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.
Mathematical and numerical modelling of physical systems undergoing impacts 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
A new research direction pursued by the team is the mathematical modelling of vascular blood flows in arteries. Together with the startup Nurea (http://
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 

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.
Contact: Florian Bernard
Kinetic Octree Parallel PolyAtomic
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
Contact: Florian Bernard
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
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
We present below results concerning the application of the hybrid FOM/ROM method to a realistic problem in CFD.
The purpose of the study is to investigate the behavior of the flow past a car for several front bumper configurations. We here resort to FreeForm Deformation (FFD) based on two parameters to determine a satisfactory parametrization of all possible configurations, and we consider a steady RANS solver at
Figure (left) shows the domain decomposition; in the blue region we solve the FullOrder model, while in the outer region we rely on a PODGalerkin Reduced Order model.
The partitioning is obtained adaptively using the algorithm described in .
Figure (right) shows the flow prediction error relative to dynamic pressure for the worstcase parameter:
the proposed method leads to
We are interested in the development of numerical models for phenomena involving fluid flows and elastic material deformations. We pursue a monolithic approach, which describes the behavior of each material (gas, liquid or solid) through a system of conservation laws and appropriate constitutive relationships. Our method is designed to handle both highMach and lowMach regimes.
It is wellknown that Godunovtype schemes are inadequate for lowMach problems: first, they introduce an eccessive amount of numerical artificial viscosity; second, they require the enforcement of a CFL stability condition which leads to unpractical time steps. For this reason, we resort to the relaxation method proposed in , to derive a novel discretization scheme which can be applied to problems characterized by a broad range of Mach numbers. As opposed to , we propose in to treat the advective term implicitly.
Figure shows results for a quasi 1D de Laval nozzle problem in water:
the flow is lowMach and almost incompressible. In the present simulation, we impose at the inlet the total pressure
Hamid Kellay (LOMA) performs a physical experiment using a half soap bubble heated at the equator. This device allows to study thermal convection and the movement of large scale structures on the surface of the bubble. The results show strong similarities with atmospheric flows on the earth. In particular large vortical structures on the half bubble and tropical cyclones in the atmosphere have the same dynamics.
Using a stereographic transform we solve NavierStokes equations on the half bubble and get very good agreement with the experiment. In addition we find that the Nusselt and Reynolds numbers varify scaling laws quite close to the scaling law given in the literature for RayleighBénard convection:
Adding the rotation like on the earth we show that the rotation changes the nature of turbulent fluctuations and a new scaling regime is obtained for the temperature field.
Leading team of the regional project "Investigation and Modeling of Suspensions with the LOMA and LOF labs in Bordeaux.
We are part of the GDR AMORE on ROMs.
The team organized a conference in honour of CharlesHenri Bruneau. Several scientific presentations have been given by his many collaborators and friends.
This conference was organized over two halfdays: the afternoon of September 13 and the morning of September 14. The presentations are in a 30minute format.
https://
The team organized a halfday workshop on numerical modelling of swimming on December 12th 2018. Participants: Patrick Babin (MRGM), Michel Bergmann, Afaf Bouharguane, Marie Couliou (ONERA), Hamid Kellay (LOMA), Angelo Iollo, Olivier Marquet (ONERA).
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 was invited as plenary speaker to SIMAI 2018,
https://
Angelo Iollo was invited to Gran Sasso Science Institute for the Intensive Week on Fluids and Waves https://
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).
PhD: Federico Tesser. Parallel solver for the Poisson equation on a hierarchy of superimposed meshes, under a Python framework, University of Bordeaux and Insubria University. 11/09/2018. Advisors: Michel Bergmann, Angelo Iollo.
PhD: Claire Taymans. Solving Incompressible NavierStokes Equations on Octree grids : towards Application to Wind Turbine Blade Modelling, University of Bordeaux. 28/09/2018. Advisors: Michel Bergmann, Angelo Iollo.
PhD: Baptiste Lambert. Modelling and numerical simulation of interactions in particleladen flows, University of Bordeaux. 17/10/2018. Advisors: Michel Bergmann, Lisl Weynans.
PhD: Emanuela Abbate. Numerical methods for the simulation of lowMach phenomena in continuum mechanics, University of Bordeaux and Insubria University. 19/12/2018. Advisors: Angelo Iollo, Gabriella Puppo.
PhD in progress: Michele Giuliano Carlino. Fluidstructure models on Chimera grids. 01/10/2018. Advisors: Michel Bergmann, Angelo Iollo.
PhD in progress: Sebastien Riffaud. Convergence between data and numerical models. Advisor: Angelo Iollo.
PhD in progress: Antoine Fondanèche. Monolithic fluidstructure modeles on parallel hierarchical grids. 01/09/2018. Advisor: Michel Bergmann, Angelo Iollo.
PhD in progress: Luis Ramos Benetti. Monolithic fluidstructure modeles on parallel hierarchical grids. 01/10/2017. Advisor: Michel Bergmann, Angelo Iollo.
PhD in progress: Mathias Braun. Reducedorder modelling for increased resilience of water distribution networks. 01/10/2015. Advisors: Angelo Iollo, Iraj Mortazavi, Olivier Piller.
PhD in progress: Numerical simulation and modeling of zebra fish swimming for the study of human diseases of genetic and toxicological origin. 01/10/2015. Advisors: Afaf Bouharguane, Patrick Babin.
Angelo Iollo has been reviewer of the PhD thesis of Nicola Pozzi Numerical Modeling and Experimental Testing of a Pendulum Wave Energy Converter (PeWEC), Politecnico di Torino, DIMEAS, May 2018.
Tommaso Taddei has partecipated to the PhD thesis of Nicolas Cagniart A few nonlinear approaches in model order reduction, Sorbonne University, LJLL, November 2018.
Afaf Bouharguane has presented her research at the event Unithé ou Café at Inria Bordeaux, November 2018.
Michel Bergmann, "Modéliser et optimiser les énergies renouvelables". Stand for the 10th year anniversary of Inria Bordeaux South West centre, October 13th 2018.