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This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator permits to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated. Flat and non-flat domains are considered.

This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel-Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, long-time stability, Borel summation, divergent series., 33 pages, 18 figures

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This paper deals with the construction of reduced order models (ROMs) for the simulation of the interaction between a fluid and a rigid solid with imposed rotation velocity. The approach is a follows. First, we derive a monolithic description of the fluid/structure interaction by extending the Navier-Stokes equations from the fluid domain to the solid (rotor) domain similarly to the fictitious-domain approach. Second, we build a ROM by a proper orthogonal decomposition (POD) of the resulting multi-phases flow. This method consists in (i) constructing an optimal albeit empirical spatial basis for a very small sub-space of the solution space, and (ii) projecting the governing equations on this reduced basis. Third, we cope with the reconstruction of the high-dimensional velocity field needed to evaluate the imposed velocity constraint by a POD of the solid membership function. Fourth, we use state of the art method to interpolate between available POD bases to build the proposed POD-ROM for a range of parameters values. The proposed method is applied to an academic configuration and proves efficient in the reconstruction of the velocity in both the fluid and solid domains while substantially reducing the computational cost.

An adaptation of the Inverse Distance Weighting (IDW) method to the Grassmann manifold is carried out for interpolation of parametric POD bases. Our approach does not depend on the choice of a reference point on the Grassmann manifold to perform the interpolation, moreover our results are more accurate than those obtained in [ 7 ]. In return, our approach is not direct but iterative and its relevance depends on the choice of the weighting functions which are inversely proportional to the distance to the parameter. More judicious choices of such weighting functions can be carried out via kriging technics [ 23 ], this is the subject of a work in progress.

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This work deals with the computation of Hopf bifurcation points in the framework of two-dimensional fluid flows. These bifurcation points are determined by using a Hybrid method [1] which associates an indicator curve and a Newton method. The indicator provides initial values for the Newton method. As the calculus of this indicator is time consuming, we suggest using an algorithm to save computational time. This algorithm alternates reduced order and full size step resolution which are all carried out by using a pertubation method. Hence, the computed vectors on the full size problem are used to define the reduced order model. As the low-dimensional model has a finite validity range, we propose a simple criterion which makes it possible to know when the basis has to be updated. The latter phase is carried out by going through a new full step which permits to build a new basis and, thus, compute a supplementary part of the indicator curve. Some numerical tests, such as the classical lid-driven cavity or the flow in a channel, permit to fix the optimum values of the parameters for the proposed method. The objective of this study is to save computational time without modifying the performance of the Hybrid method initially introduced in Ref. [1] . These numerical methods are applied to 2D fluid flows (flow in a channel and the 2D lid-driven cavity). Our conclusions, therefore, hold only for these kinds of problem.

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A time integration method of a dynamical system, based on a (possibly divergent) time series decomposition of the solution, followed by a Borel summation using an inverse factorial series, is examined. The stress is put on the robustness of inverse factorial series algorithm, compared to the Borel-Pade one, to compute the Borel sum of the solution.

Monte-Carlo simulation is used to study the surface recombination effects on the electron beam induced current technique. The current is collected by a nanoscale electrode. We choose three values for the surface recombination velocity reflecting the sample surface states; zero, infinite value and an intermediate one. From a probabilistic point of view, each of the three surface velocities is described by a collection probability related to the surface effects and to the electrode size. Even if probabilities corresponding to zero and intermediate velocity are chosen to be close one to the other, the results are different. For intermediate velocity the collection efficiency profile is similar to that corresponding to the infinite velocity. Both profiles narrow with diminution of their maximum. At electron beam energies higher than 4 keV the profiles are superimposed because of the large penetration depth, and the smallness of the electrode and its depletion zone.

This paper presents numerical study of an oval-sail, a bluff-body equipped with a grid all along the span. Suction based flow control is applied to this body that is developed for wind assisted ship propulsion. First, a choice of numerical turbulence model is discussed through results of an oval-sail without suction. Three turbulence models are applied: the Ri j SSG, the Ri j EBRSM and the v2 f model. Then, computations are performed for the oval-sail fitted with suction grid. These last simu- lations are carried out with the low-Reynolds-number Ri j EBRSM turbulence model. The influence of the grid geometry on the oval-sail aerodynamic performances is highlighted. All simulations are carried out for the sail set at zero incidence. The Reynolds number based on the free stream velocity and the profile chord is Re = 5105. Results are compared to available experimental data.

In this manuscript, we present optimal sensitivity results of eigenvalues and eigenspaces with respect to self-adjoint compact operators. We show that while eigenvalues depend in a Lipschitzian way in compact operators, the eigenspaces are only locally Lipschitz. Our results generalize to arbitrary dimension eigenspaces the results obtained in [19] for one-dimensional eigenspaces sensitivity and thus simplify the celebrate results by Davis and Kahan [6] developed for general Hermitian operator perturbations. Moreover, Proper Orthogonal Decomposition bases sensitivity is carried out in the case of time-interval perturbations, spatial perturbations (Gappy-POD) or parameter perturbations.

The present paper is Part 2 of a two parts paper on flow around vibratingwind turbine airfoils. The first part of the paper dealt with a forced oscillatingairfoil. Part 2 focuses on free vibrating airfoils. The flow induced vibrationson two airfoils used for wind turbine blades are investigated by applying afluid structure interaction approach. A commercial Computational FluidDynamics (CFD) code is coupled to a computational structural programthat solves the dynamic equations of the airfoil oscillations. The fluidgoverning equations are described in the Arbitrary Lagrangian Euleriancoordinates and solved with a moving mesh. A straightforward meshingtechnique is implemented in a subroutine called by the CFD code at eachtime step for updating the grid. The method is applied to a free pitchoscillating airfoil and to combined pitch and vertical oscillations known asthe flutter instability.

The aerodynamic performances of a roof mounted small vertical axis wind turbine are evaluated using a Computational Fluid Dynamics (CFD) code and a computer program based on the Blade Element Momentum theory. The CFD code solves the incompressible Reynolds Averaged Navier-Stokes (RANS) equations for the simulation of the flow around a building with a roof mounted wind turbine. These computations are performed with a 3D grid, assuming steady and fully turbulent flow. The CFD results are used to derive the wind velocity vector approaching the wind turbine. Then the aerodynamic performances of the roof mounted wind turbine are determined using a Multiple Stream tube Theory. Beforehand, to check the reliability of the CFD computations, the simulations of the flow around reference buildings are performed and the results are compared with reference data.

This paper describes the Reduced Order Modeling (ROM) for fluid rigid body interaction problem and discusses Proper Orthogonal Decomposition (POD) utilisation. The principal difficulty for using POD being the moving domains, a referenced fixed domain is introduced. The POD is applied for the velocity field obtained on the fixed domain. Then a method to reduce dynamical system in rigid body fluid interaction is developed. This method uses the fictitious domain method approach, which consists in treating the entire fluid-solid domain as a fluid. The rigid body is considered as a fluid, by using a high viscosity which can play the role of a penalisation factor of the rigidity constraint, and by adding a distributed Lagrange multiplied associated to this constraint in the weak formulation. The fluid flow problem is then formulated on the reference domain and POD modes are used in the weak formulation. The results are compared with computational solution and discussed.

This paper proposes an advanced energy management strategy for a stand-alone hybrid energy system. The considered hybrid system includes a photovoltaic panel, a fuel cell, an electrolyzer, a battery bank and a supercapacitor. The proposed power management system aims to control the energy flow within the system and decides the amount of the load power shared with each power source. The system control is implemented in two parts: a central power flow controller that provides overall control of the power system and a local loads controller, which controls the different loads according to the energy balance of the system. The hybrid power system has been tested by simulation using models implemented in Matlab/Simulink software. The simulation is performed over a short and a long period of time in order to evaluate the performance of the dynamic controllers and the effectiveness of the management strategy. For a long simulation period, a house located in the province of Batna ( 35 ° 33 ′ N 6 ° 10 ′ E) has been taken as a case study with a deep study on real load profile for an average house with all required weather data with respect to the location. The simulation results confirm the efficiency of the proposed control strategy, as it increases the reliability of the system and improves its energy balance.

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A time integration method for the resolution of ordinary and partial dierential equations is proposed. The method consists in computing a formal solution as a (eventually divergent) time series. Next, the Borel resummation method is applied to deduce an (sectorial) analytical solution. The speed and spectral properties of the scheme are analyzed through some examples. Applications to fluid mechanics are presented. ResumOn propose une methode numerique d'integration temporelle d'´ equations di´ erentielles ou aux derivees partielles. Cette methode consiste d'abordcalculer une solution sous forme de serie formelle, dont le rayon de convergence peutnul. Ensuite, la methode de resommation de Borel- Laplace est utilisee pour deduire une solution analytique (dans un secteur) de l'´ equation. La rapidite et les proprietes geometriques du schema sont analyseestravers quelques exemples. Des applications en mecanique des fluides sont presentees.

In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. A POD basis associated with a set of reference values of the characteristic parameters is considered. From this basis, a parametric reduced order model (ROM) projecting the initial equation is constructed. A mathematical a priori estimate of the parametric squared L2-error induced by this projection is developed. This later estimate is based on both, the parametric behavior of the squared L2-ROM-error thanks to the resolution of a Ricatti differential inequality in the parametric ROM-error, and the convergence rate of the parametric ROM to the full problem, via the augmentation of the basis dimension. Indeed, under restrictive conditions on the solutions regularity of such equations, we are able to precise the slope of the logarithm of the squared L2-norm of the ROM error, as a function of the logarithm of the basis modes number. Numerical experiments of our theoretical estimate, are presented for the 2D Navier-Stokes equations in the case of an unsteady and incompressible fluid flow in a channel around a circular cylinder. A mathematical a priori estimate of the parametric squared L2-error induced by the model reduction by POD is developped for a parameterized quasi-nonlinear parabolic equation. This estimate is obtained thanks to the resolution of a Ricatti differential inequality.

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Fluid-Structure Interactions are present in a large number of systems of nuclear power plants and nuclear on-board stoke-holds. Particularly in steam generators, where tube bundles are submitted to cross-flow which can lead to structure vibrations. We know that numerical studies of such a complex mechanism is very costly, that is why we propose the use of reduced-order methods in order to reduce calculation times and to make easier parametric studies for such problems. We use the multiphase-POD approach, initially proposed by Liberge (E. Liberge; POD-Galerkin Reduction Models for Fluid-Structure Interaction Problems, PhD Thesis, Universite de La Rochelle, 2008). This method is an adaptation of the classical POD approach to the case of a moving structure in a flow, considering the whole system (fluid and structure) as a multiphase domain. We are interested in the case of large displacements of a structure moving in a fluid, in order to observe the ability of the multiphase-POD technique to give a satisfying solution reconstruction. We obtain very interesting results for the case of a single circular cylinder in cross-flow (lock-in phenomenon). Then we present the application of the method to a case of confined cylinders in large displacements too. Here again, results are encouraging. Finally, we propose to go further presenting a first step in parametric studies with POD-Galerkin approach. We only consider a flowing-fluid around a fixed structure and the Burgers’ equation. A future work will consist in applications to fluid-structure interactions.

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International audience; The symmetry group of the non-isothermal Navier-Stokes equations is used to develop physics-preserving turbulence models for the subgrid stress tensor and the subgrid heat flux. The Reynolds analogy is not used. The theoretical properties of the models are investigated. In particular, their compatibility with the scaling laws of the flow are proven. A numerical test, in the configuration of an air flow in a ventilated and differentially heated room is presented.

Tube bundles in steam boilers of nuclear power plants and nuclear on-board stokehold are known to be exposed to high levels of vibrations under flowing fluid. This coupled fluid-structure problem is still a challenge for engineers, first because of the difficulty to fully understand it, second because of the complexity for setting it up numerically. Although numerical techniques could help the understanding of such a mechanism, a complete simulation of a fluid past a whole elastically mounted tube bundle is currently out of reach for engineering purposes. To get round this problem, the use of a reduced-order model has been proposed with the introduction of the widely used Proper Orthogonal Decomposition (POD) method for a flow past a fixed structure [M. Pomare`de, E. Liberge, A. Hamdouni, E.Longatte, & J.F. Sigrist - Simulation of a fluid flow using a reduced-order modelling by POD approach applied to academic cases; PVP2010, July 18–22, Seattle]. Interesting results have been obtained for the reconstruction of the flow. Here a first step is to propose to consider the case of a flow past a fixed tube bundle configuration in order to check the good reconstruction of the flow. Then, an original approach proposed by Liberge (E. Liberge; POD-Galerking Reduction Models for Fluid-Structure Interaction Problems, PhD Thesis, Universite´ de La Rochelle, 2008) is applied to take into account the fluid-structure interaction characteristic; the so-called “multiphase” approach. This technique allows applying the POD method to a configuration of a flow past an elastically mounted structure. First results on a single circular cylinder and on a tube bundle configuration are encouraging and let us hope that parametric studies or prediction calculations could be set up with such an approach in a future work.

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This paper presents reduced order modelling (ROM) in fluid–structure interaction (FSI). The ROM via the proper orthogonal decomposition (POD) method has been chosen, due to its efficiency in the domain of fluid mechanics. POD-ROM is based on a low-order dynamical system obtained by projecting the nonlinear Navier–Stokes equations on a smaller number of POD modes. These POD modes are spatial and temporally independent. In FSI, the fluid and structure domains are moving, owing to which the POD method cannot be applied directly to reduce the equations of each domain. This article proposes to compute the POD modes for a global velocity field (fluid and solid), and then to construct a low-order dynamical system. The structure domain can be decomposed as a rigid domain, with a finite number of degrees of freedom. This low-order dynamical system is obtained by using a multiphase method similar to the fictitious domain method. This multiphase method extends the Navier–Stokes equations to the solid domain by using a penalisation method and a Lagrangian multiplier. By projecting these equations on the POD modes obtained for the global velocity field, a nonlinear low-order dynamical system is obtained and tested on a case of high Reynolds number.

International audience; We propose a new approach for moving frame construction that allows to make finite difference scheme invariant. This approach takes into account the order of accuracy and guarantees numerical properties of invariant schemes that overcome those of classical schemes. Benefits obtained with this process are illustrated with the Burgers equation. To cite this article: A. Chhay, A. Hamdouni, C. R. Mecanique 333 (2009). Résumé Une construction nouvelle des schémas invariants utilisant les repères mobiles. On propose une procédure nouvelle de construction des repères mobiles permettant de rendre invariant les schémas de discrétisation en différences finies. Elle prend en compte l'ordre de consistance et garantit aux schémas invariants de meilleures performances que celles des schémas classiques. On illustre les performances de cette approche sur l'exemple de l'´ equation de Burgers. Pour citer cet article : M. Chhay, A. Hamdouni, C. R. Mecanique 333 (2009). Version française abrégée Les méthodes numériques construites afin de préserver certaines propriétés liéesliéesà la structure géométrique deséquationsdeséquations s'appelle les intégrateurs géométriques. Elles permettent de traduire naturellement le com-portement qualitatif des solutions ainsi que de réduire les instabilités numériques. En particulier, les Email addresses: nchhay01@univ-lr.fr (Marx Chhay), aziz.hamdouni@univ-lr.fr (Aziz Hamdouni). schémas invariants permettent de conserver le groupe de symétrie deséquationsdeséquations et de réduire les erreurs numériques. Une méthode de construction de tels schémas a ´ eté développée par M. Fels et P. J. Olver. Elle est basée sur le concept de repère mobile. Dans cette procédure, la qualité des solutions numériques d'un schéma invariant estentì erement conditionnée par le choix du repère mobile associé au groupe de symétrie. Ce choix est déterminé par le procédé de normalisation d' ´ E. Cartan qui permet de ramener la détermination du répère mobile associéassociéà un groupe continue au choix d'une section transverse de l'orbite d'unélémentunélément. Ce procédé possède l'avantage d'exhiber une famille importante de schémas invariant mais ne garantit pas au schéma obtenu des qualités numériques meilleures que le schéma d'origine. Nous pro-posons une méthode de construction nouvelle des schémas invariants utilisant les repères mobiles. Cette méthode peutêtrepeutêtre décrite sous la forme algorithmique suivante : (i) On considère un schéma discrétisant une EDP et le groupe de symétrie dépendant de paramètres réels de l'EDP, (ii) on transforme le schéma afin d'obtenir un schéma paramétrisé, (iii) on suppose une forme algébrique des paramètres en fonction de coefficients réels, (iv) on calcule les conditions d'´ equivariance afin que les paramètres de transformation deviennent des repères mobiles, (v) enfin, on calcule les conditions sur les coefficients réels pour que le schéma transformé soit d'un ordre de précision fixé. Ainsi on obtient un schéma invariant dont l'ordre de consistance est déterminé. On illustre les performances de cette approché a travers la construction d'un schéma invariant pour l'´ equation de Burgers.

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In this paper we present a method of construction of invariant numerical schemes for partial differential equations. The resulting schemes preserve the Lie-symmetry group of the continuous equation and they are at least as accurate as the original scheme. The improvement of the numerical properties thanks to the Lie-symmetry preservation is illustrated on the exam-ple of the Burgers equation.

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Colloque avec actes et comité de lecture. Internationale.; International audience; On propose une procédure nouvelle de construction des repères mobiles permettant de rendre invariant les schémas de discrétisation en différences finies. Elle prend en compte l’ordre de consistance et garantit aux schémas invariants de meilleures performances que celles des schémas classiques. On illustre les performances de cette approche sur l’exemple de l’équation de Burgers.

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International audience; Cet article s'intéresse à la réduction de modèle en Interaction Fluide Structure. Trois méthodes utilisant de manière différentes la POD sont comparées. La première, utilisée en aéroélasticité, applique la POD de la même manière qu'en mécanique des fluide. Une seconde méthode utilise une formulation multiphasique du problème d'interaction fluide structure pour l'écriture du modèle réduit. La dernière combine POD et décomposition de domaine.