Keywords
 A6. Modeling, simulation and control
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.5. Multiphysics modeling
 A6.2.1. Numerical analysis of PDE and ODE
 A6.3.1. Inverse problems
 A6.3.2. Data assimilation
 A6.3.4. Model reduction
 A6.5.1. Solid mechanics
 A6.5.2. Fluid mechanics
 A9.2. Machine learning
 B2.2.1. Cardiovascular and respiratory diseases
 B4.2. Nuclear Energy Production
 B4.3.2. Hydroenergy
 B4.3.3. Wind energy
 B5.2.3. Aviation
 B5.2.4. Aerospace
 B5.5. Materials
1 Team members, visitors, external collaborators
Research Scientists
 Michel Bergmann [Inria, Researcher, HDR]
 Tommaso Taddei [Inria, Researcher]
Faculty Members
 Angelo Iollo [Team leader, Univ de Bordeaux, Professor, HDR]
 Afaf Bouharguane [Univ de Bordeaux, Associate Professor]
PostDoctoral Fellows
 Antoine Fondaneche [Inria, from Oct 2021]
 Nishant Kumar [Inria, from Oct 2021]
 Gwladys Ravon [Inria]
 Sebastien Riffaud [Inria, from Apr 2021 until Oct 2021]
 Lei Zhang [Inria, until Sep 2021]
PhD Students
 Eki Agouzal [Inria, CIFRE, from Mar 2021]
 Beatrice Battisti [Ecole Polytechnique de Turin]
 Michele Giuliano Carlino [Inria]
 Antoine Fondaneche [Univ de Bordeaux, until Aug 2021]
 Caroline Anna Lise Andrea Le Guern [Inria, from Oct 2021 until Nov 2021]
 Karl Maroun [Univ de Poitiers, from Nov 2021]
 Thomas Philibert [Ecole Polytechnique de Turin]
 Michele Romanelli [ONERA, from Oct 2021]
 Ludovica Saccaro [Inria]
 Giulia Sambataro [ANDRA]
 Alexis Tardieu [Univ de Bordeaux, from Oct 2021]
Interns and Apprentices
 Pascal Engelibert [Univ de Bordeaux, from Jun 2021 until Jul 2021]
Administrative Assistant
 AnneLaure Gautier [Inria]
Visiting Scientists
 Andrea Christine Thomann [ University of Insubria in Como (Italy), from Sep 2021 until Oct 2021]
 Elena Travaglia [Ecole Polytechnique de Turin, Oct 2021]
External Collaborator
 Majdi Azaiez [Institut National Polytechnique de Bordeaux, HDR]
2 Overall objectives
2.1 Multiphysics numerical modeling
2.1.1 Reducedorder models: convergence between PDE models and data
Unprecedented opportunities exist to directly use already collected computational or experimental data to improve and build predictive models that can be used online for the simulation of parametric problems, robust design, and control in science and engineering. In this regard, our goal is to combine mechanistic causal models based on partial differential equations (PDEs) with large data sets to reduce the marginal cost of predictions.
Reducedorder models (ROMs) are our main tool for this purpose. ROMs are parametric mathematical models derived from the full set of PDEs using previously computed solutions. In many applications, the solution space turns out to be lowdimensional, so one can trade a minimal loss of accuracy for speed and scalability. ROMs counteract the curse of dimensionality by significantly reducing computational complexity.
Overall, ROMs have reached a certain level of maturity during the last decade, allowing their implementation in largescale industrial codes, mainly in structural mechanics. Nevertheless, some hard points stand. Parametric problems governed by strong advection fields or sensibly compactsupport solutions such as moving shocks suffer from a limited possibility of dimensional reduction and, at the same time, insufficient generalization of the model (outofsample solutions). The main reason for this is that the solution space is usually approximated by an affine or linear representation, which is intrinsically broad band for such problems.
We have worked on the development of model order reduction (MOR) techniques for nonlinear, advectiondominated problems, with emphasis on projectionbased Galerkin and PetrovGalerkin ROMs. First, we worked on the development of effective sampling strategies to reduce training costs. Second, we developed nonlinear, registrationbased approximation techniques, to overcome limitations of linear approximation methods (e.g., proper orthogonal decomposition, POD) to deal with strong advection fields. Third, we developed hybrid formulations that combine reducedorder and fullorder models to deal with complex flow features and/or complex parameterizations.
2.1.2 Schemes for Hierarchical meshes, multiphysics and asymptotic limits
The schemes we have developed aim at simulating complex multiphysics phenomena through appropriate PDE modeling, automatic implicit geometry representation (level sets), hierarchical Cartesian schemes (quadoctrees), parallel simulations, and accurate treatment of boundaries. Discretization schemes on hierarchical meshes allow multiscale solution of PDEs on nonbodyfitted meshes with a drastic reduction in computational setup overhead. The key idea is to use an octree mesh to approximate the solution fields, while the geometry is captured by level set functions. The boundary conditions are determined by appropriate interpolation methods to achieve sufficient accuracy. This approach eliminates the need for boundary conforming meshes, which require timeconsuming and errorprone mesh generation procedures, and opens the door to easy parallel simulation of very complex geometries.
One of the limitations of this approach is that a mesh with a fixed aspect ratio is not optimal for very anisotropic fields such as boundary layers. For such cases, we explored the idea of using a bodyfitted grid near the immersed obstacles and a hierarchical mesh in the background. Essentially, we use the techniques we have developed to impose boundary conditions on nonbodymatched meshes further from the boundary, where the solution is smoother and more isotropic. Our current investigations build on discontinuous Galerkin (DG) methods / ADER approaches to combine efficient interpolation strategies at the grid interfaces and compact reconstruction of the data at the grid level.
Part of our activity has been dedicated to improve schemes for all Mach number flows in both fluid dynamics and continuum mechanics. Phenomena of interest involve fluid flows and elastic materials whose deformations are investigated within a monolithic Eulerian framework. With this approach any material (gas, liquid or solid) can be described with the same system of conservation equations and a suitable general formulation of the constitutive law.
These schemes are accurate in computing steady state solutions as well as in approximating material wave propagation in various Mach regimes and different materials. We are presently studying methods to overcome the need to solve for auxiliary relaxation variables while preserving the properties of the linearly implicit schemes. To achieve this, we split the stiff relaxation source terms from the fluxes and then reformulate the homogeneous part in an elliptic form.
3 Research program
Coherently with our investigation approach, we will start from applications to identify key methodological problems, study those problems and go back to actual code implementation.
3.1 Numerical models
We aim to further develop automated modelorder reduction (MOR) procedures for largescale systems in computational mechanics — here, automated refers to the ability to complete the analysis with minimal user intervention. First, we wish to combine nonlinear MOR with mesh adaptation to simultaneously learn rapid and reliable ROMs and effective highfidelity discretizations over a range of parameters. Second, we wish to develop componentbased MOR procedures to build interoperable components for steady and unsteady nonlinear PDEs: towards this end, we should develop efficient localized training procedures to build local ROMs for each archetype component, and also domain decomposition techniques to glue together the local models for prediction. We also wish to develop and analyze hybrid approaches that combine and merge firstprinciple models with datafit models, and also fullorder and reducedorder models for prediction of global engineering quantities of interest.
These objectives, hence, do not substantially differ in the long term from the previously but some of the methods that we develop can be complemented by available tools form machine learning, like solution clustering, optimal sampling, classification of the solutions... In this respect, a leap forward in industrial applications that we will pursue is without doubts the possibility of capitalizing on previous experience drawn from already acquired simulations to build nonintrusive models combining nonlinear interpolations and nonlinear regression. New perspectives in this direction are offered by the Chair OneraNouvelle Aquitaine. This Chair (Angelo Iollo and Denis Sipp of Onera are the PIs) was prefunded in late 2021 and is endowed with 6 PhD grants and 2 post docs. This framework partly funded by Onera, the french aerospace lab, and the region Nouvelle Aquitaine will be dedicated to several aspects of reduced modeling and data driven models involving research and industrial partners. One PhD thesis funded by the Chair has already started end 2021 dedicated to data driven walllaw models for turbulent simulations (Michele Romanelli). The goal is to apply deep learning approaches to approximate wall laws for airfoils. In this direction, with the PhD of Thomas Philibert funded by Politecnico di Torino in codirection with our team, we will be investigating datadriven models that intrinsically respect physical constraints for turbulence modeling. Two additional PhD thesis funded by the OneraNouvelle Aquitain Chair are foreseen in 2022 dedicated to distributed nonlinear interpolations and surrogate models (response surface, clustering, active subspaces).
As for approximation schemes for PDEs, with respect to the previous evaluation, emphasis will be put on the representation of the solution in each computational cell by adopting a DG / ADER approach to improve accuracy at the level jumps. This approach will be complemented with a Chimera grid at the boundaries in order to improve accuracy by a body fitted mesh avoiding grid generation complexity for a general, possibly varying, geometrical topology. The thesis of Alexis Tardieu started in October 2021 and funded by the university of Bordeaux will study this approach. Still in this direction we will continue to study asymptotic scheme for multimaterials, towards a unified approach for compressible and incompressible materials.
In parallel, we will continue our exploration of schemes that contour the problem of accuracy and time stepping in the asymptotic regimes such as low and high mach numbers for multimaterial flows, the Graal being an asymptotic preserving scheme that is able to capture phenomena at the time scale of the fast waves and of the material waves with the same accuracy, exclusively choosing the appropriate timescale.
3.2 Applications
For energy applications, we will continue our investigations on wave energy converters and windturbines. Relative to wave energy converters, we are developing multifidelty models coupling the incompressible NavierStoke equations (NSE) around the floater with a Proper Orthogonal (POD) Reduced Order Model elsewhere. In April 2022, a two year post doc, funded by Inria and the Region Nouvelle Aquitaine, will continue in that direction replacing the POD model with an asymptotic one like shallow water equation or Boussinesq model. A similar approach (NSEPOD coupling) is also considering for windturbines. In the two year post doc of Nishant Kumar, founded by the InriaIfpEN program and started in October 2021, we implement a multifidelity approach within the SOWFA framework. In the PhD of Caroline Le Guern, funded by the InriaIfpEN program and started in December 2021, we are studying the fluidstructure interactions of a new generation of large windturbines (250 meter rotor) using the Deeplines software codeveloped by IfpEN. The collaboration with EDF is likely to be continued in the direction of multifidelity models (HFMROM) in the Telemac Mascaret code.
Within the ARIA project, in collaboration with Nurea and the biomechanics lab of the Politecnico di Torino, we will investigate the idea of data augmentation starting from a given aneurysm database. We will construct statistically relevant synthetic aneurysms that can provide both heterogeneity and closeness to reality to test new biomarkers for aneurysm rupture. The thesis of Ludovica Saccaro funded by Inria is dedicated to this subject.
We are proposing a DFGCNRS project (with Wolfgang Schröder, RWTH Aachen University) related the Finite Size Objects Interacting in Turbulent Flows, and more precisely on the development of particleinteraction models for spheroidal particles in turbulent free jets. We are also participating in proposing a European Project around Green Aviation/Airport with focus on reduced order models, gappy data and surrogate models.
We will also increase our interactions with biological and physicists partners. In the ANR DRAGON, we will build a digital twin of a snake swimming based on data obtained by biologists.
The sofware development will be continued. We will pursue the development of the NEOS library: NEOS will be distributed in open source LGPL3.0. The HIWIND software will be rewritten based on NEOS library.
4 Application domains
4.1 Energy conversion
We apply the methods developed in our team to the domain of wind engineering and seawave converters. In Figure 1, we show results of a numerical model for a seawave energy converter. We here rely on a monolithic model to describe the interaction between the rigid floater, air and water; material properties such as densities, viscosities and rigidity vary across the domain. The appropriate boundary conditions are imposed at interfaces that arbitrarily cross the grid using adapted schemes built thanks to geometrical information computed via level set functions 37. The background method for fluidstructure interface is the volume penalization method 28 where the level set functions is used to improve the degree of accuracy of the method 4 and also to follow the object. The underlined mathematical model is unsteady, and three dimensional; numerical simulations based on a grid with $\mathcal{O}\left({10}^{8}\right)$ degrees of freedom are executed in parallel using 512 CPUs.
In the context of the Aerogust (Aeroelastic gust modelling) European project, together with Valorem, we investigated the behavior of wind turbine blades under gust loading. The aim of the project was to optimize the design of wind turbine blades to maximize the power extracted. A meteorological mast (Figure 2(a)) has been installed in March 2017 in Brittany to measure wind onsite: data provided by the mast have been exploited to initialize the mathematical model. Due to the large cost of the fullorder mathematical model, we relied on a simplified model 35 to optimize the global twist. Then, we validated the optimal configuration using the fullorder Cartesian model based on the NaSCar solver. Figure 2(b) shows the flow around the optimized optimized wind turbine rotor.
4.2 Schemes for turbulent flow simulations using Octrees
We have initially developed and tested a 3D firstorder Octree code for unsteady incompressible NavierStokes equations for full windmill simulations with an LES model and wall laws. We have validated this code on Occigen for complex flows at increasing Reynolds numbers. This step implied identifying stable and feasible schemes compatible with the parallel linear Octree structure. The validation has been conducted with respect to the results of a fully Cartesian code (NaSCAR) that we run on Turing (with significantly more degrees of freedom) and with respect to experimental results.
Subsequently, we have developed a secondorder Octree scheme that has been validated on Occigen for a sphere at a moderate Reynolds number ($\mathrm{Re}=500$), see Table 1. Then, for a cylinder at ($\mathrm{Re}=140000$) (Figures 3(a) and 3(b)), close to real applications, we have preliminary validation results for the secondorder scheme with respect to experimental drag coefficient (Table 2). Additional resources will be asked on Occigen to complete the study.
Mesh  $\Delta {x}_{\mathrm{min}}$  number of cells  ${C}_{\mathrm{D}}$ (1storder scheme)  ${C}_{\mathrm{D}}$ (2ndorder scheme) 
1  $0.094$  $0.72\xb7{10}^{5}$  N.A.  $0.526$ 
2  $0.047$  $4.9\xb7{10}^{5}$  $0.595$  $0.522$ 
3  $0.023$  $4.7\xb7{10}^{6}$  $0.546$  $0.492$ 
4  $0.012$  $37.6\xb7{10}^{6}$  $0.555$  $0.496$ 
Case  ${C}_{\mathrm{D}}$ 
Octree, 1storder scheme  $1.007$ 
Octree, 2ndorder scheme  $1.157$ 
Cartesian  $1.188$ 
Experimental estimate 32  $1.237$ 
4.3 Vascular flows
A new research direction pursued by the team is the mathematical modelling of vascular blood flows in arteries. Together with the startup Nurea and the surgeon Eric Ducasse, we aim at developing reliable and automatic procedures for aneurysm segmentation and for the prediction of aneurysm rupture risk. Our approach exploits two sources of information: (i) numerical simulations of blood flows in complex geometries, based on an octree discretization, and (ii) computed tomography angiography (CTA) data. Figure 4 shows the force distribution on the walls of the abdominal aorta in presence of an aneurysm; results are obtained using a parallelized hierarchical Cartesian scheme based on octrees.
4.4 Fluidstructure interactions using Eulerian nonlinear elasticity models
Mathematical and numerical modeling of continuum systems undergoing extreme regimes is challenging due to the presence of large deformations and displacements of the solid part, and due to the strongly nonlinear behavior of the fluid part. At the same time, proper experiments of impact phenomena are particularly dangerous and require expensive facilities, which make them largely impractical. For this reason, there is a growing interest in the development of predictive models for impact phenomena.
In MEMPHIS, we rely on a fully Eulerian approach based on conservation laws, where the different materials are characterized by their specific constitutive laws, to address these tasks. This approach was introduced in 34 and subsequently pursued and extended in 36, 33, 29, 31 and 9. In Figure 5, we show the results of the numerical simulation of the impact of a copper projectile immersed in air over a copper shield. Results are obtained using a fully parallel monolithic Cartesian method, based on a ${4000}^{2}$ fixed Cartesian grid. Simulations are performed on a cluster of 512 processors, and benefits from the isomorphism between grid partitioning and processor topology.
In figure 6, we show the results of a three dimensional simulation of a cardiac pump (LVAD, left ventricule assisted device).
Other examples are given in the sections dedicated to the new results.
5 Social and environmental responsibility
One main focus of the team is the renewable energy. We develop mathematical models and numerical methods to study problems related to renewable energies.
5.1 Impact of research results
We are studying two types of green energy extractors.
The first one is around wave energy converters (WECs). For this application, we are working with the PoliTO (Torino, Italy) to model the ISWEC, and we are also starting to work with a Bordeauxbased startup for another device for extracting the energy from the waves via an InriaTech project and a Nouvelle Aquitaine Regional Project submitted by Memphis in collaboration with the CARDAMOM team.
The second one is around wind energy, and in particular wind turbines. In the past, we have supervised two PhD CIFRE thesis with VALOREMValeol, and are currently working with them in a European RISE ARIA project led by Memphis. We are also starting to work with IFPEN around the aeroelastic modeling of large wind turbines and the study and optimization of turbines farms in the framework of the joint laboratory InriaIFPEN with a thesis funded by IFPEN and a postdoc funded by Inria (which will start in October 2021).
In conjunction with these activities we investigate with ANDRA, the national agency for storage of nuclear waste, reduced models allowing efficient and accurate simulations for deep geological storage planning. This activity is concretized with the PhD thesis of Giulia Sambataro.
6 Highlights of the year
Optimal transport on quadtree mesh
The idea of the optimal mass transportation is to find an optimal mapping which realizes the transfer of a density to another one. This modeling tool finds its application in ROMs, as discussed above, and in fluidstructure interactions problems as dicussed in the applications. In essence, it provides a rational approach to nonlinearly interpolate distributed data. Our research aim at the numerical development of algorithms to solve the L2 MongeKantorovich problems. In particular, in 30 two computational algorithms based on extension of the Newton method have been proposed to solve the optimal mass transfer problem for compactly supported densities. These schemes have been shown to be computationally efficient for data that are relatively close in the sense of the Wasserstein distance, but they seem to be less robust when the data are far away because they also require a continuation approach. Thus, we first fixed this problem by modifying the previous schemes and then we used the numerical tools developed within Memphis (NEOS) to solve the problem on quadtree meshes. The combination of these two developments allows the numerical implementation of extremely fast schemes for solving the L2 MongeKantorovich problem. A publication is in preparation.
7 New results
7.1 FluidFluidStructure interactions
The following results are obtained with the software NaSCar based on our previous works 4, 5, 6, 10. These numerical simulations have been carried out on the cluster CURTA from the mesocentre MCIA, using 384 CPUs over 24 hours. Periodic boundary conditions are imposed on the horizontal boundaries. A special treatment is perfomed on the triple line, defined as being the line separeting the three phases, i.e. warter, air and the structure. A model derived form the Cox's one is used. An example of the snake swimming on the water surface is shown in Figure 7. The generation of the gravity waves are in good agreements with experiments and nature observations.
7.2 Componentbased model order reduction for radioactive waste management
At the end of their cycle, radioactive materials are placed in arrays of cylindrical boreholes (dubbed alveoli) deep underground; due to the large temperatures of the radioactive waste, the thermal flux generated by the alveoli drives a complex timedependent phenomenon which involves the thermal, hydraulic and mechanical (THM) response of the medium. The role of simulations is to predict the longterm system response and ultimately assess the impact of the repository site to the surrounding areas: Figure 8(a) shows a typical system configuration considered for numerical investigations. Due to the complex nature of the equations (a system of five coupled nonlinear timedependent threedimensional equations) and due to the uncertainty in several parameters of the model and on boundary conditions, MOR techniques are important to reduce the computational burden associated with thorough parametric studies.
The PhD project of Giulia Sambataro aims to devise a rapid and reliable componentbased MOR technique for THM systems, for radioactive waste management applications. Domain decomposition methods are required to deal with varying numbers of alveoli, and also to reduce the training costs of the ROM by avoiding fullscale offline simulations. During the first year of her PhD, Giulia has worked on a twodimensional hydromechanical (HM) model problem and has studied the performance of projectionbased hyperreduced MOR strategies. Figure 8(b) shows the convergence of the relative solution error with respect to the size $N$ of the ROM and highlights the impact of hyperreduction, which reduces the "integration domain" to improve efficiency.
7.3 Registration methods for advectiondominated PDEs
A major issue of stateoftheart MOR techniques based on linear approximation spaces is the inability to deal with parameterdependent sharp gradients, which characterize the solutions to advectiondominated problems. To address this issue, we propose a registration technique to align local features in a fixed reference domain. In computer vision and pattern recognition, registration refers to the process of finding a transformation that aligns two datasets; here, registration refers to the process of finding a parametric spatiotemporal transformation that improves the linear compressibility of the solution manifold.
A registration procedure has been proposed in 38, 16 and then further developed in 40, 39. In particular, in 40, we considered the application to onedimensional applications in hydraulics; in an ongoing collaboration with EDF, we aim to extend the approach to twodimensional steady and unsteady problems. Figure 9 shows results for a SaintVenant problem (flow past a bump): Figures 9(a) and 9(b) show the free surface $z$ for two different parameters and two time instants, while Figure 9(c) shows the behavior of the outofsample projection error associated with a snapshotbased POD space with and without registration. We observe that registration is key to improve performance of linear compression strategies such as POD.
In 25, Iollo and Taddei proposed a general (i.e., independent of the underlying PDE) nonlinear interpolation technique based on optimal transportation of Gaussian models of coherent structures of the flow. Given the domain $\Omega $ and the states ${U}_{0},{U}_{1}:\Omega \to \mathbb{R}$, we aim to determine an interpolation $\widehat{U}:[0,1]\times \Omega \to \mathbb{R}$ such that $\widehat{U}(0,\xb7)={U}_{0}$ and $\widehat{U}(1,\xb7)={U}_{1}$. The key features of the approach are (i) a scalar testing function that selects relevant features of the flow; (ii) an explicit mapping procedure that exploits explicit formulas valid for Gaussian distributions; (iii) a nonlinear interpolation dubbed “convex displacement interpolation” to define $\widehat{U}$. The mapping built at step (ii) might not satisfy the bijectivity constraint in $\Omega $: to address this issue, a nonlinear projection procedure over a space of admissible maps based on registration is proposed.
Figure 10 illustrates performance of our procedure for a compressible inviscid flow past a NACA0012 profile at angle of attack 4o for varying freestream Mach number between $Ma=0.77$ and $Ma=0.83$. Figures 10(a) and 10(b) show the fluid density for $Ma=0.77$ and $Ma=0.83$, while Figure 10(c) shows an interpolation for an intermediate Mach number: we observe that the nonlinear interpolation smoothly deforms the shock attached to the airfoil. Figure 10(d) compares performance of the nonlinear interpolation with the linear convex interpolation ${\widehat{U}}^{\mathrm{co}}\left(s\right)=(1s){U}_{0}+s{U}_{1}$: we observe that the proposed nonlinear interpolation is significantly more accurate than linear interpolation, for the same amount of highfidelity information.
7.4 Numerical modeling of a real zebrafish swimming: an effort test
Zebrafish is used by biologists as an animal model to study the effects of neurotoxicants and drugs on locomotion and develop pharmacological treatments. Very few fish swimming simulations have been derived from real body deformations to investigate the complex and stereotyped escape response of zebrafish and support animal experimentation. An experimentdriven numerical approach has been developed to model the body deformations from experimental imaging and perform threedimensional (3D) numerical simulations to compute the actual energetic performances. To this end, a novel 3D zebrafish shape described by $300\times 180$ Lagrangian markers, was reconstructed based on 1,600 transverse slices and deformed according to experimental data (see figure 11). As a first application, three escape locomotion were recorded and simulated across six highviscosity fluids in the range of $0.83$ mPa.s to 15 mPa.s. In addition to kinematic data such as traveled distance, velocity, and bending amplitude, the expended energy and cost of transport were computed based on the power output. Eventually, fictitious simulations were performed by combining body deformations and fluid viscosity, especially for challenging the experimental escape motions. Such simulations have revealed energetic expenditure could be emphasized by increasing fluid viscosity. Therefore, those preliminary results provided new insights for the implementation of an effort test, involving zebrafish experiments, viscous fluids, and energetic performances from numerical simulations. The main goal of effort tests designed by biologists is to highlight zebrafish swimming performances altered by genetics or chemical compounds, by computing the power output expended in higherviscosity fluids.
7.5 Fluidstructure interactions on AMR enabled quadree grids
A versatile fully Eulerian method has been developed for the simulation of fluidstructure interaction problems in two dimensions, involving stiff hyperelastic materials 19. The unified single continuum model is solved in a monolithic way using a quadtreebased Finite Volume scheme, built on very compact discretizations. In the context of fictitious domain methods, the geometry of a structure is captured through a levelset formalism, which enables to define a diffuse fluidstructure interface.
The numerical method has been validated with respect to the literature and the benefits obtained in terms of computational costs through the use of dynamic adaptive meshes has been highlighted. The low impact of coarsening on the structure deformation has been emphasized and the results suggest that the numerical method offers a valuable compromise between accuracy and feasibility of the simulation. As depicted in Figure 12, the simulation of a twodimensional axisymmetric flow in a cardiac assist device (LVAD geometry) has finally been proposed as a biomedical application. One paper is submitted.
7.6 An ADER approach for the incompressible NavierStokes model on Overset Grids
One of the difficulties in the simulation of a fluid flow problem is the representation of the computational domain with a unique block mesh. As a matter of fact, not only the geometry could be particularly complex in itself, but it could change during the simulation and this necessary involves an in itinere geometrical adaptation of the mesh, with consequent high computational costs. One of the ways to overcome this problem is to use multiple overlapping mesh blocks that together define a Chimera or overset grid. Once the different mesh blocks are generated, they are properly composed by the creation of holes and, consequently, an overlapping zone between two overlapping blocks is defined. The overlapping zone is necessary for the the communication and data transfer from one mesh to another through an appropriate definition of local stencils of cells, both within and at the border of the individual blocks. In our simulations, the Chimera grid is composed of a background and a foreground mesh (see Figure 13). In particular, the foreground mesh can move and deform. The NavierStokes equations for incompressible fluids are solved through a projection method (ChorinTemam) using second order schemes 20, 21. Solution at two different time instances of an advectiondiffusion PDE over an evolving nontrivial domain discretized through an overset grid are preesnted in Figure 13.
7.7 Projectionbased reducedorder models for unsteady compressible flows
An analysis of calibration for reducedorder models (ROMs) is done in 17. The Galerkin and leastsquares PetrovGalerkin (LSPG) methods are tested on compressible flows involving a disparity of temporal scales. A novel calibration strategy is proposed for the LSPG method and two test cases are analyzed. The first consists of a subsonic airfoil flow where boundary layer instabilities are responsible for trailingedge noise generation and the second comprises a supersonic airfoil flow with a transient period where a detached shock wave propagates upstream at the same time that shockvortex interaction occurs at the trailing edge. Results show that calibration produces stable and longtime accurate Galerkin and LSPG ROMs for both cases investigated. The impact of hyperreduction is tested on LSPG models via an accelerated greedy missing point estimation (MPE) algorithm. The location for hyperreduction points is depicted in Figure 14. For the first case investigated, LSPG solutions obtained with hyperreduction show good comparison with those obtained by the full order model. However, for the supersonic case the transient features of the flow need to be properly captured by the sampled points of the accelerated greedy MPE method. Otherwise, the dynamics of the moving shock wave are not fully recovered. The impact of different timemarching schemes is also assessed and, differently than reported in literature, Galerkin models are shown to be more accurate than those computed by LSPG when the nonconservative form of the NavierStokes equations is solved. For the supersonic case, the Galerkin and LSPG models (without hyperreduction) capture the overall dynamics of the detached and oblique shock waves along the airfoil. However, when shockvortex interaction occurs at the trailingedge, the Galerkin ROM is able to capture the highfrequency fluctuations from vortex shedding while the LSPG presents a more dissipative solution, not being able to recover the flow dynamics.
7.8 Aortic aneurysms: automatic segmentation and registration
We developed a new artificial neural network to automatically segment aortic aneurysm. The main idea with this approach was to consider each pixel of the image individually and to see if a model could learn how to categorize it as lumen only from its own intensity and the intensity of its 26 neighbors. We tested different types of input (values, means, variances...) and architecture: a sequential model was retained. For the input, each sample is a vector of 27 intensity values. Only pixels whose intensity is between 100 and 700 are kept for training and prediction.
The second axis of development concerned registration. When a patient have several scans taken at different times, the segmentations are not in the same frame so any comparison would be complicated. The objective was to bring the second segmentation in the frame of the first one. We tested different pointsbased approaches: register the centerline of the segmentation or the geometry; consider only the lumen or the entire aneurysm. The best results were obtained with the surface of the aneurysm and the iterative closest point algorithm. Once the registration is performed (Figure 15) we can visualize how the aneurysm has evolved.
8 Bilateral contracts and grants with industry
8.1 Bilateral Contracts with Industry
EDF ROM HYDRAU dates: 15/08/2020 au 15/12/2021
9 Partnerships and cooperations
9.1 International initiatives
9.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program
MARE

Title:
Multiscale Accurate Reducedorder model Enablers

Duration:
2019 > 2022

Coordinator:
Angelo Iollo

Partners:
VNU University of Engineering and Technology, Stanford (United States)

Inria contact:
Angelo Iollo

Summary:
Reducedorder models (ROMs) are simplified mathematical models derived from the full set of partial differential equations governing the physics of the phenomenon of interest. We focus on ROMs that are datadriven as they are based on relevant solution data previously obtained. In particular we will focus on multiscale adaptive models where the large scales are governed by a PDE and the small scales are described by data driven models. To do that we will leverage on tools from data geometry, numerical PDEs and machine learning.
9.1.2 Inria International Partners
Declared Inria International Partners Politecnico di Torino (Giovanni Bracco, Francesco Larocca), with two cotutelle PhDs: Beatrice BATTISTI (Sujet : Multifidelity Multiscale numerical modelling og wave energy converters farms)
Thomas PHILIBERT (Sujet : Datadriven models for aerospace propulsion)
Informal International Partners La Sapienza Universita' di Roma (Gabriella Puppo, Dipartimento di Matematica) Università di Pisa (UNIPI) (Simone Camarri, Department of Civil and Industrial Engineering)
9.2 International research visitors
9.2.1 Visits of international scientists
 Andrea Christine Thomann University of Insubria in Como (Italy), from Sep 2021 until Oct 2021.
 Elena Travaglia Ecole Polytechnique de Turin, Oct 2021.
9.2.2 Visits to international teams
Michel Bergmann: OPTIMAD, Turin (ARIA) : 16/11/202120/11/2021.
9.3 European initiatives
9.3.1 FP7 & H2020 projects
ARIA

Title:
Accurate Roms for Industrial Applications

Duration:
2019  2023

Partners:
 STANFORD: Aeronautics & Astronautics Department (United States)
 ESTECO SpA: Department of Research and Development (Italy)
 Politecnico di Milano: Laboratory for Modeling and Scientific Computing (MOX) (Italy)
 Politecnico di Torino: Department of Science and Mathematics (Italy)
 Scuola Internazionale Superiore di Studi Avanzati di Trieste: Mathematics Area/mathLab (Italy)
 Universidad de Sevilla: Instituto de Matemáticas de la Universidad de Sevilla (IMUS) (Spain)
 University of South Carolina: Department of Mathematics (United States)
 VALOREM SAS: R & D Department (France)
 Virginia Tech  Department of Mathematics (United States)
 Virtualmechanics, S.L. (Spain)
 Optimad: R& D Department (Italy)
 Volkswagen AG: Group Research, Vehicle Technology (Germany)

Inria contact:
contact@risearia.eu

Summary:
The project Accurate Roms for Industrial Applications aims at developing an array of mathematical methods for constructing predictive reducedorder models (ROMs) with guaranteed accuracy, robustness, reliability and efficiency for applications involving complex physical phenomena. New approaches to this challenge are proposed here with a focus on the Euler and Navier–Stokes equations of fluid flow, two of the most challenging continuum models with an extraordinary rich range of industrial applications. The mathematical modeling and solution of the Euler and NavierStokes equations is sometimes cited as the greatest challenge in continuum modeling of physical phenomena. This topic is selected as our principal focus because of its intrinsic importance, but also because the mathematical methods developed in addressing this very challenging task may well have an impact on other fields of knowledge. We plan to tackle these challenging objectives in this staff exchange program by combining the unique expertise of our extended research team whose members have made significant progress in ROM research during the past decade. This academic expertise is crossfertilized by the exchange with knowledge intensive SMEs ans start up and well established industrial partners that will benefit from the scientific and technological results of the team and will challenge the solutions found with applications in real world problems.
9.3.2 Collaborations with Major European Organizations
EDF ROM HYDRAU : dates: 15/08/2020 au 15/12/2021.
9.4 National initiatives
 ANR (national agency for research funding) DRAGON2. Partners : CNRS/Université de Poitiers/Inria. 27 k€+ 1 PhD. The goal is study the aquatic swimming a several snakes using biomimetism and bioinspiration. In this project, we have experimental data for snake swimming, and we are building a numerical twin to compute integral quantities like the efficiency. Reinforcement learning is also considered to optimize the snake swimming.
9.5 Regional initiatives
 Contrat Région Nouvelle Aquitaine MODEM " MODélisation multiéchelles et multifidélités pour l'extraction d'Energies Marines". 48k€. Description: the goal of this project is to develop a multifidelity approach for the modeling and the optimization of Wave Energey Converters (WECs). We compute the fluid structure interaction problem in a close neighbourhood of the WEC using accurate but expensive non linear NavierStokes equations. The far field is computed using using a simpler linear or weakly non linear reduced order model, like the shallow water or Boussinesq models.
10 Dissemination
10.1 Promoting scientific activities
Angelo Iollo: 2019. Nominated at the council of "Département de Sciences du Numérique et Ingénierie" of the University of Bordeaux.
10.1.1 Scientific events: organisation
Tommaso TADDEI has coorganized CEMRACS 2021 on data assimilation and model reduction in highdimensional problems.
Michel Bergmann & Tommaso Taddei. Organisation of the minisymposium at the SIAM CSE conference, 2021 (with K. Smetana).
Reviewer  Reviewing Activities
The members of the team were reviewers for: Journal of Computational Physics, International Journal of CFD, Journal of Nonlinear Analysis B, ASME Journal of Computational and Nonlinear Dynamics, Journal of Fluid Mechanics, Acta Mechanica, AIAA Journal, International Journal Numerical Methods in Fluids, Computers & Fluids, Journal of Engineering Mathematics, European Journal of Mechanics / B Fluids, Journal Europeen de Systemes Automatises, Applied Mathematics and Computation. Nuclear Science and Engineering, Computer Methods in Applied Mechanics and Engineering, Journal of Theoretical Biology, Computational Optimization and Applications, Applied science, Meccanica, SIAM journal on scientific computing, SIAM journal on uncertainty quantification, Advances in Computational Mathematics.
10.1.2 Journal
Michel Bergmann is Topic Editor in FLUIDS.
Michel Bergmann is Guest Editor in Scientific Computing in Fluids, FLUIDS (2021)
Michel Bergmann is Guest Editor in DataDriven Modeling and Optimization in Fluid Dynamics: From PhysicsBased to Machine Learning Approaches, Frontiers in Physics (20212022)
10.1.3 Invited talks
Angelo Iollo is invited in international conferences or workshops:
 November 89, 2021. Année de la mécanique. Séminaire à IHP: Mathématiques et Mécanique. Parameterized flows: examples of convergence between data and computational science. .
 June, 17th, 2021. Euromech Conference on Machine learning methods for prediction and control of turbulent flows. “Parameterized flows: examples of convergence between data and computational science”.
10.2 Teaching  Supervision  Juries
10.2.1 Teaching
Two members of the team are Professor (Angelo Iollo) or Assistant Professor (Afaf Bouharguane) at Université de Bordeaux and have teaching duties, which consist in courses and practical exercises in numerical analysis and scientific computing. Michel Bergmann (CR) also teaches around 64 hours per year (practical exercises in programming for scientific computing). Tommaso Taddei (CR) also teaches around 50 hours per year (practical exercises in numerical analysis and scientific computing).
10.2.2 Supervision
 20182021. Antoine Fondanèche. Bourse Université de Bordeaux. Interaction fluidestructure dans les endoprothèses et autres dispositifs vasculaires actifs. Advisors: Michel Bergmann, Angelo Iollo.
 20212024. Eki AGOUZAL Cifre. (Industrial thesis), EDF. Estimation de l'état mécanique d'enceintes de confinement par assimilation de données provenant de la maquette Vercors. Advisors: Michel Bergmann, Tommaso Taddei.
 20202023. Beatrice BATTISTI, Politecnico di Torino. Multifidelity multiscale numerical modelling of wave energy converters farms. Advisors: Michel Bergmann, Giovanni Bracco.
 20182021. Michele Giuliano CARLINO, Inria. Fluidstructure Interactions models on Chimera grids. Advisors: Michel Bergmann, Angelo Iollo.
 Caroline LE GUERN, Ifpen. Modelisation de l'aeroelasticité en grandes transformations par un couplage partitioné: application aux grandes eoliennes. Advisor: Michel Bergmann.
 20212024. Karl MAROUN, Université de Poitiers (ANR Dragon2). Biomimetism and bioinspiration of a snakes's anguilliform swimming: numerical twin and machine learning. Advisors: Michel Bergmann, Philippe Traoré.
 20202023. Thomas PHILIBERT, Politecnico di Torino. Convergence between models and data for the simulation of turbulent flows in aeronautical propulsion. Advisors: Angelo Iollo, Francesco Larocca.
 20212024. Michele ROMANELLI, Chaire ONERA. Lois de Paroi à Apprentissage Profond pour Simulations Aérodynamiques. Advisors: Héloïse Beaugendre, Michel Bergmann.
 20202023. Ludovica SACCARO, Inria. Identification of mechanical biomarkers for assessing the risk of rupture of abdominal aortic aneurysms. Advisor: Angelo Iollo.
 20212024. Alexis TARDIEU, Université de Bordeaux. Approches Galerkin Discontinues spatiotemporelles sur des mailles hiérarchiques pour les équations de NavierStokes incompressibles. Advisors: Angelo Iollo, Afaf Bouharguane.
 20192022. Giulia SAMBATARO, Andra. Componentbased reduction strategies for THM equations. Advisors: Angelo Iollo, Tommaso Taddei.
10.2.3 Juries
Michel Bergmann: membre of the Phd jury of Luca Berti « méthodes numériques et optimisation pour la micronatation», Université de Strasbourg, 13 décembre 2021.
Angelo Iollo:
Rapporteur de la thèse de Matteo Zancanaro, SISSA département de mathématiques, «Model order reduction for compressible turbulent flows: hybrid approaches in physics and geometry parametrization», september 24th, 2021. Trieste.
Rapporteur de la thèse en mécanique de Nan Deng, “Deep meanfield modelling for successive bifurcations exemplified for the fluidic pinball”, 27 septembre 2021. ENSTA Paris.
Président du jury de thèse mention mathématiques de Stefano Pezzano, « Méthode de Galerkin Discontinue Isogéométrique avec domaines dépendants du temps », 13 septembre 2021, Inria Sophia Antipolis, Université de Nice.
10.3 Popularization
In September 2021, Tommaso TADDEI has presented his research at the event Unithé ou Café at Inria Bordeaux.
11 Scientific production
11.1 Major publications
 1 articleAn allspeed relaxation scheme for gases and compressible materials.Journal of Computational Physics3512017, 124
 2 articleFluidsolid Floquet stability analysis of selfpropelled heaving foils.Journal of Fluid Mechanics9102021, A28
 3 articleEnablers for robust POD models.Journal of Computational Physics22822009, 516538
 4 articleAn accurate cartesian method for incompressible flows with moving boundaries.Communications in Computational Physics1552014, 12661290
 5 articleBioinspired swimming simulations.Journal of Computational Physics3232016, 310  321
 6 articleModeling and simulation of fishlike swimming.Journal of Computational Physics23022011, 329  348
 7 articleAccurate Asymptotic Preserving Boundary Conditions for Kinetic Equations on Cartesian Grids.Journal of Scientific Computing2015, 34
 8 articleNumerical solution of the MongeKantorovich problem by density liftup continuation.ESAIM: Mathematical Modelling and Numerical Analysis4961577November 2015
 9 articleA Cartesian Scheme for Compressible Multimaterial Models in 3D.Journal of Computational Physics3132016, 121143
 10 articleEnablers for highorder level set methods in fluid mechanics.International Journal for Numerical Methods in Fluids79December 2015, 654675
11.2 Publications of the year
International journals
 11 articleFluid–solid Floquet stability analysis of selfpropelled heaving foils.Journal of Fluid Mechanics9102021, A28
 12 articleA Cartesian Method with SecondOrder Pressure Resolution for Incompressible Flows with Large Density Ratios.Fluids611November 2021, 402
 13 articleThe DGDD Method for ReducedOrder Modeling of Conservation Laws.Journal of Computational Physics4372021
 14 articleA discretizethenmap approach for the treatment of parameterized geometries in model order reduction.Computer Methods in Applied Mechanics and EngineeringOctober 2021
 15 articleRegistrationbased model reduction in complex twodimensional geometries.Journal of Scientific ComputingAugust 2021
 16 articleSpacetime registrationbased model reduction of parameterized onedimensional hyperbolic PDEs.ESAIM: Mathematical Modelling and Numerical Analysis551January 2021, 99130
 17 articleCalibration of projectionbased reducedorder models for unsteady compressible flows.Journal of Computational Physics433May 2021, 110196
Doctoral dissertations and habilitation theses
 18 thesisADER scheme on Overset Grids with Compact Transmission and Hyperreduction : Application to Incompressible NavierStokes Equations.Université de BordeauxDecember 2021
 19 thesisFluidstructure interaction in an active vascular device.Université de BordeauxSeptember 2021
Reports & preprints
 20 miscSecond order ADER scheme for advectiondiffusion on moving overset grids with a compact transmission condition.October 2021
 21 reportADER scheme for incompressible NavierStokes equations on Overset grids with a compact transmission condition.RR9414Inria & Labri, Univ. Bordeaux; OptimadJune 2021, 32
 22 miscAn Eulerian finitevolume approach of fluidstructure interaction problems on quadtree meshes.December 2021
 23 miscRegistrationbased model reduction of parameterized twodimensional conservation laws.November 2021
 24 miscA projectionbased model reduction method for nonlinear mechanics with internal variables: application to thermohydromechanical systems.November 2021
 25 miscMapping of coherent structures in parameterized flows by learning optimal transportation with Gaussian models.November 2021
 26 miscInferring characteristics of bacterial swimming in biofilm matrix from timelapse confocal laser scanning microscopy.January 2022
 27 miscAutomatic branch detection of the arterial system from abdominal aortic segmentation.January 2022
11.3 Cited publications
 28 articleA penalization method to take into account obstacles in a incompressible flow.Numerische Mathematik8141999, 497520
 29 articleExact and approximate solutions of Riemann problems in nonlinear elasticity.Journal of Computational Physics228182009, 70467068
 30 articleNumerical solution of the MongeKantorovich problem by density liftup continuation.ESAIM: M2AN4962015, 15771592
 31 articleA Cartesian scheme for compressible multimaterial models in 3D.Journal of Computational Physics3132016, 121143URL: http://www.sciencedirect.com/science/article/pii/S0021999116000966
 32 articleAn experimental study of entrainment and transport in the turbulent near wake of a circular cylinder.Journal of fluid mechanics1361983, 321374
 33 articleModelling wave dynamics of compressible elastic materials.Journal of Computational Physics22752008, 29412969
 34 bookElements of continuum mechanics.Nauka Moscow1978
 35 phdthesisConstruction d'une chaîne d'outils numériques pour la conception aérodynamique de pales d'éoliennes.Université de Bordeaux2014
 36 articleA Conservative ThreeDimensional Eulerian Method for Coupled SolidFluid Shock Capturing.Journal of Computational Physics18312002, 2682
 37 bookLevel Set Methods and Fast Marching Methods.Cambridge University Press, Cambridge, UK1999
 38 articleA registration method for model order reduction: data compression and geometry reduction.SIAM Journal on Scientific Computing4222020, A997A1027
 39 articleRegistrationbased model reduction in complex twodimensional geometries.submitted to Journal of Scientific Computing2021
 40 articleSpacetime registrationbased model reduction of parameterized onedimensional hyperbolic PDEs.ESAIM: Mathematical Modelling and Numerical Analysis (accepted)2020