Team geometrica

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
Other Grants and Activities

Section: New Results

Keywords : Isotropic meshing, anisotropic meshing, triangle meshing, tetrahedral meshing, level sets.

Mesh generation

A generic software design for Delaunay refinement meshing

Participants : Laurent Rineau, Mariette Yvinec.

We presentĀ [78] a generic software designed to implement meshing algorithms based on the Delaunay refinement paradigm. Such a meshing algorithm is generally described through a set of rules guiding the refinement of mesh elements. The central item of the software design is a generic class, called a mesher level, that is able to handle one of the rules guiding the refinement process. Several instantiations of the mesher level class can be stacked and tied together to implement the whole refinement process. As shown in this paper, the design is flexible enough to implement all currently known mesh generation algorithms based on Delaunay refinement. In particular it can be used to generate meshes approximating smooth or piecewise smooth surfaces, as well as to mesh three dimensional domains bounded by such surfaces. It also adapts to algorithms handling small input angles and various refinement criteria. This design highly simplifies the task of implementing Delaunay refinement meshing algorithms. It has been used to implement several meshing algorithms in the cgal library.

A Lagrangian approach to dynamic interfaces through kinetic triangulations

Participants : Jean-Daniel Boissonnat, Jean-Philippe Pons.

We proposeĀ [44] a robust and efficient Lagrangian approach for modeling dynamic interfaces between different materials undergoing large deformations and topology changes, in two dimensions. Our work brings an interesting alternative to popular techniques such as the level set method and the partical level set method, for two-dimensional and axisymmetric simulations. The principle of our approach is to maintain a two-dimensional triangulation which embeds the one-dimensional polygonal description of the interfaces. Topology changes can then be detected as inversions of the faces of this triangulation. Each triangular face is labeled with the type of material it contains. The connectivity of the triangulation and the labels of the faces are updated consistently during deformation, within a neat framework developed in computational geometry: kinetic data structures. Thanks to the exact computation paradigm, the reliability of our algorithm, even in difficult situations such as shocks and topology changes, can be certified. We demonstrate the applicability and the efficiency of our approach with a series of numerical experiments in two dimensions. Finally, we discuss the feasibility of an extension to three dimensions.

Figure 3. Application to brain segmentation: (a) MR image, (b-g) different stages of the evolution, (h) labeled triangulation of the final shape.


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