Martin Skrodzki and Konrad Polthier,
Turing-Like Patterns Revisited: A Peek Into The Third Dimension
To appear in: Proceedings of Bridges, 2017
Beginning with their introduction in 1952 by Alan Turing, Turing-like patterns have inspired research in several different fields. One of these is the field of cellular automata, which have been utilized to create Turing-like patterns by David A. Young and others. In this paper we provide a generalization of these patterns to the third dimension. Several visualizations are given to illustrate the created models.
[pdf, 8 MB]
Faniry Razafindrazaka and Konrad Polthier,
Optimal Base Complexes for Quadrilateral Meshes
In: Computer Aided Geometric Design, pp. 63-74, 2017
In this paper we give an explicit algorithm to optimize the global structure of quadrilateral meshes i.e base complexes, using a graph perfect matching. The approach consists of constructing a special graph over the singularity set of the mesh and finding all quadrilateral based complex subgraphs of that graph. We show by construction that there is always an optimal base complex to a given quadrilateral mesh relative to coarseness versus geometry awareness. Local structures of the mesh induce extra constraints which have been previously ignored but can give a completely different layout. These are diagonal, multiple and close to zero length edges. We give an efficient solution to solve these problems and improve the computation speed. Generally all base complex optimization schemes are bounded by the topology of the singularities, we explore the space of layouts encoded in the graph to identify removable singularities of the mesh while simultaneously optimize the base complex.
[pdf, 15 MB][BibTex]
Konstantin Poelke and Konrad Polthier,
Discrete Topology-Revealing Vector Fields on Simplicial Surfaces with Boundary
In: TopoInVis 2017 proceedings
We present a discrete Hodge-Morrey-Friedrichs decomposition for piecewise constant vector fields on simplicial surfaces with boundary which is structurally consistent with the smooth theory. In particular, it preserves a deep linkage between metric properties of the spaces of harmonic Dirichlet and Neumann fields and the topology of the underlying geometry, which reveals itself as a discrete de Rham theorem and a certain angle between Dirichlet and Neumann fields. We illustrate and discuss this linkage on several geometries.
[Extended Abstract, pdf, 1 MB][BibTex]
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Proceedings of the Forum "Math-for-Industry" 2015.
Editors: Bob Anderssen, Philip Broadbridge, Yasuhide Fukumoto. Naoyuki Kamiyama, Yoshihiro Mizoguchi, Konrad Polthier, Osamu Saeki
Georg Glaeser and Konrad Polthier
2. Aufl. 2010. Nachdruck 2014, XII, 340 S.
Table of content (pdf)
Proceedings of the Forum of Mathematics for Industry 2013. Series: Mathematics for Industry, Vol. 1. Wakayama, M., Anderssen, R.S., Cheng, J., Fukumoto, Y., McKibbin, R., Polthier, K., Takagi, T., Toh, K.-C. (Eds.). 2014, XV, 369 p. 113 illus., 84 illus. in color.
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Hodge-Type Decompositions for Piecewise Constant Vector Fields on Simplicial Surfaces and Solids with Boundary. PhD Thesis, Freie Universität Berlin, 2017
This dissertation develops a theory for Hodge-type decompositions of piecewise constant vector fields (PCVF) on simplicial surfaces and solids with boundary in R3 which is structurally consistent with the corresponding results in the smooth world. First, the necessary differential-geometric and topological foundations of Hodge decomposition statements on smooth manifolds with boundary are reviewed, with a particular focus on a recent result by Shonkwiler which classifies certain harmonic fields as cohomology representatives for either the cohomology induced by the boundary components, or the "inner topology" of the manifold. Next, based on linear Lagrange, Crouzeix-Raviart and Nédélec elements, a discrete Hodge-Morrey-Friedrichs decomposition for PCVFs on simplicial surfaces and solids with boundary is derived. Of particular importance are the spaces H_N and H_D of discrete harmonic Neumann and Dirichlet fields, respectively, as they represent certain cohomology classes of the manifold and are therefore deeply linked to the topology of the simplicial manifold.
For surfaces with boundary that come from a closed surface of genus g=0, one obtains a complete, L2-orthogonal five-term decomposition where both the spaces H_N and H_D appear as L2-orthogonal subspaces. On the other hand, if g>0, these two spaces are no longer orthogonal to each other, and there are now two orthogonal decompositions - one involving H_N, the other one involving H_D.
A deep result in the smooth world states that the spaces of Neumann and Dirichlet forms always have a trivial intersection, but the corresponding result for the discrete spaces does not hold in general.
Surprisingly, at this stage the combinatorics of the triangulating grid comes into play, and it becomes a matter of how the triangulation connects topologically rich regions with the boundary components to decide whether the statement holds true or not. Based on results by Lovász and Benjamini on weighted networks, a criterion is derived that guarantees the validity of the trivial-intersection result. Next, convergence of the derived decomposition statements is proved under the assumption that the simplicial geometry converges metrically against a smooth geometry. To this end, a seminal convergence result by Dodziuk on the approximation of smooth Hodge decomposition components by Whitney forms is generalized to include the refined discrete decompositions. One obtains convergence with respect to the L2-norm which is of linear order in the mesh size.
The dissertation concludes with two central applications of this discrete Hodge theory: the computation of harmonic cohomology representatives and the computational decomposition of a given PCVF.
For both applications, algorithms are presented and evaluated on various test models and the numerical aspects of the involved solving steps are discussed.
On Isoparametric Catmull-Clark Finite Elements for Mean Curvature Flow. PhD Thesis, Freie Universität Berlin, 2016
In this thesis, we deal with the study of subdivision finite element methods for the solution of differential equations on curved surfaces based on the Catmull-Clark subdivisions. The focus is on quadrangular control grids and the characteristic parameterization of the limit surfaces. These are descried using the generalized B-spline basis functions of the Catmull-Clark type. In particular, we present a new finite element approach compatible with the classical definition of the subdivision surfaces. Compared to the previously used natural finite elements, the form of the grid and, consequently, the stability of the limit surface remains resistant. This is achieved as the characteristic finite elements inherit the continuity properties of the subdivision surfaces classically generated by grid refinements.
Faniry H. Razafindrazaka
Quad Layout Generation and Symmetric Tilings of Closed Surfaces. PhD Thesis, Freie Universität Berlin, 2016
This thesis concerns two fundamental concepts in surface topology. The first part proposes a solution of the problem of generating an all quadrilateral patch layout on a given surface. We approach the problem from a combinatorial graph optimization point of view. Mainly, finding a nice quad layout of a given surface is equivalent to solving a minimum weight perfect matching problem with additional quad guarantee constraints. The results are of high quality in terms of coarseness and alignment to important features of the geometry which can be used for wide range of applications such as hierarchical subdivision or high order surface fitting. The second part suggests an algorithm to symmetrically generate high genus surfaces suitable for space models of regular maps. It is based on a novel identification in hyperbolic space to derive directly the tubular neighborhood of the edge of a tiling directly the the hyperbolic representation followed by a spring relaxation procedure with intersection-free guarantee. We succeed to produce new embeddings of regular maps ranging from genus 5 to 85.
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