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]
 PaulLouis George [Inria (pnp), Emeritus]
 David Marcum [MSU, Advanced Research Position]
 Julien Vanharen [INRIA, Senior Researcher, from Oct 2022]
PhD Students
 Sofiane Benzait [CEA]
 Francesco Clerici [INRIA]
 Eloi Guilbert [INRIA, from Sep 2022]
 Lucien Rochery [INRIA]
 Lucille Marie Tenkès [Inria, until Aug 2022]
Technical Staff
 Matthieu Maunoury [INRIA, Engineer, SED / Gamma]
 Cosimo Tarsia Morisco [INRIA, Engineer]
Administrative Assistant
 Maria Ronco [INRIA]
2 Overall objectives
Numerical simulation has been booming over the last thirty years, thanks to increasingly powerful numerical methods, computeraided 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 wellknown software Ghs3d29, 30 which has been distributed worldwide so far and in hexahedral meshing with the software Hexotic38, 39 which was the first automated full hex mesher. The team has also worked on surface meshers with Yams26 and BLSurf23 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 metricbased 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 NavierStokes equations 20, and its extension to unsteady problems with moving geometries 22 leading to the development of several softwares Feflo.a/AMGLib, Wolf, Metrix, WolfInterpol. A significant accomplishment was the highfidelity prediction of the sonic boom emitted by supersonic aircraft 21. 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 multithread library LPlib and the efficient management of memory using space filling curves, and the generation of large meshes (a billion of elements) 36. Theoretical work on highorder meshes has been also done 28.
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 nonautomated step of the design pipeline and, thus, the most human time consuming
 Highperformance computing (HPC) for all tools included in the design loop
 The cost in euros of a numerical simulation
 Certification of highfidelity 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 hightech projects with high added value (Boeing, Safran, DassaultAviation, Ariane Group, ...) consider their design pipeline inside a HPC framework. Indeed, they are performing complex numerical simulations on complex geometries on a dailybasis, and they aim at using this in a shapeoptimization 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 (multithreaded), 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 windtunnel 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 (1D2D), to 150 euros for Reynoldsaveraged NavierStokes (3D) stationary models, or up to 15 000 euros for unsteady models like LES or LatticeBoltzmann 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 highfidelity 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 40 and HighLift Prediction Workshops 44, 43. 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" 45 and the study dedicated to mesh adaptation 41.
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 geometrymesh 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 highorder meshadaptive 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)
geometrical adaptive modeling,

ii)
solution adaptation,

iii)
highorder,

iv)
multielements (structured or not), and

v)
using hybrid scalable parallelism.
Note that items $i)$ to $iv)$ are based on the wellposed metricbased theoretical framework. Moreover, Gamma will continue to work on robust flow solvers, solving the turbulent NavierStokes equations from second order using Finite Volume  Finite Element numerical scheme to higherorder using Flux Reconstruction (FR) method.
The combination of adaptation  highorder  multielements coupled with appropriate error estimates is for the team the way to go to reduce the cost of numerical simulations while ensuring highfidelity in a fully automated framework.
3 Research program
The main axes are:
 Geometric Modeling:
 Highfidelity discrete CAD kernel.
 Continuous parametric CAD kernel.
 Enhanced Generic Meshing Algorithm:
 Adaptation (extreme anisotropy, metricaligned, metricorthogonal).
 Highorder (tetrahedra, hexahedra, boundary layer, adapted).
 Larges meshes (tetrahedra, hexahedra, adapted).
 Moving mesh methods for moving geometries.
 Toward Certified Solutions to the NavierStokes Equations:
 Flow solver and adjoints (Finite Volumes, Finite Elements, Flux Reconstruction).
 Error estimates and correctors.
 Advanced Mesh and Solution Visualisation:
 Pixel exact rendering (HighOrder mesh, HighOrder solution).
 Preprocessing and postprocessing.
4 Application domains
Our research in mesh generation, mesh adaptation and certification of the Numerical Simulation Pipeline finds applications in several different domains such as aviation and aerospace but also all fields where computation and simulation are used: fluid mechanics, solid mechanics, solving wave equations (acoustic, electromagnetism...), energy or biomedical.
5 New software and platforms
5.1 New software
5.1.1 GHS3D

Keywords:
Tetrahedral mesh, Delaunay, Automatic mesher

Functional Description:
GHS3D is an automatic volume mesher
 URL:

Contact:
Frederic Alauzet

Participants:
Paul Louis George, Adrien Loseille, Frederic Alauzet
5.1.2 HEXOTIC

Keywords:
3D, Mesh generation, Meshing, Unstructured meshes, Octree/Quadtree, Multithreading, 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:

Contact:
Loic Marechal

Participant:
Loic Marechal

Partner:
Distene
5.1.3 FEFLOAREMESH

Keywords:
Scientific calculation, Anisotropic, Mesh adaptation

Functional Description:
FEFLOAREMESH is intended to generate adapted 2D, surface and volume meshes by using a unique cavitybased operator. The metricaligned or metricorthogonal approach is used to generate high quality surface and volume meshes independently of the anisotropy involved.
 URL:

Contact:
Adrien Loseille

Participants:
Adrien Loseille, Frederic Alauzet, Rémi Feuillet, Lucien Rochery, LucilleMarie Tenkes
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, isolines 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:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.5 Wolf

Keyword:
Scientific calculation

Functional Description:
Numerical solver for the Euler and compressible NavierStokes 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:

Contact:
Frederic Alauzet

Participants:
Frederic Alauzet, Adrien Loseille, Rémi Feuillet, LucilleMarie Tenkes, Francesco Clerici, Cosimo Tarsia Morisco
5.1.6 WolfBloom

Keyword:
Scientific calculation

Functional Description:
WolfBloom 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. Meshconnectivity optimizations are performed to control volume mesh element quality.
 URL:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, David Marcum, Frederic Alauzet
5.1.7 WolfElast

Keyword:
Scientific calculation

Functional Description:
WolfElast is a linear elasticity solver using the P1 to P3 FiniteElement method. The Young and Poisson coefficient can be parametrized. The linear system is solved using the Conjugate Gradient method with the LUSGS preconditioner.
 URL:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.8 WolfInterpol

Keyword:
Scientific calculation

Functional Description:
WolfInterpol 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. WolfInterpol also extract solutions along lines or surfaces.
 URL:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.9 WolfMovMsh

Keyword:
Scientific calculation

Functional Description:
WolfMovMsh is a moving mesh algorithm coupled with meshconnectivity 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. Highorder meshes are also handled
 URL:

Contact:
Paul Louis George

Participants:
Adrien Loseille, Frederic Alauzet
5.1.10 WolfNsc

Keyword:
Scientific calculation

Functional Description:
WolfNsc is numerical flow solver solving steady or unsteady turbulent compressible Euler and NavierStokes equations. The available turbulent models are the SpalartAlmaras and the Menter SST komega. 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:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.11 WolfShrimp

Keywords:
Scientific calculation, Domain partitionning

Scientific Description:
WolfShrimp 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: Hilbertbased, BFS, BFS with restart. It can work with or without weight function and can correct the partitions to have only one connected component.

Functional Description:
WolfShrimp 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: Hilbertbased, BFS, BFS with restart. It can work with or without weight function and can correct the partitions to have only one connected component.
 URL:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.12 WolfSpyder

Keyword:
Scientific calculation

Functional Description:
WolfSpyder is a metricbased highorder mesh quality optimizer using vertex smoothing and edge/face swapping.
 URL:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.13 WolfXfem

Keyword:
Scientific calculation

Functional Description:
WolfXfem is a tool providing the mesh of the intersection between a surface mesh and a volume mesh.
 URL:

Contact:
Frederic Alauzet

Participants:
Adrien Loseille, Frederic Alauzet
5.1.14 ViZiR4

Name:
ViZiR4

Keywords:
Visualization, Pixelexact rendering, Instant rendering, High order methods

Functional Description:
Its main features are:  Light, simple and interactive visualization software.  Surface and volume (tetrahedra, pyramids, prisms, hexahedra) meshes.  Pixel exact rendering of highorder solutions on straight elements.  Almost pixel exact rendering on curved elements (highorder meshes).  Postprocessing tools, such as picking, isolines, clipping, capping.
 URL:
 Publications:

Contact:
Adrien Loseille

Participants:
Adrien Loseille, Matthieu Maunoury, Frederic Alauzet
6 New results
6.1 Numerical simulations on GPU with the GMlib v3.0 library
Participants: Loïc Maréchal [correspondant], Julien Vanharen.
The whole library was completely rewritten to implement an automatic finiteelement 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.
6.2 High Order Meshing: from a straight mesh to a curved one
Participants: Loïc Maréchal [correspondant].
Works continued on P1toPk, a software that transform any first order hybrid mesh (triangles, quads, tets, pyramids, prisms and hexes) into a second order one while respecting a prescribe surface curvature. Efforts were made on boundary layers curving, which was challenging because jacobian validity is harder to guarantee as the elements get highly stretched, and a lot effort were also made to speed up the code by optimizing mathematical operations and parallelizing them. The code is now mature enough to be sent to industrial users for real life usage and we are waiting for valuable feedback in the present year.
6.3 Pixelexact rendering for highorder meshes and solutions
Participants: Matthieu Maunoury [correspondant], Adrien Loseille.
We are developing ViZiR 4, a visualization software with pixel exact rendering to address the highorder visualization challenges 25, 15. ViZiR 4 is bundled as a light, simple and interactive highorder 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 (without extra subdivision or ray casting) and almost pixel exact rendering on curved elements (highorder 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. Unlike other visualization software (ParaView 33, TecPlot 34, FieldView 35, Ensight 32, Medit 27, Vizir (OpenGL legacy based version) 37, Gmsh 31), there is no subdivision process that is expensive nor visualization error that has to be controlled. 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. 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 postprocessing tools, such as picking, hidding surfaces, isolines, clipping, capping, are integrated to enable on the fly the analysis of the numerical results.
6.4 Fast highorder tetrahedral mesh correction and metricbased curving for P2 cavity remeshing
Participants: Lucien Rochery [correspondant], Adrien Loseille.
We aim to deal with three main topics around highorder mesh adaptation with applications to classic a posteriori curving.
One, a new method to untangle highorder Bézier meshes is introduced. It maximizes directly the minimum control coefficient over the mesh. Under some conditions, the $maxmin$ problem can be recast as a linear program, solvable exactly by specialized algorithms. The best possible position of an edge control point is obtained in as long as to compute and differentiate dependent control coefficients five times.
Two, Riemannian edge length minimization is used to prescribe metricbased meshinterior curvature. This generalizes weakly the unit mesh definition on linear meshes, and is very fast as only edge scale problems are solved. Through metric gradation on surface metrics, surface curvature is naturally propagated to the interior by length minimization. Similarly, the meshintrinsic metric can be used to curve boundary layers. In realistic settings, a socalled back mesh holding the discrete metric field is kept unchanged throughout remeshing. This prevents anisotropy loss through repeated interpolation. It also leads to the metric at a point $P$ within an element $K$ being the result of two interpolations. First, at the control points of $K$ using the back mesh. Then, at $P$ using those. This metric at $P$ is differentiated, in order to provide derivatives of anisotropic geometric quantities.
Three, the cavity operator if extended to ${P}^{2}$ meshes. It rewrites topological changes (insertions, collapses, generalized swaps) as element deletions followed by point starring. The ${P}^{2}$ cavity operator is modular and distinguishes between two types of curvature: prescribed and necessary. Here, CAD/${P}^{3}$ surrogate projection and Riemannian length minimization are used for the surface and interior respectively.
Necessary curvature results from subsequent highorder untangling, as prescribed curvature is not expected to yield valid cavities.
Numerical results focus on a posteriori curving of difficult cases. Boundary control points are projected. Metricbased curvature is used to propagate surface curvature, curve boundary layers, and follow natural curvature of a metric field resulting from CFD adaptation. The simplexbased Jacobian smoother corrects the resulting meshes. Examples are based on 3D realworld geometries encountered in Computational Fluid Dynamics (CFD). This framework allow us to curve highly anisotropic meshes with around 10 million elements within minutes.
6.5 Unstructured anisotropic mesh adaptation for 3D RANS turbomachinery applications
Participants: Frédéric Alauzet [correspondant], Adrien Loseille, Julien Vanharen.
We aim 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 meshadaptive 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 geometric entities 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 featurebased error estimate based on the standard multiscale 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 Mixedelement mesh adaptation for CFD simulations
Participants: Frédéric Alauzet [correspondant], Julien Vanharen, Adrien Loseille, Cosimo Tarsia Morisco.
Due to their various nature, physical phenomena that we seek to capture in CFD simulations may have specific specific mesh requirements. For example, to solve the boundary layer, some numerical schemes favor structured meshes respecting alignment with the boundary of the domain, while these constraints are not necessary elsewhere. Our approach is to use the techniques of metricbased mesh adaptation to generate a mixedelement mesh that can fulfill these different mesh requirements. This approach is based on the metricorthogonal pointplacement, creating some structured parts from the intrinsic directional information bore by the metricfield. Some unstructured areas may remain where structure is not needed. The main goals of this work are to improve the orthogonality of the output mesh and its alignment with the metric field. This work has three main axes. First, we have improved the preprocessing gradation step to smooth the metric field and improve the orthogonality of the final mesh. Then, we have studied two methods to obtain quadrilaterals: one using an a priori quadrilaterals recombination, the other detecting straightforwardly the orthogonal patterns during the remeshing step. Finally, the work on the solver Wolf has been carried on and corrected to perform robust and accurate simulations on mixedelement meshes. These three developments were embodied in a mixedelement adaptation loop. The first two topics are detailed in what follows.
Enhanced 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 and shows abrupt size variations. In standard mesh adaptation, it leads to lowquality elements. In orthogonal mesh adaptation, it additionally breaks the alignment and the structure of the output mesh. An additional step to smooth the input metric field is therefore required. In the context of mixedelement mesh adaptation, this gradation correction process has been modified to improve the number and the quality of the quadrilaterals in the final mesh. Further developments have been considered on this topic, in particular to increase the robustness of the method. Results have been published in 47.
A posteriori and a priori mesh generation
Metricorthogonal pointplacement is currently used to generate quasistructured meshes with rightangled 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 pointplacement information. A framework was established to set up this method. Developments and preliminary results were presented in 48.
In order to obtain a correct metric field on hybrid meshes, a robust hybrid solver is mandatory. When dealing with 2D (3D) elements different from triangles (tetrahedra), the most tricky aspect is the gradient formulation. This is due to the fact that within a Finite Elements interpolation framework, the gradient on an element with more than three nodes (i.e. not simplicial complex) is not elementwise constant. This brings many added difficulties to the flux balance computation. A first attempt at performing inviscid and laminar simulations on hybrid meshes was to approximate gradients on quadrilaterals with its isobarycenter values 48. The extension of this formulation to turbulent flows, however highlighted a lack of efficiency and robustness. For this reason, a APFE (APproximated FiniteElement) method 42 has been implemented and as well extended to a implicit time integration scheme. This approach turns out to be very efficient and robust in many fullystructured mesh verification cases. The extension to 3D cases (prisms and pyramids) is ongoing. Details can be found in 17.
An extension of the VertexCentered MixedElementVolume MUSCL scheme to 3D mixedelement meshes is proposed for the convective fluxes. This scheme involves a clever exploitation of the FE gradients, which can be efficiently implemented as ${P}^{1}$ FE gradients of particular subtetrahedra inside prismatic and/or pyramidal elements. Conversely, diffusive fluxes are discretized using an original extension of the APproximated Finite Element (APFE) method to 3D elements. This method could fit for any element, as long as a robust FE gradient and a dual volume with planar dual facets is provided.
The combination of the proposed discretization strategies for convective and diffusive fluxes shows a certain robustness for regular and not regular mixed element meshes as well interesting results for turbulent flows. The two dimensional analysis highlighted the influence of gradient formulation in source terms for highly anisotropic meshes. This aspect is under investigation in 3D. On the whole, the methodology proposed seems to be a promising candidate to tackle mixedelement adaptation in some future works. More details in 46.
6.7 Coupled flow and adjoint solver
Participants: Francesco Clerici [correspondant], Frédéric Alauzet.
When solving the RANS equations, usually one decouples the equations relative to the meanflow and the equations relative to turbulence. This division provides two separated systems to be solved at each time step, one relative to the meanflow and the other relative to the turbulence. This presents two main drawbacks: in the flow solver, the Jacobian of the system lacks of the terms bounding the meanflow and the turbulence, and this can slow down the residual convergence. The second drawback regards the adjoint problem, which consists into a linear system assembled with the transpose of the Jacobian matrix of the residuals and, on the righthand side, the derivative of an aeronautical coefficient with respect to the flow variables. A Jacobian missing the coupling terms between the meanflow and the turbulence provides a null adjoint turbulent viscosity, and this is a limitation in the development of more complex discretization error estimates. We have therefore developed a 2D version of the coupled flow and adjoint solver which includes in the Jacobian the coupling terms between the meanflow and the turbulence. When have tested this method on the 2D geometry provided for the 4th CFD AIAA High Lift Prediction Workshop, and the result provided several features to emerge, such as a high mesh refinement inside the boundary layer of the leading edges, and inside the regions of high turbulence destruction. The work is pursued in collaboration with Philippe Spalart (Boeing), and is presented in 24.
6.8 Turbulent error estimate
Participants: Francesco Clerici [correspondant], Frédéric Alauzet.
Goaloriented mesh adaptation is a methodology used to adapt the mesh in order to minimize the discretization error commited on a functional depending on the solution. As an intermediate step, one finds an upper bound to such discretization error taking the form of a weighted sum of the interpolation errors of the solution, and such upper bound is called error estimate. Regarding Wolf and the RANS equations, up to now we have focused only on the meanflow part of such an error estimate, meaning that the terms coming from turbulence have been neglected. The scope of this work is to enrich the error estimate with the information coming from the turbulence. In particular, the methodology has been tested on the 2D geometry provided for the 4th CFD AIAA High Lift Prediction Workshop, providing high mesh refinements on the boundaries of the turbulent regions. Also this work is pursued in collaboration with Philippe Spalart (Boeing), and is presented in 24.
7 Bilateral contracts and grants with industry
7.1 Bilateral contracts with industry
Participants: Frédéric Alauzet [correspondant], Adrien Loseille.
 Safran Tech
 Ariane Group
 Lemma
8 Partnerships and cooperations
8.1 European initiatives
8.1.1 Horizon Europe
NEXTAIR
Participants: Julien Vanharen [correspondant], Frédéric Alauzet.
NEXTAIR project on cordis.europa.eu

Title:
NEXTAIR  multidisciplinary digital  enablers for NEXTgeneration AIRcraft design and operations

Duration:
From September 1, 2022 to August 31, 2025

Partners:
 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE (INRIA), France
 THE UNIVERSITY OF SHEFFIELD (USFD), United Kingdom
 IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE (Imperial), United Kingdom
 AIRBUS OPERATIONS SAS (AIRBUS OPERATIONS), France
 ETHNICON METSOVION POLYTECHNION (NATIONAL TECHNICAL UNIVERSITY OF ATHENS  NTUA), Greece
 SAFRAN SA, France
 UNIVERSITA DEGLI STUDI DI CAGLIARI (UNICA), Italy
 OFFICE NATIONAL D'ETUDES ET DE RECHERCHES AEROSPATIALES (ONERA), France
 DEUTSCHES ZENTRUM FUR LUFT  UND RAUMFAHRT EV (DLR), Germany
 FUNDACION CENTRO DE TECNOLOGIAS DE INTERACCION VISUAL Y COMUNICACIONES VICOMTECH (VICOM), Spain
 DASSAULT AVIATION, France
 ASOUTI V & SIA OE, Greece
 OPTIMAD ENGINEERING SRL (Optimad srl), Italy
 IRT ANTOINE DE SAINT EXUPERY, France
 ERDYN CONSULTANTS SARL, France
 ROLLSROYCE PLC, United Kingdom

Inria contact:
Pietro Congedo
 Coordinator:

Summary:
Radical changes in aircraft configurations and operations are required to meet the target of climateneutral aviation. To foster this transformation, innovative digital methodologies are of utmost importance to enable the optimisation of aircraft performances.
NEXTAIR will develop and demonstrate innovative design methodologies, datafusion techniques and smart healthassessment tools enabling the digital transformation of aircraft design, manufacturing and maintenance. NEXTAIR proposes digital enablers covering the whole aircraft lifecycle devoted to ease breakthrough technology maturation, their flawless entry into service and smart health assessment. They will be demonstrated in 8 industrial test cases, representative of multiphysics industrial design, maintenance problems and environmental challenges and interest for aircraft and engines manufacturers.
NEXTAIR will increase highfidelity modelling and simulation capabilities to accelerate and derisk new disruptive configurations and breakthrough technologies design. NEXTAIR will also improve the efficiency of uncertainty quantification and robust optimisation techniques to effectively account for manufacturing uncertainty and operational variability in the industrial multidisciplinary design of aircraft and engine components. Finally, NEXTAIR will extend the usability of machine learningdriven methodologies to contribute to aircraft and engine components' digital twinning for smart prototyping and maintenance.
NEXTAIR brings together 16 partners from 6 countries specialised in various disciplines: digital tools, advanced modelling and simulation, artificial intelligence, machine learning, aerospace design, and innovative manufacturing. The consortium includes 9 research organisations, 4 leading aeronautical industries providing digitalphysical scaled demonstrator aircraft and engines and 2 highTech SME providing expertise in industrial scientific computing and data intelligence.
9 Scientific production
9.1 Major publications
 1 articleHigh Order Sonic Boom Modeling by Adaptive Methods.Journal Of Computational Physics2292010, 561593URL: http://dx.doi.org/10.1016/j.jcp.2009.09.020
 2 articleMagnetic cage and rope as the key for solar eruptions.Nature5547691February 2018, 211  215
 3 unpublishedAn a priori anisotropic GoalOriented Error Estimate for Viscous Compressible Flow and Application to Mesh Adaptation.November 2018, working paper or preprint
 4 bookMeshing, Geometric Modeling and Numerical Simulation 1.John Wiley & Sons, Inc.October 2017
 5 articleMetricbased anisotropic mesh adaptation for 3D acoustic boundary element methods.Journal of Computational Physics372November 2018, 473  499
 6 articleOn pixelexact rendering for highorder mesh and solution.Journal of Computational Physics424January 2021, 109860
 7 bookMeshing, Geometric Modeling and Numerical Simulation 3.1Wiley; WileyNovember 2020
 8 inbookMesh Generation and Mesh Adaptivity: Theories and Techniques.Encyclopedia of Computational Mechanics, Volume 1: FundamentalsISBN 0470846992, E. Stein, R. de Borst and T.J.R. Hughes ed., 2nd edition 2008Wiley InterScience2004, 17497523
 9 inproceedingsOptimal 3D Highly Anisotropic Mesh Adaptation based on the Continuous Mesh Framework.18th International meshing roundtableSaltLake City, UT, USASpringer2009, URL: http://dx.doi.org/10.1007/9783642043192_20
 10 articleFully anisotropic goaloriented mesh adaptation for 3D steady Euler equations.Journal Of Computational Physics2292010, 28662897URL: http://dx.doi.org/10.1016/j.jcp.2009.12.021
 11 inproceedingsVizir: Highorder mesh and solution visualization using OpenGL 4.0 graphic pipeline.2018  AIAA Aerospace Sciences Meeting, AIAA SciTech Forumkissimmee, United StatesJanuary 2018, 113
 12 incollectionRobust Boundary Layer Mesh Generation.Proceedings of the 21st International Meshing RoundtableSpringer Berlin Heidelberg2012, 493511
 13 inproceedingsNumerical Simulation of contrail formation on the Common Research Model wing/body/engine configuration.AIAA Aviation and Aeronautics Forum and Exposition 2018Atlanta, United StatesJune 2018
9.2 Publications of the year
International peerreviewed conferences
 14 inproceedingsSome progress on CFD high lift prediction using metricbased anisotropic mesh adaptation.AIAA SCITECH 2022 ForumAIAA SCITECH 2022  ForumSan Diego, United StatesAmerican Institute of Aeronautics and AstronauticsJanuary 2022, 0388
 15 inproceedingsUsing ViZiR 4 to analyze the 4th AIAA CFD High Lift Prediction Workshop Simulations.AIAA SciTech 2022 ForumSan Diego / Virtual, United StatesAmerican Institute of Aeronautics and AstronauticsJanuary 2022
 16 inproceedingsSimulation of massively separated flows using hybrid turbulence models ans mesh adaptation.DLES13 2022  Direct and LargeEddy Simulation 13Udine, ItalyOctober 2022
 17 inproceedingsQuasistructured anisotropic quaddominant mesh adaptation using metricorthogonal approach.SCITECH 2022  AIAA SciTech ForumSan Diego, United StatesAmerican Institute of Aeronautics and AstronauticsJanuary 2022
Scientific books
 18 bookMesh Adaptation for Computational Fluid Dynamics 1: Continuous Riemannian Metrics and Featurebased Adaptation.Numerical methods in engineering seriesWileyIste2022
Reports & preprints
 19 misc3D NitscheXFEM method for fluidstructure interaction with immersed thinwalled solids.December 2022
9.3 Cited publications

20
articleA decade of progress on anisotropic mesh adaptation for Computational Fluid Dynamics.
722016, 1339 
21
articleHigh Order Sonic Boom Modeling by Adaptive Methods.
2292010, 561593 
22
articleMetricbased anisotropic mesh adaptation for threedimensional timedependent problems involving moving geometries.
3312017, 157187  23 articleParametric surface meshing using a combined advancingfront  generalizedDelaunay approach.International Journal for Numerical Methods in Engineering49122000, 233259
 24 articleCoupled adjoint solver and turbulent error estimate for anisotropic mesh adaptation in highfidelity RANS simulations..AIAA J.2022
 25 articleOn pixelexact rendering for highorder mesh and solution.Journal of Computational Physics424January 2021, 109860
 26 inproceedingsAbout surface remeshing.Proceedings of the 9th international meshing roundtableNew Orleans, LO, USA2000, 123136
 27 miscMedit: An interactive mesh visualization software, INRIA Technical Report RT0253.2001
 28 articleConstruction of tetrahedral meshes of degree two.International Journal for Numerical Methods in Engineering9092012, 1156,1182

29
article``Ultimate'' robustness in meshing an arbitrary polyhedron.
5872003, 10611089 
30
articleAutomatic mesh generator with specified boundary.
921991, 269288  31 articleGmsh: A 3D finite element mesh generator with builtin pre and postprocessing facilities.International Journal for Numerical Methods in Engineering79112009, 13091331
 32 miscEnsight.
 33 miscParaView.
 34 miscTecPlot.
 35 miscFieldView.
 36 articleUnique cavitybased operator and hierarchical domain partitioning for fast parallel generation of anisotropic meshes.ComputerAided Design852017, 5367
 37 miscAn introduction to Vizir: an interactive mesh visualization and modification software.2016
 38 inproceedingsA new approach to octreebased hexahedral meshing.2001, 209221

39
inproceedingsAdvances in OctreeBased AllHexahedral Mesh Generation: Handling Sharp Features.
18Salt Lake City, UT, USA2009, 6584 
40
inproceedingsResults from the 3rd Drag Prediction Workshop using NSU3D unstructured mesh solver.
45AIAA20070256, Reno, NV, USAJan 2007 
41
inproceedingsUnstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Toward CFD Vision 2030.
46 20163323, Washington, D.C., USA2016  42 articleDiscretisation of diffusive fluxes on hybrid grids.Journal of Computational Physics22952010, 14251447
 43 articleSummary of the first AIAA CFD HighLift Prediction Workshop.Journal of Aircraft4862011, 20682079
 44 articleOverview and Summary of the Second AIAA High Lift Prediction Workshop.Journal of Aircraft5242015, 10061025
 45 techreportCFD Vision 2030 Study: A path to revolutionary computational aerosciences.NASAMarch 2014
 46 inproceedingsExtension of the VertexCentered MixedElementVolume MUSCL scheme to mixedelement meshes.AIAA SCITECH 2023 Forum2023, 0832
 47 incollectionSize Gradation Control for Anisotropic Hybrid Meshes.Numerical Geometry, Grid Generation and Scientific Computing143Lecture Notes in Computational Science and EngineeringSpringer International PublishingMay 2021, 127139
 48 inproceedingsHybrid anisotropic mesh adaptation using metricorthogonal approach.AIAA Scitech 2021 ForumVirtual, United StatesAmerican Institute of Aeronautics and AstronauticsJanuary 2021