Séminaires

Séminaires en 2022:

 Le 8 juillet à 14h : Lucille-Marie Tenkes (INRIA Gamma)

Titre:

"Méthodes de génération de maillages quad-dominants pour des applications en mécanique des fluides numérique"

Résumé:
Les simulations numériques en mécanique des fluides visent à capturer avec précision des phénomènes de diverses natures. La qualité de la simulation dépend grandement de la représentation discrète du domaine de calcul, appelée maillage, qui lui sert de support. En particulier, certains phénomènes sont mieux détectés si le maillage présente des caractéristiques spécifiques. Par exemple, il est généralement attendu d'un maillage de couche limite qu'il soit structuré et aligné avec la frontière du domaine, et idéalement constitué de quadrilatères ou d'hexaèdres. L'alignement des éléments du maillage avec l'écoulement est aussi un facteur d'amélioration des simulations. La génération de maillages structurés purement quadrilateraux ou hexaédriques se heurte à des obstacles divers: il est plus difficile, avec ce type d'éléments, de concilier alignement et structure, d'approcher avec précision tout type de géométries, et de capturer des phénomènes anisotropes. Dans ce contexte, on présente une méthode de génération de maillages multi-éléments, présentant des zones structurées formées de quadrilatères, et des zones non-structurés lorsque cette contrainte n'est pas nécessaire. 

Pour construire automatiquement de tels maillages quad-dominants, la méthode proposée s'appuie sur les outils et techniques de l'adaptation de maillage, qui repose sur le calcul d'un champ de métrique. En exploitant les directions intrinsèques du champ de métrique, à travers les méthodes « métrique-aligné » ou « métrique-orthogonal », des maillages présentant localement une certaine structure et orthogonalité peuvent être générés. 

On développe cette idée pour mettre en place une boucle d'adaptation étendue à des maillages quad-dominants. 

Des travaux ont été menés en premier lieu sur la génération de maillage, exploitant la méthode métrique-orthogonale. En particulier, on propose une méthode d'appariement se reposant sur les particularités des maillages métrique-orthogonaux.

Dans un second-temps, on a démontré l'impact majeur du champ de métrique sur la qualité des maillages quadrilatéraux et alignés, et amélioré les méthodes de lissage du champ de métrique existantes pour obtenir des maillages quad-dominants de meilleure qualité. Ensuite, on présente les modifications nécessaires à un solveur Volumes Finis pour permettre des calculs sur maillages multi-éléments. Les modifications concernent en particulier la discrétisation des termes convectifs et visqueux, et notamment le calcul des gradients. En application de ces méthodes, des calculs adaptatifs sur maillages multi-éléments ont été réalisés pour des écoulements Euler, laminaires et turbulents.

le 23 septembre à 14h: Prof. A. Lozinski (Univ. de Besançon)

φ-FEM: a fictitious domain approach achieving optimal convergence without non standard numerical integration Alexei Lozinski Laboratoire de Math´ematiques de Besan¸con Univ. Bourgogne Franche-Comt´e Geometrically unfitted methods, i.e. the numerical methods using the computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely used in the computational mechanics for decades. Their classical variants (Immersed Boundary or Fictitious Domain methods) are easy to implement but can suffer from poor accuracy. More recent approaches, like XFEM and CutFEM enjoy the optimal accuracy at the price of a considerable sophistication wrt the implementation: they introduce the integrals on the parts of the mesh cells cut by the boundary (the cut cells) so that a non trivial numerical integration is required. In this talk, we present an alternative unfitted method – φ-FEM, first introduced in [1, 2] for the Poisson equation with Dirichlet or Neumann boundary conditions, then extended to some less academic problems in [3]. As in CutFEM/XFEM, we suppose that the physical domain Ω is embedded into a simple background mesh and we introduce the active computational mesh Th, getting rid of the cells lying entirely outside Ω. The general procedure is as follows: we extend the governing equations from Ω to Ωh (the domain of Th) and write down a formal variational formulation on Ωh without taking into account the boundary conditions on ∂Ω; we then impose the boundary conditions on ∂Ω using an appropriate ansatz or additional variables, explicitly involving the level set φ that defines the actual boundary by Ω = {φ < 0}; finally, we add an appropriate stabilization to guarantee coercivity/stability on the discrete level. This approach allows us to achieve the optimal accuracy using the usual FE spaces and the usual numerical integration: all the integrals in φ-FEM can be computed by standard quadrature rules on entire mesh cells and on entire facets. FE of any order can be straightforwardly used. The geometry is naturally taken into account with the needed optimal accuracy: it suffices to approximate the levelset by piecewise polynomials of sufficiently high degree. Moreover, φ-FEM is designed so that the matrices of the discrete problems are reasonably conditioned, i.e. their condition numbers are of the same order as those produced by a standard fitting FEM on a mesh of comparable size. This is a joint work with Michel Duprez (Inria, Strasbourg) and Vanessa Lleras (IMAG, univ. Montpellier). References [1] M. Duprez and A. Lozinski. φ-FEM: a finite element method on domains defined by level-sets. SIAM J. Numer. Anal., 58(2):1008–1028, 2020. [2] M. Duprez, V. Lleras, A. Lozinski. A new φ-FEM approach for problems with natural boundary conditions. Numer. Methods Partial Differ. Equ., 2021 (to appear). [3] S. Cotin, M. Duprez, V. Lleras, A. Lozinski, K. Vuillemot. φ-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer. 2021 (preprint)

45eme Congrès National d'Analyse Numérique,

 Evian les bains, du 13 juin au 15 juin à Evian-les-bains

 

 

20 mai à 14h : Ludovic Martaud, Université de Nantes

Une famille de schémas numériques volumes finis d'ordre élevé vérifiant une stabilité entropique globale sur maillage 2D non structurés.

 

Bien qu'elles n'assurent pas l'unicité des solutions en deux dimensions d'espace, les inégalités d'entropies associées à un système de lois de conservation représentent un critère de stabilité important qu'il convient de restituer au niveaux discret par des méthodes numériques appropriées. De nombreux schéma d'ordres élevés ont été construits en ce sens conduisant la plupart du temps à des inégalités d'entropie semi discrète (DG, ENO/WENO, ...).

Dans cet exposé, nous présenterons des schémas volumes finis d'ordre élevé vérifiant une inégalité d'entropie discrète globale sur maillages non structurés 2D. Ces schémas reposent sur une discrétisation pondérée et d'ordre élevé des différents termes ; la pondération étant choisie afin d'assurer la stabilité entropique attendue. Des cas tests seront présentés afin d'illustrer la précision et la stabilité des schémas ainsi construits.

 

15 avril : J.M. Loubès (IMT) à 14h

Challenges liés au biais et à la robustesse en IA

Dans cet exposé nous nous intéresserons aux méthodes permettant de détecter des biais en IA et aux approches permettant d'y faire face. Pour cela, je présenterai les méthodes basées sur le transport optimal de mesures.

 Ludovic Chamoin (ENS Paris Saclay)=> séminaire remis à une date ultérieure 

Multiscale computations with MsFEM: adaptive strategy,
coupling with model reduction, and use for data assimilation

Abstract: The Multiscale Finite Element Method (MsFEM) is a powerful numerical method in the
context of multiscale analysis. It uses basis functions which encode details of the fine
scale description, and performs in a two-stage procedure: (i) offline stage in which
basis functions are computed solving local fine scale problems; (ii) online stage in
which a cheap Galerkin approximation problem is solved using a coarse mesh.
However, as in other numerical methods, a crucial issue is to certify that a prescribed
accuracy is obtained for the numerical solution. In the present work, we propose an a
posteriori error estimate for MsFEM using the concept of Constitutive Relation Error
(CRE) based on dual analysis. It enables to effectively address global or goal-oriented
error estimation, to assess the various error sources, and to drive robust adaptive
algorithms. We also investigate the additional use of model reduction inside the
MsFEM method in order to further decrease numerical costs. We particularly focus on
the use of the Proper Generalized Decomposition (PGD) for the effective computation
of multiscale basis functions, in a multi-query MsFEM framework that requests the
solution of parametric local fine-scale problems. Eventually, the interest of MsFEM for
data assimilation is investigated. We particularly show how this multiscale method can
be beneficially used when performing inverse analysis from digital image correlation
on a known material microstructure.

 

 

juin Stéphane Descombes, Université Nice Sophia Antipolis

Séminaires en 2021-2022

11  mars : H. Haddar (INRIA-CMAP)INRIA, project-team IDEFIX, ENSTA Paris Tech, Institut Polytechnique de Paris, Palaiseau

Interplay between imaging algorithms and the analysis of interior transmission problems

A large progress has been made in the last two decades on the analysis of so-called sampling methods (the linear sampling method, the factorization method, the generalized linear sampling method, etc...) which offered an original way to handle the imaging problem for inverse scattering problem without the need for any linearization assumption nor a forward solver. Beyond its practical aspect, this class of methods triggered interests for the theoretical analysis of an associated boundary value problem, the so-called interior transmission problem (itp) and its associated (non linear and non self adjoint) spectral problem. The latter also revealed an interesting aspect of the duality between invisibility and resonance for penetrable media [1]. After a rapid survey of these elements, I would like to point out, through two examples, how new algorithms adapted to challenging configurations have been proposed exploiting this close relation with the itp. The first example is the imaging of dense crack network. Although the problem seems to have nothing to do with the notion of transmission eigenvalues, the latter helped the design of an indicator function for the crack density, providing better results than classical approaches [2]. The second example is differential imaging. Monitoring a media through successive measurement campaigns offers the possibility to image newly born defects. The design of an indicator function capable of revealing these additional defects relies on the comparison between solutions to the interior transmission problem. Application of this procedure to cracks imaging in fracture elastic media has been applied in [3]. In my talk I will rather present a closely related problematic which is the imaging of defects in a periodic media, assuming that the periodic structure is not known a priori. For this case a single measurement campaign is needed. We show how to build an indicator function for the defect independently from the any prior reconstruction of the background. In an earlier work, we gave justification of this procedure assuming that the defect also have some (larger) periodicity scale [4]. We take the opportunity of this talk to present some recent elements of the analysis that helped removing this assumption.

References [1] F. Cakoni, D. Colton, H. Haddar, Transmission Eigenvalues, Notices of the American Mathematical Society 68, 09, October 2021, p. 1499–1510, [2] L. Audibert, L. Chesnel, H. Haddar, K. Napal, Qualitative indicator functions for imaging crack networks using acoustic waves, SIAM Journal on Scientific Computing, 2021, [3] F. Pourahmadian, H. Haddar, Differential tomography of micromechanical evolution in elastic materials of unknown micro/macrostructure, SIAM Journal on Imaging Sciences 13, 3, August 2020, p. 1302–1330, [4] F. Cakoni, H. Haddar, T.-P. Nguyen, New interior transmission problem applied to a single Floquet–Bloch mode imaging of local p

28 janvier2022 à 14h  : R. Loubère (IMB)

"Un paradigme pour construire des méthodes numériques de précision élevée

--- MOOD ---Design, mise en applications"


Solving a system of PDEs with a high accurate numerical scheme is usually well achieved if the solution is extremely regular.

Unfortunately, in the case of an hyperbolic  system of in presence of source terms or steep gradients, this assumption is often violated.
Then  "high accuracy" (for smooth solutions) often leads to "high troubles"  close to irregular ones, generation
of Gibbs phenomena, oscillations, lacks of admissibility, NaN (Not-a-Number) and ultimately code crash.

Therefore most high accurate numerical methods (FV, DG, FD, FE...) add some sort of artificial dissipation to avoid those phenomena.

Then the tricky questions to answer are: where? and how much?
In this talk we will propose a solution for FV and DG based on an 'a posteriori' check of the solution and
a recomputation with more dissipative schemes up to the validity of the numerical solution is achieved.
The technique is based on three ingredients: a robust and trustable (parachute) scheme, an ordered cascade of numerical schemes to test successively, and, a Detection procedure to (in)validate a candidate numerical solution.
In this talk we will review the general idea and present several contexts of use by different groups of researchers. A large set of (convincing) numerical results will be presented.

10 décembre : C. Coreixas (Cerfacs)

"Lattice Boltzmann methods: Weakly compressible formulations and beyond"

�� " Boltzmann equation (BE) represents the balance between transport and collision of particles, through the evolution of their velocity distribution function, usually denoted f(x,�,t). At the macroscopic level, f(x,�,t) can be understood as the number of particles at location x, time t, and with a given velocity �. Even if several approaches have been proposed to directly solve BE, it is far more efficient to solve its discrete velocity formulation, which considers only a finite number of speeds and orientations for the propagation of particles [1]. Contrarily to most discrete velocity models, the lattice Boltzmann method (LBM) relies on a very limited number of discrete velocities. The latter are chosen according to quadrature rules (or moment matching approaches) to ensure the asymptotic convergence towards Euler, Navier-Stokes-Fourier (NSF), or higher-order sets of equations [2,3]. Numerically speaking, LBMs benefit from a very efficient numerical scheme which makes them of particular interest for high-performance computing based on either CPUs or GPUs [4,5]. The latter scheme relies on Cartesian grids with octree-based refinement methodologies [6], kinetic boundary conditions [7], and it can easily include subgrid-scale and wall modelings [8,9] to simulate realistic flow conditions past complex geometries [10]. All of this explains why, during the past three decades, LBMs have gradually emerged as an interesting alternative for direct numerical simulations (DNS) and large eddy simulations (LES) of weakly compressible fluid flows, and beyond [11]. Nevertheless, LB solvers have a number of practical limitations. As an example, due to their inherent couping with Cartesian grids, it is difficult to properly simulate in an efficient manner wall-bounded turbulent flows even with advanced wall models. In addition, while LBMs can compete with NS solvers for DNS and LES, it is likely to be of no use when state-of-the-art results are obtained with (unsteady) Reynolds averaged NS solvers.� Finally, by relying on an explicit time-marching approach, LBMs are of little interest for steady-state flow simulations as compared to time-implicit NSF solvers. � In the end, to properly understand what are the pros and cons of LBMs, it is necessary to dive into their derivation. This talk then aims at recalling the main steps behind the derivation of weakly compressible LBMs, as well as, more advanced models. Particular attention is further paid to fully compressible formulations, as they have recently regained interest from both academic and industrial groups [12-16]. [1] Mieussens, A survey of deterministic solvers for rarefied flows, AIP Conf. Proc., 2014. [2] Shan &amp; He, Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett., 1998. [3] Shan et al., Kinetic theory representation of hydrodynamics: A way beyond the Navier Stokes equation, J. Fluid Mech., 2006. [4] Schornbaum &amp; R�de, Massively parallel algorithms for the lattice Boltzmann method on non-uniform grids, SIAM J. Sci. Comput., 2016. [5] Latt et al., Cross-platform programming model for many-core lattice Boltzmann simulations, PLoS One, 2021. [6] Astoul, Towards improved lattice Boltzmann aeroacoustic simulations with non-uniform grids: application to landing gears noise prediction, Aix-Marseille Universit�, 2021. [7] Kr�ger et al., The Lattice Boltzmann Method: Principles and Practice (Chap 5), Springer International Publishing, 2017. [8] Sagaut, Toward advanced subgrid models for lattice-Boltzmann-based large-eddy simulation: Theoretical formulations, Comput. Math. Appl., 2010. [9] Malaspinas &amp; Sagaut, Wall model for large-eddy simulation based on the lattice Boltzmann method, J. Comput. Phys., 2014. [10] Manoha &amp; Caruelle, Summary of the LAGOON solutions from the Benchmark problems for Airframe Noise Computations-III Workshop, 21st AIAA/CEAS Aeroacoustics Conference, 2015. [11] Succi, Lattice Boltzmann 2038, Europhys. Lett., 2015. [12] Guo &amp; Shu, Lattice Boltzmann method and its applications in engineering (Chaps 5 &amp; 6), World Scientific, 2013. [13] Fares et al., Validation of a lattice-Boltzmann approach for transonic and supersonic flow simulations, 52nd AIAA Aerospace Sciences Meeting, 2014. [14] Latt et al., Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria, Phil. Trans. R. Soc. A, 2020. [15] Coreixas &amp; Latt, Compressible lattice Boltzmann methods with adaptive velocity stencils: An interpolation-free formulation, Phys. Fluids, 2020. [16] Renard, Hybrid lattice Boltzmann Method for compressible flows, Aix-Marseille Universit�, 2021. "

19 novembre à partir de 14:00, Frédéric Alauzet (INRIA/Gamma)

Impact de l’adaptation de maillage pour les écoulements turbulents en aéronautique et turbomachine. 
Vers la certification des solutions numériques ?
 
21 octobre, A. Finel (LEM/ONERA)

Méthodes de champs de phase et solveurs mécaniques discrets.

 
10 septembre Richard Dwight (TU Delft)

Reconstruction of turbulent flows at high Reynolds number using data assimilation techniques