Willmore Flow Research Articles

On geometrically constrained variational problems of the Willmore functional I: The Lagrangian-Willmore problem

In this paper, we study a kind of geometrically constrained variational problem of the Willmore functional. A surface l : Σ → C is called a Lagrangian–Willmore surface (in short, a LW surface) or a Hamiltonian–Willmore surface (in short, a HW surface) if it is a critical point of the Willmore functional under Lagrangian deformations or Hamiltonian deformations, respectively. We extend the L∞ estimates of the second fundamental form of Willmore surfaces to both HW and LW surfaces and thus get a gap theorem for both HW and LW surfaces. To investigate the existence of HW surfaces we introduce a sixth-order flow which is called by us the Hamiltonian–Willmore flow (in short, the HW flow) decreasing the Willmore energy and we prove that this flow is well posed.

Numerical solution for the anisotropic Willmore flow of graphs

The 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.

A gap theorem for Willmore tori and an application to the Willmore flow

In 1965, Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R3 is at least 2π2 and attains this minimal value if and only if the torus is a Möbius transform of the Clifford torus. This was recently proved by Marques and Neves (2012). In this paper, we show for tori that there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.

On quasilinear parabolic evolution equations in weighted L p -spaces II

Our study of abstract quasi-linear parabolic problems in time-weighted L p -spaces, begun in Köhne et al. (J Evol Equ 10:443–463, 2010), is extended in this paper to include singular lower-order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell–Stefan diffusion, and to geometric evolution equations like the surface diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.

Colliding interfaces in old and new diffuse-interface approximations of Willmore-flow

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.

Lifespan theorem and gap lemma for the globally constrained Willmore flow

We study a fourth-order flow, which can be seen as a globally constrained Willmore flow. We obtain a lower bound on the lifespan of the smooth solution, which depends on the concentration of curvature for the initial surface and the constrained term. We also give a gap lemma for this flow, which is an important lemma in the study of the blowup analysis.

A unified method for hybrid subdivision surface design using geometric partial differential equations

Surface subdivision gains popularity in surface design owing to its flexible applications for geometry with complicated topology and simple computational scheme, and the geometric partial differential equation (GPDE) method is an advanced technology for constructing high-quality smooth surfaces. In this paper, we composite these two ingredients to form a unified method for freeform surface design. We choose the mean curvature flow and Willmore flow as our driven GPDEs, and the finite element method coupled with a hybrid Loop and Catmull–Clark subdivision algorithm as the numerical simulation method. This research presents a novel technique to evaluate the finite element basis functions and the first attempt for constructing the GPDE subdivision surface with hybrid control meshes consisting of triangles and quadrilaterals. Numerical experiments show that the construction method is efficient and robust, yielding high-quality hybrid subdivision surfaces.

Robust fairing via conformal curvature flow

We present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and naturally preserves the quality of the input mesh. The main insight is that Willmore flow becomes remarkably stable when expressed in curvature space -- we develop the precise conditions under which curvature is allowed to evolve. The practical outcome is a highly efficient algorithm that naturally preserves texture and does not require remeshing during the flow. We apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces. We also present a new algorithm for length-preserving flow on planar curves, which provides a valuable analogy for the surface case.

Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation

We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in this context, and the sharp interface limit yields a quenched mean-curvature-driven normal velocity at O ( ε −1 ), whereas on the longer O ( ε −2 ) time scale, it leads to a total surface area preserving Willmore flow. In particular, for space dimension n =2, the constrained Willmore flow drives collections of spherically symmetric vesicles to a common radius, whereas for n =3, the radii are constant, and for n ≥4 the largest vesicle dominates.

Branched Willmore spheres

Abstract In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in ℝ3 with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to the analysis of singularities of the Willmore flow of non-Willmore spheres with Willmore energy 𝒲 ( f ) ≤ 16 π $ \mathcal {W} (f) \le 16\pi $ .

A nested variational time discretization for parametric Willmore flow

A novel variational time discretization of isotropic and anisotropic Willmore flow combined with a spatial parametric finite element discretization is applied to the evolution of polygonal curves and triangulated surfaces. In the underlying natural approach for the discretization of gradient flows a nested optimization problem has to be solved at each time step. Thereby, an outer variational problem reflects the time discretization of the actual Willmore flow and involves an approximate L^2 -distance between two consecutive time steps and a fully implicit approximation of the Willmore energy. The mean curvature needed to evaluate the integrant of the latter energy is replaced by the time discrete, approximate speed from an inner, fully implicit variational scheme for mean curvature motion. To solve the resulting PDE constrained optimization problem at every time step duality techniques from PDE optimization are applied. Computational results underline the robustness of the new scheme, in particular with respect to large time steps, and show applications to surface restoration and blending.

Modeling and Computing of Deformation Dynamics of Inhomogeneous Biological Surfaces

Thin elastic surfaces frequently contain molecules influencing their mechanical properties. Such structures occur at different scales in biological systems. Prominent examples are bilayer membranes and cell tissues. We introduce a continuous dynamical model of deformation of lateral inhomogeneous surfaces, using the example of artificial membranes. In agreement with experimental observations, the membrane consists of different molecular species undergoing lateral phase separation and influencing the mechanical properties of the membrane. The model is based on minimization of free energy leading to a nonlinear PDE system of fourth order, related to the Willmore flow and the Cahn--Hilliard equation. The novelty of our work is in modeling and computing of time-dependent dynamics of the system, which requires local mass conservation. Parametric finite element simulations show how dynamics result in diverse equilibrium patterns. The presented simulations are in good agreement with recent theoretical and experi...

Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing

A number of successful variational models for processing planar images have recently been generalized to three-dimensional (3D) surface processing. With this new dimensionality, the amount of numerical computations to solve the minimization of such new 3D formulations naturally grows up dramatically. Though the need of computationally fast and efficient numerical algorithms able to process high resolution surfaces is high, much less work has been done in this area. Recently, a two-step algorithm for the fast solution of the total curvature model was introduced in Tasdizen, Whitaker, Burchard and Osher [Geometric surface processing via normal maps, ACM Trans. Graph. 22(4) (2003), pp. 1012–1033]. In this paper, we generalize and modify this algorithm to the solution of analogues of the mean curvature model of Droske and Martin Rumpf [A level set formulation for Willmore flow, Interfaces Free Bound. 6(3) (2004), pp. 361–378] and the Gaussian curvature model of Elsey and Esedoḡlu [Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul. 7(4) (2009), pp. 1549–1573]. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.

Introducing Willmore Flow Into Level Set Segmentation of Spinal Vertebrae

Segmentation of spinal vertebrae in 3-D space is a crucial step in the study of spinal related disease or disorders. However, the complexity of vertebrae shapes, with gaps in the cortical bone and boundaries, as well as noise, inhomogeneity, and incomplete information in images, has made spinal vertebrae segmentation a difficult task. In this paper, we introduce a new method for an accurate spinal vertebrae segmentation that is capable of dealing with noisy images with missing information. This is achieved by introducing an edge-mounted Willmore flow, as well as a prior shape kernel density estimator, to the level set segmentation framework. While the prior shape model provides much needed prior knowledge when information is missing from the image, and draws the level set function toward prior shapes, the edge-mounted Willmore flow helps to capture the local geometry and smoothes the evolving level set surface. Evaluation of the segmentation results with ground-truth validation demonstrates the effectiveness of the proposed approach: an overall accuracy of 89.32±1.70% and 14.03±1.40 mm are achieved based on the Dice similarity coefficient and Hausdorff distance, respectively, while the inter- and intraobserver variation agreements are 92.11±1.97%, 94.94±1.69%, 3.32±0.46, and 3.80±0.56 mm.

A Phase-Field Approximation of the Willmore Flow with Volume and Area Constraints

The well-posedness of a phase-field approximation to the Willmore flow with area and volume constraints is established when the functional approximating the area has no critical point satisfying the two constraints. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by a variational minimization principle. The main difficulty stems from the nonlinearity of the area constraint.

Superresolution of Images Using Area Preserving Geometric Evolution Laws

PDE-based approaches are used to obtain superresolution of images. Using level set method for area constraint geometric evolution laws allows to remove pixelation in images without removing small-scale features. We compare mean curvature flow, surface diffusion, and Willmore flow for this purpose.

A phase-field approximation of the Willmore flow with volume constraint

The well-posedness of a phase-field approximation to the Willmore flow with volume constraint is established. The existence proof relies on the underlying gradient flow structure of the problem: the time discrete approximation is solved by a variational minimization principle.

Error analysis for the approximation of axisymmetric Willmore flow by $C^1$-finite elements

We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radius function. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ piecewise cubic C^1 -finite elements for the one-dimensional approximation in space.We prove optimal error bounds in Sobolev norms for the solution and its time derivative and present numerical test examples.

Modeling and computation of two phase geometric biomembranes using surface finite elements

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L 2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg–Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen–Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H 1 spaces, and we employ triangulated surfaces and H 1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

Lifespan theorem for constrained surface diffusion flows

We consider closed immersed hypersurfaces in \({\mathbb R^{3}}\) and \({\mathbb R^4}\) evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for which a smooth solution exists, and a small upper bound on a power of the total curvature during this time. By phrasing the theorem in terms of the concentration of curvature in the initial surface, our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows.