Variational discretization of axisymmetric curvature flows

Barrett, John W.; Garcke, Harald; Nürnberg, Robert

doi:10.1007/s00211-018-1013-z

Cited by 20 publications

(20 citation statements)

References 43 publications

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“…A precise derivation of (2.11) in the context of a weak formulation of (2.9) can be found in Appendix A of [9]. We note that for the singular fraction in (2.7) it follows from (2.11) and (2.10), on recalling (2.6), that lim…”

Section: Generating Curvementioning

confidence: 94%

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Stable approximations for axisymmetric Willmore flow for closed and open surfaces

2021

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For a hypersurface in $\mathbb R^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.

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Section: Generating Curvementioning

confidence: 94%

“…Earlier results on the numerical approximation of geometric evolution problems in the axisymmetric setting can be found in [7,9], as well as on the surprisingly closely related problem of curve evolutions in Riemannian manifolds, see [8,10]. There appears to be little numerical analysis for such evolution problems in the literature.…”

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confidence: 99%

Stable approximations for axisymmetric Willmore flow for closed and open surfaces

2021

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“…76)seeBarrett et al (2018a), where we have noted (2.16), (2.12) and (2.13). Hence (2.76) differs from the curvature flow (2.22) for (2.5d) by a space-dependent weighting factor.…”

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confidence: 88%

“…3.8. Using the techniques from Barrett et al (2018a), it is straightforward to adapt the presented schemes to deal with open curves, with fixed endpoints. These schemes then allow to compute approximations to geodesics in the hyperbolic plane, for example.…”

Section: Theoremmentioning

confidence: 99%

Numerical approximation of curve evolutions in Riemannian manifolds

Barrett

Garcke

Nürnberg

2019

IMA Journal of Numerical Analysis

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We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e. conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional surface in R d , d ≥ 3. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate a scheme for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in R d . Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations. 1 2 .With the help of the splitting, we propose in Section 3 a semi-implicit scheme for which stability can be shown.The outline of this paper is as follows. In Section 2 we derive the governing equations for curvature flow, curve diffusion and elastic flow, provide weak formulations and relate the introduced flows to geometric evolution equations for axisymmetric hypersurfaces. In Section 3 we introduce finite element approximations and show existence and uniqueness as well as stability results. Section 4 is devoted to several numerical results, which

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“…Since the pioneering work of Brakke [8] and Huisken [20] many results have been shown for mean curvature flow and we refer to [24] and the references therein for more information about the subject. The case of rotationally symmetric evolutions lead to spatially onedimensional problems and due to the reduced complexity this situation has been studied by several authors analytically [1,13,21,22,25] as well as numerically [7,27].…”

Section: Introductionmentioning

confidence: 99%

On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

Garcke

Matioc

2020

J. Evol. Equ.

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We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing the evolution of the points of the evolving surface that lie on the rotation axis. For the fully nonlinear and degenerate parabolic problem we establish the well-posedness property in the setting of classical solutions. Besides we prove that the problem features the effect of parabolic smoothing.2010 Mathematics Subject Classification. 35K55, 53C44, 35R35, 35K93.

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Variational discretization of axisymmetric curvature flows

Cited by 20 publications

References 43 publications

Stable approximations for axisymmetric Willmore flow for closed and open surfaces

Stable approximations for axisymmetric Willmore flow for closed and open surfaces

Numerical approximation of curve evolutions in Riemannian manifolds

On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

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