , and , Numerical approximation of gradient flows for closed curves in ℝ$$. IMA J. Numer. Anal. 30 (2010) 4–60.
[5] , and , Finite element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 42 (2004) 738–772.
[6] , Variational approximation of functionals with curvatures and related properties. J. Convex Anal. 4 (1997) 91–108.
[7] , and , Semicontinuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20 (1993) 247–297.
[8] and , Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 839–880.
[9] and , On the approximation of the elastica functional in radial symmetry. Calc. Var. Partial Differ. Equ. 24 (2005) 1–20.
[10] and , A varifolds representation of the relaxed elastica functional. J. Convex Anal. 14 (2007) 543–564.
[11] and , Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 43–67.
[12] , , and , Comparison study for level set and direct Lagrangian methods for computing Willmore flow of closed planar curves. Comput. Vis. Sci. 12 (2009) 307–317.
[13] , and , Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E 72 (2005).
[14] , and , Parametric FEM for geometric biomembranes. J. Comput. Phys. 229 (2010) 3171–3188.
[15] , and , Volume reconstruction from slices. SIAM J. Imag. Sci. 10 (2017) 2326–2358.
[16] , and , Phase-field approximations of the Willmore functional and flow. Numer. Math. 131 (2015) 115–171.
[17] and , Saddle-shaped solutions of bistable diffusion equations in all of \$$. J. Eur. Math. Soc. (JEMS) 11 (2009) 819–943.
[18] , and , A gradient bound for entire solutions of quasi-linear equations and its consequences. Comm. Pure Appl. Math. 47 (1994) 1457–1473.
[19] and , Dynamic model and stationary shapes of fluid vesicles. Eur. Phys. J. E 20 (2006) 37–45.
[20] , and , A gradient flow for open elastic curves with fixed length and clamped ends. Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017) 1031–1066.
[21] and , A Willmore-Helfrich L2-flow of curves with natural boundary conditions. Commun. Anal. Geom. 22 (2014) 617–669.
[22] , and , Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 43 (1992) 984–998.
[23] , and , Shape and orientation matter for the cellular uptake of nonspherical particles. Nano Lett. 14 (2014) 687–693.
[24] Some remarks on Γ-convergence and least squares method. In Composite media and homogenization theory (Trieste, 1990). Vol. 5 of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA (1991) 135–142.
[25] , and , Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232.
[26] , and , Phase field models for thin elastic structures with topological constraint. Arch. Ration. Mech. Anal. 223 (2017) 693–736.
[27] , and , Confined elastic curves. SIAM J. Appl. Math. 71 (2011) 2205–2226.
[28] and , A level set formulation for Willmore flow. Interfaces Free Bound. 6 (2004) 361–378.
[29] , and , A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198 (2004) 450–468.
[30] , and , Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212 (2006) 757–777.
[31] , and , Evolution of elastic curves in ℝ$$: existence and computation. SIAM J. Math. Anal. 33 (2002) 1228–1245.
[32] and , Modeling and computation of two phase geometric biomembranes using surface finite elements. J. Comput. Phys. 229 (2010) 6585–6612.
[33] , and , Colliding interfaces in old and new diffuse-interface approximations of Willmore-flow. Commun. Math. Sci. 12 (2014) 125–147.
[34] and , Phase-field approximation of the willmoreflow (2020).
[35] , and , A phase field based PDE constrained optimization approach to time discrete Willmore flow. Int. J. Numer. Anal. Model. 10 (2013) 116–138.
[36] , A catalog of special plane curves. Dover books on advanced mathematics. Dover Publications (1972).
[37] and , A unique graph of minimal elastic energy. Trans. Amer. Math. Soc. 359 (2007) 2021–2041.
[38] and , Propagation of fronts in a nonlinear fourth order equation. Eur. J. Appl. Math. 11 (2000) 203–213.
[39] , and , Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E 79 (2009) 0311926.
[40] , The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 357–383.
[41] , A gradient bound and a Liouville theorem for nonlinear Poisson equations. Commun. Pure Appl. Math. 38 (1985) 679–684.
[42] and , Un esempio di Γ-convergenza. Boll. Un.Mat. Ital. B 14 (1977) 285–299.
[43] and , Some entire solutions in the plane of nonlinear Poisson equations. Boll. Un. Mat. Ital. B 17 (1980) 614–622.
[44] , A higher order asymptotic problem related to phase transitions. SIAM J. Math. Anal. 37 (2005) 712–736.
[45] , Gamma-convergence results for phase-field approximations of the 2D-Euler elastica functional. ESAIM: COCV 19 (2013) 740–753.
[46] and , A diffuse-interface model accounting for elastic membrane energies with particle–membrane interaction. Inpreparation (2020).
[47] and , On a modified conjecture of De Giorgi. Math. Z. 254 (2006) 675–714.
[48] , An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7 (2005) 229–239.
[49] and , AMDiS: adaptive multidimensional simulations. Comput. Visual. Sci. 10 (2007) 57–67.
[50] , Asymptotic analysis of phase field formulations of bending elasticity models. SIAM J. Math. Anal. 39 (2008) 1367–1401.
[51] and , Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56 (2008) 347–371.
[52] , and , Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models. J. Comput. Phys. 316 (2016) 21–38.
[53] , Riemannian geometry. Oxford Science Publications, The Clarendon Press Oxford University Press, New York (1993).
[54] , The diffuse interface approximation of the willmore functional in configurations with interacting phaseboundaries (2018).
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