We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of user-defined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone. A careful treatment of boundaries and their features is presented, offering a versatile framework for designing smoothly graded tetrahedral meshes.

Raffinement et optimisation de Delaunay pour la génération de maillages tétraédriques isotropes Résumé : Ce rapport présente une méthode de génération de maillages tétraédriques isotropes pour des domaines 3D bornés par des surfaces lisses par morceaux. Le principe consiste à entrelacer raffinement de Delaunay et optimisation de Delaunay pour maximiser la qualité des maillages tout en satisfaisant un ensemble de critères définis par l'utilisateur. Ce procédé est expérimentalement plus parcimonieux en nombre de points de Steiner que le raffinement seul, et produit des maillages de meilleure qualité que l'optimisation appliquée comme post-traitement. Un traitement particulier est résérvé à la gestion des bords et des arêtes vives pour obtenir un cadre générique pour la génération de maillages.

Anisotropic simplicial meshes are triangulations with elements elongated along prescribed directions. Anisotropic meshes have been shown well suited for interpolation of functions or solving PDEs. They can also significantly enhance the accuracy of a surface representation. Given a surface S endowed with a metric tensor field, we propose a new approach to generate an anisotropic mesh that approximates S with elements shaped according to the metric field. The algorithm relies on the well-established concepts of restricted Delaunay triangulation and Delaunay refinement and comes with theoretical guarantees. The star of each vertex in the output mesh is Delaunay for the metric attached to this vertex. Each facet has a good aspect ratio with respect to the metric specified at any of its vertices. The algorithm is easy to implement. It can mesh various types of surfaces like implicit surfaces, polyhedra, or isosurfaces in 3D images. It can handle complicated geometries and topologies, and very anisotropic metric fields.

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Summary. Isotropic tetrahedron meshes generated by Delaunay triangulations are known to contain a majority of well-shaped tetrahedra, as well as spurious sliver tetrahedra. As the slivers hamper stability of numerical simulations we aim at removing them while keeping the triangulation Delaunay for simplicity. The solution which explicitly perturbs the slivers through random vertex relocation and Delaunay connectivity update is very effective but slow. In this paper we present a perturbation algorithm which favors deterministic over random perturbation. The added value is an improved efficiency and effectiveness. Our experimental study applies the proposed algorithm to meshes obtained by Delaunay refinement as well as to carefully optimized meshes.

We tackle the problem of constructing 2D centroidal Voronoi tessellations with constraints through an efficient and robust construction of bounded Voronoi diagrams, the pseudo-dual of the constrained Delaunay triangulation. We exploit the fact that the cells of the bounded Voronoi diagram can be obtained by clipping the ordinary ones against the constrained Delaunay edges. The clipping itself is efficiently computed by identifying for each constrained edge the (connected) set of triangles whose dual Voronoi vertices are hidden by the constraint. The resulting construction is amenable to Lloyd relaxation so as to obtain a centroidal tessellation with constraints.

Updating a Delaunay triangulation when its vertices move is a bottleneck in several domains of application. Rebuilding the whole triangulation from scratch is surprisingly a very viable option compared to relocating the vertices. This can be explained by several recent advances in efficient construction of Delaunay triangulations. However, when all points move with a small magnitude, or when only a fraction of the vertices move, rebuilding is no longer the best option. This paper considers the problem of efficiently updating a Delaunay triangulation when its vertices are moving under small perturbations. The main contribution is a set of filters based upon the concept of vertex tolerances. Experiments show that filtering relocations is faster than rebuilding the whole triangulation from scratch under certain conditions.

Consider a flat-fading Multiple-Input Multiple-Output (MIMO) system. The near-optimum detection problematic has been shown to be efficiently solved through QR Decomposition (QRD)-based detectors. However, this step is demanding, and leads to wasting a large part of the system resource in case of a straightforward multi-carrier extension. In this paper, a solution that provides the QRD output -by advantageously considering the correlation between adjacent channels -is proposed. Originally, it relies on the matrix perturbation theory and induces a strong complexity reduction. Simulation results show that the proposed solution significantly reduces the implementation complexity while maintaining good performance.