A04
Integrating Discrete Geometries and Finite Element Spaces

Building Bridges Between Discretization in Computational Geometry and in Finite Element Analysis

Finite element methods are in every day use in engineering and modelling. The main idea with finite elements is to discretize objects such as machine parts or architectural elements in order to then simulate the movement and behaviour of these objects via discrete computations. Project A04 aims to link experiences from those applications of scientific computing with ideas from discrete geometry to improve the integration of technologies.

Scientific Details+

In recent years computational geometry and numerical analysis have been put in close contacts in fields such as computer aided design (CAD) and scientific computing. We will investigate the different approaches to discretization in differential respectively computational geometry and in finite element analysis. Although some attempts have been made to put concepts on a common ground the focus in geometry has traditionally been on the discretization of shapes and exact integrability while the view in finite element analysis was directed more on the discretization of function spaces and approximation issues. The research of this project will concentrate on an improved linking of both aspects of discretization techniques to gain better insight and improve integration of technologies.

The project is based on combined efforts in discrete differential geometry and finite element methods for geometric partial differential equations. Especially, discretizations using polyhedral surfaces and piecewise linear functions on them proved to be very successful in both theory and applications. This development led to the creation of counterparts of geometric and metric properties of smooth surfaces on polyhedral surfaces and to insights on convergence properties of them. Some of these questions, e.g. the convergence of surface area, are already more 100 years old (cf the Lantern of Schwarz). A geometric view onto finite element spaces of piecewise linear functions helped to develop a consistent theory of discrete differential forms on polyhedral surfaces, where discrete analogs to important theorems, like the Hodge decomposition hold exactly (and not only in the limit of refinement). For applications in engineering, computer aided design, and computer graphics the discretization of the Laplace-Beltrami operator of surface to polyhedral surfaces is a prominent examples. It has been used in various applications including physical simulation, parametization, geometric modeling, shape analysis, and surface optimization.

The long term aim of this project is to extend the geometric view onto finite element constructions to higher order elements, constructed from subdivision processes as well as from NURBS surface representations. As one of the starting points we will consider the geometric problems of constructing minimal and cmc surfaces in this setting.

A related recent development in the computer aided design community is isogeometric analysis, where also NURBS and subdivision schemes are used to build discrete function spaces, e.g. finite element spaces. Since the focus in isogeometric analysis is less on differential geometric problems we see interesting contact points.

Publications+

Papers
  • Konrad Polthier and Faniry Razafindrazaka.
    Discrete Geometry for Reliable Surface Quad-Remeshing.
    In Robert S. Anderssen, Philip Broadbridge, Yasuhide Fukumoto, Kenji Kajiwara, Tsuyoshi Takagi, Evgeny Verbitskiy, and Masato Wakayama, editors, Applications + Practical Conceptualization + Mathematics = fruitful Innovation, volume 11 of Mathematics for Industry, pages 261–275. Springer Japan, 2016.
    doi:10.1007/978-4-431-55342-7_22.
  • Faniry H. Razafindrazaka, Ulrich Reitebuch, and Konrad Polthier.
    Perfect Matching Quad Layouts for Manifold Meshes.
    Computer Graphics Forum (proceedings of EUROGRAPHICS Symposium on Geometry Processing), 2015.
    URL: http://www.mi.fu-berlin.de/en/math/groups/ag-geom/publications/db/2015_RRP_PerfectMatchingQuadLayoutsForManifoldMeshes_New.pdf.
  • Faniry Razafindrazaka and Konrad Polthier.
    Realization of Regular Maps Large Genus.
    In Valerio Pascucci Janine Bennet, Fabien Vivodtzev, editor, Topological and Statistical Methods for Complex Data, pages 239–252. Springer Berlin Heidelberg, 2015.
    URL: http://page.mi.fu-berlin.de/faniry/files/faniryTSCD2015.pdf, doi:10.1007/978-3-662-44900-4_14.
  • Niklas Krauth, Matthias Nieser, and Konrad Polthier.
    Differential-Based Geometry and Texture Editing with Brushes.
    J. Math. Imaging Vis., 48(2):359–368, February 2014.
    doi:10.1007/s10851-013-0443-6.
  • Zoi Tokoutsi Konstantin Poelke and Konrad Polthier.
    Complex Polynomial Mandalas and their Symmetries.
    In George Hart Gary Greenfield and Reza Sarhangi, editors, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, 433–436. Phoenix, Arizona, 2014. Tessellations Publishing.
    URL: http://archive.bridgesmathart.org/2014/bridges2014-433.html.
  • Faniry Razafindrazaka and Konrad Polthier.
    Regular Surfaces and Regular Maps.
    In George Hart Gary Greenfield and Reza Sarhangi, editors, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, 225–234. Phoenix, Arizona, 2014. Tessellations Publishing.
    URL: http://archive.bridgesmathart.org/2014/bridges2014-225.html.
  • Christoph von Tycowicz, Christian Schulz, Hans-Peter Seidel, and Klaus Hildebrandt.
    An Efficient Construction of Reduced Deformable Objects.
    ACM Trans. Graph., 32(6):213:1–213:10, November 2013.
    doi:10.1145/2508363.2508392.
  • Faniry Razafindrazaka and Konrad Polthier.
    Regular Map Smoothing.
    IMAGEN-A, March 2013.
    URL: http://munkres.us.es/Volume3/Volumen3/N_5_files/5.4.pdf.
  • E. Nava-Yazdani and K. Polthier.
    De Casteljauʼs algorithm on manifolds.
    Computer Aided Geometric Design, 30(7):722 – 732, 2013.
    URL: http://www.sciencedirect.com/science/article/pii/S0167839613000551, doi:10.1016/j.cagd.2013.06.002.
  • Faniry Razafindrazaka and Konrad Polthier.
    The 6-ring.
    In George W. Hart and Reza Sarhangi, editors, Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, 279–286. Phoenix, Arizona, 2013. Tessellations Publishing.
    URL: http://archive.bridgesmathart.org/2013/bridges2013-279.html.
  • Hao Pan, Yi-King Choi, Yang Liu, Wenchao Hu, Qiang Du, Konrad Polthier, Caiming Zhang, and Wenping Wang.
    Robust Modeling of Constant Mean Curvature Surfaces.
    ACM Trans. Graph., 31(4):85:1–85:11, July 2012.
    doi:10.1145/2185520.2185581.
  • K. Poelke and K. Polthier.
    Domain Coloring of Complex Functions: An Implementation-Oriented Introduction.
    IEEE Computer Graphics and Applications, 32(5):90–97, 2012.
    doi:10.1109/MCG.2012.100.
  • Matthias Nieser, Jonathan Palacios, Konrad Polthier, and Eugene Zhang.
    Hexagonal Global Parameterization of Arbitrary Surfaces.
    IEEE Transactions on Visualization and Computer Graphics, 18(6):865–878, 2012.
    doi:10.1109/TVCG.2011.118.

Books
  • Masato Wakayama, Robert S. Anderssen, Jin Cheng, Yasuhide Fukumoto, Robert McKibbin, Konrad Polthier, Tsuyoshi Takagi, and Kim-Chuan Toh, editors.
    The Impact of Applications on Mathematics. Proceedings of the Forum of Mathematics for Industry 2013, Japan, 2014. Springer.
  • Georg Glaeser and Konrad Polthier.
    Immagini Della Matematica.
    Springer, Italia, 2013. ISBN 978-88-6030-619-7.
  • Georg Glaeser and Konrad Polthier.
    Surprenantes images de mathématiques.
    Belin, 2013. ISBN 978-27-0115-695-8. Janie Molard(Übersetzerin).
  • Georg Glaeser and Konrad Polthier.
    Wiskunde in beeld.
    Uitgevrij Veen Magazines BV, 2012. ISBN 978-90-8571-250-3.
  • Georg Glaeser and Konrad Polthier.
    Bilder der Mathematik.
    Springer Spektrum, 2 edition, 2010. ISBN 978-3-662-43416-1. Nachdruck 2014.

Team+

Prof. Dr. Folkmar Bornemann   +

Projects: B03
University: TU München
E-Mail: bornemann[at]tum.de
Website: http://www-m3.ma.tum.de/Allgemeines/FolkmarBornemann


Prof. Dr. Konrad Polthier   +

Projects: C05
University: FU Berlin
E-Mail: konrad.polthier[at]fu-berlin.de
Website: http://page.mi.fu-berlin.de/polthier/