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The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation.

Landslide susceptibility indices were calculated and landslide susceptibility maps were generated for the Oudka, Morocco, study area using a geographic information system. The spatial database included current landslide location, topography, soil, hydrology, and lithology, and the eight factors related to landslides (elevation, slope, aspect, distance to streams, distance to roads, distance to faults, lithology, and Normalized Difference Vegetation Index [NDVI]) were calculated or extracted. Logistic regression (LR), multivariate adaptive regression spline (MARSpline), and Artificial Neural Networks (ANN) were the methods used in this study to generate landslide susceptibility indices. Before the calculation, the study area was randomly divided into two parts, the first for the establishment of the model and the second for its validation. The results of the landslide susceptibility analysis were verified using success and prediction rates. The MARSpline model gave a higher success rate (AUC (Area Under The Curve) = 0.963) and prediction rate (AUC = 0.951) than the LR model (AUC = 0.918 and AUC = 0.901) and the ANN model (AUC = 0.886 and AUC = 0.877). These results indicate that the MARSpline model is the best model for determining landslide susceptibility in the study area.

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 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.

International audience; We compare the performance of two algorithms of computing the Borel sum of a time power series. The first one uses Padé approximants in Borel space, followed by a Laplace transform. The second is based on factorial series. These algorithms are incorporated in a numerical scheme for time integration of differential equations.

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.

International audience; Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic inte-grators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

From mathematical and numerical points of view, we study the sensitivity with respect to parametric evolutions, of the error obtained by approximating a given parametric partial differential equation using a proper orthogonal decomposition (POD) basis determined once and for all, with association to a fixed parameter. More precisely, we will be considering the one-dimensional Burgers equation with a source term.

In this work, we present contributions concerning a mathematical study of the sensitivity of a reduced order model (ROM) by the proper orthogonal decomposition (POD) technique applied to a quasi-linear parabolic equation. In particular, we apply our theoretical study to the Navier–Stokes equations for a 2D incompressible fluid flow. We present a numerical test of our theoretical result, for an unsteady fluid flow in a channel around a circular cylinder.

We are interested in the mathematical study of the sensitivity of a reduced order model (ROM) of a particular single-parameterised quasi-linear equation, via the parametric evolution. More precisely, the ROM of interest is obtained in two different ways: First, we reduce the complete parametric equation using a proper orthogonal decomposition (POD) basis computed at a given reference value of the parameter, and second the parametric ROM is obtained by an expanded POD basis associated this time to a reference solution and its parametric derivative. The second case of our study was considered in a nearly similar way in Ito and Ravindran (1998), but in the context of the reduced basis (RB) method of the Navier–Stokes equations reduction. Indeed, the authors, Ito and Ravindran (1998) proposed to use an expanded set of basis functions, including solution flows for different values of the Reynolds number and their associated first-order derivatives with respect to this parameter. Beside this work, our second st...

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.

International audience; Borel's technique of divergent series resummation is transformed into a numerical code and used as a time integration scheme. It is applied to the resolution of regular and singular problems arising in fluid mechanics. Its efficiency is compared to those of classical discretization schemes.

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.

In this paper, we present the study of an asymptotic linear one-dimensional model for thin-walled beams with open strongly curved cross-section, which is different from the classical Vlassov model. After recalling the main steps of the asymptotic procedure leading to the equilibrium equations of the thin-walled beam model, an analytical resolution of the twist equation is performed. Then comparisons with experimental measurements and with the Vlassov model are carried out. The relevance of the asymptotic thin-walled beam model presented is put in a prominent position.

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.

In this paper, an implicit coupling algorithm for fluid–structure interaction problems with under-time steps for the solid is presented. Its implementation on two configurations is achieved by using the CASTEM finite-elements code. First, the free oscillations of a cylinder in an annular fluid domain where its movement is determined by the coupled fluid–solid action is considered in the case of viscous fluid. It should be noted that the implicit coupling algorithm gives the best prediction of the structure oscillations. The under-time steps for the solid are introduced in order to obtain better results. Then, an application whose final objective is to model a floating barrage is studied. The main goal of this application is to predict the displacements of a ring completely immersed and anchored by a cable to the lower boundary of the fluid domain. The finite-element discretization of the Navier–Stokes equations in the ALE formulation is used

International audience; The two-dimensional Burgers' equations are solved here using the A Priori Reduction method. This method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved. The method is called a priori because it does not need any prior knowledge of the solution, which is not the case if the standard Karhunen-Loéve decomposition is used. The accuracy of the APR method is compared with the standard Newton-Raphson scheme and with results from the literature. The APR basis is also compared with the Karhunen-Loéve basis.

International audience; Lie-symmetry based integrators are constructed in order to preserve the local invariance properties of the equations. The geometrical methods leading to discretized equations for numerical computations involve many different concepts. Therefore they give rise to numerical schemes that vary in the accuracy, in the computational cost and in the implementation. In this paper a comparison is made between some alternative Lie-symmetry based methods illustrated on the example of the Burgers equation. The importance of the symmetry preservation is numerically highlighted.

A linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods is presented. A dimensional analysis of the linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions. In the case of low force levels, we obtain a one dimensional model whose kinematics, traction and twist equations correspond to Vlassov ones. However this model couples twist and bending effects in the bending equations, at the difference from Vlassov model where the twist angle and the bending displacement are uncoupled. Запропоновано отриману асимптотичним методом одновимірну модель для тонкостінного стержня з відкритим сильно скривленим поперечним перерізом, яка враховує взаємо- зв’язок між скручуванням та згином. За допомогою аналізу розмірностей в лінійних тривимірних рівняннях рівноваги знайдено безрозмірні величини, які характеризують геометрію стержня та рівень прикладених сил. Для заданого рівня сил методом асимптотичного розкладу отримані порядок змі- щень та відповідна одновимірна модель. У випадку низького рівня сил отримано одновимірну модель, кінематичні рівняння, рівняння кручення та згину відповідають моделі Власова. Однак ця модель враховує в рівняннях згину взаємодію між згином і крученням на відміну від моделі Власова, яка таку взаємодію не враховує. Запропоновано отриману асимптотичним методом одновимірну модель для тонкостінного стержня з відкритим сильно скривленим поперечним перерізом, яка враховує взаємо- зв’язок між скручуванням та згином. За допомогою аналізу розмірностей в лінійних тривимірних рівняннях рівноваги знайдено безрозмірні величини, які характеризують геометрію стержня та рівень прикладених сил. Для заданого рівня сил методом асимптотичного розкладу отримані порядок зміщень та відповідна одновимірна модель. У випадку низького рівня сил отримано одновимірну модель, кінематичні рівняння, рівняння кручення та згину відповідають моделі Власова. Однак ця модель враховує в рівняннях згину взаємодію між згином і крученням на відміну від моделі Власова, яка таку взаємодію не враховує.

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 example of the Burgers equation.

The objective of this paper is to compare the performance of different estimators of Hurst index for multifractional Brownian motion (mBm), namely, Generalized Quadratic Variation (GQV) Estimator, Wavelet Estimator and Linear Regression GQV Estimator. Both estimators are used in the real financial dataset Nasdaq time series from 1971 to the 3rd quarter of 2009. Firstly, we review definitions, properties and statistical studies of fractional Brownian motion (fBm) and mBm. Secondly, a numerical artifact is observed: when we estimate the time varying Hurst index H(t) for an mBm, sampling fluctuation gives the impression that H(t) is itself a stochastic process, even when H(t) is constant. To avoid this artifact, we introduce sparse modelling for mBm and apply it to Nasdaq time series.

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.

Given buses of different types arriving at a depot during the evening, the bus parking problem consists of assigning these buses to parking slots in such a way that they can be dispatched adequately to the next morning routes without moving them between their arrivals and departures. In practice, the bus arrival times deviate stochastically from the planned schedule. In this paper, we introduce for this problem two solution approaches that produce solutions which are robust to variations in the arrival times. The first approach considers that each arrival can deviate from its planned arrival order (sooner or later) by at most k positions, where k is a predefined parameter. In the second approach, the objective aims at minimizing the expectation of a function positively correlated with the number of buses that make the planned solution infeasible because they arrive too late or too early. In both approaches, the problem is modeled as an integer linear program that can be solved by a commercial mip solver. Computational results obtained on instances derived from a real-world dataset are reported.

In a transit authority bus depot, buses of different types arrive in the evening to be parked in the depot for the night, and then dispatched in the morning to a set of routes, each of which requests a specific bus type. A type mismatch occurs when the requested type is not assigned to a morning route. We consider the problem of assigning the buses to the depot parking slots such that the number of mismatches is minimized, under the constraint that the buses cannot be repositioned overnight. As in Hamdouni et al. [Dispatching buses in a depot using block patterns. Technical Report, Les Cahiers du GERAD G-2004-51, HEC Montreal, Montreal, Canada, 2004, Transportation Science, to appear], we seek robust solutions by assigning a block pattern to each depot. This pattern partitions the lane into at most two blocks, each block containing buses of a given type. Since it may not be possible to respect the selected block patterns, the problem also involves a second objective which is to minimize the discrepancy between the bus type assignment to the parking slots and the block patterns. In this paper, we first study the simplified case where only the second objective is taken into account. We model this simplified problem as an integer linear program and show that practical instances of it can easily be solved using a commercial MIP solver. Then we formulate the general case as an extension of the simplified model and propose to solve it with a Benders decomposition approach embedded in a branch-and-bound procedure. This procedure is required because the Benders decomposition yields a subproblem with integrality constraints. Of particular interests, we develop strong pruning criteria and an innovative branching strategy that imposes decisions on the master problem variables which already take integer values. Computational results for the general case are also reported.

Navier–Stokes equations (NS) admit transformations which transform a solution to another solution (galilean transformation, scaling transformation, …). They also admit viscosity dependent transformations which transform a solution to a solution of another NS with different viscosity. These particular transformations are called symmetries of NS. Each of them has a physical role (such as conservation laws, …). A consistent turbulence model should then remain invariant under these symmetry transformations. Unfortunately, this is not the case of several models. In this article, a class of subgrid-scale models preserving the symmetries of NS is built. This class is then refined such that the models respect the second law of thermodynamics. One of the simplest models of the class is tested to the flow in a ventilated room. Better results than those provided by Smagorinsky and dynamic models are obtained.

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions.

In this article, we present an adaptive method for solving transfer equations. The method consists in projecting the discretized problem on a basis we have defined in order to obtain a reduced model that can be quickly and accurately solved with classic numerical schemes. The originality of the methods stays in the way of the basis is constructed. At each iteration of computation, the basis is adapted: first the old basis is improved using a Karhunen-Loeve decomposition whereas in a second phase the improved basis is expanded with Krylov vectors. The example we study is the one-dimension Burgers’ equation. The results we obtained were compared to the Newton-Raphson method: whereas the accuracy is not better than the Newton-Raphson method, we show that the computationnal time is drastically reduced. In addition, the basis we obtain shows a great ability to represent the long-time dynamics of the system, as shown in the last part of the paper.

In order to preserve the physical properties of the flow (scaling laws, conservation laws, …) during the simulation, a class of subgrid models respecting the symmetry group of the Navier–Stokes equations is built. The class is then refined such that models satisfy the second law of thermodynamics and are suited to take into account the inverse energy cascade. A simple model belonging to the class is tested and a better result than those provided by Smagorinsky and dynamic models is obtained. To cite this article: D. Razafindralandy, A. Hamdouni, C. R. Mecanique 333 (2005).

In this Note we present a justification of the kinematic assumptions for thin-walled rods with shallow profile. These assumptions are fundamental to writing the one-dimensional equilibrium equations for such structures. The obtained kinematics are different from the Vlassov case, which is only valid for strongly curved profiles. They are also different from the that classically used in shell theory. The justification given in this Note is based on an asymptotic approach. To cite this article: L. Grillet et al., C. R. Mecanique 333 (2005).

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions. To cite this article: L. Grillet et al., C. R. Mecanique 332 (2004).

The two-dimensional flow along an inclined plate may be detached or reattached to the plate by Coanda effect. Experimentally, we explore the influence of the inclination angle and of the Reynolds number on the attachment and detachment phenomena, and on the hysteresis loop. A proper orthogonal decomposition (POD) of the flow is applied to a LES simulation resulting data. A low-dimensional dynamical model is obtained using by Galerkin projection of the Navier–Stokes equations upon the POD basis functions. We show that this model represents qualitatively the characteristics of the flow.

The purpose of this article is to develop an asymptotic Eulerian approach in plate and shell theory valid for large displacements. With this Eulerian approach, the dead load assumption generally used with Lagrangian approaches can be dropped. We also obtain a membrane model which takes into account following forces, whose direction can vary during large displacements.

We propose in this paper to generalize to elastic–plastic plates the constructive asymptotic approach developed by the authors for elastic plates and shells. A dimensional analysis of three-dimensional equations makes appear dimensionless numbers characterizing the problem. Then using the geometric and mechanical data, these dimensionless numbers are linked to the perturbation parameter (the relative thickness of the plate) to obtain a one scale problem. For standard generalized materials with strong strain hardening, subjected to usual force levels, the asymptotic expansion of equations leads to an elastic–plastic plate model valid for moderate displacements. The constructive approach developed enables to specify the domain of validity of the asymptotic elastic–plastic plate model obtained thanks to the dimensional numbers introduced.

This paper deals with the implementation of a volume penalization technique in a lattice Boltzmann model, in order to compute flows around obstacles. The penalization term is introduced into the lattice Boltzmann equation via a forcing term. Numerical examples (flows over a square obstacle, and past a circular cylinder) show that a good agreement with results from the literature was obtained.

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.

We present in this paper a new constructive method of classification of two-dimensional plate models from the applied forces level and the geometrical data. This approach which uses asymptotic methods is based on a dimensional analysis of the non-linear equilibrium equations. This dimensional analysis leads to dimensionless numbers which reflect the geometry of the structure and the applied forces. For a given forces level, the order of magnitude of the displacements and the corresponding two-dimensional model are deduced by asymptotic expansion of the three-dimensional equations. For decreasing forces level, we obtain successively the non-linear membrane model, another membrane model, the usual non-linear plate model and the linear Kirchhoff–Love model.

Using the polar decomposition RU of the deformation gradient of a shell midsurface and studying the variations of R, we introduce new shell variables. These variables simplify remarkably the differential calculus over a surface. So, we obtain an explicit form of the compatibility conditions satisfied by the membrane strain tensor E and the curvature strain tensor K. Moreover, these equations are decomposed into a first order differential system. To end the article we show how to obtain the deformed surface when the strain tensors are given.

Resume Nous proposons ici une nouvelle justification du modele bidimensionnel non lineaire de plaque mince par developpement asymptotique direct des equations d'equilibre non lineaires. L'adimensionnalisation des equations d'equilibre fait apparaitre naturellement des nombres sans dimension caracteristiques de la nature physique du probleme, mesurables par l'ingenieur, et permettant de definir plus precisement le domaine de validite du modele bidimensionnel non lineaire obtenu.

Separated representations allow substituting the resolution of 3D models by a sequence of three one-dimensional problems [1]. This route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist. In a former work [2], we explored the route of immersing the domain into a hexahedral domain, and then use a kind of penalty method to solve the model whilst enforcing the boundary conditions. In the present work, we are analyzing two alternative routes. The first one consists of using an inverse technique in order to compute the boundary conditions on the border of the hexahedra in which the complex domain is immersed. The second one consists in solving the model in a regular domain for a number of geometrical parameters considered as extra-coordinates from which we could have access to the solution in any geometry resulting from a choice of those parameters.

International audience; In this article, we present some investigations about the solving of transfer equations by reduced-order models (ROM). We introduce a ROM, the a priori reduction (APR), and we present the results obtained for the 2D unsteady convection-diffusion equation and the 1D Burgers equation. The APR approach is then compared with the Karhunen-Loève decomposition and some properties of this method are emphasized. We show that the computation time necessary for solving these transfer equations is reduced, whereas the accuracy is of the same order of magnitude, in comparison with the solution obtained for the full model with classical methods. At last it is noticed that the APR method is an efficient way to correct the long term behavior of low order dynamical systems.

This paper introduces a novel technique that is developed to separate the audio sources from a single mixture. Indeed, audio signals and, in particular, musical signals can be well approximated by a sum of damped sinusoidal (modal) components. Based on this representation, Empirical Mode Decomposition (EMD) is employed to extract Intrinsic Mode Functions (IMFs) for audio mixture signal. By applying PCA (Principal Component Analysis) to the extracted components, we find uncorrelated components which are the artificial observations. Then we obtain independent components by applying Independent Component Analysis (ICA) to the uncorrelated components. A k-means clustering algorithm is introduced to group the independent basis vectors into the number of component sources inside the mixture.

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.

International audience; A numerical simulation of the flow past a circular cylinder which is able to oscillate transversely to the incident stream is presented in this paper for a fixed Reynolds number equal to 100. The 2D Navier-Stokes equations are solved by a finite volume method with an industrial CFD code in which a coupling procedure has been implemented in order to obtain the cylinder displacement. A preliminary work is first conducted for a fixed cylinder to check the wake characteristics for Reynolds numbers smaller than 150 in the laminar regime. The Strouhal frequency $f_S$ and the aerodynamic coefficients are thus controlled among other parameters. Simulations are then performed with forced oscillations characterized by the frequency ratio $F = f_0 /f_S$, where $f_0$ is the forced oscillation frequency, and by the adimensional amplitude A. The wake characteristics are analyzed using the ti me series of the fluctuating aerodynamic coefficients and their power spectral densities (PSD). The frequency content is then linked to the shape of the phase portraits and to the vortex shedding mode. By choosing interesting couples $(A, F)$, different vortex shedding modes have been observed, which are similar to those of the Wil-liamson-Roshko map. A second batch of simulations involving free vibrations (so-called vortex-induced vibrations or VIV) is finally carried out. Oscillations of the cylinder are now directly induced by the vortex shedding process in the wake and therefore, the time integration of the motion is realized by an explicit staggered algorithm which provides the cylinder displacement according to the aerodynamic charges exerted on the cylinder wall. Amplitude and frequency response of the cylinder are thus investigated over a wide range of reduced velocities to observe the different phenomena at stake. In particular, the vortex shedding modes have also been related to the frequency response observed and our results at Re = 100 show a very good agreement with other studies using different numerical approaches.