Traditionally, the scope of computational geometry research has been limited to manipulation of geometric elements in the Euclidean space R^{d}.
This one day event will bring together researchers from various application areas working on simulations, having in common that the domains of their simulation are often periodic spaces. The periodicity avoids special treatment at the domain boundaries. Additionally, we will present the most recent advances in CGAL concerning 3D Delaunay triangulations and Voronoi diagrams in periodic spaces. The workshop is a prospective workshop, as the periodic triangulations are work in progress, and hence not yet integrated in the CGAL distribution. However, they are already robust and fast enough to be exposed and tested by potential real world users.
The workshop has two main goals: To share experiences and to identify common algorithmic needs as far as the the geometric foundation is concerned, that is by abstracting from the fact that the domain models the universe, a composite material in a nuclear reactor, or huge molecules floating in water.
10:00  10:15 
Welcome 

10:15  11:45 
Presentations 

10:15  10:45 
CGAL 3D Triangulations in Periodic Spaces Naively extending the Delaunay triangulation algorithm to periodic triangulations would require to compute a triangulation of infinitely many points. However, exploiting the periodicity it is possible to compute a triangulation using only a finite number of periodic copies of each input point. We describe how we handle this issue using even only one periodic copy of each point, whenever this is possible. We formulate a condition on the point set that is efficiently verified by our implementation. Additionally we provide some heuristics for speedup. To the best of our knowledge there is no implementation available using this approach. Unlike other available implementations we came to know about we guarantee to correctly/exactly compute the periodic Delaunay triangulation for a point set without having the need of any special boundary treatment. For topological reasons the interface requires some changes, however we maintain a maximum part of the interface of the CGAL 3D Triangulation package. A video about this software is available
here
(the image on the right illustrates the 2D case).


10:45  11:15 
Periodicity and the Design of Bone Scaffolds One design strategy involves constructing a regular bone scaffold based on a parametric unit cell. Because such a unit cell provides a complete geometrical description of the bone scaffold while being relatively small, it allows costeffective determination of local and apparent material properties using numerical simulations, on the condition that appropriate boundary conditions are applied. In this talk, the suitability of periodic boundary conditions will be discussed, together with the practical requirements that they impose on mesh generation. In addition, open questions will be raised concerning the characterization of topological properties based on periodic unit cells. 

11:15  11:45 
Multiscale mechanics of granular materials To obtain detailed information at the microscale, Discrete Element Method simulations are frequently used, in which the motion of a large set of particles is determined by numerically integrating the equations of motion. To suppress wall effects and to maintain spatially homogeneous deformation, such simulations are often performed in threedimensional periodic space. To study characteristics and statistics of the deformation at the microscale, Delaunay tessellations are used. In addition, adaptive multiphase computational fluid dynamics methods, with hierarchical datastructures, are being developed that depend on fast Delaunay tessellations. 

12:00  13:30 
Lunch 

13:30  14:30 
Presentations 

13:30  14:00 
Periodic Computer Representations of the Cosmic Structure When simulating the evolution of cosmic structure we therefore need to guarantee that the simulation volume is representative. This is most straightforwardly accomplished by specifying volumes with periodic boundary conditions. Effectively it means that the simulation ignores the influence of primordial density and velocity fluctuations with wavenumbers smaller than the fundamental wavelength of the simulation box. One of the main instruments for studying the formation of structure
are Nbody simulations, in which the cosmic mass distribution is
reflected in a representative spatial distribution of discrete particles.
I will describe how Delaunay and Voronoi tessellations are powerful
instruments for analyzing and following the multiscale and strongly
patterned nature of spatial structure on Megaparsec scales.


14:00  14:30 
Radiative Transfer Using Unstructured Grids However, it has become clear that traditional hierarchical AMR techniques have many problems, and slowly but surely the astrophysical community is getting used to the idea of using fully unstructured grids to model complex astrophysical problems. Since space is effectively infinite, to limit the computational domain to finite size for hydrodynamic calculations, periodic boundary conditions are often invoked. While they are not necessarily realistic, they appear to be much less troubling than fixed wall or outflow boundary conditions. To perform the corresponding radiative transfer for such simulations requires that also the radiative transfer model can make use of such boundary conditions. In particular for cosmological simulations this is of particular use. I will mainly present how radiative transfer is done or will be done on Delaunay and/or Voronoi grids, and I will then briefly outline the pros and cons of using periodic boundary conditions for this problem. 

14:30  15:00 
Coffee Break 

15:00  16:45 
Discussion 

16:45  17:00 
Closing 
The number of participants is limited. You can register for the workshop by sending a mail to the organizers. There is no registration fee. Participants have to pay for their lunch at Inria.
The following hotels in Antibes are located at walking distance from both a bus station to INRIA, and the scenic center of Antibes, as well as the sea. We refer to the hotels web sites for more details.
Chrys Hotel***, Le Collier**, Hôtel de l'Étoile**, Hôtel Josse***. The latter is located in Juan les Pins, in front of the sea, but further away from bus stations.
From Nice and/or the airport take the bus Line230 which provides regular shuttles between Nice, the airport NiceCôte d'Azur and Sophia Antipolis (40 mn). Ask the driver for the INRIA bus stop.
From Antibes take the Envibus (the name of the bus stop is Gare Sncf  Passerelle). On Line 11 get off at the bus stop INRIA. On Line 1 get off at the bus stop IUT. On Line 100 (the ride is free) get off at bus stop Templiers, then walk a little bit.
This workshop is organized by Monique Teillaud, INRIA, Manuel Caroli, INRIA, and Andreas Fabri, GeometryFactory. The workshop is partially supported by the French Research Agency project Triangles.