Carte orthotropes pour la génération de maillages

école doctorale

équipe

PIXEL

contexte

***Orthotropic map applications for remeshing*** In a modern version of the mapping problem, surfaces are represented by triangle meshes and the flat representation is not only used for information storage (texture, normal). In particular, by computing a cleverly constrained map to the plane and by overlying a regular grid in parameter space, we obtain a decomposition of the original surface into quadrangles (see Figure 1). This transformation of triangle mesh to a quadrangle mesh proves to be quite challenging but very useful in practice. Anisotropic remeshing. The objective of quad (or hex) remeshing is to generate a mesh that accurately approximate a target geometry while maintaining a fixed number of elements. When approximating a surface using a quad mesh, theoretical findings indicate that the edges should align with the principal (orthogonal) curvature directions, and the aspect ratio of the elements should be in proportion to the ratio of the principal curvatures as in the inset figure. This result is a quite intuitive because regions with high curvature demand smaller elements for a precise approximation. Similarly, to reduce numerical errors in numerical simulations, the mesh should be denser in areas where the expected solution exhibits significant variations and less dense in areas where the solution is nearly flat. The theory suggests that the most accurate quad (or hex) mesh must have edges aligned with the (orthogonal) eigenvectors of the function's Hessian. Clearly, both aspects of the approximation problem can be addressed by meshes with rectangular (or rectangular cuboid) elements, which can be extracted from an orthotropic map. Numerical simulation. It is well-known that the finite element method exhibits improved convergence properties when elements approach perfect squares or cubes, and the convergence is not guaranteed in the presence of non-convex elements. By enforcing rectangular elements through orthotropic mappings, we not only avoid non-convexity but also fulfill the requirements for the optimal convergence of the popular finite elements.

spécialité

Informatique

laboratoire

LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications