Processing Elastic Surfaces and Related Gradient Flows

Surface processing tools and techniques have a long history in the fields of computer graphics, computer aided geometric design and engineering. In this thesis we consider variational methods and geometric evolution problems for various surface processing applications including surface fairing, surface restoration and surface matching. Geometric evolution problems are often based on the gradient flow of geometric energies. The Willmore functional, defined as the integral of the squared mean curvature over the surface, is a geometric energy that measures the deviation of a surface from a sphere. Therefore, it is a suitable functional for surface restoration, where a destroyed surface patch is replaced by a smooth patch defined as the minimizer of the Willmore functional with boundary conditions for the position and the normal at the patch boundary. However, using the Willmore functional does not lead to satisfying results if an edge or a corner of the surface is destroyed. The anisotropic Willmore energy is a natural generalization of the Willmore energy which has crystal-shaped surfaces like cubes or octahedra as minimizers. The corresponding L2-gradient flow, the anisotropic Willmore flow, leads to a fourth-order partial differential equation that can be written as a system of two coupled second second order equations. Using linear Finite Elements, we develop a semi-implicit scheme for the anisotropic Willmore flow with boundary conditions. This approach suffer from significant restrictions on the time step size. Effectively, one usually has to enforce time steps smaller than the squared spatial grid size. Based on a natural approach for the time discretization of gradient flows we present a new scheme for the time and space discretization of the isotropic and anisotropic Willmore flow. The approach is variational and takes into account an approximation of the L2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the anisotropic Willmore functional at the new time step. To evaluate the anisotropic Willmore energy on the unknown surface of the next time step, we first ask for the solution of an inner, secondary variational problem describing a time step of anisotropic mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the anisotropic mean curvature vector and enters the approximation of the actual anisotropic Willmore functional. The resulting two step time discretization of the Willmore flow is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size. The Willmore functional of a surface is referred to as the elastic surface energy. Another interesting application of modeling elastic surfaces as minimizers of elastic energies is surface matching, where a correspondence between two surfaces is subject of investigation. There, we seek a mapping between two surfaces respecting certain properties of the surfaces. The approach is variational and based on well-established matching methods from image processing in the parameter domains of the surfaces instead of finding a correspondence between the two surfaces directly in 3D. Besides the appropriate modeling we analyze the derived model theoretically. The resulting deformations are globally smooth, one-to-one mappings. A physically proper morphing of characters in computer graphic is capable with the resulting computational approach

Variational Discretization of Higher Order Geometric Gradient Flows Based on Phase Field Models

In this thesis a phase field based nested variational time discretization for Willmore flow is presented. The basic idea of our model is to approximate the mean curvature by a time-discrete, approximate speed of the mean curvature motion. This speed is computed by a fully implicit time step of mean curvature motion, which forms the inner problem of our model. It is set up as a minimization problem taking into account the concept of natural time discretization. The outer problem is a variational problem balancing between the L2-distance of the surface at two consecutive time steps and the decay of the Willmore energy. This is a typical ansatz in case of natural time discretization as it is used in the inner problem. Within the Willmore energy the mean curvature is approximated as mentioned above. Consequently our model is a nested variational and leads to a PDE constraint optimization problem to compute a single time step. It allows time steps up to the size of the spatial grid width. A corresponding parametric version of this model based on finite elements on a triangulation of the evolving geometry was investigated by Olischläger and Rumpf. In this work we derive the corresponding phase field version and prove the existence of a solution. Since biharmonic heat flow is a linear model problem for our nested time discretization of Willmore flow we transfer our model to the linear case. Moreover we present error estimates for the fully discrete biharmonic heat flow and validate them numerically. In addition we compare our model with the semi-implicit phase field scheme for Willmore flow introduced by Du et al. which leads to the result that our nested variational method is significantly more robust. An application of our nested time discretized Willmore model consists in reconstructing a hypersurface corresponding to a given lower-dimensional apparent contour or Huffman labeling. The apparent contour separates the regions where the number of intersections between the hypersurface and the projection ray is constant and the labeling which specifies these intersection numbers is called Huffman labeling. For reconstructing the hypersurface we minimize a regularization energy consisting of the scaled area and Willmore energy subject to the constraint that the Huffman labeling of the minimizing surface equals the given Huffman labeling almost everywhere. To solve the corresponding phase field problem we use an algorithm alternating the minimizes of the regularization and mismatch energy. Moreover we use a multigrid ansatz. In most parts of this work our nested variational problem is solved by setting up the corresponding Lagrange equation and solving the resulting saddle point problem. An alternative is presented in the last part of this work. It deals with the problem of solving the linear model problem as well as our nested variational problem with an Augmented Lagrange method

A p-ADAPTIVE LOCAL DISCONTINUOUS GALERKIN LEVEL SET METHOD FOR WILLMORE FLOW

International audienceThe level set method is often used to capture interface behavior in two or three dimensions. In this paper, we present a combination of local discontinuous Galerkin (LDG) method and level set method for simulating Willmore flow. The LDG scheme is energy stable and mass conservative, which are good properties comparing with other numerical methods. In addition, to enhance the efficiency of the proposed LDG scheme and level set method, we employ a p-adaptive local discontinuous Galerkin technique, which applies high order polynomial approximations around the zero level set and low order ones away from the zero level set. A major advantage of the level set method is that the topological changes are well defined and easily performed. In particular, given the stiffness of Willmore flow, a high order semi-implicit Runge-Kutta method is employed for time discretization, which allows larger time step. The equations at the implicit time level are linear, we demonstrate an efficient and practical multi-grid solver to solve the equations. Numerical examples are given to illustrate the combination of the LDG scheme and level set method provides an efficient and practical approach when simulating the Willmore flow

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

On a new conformal functional for simplicial surfaces

We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing the squared local conformal discrete Willmore energy $W$ this functional also minimizes local differences of the angles $\beta$. We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are observed. In particular for most of randomly generated simplicial polyhedra the minimizers of $W_2$ and $W_{2,w}$ are inscribed polyhedra. We demonstrate also some applications in geometry processing, for example, a conformal deformation of surfaces to the round sphere. A partial theoretical explanation through quadratic optimization theory of some observed phenomena is presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and Surfaces, 8th International Conference", June 201

An energy-stable parametric finite element method for the planar Willmore flow

We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first one involves the coupling of the outward unit normal vector $\boldsymbol{n}$ and the normal velocity $V$, and the second one concerns the time derivative of the mean curvature $\kappa$. Based on them, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully-discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate its efficiency and validate its unconditional energy stability