Interaction Between Design and Functionality

For several years already, neither automotive engineering nor life science, architecture or the producers of animation films have been able to work without geometry processing. In the past few years, it has moved into new markets and lastingly changed our communication behaviour. Methods of applied mathematics and resulting efficient and robust algorithms form the basis and driving force of that development.

A team of the DFG Research Center MATHEON is involved and plays a decisive role in that work. Within the framework of a project entitled “GEOMEC – Discrete Geometrical Structural Mechanics for Applications in Virtual and Extended Reality” that is promoted by the Federal Ministry of Education and Research (BMBF), the team of Prof. Ulrich Pinkall of MATHEON has been awarded the subproject “Elasticity with Discrete Differential Geometry”. The joint project is coordinated by Junior Professor Max Wardetzky of the Georg-August Universität Göttingen.

Up to now, physical aspects have not been taken into account to their full extent in computer aided geometric design, which has often led to a costly gap between design and functionality in today's production processes. GEOMEC shall close that gap. For example, a designer would like to understand what influence the change of parameters will have on a physical system. At the same time, however, the computing complexity of an exact simulation does still require a considerable expenditure of time and does not allow exact navigation. So it is the aim of the project to overcome that limitation by completely new methods and to be able to describe the physical trajectories in real time and on rough scales. This approach is in particular of importance because it is the extraction of essential information what is of interest to the designer, but never every detail of the actual physical trajectory.

There are being developed new interactive processes and methods that allow to directly integrate the physical-functional aspects into the process of shaping and design. This way, GEOMEC is setting completely new trends in virtual product development and rapid prototyping. The innovative content of the new methods is demonstrated using a selection of concrete applications from the automotive industry as examples.

From a mathematical point of view, these objectives will be achieved by a fusion of methods of structural mechanics with the latest findings in discrete differential geometry (DDG). DDG is located at the interface between numerics and classical differential geometry. It is its approach to maintain and imitate axioms, invariants and fundamental properties of established classical differential geometry at a structural level. Here it has turned out that sophisticated and profound properties of the classical theory often find an amazingly simple and intuitive correspondence in the discrete.

This approach is based on a number of working projects realized by some scientists involved in GEOMEC, such as in particular Ulrich Pinkall and Max Wardetzky, at the MATHEON at an earlier time already. Some of those research projects also relate to the animation of large motion pictures. Realtime simulations of flexible structures considering physical-functional behaviour have caused revolutionary changes especially in the film industry so far – such as in the animation of virtual characters or the simulation of hair, animal coats or fabric close to reality.

But also outside computer graphics, the methods of discrete differential geometry have an extensive potential for industrial applications. Respective examples include digital prototype design, assembly simulation or construction space simulation in automotive engineering, virtual surgery planning or catheter simulation in medicine, fabric simulation and virtual catwalk in the fashion industry, sail simulation in boat building, simulation of flexible cables and tubes in machine and plant construction, rotor blade dynamics of helicopters in aeronautics, but also the simulation of rotors of wind turbines in connection with overall-system simulation using methods of multibody dynamics in the field of renewable energy.

MATHEON-Professor Ulrich Pinkall is responsible for the GEOMEC subproject “Elasticity with Discrete Differential Geometry”. Further subprojects are headed by Prof. Marc Alexa, Technische Universität Berlin, School of Electrical Engineering and Computer Science, Prof. Arnd Meyer, Technische Universität Chemnitz, Faculty of Mathematics, Junior Prof. Max Wardetzky, Georg–August Universität Göttingen, Institute for Numerical and Applied Mathematics as well as Dr. Joachim Linn, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik. Volkswagen AG is the cooperation partner.

Further information: Prof. Max Wardetzky, Tel: 0551 3922235, E-mail: wardetzky@math.uni-goettingen.de and Prof. Ulrich Pinkall, Tel.: 030 31424607, E-mail: pinkall@math.tu-berlin.de