Enablers for robust POD models

Bergmann M. & Iollo A.

J. Comput. Physics,  230 (2), 2011.

Abstract:  Modeling and simulation of two-dimensional flows past deformable bodies are considered. The incompressible Navier-Stokes equations are discretized in space onto a fixed cartesian mesh and the displacement of deformable objects through the fluid is taken into account using a penalization method. The interface between the solid and the fluid is tracked using a level-set description so that it is possible to simulate several bodies freely evolving in the fluid. As an illustration of the methods, fish-like locomotion is analyzed in terms of propulsion efficiency.  Underwater maneuvering and school swimming are also explored.

Enablers for robust POD models

Bergmann M., Bruneau C.-H. & Iollo A.

J. Comput. Physics, 228 (2), 2009.

Abstract: This paper focuses on improving the stability as well as the approximation properties of Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD). The ROM is obtained by seeking a solution that lives in the POD subspace and at the same time minimizes the Navier-Stokes residuals. A modified ROM that directly incorporates the pressure term is proposed. The ROM stabilization makes use of methods based on the fine scale equations. The solution to these equations are approximated using the residuals of the Navier-Sokes equations. The improvement of the POD subspace is performed thanks to an hybrid method that couples direct numerical simulations (DNS) and reduced order model simulations.  The methods proposed are tested on the two-dimensional confined square cylinder wake flow in laminar regime.

Numerical methods for low-order modeling of fluid flows based on POD

Weller J., Lombardi E., Bergmann M. & Iollo A.

International Journal for Numerical Methods in Fluids, 63, 249-268 (2010).

Abstract: This articles explores some numerical alternatives that can be exploited to derive efficient low-order models of the Navier-Stokes equations. It is shown that an optimal solution sampling can be derived using appropriate norms of the Navier-Stokes residuals. Then the classical Galerkin approach is derived in the context of a residual minimization method that is similar to variational multiscale modeling. Finally, calibration techniques are reviewed and applied to the computation of unsteady aerodynamic forces. Examples pertaining to both non-actuated and  actuated flows are shown.

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models 

Bergmann M., & Cordier L.

Abstract: In this paper we investigate the optimal control approach for the active control of the circular cylinder wake flow considered in the laminar regime (Re = 200). The objective is the minimization of the total mean drag where the control function is the time harmonic angular velocity of the rotating cylinder. When the Navier-Stokes equations are used as state equation, the discretization of the optimality system leads to large scale discretized optimization problems that represent a tremendous computational task. In order to reduce the number of state variables during the optimization process, a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) is then derived to be used as state equation. Since the range of validity of the POD ROM is generally limited to the vicinity of the design parameters in the control parameter space, we propose to use the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reduced-order models during the optimization process. Benefiting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with the Navier-Stokes equations. A lot of computational work is indeed saved because the optimization process is now based only on low-fidelity models. The key enablers to an accurate and robust POD ROM for the pressure and velocity fields are the extension of the POD basis functions to the pressure data, the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of different non-equilibrium modes. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% for reduced numerical costs (a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly).

Note on the determination of the ignition point in forest fires propagation using a control algorithm

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Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition

Abstract: In this paper we investigate the optimal control approach for the active control of the circular cylinder wake flow considered in the laminar regime (Re = 200). The objective is the minimization of the mean total drag where the control function is the time harmonic angular velocity of the rotating cylinder. When the Navier-Stokes equations are used as state equations, the discretization of the optimality system leads to large scale discretized optimization problems that represent a tremendous computational task. In order to reduce the number of state variables during the optimization process, a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) is then derived to be used as state equation. Since the range of validity of the POD ROM is generally limited to the vicinity of the design parameters in the control parameter space, we propose to use the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reduced-order models during the optimization process. Benefiting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with the Navier-Stokes equations. A lot of computational work is indeed saved because the optimization process is now based only on low-fidelity models. The key enablers to an accurate and robust POD ROM for the pressure and velocity fields are the extension of the POD basis functions to the pressure data, the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of different non-equilibrium modes. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% for reduced numerical costs (a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly).

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Abstract: In this Brief Communication, we determine an approximate relation that gives the mean time power required to control the wake flow downstream from a circular cylinder. The control law is the sinusoidal tangential velocity imposed on whole or part of the cylinder surface. The mean control power thus depends on four parameters: the amplitude and the Strouhal number of forcing, the control angle that defines the controlled upstream part of the cylinder, and the Reynolds number. This relation indicates that the control power grows like the square of the forcing amplitude, like the square root of the forcing Strouhal number, linearly with the control angle and varies like the inverse of the square root of the Reynolds number. We show that the values obtained with this approximate relation are in very good agreement with the corresponding values given numerically. Finally, the energetic efficiency of the control is discussed. We claimed that the most energetically efficient control law corresponds a priori to low forcing amplitudes applied to a restricted upstream part of the cylinder for relatively high values of the Reynolds number.

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Abstract: In this Brief Communication we are interested in the maximum mean drag reduction that can be achieved under rotary sinusoidal control for the circular cylinder wake in the laminar regime. For a Reynolds number equal to 200, we give numerical evidence that partial control restricted to an upstream part of the cylinder surface may considerably increase the effectiveness of the control. Indeed, a maximum value of relative mean drag reduction equal to 30% is obtained when applying a specific sinusoidal control to the whole cylinder, where up to 75% of reduction can be obtained when the same control law is applied only to a well-selected upstream part of the cylinder. This result suggests that a mean flow correction field with negative drag is observable for this controlled flow configuration. The significant thrust force that is locally generated in the near wake corresponds to a reverse von Karman vortex street as commonly observed in fish-like locomotion or flapping wing flight. Finally, the energetic efficiency of the control is quantified by examining the power saving ratio: it is shown that our approach is energetically inefficient. However, it is also demonstrated that for this control scheme the improvement of the effectiveness generally occurs along with an improvement of the efficiency.

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Optimal rotary control of the cylinder wake using POD reduced order model

Bergmann M., Cordier L. & Brancher J.-P.

Phys. fluids 17 (9), 2006.

Abstract: In this paper we investigate the optimal control approach for the active control and drag optimization of incompressible viscous flow past circular cylinders. The control function is the time angular velocity of the rotating cylinder. The wake flow is solved in the laminar regime Re=200 with a finite-element method. Due to the CPU and memory costs related to the optimal control theory, a proper orthogonal decomposition (POD) reduced-order model (ROM) is used as the state equation. The key enablers to an accurate and robust POD ROM are the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem and the use of a snapshot ensemble for POD based on chirp-forced transients. Since the POD basis represents only velocities, we minimize a drag-related cost functional characteristic of the wake unsteadiness. The optimization problem is solved using Lagrange multipliers to enforce the constraints. 25% of relative drag reduction is found when the Navier-Stokes equations are controlled using a harmonic control function deduced from the optimal solution determined with the POD ROM. Earlier numerical studies concerning mean drag reduction are confirmed: it is shown, in particular, that without a sufficient penalization of the control input, our approach is energetically inefficient. The main result is that cost-reduction factors of 100 and 760 are obtained for the CPU time and the memory, respectively. Finally, limits of the performance of our approach are discussed.

Abstract: L'objectif de cette étude est de minimiser, par rotation sinusoïdale, le coefficient de traîee moyen d’un cylindre circulaire en régime laminaire. Une procédure d’optimisation couplant modèle réduit par POD et méthode à  région de confiance (TRPOD) est utilisée. Cette approche conduit à une réduction du coefficient de traînée de 30% pour un coût de calcul limité.  - The objective of this study is the mean drag minimization under rotary control of the cylinder wake in the laminar regime. The optimization problem is solved by a procedure that couples POD Reduced-Order Models (POD ROM) and trust-region method (TRPOD). Finally, 30% of relative mean drag reduction is found for reduced numerical costs.

Bergmann M. & Cordier L.

Journal of Fluid Mechanics, under review.

Abstract: This paper is a numerical investigation of laminar wake flows generated by a circular cylinder performing harmonic rotary oscillations. The two forcing parameters are the amplitude and the frequency of the sinusoidal tangential velocity of the cylinder. The modification of the dynamics of the forced wake is explored for a large range of amplitudes and frequencies. The evolution of the mean drag coefficient with the variation of the forcing parameters is investigated thoroughly. Depending on the values of the forcing parameters, we classified the flow into two categories: lock-on flows when the whole wake is synchronized with the forcing frequency and non lock-on flows otherwise. Moreover, we highlighted the existence of an effective maximal angle of rotation that minimizes the mean drag coefficient and that perfectly matches the angle defining the frontiers that separates the lock-on and the non lock-on regions. Finally, the existence of non-symmetric mean flows that generate non-zero mean lift is discussed.

On the estimation of the forest fire front position

Bergmann M., Séro-Guillaume O. & Ramezani S.

Inverse Problems, under review.

Abstract: This paper presents a method to estimate precisely a forest fire front position which is often undetectable due to the dense smoke released by the fire. Usually, a fire front position is defined by a curve of constant temperature. In this study we propose to use only some measurements of the radiative flux obtained by sensors to estimate the fire front position. The inverse method developed in this paper uses both simplex and conjugate gradient methods. The performance of this method is then demonstrated on a forest fire test case.

Bergmann M. & Cordier L.

Theoretical and Computational Fluid Dynamics, in preparation.

Abstract: Since fluid flows are commonly governed by few coherent structures it is possible to built simple low order models describing the flows for weak numerical costs. One promising solution is to use the Proper Orthogonal Decomposition (POD) to build effective Reduced Order Models (ROM). The effectiveness of such Reduced Order Models is that only few modes corresponding to the more energetic eddies (coherent structures) present in the flow are necessary. However, this property is also the main drawback of this approach because small eddies responsible of the main dissipation are not taken into account in the reduced order model leading to inaccurate solutions for the long time integration behavior. The purpose of this paper is to improve the accuracy of Reduced Order Models based on Proper Orthogonal Decomposition (POD ROM) in fluid mechanics context. The accuracy of low order models could be improved by adding artificial eddy viscosities on each POD modes in order to model the dissipative scales which are not solved by the reduced order model. The aim of this study is to determine these viscosities in an optimal way so that no other eddy viscosities could be more effective to improve the accuracy of the POD ROM. This method is tested on POD ROM for the circular cylinder wake flow. A great improvement could then be obtained, even if an unsteady forcing is apply to the cylinder. Accurate reduced order models could be also successful used in linear stability analysis to determine and predict the two first Hopf bifurcations that undergoes the flow.

Modeling and numerical simulations of 3D flows past self propelled fishes

Bergmann M. & Iollo A.

62st Annual APS/DFD Meeting
  Long Beach, California, USA, november 21-23, 2010.

Abstract: Modeling and simulation of three-dimensional flows past deformable bodies are considered. The incompressible Navier-Stokes equations are discretized in space onto a fixed cartesian mesh. The displacement of self propelled deformable objects through the fluid is computed from the Newtons laws (forces and torques computation) and is taken into account using a penalisation method. The interface between the solid and the fluid is tracked using a level-set description so that it is possible to simulate several bodies freely evolving in the fluid. The application considered is fish-like swimming . Fish maneuvers and propulsion efficiency for different swimming modes for a single fish or for a fish school are investigated.

Improvement of the POD ROM Robustness by Optimal Sampling

Bergmann M.

The 2009 Joint ASCE-ASME-SES Conference on Mechanics and Materials,
Blacksburg, Virginia, USA, June 24-27, 2009.

Abstract: This talk focuses on improving the robustness of the functional subspace built using Proper Orthogonal Decomposition (POD). Since a POD basis is able to give an optimal representation of the kinetic energy included in the snapshots database generated with some given input parameters, this same basis is not adapted to represent flow dynamics generated with other input parameters. Our aim is thus to build a POD basis that accurately represents the solution over a desired input parameter subspace by enlarging the database. We present a systematic method to sample the input parameter subspace. The basic idea is to add to the existing database, snapshots of the solution for which the POD approximation error is maximal. This is the Greedy sampling. The approach we follow is similar: it is based on finding the centroid of a region around the point where an estimate of the POD approximation error is maximal. We show numerical evidence that the Navier-Stokes residuals are a reliable estimate of the POD approximation error. Results relative to a 2D confined square cylinder wake flow are presented. The input parameter subspace is represented by an interval of Reynolds numbers that corresponds to periodical laminar flows. We show that a judicious choice of the sampling Reynolds numbers leads to a POD basis that minimizes the average approximation error on the chosen interval.

Improving Proper Orthogonal Decomposition Robustness by Optimal Sampling

Bergmann M., Iollo A.

61st Annual APS/DFD Meeting
San Antonio, Texas, USA, november 23-25, 2008.

Abstract: This talk focuses on improving the robustness of the functional subspace built using Proper Orthogonal Decomposition (POD). Since a POD basis is able to give an optimal representation of the kinetic energy included in the snapshots database generated with some given input parameters, this same basis is not adapted to represent flow dynamics generated with other input parameters. Our aim is thus to build a POD basis that accurately represents the solution over a desired input parameter subspace by enlarging the database. We present a systematic method to sample the input parameter subspace. The basic idea is to add to the existing database, snapshots of the solution for which the POD approximation error is maximal. This is the Greedy sampling. The approach we follow is similar: it is based on finding the centroid of a region around the point where an estimate of the POD approximation error is maximal. We show numerical evidence that the Navier-Stokes residuals are a reliable estimate of the POD approximation error. Results relative to a 2D confined square cylinder wake flow are presented. The input parameter subspace is represented by an interval of Reynolds numbers that corresponds to periodical laminar flows. We show that a judicious choice of the sampling Reynolds numbers leads to a POD basis that minimizes the average approximation error on the chosen interval.

Control of the cylinder wake by the Trust Region POD algorithm

Bergmann M., Cordier L. & Brancher J.-P.

77th GAMM (Gesellschaft fur Angewandte Mathematik und Mechanik)
Berlin, Germany, march 27-31, 2006.

Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models

Bergmann M., Cordier L. & Brancher J.-P.

58th Annual APS/DFD Meeting
Chicago, Illinois, USA, november 20-22, 2005.

Optimal rotary control of the cylinder wake using POD reduced order model

Bergmann M., Cordier L. & Brancher J.-P.

13th European Drag Reduction Meeting
Aussois, France, June 1-4, 2004.

Optimal control of cylinder wake using Proper Orthogonal Decomposition (POD)

Bergmann M., Cordier L. & Brancher J.-P.

56th Annual APS/DFD Meeting
East Rutherford, New Jersey, USA, november 23-25, 2003.

Amélioration de la robustesse des bases POD

Bergmann M., Lombardi E. and Iollo A.

19h Congrès Francais de Mécanique,
Marseille, august 24-28, 2009.

Abstract: Cette étude concerne l'amélioration de la robustesse du sous-espace propre construit par Décomposition Orthogonale aux valeurs Propres (POD). Puisqu'une base POD est uniquement capable de donner une représentation optimale de l'énergie incluse dans la base de données, cette même base n'est pas adaptée pour représenter une dynamique d'écoulement engendrée avec d'autres paramètres d'entrée (paramères de contrôle, nombre de Reynolds). L'objectif est donc de construire une base POD robuste capable de représenter tout un ensemble de dynamiques. Notre approche est basée sur une technique qui consiste à enrichir de facon itérative la base de donnée avec des réalisations calculées avec l'erreur commise par le modèle réduit est la plus élevée Greedy method. Des résultats issus du sillage d?un barreau dans un canal confiné seront présentés. Les paramères d'entrée seront limités à un intervalle pour le nombre de Reynolds qui correspond au régime périodique 2D. On montrera alors qu'il est possible d'améliorer l'efficacité de la méthode {\em "Greedy"}, en temps CPU en approximant l'erreur par une norme appropriée des résidus des équations de Navier-Stokes, ainsi qu'en terme d'approximation sur tout l'intervalle considéré.

Contrôle optimal par réduction de modèle POD et méthode à région de confiance du sillage laminaire d'un cylindre circulaire

Bergmann M., Cordier L. & Brancher J.-P.

Journées AUM/AFM 2006
La Rochelle, august 31 - september 1, 2006.

Contrôle optimal et sous-optimal d'écoulement décollé par réduction de modèles POD

Bergmann M., Cordier L. & Brancher J.-P.

Colloque de synthèse du GDR Contrôle des Décollements
Toulouse, France, november 7-8, 2005.
Cépaduès éditions.

Optimisation aérodynamique par modèle réduit POD et méthode à  région de confiance

Bergmann M., Cordier L. & Brancher J.-P.

17ème Congrès Français de Mécanique
Troyes, France, august 29 - september 2, 2005.

Contrôle optimal d'un modèle réduit du sillage d'un cylindre circulaire

Bergmann M., Cordier L. & Brancher J.-P.

39ème Colloque d'Aérodynamique Appliquée - AAAF
Paris, France, march 22-24, 2004.

Contrôle optimal par réduction de dynamique du sillage instationnaire d'un cylindre circulaire

Bergmann M., Cordier L. & Brancher J.-P.

16ème Congrès Français de Mécanique
Nice, France, september 1-5, 2003.