arXiv subjects

Jean-François Remacle

Publications and source records attributed to Jean-François Remacle.

28 records · Page 2Linked to original sources

arXiv2020

Quality tetrahedral mesh generation with HXT

We proposed, in a recent paper (10.1002/nme.5987), a fast 3D parallel Delaunay kernel for tetrahedral mesh generation. This kernel was however incomplete in the sense that it lacked the necessary mesh improvement tools. The present paper builds on that previous work and proposes a fast parallel mesh improvement stage that delivers high-quality tetrahedral meshes compared to alternative open-source mesh generators. Our mesh improvement toolkit includes edge removal and improved Laplacian smoothing as well as a brand new operator called the Growing SPR Cavity, which can be regarded as the mother of all flips. The paper describes the workflow of the new mesh improvement schedule, as well as the details of the implementation. The result of this research is a series of open-source scalable software components, called HXT, whose overall efficiency is demonstrated on practical examples by means of a detailed comparative benchmark with two open-source mesh generators: Gmsh and TetGen.

cs.CG↗

arXiv2020

Automatic surface mesh generation for discrete models: A complete and automatic pipeline based on reparameterization

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-to-one 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.

cs.CG↗

arXiv2020

Computing cross fields -- A PDE approach based on the Ginzburg-Landau theory

This paper proposes a method to compute crossfields based on the Ginzburg-Landau theory. The Ginzburg-Landau functional has two terms: the Dirichlet energy of the distribution and a term penalizing the mismatch between the fixed and actual norm of the distribution. Directional fields on surfaces are known to have a number of critical points, which are properly identified with the Ginzburg-Landau approach: the asymptotic behavior of Ginzburg-Landau problem provides well-distributed critical points over the 2-manifold, whose indices are as low as possible. The central idea in this paper is to exploit this theoretical background for crossfield computation on arbitrary surfaces. Such crossfields are instrumental in the generation of meshes with quadrangular elements. The relation between the topological properties of quadrangular meshes and crossfields are hence first recalled. It is then shown that a crossfield on a surface can be represented by a complex function of unit norm with a number of critical points, i.e., a nearly everywhere smooth function taking its values in the unit circle of the complex plane. As maximal smoothness of the crossfield is equivalent with minimal energy, the crossfield problem is equivalent to an optimization problem based on Ginzburg-Landau functional. A discretization scheme with Crouzeix-Raviart elements is applied and the correctness of the resulting finite element formulation is validated on the unit disk by comparison with an analytical solution. The method is also applied to the 2-sphere where, surprisingly but rightly, the computed critical points are not located at the vertices of a cube, but at those of an anticube.

math.NA↗

arXiv2019

Finding Hexahedrizations for Small Quadrangulations of the Sphere

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.

cs.CG↗

arXiv2018

One machine, one minute, three billion tetrahedra

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh.

cs.CG↗

arXiv2018

Representing three-dimensional cross fields using 4th order tensors

This paper presents a new way of describing cross fields based on fourth order tensors. We prove that the new formulation is forming a linear space in $\mathbb{R}^9$. The algebraic structure of the tensors and their projections on $\mbox{SO}(3)$ are presented. The relationship of the new formulation with spherical harmonics is exposed. This paper is quite theoretical. Due to pages limitation, few practical aspects related to the computations of cross fields are exposed. Nevetheless, a global smoothing algorithm is briefly presented and computation of cross fields are finally depicted.

cs.CG↗

arXiv2017

Robust and efficient validation of the linear hexahedral element

Checking mesh validity is a mandatory step before doing any finite element analysis. If checking the validity of tetrahedra is trivial, checking the validity of hexahedral elements is far from being obvious. In this paper, a method that robustly and efficiently compute the validity of standard linear hexahedral elements is presented. This method is a significant improvement of a previous work on the validity of curvilinear elements. The new implementation is simple and computationally efficient. The key of the algorithm is still to compute Bézier coefficients of the Jacobian determinant. We show that only 20 Jacobian determinants are necessary to compute the 27 Bézier coefficients. Those 20 Jacobians can be efficiently computed by calculating the volume of 20 tetrahedra. The new implementation is able to check the validity of about 6 million hexahedra per second on one core of a personal computer. Through the paper, all the necessary information is provided that allow to easily reproduce the results, \ie write a simple code that takes the coordinates of 8 points as input and outputs the validity of the hexahedron.

cs.CG↗

arXiv2017

High quality mesh generation using cross and asterisk fields: Application on coastal domains

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.

cs.CG↗

arXiv2016

Fast and robust mesh generation on the sphere Application to coastal domains

This paper presents a fast an robust mesh generation procedure that is able to generate meshes of the earth system (ocean and continent) in matters of seconds. Our algorithm takes as input a standard shape-file i.e. geospatial vector data format for geographic information system (GIS) software. The input is initially coarsened in order to automatically remove unwanted channels that are under a desired resolution. A valid non-overlapping 1D mesh is then created on the sphere using the Euclidian coordinates system $x,y,z$. A modified Delaunay kernel is then proposed that enables to generate meshes on the sphere in a straightforward manner without parametrization. One of the main difficulty in dealing with geographical data is the over-sampled nature of coastline representations. We propose here an algorithm that automatically unrefines coastline data. Small features are automatically removed while always keeping a valid (non-overlapping) geometrical representation of the domain. A Delaunay refinement procedure is subsequently applied to the domain. The refinement scheme is also multi-threaded at a fine grain level, allowing to generate about a million points per second on 8 threads. Examples of meshes of the Baltic sea as well as of the global ocean are presented.

cs.CG↗

arXiv2015

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.

physics.comp-ph↗