The elastic flow of curves on the sphere

Dall’Acqua, Anna; Laux, Tim; Lin, N.; Pozzi, Paola; Spener, Adrian

doi:10.1515/geofl-2018-0001

Cited by 23 publications

(28 citation statements)

References 3 publications

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“…Axisymmetric versions of geometric flows have also been treated analytically and questions regarding stability and singularity formation have been studied, see [34,28,13,18,35]. We also refer to [21,22], who discuss the relation between the axisymmetric Willmore flow and the elastic flow in hyperbolic space.…”

Section: Introductionmentioning

confidence: 99%

Finite element methods for fourth order axisymmetric geometric evolution equations

Barrett

Garcke

Nürnberg

2019

Journal of Computational Physics

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Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce and analyze several new finite element schemes for fourth order geometric evolution equations in an axisymmetric setting, and for selected schemes we will show existence, uniqueness and stability results. The presented schemes have very good mesh and stability properties, as will be demonstrated by several numerical examples.

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Section: Introductionmentioning

confidence: 99%

Finite element methods for fourth order axisymmetric geometric evolution equations

Barrett

Garcke

Nürnberg

2019

Journal of Computational Physics

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“…Elastic flow of curves in a two-dimensional Riemannian manifold (M, g) is given as the L 2 -gradient flow of the elastic energy 1 2 κ 2 g , where κ g is the geodesic curvature. It has been shown, see [8] for the general case and [12] for the hyperbolic plane, that the gradient flow of the elastic energy is given as…”

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confidence: 99%

“…In particular, let us mention that a curve is an absolute minimizer if and only if it is a geodesic. Recently the flow (1.1) was studied in [12,13], for the case of the hyperbolic plane, relying on earlier results in [15] for a flat background metric. The hyperbolic plane is a particular case of a manifold with non-positive sectional curvature, which is of particular interest as the set of free elasticae is much richer, see [17].…”

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confidence: 99%

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Stable Discretizations of Elastic Flow in Riemannian Manifolds

Barrett¹,

Garcke²,

Nürnberg³

2019

SIAM J. Numer. Anal.

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The elastic flow, which is the L 2 -gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow 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 manifold in R d , d ≥ 3. Numerical results show the robustness of the method, as well as quadratic convergence with respect to the space discretization.

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“…we have to "shift and reduce" progressively the interval where η ∈ (0, 1). More details in this respect can be found in [6,App.B.2.3]. Eventually we attain f ∈ C ∞ ([ǫ, T ]) and since ǫ was arbitrarily chosen the claim follows.…”

Section: )mentioning

confidence: 99%

Elastic flow of networks: long-time existence result

2019

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In this paper we study the L 2 -gradient flow of the penalized elastic energy on networks of q-curves in R n for q ≥ 3. Each curve is fixed at one end-point and at the other is joint to the other curves at a movable q-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourthorder non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov's theory on linear parabolic systems and Banach fixed point theorem in proper Hölder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.

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The elastic flow of curves on the sphere

Abstract: We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

Cited by 23 publications

References 3 publications

Finite element methods for fourth order axisymmetric geometric evolution equations

Finite element methods for fourth order axisymmetric geometric evolution equations

Stable Discretizations of Elastic Flow in Riemannian Manifolds

Elastic flow of networks: long-time existence result

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