Planar hexagonal meshing for architecture

published_or_final_versio

Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.Comment: Submitted to JC

Discrete conformal mappings via circle patterns

We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, that is, arrangements of circles---one for each face---with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes, we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples

CSG based automatic mesh generation using multiple element types

The objective of this thesis project is to explore a unique approach toward automatic mesh generation for finite element analysis. Current mesh generation algorithms are only applicable to a single type of domain. Countless mesh generators exist for meshing 2D regions with triangles and quadrilaterals, and mesh generators also exist which can mesh 3D regions with tetrahedra and other element types. However, not all structures are strictly 2D or 3D , and not all structures are best modeled with a single type of element. An experienced finite element analyst typically uses many types of elements when modeling a real problem. This thesis addresses this approach to meshing in an automatic manner. However, at various stages, the user has the ability to change the course of the modeler. In this thesis project, a program for automatic mesh generation has been developed on a constructive solid geometry (CSG) foundation. This program was written in object-oriented Pascal, and consists of well over 25,000 lines of code. The CSG system used was developed with PADL-2 as the guide, and allows complex geometries to be modeled as combinations of blocks and cylinders. This solid model is then broken into ID, 2D and 3D regions, or segments , using CSG-Tree segmentation logic. Each segment can then be meshed using an appropriate mesh generation technique. Thus, a single model can be meshed with multiple element types, just as an experienced analyst would do it

Lp Centroidal Voronoi Tesselation and its applications

International audienceThis paper introduces Lp -Centroidal Voronoi Tessellation (Lp -CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy). Lp -CVT is computed by a quasi-Newton optimization framework, based on closed-form derivations of the objective function and its gradient. The derivations are given for both surface meshing (Ω is a triangulated mesh with per-facet anisotropy) and volume meshing (Ω is the interior of a closed triangulated mesh with a 3D anisotropy field). Applications to anisotropic, quad-dominant surface remeshing and to hex-dominant volume meshing are presented. Unlike previous work, Lp -CVT captures sharp features and intersections without requiring any pre-tagging

Delaunay Tessellations and Voronoi Diagrams in CGAL

The Cgal library provides a rich variety of Voronoi diagrams and Delaunay triangulations. This variety covers several aspects: generators, dimensions and metrics, which we describe in Section 2. One aim of this paper is to present the main paradigms used in CGAL: Generic programming, separation between predicates/constructions and combinatorics, and exact geometric computation (not to be confused with exact arithmetic!). The first two paradigms translate into software design choices, described in Section 4, while the last covers both robustness and efficiency issues, respectively described in Sec- tion 6 and 7. Other important aspects of the Cgal library are the interface issues, be they for traversing a tessellation, or for interoperability with other libraries or languages, see Section 5. We present in Section 8 some tessellations at work in the context of surface reconstruction and mesh generation. Section 9 is devoted to some on-going and future work on periodic triangulations (triangulations in periodic spaces), and on high-quality mesh generation with optimized tessellations. Section 10 provides typical numbers in terms of efficiency and scalability for constructing tessellations, and lists the remaining weaknesses. We conclude by listing some of our directions for the future