Comparison study for Level set and Direct Lagrangian methods for computing Willmore flow of closed planar curves

Beneš, Michal; Mikula, Karol; Oberhuber, Tomáš; Ševčovič, Daniel

doi:10.1007/s00791-008-0112-2

Cited by 11 publications

(16 citation statements)

References 19 publications

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“…For numerical treatment of Willmore flow in a sharp-interface approach, we refer to [2-4, 14, 25, 32, 48]. Moreover, level set techniques have been applied in order to simulate Willmore flow in [12,28].…”

Section: Standard Diffuse Approximationmentioning

confidence: 99%

A new diffuse-interface approximation of the Willmore flow

Rätz

Röger

2021

ESAIM: COCV

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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 an 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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Section: Standard Diffuse Approximationmentioning

confidence: 99%

A new diffuse-interface approximation of the Willmore flow

Rätz

Röger

2021

ESAIM: COCV

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“…Alternatively, methods may also be based on interface capturing approaches. This includes level set representations of the curve or surface ( [36,16], see [7] for a comparison with parametric methods) and the phase field methodology ( [17,18,8,28]. For an overview we refer to [13] but we remark that the field has seen significant advances since.…”

Section: )mentioning

confidence: 99%

Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case

Pozzi

Stinner

2018

IMA Journal of Numerical Analysis

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A finite element approach to the elastic flow of a curve coupled with a diffusion equation on the curve is analysed. Considering the graph case, the problem is weakly formulated and approximated with continuous linear finite elements, which is enabled thanks to second-order operator splitting. The error analysis builds up on previous results for the elastic flow. To obtain an error estimate for the quantity on the curve a better control of the velocity is required. For this purpose, a penalty approach is employed and then combined with a generalised Gronwall lemma. Numerical simulations support the theoretical convergence results. Further numerical experiments indicate stability beyond the parameter regime with respect to the penalty term which is covered by the theory.

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“…URL: http://geraldine.fjfi.cvut.cz/~oberhuber. 1 Tel. : +420 224 358563. in image processing in image inpainting [2].…”

Section: Introductionmentioning

confidence: 99%

“…One recognizes several formulations of the Willmore flow depending of the form in which Γ (t) is expressed -the graph formulation [5,9,14], the level-set formulation [6,1] or the parametric formulation [8,1]. Also approximation by the phase-field model exists [7].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution for the anisotropic Willmore flow of graphs

Oberhuber

2015

Applied Numerical Mathematics

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Laplace-Beltrami operator Method of lines Complementary finite volume method Finite difference methodThe Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge-Kutta-Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.

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Comparison study for Level set and Direct Lagrangian methods for computing Willmore flow of closed planar curves

Cited by 11 publications

References 19 publications

A new diffuse-interface approximation of the Willmore flow

A new diffuse-interface approximation of the Willmore flow

Elastic flow interacting with a lateral diffusion process: the one-dimensional graph case

Numerical solution for the anisotropic Willmore flow of graphs

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