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 led 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 classic 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.

Gamma's research program is motivated by four grand challenges in order to achieve certification and high-fidelity in the numerical simulation pipeline. The goal is to deliver innovative and ground-breaking solutions to each step of the adaptive numerical simulation pipeline. Not surprisingly, these challenges (and the themes that result) are clearly indicated in the NASA CFD Vision 2030 Study 41 and have been mentioned recurrently during the previous evaluations of the Gamma3 team where they have been judged as very ambitious and long-term. The four grand challenges are:

1. Geometric modeling. The goal is to address geometry modeling issues and their interactions with the meshing pipeline.
To this end, Gamma will develop more versatile and robust geometry and modeling processes to be embedded within meshing tools.

2. Enhanced generic meshing algorithms. Gamma will pursue its work on state-of-the-art meshing technologies which
should fulfill these three requirements: adaptation, high-order and large size. Mesh adaptation and high-order meshing will be based on
the well-posed metric-based mathematical framework. The generation of large size mesh will be achieved with hybrid parallelism (multi-thread and MPI).

3. Toward certified numerical solutions to the Navier-Stokes equations. This research axis will focus on error estimates and robust numerical schemes (flow solver).
Gamma will primarily focus on the design of uncertainty aware RANS error estimates.
We will pursue the work on anisotropic mesh adaptation for the turbulent Navier-Stokes equations with moving geometries.
We envision to develop a new high-order mesh-adaptive solution platform which requires to design high-order error estimates and a high-order flow solver.

4. Advanced visualization of mesh and solution.
High-order representations (both on the solver and meshing sides) use higher-degree polynomials to interpolate
solution data. The challenge is to develop algorithms for pixel exact rendering of high-order meshes and solutions
which will provide the potential to reveal features that otherwise might be masked by classic visualization approaches.
The visualization software will be used for pre and post processing by interfacing all the Gamma's software components.

These four grand challenges cover the whole numerical simulation pipeline depicted below. The geometric modeling is represented by the green part, the enhanced generic meshing algorithms by the blue part, the certification of Navier-Stokes simulations by the red part, and the advanced mesh and solution visualization by the brown part. We can also see the clear interaction between each research axis.

Most of the proposed meshing developments and technologies are generic and can be applied to a broader field of applications in order to increase the impact of this research program. Flow solver developments and technologies focus specifically on CFD with applications to aerospace, turbomachinery, and defense.

The main axes are:

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.

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.

We show the ability of metric-based anisotropic mesh adaptation to accurately predict the flow field on a particularly complex test case: a film-cooled turbine vane. Film-cooled turbine blades are very complex turbomachinery geometries because they are composed of a cooling network and many different-shaped film-cooling holes which makes these geometries extremely complex to mesh with structured meshing approaches (like multi-block hexahedral meshing methods). Consequently, metric-based anisotropic mesh adaptation seems quite appropriate for such applications as it has demonstrated that it provides automation of the whole process and it allows significant gains in terms of simulation accuracy by capturing coherent flow features proper to the turbomachinery physics. The film-cooled turbine vane simulation was made possible because the mesh-adaptive solution platform was successfully extended to periodic flows and domains.

In order do to a fair comparison, the same numerical strategy must be used when dealing full-tetrahedral meshes as well as hybrid/structured meshes. For these reason, an appropriate numerical strategy was elaborated for extending the Mixed Finite Element-Finite Volume (MEV) scheme to adapted meshes composed of both triangular and quadrangular elements, called hybrid meshes 23, 22 (also named mixed-element meshes 37, 38). Particularly, in the context of mesh adaptation, a specific metric gradation process is proposed to favor the clustering of structured elements at wall boundaries. The details are given in 40.

One important challenge is the accurate discretization of turbulence. In the context of RANS simulations, we investigated the numerical behavior of the Spalart-Allmaras SA-neg-QCR2000-R one-equation turbulence model for the anisotropic mesh adaptation process. The solutions obtained on the adapted unstructured meshes are compared with those obtained on some "best-practice" structured/hybrid meshes. Same numerical schemes are adopted to perform a fair comparison. Three test cases of the High-Fidelity Prediction Workshop 8 are considered: a two-dimensional subsonic flow past a Joukowski airfoil, a subsonic three-dimensional flow past the wing-body configuration developed for the High-Lift Prediction Workshop and a subsonic three-dimensional flow past an extruded NACA 0012 wing in a tunnel.

We are developing ViZiR 4, a visualization software with pixel exact rendering to address the high-order visualization challenges 28, 9.
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 (without extra subdivision or ray casting) 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.
Unlike other visualization software (ParaView 20, TecPlot 21, Medit 11, Vizir (OpenGL legacy based version) 30, Gmsh 18), 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 visualization 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.

The development of the libHOM library and P1toPk mesh converter continued during my stay at Dassault-Systemes.
The libHOM library handles many basic geometrical and topological operations on arbitrary order elements and is used by P1toPk which is an end-user software that converts a hybrid P1 volume mesh into a curved P2 volume mesh.
Both programs were started during the collaboration between Distene and INRIA and development continued at Dassault-Systemes after they acquired Distene.
P1toPk has been integrated as a module in the common numerical simulation toolbox that is shared across all 3DS's software products.
The work consisted mostly in validation on industrial test cases and the development of dedicated quality criteria for specific solvers like Abaqus (C3D10 quality model).

In order to design the new transport devices answering to the climatic challenge, the needs in High-Fidelity CFD simulation are increasing. High-Fidelity CFD simulation demands: high-fidelity models like RANS and hybrid RANS/LES,high-fidelity approximations with a strong control of the approximation error. This work focuses on the combination of design with anisotropic mesh adaptation. The key components are: unstructured mesh for CFD, use of adjoint state for design, use of mesh adaptation 1, 2.

We are developing new algorithms to solve the adjoint problem that may struggle to converge. Adjoint resolution is a key step in the process of goal-oriented mesh adaptation as the error estimate and the metric computed for the remeshing depends of the adjoint solution. Currently, the method used is an iterative GMRES preconditioned with a symmetric Gauss-Seidel method (SGS). New algorithms have been implemented such as partitioning method or incomplete LU factorization with k levels of fills in (ILU(k)) based on the connectivity of the mesh. Some methods have shown some interest but need to be pushed further especially method that helps recover the parallel implementation of the solver like restricted Scharz additive method or point implicit method.

We performed a deep accuracy analysis of a Mixed Element Volume (MEV) numerical scheme 42, 3, namely the V4 scheme 7, by studying it on the 2D linear advection equation and comparing the results between structured and unstructured triangular meshes of different regularity. The scheme in fact achieves third order accuracy on this equation with cartesian Friedrichs-Keller triangular meshes, and second order accuracy on the 3D Euler equations with anisotropic adapted meshes 6. A solver was specifically implemented for this analysis, in order to compare the results between structured Friedrichs-Keller meshes and unstructured Delaunay ones. After that we considered more irregular unstructured meshes obtained by taking the ones used previously and perturbing them at different levels. These new meshes were then tested in order to compare the accuracy, in particular with another convergence study, obtaining a second order accuracy. An analysis of the local truncation error followed, by means of an algorithm that computes automatically the coefficients of the derivatives in the error terms. These coefficients were computed for all the families of meshes studied, in an attempt to explain the global errors computed through the local ones. The results were a third order of local accuracy for the regular meshes, and a first order for the perturbed ones, in opposition to the second order for the global error. This difference will be investigated with further studies.

The mixing plane method is used typically on turbomachinery in order to model the interaction between a rotor (rotating part) and a stator (static part). Each row (stator, rotor) has its own computation which is coupled with his surroundings rows with the mixing plane. The idea is to average quantities along the pitch direction at the outlet of a row, those averaged values are transferred to the next row. The same process is done at the inlet of a row and transferred at the previous row. Associating mesh adaptation with mixing planes requires that the position and the distribution radial discretization must be automatic and consistent with the current adapted mesh. Therefore, after each mesh adaptation, a specific radial discretization for the mixing plane is built based on the current adapted mesh size. These radial discretizations are different on either side of the mixing plane as the adapted meshes do not match on either side. It allows removing the dependency of the results on the relative position between the rotor and the stator. Thus, RANS simulation can be used, even if the problem is naturally unsteady.

Since COVID crisis the air traffic is getting back to normal with a growing trend. Aeronautical engines' manufacturer did realize this as a critical time for aviation and huge effort should be made to reduce our environmental impact. For instance, The Advisory Council for Aeronautics Research in Europe aka ACARE 2050 objectives set a reduction of 75% in production of CO2 and of 90% of NOx. This objective brings to the design of new and groundbreaking parts and requires more efficient and complex numerical tools. To cope with new challenges an increase in simulation complexity and representativity is needed. It leads industrials like Safran to wonder if current numerical toolchains can still address those issues or should we optimize it. Parts, like propellers and turbines are very complex with a large number of technological effect, taking into account all physical aspect is often not possible since it leads to an exponential growth of the number of degrees of freedom with no convergence or accuracy guarantee. Thus designers simplify the geometry to use their traditional old simulation methods. For this reason, SAFRAN is pushing to develop new numerical toolchains that allows to reach better accuracy and scalability on the key design parameters. The recent studies in this sense (4 is an example of application of remeshing technique with a turbine) highlights how the combination of an anisotropic meshing technique, a robust and accurate solver and the use of a remeshing technology would bring an improvement on physical quantities accuracy and reliability. The need of testing integrated parts, like propellers installed on wide body aircraft or multi-stage turbomachine, shows the interest for industrials of scaling the numerical toolchain to exascale machines in order to have accurate results to real world design problems.

Nowadays, industrials such as Airbus, Safran or ArianeGroup, are facing major technological challenges, which include the design
of the Airbus A350 high-lift configuration, the cooling system of turbomachinery involving
microperforated panels, the Ariane 6 launch vehicle or the consideration of ice accretion on aircraft wings.
All of these challenges imply an intensive use of Computational Fluid Mechanics (CFD) in order to alleviate
design and manufacturing costs and environmental impacts.
However, three technological roadblocks need to be overwhelmed in the coming decades
to treat the previous quoted industrial applications: the ability to handle very complex geometries,
to predict unsteady turbulent flows with a high-fidelity and very quickly,
which necessitates the High-Performance Computing (HPC).
The application proposed in this project is out of reach with the present technology.
Hence, disruptive methods need to be proposed to maximize the socio-economic impact.
The first challenge requires to use the automatic generation of tetrahedra meshes,
which is developed in our team for more than thirty years
12, 35, 19, 10, 13, 14, 15, 16, 17, 26, 25, 24, 39, 36, 27, 29.
Indeed, it is for instance impossible to generate a structured mesh around a real landing gear
and even if it would be possible, it would require an enormous amount of time and resources
compared with our automatic unstructured mesh generator.
The last two challenges deal with the importance of having accurate and fast numerical methods
on unstructured tetrahedra meshes to capture unsteady turbulent three-dimensional flows.
Last but not least, these challenges have to be simultaneously addressed and impose strong
constraints on the whole computational workflow.
Moreover, anisotropic mesh adaptation, developed in our team
33, 31, 32, 34
works like a catalyst since it allows to:

Our simulation suite is made of two main softwares: Feflo.a and Wolf. Feflo.a is our robust anisotropic local remeshing software for three-dimensional volume and parametric surface mesh generation conforming in sizes and orientations to a prescribed input metric field. It is based on a unique cavity-based operator 34, 43. It also includes many components required to generate an initial mesh or generate highly-adapted meshes. It also handles non-manifold geometries and boundary layer mesh generation. Wolf is a mixed Finite Volume Finite Element flow solver for the compressible Euler and Navier Stokes equations with or without moving geometries. It also solves the steady and unsteady adjoint problems. It achieves second-order accuracy in space and up to fourth-order accuracy in time with explicit or implicit schemes. These suite has demonstrated its incredible potential and provided substantial breakthroughs in industrial applications 44, 43, 5.

It is absolutely crucial to keep in mind that the challenge to handle highly anisotropic unstructured meshes arise from the connectivity table needed to represent an unstructured mesh which induces loop indirections and load imbalances during computations and from the anisotropy which invalidates most of the standards geometric algorithms. Very simple standard algorithm such as a cut plane in an anisotropic tetrahedra mesh should be revised to efficiently run on a distributed memory architecture. Furthermore, the GPU algorithm has very little to do with the CPU one which notably increases the development cost to obtain a portable solution.

To continue and improve the capabilities of our suite, we propose a new library GMlib to specifically address the issue of handling anisotropic unstructured meshes on GPU and an industrial test case in agreement with the Safran Tech's roadmap.

Radical changes in aircraft configurations and operations are required to meet the target of climate-neutral 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, data-fusion techniques and smart health-assessment tools enabling the digital transformation of aircraft design, manufacturing and maintenance. NEXTAIR proposes digital enablers covering the whole aircraft life-cycle 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 multi-physics industrial design, maintenance problems and environmental challenges and interest for aircraft and engines manufacturers.

NEXTAIR will increase high-fidelity 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 multi-disciplinary design of aircraft and engine components. Finally, NEXTAIR will extend the usability of machine learning-driven 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 digital-physical scaled demonstrator aircraft and engines and 2 high-Tech SME providing expertise in industrial scientific computing and data intelligence.