Konstantin Poelke and Konrad Polthier,
Boundary-aware hodge decompositions for piecewise constant vector fields
In: Computer-Aided Design (78), pp. 126-136, 2016
1st price at SPM 2016 best paper awards
We provide a theoretical framework for discrete Hodge-type decomposition theorems of piecewise constant vector fields on simplicial surfaces with boundary that is structurally consistent with decomposition results for differential forms on smooth manifolds with boundary. In particular, we obtain a discrete Hodge–Morrey–Friedrichs decomposition with subspaces of discrete harmonic Neumann fields and Dirichlet fields, which are representatives of absolute and relative cohomology and therefore directly linked to the underlying topology of the surface. In addition, we discretize a recent result that provides a further refinement of the spaces of discrete harmonic Neumann and Dirichlet fields, and answer the question in which case one can hope for a complete orthogonal decomposition involving both spaces at the same time. As applications, we present a simple strategy based on iterated L2-projections to compute refined Hodge-type decompositions of vector fields on surfaces according to our results, which give a more detailed insight than previous decompositions. As a proof of concept, we explicitly compute harmonic basis fields for the various significant subspaces and provide exemplary decompositions for two synthetic vector fields.
Anna Wawrzinek and Konrad Polthier,
Integration of generalized B-spline functions on Catmull–Clark surfaces at singularities
In: Computer-Aided Design (78), pp. 60-70, 2016
Subdivision surfaces are a common tool in geometric modeling, especially in computer graphics and computer animation. Nowadays, this concept has become established in engineering too. The focus here is on quadrilateral control grids and generalized B-spline surfaces of Catmull–Clark subdivision type. In the classical theory, a subdivision surface is defined as the limit of the repetitive application of subdivision rules to the control grid. Based on Stam′s idea, the labour-intensive process can be avoided by using a natural parameterization of the limit surface. However, the simplification is not free of defects. At singularities, the smoothness of the classically defined limit surface has been lost. This paper describes how to rescue the parameterization by using a subdivision basis function that is consistent with the classical definition, but is expensive to compute. Based on this, we introduce a characteristic subdivision finite element and use it to discretize integrals on subdivision surfaces. We show that in the integral representation the complicated parameterization reduces to a decisive factor. We compare the natural and the characteristic subdivision finite element approach solving PDEs on surfaces. As model problem we consider the mean curvature flow, whereby the computation is done on the step-by-step changing geometry.
Konrad Polthier and Faniry Razafindrazaka
Discrete Geometry for Reliable Surface Quad-Remeshing. In: Multiple Shooting and Time Domain Decomposition Methods. Springer. 2015
In this overview paper we will glimpse how new concepts from discrete differential geometry help to provide a unifying vertical path through parts of the geometry processing pipeline towards a more reliable interaction. As an example, we will introduce some concepts from discrete differential geometry and the QuadCover algorithm for quadrilateral surface parametrization. QuadCover uses exact discrete differential geometric concepts to convert a pair (simplicial surface, guiding frame field) to a global quad-parametrization of the unstructured surface mesh. Reliability and robustness is an omnipresent issue in geometry processing and computer aided geometric design since its beginning. For example, the variety of incompatible data structures for geometric shapes severely limits a reliable exchange of geometric shapes among different CAD systems as well as a unifying mathematical theory. Here the integrable nature of the discrete differential geometric theory and its presented application to an effective remeshing algorithm may serve an example to envision an increased reliability along the geometry processing pipeline through a consistent processing theory.
[preprint pdf, 10.7 MB] [BibTex]
Sebastian Götschel, Christoph von Tycowicz, Konrad Polthier, Martin Weiser
Reducing Memory Requirements in Scientific Computing and Optimal Control. In: Multiple Shooting and Time Domain Decomposition Methods. Springer. 2015
This paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well-known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.
[preprint pdf, 828 KB] [BibTex]
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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.
Editors: Peter Deuflhard (Konrad-Zuse-Zentrum, Berlin, Germany); Martin Grötschel (Konrad-Zuse-Zentrum; Berlin, Germany); Dietmar Hömberg (Technische Universität Berlin, Germany); Ulrich Horst (Humboldt-Universität zu Berlin, Germany); Jürg Kramer (Humboldt-Universität zu Berlin, Germany); Volker Mehrmann; Konrad Polthier (Freie Universität Berlin, Germany); Frank Schmidt (Konrad-Zuse-Zentrum, Berlin, Germany); Christof Schütte (Freie Universität Berlin, Germany); Martin Skutella; Jürgen Sprekels (Weierstraß Institut für Angewandte Analysis und Stochastik, Berlin, Germany). EMS Series in Industrial and Applied Mathematics Vol. 1. April 2014, 466 pages
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