Cross fields are auxiliary in the generation of quadrangular meshes. A method to generate cross fields on surface manifolds is presented in this paper. Algebraic topology constraints on quadrangular meshes are first discussed. The duality between quadrangular meshes and cross fields is then outlined, and a generalization to cross fields of the Poincaré-Hopf theorem is proposed, which highlights some fundamental and important topological constraints on cross fields. A finite element formulation for the computation of cross fields is then presented, which is based on Ginzburg-Landau equations and makes use of edge-based Crouzeix-Raviart interpolation functions. It is first presented in the planar case, and then extended to a general surface manifold. Finally, application examples are solved and discussed.

Figure 1: Three example models of HexMe (https://hexme.algohex.eu): The tetrahedral meshes faithfully represent feature points, curves (depicted in blue), and surfaces of the underlying CAD primitives.

Triangulations are an ubiquitous input for the finite element community. However, most raw triangulations obtained by imaging techniques are unsuitable as-is for finite element analysis. In this paper, we give a robust pipeline for handling those triangulations, based on the computation of a one-to-one parametrization for automatically selected patches of input triangles, which makes each patch amenable to remeshing by standard finite element meshing algorithms. Using only geometrical arguments, we prove that a discrete parametrization of a patch is one-to-one if (and only if) its image in the parameter space is such that all parametric triangles have a positive area. We then derive a non-standard linear discretization scheme based on mean value coordinates to compute such one-toone parametrizations, and show that the scheme does not discretize a Laplacian on a structured mesh. The proposed pipeline is implemented in the open source mesh generator Gmsh, where the creation of suitable patches is based on triangulation topology and parametrization quality, combined with feature edge detection. Several examples illustrate the robustness of the resulting implementation.

HEXME consists of tetrahedral meshes with tagged features, and of a workflow to generate them. The main purpose of HEXME meshes is to enable consistent and fair evaluation of hexahedral meshing algorithms and related techniques. The tetrahedral meshes have been generated with Gmsh, starting from 63 computer-aided design (CAD) models coming from various databases. To highlight and label the various and challenging aspects of hexahedral mesh generation, the CAD models are classified into three categories: simple, nasty and industrial. For each CAD model, we provide three kinds of tetrahedral meshes. The mesh generation yielding those 189 tetrahedral meshes is defined thanks to Snakemake, a modern workflow management system, which allows us to define a fully automated, extensible and sustainable workflow. It is possible to download the whole dataset or to pick some meshes by browsing the online catalog. Since there is no doubt that the hexahedral meshing techniques are going to progress, the HEXME dataset is also built with evolution in mind. A public GitHub repository hosts the HEXME workflow, in which external contributions and future releases are possible and encouraged.

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This paper presents a method to generate high quality triangular or quadrilateral meshes that uses direction fields and a frontal point insertion strategy. Two types of direction fields are considered: asterisk fields and cross fields. With asterisk fields we generate high quality triangulations, while with cross fields we generate right-angled triangulations that are optimal for transformation to quadrilateral meshes. The input of our algorithm is an initial triangular mesh and a direction field calculated on it. The goal is to compute the vertices of the final mesh by an advancing front strategy along the direction field. We present an algorithm that enables to efficiently generate the points using solely information from the base mesh. A multi-threaded implementation of our algorithm is presented, allowing us to achieve significant speedup of the point generation. Regarding the quadrangulation process, we develop a quality criterion for right-angled triangles with respect to the local cross field and an optimization process based on it. Thus we are able to further improve the quality of the output quadrilaterals. The algorithm is demonstrated on the sphere and examples of high quality triangular and quadrilateral meshes of coastal domains are presented.

Full hexahedral meshes are required to be as regular as possible, which means that the local topology has to be constant almost everywhere. This constraint is usually modelled by 3D frames. A 3D frame consists of three mutual orthogonal (unit) vectors, defining a local basis. 3D frame fields are auxiliary for hexahedral mesh generation. Computation of 3D frame fields is an active research field. There mainly exist three ways to represent 3D frames: combination of rotations, spherical harmonics and fourth order tensor. We propose here a representation carried out by the special unitary group. The article strongly relies on [1]. We first describe the rotations with quaternions, [1, §13-15]. We define and show the isomorphism between unit quaternions and the special unitary group, [1, §16]. The frame field space is identified as the quotient group of rotations by the octahedral group, [1, §20]. The invariant forms of the vierer, tetrahedral and octahedral groups are successively built, without using homographies [1, §39]. Modifying the definition of the isomorphism between unit quaternions and the special unitary group allows to use the invariant forms of the octahedral group as a unique parameterization of the orientation of 3D frames. The parameterization consists in three complex values, corresponding to a coordinate of a variety which is embedded in a three complex valued dimensional space. The underlined variety is the model surface of the octahedral group, which can be expressed with an implicit equation. We prove that from a coordinate of the surface, we may identify all the quaternions giving the corresponding 3D frames. We show that the euclidean distance between two coordinates does not correspond to the actual distance of the corresponding 3D frames. We derive the expression of three components of a coordinate in the case of frames sharing an even direction. We then derive a way to ensure that a coordinate corresponds to the special unitary group. Finally, the attempted numerical schemes to compute frame fields are given.

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