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.
We introduce the scale axis transform, a new skeletal shape representation for bounded open sets O ⊂ R d . The scale axis transform induces a family of skeletons that captures the important features of a shape in a scale-adaptive way and yields a hierarchy of successively simplified skeletons. Its definition is based on the medial axis transform and the simplification of the shape under multiplicative scaling: the s-scaled shape Os is the union of the medial balls of O with radii scaled by a factor of s. The s-scale axis transform of O is the medial axis transform of Os, with radii scaled back by a factor of 1/s. We prove topological properties of the scale axis transform and we describe the evolution s → Os by defining the multiplicative distance function to the shape and studying properties of the corresponding steepest ascent flow. All our theoretical results hold for any dimension. In addition, using a discrete approximation, we present several examples of two-dimensional scale axis transforms that illustrate the practical relevance of our new framework.
The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a well-shaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other by convex duality or embedding. Moreover, we can always compute them indirectly as power diagrams in primal or dual spaces, or directly after linearization in an extra-dimensional space as the projection of a Euclidean polytope. Finally, our paper proposes to generalize Bregman divergences to higherorder terms, called k-jet Bregman divergences, and touch upon their Voronoi diagrams.
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 meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations have been shown to be particularly well suited for interpolation of functions or numerical modeling. We propose a new approach to anisotropic mesh generation, relying on the notion of locally uniform anisotropic mesh. A locally uniform anisotropic mesh is a mesh such that the star around each vertex v coincides with the star that v would have if the metric on the domain was uniform and equal to the metric at v. This definition allows to define a simple refinement algorithm which relies on elementary predicates, and provides, after completion, an anisotropic mesh in dimensions 2 and 3.A practical implementation has been done in the 2D case.
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