We aim at a step change in multiphysics numerical modeling by developing two fundamental enablers:
reducedorder models;
hierarchical Cartesian schemes.
Reducedorder models (ROMs) are simplified mathematical models derived from the full set of PDEs governing the physics of the phenomenon of interest. ROMs can be obtained exploiting first principles or be datadriven. With ROMs one trades accuracy for speed and scalability, and counteracts the curse of dimensionality of traditional highfidelity solvers by significantly reducing the computational complexity. ROMs represent an ideal building block for systems with realtime requirements, like interactive decision support systems that offer the possibility to rapidly explore various alternatives.
Hierarchical Cartesian schemes allow the multiscale solution of PDEs on non bodyfitted meshes with a drastic reduction of the computational setup overhead. These methods are easily parallelizable and they can efficiently be mapped to highperformance computer architectures. They avoid dealing with grid generation, a prohibitive task when the boundaries are moving and the topology is complex and unsteady.
Massive parallelization and rethinking of numerical schemes will allow the use of mathematical models for a broader class of physical problems. For industrial applications, there is an increasing need for rapid and reliable numerical simulators to tackle design and control tasks. To provide a concrete example, in the design process of an aircraft, the flight conditions and manoeuvres, which provide the largest aircraft loads, are not known a priori. Therefore, the aerodynamic and inertial forces are calculated for a large number of conditions to give an estimate of the maximum loads, and hence stresses, that the structure of the detailed aircraft design might experience in service. As a result, the number of simulations required for a realistic design problem could easily be in the order of tens of millions. Even with simplistic models of the aircraft behavior this is an unfeasible number of separate simulations. However, engineering experience is used to identify the most likely critical load conditions, meaning that approximately hundreds of thousands simulations are required for conventional aircraft configurations. Furthermore, these analyses have to be repeated every time that there is an update in the aircraft structure.
Compared to existing approaches for ROMs , our interest will be focused on two axes. On the one hand, we start from the consideration that small, highly nonlinear scales are typically concentrated in limited spatial regions of the full simulation domain. So for example, in the flow past a wing, the highly nonlinear phenomena take place in the proximity of the walls at the scale of a millimeter, for computational domains that are of the order of hundreds of meters. Based on these considerations, we propose in a multiscale model where the large scales are described by farfield models based on ROMs and the small scales are simulated by highfidelity models. The whole point for this approach is to optimally decouple the far field from the near field.
A second characterizing feature of our ROM approach is nonlinear interpolation. We start from the consideration that dynamical models derived from the projection of the PDE model in the reduced space are neither stable to numerical integration nor robust to parameter variation when hard nonlinear multiscale phenomena are considered.
However, thanks to Proper Orthogonal Decomposition (POD) , , we can accurately approximate large solution databases using a lowdimensional base. Recent techniques to investigate the temporal evolution of the POD modes (Koopman modes , , Dynamic Mode Decomposition ) and allow a dynamic discrimination of the role played by each of them. This in turn can be exploited to interpolate between modes in parameter space, thanks to ideas relying on optimal transportation , that we have started developing in the FP7 project FFAST and H2020 AEROGUST.
We intend to conceive schemes that will simplify the numerical approximation of problems involving complex unsteady objects together with multiscale physical phenomena. Rather than using extremely optimized but nonscalable algorithms, we adopt robust alternatives that bypass the difficulties linked to grid generation. Even if the mesh problem can be tackled today thanks to powerful mesh generators, it still represents a severe difficulty, in particular when highly complex unsteady geometries need to be dealt with. Industrial experience and common practice shows that mesh generation accounts for about 20% of overall analysis time, whereas creation of a simulationspecific geometry requires about 60%, and only 20% of overall time is actually devoted to analysis. The methods that we develop bypass the generation of tedious geometrical models by automatic implicit geometry representation and hierarchical Cartesian schemes.
The approach that we plan to develop combines accurate enforcement of unfitted boundary conditions with adaptive octree and overset grids. The core idea is to use an octree/overset mesh for the approximation of the solution fields, while the geometry is captured by level set functions , and boundary conditions are imposed using appropriate interpolation methods , , . This eliminates the need for boundaryconforming meshes that require timeconsuming and errorprone mesh generation procedures, and opens the door for simulation of very complex geometries. In particular, it will be possible to easily import the industrial geometry and to build the associated level set function used for simulation.
Hierarchical octree grids offer several considerable advantages over classical adaptive mesh refinement for bodyfitted meshes, in terms of data management, memory footprint and parallel HPC performance. Typically, when refining unstructured grids, like for example tetrahedral grids, it is necessary to store the whole data tree corresponding to successive subdivisions of the elements and eventually recompute the full connectivity graph. In the linear octree case that we develop, only the tree leaves are stored in a linear array, with a considerable memory advantage. The mapping between the tree leaves and the linear array as well as the connectivity graph is efficiently computed thanks to an appropriate spacefilling curve. Concerning parallelization, linear octrees guarantee a natural load balancing thanks to the linear data structure, whereas classical unstructured meshes require sophisticated (and moreover time consuming) tools to achieve proper load distribution (SCOTCH, METIS etc.). Of course, using unfitted hierarchical meshes requires further development and analysis of methods to handle the refinement at level jumps in a consistent and conservative way, accuracy analysis for new finitevolume or finitedifference schemes, efficient reconstructions at the boundaries to recover appropriate accuracy and robustness. These subjects, that are currently virtually absent at Inria, are among the main scientific challenges of our team.
We apply the methods developed in our team to the domain of wind engineering and seawave converters.
In Figure , 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 .
The background method for fluidstructure interface is the volume penalization method where the level set functions is used to improve the degree of accuracy of the method
and also to follow the object.
The underlined mathematical model is unsteady, and
three dimensional; numerical simulations based on a grid
with
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 (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 to optimize the global twist. Then, we validated the optimal configuration using the fullorder Cartesian model based on the NaSCar solver. Figure (b) shows the flow around the optimized optimized wind turbine rotor.
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 (
Mesh 

number of cells  
1 


N.A. 

2 




3 




4 




Case 

Octree, 1^{st}order scheme 

Octree, 2^{nd}order scheme 

Cartesian 

Experimental estimate 

A new research direction pursued by the team is the mathematical modelling of vascular blood flows in arteries. Together with the startup Nurea (http://
Mathematical and numerical modelling 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 behaviour 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 and subsequently
pursued and extended in , , ,
In Figure , 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
The full reconstruction of a 3D larval zebrafish (5 days post fertilization) was realized using a serialsection electron microscopy data set combined with the technique of levelset and optimal transportation for shape interpolation. From an experimental video of zebrafish escape swimming, the kinematics of the swimming is extracted removing both translational and rotating displacements. Based on this videoextracted body deformation, 3D zebrafish snapshots of the body surface were generated deforming the 3D model according to the midline motion. The escape response of the zebrafish larva has been simulated using the NaSCar solver. The numerical simulation of the hydrodynamic zebrafishlocomotion provides a full range of the energetic performance performed by the larva during an escape response that are used by the MRGM biology lab in Bordeaux for toxicology evaluations. See figures and .
Keywords: 3D  Elasticity  MPI  Compressible multimaterial flows
Functional Description: The code is written in fortran 95 with a MPI parallelization. It solves equations of conservation modeling 3D compressible flows with elastic models as equation of state.
Authors: Alexia De Brauer, Florian Bernard, Yannick Gorsse, Thomas Milcent and Angelo Iollo
Partners: CNRS  Université Bordeaux 1
Contact: Florian Bernard
Kinetic Octree Parallel PolyAtomic
Keywords: C++  3D  MPI
Functional Description: KOPPA is a C++/MPI numerical code solving a large range of rarefied flows from external to internal flows in 1D, 2D or 3D. Different kind of geometries can be treated such as moving geometries coming from CAO files or analytical geometries. The models can be solved on Octree grids with dynamic refinement.
Participant: Florian Bernard
Partners: Université de Bordeaux  INP Bordeaux  CNRS
Contact: Angelo Iollo
URL: https://
NavierStokes Cartesian
Keywords: HPC  Numerical analyse  Fluid mechanics  Langage C  PETSc
Scientific Description: NaSCar can be used to simulate both hydrodynamic biolocomotion as fish like swimming and aerodynamic flows such wake generated by a wind turbine.
Functional Description: This code is devoted to solve 3Dflows in around moving and deformable bodies. The incompressible NavierStokes equations are solved on fixed grids, and the bodies are taken into account thanks to penalization and/or immersed boundary methods. The interface between the fluid and the bodies is tracked with a level set function or in a Lagrangian way. The numerical code is fully second order (time and space). The numerical method is based on projection schemes of ChorinTemam's type. The code is written in C language and use Petsc library for the resolution of large linear systems in parallel.
NaSCar can be used to simulate both hydrodynamic biolocomotion as fish like swimming and aerodynamic flows such wake generated by a wind turbine.
Participant: Michel Bergmann
Partner: Université de Bordeaux
Contact: Michel Bergmann
NavierStokespenalization
Keywords: 3D  Incompressible flows  2D
Functional Description: The software can be used as a black box with the help of a data file if the obstacle is already proposed. For new geometries the user has to define them. It can be used with several boundary conditions (Dirichlet, Neumann, periodic) and for a wide range of Reynolds numbers.
Partner: Université de Bordeaux
Contact: CharlesHenri Bruneau
Reducedorder models are attractive method to decrease significantly the computational cost of the simulations. However, the ability of reducedorder models to accurately approximate solutions containing strong convection, sharp gradients or discontinuities can be challenging. The discontinuous Galerkin domain decomposition (DGDD) reduced model for systems of conservation laws couples at the discrete level subdomains of highfidelity polynomial approximation to regions of lowdimensional resolution as shown in figure .
In this approach, the highdimensional model solves the equations where a given degree of accuracy is required, while the reducedorder model approximates the solution elsewhere. Since the highdimensional model is used in a small part of the domain, the computational cost is significantly reduced. To perform the coupling, we develop a reducedorder model based on Proper Orthogonal Decomposition in the offline stage and on discontinuous Galerkin method in the online stage instead of the standard Galerkin method. In this way, the domain decomposition is applied transparently thought the numerical fluxes. We investigate the prediction of unsteady flows over a NACA 0012 airfoil. The results demonstrate the accuracy of the proposed method and the significant reduction of the computational cost. In the figures ( and ) we show examples of predictions obtained by the loworder model compared to the actual solutions as a function of the Mach number at infintiy and the angle of attack. In the last figure we present the overall spacetime
Starting from February 2019, AMIES granted a oneyear contract engineer in close collaboration with the Nurea startup.
The main objective of the project is to improve the quality and the robustness of automatic segmentation of aortic aneurism. An important part of the work was to decide if patient data needed to be preprocessed or not. To do so a criteron was developed to apply or not a smoothing filter. Several filters were tested and their performance were compared in order to choose the filter the more appropriate to our problem.
Another issue was the bones wrongly taken into the segmentation. A cleaning function was created to deal with it and remove the bones from the segmentation. An usual issue when working with medical images is to deal with the gradient of intensity. Existing tools need to be adapted to take into account these variations within images. This is currently worked out.
The code is implemented in C++ and mostly relies on itk and vtk libraries. An example in figure .
We develop a versatile fully Eulerian method for the simulation of fluidstructure interactions. In the context of a monolithic approach, the whole system is modeled through a single continuum model. The equations are numerically solved using a finitevolume scheme with a compact stencil on AMR enabled quadtree grids where the dynamic refinement is adapted in time to the fluidstructure system.
The geometry is followed using a levelset formulation. In the Eulerian representation, a smooth Heaviside function is defined according to a levelset function on the cartesian mesh to distinguish between fluid and elastic phases. The temporal deformation of the structure is described according to the backward characteristics which are employed to express the Cauchy stress of a twoparameter hyperelastic MooneyRivlin material. This model is particularly adapted to elastomeric materials undergoing large deformations, see figure .
One of the difficulties in the simulation of a fluid flow problem is the representation of the computational domain with a static 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 a consequent high computational cost. 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 (hole cutting) and, consequently, an overlapping zone between two overlapping blocks is defined. Figure shows a Chimera grid in the computational domain
The NavierStokes equations for incompressible flows are going to be approximated through a projection method (ChorinTemam), for this reason we have studied two Finite Volume (FV) solvers, for the pure diffusive equation and the unsteady convectivediffusive equation, on Chimera configurations. The first numerical experiments were conducted on 2D problems. The order of convergence of the error of the mismatch between the exact solution and its FV approximation in
Motivated by recent studies and the locomotion of animal groups for robotics, we investigated the influence of hydrodynamic interactions on the collective propulsion of flapping wings. We studied the horizontal locomotion of an infinite array of flapping wings separated by a constant gap using unsteady nonlinear simulations. Two control parameters were explored: the flapping frequency and the gap between the wings. Results obtained for different gaps at a fixed frequency are shown in Figure . We first observe that for a very large spacing between the wings — greater than 20 times the chord of a wing — the interaction effects are no longer present (Figure (b)) and the average speed of the system tends to the speed of a single wing. For lower gaps, the average speed may become lower or higher than that of a single wing. For certain gaps, one can find two different stable solutions: one at higher propulsion speed and the other lower than a single wing. This phenomenon has already been observed for a fixed spacing and different frequencies of movement. The stability of these solutions is linked to the interaction between the vortex wake generated by the previous wings and the vortex ejected at the leading or trailing edge of the considered wing (Figure (a)). We remark that propulsive efficiency is higher for the collective case both in faster and slower solutions. Understanding the key mechanisms responsible for the stable solutions will provide directions to control strategies aiming to optimize the wing horizontal speed.
As part of the ongoing team effort on ROMs, we work on the development of automatic registration procedures for model reduction. In computer vision and pattern recognition, registration refers to the process of finding a spatial transformation that aligns two datasets; in our work, registration refers to the process of finding a parametric transformation that improves the linear compressibility of a given parametric manifold. For advectiondominated problems, registration is motivated by the inadequacy of linear approximation spaces due to the presence of parameterdependent boundary layers and travelling waves.
In , we proposed and analysed a computational procedure for stationary PDEs and investigated performance for twodimensional model problems. In Figure , we show slices of the parametric solution for three different parameters before (cf. Left) and after (cf. Right) registration: we observe that the registration procedure is able to dramatically reduce the sensitivity of the solution to the parameter value
We aim to extend the approach to a broad class of nonlinear steady and unsteady PDEs: in October 2019, we funded a 16month postdoc to work on the reduction of hyperbolic systems of PDEs. We are also collaborating with EDF (departments PERICLES and LNHE) to extend the approach to the SaintVenant (shallow water) equations.
Despite being relevant in many natural and industrial processes, suspensions of nonspherical particles have been largely underinvestigated compared to the extensive analyses made on the gravitydriven motions of spherical particles. One of the main reasons for this disparity is the difficulty of accurately correcting the shortrange hydrodynamic forces and torques acting on complex particles. These effects, also known as lubrication, are essential to the suspension of the particles and are usually poorly captured by direct numerical simulation of particleladen flows. We have proposed a partitioned VPDEM (Volume Penalization method  Discrete Element Method) solver which estimates the unresolved hydrodynamic forces and torques. Corrections are made locally on the surface of the interacting particles without any assumption on the particle global geometry. This is an extension of our previous work . Numerical validations have been made using ellipsoidal particles immersed in an incompressible NavierStokes flow.
Self organization of groups of several swimmers is of interest in biological applications. One of the main question is to determine if the possible organization comes from an uncontrolled or a controlled swimming behavior. This work has been motivated by the recent studies of Hamid Kellay (LOMA). Hamid Kellay has presented his results during a small workshop we organized earlier this year between members of the Memphis team (modeling and numerical methods), the MRGM (experimental zebrafishes swimming), the LOMA (interaction between selfpropelled particles) and the ONERA (flapping wings).
The collision model developed in the previous section has been developed for concave interactions like spheresphere or in the limit of sphereplane wall. For non concave interactions, we have derived a simple approximation considering locally convexity as being a plane wall. We have performed numerical simulations of the interaction of several self propelled swimmers in a porous enclosure (see figure ). Fishes are organized in small groups that are able to put into motion the enclosure. This behavior is similar to the one observed in the experimental setup of Hamid Kellay.
40kEuro contract for a study on the development of projectionbased reduction strategies for the shallowwater equations, for applications in Hydraulics.
36kEuro contract for the development of a projectionbased reduced model for a thermohydraulicmechanical (THM) system.
We are part of the GDR AMORE on ROMs.
The overarching objective of ARIA (Accurate Roms for Industrial Applications) project is to form an international and intersectoral network of organizations working on a joint research program in numerical modelling, specifically in the fields of model reduction and convergence between data and models. Memphis team is ccordinating this 926KEuro project. 7 industrial partners are involved (VW, Valorem, Optimad, IEFluids, VirtualMech, Nurea, Esteco), 5 EU academic partners (Inria, Université de Seville, Poitecnico di Milano, Politecnico di Torino, SISSA) and 3 universities in the USA: Stanford University, Virginia Tech and University of South Carolina.
Inria@SiliconValley
Associate Team involved in the International Lab:
Title: Multiscale Accurate Reducedorder model Enablers
International Partner (Institution  Laboratory  Researcher):
Stanford (United States)  VNU University of Engineering and Technology  Charbel Farhat
Start year: 2019
See also: https://
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.
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 Européen de Systèmes Automatisés, 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.
Angelo Iollo
June 5th, 2019. Journées scientifiques Inria, Lyon.
https://project.inria.fr/journeesscientifiques2019/francaisprogramme/.
May 2019. CIMPA School, Tunis. Science des données pour l’ingénierie et la technologie.
https://www.cimpa.info/fr/node/6217.
March 2019. Conférencier invité à la conférence « Fluidstructure interaction», Politecnico di Milano, Milano, 18/320/3/2019.
http://www1.mate.polimi.it/ gazzola/fs.html.
Tommaso Taddei
November 2019. MORTECH 2019, Paris.
https://mortech2019.sciencesconf.org/
Angelo Iollo is an expert for the European Union for the program FET OPEN.
Four members of the team are Professors or Assistant Professors at Bordeaux University 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).
20192022. Giulia Sambataro. Bourse ANDRA. Componentbased reduction strategies for THM equations. Advisors: Angelo Iollo, Tommaso Taddei.
20182021. Michele Giuliano Carlino. Bourse Inria. Fluidstructure models on Chimera grids. Advisors: Michel Bergmann, Angelo Iollo.
20182021. Antoine Fondanèche. Bourse UB. Monolithic fluidstructure modeles on parallel hierarchical grids. Advisors: Michel Bergmann, Angelo Iollo.
20172020. Sebastien Riffaud. Convergence between data and numerical models. Advisor: Angelo Iollo.
20172020. Luis Ramos Benetti. Bourse ERC Aeroflex (O. Marquet, ONERA). Monolithic fluidstructure modeles on parallel hierarchical grids. Advisors: Michel Bergmann, Angelo Iollo.
Angeo Iollo: reviewer of 3 PhD theses, president of one PhD jury, member of one PhD jury, in France and abroad.
Michel Bergmann: reviewer of 2 PhD theses.