# Projects

### DFG SFB/Transregio 109 - Discretization in Geometry and Dynamics

DFG SFB/Transregio 109 - Project C05

In project C05 “Computational and structural aspects of point set surfaces”, we develop discrete differential geometric representations for point set surfaces and effective computational algorithms. Instead of first reconstructing a triangle based mesh, our operators act directly on the point set data. The concepts have contact to meshless methods and ansatz spaces of radial basis functions. As proof of concept of our theoretical investigations we transfer and implement key algorithms from surface processing, for example, for surface parametrization and for feature aware mesh filtering on point set surfaces.

DFG SFB/Transregio 109 - Project A04

This 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. A geometric view onto finite element spaces of piecewise linear functions helped to develop a consistent theory of discrete differential forms on polyhedral surfaces.

### DFG Research Unit

Project Polyhedral Surfaces - Discrete Implicit Surfaces

A discrete implicit surface is considered as the zero set of a scalar-valued function on an ambient 3-dimensional simplicial complex. In contrast to the discrete geometry of simplicial surfaces, the differential operators of implicit surfaces live on the ambient grid. The purpose of this project is to derive a discrete differential geometric toolbox for implicit surfaces similar to (explicit) discrete surfaces. This includes the development of discrete differential operators for implicit surfaces including curvature operators, and the study of curvature flows. A major application is the computation of (parametric) discrete minimal surfaces whose topology is a priori unknown.

### Matheon

Project F4 - Geometric Shape Optimization

Geometric shape optimization is concerned with novel discrete surface energies and computational geometry algorithms for processing and optimizing polyhedral meshes in industrial applications such as computer aided design (CAD) and computer graphics. The main focus of this project addresses the development of efficient mesh processing algorithms and robust modeling tools. The mathematical tools are novel discrete differential and curvature operators.

Project F6 - Multilevel Methods on Manifold Meshes

This project addresses the strong needs for multilevel algorithms in industrial applications and in computer graphics where large surface meshes must be efficiently processed. Typical applications are solutions of PDEs on surfaces, surface optimization, and automatic mesh parametrization. Funding is provided by the DFG Research Center MATHEON - "Mathematics for key technologies".

Project F9 - State Trajectory Compression in Optimal Control

For large, nonlinear, and time dependent PDE constrained optimization problems with 3D spatial domain, reduced methods are a viable algorithmic approach. The computation of reduced gradients by adjoint methods requires the storage of 4D data, which can be quite expensive from both a capacity and bandwith point of view. This project investigates lossy compression schemes for storing the state trajectory, based on hierarchical interpolation in adaptively refined meshes as a general predictor.

### Industry Cooperations

The aim of this project is the development of state of the art visualization methods for semantically marked-up data in interdisciplinary applications, with a focus on data that comes from analyzed and preprocessed patent documents using the Teles "Facts Screening & Transforming Processor" (FSTP) strategy. This includes the development of geometrically motivated algorithms on networks with a view towards their efficient implementation and interface design to provide real-time user interactivity. Inspiration comes from hyperbolic and complex geometry as well as from unconventional techniques in information visualization.

Design Research Exchange (DRX 2012)

The first Design Research Exchange (DRX 2012) takes place in Berlin 16.07.12 through 07.09.12. Participants of the DRX 2012 include four invited DRX Experts and nine invited DRX Researchers with backgrounds in architecture, engineering, mathematics, and computer programming. The topic of exploration for the DRX 2012 is Minimal Surface High-Rise Structures. The DRX 2012 Opening (16 July) and Final Exhibition (7 September) comprise two public events.

### Spin-offs

Mit neuen Algorithmen zum eigenen Design Allein in Deutschland geben bereits über 500 Unternehmen ihren Kunden die Möglichkeit, sich im Internet ihr eigenes Produkt zu entwerfen und zu bestellen. Wer selbst zum Designer wird, möchte allerdings vor der Bestellung möglichst genau wissen, wie das Produkt aussehen wird. Daran hapert es bei vielen Anbietern noch. Das Gründungsvorhaben designby.me will dieses Problem mit "JavaView" lösen, einer Software für 3D-Darstellung, die am ...

The project Trinckle aims to enable private customers to benefit from the advantages of modern 3D printing technology. Within an online marketplace, people will be able to share and personalize 3D models and products that then can be ordered from the Trinckle printout service. In order to improve the access to high quality 3D models, the Trinckle team is developing a novel software system, that enables customers to generate individual 3D models for sharing or printing without the need of any ...

Mathematics is everywhere: in mobile phones, MP3 players, and computers, just to name a few; it is a fundament of science and technology. Still its reputation could be better: it is usually regarded as an abstract subject, tedious and too dry to be comprehensible, with its uses often remaining often a mystery. vismath, founded in March 2010 and supported by an EXIST-grant, aims at a change: presenting mathematics in an inspiring and vivid way, accessible for everyone and unveiling its effective applications in today’s world.