2.1 Multi-physics numerical modeling

We aim at a step change in multi-physics numerical modeling by developing two fundamental enablers:

• reduced-order models;
• hierarchical Cartesian schemes.

Reduced-order 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 data-driven. With ROMs one trades accuracy for speed and scalability, and counteracts the curse of dimensionality of traditional high-fidelity solvers by significantly reducing the computational complexity. ROMs represent an ideal building block for systems with real-time requirements, like interactive decision support systems that offer the possibility to rapidly explore various alternatives.

Hierarchical Cartesian schemes allow the multi-scale solution of PDEs on non body-fitted meshes with a drastic reduction of the computational setup overhead. These methods are easily parallelizable and they can efficiently be mapped to high-performance computer architectures. They avoid dealing with grid generation, a prohibitive task when the boundaries are moving and the topology is complex and unsteady.

3.1 Reduced-order models

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  33, 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 non-linear 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 29 a multi-scale model where the large scales are described by far-field models based on ROMs and the small scales are simulated by high-fidelity 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 non-linear 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 non-linear multi-scale phenomena are considered.

However, thanks to Proper Orthogonal Decomposition (POD) 39, 45, 28 and 3 we can accurately approximate large solution databases using a low-dimensional base. Recent techniques to investigate the temporal evolution of the POD modes (Koopman modes  40, 26, Dynamic Mode Decomposition  43) 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  50, 30 that we have started developing in the FP7 project FFAST and H2020 AEROGUST.

3.2 Hierarchical Cartesian schemes

We intend to conceive schemes that will simplify the numerical approximation of problems involving complex unsteady objects together with multi-scale physical phenomena. Rather than using extremely optimized but non-scalable 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 simulation-specific 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  44, 38 and boundary conditions are imposed using appropriate interpolation methods  25, 49, 42. This eliminates the need for boundary-conforming meshes that require time-consuming and error-prone 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 body-fitted 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 space-filling 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 finite-volume or finite-difference 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.

4.1 Energy conversion

We apply the methods developed in our team to the domain of wind engineering and sea-wave converters. In Figure 1, we show results of a numerical model for a sea-wave 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 44. The background method for fluid-structure interface is the volume penalization method 25 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 $𝒪\left({10}^8\right)$ 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 on-site: data provided by the mast have been exploited to initialize the mathematical model. Due to the large cost of the full-order mathematical model, we relied on a simplified model 36 to optimize the global twist. Then, we validated the optimal configuration using the full-order 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 first-order Octree code for unsteady incompressible Navier-Stokes 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 second-order 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 second-order scheme with respect to experimental drag coefficient (Table 2). Additional resources will be asked on Occigen to complete the study.

  

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 start-up Nurea (http://nurea-soft.com/) 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 Fluid-structure interactions using Eulerian non-linear 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 non-linear 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 35 and subsequently pursued and extended in 41, 34, 27, 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.

5.1 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 Bordeaux-based start-up for another device for extracting the energy from the waves via an Inria-Tech 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 VALOREM-Valeol, 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 Inria-IFPEN with a thesis funded by IFPEN and a post-doc 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.

7.1 New software

8.1 Fluid-Fluid-Structure interactions

The following results are obtained with the software NaSCar based on our previous works 4, 5, 6, 10. These numerical simulations were performed on the cluster CURTA from the mesocentre MCIA, using 384 CPUs over 48 hours. Periodic boundary conditions are used 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 bodies. A model derived form the Cox's model is used. The first example is the sea cruise of a boat. The sea state is artificially generated, but we can see that the waves generated by the boat are realistic (see figure 7).

8.2 Component-based 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 time-dependent phenomenon which involves the thermal, hydraulic and mechanical (THM) response of the medium. The role of simulations is to predict the long-term system response and ultimately assess the impact of the repository site to the surrounding areas: Figure 9(a) shows a typical system configuration considered for numerical investigations. Due to the complex nature of the equations (a system of five coupled nonlinear time-dependent three-dimensional 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 component-based 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 full-scale offline simulations. During the first year of her PhD, Giulia has worked on a two-dimensional hydro-mechanical (HM) model problem and has studied the performance of projection-based hyper-reduced MOR strategies. Figure 9(b) shows the convergence of the relative solution error with respect to the size $N$ of the ROM and highlights the impact of hyper-reduction, which reduces the "integration domain" to improve efficiency.

8.3 Registration methods for advection-dominated PDEs

A major issue of state-of-the-art MOR techniques based on linear approximation spaces is the inability to deal with parameter-dependent sharp gradients, which characterize the solutions to advection-dominated 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 spatio-temporal transformation that improves the linear compressibility of the solution manifold.

A registration procedure has been proposed in 46, 18 and then further developed in 48, 47. In particular, in 48, we considered the application to one-dimensional applications in hydraulics; in an ongoing collaboration with EDF, we aim to extend the approach to two-dimensional steady and unsteady problems. Figure 10 shows results for a Saint-Venant problem (flow past a bump): Figures 10(a) and 10(b) show the free surface $z$ for two different parameters and two time instants, while Figure 10(c) shows the behavior of the out-of-sample projection error associated with a snapshot-based POD space with and without registration. We observe that registration is key to improve performance of linear compression strategies such as POD.

$t=1.5$

$t=3$

$t=3$

8.4 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 experiment-driven numerical approach has been developed to model the body deformations from experimental imaging and perform three-dimensional (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 high-viscosity 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 higher-viscosity fluids.

Top and side views

3D reconstruction

8.5 Fluid-structure interactions on AMR enabled quadree grids

A versatile fully Eulerian method has been developed for the simulation of fluid-structure interaction problems in two dimensions, involving stiff hyper-elastic materials. The unified single continuum model is solved in a monolithic way using a quadtree-based Finite Volume scheme, built on very compact discretizations. In the context of fictitious domain methods, the geometry of a structure is captured through a level-set formalism, which enables to define a diffuse fluid-structure interface. The numerical model has been validated with respect to the literature and an example has been presented in order to compare the computational costs of simulations performed on uniform and dynamic adaptive grids. From the 1D asymptotic analysis of the discretization scheme, recent developments have led to a stabilization of the Finite Volume method, which is needed to take into account very stiff materials. The simulation of a two dimensional axisymmetric flow in a complex cardiac pump geometry is proposed as a biomedical application (see figure 12).

8.6 An ADER approach to advection diffusion equations 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 Navier-Stokes equations for incompressible fluids are solved through a projection method (Chorin-Temam), thus we propose in the sequel a finite volume (FV) method for the steady diffusion equation and the unsteady advection-diffusion (AD) equation on a Chimera configuration. Both methods are second order accurate  21.

8.7 Modeling and numerical simulation of ellipsoidal particle-laden flows and self propelled swimmers in a porous enclosure

Despite being relevant in many natural and industrial processes, suspensions of non-spherical particles have been largely under-investigated compared to the extensive analyses made on the gravity-driven motions of spherical particles. One of the main reasons for this disparity is the difficulty of accurately correcting the short-range 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 particle-laden flows. We have proposed a partitioned VP-DEM (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 3717. Numerical validations have been made using ellipsoidal particles immersed in an incompressible Navier-Stokes 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).

The collision model developed in the previous section has been developed for concave interactions like sphere-sphere or in the limit of sphere-plane wall. For non concave interactions, we have derived a simple approximation considering locally convexity as being a plane wall. Recent 3D results are obtained for a porous and impermeable enclosure (see Fig. 14). The numerical results are similar to those obtained in the experimental ones of Hamid Kellay's group.

8.8 Projection-based reduced-order models for unsteady compressible flows

An analysis of calibration for reduced-order models (ROMs) is done in 19. The Galerkin and least-squares Petrov-Galerkin (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 trailing-edge 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 shock-vortex interaction occurs at the trailing edge. Results show that calibration produces stable and long-time accurate Galerkin and LSPG ROMs for both cases investigated. The impact of hyper-reduction is tested on LSPG models via an accelerated greedy missing point estimation (MPE) algorithm. The location for hyper-reduction points is depicted in Figure 15. For the first case investigated, LSPG solutions obtained with hyper-reduction 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 time-marching 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 non-conservative form of the Navier-Stokes equations is solved. For the supersonic case, the Galerkin and LSPG models (without hyper-reduction) capture the overall dynamics of the detached and oblique shock waves along the airfoil. However, when shock-vortex interaction occurs at the trailing-edge, the Galerkin ROM is able to capture the high-frequency fluctuations from vortex shedding while the LSPG presents a more dissipative solution, not being able to recover the flow dynamics.

8.9 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 points-based 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 16) we can visualize how the aneurysm has evolved.

9.1 Bilateral Contracts with Industry

EDF ROM HYDRAU dates: 15/08/2020 au 15/12/2021

10.1 International Initiatives

10.2 International Research Visitors

10.3 European Initiatives

10.4 National Initiatives

We are part of the GDR AMORE on ROMs.

PEPS DYNAMO : dates : 01/03/2019 au 30/04/2020

10.5 Regional Initiatives

RNA 19 MDPAA (Post-doc of Gwladys Ravon, $50\%$) : dates : 14/01/2019 au 13/01/2024

Projet IDEX EMERGENCE : dates : 01/12/2019 au 31/12/2021

11.1 Promoting Scientific Activities

Angelo Iollo: Responsible of the Applied Mathematics Master of the University of Bordeaux. From september 2016 up septembre 2020.

Angelo Iollo: 2019-. Nominated at the council of "Département de Sciences du Numérique et Ingénierie" of the université de Bordeaux.

12.1 Major publications

• 1 articleE. Abbate, A. Iollo and G. Puppo. An all-speed relaxation scheme for gases and compressible materialsJournal of Computational Physics3512017, 1-24
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• 2 articleLuisL. Benetti Ramos, OlivierO. Marquet, MichelM. Bergmann and AngeloA. Iollo. Fluid--solid Floquet stability analysis of self-propelled heaving foilsJournal of Fluid Mechanics9102021, A28
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• 3 articleM. Bergmann, Charles-HenriC.-H. Bruneau and A. Iollo. Enablers for robust POD modelsJournal of Computational Physics22822009, 516--538
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• 4 articleM. Bergmann, J. Hovnanian and A. Iollo. An accurate cartesian method for incompressible flows with moving boundariesCommunications in Computational Physics1552014, 1266--1290
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• 5 articleM. Bergmann and A. Iollo. Bioinspired swimming simulationsJournal of Computational Physics3232016, 310 - 321
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• 6 articleM. Bergmann and A. Iollo. Modeling and simulation of fish-like swimmingJournal of Computational Physics23022011, 329 - 348
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• 7 articleF. Bernard, A. Iollo and G. Puppo. Accurate Asymptotic Preserving Boundary Conditions for Kinetic Equations on Cartesian GridsJournal of Scientific Computing2015, 34
• 8 article A. Bouharguane, A. Iollo and L. Weynans. Numerical solution of the Monge--Kantorovich problem by density lift-up continuation ESAIM: Mathematical Modelling and Numerical Analysis 49 6 1577 November 2015
• 9 articleA. De Brauer, A. Iollo and T. Milcent. A Cartesian Scheme for Compressible Multimaterial Models in 3DJournal of Computational Physics3132016, 121-143
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• 10 articleF. Luddens, M. Bergmann and L. Weynans. Enablers for high-order level set methods in fluid mechanicsInternational Journal for Numerical Methods in Fluids79December 2015, 654-675
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12.2 Publications of the year

12.3 Cited publications

• 25 articleP. Angot, Charles-HenriC.-H. Bruneau and P. Fabrie. A penalization method to take into account obstacles in a incompressible flowNumerische Mathematik8141999, 497-520
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• 26 article SS. Bagheri. Koopman-mode decomposition of the cylinder wake Journal of Fluid Mechanics 2013
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• 27 articleP.T.P. Barton, D. Drikakis, E. Romenski and V.A.V. Titarev. Exact and approximate solutions of Riemann problems in non-linear elasticityJournal of Computational Physics228182009, 7046-7068
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• 28 articleM. Bergmann, Charles-HenriC.-H. Bruneau and A. Iollo. Enablers for robust POD modelsJournal of Computational Physics22822009, 516--538
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• 29 articleMichelM. Bergmann, AndreaA. Ferrero, AngeloA. Iollo, EdoardoE. Lombardi, AngelaA. Scardigli and HaysamH. Telib. A zonal Galerkin-free POD model for incompressible flowsJournal of Computational Physics3522018, 301--325
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• 30 articleA. Bouharguane, A. Iollo and L. Weynans. Numerical solution of the Monge-Kantorovich problem by density lift-up continuationESAIM: M2AN4962015, 1577-1592
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• 31 articleA. DeA. Brauer, A. Iollo and T. Milcent. A Cartesian scheme for compressible multimaterial models in 3DJournal of Computational Physics3132016, 121-143URL: http://www.sciencedirect.com/science/article/pii/S0021999116000966
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• 32 articleBB. Cantwell and DD. Coles. An experimental study of entrainment and transport in the turbulent near wake of a circular cylinderJournal of fluid mechanics1361983, 321--374
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• 33 incollection L. Cordier and M. Bergmann. Two typical applications of POD: coherent structures eduction and reduced order modelling Lecture series 2002-04 on post-processing of experimental and numerical data Von Kármán Institute for Fluid Dynamics 2002
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• 34 articleS.L.S. Gavrilyuk, N. Favrie and R. Saurel. Modelling wave dynamics of compressible elastic materialsJournal of Computational Physics22752008, 2941-2969
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• 35 book S.K.S. Godunov. Elements of continuum mechanics Nauka Moscow 1978
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• 36 phdthesis XX. Jin. Construction d'une chaîne d'outils numériques pour la conception aérodynamique de pales d'éoliennes Université de Bordeaux 2014
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• 37 articleB. Lambert, L. Weynans and M. Bergmann. Local lubrication model for spherical particles within incompressible Navier-Stokes flowsPhys. Rev. E973Mar 2018, 033313URL: https://link.aps.org/doi/10.1103/PhysRevE.97.033313
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• 38 articleF. Luddens, M. Bergmann and L. Weynans. Enablers for high-order level set methods in fluid mechanicsInternational Journal for Numerical Methods in Fluids79December 2015, 654-675
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• 39 articleJLJ. Lumley, AMA. Yaglom and VIV. Tatarski. Atmospheric turbulence and wave propagationThe structure of inhomogeneous turbulence, AM Yaglom & VI Tatarski1967, 166--178
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• 40 article IgorI. Mezić. Spectral Properties of Dynamical Systems, Model Reduction and Decompositions Nonlinear Dynamics 41 1 2005
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• 41 articleG.H.G. Miller and P. Colella. A Conservative Three-Dimensional Eulerian Method for Coupled Solid-Fluid Shock CapturingJournal of Computational Physics18312002, 26-82
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• 42 articleR. Mittal and G. Iaccarino. Immersed boundary methodsAnnu. Rev. Fluid Mech.372005, 239-261
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• 43 articleP. J.P. Schmid. Dynamic mode decomposition of numerical and experimental dataJournal of Fluid Mechanics656008 2010, 5-28
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• 44 book J. A.J. Sethian. Level Set Methods and Fast Marching Methods Cambridge University Press, Cambridge, UK 1999
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• 45 articleL. Sirovich. Turbulence and the dynamics of coherent structuresQuarterly of Applied MathematicsXLV31987, 561-590
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• 46 articleTommasoT. Taddei. A registration method for model order reduction: data compression and geometry reductionSIAM Journal on Scientific Computing4222020, A997--A1027
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• 47 article TommasoT. Taddei and LeiL. Zhang. Registration-based model reduction in complex two-dimensional geometries submitted to Journal of Scientific Computing 2021
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• 48 article TommasoT. Taddei and LeiL. Zhang. Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs ESAIM: Mathematical Modelling and Numerical Analysis (accepted) 2020
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• 49 articleKunihikoK. Taira and TimT. Colonius. The immersed boundary method: a projection approachJournal of Computational Physics22522007, 2118-2137
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• 50 book C. Villani. Topics in optimal transportation American Mathematical Society 2003
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