2020
Activity report
Project-Team
GAMMA
RNSR: 201923242K
Research center
Team name:
Adaptive Mesh Generation and Advanced numerical Methods
Domain
Applied Mathematics, Computation and Simulation
Theme
Numerical schemes and simulations
Creation of the Project-Team: 2019 June 01

Keywords

  • A6.2. Scientific computing, Numerical Analysis & Optimization
  • A6.2.7. High performance computing
  • A6.2.8. Computational geometry and meshes
  • A6.5.1. Solid mechanics
  • A6.5.2. Fluid mechanics
  • B5.2.3. Aviation
  • B5.2.4. Aerospace
  • B9.5.1. Computer science
  • B9.5.2. Mathematics
  • B9.5.3. Physics
  • B9.5.5. Mechanics

1 Team members, visitors, external collaborators

Research Scientists

  • Frederic Alauzet [Team leader, Inria, Senior Researcher, HDR]
  • Paul-Louis George [Inria, Emeritus]
  • Adrien Loseille [Inria, Researcher, HDR]

PhD Students

  • Sofiane Benzait [CEA]
  • Francesco Clerici [Inria]
  • Rémi Feuillet [Inria, until Mar 2020]
  • Lucien Rochery [Inria]
  • Lucille Marie Tenkes [Inria]

Technical Staff

  • Matthieu Maunoury [Inria, Engineer]
  • Julien Vanharen [Inria, Engineer, until May 2020]

Administrative Assistant

  • Maria Agustina Ronco [Inria]

External Collaborators

  • Rémi Feuillet [Siemens Industry Software, from Apr 2020]
  • David Marcum [Mississippi State University, Starkville - USA]
  • Loic Marechal [Distene]

2 Overall objectives

Numerical simulation has been booming over the last thirty years, thanks to increasingly powerful numerical methods, computer-aided design (CAD) and the mesh generation for complex 3D geometries, and the coming of supercomputers (HPC). The discipline is now mature and has become an integral part of design in science and engineering applications. This new status has lead scientists and engineers to consider numerical simulation of problems with ever increasing geometrical and physical complexities. A simple observation of this chart

shows: no mesh = no simulation along with "bad" mesh = wrong simulation. We have concluded that the mesh is at the core of the classical computational pipeline and a key component to significant improvements. Therefore, the requirements on meshing methods are an ever increasing need, with increased difficulty, to produce high quality meshes to enable reliable solution output predictions in an automated manner. These requirements on meshing or equivalent technologies cannot be removed and all approaches face similar issues.

In this context, Gamma team was created in 1996 and has focused on the development of robust automated mesh generation methods in 3D, which was clearly a bottleneck at that time when most of the numerical simulations were 2D. The team has been very successful in tetrahedral meshing with the well-known software Ghs3d28, 29 which has been distributed worldwide so far and in hexahedral meshing with the software Hexotic49, 50 which was the first automated full hex mesher. The team has also worked on surface meshers with Yams20 and BLSurf12 and visualization with Medit. Before Medit, we were unable to visualize in real time 3D meshes !

In 2010, Gamma3 team has replaced Gamma with the choice to focus more on meshing for numerical simulations. The main goal was to emphasize and to strengthen the link between meshing technologies and numerical methods (flow or structure solvers). The metric-based anisotropic mesh adaptation strategy has been very successful with the development of many error estimates, the generation of highly anisotropic meshes, its application to compressible Euler and Navier-Stokes equations 5, and its extension to unsteady problems with moving geometries 8 leading to the development of several softwares Feflo.a/AMG-Lib, Wolf, Metrix, Wolf-Interpol. A significant accomplishment was the high-fidelity prediction of the sonic boom emitted by supersonic aircraft 6. We were the first to compute a certified aircraft sonic boom propagation in the atmosphere, thanks to mesh adaptation. The team has started to work on parallelism with the development of the multi-thread library LPlib and the efficient management of memory using space filling curves, and the generation of large meshes (a billion of elements) 43. Theoretical work on high-order meshes has been also done 27.

Today, numerical simulation is an integral part of design in engineering applications with the main goal of reducing costs and speeding up the process of creating new design. Four main issues for industry are:

  • Generation of a discrete surface mesh from a continuous CAD is the last non-automated step of the design pipeline and, thus, the most human time consuming
  • High-performance computing (HPC) for all tools included in the design loop
  • The cost in euros of a numerical simulation
  • Certification of high-fidelity numerical simulations by controlling errors and uncertainties.

Let us now discuss in more details each of these issues.

Generating a discrete surface mesh from a CAD geometry definition has been the numerical analysis Achille's heel for the last 30 years. Significant issues are far too common and range from persistent translation issues between systems that can produce ill defined geometry definitions to overwhelming complexity for full configurations with all components. A geometry definition that is ill defined often does not perfectly capture the geometry's features and leads to a bad mesh and a broken simulation. Unfortunately, CAD system design is essentially decoupled from the needs of numerical simulation and is largely driven by the those of manufacturing and other areas. As a result, this step of the numerical simulation pipeline is still labor intensive and the most time consuming. There is a need to develop alternative geometry processes and models that are more suitable for numerical simulations.

Companies working on high-tech projects with high added value (Boeing, Safran, Dassault-Aviation, Ariane Group, ...) consider their design pipeline inside a HPC framework. Indeed, they are performing complex numerical simulations on complex geometries on a daily-basis, and they aim at using this in a shape-optimization loop. Therefore, any tools added to their numerical platform should be HPC compliant. This means that all developments should consider hybrid parallelism, i.e., to be compatible with distributed memory architecture (MPI) and shared memory architecture (multi-threaded), to achieve scalable parallelism.

One of the main goals of numerical simulation is to reduce the cost of creating new designs (e.g reduce the number of wind-tunnel and flight tests in the aircraft industry). The emergence of 3D printers is, in some cases, making tests easier to perform, faster and cheaper. It is thus mandatory to control the cost of the numerical simulations, in other word, it is important to use less resources to achieve the same accuracy. The cost takes into account the engineer time as well as the computing resources needed to perform the numerical simulation. The cost for one simulation can vary from 15 euros for simple models (1D-2D), to 150 euros for Reynolds-averaged Navier-Stokes (3D) stationary models, or up to 15 000 euros for unsteady models like LES or Lattice-Boltzmann 1. It is important to know that a design loop is equivalent to performing between 100 and 1 000 numerical simulations. Consequently, the need for more efficient algorithms and processes is still a key factor.

Another crucial point is checking and certification of errors and uncertainties in high-fidelity numerical simulations. These errors can come from several sources:

  • i) modeling error (for example via turbulence models or initial conditions),
  • ii) discretization error (due to the mesh),
  • iii) geometry error (due to the representation of the design) and
  • iv) implementation errors in the considered software.

The error assessment and mesh generation procedure employed in the aerospace industry for CFD simulations relies heavily on the experience of the CFD user. The inadequacy of this practice even for geometries frequently encountered in engineering practice has been highlighted in studies of the AIAA 2 CFD Drag Prediction Workshops 53 and High-Lift Prediction Workshops 68, 67. These studies suggest that the range of scales present in the turbulent flow cannot be adequately resolved using meshes generated following what is considered best present practices. In this regard, anisotropic mesh adaptation is considered as the future, as stated in the NASA report "CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences" 70 and the study dedicated to mesh adaptation 59.

These preoccupations are the core of the Gamma project scientific program. To answer the first issue, Gamma will focus on designing and developing a geometry modeling framework specifically intended for mesh generation and numerical simulation purposes. This is a mandatory step for automated geometry-mesh and mesh adaptation processes with an integrated geometry model. To answer the last three issues, the Gamma team will work on the development of a high-order mesh-adaptive solution platform compatible with HPC environment. To this end, Gamma will pursue its work on advanced mesh generation methods which should fulfill the following capabilities:

  • i) geometric adaptive,
  • ii) solution adaptive,
  • iii) high-order,
  • iv) multi-elements (structured or not), and
  • v) using hybrid scalable parallelism.

Note that items i) to iv) are based on the well-posed metric-based theoretical framework. Moreover, Gamma will continue to work on robust flow solvers, solving the turbulent Navier-Stokes equations from second order using Finite Volume - Finite Element numerical scheme to higher-order using Flux Reconstruction (FR) method.

The combination of adaptation - high-order - multi-elements coupled with appropriate error estimates is for the team the way to go to reduce the cost of numerical simulations while ensuring high-fidelity in a fully automated framework.

3 Research program

The main axes are:

  • Geometric Modeling:
    • High-fidelity discrete CAD kernel.
    • Continuous parametric CAD kernel.
  • Enhanced Generic Meshing Algorithm:
    • Adaptation (extreme anisotropy, metric-aligned, metric-orthogonal).
    • High-order (tetrahedra, hexahedra, boundary layer, adapted).
    • Larges meshes (tetrahedra, hexahedra, adapted).
    • Moving mesh methods for moving geometries.
  • Toward Certified Solutions to the Navier-Stokes Equations:
    • Flow solver and adjoints (Finite Volumes, Finite Elements, Flux Reconstruction).
    • Error estimates and correctors.
  • Advanced Mesh and Solution Visualisation:
    • Pixel exact rendering (High-Order mesh, High-Order solution).
    • Pre-processing and post-processing.

4 Application domains

Applied Mathematics, Computation and Simulation.

5 New software and platforms

5.1 New software

5.1.1 FEFLOA-REMESH

  • Keywords: Scientific calculation, Anisotropic, Mesh adaptation
  • Functional Description: FEFLOA-REMESH is intended to generate adapted 2D, surface and volume meshes by using a unique cavity-based operator. The metric-aligned or metric-orthogonal approach is used to generate high quality surface and volume meshes independently of the anisotropy involved.
  • URL: https://pages.saclay.inria.fr/adrien.loseille/index.php?page=softwares
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Adrien Loseille
  • Participants: Adrien Loseille, Frédéric Alauzet, Rémi Feuillet, Lucien Rochery, Lucille Marie Tenkes

5.1.2 GHS3D

  • Keywords: Tetrahedral mesh, Delaunay, Automatic mesher
  • Functional Description: GHS3D is an automatic volume mesher
  • URL: http://www.meshgems.com/volume-meshing.html
  • Authors: Paul Louis George, Houman Borouchaki, Éric Saltel, Adrien Loseille, Frédéric Alauzet, Frederic Hecht
  • Contact: Paul Louis George
  • Participants: Paul Louis George, Adrien Loseille, Frédéric Alauzet

5.1.3 HEXOTIC

  • Keywords: 3D, Mesh generation, Meshing, Unstructured meshes, Octree/Quadtree, Multi-threading, GPGPU, GPU
  • Functional Description:

    Input: a triangulated surface mesh and an optional size map to control the size of inner elements.

    Output: a fully hexahedral mesh (no hybrid elements), valid (no negative jacobian) and conformal (no dangling nodes) whose surface matches the input geometry.

    The software is a simple command line that requires no knowledge on meshing. Its arguments are an input mesh and some optional parameters to control elements sizing, curvature and subdomains as well as some features like boundary layers generation.

  • URL: https://team.inria.fr/gamma/gamma-software/hexotic/
  • Contact: Loic Marechal
  • Participant: Loic Marechal
  • Partner: Distene

5.1.4 Metrix

  • Name: Metrix: Error Estimates and Mesh Control for Anisotropic Mesh Adaptation
  • Keywords: Meshing, Metric, Metric fields
  • Functional Description: Metrix is a software that provides by various ways metric to govern the mesh generation. Generally, these metrics are constructed from error estimates (a priori or a posteriori) applied to the numerical solution. Metrix computes metric fields from scalar solutions by means of several error estimates: interpolation error, iso-lines error estimate, interface error estimate and goal oriented error estimate. It also contains several modules that handle meshes and metrics. For instance, it extracts the metric associated with a given mesh and it performs some metric operations such as: metric gradation and metric intersection.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.5 VIZIR

  • Name: Interactive visualization of hybrid, curbed and high-order mesh and solution
  • Keyword: Mesh
  • Functional Description: Vizir is a light, simple and interactive mesh visualization software, including: (i) A curved meshes visualizator: it handles high order elements and solutions, (ii) Hybrid elements mesh visualization (pyramids, prisms, hexahedra), (iii) Solutions visualization : clip planes, capping, iso-lines, iso-surfaces.
  • URL: http://vizir.inria.fr
  • Publication: hal-01686714
  • Author: Adrien Loseille
  • Contact: Adrien Loseille
  • Participants: Adrien Loseille, Rémi Feuillet, Matthieu Maunoury

5.1.6 Wolf

  • Keyword: Scientific calculation
  • Functional Description: Numerical solver for the Euler and compressible Navier-Stokes equations with turbulence modelling. ALE formulation for moving domains. Modules of interpolation, mesh optimisation and moving meshes. Wolf is written in C++, and may be later released as an opensource library. FELiScE was registered in July 2014 at the Agence pour la Protection des Programmes under the Inter Deposit Digital Number IDDN.FR.001.340034.000.S.P.2014.000.10000.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Frédéric Alauzet, Adrien Loseille, Rémi Feuillet, Lucille Marie Tenkes, Francesco Clerici

5.1.7 Wolf-Bloom

  • Keyword: Scientific calculation
  • Functional Description: Wolf-Bloom is a structured boundary layer mesh generator using a pushing approach. It start from an existing volume mesh and insert a structured boundary layer by pushing the volume mesh. The volume mesh deformation is solved with an elasticity analogy. Mesh-connectivity optimizations are performed to control volume mesh element quality.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: David Marcum, Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Adrien Loseille, David Marcum, Frédéric Alauzet

5.1.8 Wolf-Elast

  • Keyword: Scientific calculation
  • Functional Description: Wolf-Elast is a linear elasticity solver using the P1 to P3 Finite-Element method. The Young and Poisson coefficient can be parametrized. The linear system is solved using the Conjugate Gradient method with the LUSGS preconditioner.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.9 Wolf-Interpol

  • Keyword: Scientific calculation
  • Functional Description: Wolf-Interpol is a tool to transfer scalar, vector and tensor fields from one mesh to another one. Polynomial interpolation (from order 2 to 4) or conservative interpolation operators can be used. Wolf-Interpol also extract solutions along lines or surfaces.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contacts: Frédéric Alauzet, Paul Louis George
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.10 Wolf-MovMsh

  • Keyword: Scientific calculation
  • Functional Description: Wolf-MovMsh is a moving mesh algorithm coupled with mesh-connectivity optimization. Mesh deformation is computed by means of a linear elasticity solver or a RBF interpolation. Smoothing and swapping mesh optimization are performed to maintain good mesh quality. It handles rigid bodies or deformable bodies, and also rigid or deformable regions of the domain. High-order meshes are also handled
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Paul Louis George
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.11 Wolf-Nsc

  • Keyword: Scientific calculation
  • Functional Description: Wolf-Nsc is numerical flow solver solving steady or unsteady turbulent compressible Euler and Navier-Stokes equations. The available turbulent models are the Spalart-Almaras and the Menter SST k-omega. A mixed finite volume - finite element numerical method is used for the discretization. Second order spatial accuracy is reached thanks to MUSCL type methods. Explicit or implicit time integration are available. It also resolved dual (adjoint) problem and compute error estimate for mesh adaptation.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.12 Wolf-Shrimp

  • Keyword: Scientific calculation
  • Functional Description: Wolf-Shrimp is a generic mesh partitioner for parallel mesh generation and parallel computation. It can partition planar, surface (manifold and non manifold), and volume domain. Several partitioning methods are available: Hilbert-based, BFS, BFS with restart. It can work with or without weight function and can correct the partitions to have only one connected component.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.13 Wolf-Spyder

  • Keyword: Scientific calculation
  • Functional Description: Wolf-Spyder is a metric-based high-order mesh quality optimizer using vertex smoothing and edge/face swapping.
  • URL: https://pages.saclay.inria.fr/frederic.alauzet/software.html
  • Authors: Adrien Loseille, Frédéric Alauzet
  • Contact: Frédéric Alauzet
  • Participants: Adrien Loseille, Frédéric Alauzet

5.1.14 Wolf-Xfem

6 New results

6.1 Books on Meshing, Geometric Modeling and Numerical Simulation

Participants: Paul Louis George, Frédéric Alauzet, Adrien Loseille, Loïc Maréchal.

The third volume on Meshing, Geometric Modeling and Numerical Simulation 3 was published in 2020 (see Fig. 1) after 11, 25. These books are also written in French (see Fig. 2) 10, 24, 23.

IMG/gamma_book1
IMG/gamma_book2
IMG/gamma_book3
Figure 1: The three volumes of Meshing, Geometric Modeling and Numerical Simulation
IMG/Maillage-modélisation-géométrique-et-simulation-numérique-1
IMG/Maillage-modélisation-géométrique-et-simulation-numérique-2
IMG/Maillage-modélisation-géométrique-et-simulation-numérique-3
Figure 2: The three volumes of Meshing, Geometric Modeling and Numerical Simulation in French

6.2 Numerical simulations on GPU with the GMlib v3.0 library

Participants: Loïc Maréchal, Julien Vanharen.

The whole library was completely rewritten to implement an automatic finite-element shader generation that converts a simple user source code into an OpenCL source that is in compiled on the GPU at run time. The library handles all meshing data structures, from file reading, renumbering and vectorizing for efficient access on the GPU, and transfer to the graphic card, all automatically and transparently. With this framework, the user can focus on the calculation part of the code, known as kernel, as all the rest is taken care of by the library. The OpenCL language was chosen as it is hardware agnostic and runs on any GPU (Intel, Nvidia and AMD) and can also use the multicore and vector capacities of modern CPUs.

Julien Vanharen developed a basic heat solver using the v3.0 as a test case so we could validate the software with various boundary conditions, calculation scheme, unstructured meshes and different memory access patterns with success. Even with basic calculation which does not stress the full GPU's power, we achieved two orders of magnitude greater speed against a single CPU core and one order of magnitude compared to a multithreaded implementation. As Julien moved to ONERA, we plan on setting up a collaboration between the two teams to implement more complex HPC codes.

6.3 Pixel-exact rendering for high-order meshes and solutions

Participants: Adrien Loseille, Rémi Feuillet, Matthieu Maunoury.

Classic visualization software like ParaView 34, TecPlot 35, FieldView 40, Ensight 33, Medit 21, Vizir (OpenGL legacy based version) 46, Gmsh 30, ...historically rely on the display of linear triangles with linear solutions on it. More precisely, each element of the mesh is divided into a set of elementary triangles. At each vertex of the elementary triangle is attached a value and an associated color. The value and the color inside the triangle is then deduced by a linear interpolation inside the triangle. With the increase of high-order methods and high-order meshes, these softwares adapted their technology by using subdivision methods. If a mesh has high-order elements, these elements are subdivided into a set of linear triangles in order to approximate the shape of the high-order element 75. Likewise, if a mesh has a high-order solution on it, each element is subdivided into smaller linear triangles in order to approximate the rendering of the high-order solution on it. The subdivision process can be really expensive if it is done in a naive way. For this reason, mesh adaptation procedures 62, 51, 52 are used to efficiently render high-order solutions and high-order elements using the standard linear rendering approaches. Even when optimized these approaches do have a huge RAM memory footprint as the subdivision is done on CPU in a preprocessing step. Also the adaptive subdivision process can be dependent on the palette (i.e. the range of values where the solution is studied) as the color only vary when the associated value is in this range. In this case, a change of palette inevitably imposes a new adaptation process. Finally, the use of a non conforming mesh adaptation can lead to a discontinuous rendering for a continuous solution.

Other approaches are specifically devoted to high-order solutions and are based on ray casting 57, 58, 60. The idea is for a given pixel, to find exactly its color. To do so, for each pixel, rays are cast from the position of the screen in the physical space and their intersection with the scene determines the color for the pixel. If high-order features are taken into account, it determines the color exactly for this pixel. However, this method is based on two non-linear problems: the root-finding problem and the inversion of the geometrical mapping. These problems are really costly and do not compete with the interactivity of the standard linear rendering methods even when these are called with a subdivision process unless they are done conjointly on the GPU. However, synchronization between GPU and OpenGL buffer are non-trivial combination.

The proposed method intends to be a good compromise between both methods. It does guarantee pixel-exact rendering on linear elements without extra subdivision or ray casting and it keeps the interactivity of a classical method. Moreover, the subdivision of the curved entities is done on the fly on GPU which leaves the RAM memory footprint at the size of the loaded mesh.

We are developing a software, ViZiR 4, with exact non linear solution rendering to address the high-order visualization challenge 2. ViZiR 4 is bundled as a light, simple and interactive high-order meshes and solutions visualization software. It is based on OpenGL 4 core graphic pipeline. The use of OpenGL Shading Language (GLSL) allows to perform pixel exact rendering of high order solutions on straight elements and almost pixel exact rendering on curved elements (high-order meshes). ViZiR 4 enables the representation of high order meshes (up to degree 4) and high order solutions (up to degree 10) with pixel exact rendering. Furthermore, in comparison with standard rendering techniques based on legacy OpenGL, the use of OpenGL 4 core version improves the speed of rendering, reduces the memory footprint and increases the flexibility. Many post-processing tools, such as picking, hidding surfaces, isolines, clipping, capping, are integrated to enable on the fly the analysis of the numerical results.

6.4 High-order mesh generation

Participants: Frédéric Alauzet, Adrien Loseille, Rémi Feuillet, Dave Marcum, Lucien Rochery.

For years, the resolution of numerical methods has consisted in solving Partial Derivative Equations by means of a piecewise linear representation of the physical phenomenon on linear meshes. This choice was merely driven by computational limitations. With the increase of the computational capabilities, it became possible to increase the polynomial order of the solution while keeping the mesh linear. This was motivated by the fact that even if the increase of the polynomial order requires more computational resources per iteration of the solver, it yields a faster convergence of the approximation error 3  74 and it enables to keep track of unsteady features for a longer time and with a coarser mesh than with a linear approximation of the solution. However, in  14, 39, it was theoretically shown that for elliptic problems the optimal convergence rate for a high-order method was obtained with a curved boundary of the same order and in  9, evidence was given that without a high-order representation of the boundary the studied physical phenomenon was not exactly solved using a high-order method. In  78, it was even highlighted that, in some cases, the order of the mesh should be of a higher degree than the one of the solver. In other words, if the used mesh is not a high-order mesh, then the obtained high-order solution will never reliably represent the physical phenomenon.

Based on these issues, the development of high-order mesh generation procedures appears mandatory 1. To generate high-order meshes, several approaches exist. The first approach was tackled twenty years ago  16 for both surface and volume meshing. At this moment the idea was to use all the meshing tools to get a valid high-order mesh. The same problem was revisited a few years later in  69 for bio-medical applications. In these first approaches and in all the following, the underlying idea is to use a linear mesh and elevate it to the desired order. Some make use of a PDE or variational approach to do so  4, 61, 18, 54, 73, 76, 31, others are based on optimization and smoothing operations and start from a linear mesh with a constrained high-order curved boundary in order to generate a suitable high-order mesh 38, 22, 71. Also, when dealing with Navier-Stokes equations, the question of generating curved boundary layer meshes (also called viscous meshes) appears. Most of the time, dedicated approaches are set-up to deal with this problem  55, 37. In all these techniques, the key feature is to find the best deformation to be applied to the linear mesh and to optimize it. The prerequisite of these methods is that the initial boundary is curved and will be used as an input data. A natural question is consequently to study an optimal position of the high-order nodes on the curved boundary starting from an initial linear or high-order boundary mesh. This can be done in a coupled way with the volume  64, 72 or in a preprocessing phase  65, 66. In this process, the position of the nodes is set by projection onto the CAD geometry or by minimization of an error between the surface mesh and the CAD surface. Note that the vertices of the boundary mesh can move as well during the process. In the case of an initial linear boundary mesh with absence of a CAD geometry, some approaches based on normal reconstructions can be used to create a surrogate for the CAD model  75, 32. Finally, a last question remains when dealing with such high-order meshes: Given a set of degrees of freedom, is the definition of these objects always valid ? Until the work presented in  27, 36, 26, no real approach was proposed to deal in a robust way with the validity of high-order elements. The novelty of these approaches was to see the geometrical elements and their Jacobian as Bézier entities. Based on the properties of the Bézier representation, the validity of the element is concluded in a robust sense, while the other methods were only using a sampling of the Jacobian to conclude about its sign without any warranty on the whole validity of the elements.

In this context, several issues have been addressed : the analogy between high-order and Bézier elements, the development of high-order error estimates suitable for parametric high-order surface mesh generation and the generalization of mesh optimization operators and their applications to curved mesh generation, moving-mesh methods, boundary layer mesh generation and mesh adaptation.

Metric fields are the link between particular error estimates - be they for low-order 41, 42 or high-order methods 15, for the solution of a PDE 44 or a quantity of interest derived from it such as drag or lift 45 - and automatic mesh adaptation. In the case of linear meshes, a metric field locally distorts the measure or distance such that, when the mesh adaptation algorithm has constructed an uniform mesh in the induced Riemannian space, it is strongly anisotropic in the usual Euclidean (physicial) space. As such, anisotropy arrises naturally, without it ever being explicitely sought by the (re)meshing algorithm.

We seek to extend these principles of metric-based P1 adaptation to high-order meshes. In particular, we expect the meshing process to naturally recover curvature from the variations of the metric field, very much like P1 remeshing recovers anisotropy from local values of the metric field. As such, curvature must be the consequence of a simple geometric property computed in the Riemannian space, like anisotropy is the consequence of unitness in a space where distances are distorted. Therefore, we propose Riemannian edge length minimization (or geodesic seeking as in 77) as the driver for metric field curvature recovery.

The metric field's own intrinsic curvature may derive from any error estimate, be it boundary approximation error 19, 17 or an interpolation error estimate. So far, interpolation error estimates on high-order elements are limited to isotropy (13 in L2 and 56 in L1 norms) or require that the curvature of the element be bounded, essentially establishing a range where it may be considered linear 7.

If genericity with regards to error estimation is achieved through the use of a metric field, robustness and modularity of the general remeshing algorithm may be derived from the use of a single topological operator such as the cavity operator 43, 47, 48. This is the reason why we chose to extend the original P1 operator to work with P2 meshes as input and output.

Work has begun on the new P2 cavity operator. It is based, for the volume, on a purely metric-based curving procedure - that remains consistent with log-euclidean metric interpolation - and, for the surface, on CAD or CAD surrogate (typically P3) projection. Preliminary results have been obtained on complex geometries, showing volume curvature recovery at an acceptable CPU cost. This has lead to a communication at AIAA SciTech 2020 63.

6.5 Unstructured anisotropic mesh adaptation for 3D RANS turbomachinery applications

Participants: Frédéric Alauzet, Loïc Frazza, Adrien Loseille, Julien Vanharen.

The scope of this paper is to demonstrate the viability and efficiency of unstructured anisotropic mesh adaptation techniques to turbomachinery applications. The main difficulty in turbomachinery is the periodicity of the domain that must be taken into account inthe solution mesh-adaptive process. The periodicity is strongly enforced in the flow solver using ghost cells to minimize the impact on the source code. For the mesh adaptation, the local remeshing is done in two steps. First, the inner domain is remeshed with frozen periodic frontiers, and, second, the periodic surfaces are remeshed after moving geometrice ntities from one side of the domain to the other. One of the main goal of this work is to demonstrate how mesh adaptation, thanks to its automation, is able to generate meshes that are extremely difficult to envision and almost impossible to generate manually. This study only considers feature-based error estimate based on the standard multi-scale Lpinterpolation error estimate. We presents all the specific modifications that have been introduced in the adaptive process to deal with periodic simulations used for turbomachinery applications. The periodic mesh adaptation strategy is then tested and validated on the LS89 high pressure axial turbine vane and the NASA Rotor 37 test cases.

6.6 Hybrid mesh adaptation for CFD simulations

Participants: Frédéric Alauzet, Lucille Marie Tenkès, Julien Vanharen.

CFD simulations aim at capturing several phenomena of various natures. Therefore, these phenomena have very different mesh requirements. For example, most numerical schemes for the boundary layer require a structured mesh, that is aligned with the boundary of the domain. We use the techniques of mesh metric-based mesh adaptation to generate a hybrid mesh that can fulfill these different mesh requirements. This approach is based on the metric-orthogonal point-placement, creating some structured parts from the intrinsic directional information bore by the metric-field. Some unstructured areas may remain where structure is not required. The main goals of this work in progress are to improve the orthogonality of the output mesh and its alignment with the metric field. This work can fall into three parts. First, we have re-designed the preliminary step of size gradation correction. Then, we have studied two hybrid mesh generation processes methods: one using an a priori quadrilaterals recombination, the other building straightforwardly the quadrilaterals during the re-meshing step. Finally, some modifications have been added to the solver Wolf to perform simulations on hybrid meshes. Wolf is not able to run simulations of inviscid and viscous (laminar and turbulent) flows on two dimensional hybrid meshes. The two first topics are detailed in what follows.

Hybrid aware metric gradation correction

The previously described generation method highly relies on the metric field. However, a metric field computed from a solution during the adaptation process is most of the time quite messy, with for example strong size variations. In standard mesh adaptation, it leads to low-quality elements. In orthogonal mesh adaptation, it additionally breaks the alignment and the structure of the output mesh. In both cases, a step has been added to smooth the input metric field. In the context of hybrid mesh adaptation, this gradation correction process has been re-designed to improve the number and the quality of the quadrilaterals in the output mesh.

A posteriori and a priori mesh generation

Metric-orthogonal point-placement is currently used to generate quasi-structured meshes with right-angled triangles where the metric is the most anisotropic and unit triangles elsewhere. The aim of this work is to recover some quadrilaterals in the structure. To do so, two approaches can be considered: an a posteriori quadrilateral recombination based on geometrical criteria, and an a priori quadrilateral detection. The latter is more straightforward because it uses directly the point-placement information.

6.7 Anisotropic mesh adaptation for fluid-structure interactions

Participants: Frédéric Alauzet, Adrien Loseille, Julien Vanharen.

A new strategy for mesh adaptation dealing with Fluid-Structure Interaction (FSI) problems is presented using a partitioned approach. The Euler equations are solved by an edge-based Finite Volume solver whereas the linear elasticity equations are solved by the Finite Element Method using the Lagrange P1 elements. The coupling between both codes is realized by imposing boundary conditions. Small displacements of the structure are assumed and so the mesh is not deformed. The computation of a well-documented FSI test case is finally carried out to perform validation of this new strategy.

6.8 Convergence improvement of the flow solver on highly anisotropic meshes

Participants: Francesco Clerici, Frédéric Alauzet.

When using anisotropic mesh adaptation in computational fluid dynamics, the interactions occurring among complex geometries, high gradients (such as boundary layers and shocks) present some drawbacks, such as stallations and oscillations in the global residual convergence, specially when one makes use of slope limiters. In particular, we studied how the presence of a slope limiter affects the overall convergence of a simulation of the Navier-Stokes equations making use of anisotropic mesh adaptation and the Spalart-Allmaras turbulence model, and we have shown several techniques to reduce such such undesirable effects. With this regard, we successfully tested the freezeing of the slope limiter and the CFL reduction inside the slope limiter-oscillating vertices of the mesh, and then we tested the same methodologies inside the shockwaves generated by transonic flows.

6.9 Pseudo-transient adjoint continuation

Participants: Francesco Clerici, Frédéric Alauzet.

In aeronautical engineering, anisotropic mesh adaptation is used to predict accurately dimensionless quantities such as the lift and the drag coefficients, and, in general, functionals depending on the solution field. Anyway, in order to get the optimal adapted mesh with respect to the accuracy of a goal functional, it is necessary to solve an adjoint system providing the sensitivity of the goal functional with respect to the residuals of the equations. The linear system associated to the adjoint problem revealed to be stiff for RANS equations with a standard solver such as the GMRES preconditioned with several SGS iterations, and hence an alternative method has been developed, which is based on the transient simulation of the RANS adjoint state, starting from a previous valid solution.

7 Bilateral contracts and grants with industry

7.1 Bilateral contracts with industry

  • Boeing
  • Safran Tech

7.2 Bilateral grants with industry

  • Projet RAPID DGA

ANR IMPACTS 2018-2021

Ideal Mesh generation for modern solvers and comPuting ArchiteCTureS.

  • Coordinateur : Adrien Loseille
  • The rapid improvement of computer hardware and physical simulation capabilities has revolutionized science and engineering, placing computational simulation on an equal footing with theoretical analysis and physical experimentation. This rapidly increasing reliance on the predictive capabilities has created the need for rigorous control of numerical errors which strongly impact these predictions. A rigorous control of the numerical error can be only achieved through mesh adaptivity. In this context, the role of mesh adaptation is prominent, as the quality of the mesh, its refinement, and its alignment with the physics are major contributions to these numerical errors. The IMPACTS project aims at pushing the envelope in mesh adaptation in the context of large size, very high fidelity simulations by proposing a new adaptive mesh generation framework. This framework will be based on new theoretical developments on Riemannian metric-field and on innovative algorithmic developments coupling a unique cavity-operator with an advancing-point techniques in order to produce high quality hybrid, curved and adapted meshes.

8 Scientific production

8.1 Publications of the year

International journals

Scientific books

  • 3 book Paul LouisP. George, FrédéricF. Alauzet, AdrienA. Loseille and LoïcL. Maréchal. 'Meshing, Geometric Modeling and Numerical Simulation 3'. 1 November 2020

8.2 Cited publications

  • 4 articleR. Abgrall, C. Dobrzynski and A. Froehly. 'A method for computing curved meshes via the linear elasticity analogy, application to fluid dynamics problems'.International Journal for Numerical Methods in Fluids7642014, 246--266
  • 5 articleF. Alauzet and A. Loseille. 'A decade of progress on anisotropic mesh adaptation for Computational Fluid Dynamics'.722016, 13-39
  • 6 articleF. Alauzet and A. Loseille. 'High Order Sonic Boom Modeling by Adaptive Methods'.2292010, 561-593
  • 7 book ThomasT. Apel. 'Anisotropic finite elements: local estimates and applications'. 3 Teubner Stuttgart 1999
  • 8 articleN. Barral, G. Olivier and F. Alauzet. 'Metric-based anisotropic mesh adaptation for three-dimensional time-dependent problems involving moving geometries'.3312017, 157-187
  • 9 articleF. Bassi and S. Rebay. 'High-order accurate discontinuous finite element solution of the 2D Euler equations'.Journal of computational physics13821997, 251--285
  • 10 book H. Borouchaki and P.~L.P. George. 'Maillage, modélisation géométrique et simulation numérique 1: Fonctions de forme, triangulations et modélisation géométrique'. 1 ISTE Group 2017
  • 11 book H. Borouchaki and P.~L.P. George. 'Meshing, Geometric Modeling and Numerical Simulation 1: Form Functions, Triangulations and Geometric Modeling'. John Wiley & Sons 2017
  • 12 articleH. Borouchaki, P. Laug and P.L.P. George. 'Parametric surface meshing using a combined advancing-front -- generalized-Delaunay approach'.International Journal for Numerical Methods in Engineering491-22000, 233-259
  • 13 article LorenzoL. Botti. 'Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces'. Journal of Scientific Computing 52 09 2012
  • 14 incollectionP. G.P. Ciarlet and P.-A. Raviart. 'The combined effect of curved boundaries and numerical integration in isoparametric finite element methods'.The mathematical foundations of the finite element method with applications to partial differential equationsElsevier1972, 409--474
  • 15 articleOlivierO. Coulaud and AdrienA. Loseille. 'Very High Order Anisotropic Metric-Based Mesh Adaptation in 3D'.Procedia Engineering16325th International Meshing Roundtable2016, 353 - 365URL: http://www.sciencedirect.com/science/article/pii/S187770581633380X
  • 16 inproceedingsS. Dey, R. M.R. O'bara and M. S.M. Shephard. 'Curvilinear Mesh Generation in 3D'.Proceedings of the 7th International Meshing Roundtable1999, 407--417
  • 17 inproceedings RémiR. Feuillet, OlivierO. Coulaud and AdrienA. Loseille. 'Anisotropic Error Estimate for High-order Parametric Surface Mesh Generation'. 28th International Meshing Roundtable Buffalo, NY, United States October 2019
  • 18 articleM. Fortunato and P.-O. Persson. 'High-order Unstructured Curved Mesh Generation Using the Winslow Equations'.J. Comput. Phys.307February 2016, 1--14
  • 19 misc Pascal J.P. Frey. 'About Surface Remeshing'. 2000
  • 20 inproceedingsP.J.P. Frey. 'About surface remeshing'.Proceedings of the 9th international meshing roundtableNew Orleans, LO, USA2000, 123-136
  • 21 misc P. J.P. Frey. 'Medit: An interactive mesh visualization software, INRIA Technical Report RT0253'. 2001
  • 22 incollectionA. Gargallo-Peiró, X. Roca, J. Peraire and J. Sarrate. 'Defining quality measures for mesh optimization on parameterized CAD surfaces'.Proceedings of the 21st International Meshing RoundtableSpringer2013, 85--102
  • 23 book Paul LouisP. George, FrédéricF. Alauzet, AdrienA. Loseille and LoïcL. Maréchal. 'Maillage, modélisation géométrique et simulation numérique 3: Stockage, transformation, utilisation et visualisation de maillage'. 4 ISTE Group 2020
  • 24 book P.~L.P. George, H. Borouchaki, F. Alauzet, P. Laug, A. Loseille and L. Maréchal. 'Maillage, modélisation géométrique et simulation numérique 2: Métriques, maillages et adaptation de maillages'. 2 ISTE Group 2018
  • 25 book P.~L.P. George, H. Borouchaki, F. Alauzet, P. Laug, A. Loseille and L. Maréchal. 'Meshing, Geometric Modeling and Numerical Simulation 2: Metrics, Meshes and Mesh Adaptation'. John Wiley & Sons 2019
  • 26 articleP.~L.P. George, H. Borouchaki and N. Barral. 'Geometric validity (positive jacobian) of high-order Lagrange finite elements, theory and practical guidance'.Engineering with computers3232016, 405--424
  • 27 articleP.~L.P. George and H. Borouchaki. 'Construction of tetrahedral meshes of degree two'.International Journal for Numerical Methods in Engineering9092012, 1156,1182
  • 28 articleP.L.P. George and H. Borouchaki. '``Ultimate'' robustness in meshing an arbitrary polyhedron'.5872003, 1061-1089
  • 29 articleP.L.P. George, F. Hecht and E. Saltel. 'Automatic mesh generator with specified boundary'.921991, 269-288
  • 30 articleC. Geuzaine and J.-F. Remacle. 'Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities'.International Journal for Numerical Methods in Engineering79112009, 1309-1331
  • 31 articleR. Hartmann and T. Leicht. 'Generation of unstructured curvilinear grids and high-order discontinuous Galerkin discretization applied to a 3D high-lift configuration'.International Journal for Numerical Methods in Fluids8262016, 316-333
  • 32 articleJ. Ims and Z. J.Z. Wang. 'Automated low-order to high-order mesh conversion'.Engineering with Computers351Jan 2019, 323--335
  • 33 misc AnsysA. Inc.. 'Ensight'.
  • 34 misc KitWareK. Inc.. 'ParaView'.
  • 35 misc TecPlotT. Inc.. 'TecPlot'.
  • 36 articleA. Johnen, J.-F. Remacle and C. Geuzaine. 'Geometrical validity of curvilinear finite elements'.Journal of Computational Physics233152013, 359-372
  • 37 inproceedingsS. L.S. Karman. 'Curving for Viscous Meshes'.27th International Meshing RoundtableChamSpringer International Publishing2019, 303--325
  • 38 incollection S. L.S. Karman, J T.J. Erwin, R. S.R. Glasby and D. Stefanski. 'High-Order Mesh Curving Using WCN Mesh Optimization'. 46th AIAA Fluid Dynamics Conference AIAA AVIATION Forum American Institute of Aeronautics and Astronautics 2016
  • 39 articleM. Lenoir. 'Optimal isoparametric finite elements and error estimates for domains involving curved boundaries'.SIAM journal on numerical analysis2331986, 562--580
  • 40 misc IntelligentI. Light. 'FieldView'.
  • 41 articleAdrienA. Loseille and FrédéricF. Alauzet. 'Continuous mesh framework part I: well-posed continuous interpolation error'.SIAM Journal on Numerical Analysis4912011, 38--60
  • 42 articleAdrienA. Loseille and FrédéricF. Alauzet. 'Continuous mesh framework part II: validations and applications'.SIAM Journal on Numerical Analysis4912011, 61--86
  • 43 articleAdrienA. Loseille, FrédéricF. Alauzet and VictorienV. Menier. 'Unique cavity-based operator and hierarchical domain partitioning for fast parallel generation of anisotropic meshes'.Computer-Aided Design852017, 53-67
  • 44 phdthesis AdrienA. Loseille. 'Anisotropic 3D hessian-based multi-scale and adjoint-based mesh adaptation for Computational fluid dynamics: Application to high fidelity sonic boom prediction.'. Université Pierre et Marie Curie - Paris VI December 2008
  • 45 articleA. Loseille, A. Dervieux and F. Alauzet. 'Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations'.Journal of Computational Physics22982010, 2866 - 2897URL: http://www.sciencedirect.com/science/article/pii/S0021999109007001
  • 46 misc A. Loseille, H. Guillard and A. Loyer. 'An introduction to Vizir: an interactive mesh visualization and modification software'. 2016
  • 47 inbook AdrienA. Loseille and RainaldR. Lohner. 'Cavity-Based Operators for Mesh Adaptation'. 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition URL: https://arc.aiaa.org/doi/abs/10.2514/6.2013-152
  • 48 inbook AdrienA. Loseille. 'Recent Improvements on Cavity-Based Operators for RANS Mesh Adaptation'. 2018 AIAA Aerospace Sciences Meeting URL: https://arc.aiaa.org/doi/abs/10.2514/6.2018-0922
  • 49 inproceedingsL. Maréchal. 'A new approach to octree-based hexahedral meshing'.2001, 209--221
  • 50 inproceedingsL. Maréchal. 'Advances in Octree-Based All-Hexahedral Mesh Generation: Handling Sharp Features'.18Salt Lake City, UT, USA2009, 65-84
  • 51 articleM. Maunoury, C. Besse, V. Mouysset, S. Pernet and P.-A. Haas. 'Well-suited and adaptive post-processing for the visualization of hp simulation results'.Journal of Computational Physics3752018, 1179 - 1204
  • 52 phdthesis M. Maunoury. 'Méthode de visualisation adaptée aux simulations d'ordre élevé. Application à la compression-reconstruction de champs rayonnés pour des ondes harmoniques.'. Université de Toulouse February 2019
  • 53 inproceedings D.J.D. Mavriplis. 'Results from the 3rd Drag Prediction Workshop using NSU3D unstructured mesh solver'. 45 AIAA-2007-0256, Reno, NV, USA Jan 2007
  • 54 articleD. Moxey, D. Ekelschot, Ü. Keskin, S.J.S. Sherwin and J. Peirò. 'High-order curvilinear meshing using a thermo-elastic analogy'.Computer-Aided Design722016, 130 - 139
  • 55 articleD. Moxey, M.D.M. Green, S.J.S. Sherwin and J. Peiró. 'An isoparametric approach to high-order curvilinear boundary-layer meshing'.Computer Methods in Applied Mechanics and Engineering2832015, 636 - 650
  • 56 articleDavidD. Moxey, Shankar PS. Sastry and Robert MR. Kirby. 'Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement'.Journal of Scientific Computing7822019, 1045--1062
  • 57 articleB. Nelson, R. Haimes and R.~M.R. Kirby. 'GPU-Based Interactive Cut-Surface Extraction From High-Order Finite Element Fields'.IEEE Transactions on Visualization and Computer Graphics17122011, 1803--11
  • 58 articleB. Nelson, E. Liu, R. M.R. Kirby and R. Haimes. 'ElVis: A System for the Accurate and Interactive Visualization of High-Order Finite Element Solutions'.IEEE Transactions on Visualization and Computer Graphics18122012, 2325-2334
  • 59 inproceedings M. Park, J. Krakos, T.R.T. Michal, A. Loseille and J. Alonso. 'Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Toward CFD Vision 2030'. 46 2016-3323, Washington, D.C., USA 2016
  • 60 incollectionJ. Peiró, D. Moxey, B. Jordi, S. J.S. Sherwin, B.W.B. Nelson, R. M.R. Kirby and R. Haimes. 'High-Order Visualization with ElVis'.Notes on Numerical Fluid Mechanics and Multidisciplinary DesignSpringer International Publishing2015, 521--534
  • 61 inproceedingsP.-O. Persson and J. Peraire. 'Curved mesh generation and mesh refinement using Lagrangian solid mechanics'.47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition2009, 949
  • 62 articleJ.-F. Remacle, N. Chevaugeon, E. Marchandise and C. Geuzaine. 'Efficient visualization of high-order finite elements'.International Journal for Numerical Methods in Engineering6952007, 750-771
  • 63 inproceedingsLucienL. Rochery and AdrienA. Loseille. 'P2 cavity operator and Riemannian curved edge length optimization: a path to high-order mesh adaptation'.AIAA Scitech 2021 Forum2021, 1781
  • 64 articleE. Ruiz-Gironès, X. Roca and J. Sarrate. 'High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation'.Computer-Aided Design7223rd International Meshing Roundtable Special Issue: Advances in Mesh Generation2016, 52 - 64
  • 65 articleE. Ruiz-Gironès, J. Sarrate and X. Roca. 'Defining an L 2 -disparity Measure to Check and Improve the Geometric Accuracy of Non-interpolating Curved High-order Meshes'.Procedia Engineering1242015, 122--134
  • 66 articleE. Ruiz-Gironès, J. Sarrate and X. Roca. 'Generation of curved high-order meshes with optimal quality and geometric accuracy'.Procedia engineering1632016, 315--327
  • 67 articleC.L.C. Rumsey, J.P.J. Slotnick, M. Long, R.A.R. Stuever and T.R.T. Wayman. 'Summary of the first AIAA CFD High-Lift Prediction Workshop'.Journal of Aircraft4862011, 2068-2079
  • 68 articleC.L.C. Rumsey and J.P.J. Slotnick. 'Overview and Summary of the Second AIAA High Lift Prediction Workshop'.Journal of Aircraft5242015, 1006-1025
  • 69 articleS. J.S. Sherwin and J. Peiró. 'Mesh generation in curvilinear domains using high-order elements'.International Journal for Numerical Methods in Engineering5312002, 207-223
  • 70 techreport J. Slotnick, A. Khodadoust, J. Alonso, D. Darmofal, W. Gropp, E. Lurie and D. Mavriplis. 'CFD Vision 2030 Study: A path to revolutionary computational aerosciences'. NASA March 2014
  • 71 articleT. Toulorge, C. Geuzaine, J.-F. Remacle and J. Lambrechts. 'Robust untangling of curvilinear meshes'.Journal of Computational Physics2542013, 8 - 26
  • 72 articleT. Toulorge, J. Lambrechts and J.-F. Remacle. 'Optimizing the geometrical accuracy of curvilinear meshes'.Journal of Computational Physics3102016, 361 - 380
  • 73 articleM. Turner, J. Peirò and D. Moxey. 'A Variational Framework for High-order Mesh Generation'.Procedia Engineering163Supplement C25th International Meshing Roundtable2016, 340 - 352
  • 74 phdthesis J. Vanharen. 'High-order numerical methods for unsteady flows around complex geometries'. Université de Toulouse 2017
  • 75 inproceedingsA. Vlachos, P. Jörg, C. Boyd and J.~L.J. Mitchell. 'Curved PN Triangles'.Proceedings of the 2001 Symposium on Interactive 3D Graphics2001, 159-166
  • 76 articleZ. Q.Z. Xie, R. Sevilla, O. Hassan and K. Morgan. 'The generation of arbitrary order curved meshes for 3D finite element analysis'.Computational Mechanics513Mar 2013, 361--374
  • 77 inproceedingsRuiliR. Zhang, AmauryA. Johnen and Jean-FrançoisJ.-F. Remacle. 'Curvilinear mesh adaptation'.International Meshing RoundtableSpringer2018, 57--69
  • 78 inproceedings P. Zwanenburg and S. Nadarajah. 'On the Necessity of Superparametric Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries'. 23rd AIAA Computational Fluid Dynamics Conference 2017