Section: New Results

Geomety processing

Hexahedral Meshes: Generation, Simulation, Evaluation

Participants: Maxence Reberol, Nicolas Ray, Dmitry Sokolov, Bruno Lévy.

We continued working on the generation of the so-called hexahedral (or hexahedral-dominant) meshes. It is believed that these meshes are more efficient (both in terms of required space and computational time) for certain physics and numerical simulation. However, they are much more difficult to generate, and no fully automatic method currently exists. It is a huge problem in the industry, that uses days/weeks/months of manual work to generate them, because they are preferred in certain domains (fluids, mechanics, wave propagation). In this research, we aim at answering the following questions:

• How can we generate a hexahedral (or hex-dominant) mesh in a fully automatic manner?

• How can we evaluate the quality of this mesh, suitability for numerical simulation?

In the context of Maxence Reberol's Ph.D. thesis, we developed new algorithms to answer both questions. To answer the first question, in [17], building on our previous results based on global parameterization, we proposed a method to mesh the globality of the domain, by isolating the singular zones of the parameterization and meshing them with a separate algorithm. To answer the second question, in [16], we proposed a new method to estimate the distance between two Finite Element simulations obtained from two different computational meshes / function bases. We started using our algorithm to compare the rate of convergence of the method as a function of element size h with different PDEs (Poisson, linear elasticity) using different function bases (tetrahedral: P1, P2, P3, hexahedral: Q1, Q2, Q3).

Surface Reconstruction

Participants: Dobrina Boltcheva, Bruno Lévy.

We developed a new algorithm [7] for surface reconstruction. Our algorithm is equivalent to Delaunay-based reconstruction, it computes the Delaunay triangulation restricted to an object computed from the input pointset. The object is a set of disks centered on the input points and perpendicular to estimated normals. The algorithm is fast and memory efficient, because the only used global data structure is a Kd-tree. Applications are demonstrated with a parallel implementation on a multicore processor, and a version for hand-held devices.

Geometric Algorithms for 3D modeling in Geo-sciences

Participants: Bruno Lévy.

We developed RingMesh [13], an application layer around our Geogram library, specialized for 3D modeling in Geo-sciences. The RingMesh library uses the mesh data structure and basic algorithms in Geogram to offer 3D modeling primitives well-suited to geosciences. It has datatypes to efficiently represent complicated 3D models of the underground, with the topological relationships between the interfaces (horizons and faults), as well as interfaces to 3D mesh generation softwares.