arXiv subjects

Jean-Francois Remacle

Publications and source records attributed to Jean-Francois Remacle.

8 recordsLinked to original sources

arXiv2024

A Delaunay Refinement Algorithm for the Particle Finite Element Method applied to Free Surface Flows

This paper proposes two contributions to the calculation of free surface flows using the particle finite element method (PFEM). The PFEM is based on a Lagrangian approach: a set of particles defines the fluid. Then, unlike a pure Lagrangian method, all the particles are connected by a triangular mesh. The difficulty lies in locating the free surface from this mesh. It is a matter of deciding which of the elements in the mesh are part of the fluid domain, and to define a boundary - the free surface. Then, the incompressible Navier-Stokes equations are solved on the fluid domain and the particles' position is updated using the resulting velocity vector. Our first contribution is to propose an approach to adapt the mesh with theoretical guarantees of quality: the mesh generation community has acquired a lot of experience and understanding about mesh adaptation approaches with guarantees of quality on the final mesh. We use here a Delaunay refinement strategy, allowing to insert and remove nodes while gradually improving mesh quality. We show that this allows to create stable and smooth free surface geometries. Our PFEM approach models the topological evolution of one fluid. It is nevertheless necessary to apply conditions on the domain boundaries. When a boundary is a free surface, the flow on the other side is not modelled, it is represented by an external pressure. On the external free surface boundary, atmospheric pressure can be imposed. Nevertheless, there may be internal free surfaces: the fluid can fully encapsulate cavities to form bubbles. The pressure required to maintain the volume of those bubbles is a priori unknown. We propose a multi-point constraint approach to enforce global incompressibility of those empty bubbles. This approach allows to accurately model bubbly flows that involve two fluids with large density differences, while only modelling the heavier fluid.

cs.CE↗

arXiv2023

The eXtreme Mesh deformation approach (X-MESH) for the Stefan phase-change model

The eXtreme Mesh deformation approach (X-MESH) is a new paradigm to follow sharp interfaces without remeshing and without changing the mesh topology. Even though the mesh does not change its topology, it can follow interfaces that do change their topology (nucleation, coalescence, splitting). To make this possible, the key X-MESH idea is to allow elements to reach zero measure. This permits interface relaying between nodes as well as interface annihilation and seeding in a time continuous manner. The paper targets the Stefan phase change model in which the interface (front) is at a given temperature. Several examples demonstrate the capability of the approach.

cs.CE↗

arXiv2022

Hex-Mesh Generation and Processing: a Survey

In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.

cs.GR↗

arXiv2020

Quaternionic octahedral fields: SU(2) parameterization of 3D frames

3D frame fields are auxiliary for hexahedral mesh generation. 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 \cite{du1964homographies}. We first describe the rotations with quaternions, \cite[§13-15]{du1964homographies}. We define and show the isomorphism between unit quaternions and the special unitary group, \cite[§16]{du1964homographies}. The frame field space is identified as the quotient group of rotations by the octahedral group, \cite[§20]{du1964homographies}. The invariant forms of the vierer, tetrahedral and octahedral groups are successively built, without using homographies \cite[§39]{du1964homographies}. 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.

math.AT↗

arXiv2019

Multiple Approaches to Frame Field Correction for CAD Models

Three-dimensional frame fields computed on CAD models often contain singular curves that are not compatible with hexahedral meshing. In this paper, we show how CAD feature curves can induce non meshable 3-5 singular curves and we study four different approaches that aims at correcting the frame field topology. All approaches consist in modifying the frame field computation, the two first ones consisting in applying internal constraints and the two last ones consisting in modifying the boundary conditions. Approaches based on internal constraints are shown not to be very reliable because of their interactions with other singularities. On the other hand, boundary condition modifications are more promising as their impact is very localized. We eventually recommend the 3-5 singular curve boundary snapping strategy, which is simple to implement and allows to generate topologically correct frame fields.

cs.GR↗

arXiv2018

There are 174 Subdivisions of the Hexahedron into Tetrahedra

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian.

cs.CG↗

arXiv2018

Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.

cs.CG↗

arXiv2015

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.

math.NA↗