Paper · Dec 18, 2020

Sphere Under Advection and Mean Curvature Flow

Bryce A. Besler, Tannis D. Kemp, Nils D. Forkert, Steven K. Boyd

math.DGWider flowsarXiv:2012.10483
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Abstract

Advection and mean curvature flow is used as a model of bone microarchitecture adaptation. It is an equivalent geometric flow to prescribed mean curvature flow with an additional rate term. In order to validate numerical methods for simulating this flow and developing an inverse solver, a closed-form solution for advection and mean curvature flow of a sphere is derived.

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Submitted
Dec 18, 2020
Subjects
math.DG

Comments: 5 pages, 5 figures

Mean curvature flowNumerics & visualization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Contour Parametrization via Anisotropic Mean Curvature Flows (P. Suárez-Serrato and E. I. Velázquez Richards, 2018)8Suárez-Serra 2018 The entropy of the Angenent torus is approximately 1.85122 (Yakov Berchenko-Kogan, 2018)6Berchenko-Ko 2018 Theory and numerics for Chen's flow of curves (Matthew Cooper, Glen Wheeler and Valentina-Mira Wheeler, 2020)5Cooper 2020 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup (David Garfinkle, James Isenberg, Dan Knopf et al., 2021)2Garfinkle 2021 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)4Conboye 2023 Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus (Roman Bahadursingh, 2024)1Bahadursingh 2024 Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group (Giovanna Citti, Nicolas Dirr, Federica Dragoni et al., 2024)3Citti 2024 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick (Klaus Deckelnick and Vanessa Styles, 2026)7Deckelnick 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows

    Topics Numerics & visualization · Shared terms “numerical”

  2. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Numerics & visualization · Shared terms “numerical”, “developing”

  3. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

    Giovanna Citti, Nicolas Dirr, Federica Dragoni et al.Wider flows

    Topics Numerics & visualization

  4. Stability of piecewise flat Ricci flow

    Rory Conboye

    Topics Numerics & visualization · Shared terms “adaptation”, “numerical”

  5. Theory and numerics for Chen's flow of curves

    Matthew Cooper, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Numerics & visualization · Shared terms “numerical”

  6. The entropy of the Angenent torus is approximately 1.85122

    Yakov Berchenko-KoganWider flows

    Topics Numerics & visualization

  7. Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick

    Klaus Deckelnick and Vanessa StylesWider flows

    Topics Numerics & visualization · Shared terms “numerical”

  8. Contour Parametrization via Anisotropic Mean Curvature Flows

    P. Suárez-Serrato and E. I. Velázquez RichardsWider flows

    Topics Numerics & visualization · Shared terms “numerical”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.