DFG Research Center
Matheon
“Mathematics for key technologies”


Matheon Miniworkshop

10:00  Opening  
10:15  11:45  Peter Schröder  Discrete conformal mapping through circle packing 
Lunch  
13:15  14:45  Eitan Grinspun  Adaptive solvers for thin shells: Achievements and challenges 
15:15  15:45  Boris Springborn  Variational description of circle patterns on a sphere 
16:00  16:30  Eike Preuss  Curve optimization under spatial constraints 
16:45  17:15  Felix Kälberer  Triangle mesh compression 
10:15  11:45  Bernard Deconinck  Calculus on Riemann surfaces starting from algebraic curves 
Lunch  
13:15  13:55  Markus Schmies  Computing helicoids with handles using Schottky uniformization 
14:15  14:55  Ulrich Pinkall  Design of cylinders with constant curvature 
15:15  15:45  Tim Hoffmann & Ulrich Pinkall 
Oorange and jReality: Real time programming and experiments in virtual reality 
16:15  17:00  Steffen Weissmann  PORTAL demo 
Almost all integrable equations have solutions that are expressed in terms of Riemann's theta function. The wellknown soliton solutions are special cases of these. The theta function solutions are parametrized in a highly transcendental way by Riemann surfaces. The nature of this parametrization has long precluded the use of these solutions in applications. In this talk, I will give a status report of an ongoing research program to make the calculation of these theta function solutions effective. Our starting point is the representation of a Riemann surface as the desingularization and compactification of a plane algebraic curve. I will describe algorithms for the compution of the monodromy of this curve, the (co)homology of the Riemann surface, and its period matrix, the computation of the theta function, and finally the computation of the Abel transform on the Riemann surface. The talk will conclude with a maple demo of implementations of all these algorithms.
The mechanics of thin, flexible "shells" are key to physicallybased surface modeling, where modeling is posed as a variational problem and the modeled surface minimizes a physical energy functional. Our interest lies in models and solvers for thin shell mechanics. Traditionally, practitioners have relied on finiteelement and more recently subdivisionelement formulations. Both approaches have wellunderstood convergence properties and are amenable to treatment by adaptive solversconvergence and adaptivity are key components of an efficient, accurate, and predictable modeling framework. We will examine a successful and general approach to adaptivity in this context. Unfortunately, the traditional treatments of thinshells require complex computational machinery thus prohibitive implementation cost, motivating the formulation of simple discrete models for thin shells. Such "discrete shells" are being rapidly adopted by the computer graphics community for their visually convincing results. However, the meshingdependence of discrete shells is not yet wellunderstood. Furthermore, since establishing convergence is a prerequisite for building an adaptive solver, this promising approach cannot yet present itself in full bloom. We will discuss some initial results in studying convergence, and some of the current incompatibilities between constructions that ensure convergence and those that maintain discrete surface invariants.
Efficient computational procedures to construct mappings from (regions
of) surfaces to the plane are essential for many applications in geometry
processing and beyond. Among such mappings, conformal mappings stand out
as they locally preserve angles, one of the more important distortion
measures. Standard finite element formulations of the associated PDE
problems are well understood, though they generally loose the underlying
symmetries as they discretize a continuous notion of conformality. Instead
one can approach this problem directly in the discrete setting by defining
an appropriate notion of discrete conformality. This is were circle
packings enter. Given a simplicial 2manifold (with boundary) mesh,
construct a mapping to the unit disk which preserves certain angles
exactly and solves for the only relevant degrees of freedom, discrete area
factors, directly. With the recent formulation of circle packings as
solutions which minimize a variational energy very efficient numerical
algorithms can be brought to bear on this problem and in my talk I will
cover some of the progress we have been able to make in this realm.
Joint work with Liliya Kharevych and Boris Springborn.
Boris Springborn  .  14.12.2004. Accesses: [an error occurred while processing this directive] since 11.03 
DFG Research Center Matheon "Mathematics for key technologies" 