Subdivision in Willmore Flow Problems

Transcription

1 Subdivision in Willmore Flow Problems Jingmin Chen 1 Sara Grundel 2 Rob Kusner 3 Thomas Yu 1 Andrew Zigerelli 1 1 Drexel University, Philadelphia 2 New York University and Max Planck Institute Magdeburg 3 University of Massachusetts, Amherst November 2, 2012

2 Red Blood Cells

3 Genus 1 Lipid

4 Canham-Helfrich Model for Lipid Bilayer Let H be the mean curvature and K be the Gauss curvature. min αh 2 + βk da S subject to: 1 area constraint: 2 volume constraint: V = 1 3 S A = S S 1 da = a 0 [xî + yĵ + zˆk] da = v 0 3 bilayer area difference constraint: M = H da = m 0 S (α, β, a 0, v 0, m 0 - constants determined by physical conditions)

5 Canham-Helfrich Model for Lipid Bilayer Let H be the mean curvature and K be the Gauss curvature. min αh 2 + βk da S subject to: 1 area constraint: 2 volume constraint: V = 1 3 S A = S S 1 da = a 0 [xî + yĵ + zˆk] da = v 0 3 bilayer area difference constraint: M = H da = m 0 S (α, β, a 0, v 0, m 0 - constants determined by physical conditions)

6 Lipid Bilayer Membrane Cross-section: S H da = lim ε 0 area difference of the two ε-offset surfaces of S 2ε

7 By Gauss-Bonnet Theorem, S K da = 4π(1 g), we get the following equivalent problem. subject to: min W (C) := H 2 da C S(C) A = S 1 da = a 0 V = 1 3 S [xî + yĵ + zˆk] da = v 0 M = S H da = m 0

8 Previous Computational Work Pioneer work: Brakke s surface evolver Recent work: Driuzk, Bonita-Nochetto-Pauletti uses piecewise linear/quadratic surfaces requires technical weak formulations Also: Bobenko-Schröder defines a discrete Willmore energy on piecewise linear surfaces, respecting Möbius invariance unknown connection to the continuous Willmore energy did not explore the lipid bilayer problem

9 Our approach based on subdivision surfaces computes bona-fide Willmore energy, hence NO weak formulation computes gradient of W, A, V, M w.r.t. control data uses standard optimization solvers sped up by multiscale optimization

10 Our approach based on subdivision surfaces computes bona-fide Willmore energy, hence NO weak formulation computes gradient of W, A, V, M w.r.t. control data uses standard optimization solvers sped up by multiscale optimization

11 Our approach based on subdivision surfaces computes bona-fide Willmore energy, hence NO weak formulation computes gradient of W, A, V, M w.r.t. control data uses standard optimization solvers sped up by multiscale optimization

12 Subdivision Surface 1-4 refinement (connectivity) Subdivision rule (geometric positions) S = S(C), C R 3 #V - (coarse scale) control data

13 Subdivision Surface 1-4 refinement (connectivity) Subdivision rule (geometric positions) S = S(C), C R 3 #V - (coarse scale) control data

14 2-D Box Splines & Ordinary Rules Features of this subdivision scheme: Acts locally Shift invariant D6 Symmetric Linear Converge to C 2 spline function

15 2-D Box Splines & Subdivision

16 Surfaces of arbitrary topology Bad news: Tori are the only surfaces that can be regularly triangulated (By Euler characteristics V E + F = 2(1 g))

17 Surfaces of arbitrary topology Bad news: Tori are the only surfaces that can be regularly triangulated (By Euler characteristics V E + F = 2(1 g)) Good news: Any extraordinary vertex will be isolated in this iterative 1-4 refinement process

18 Extraordinary vertex rule Loop: β = 1 k (5 8 ( cos 2π k )2 ).

19 Patch layout

20 Computation of Willmore energy and its gradient P = C ν Φ ν = C η B η D Cν W = D Cη W D Cν C η

21 Multiscale Minimization Minimize at coarsest scale Subdivide Minimize at finer scale Subdivide Minimize at next finer scale... Feature: built-in subdivision structure guarantees (i) well-definitedness of W-energy at all scales and (ii) progressive improvement Also: 4-th order accurate

22 Uniqueness of W -energy minimization Open question Do we have uniqueness up to rigid motions? Recall the following facts: W invariant under rigid motion, scaling, and sphere inversion the 3-D Möbius group dim Möb(3) = 10 There are still 4 degrees of freedom left!

23 Uniqueness of W -energy minimization Open question Do we have uniqueness up to rigid motions? Recall the following facts: W invariant under rigid motion, scaling, and sphere inversion the 3-D Möbius group dim Möb(3) = 10 There are still 4 degrees of freedom left!

24 Uniqueness(unconstrained) For unconstrained W -energy minimization: Genus 0: round spheres A round sphere is invariant under Möbius transformations.

25 Uniqueness(unconstrained) For unconstrained W -energy minimization: Genus 0: round spheres A round sphere is invariant under Möbius transformations. Genus 1: Clifford torus (Willmore s conjecture, Marques-Neves theorem since Feb 27, 2012)

26 Uniqueness(unconstrained) For unconstrained W -energy minimization: Genus 0: round spheres A round sphere is invariant under Möbius transformations. Genus 1: Clifford torus (Willmore s conjecture, Marques-Neves theorem since Feb 27, 2012) Genus 2: Lawson s surface (Kusner s conjecture)

27 Uniqueness(constrained) By the Hurwitz 84(g-1) theorem, conclusion for the (A, V, M) constrained Willmore problem: non-uniqueness for genus 2 uniqueness problem far from settled when genus = 0, 1

28 Uniqueness(constrained) By the Hurwitz 84(g-1) theorem, conclusion for the (A, V, M) constrained Willmore problem: non-uniqueness for genus 2 uniqueness problem far from settled when genus = 0, 1

29 Open questions numerical exploration of uniqueness simulation of conformal diffusion numerical analysis of proposed subdivision approach

30 Open questions numerical exploration of uniqueness simulation of conformal diffusion numerical analysis of proposed subdivision approach Thanks for your attention!