# 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.
**

**Group:**A. Geometry-
**Principal Investigators:**Prof. Dr. Folkmar Bornemann, Prof. Dr. Konrad Polthier -
**Investigator:**Anna Wawrzinek **Universities:**FU Berlin, TU München**Term:**2012 - 2016

#### 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: **
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/

#### Anna Wawrzinek +

**University: **
FU Berlin

**E-Mail: **
anna.wawrzinek[at]fu-berlin.de