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+

Books
The Impact of Applications on Mathematics. Proceedings of the Forum of Mathematics for Industry 2013

Authors: Wakayama, Masato and Anderssen, Robert S. and Cheng, Jin and Fukumoto, Yasuhide and McKibbin, Robert and Polthier, Konrad and Takagi, Tsuyoshi and Toh, Kim-Chuan (Editors)
Date: 2014
ISBN: 978-4-431-54906-2

Immagini Della Matematica

Authors: Glaeser, Georg and Polthier, Konrad
Date: 2013
ISBN: 978-88-6030-619-7

Surprenantes images de mathématiques

Authors: Glaeser, Georg and Polthier, Konrad
Note: Janie Molard(Übersetzerin)
Date: 2013
ISBN: 978-27-0115-695-8

Wiskunde in beeld

Authors: Glaeser, Georg and Polthier, Konrad
Date: 2012
ISBN: 978-90-8571-250-3

Bilder der Mathematik

Authors: Glaeser, Georg and Polthier, Konrad
Note: Nachdruck 2014
Date: 2010
ISBN: 978-3-662-43416-1


Papers
Discrete Geometry for Reliable Surface Quad-Remeshing

Authors: Polthier, Konrad and Razafindrazaka, Faniry
In Collection: Applications + Practical Conceptualization + Mathematics = fruitful Innovation, Springer Japan
Date: 2016
DOI: 10.1007/978-4-431-55342-7_22
ISBN: 978-4-431-55341-0
Download: external

Perfect Matching Quad Layouts for Manifold Meshes

Authors: Razafindrazaka, Faniry H. and Reitebuch, Ulrich and Polthier, Konrad
Journal: Computer Graphics Forum (proceedings of EUROGRAPHICS Symposium on Geometry Processing), 34(5)
Date: 2015
Download: external

Realization of Regular Maps Large Genus

Authors: Razafindrazaka, Faniry and Polthier, Konrad
In Book: Topological and Statistical Methods for Complex Data, Springer Berlin Heidelberg
Date: 2015
DOI: 10.1007/978-3-662-44900-4_14
ISBN: 978-3-662-44900-4
Download: external

Differential-Based Geometry and Texture Editing with Brushes

Authors: Krauth, Niklas and Nieser, Matthias and Polthier, Konrad
Journal: J. Math. Imaging Vis., 48(2):359--368
Date: Feb 2014
DOI: 10.1007/s10851-013-0443-6
Download: external

Complex Polynomial Mandalas and their Symmetries

Authors: Konstantin Poelke, Zoi Tokoutsi and Polthier, Konrad
In Proceedings: Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, Tessellations Publishing
Date: 2014
ISBN: 978-1-938664-11-3
Download: external

Regular Surfaces and Regular Maps

Authors: Razafindrazaka, Faniry and Polthier, Konrad
In Proceedings: Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, Tessellations Publishing
Date: 2014
ISBN: 978-1-938664-11-3
Download: external

An Efficient Construction of Reduced Deformable Objects

Authors: von Tycowicz, Christoph and Schulz, Christian and Seidel, Hans-Peter and Hildebrandt, Klaus
Journal: ACM Trans. Graph., 32(6):213:1--213:10
Date: Nov 2013
DOI: 10.1145/2508363.2508392
Download: external

Regular Map Smoothing

Authors: Razafindrazaka, Faniry and Polthier, Konrad
Journal: IMAGEN-A, 3(5)
Date: Mar 2013
Download: external

De Casteljauʼs algorithm on manifolds

Authors: Nava-Yazdani, E. and Polthier, K.
Journal: Computer Aided Geometric Design, 30(7):722 - 732
Date: 2013
DOI: 10.1016/j.cagd.2013.06.002
Download: external

The 6-ring

Authors: Razafindrazaka, Faniry and Polthier, Konrad
In Proceedings: Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, Tessellations Publishing
Date: 2013
ISBN: 978-1-938664-06-9
Download: external

Robust Modeling of Constant Mean Curvature Surfaces

Authors: Pan, Hao and Choi, Yi-King and Liu, Yang and Hu, Wenchao and Du, Qiang and Polthier, Konrad and Zhang, Caiming and Wang, Wenping
Journal: ACM Trans. Graph., 31(4):85:1--85:11
Date: Jul 2012
DOI: 10.1145/2185520.2185581
Download: external

Domain Coloring of Complex Functions: An Implementation-Oriented Introduction

Authors: Poelke, K. and Polthier, K.
Journal: IEEE Computer Graphics and Applications, 32(5):90-97
Date: 2012
DOI: 10.1109/MCG.2012.100
Download: external

Hexagonal Global Parameterization of Arbitrary Surfaces

Authors: Nieser, Matthias and Palacios, Jonathan and Polthier, Konrad and Zhang, Eugene
Journal: IEEE Transactions on Visualization and Computer Graphics, 18(6):865-878
Date: 2012
DOI: 10.1109/TVCG.2011.118
Download: external


Team+

Prof. Dr. Folkmar Bornemann   +

Projects: A04, B03
University: TU München
E-Mail: bornemann[at]tum.de
Website: http://www-m3.ma.tum.de/Allgemeines/FolkmarBornemann


Prof. Dr. Konrad Polthier   +

Projects: A04
University: FU Berlin
E-Mail: konrad.polthier[at]fu-berlin.de
Website: http://page.mi.fu-berlin.de/polthier/


Anna Wawrzinek   +

Projects: A04
University: FU Berlin
E-Mail: anna.wawrzinek[at]fu-berlin.de