Curvature-adapted Remeshing of CAD Surfaces

A common representation of surfaces with complicated topology and geometry is through composite parametric surfaces as is the case for most CAD modelers. A challenging problem is how to generate a mesh of such a surface that well approximates the geometry of the surface, preserves its topology and important geometric features, and contains nicely shaped elements. In this work, we present an optimization-based surface remeshing method that is able to satisfy many of these requirements simultaneously. This method is inspired by the recent work of Levy \ub4 and Bonneel (Proc. 21th International Meshing Roundtable, October 2012), which embeds a smooth surface into a high-dimensional space and remesh it uniformly in that embedding space. Our method works directly in the 3d spaces and uses an embedding space in R6 to evaluate mesh size and mesh quality. It generates a curvatureadapted anisotropic surface mesh that well represents the geometry of the surface with a low number of elements. We illustrate our approach through various examples

Anisotropic Delaunay Meshes of Surfaces

Anisotropic simplicial meshes are triangulations with elements elongated along prescribed directions. Anisotropic meshes have been shown to be well suited for interpolation of functions or solving PDEs. They can also significantly enhance the accuracy of a surface repre- sentation. 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.Les maillages anisotropes simpliciaux sont des triangulations dont les éléments sont étirés suivant certaines directions imposées. Les maillages anisotropes sont connus pour être bien adaptés à l'interpolation de fonctions ou à la résolution d'équations aux dérivées partiellles. Ces maillages peuvent aussi améliorer notablement la précision de l'approximation d'une surface. Etant donnée une surface S, munie d'un champs de tenseurs qui définit la métrique en tout point de la surface, nous proposons un nouvel algorithme pour générer un maillage anisotrope qui approxime S par des triangles dont les formes s'adaptent à la métrique locale. L'algorithme repose sur les concepts bien établis de triangulation de Delaunay restreinte et de raffinement de Delaunay et offre des garanties théoriques. L'étoile de chaque sommet dans le maillage est formée par des triangles de Delaunay pour la métrique du sommet central. Chaque triangle a un bon rapport d'aspect dans la métrique attachée à chacun de ces sommets. L'algorithme est facile à programmer. Il permet de mailler diff'rents types de surfaces, comme des surfaces implicites, des polyèdres ou encores des isosurfaces dans des images 3D. L'algorithme peut traiter des surfaces de géométrie ou topologie complexe, il peut aussi prendre en compte des anisotropies très prononcées

Patient-specific anisotropic model of human trunk based on MR data

There are many ways to generate geometrical models for numerical simulation, and most of them start with a segmentation step to extract the boundaries of the regions of interest. This paper presents an algorithm to generate a patient-specific three-dimensional geometric model, based on a tetrahedral mesh, without an initial extraction of contours from the volumetric data. Using the information directly available in the data, such as gray levels, we built a metric to drive a mesh adaptation process. The metric is used to specify the size and orientation of the tetrahedral elements everywhere in the mesh. Our method, which produces anisotropic meshes, gives good results with synthetic and real MRI data. The resulting model quality has been evaluated qualitatively and quantitatively by comparing it with an analytical solution and with a segmentation made by an expert. Results show that our method gives, in 90% of the cases, as good or better meshes as a similar isotropic method, based on the accuracy of the volume reconstruction for a given mesh size. Moreover, a comparison of the Hausdorff distances between adapted meshes of both methods and ground-truth volumes shows that our method decreases reconstruction errors faster. Copyright © 2015 John Wiley & Sons, Ltd.Natural Sciences and Engineering Research Council (NSERC) of Canada and the MEDITIS training program (´Ecole Polytechnique de Montreal and NSERC)

A curvature-adapted anisotropic surface remeshing method

We present a new method for remeshing surfaces that respect the intrinsic anisotropy of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a higher dimensional space (here we use 6d). This allow us to form an isotropic mesh optimization problem in this embedded space. Starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by combining common local modifications operations, i.e., edge flip, edge contraction, vertex smoothing, and vertex insertion. All operations are applied directly on the 3d surface mesh. This method results a curvature-adapted mesh of the surface. This method can be easily adapted to mesh multi-patches surfaces, i.e., containing corner singularities and sharp features. We present examples of remeshed surfaces from implicit functions and CAD models

Unstructured mesh generation and adaptivity

An overview of current unstructured mesh generation and adaptivity techniques is given. Basic building blocks taken from the field of computational geometry are first described. Various practical mesh generation techniques based on these algorithms are then constructed and illustrated with examples. Issues of adaptive meshing and stretched mesh generation for anisotropic problems are treated in subsequent sections. The presentation is organized in an education manner, for readers familiar with computational fluid dynamics, wishing to learn more about current unstructured mesh techniques

Front-tracking finite element methods for a void electro-stress migration problem

Continued research in electronic engineering technology has led to a miniaturisation of integrated circuits. Further reduction in the dimensions of the interconnects is impeded by the presence of small cracks or voids. Subject to high current and elastic stress, voids tend to drift and change shape in the interconnect, leading to a potential mechanical failure of the system. This thesis investigates the temporal evolution of voids moving along conductors, in the presence of surface diffusion, electric loading and elastic stress. We simulate a bulk-interface coupled system, with a moving interface governed by a fourth-order geometric evolution equation and a bulk where the electric potential and the displacement field are computed. We first give a general overview about geometric evolution equations, which define the motion of a hypersurface by prescribing its normal velocity in terms of geometric quantities. We briefly describe the three main approaches that have been proposed in the literature to solve numerically this class of equations, namely parametric approach, level set approach and phase field approach. We then present in detail two methods from the parametric approach category for the void electro-stress migration problem. We first introduce an unfitted method, where bulk and interface grids are totally independent, i.e. no topological compatibility between the two grids has to be enforced over time. We then discuss a fitted method, where the interface grid is at all times part of the boundary of the bulk grid. A detailed analysis, in terms of existence and uniqueness of the finite element solutions, experimental order of convergence (when the exact solution to the free boundary problem is known) and coupling operations (e.g., smoothing/remeshing of the grids, intersection between elements of the two grids), is carried out for both approaches. Several numerical simulations, both two- and three-dimensional, are performed in order to test the accuracy of the methods.Open Acces

Surface and bulk moving mesh methods based on equidistribution and alignment

In this dissertation, we first present a new functional for variational mesh generation and adaptation that is formulated by combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive which, when employed with the moving mesh partial differential equation method, allows various theoretical properties to be proved. Numerical examples for bulk meshes demonstrate that the new functional performs comparably to a similar existing functional that is known to work well but contains an additional parameter. Variational mesh adaptation for bulk meshes has been well developed however, surface moving mesh methods are limited. Here, we present a surface moving mesh method for general surfaces with or without explicit parameterization. The development starts with formulating the equidistribution and alignment conditions for surface meshes from which, we establish a meshing energy functional. The moving mesh equation is then defined as the gradient system of the energy functional, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. A selection of two-dimensional and three-dimensional examples are presented