Bioinspired swimming simulations


Bergmann M. & Iollo A.

J. Comput. Physics 323, 310-321, 2016.

Abstract:  We present a method to simulate the flow past bioinspired swimmers starting from pictures of an actual fish. The overall approach requires i) a skeleton graph generation to get a level-set function from pictures; ii) optimal transportation to obtain the velocity on the body surface; iii) flow simulations realized with a Cartesian method based on penalization. This technique can be used to automate modeling swimming motion from data collected by biologists. We illustrate this paradigm by simulating the swimming of a mackerel fish.


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Enablers for high-order level set methods in fluid mechanics


Luddens F., Bergmann M. & Weynans L.

International Journal for Numerical Methods in Fluids 79 654-675, 2015.

Abstract:  In the context of numerical simulations of multiphysics flows, accurate tracking of an interface and consis- tent computation of its geometric properties are crucial. In this paper, we investigate a level set technique that satisfies these requirements and ensures local third-order accuracy on the level set function (near the interface) and first-order accuracy on the curvature, even for long-time computations. The method is devel- oped in a finite differences framework on Cartesian grids. As in usual level set strategies, reinitialization steps are involved. Several reinitialization algorithms are reviewed and mixed to design an accurate and fast reinitialization procedure. When coupled with a time evolution of the interface, the reinitialization procedure is performed only when there are too large deformations of the isocontours. This strategy limits the num- ber of reinitialization steps and shows a good balance between accuracy and computational cost. Numerical results compare well with usual level set strategies and confirm the necessity of the reinitialization procedure, together with a limited number of reinitialization steps. 

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Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer


Bergmann M., Iollo A. & Mittal R.

Bioinspiration & Biomimetics 9, 2014.

Abstract:   A computational model is used to examine the effect of caudal fin flexibility on the propulsive efficiency of a self-propelled swimmer. The computational model couples a penalization method based Navier–Stokes solver with a simple model of flow induced deformation and self-propelled motion at an intermediate Reynolds number of about 1000. The results indicate that a significant increase in efficiency is possible by careful choice of caudal fin rigidity. The flow-physics underlying this observation is explained through the use of a simple hydrodynamic force model and guidelines for bioinspired designs of flexible fin propulsors are proposed.


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An accurate cartesian method for incompressible flows with moving boundaries

Bergmann M. Hovnanian J. & Iollo A.

Communications in Computational Physics, 15, 1266-1290, 2014.

Abstract:   An accurate cartesianmethod is devised to simulate incompressible viscous flows past an arbitrary moving body. The Navier-Stokes equations are spatially discretized onto a fixed Cartesian mesh. The body is taken into account via the ghost-cell method and the so-called penalty method, resulting in second-order accuracy in velocity. The accuracy and the efficiency of the solver are tested through two-dimensional reference simulations. To show the versatility of this scheme we simulate a three-dimensional self propelled jellyfish prototype.

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Modeling and simulation of fish like swimming

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.

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[pdf(1Mo)] [pdf©(1Mo)] [http://]
Low-order models. Optimal sampling and linearized control strategies

Lombardi E., Bergmann M., Camarri S. & Iollo A.

Journal Européen des Systèmes Automatisés 45 (7), 2011.

Abstract:  We propose an optimal sampling strategy to build a robust low-order model. This idea is applied to the construction of a vortex wake model accurate for several regimes. In addition we explore the relationships between unstable modes and low-order modelling. An example of control based on a linearized approach is presented / Nous proposons une méthode d’échantillonnage optimale pour construire un modèle d’ordre réduit basé sur la décomposition orthogonale aux valeurs propres (POD) qui soit robuste par rapport à la variation des paramètres d’entrée. Cette méthode a été appliquée au cas de l’écoulement confiné autour d’un cylindre de section carré lorsque le nombre de Reynolds varie. Nous examinons également le lien entre les modes instables et la modélisation POD. Un exemple de contrôle basé sur une approche linéarisée est présenté.

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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.

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[pdf (1.9Mo)] [http://]
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.

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

Bergmann M., & Cordier L.

J. Comput. Physics, 227 (16), 2008.

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).

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

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

Commun. Numer. Meth. Engng.
, 24 (11), 879-896, 2008.

Abstract: This paper is devoted to the determination of the origin point in forest fires propagation using a control algorithm. The forest fires propagation are mathematically modelled starting from a reaction diffusion model. A volume of fluid (V.O.F.) formulation is also used to determine the fraction of the area which is burnt. After having developed the objective functional and its derivative, results from an optimization process based on the simplex method is presented. It is shown that the ignition point and the final time of the fire propagation are precisely recovered, even for a realistic, non-horizontal, terrain.

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[pdf (400k)] [pdf©(316k)] [http://]
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.

Mécanique & Industries 8 (2), pp. 111-118, 2007.

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.

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[pdf (3.1Mo)] [pdf©(2.3Mo)] [http://]
Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition

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

Notes on Numerical Fluid Mechanics and Multidisciplinary Design 95, 19 pages, 2007.

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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[pdf (1.8Mo)]
On the power used to control the circular cylinder drag

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

Phys. fluids 18 (8), 2006.

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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[pdf(160k)] [pdf©(135k)] [http://]
On the generation of a reverse Von Karman street for the controlled cylinder wake in the laminar regime

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

Phys. fluids 18 (2), 2006.

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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[pdf(530k)] [pdf©(452k)] [http://]
Optimal rotary control of the cylinder wake using POD reduced order model

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

Phys. fluids 17 (9), 2005.

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.

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[pdf(5.2Mo)] [pdf©(1.8Mo)] [http://]









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

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

Research Report 6758, INRIA, 2008.

Abstract: This report 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.

Download:
[pdf (1.9Mo)] [http://]
Improvement of Reduced Order Modeling based on Proper Orthogonal Decomposition

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

Research Report 6561, INRIA, 2008.

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

Download:
[pdf (7.6Mo)] [http://]
Control of the circular cylinder wake by Trust-Region methods and POD Reduced-Order Models

Bergmann M. & Cordier L.

Research Report 6552, INRIA, 2008.

Abstract: In this report we investigate the optimal control approach for the active control of the laminar circular cylinder wake flow (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 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 order 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 eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion 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.

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[pdf (6.3Mo)]  [http://]



Influence of caudal fin elasticity on swimmer propulsion

Bergmann M., Iollo A. & Mittal R.

2nd Symposium on Fluid-Structure-Sound Interactions and Control.
Hong-Kong/Macau, May 20-23, 2013.

Abstract: The aim of this study is to estimate the influence of caudal fin elasticity on swimmer propulsion. The swimmer paradigm is a simplified fish model where the fins are limited to a caudal one. This caudal fin can be either solid or elastic. The fin spine elasticity is modeled by lumped spring and dampers. The effect of the elasticity will be shown on 2D or 3D self propelled fishes.

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[Extended abstract pdf(0.5Mo)] [Acte pdf(1.7Mo)] [Slides pdf(9.1Mo)] [http://]
Numerical simulation of horizontal-axis wind turbine (HAWT)

Bergmann M. & Iollo A.

The seventh International Conference on Computational Fluid Dynamics
Mauna Lani Bay Hotel, Hawaii, Big Island, USA, July 9-13, 2012.


Abstract: The aim of this study is to estimate the wind power that can be extracted by an horizontal-axis wind turbine (HAWT) as a function of upstream wind. The incompressible Navier-Stokes equations are solved on a fixed cartesian mesh via a second-order accurate scheme in space and time. The turning blades and the mast are modeled by a penalization term in the governing equations within a collocated Chorin-Temam fractional time integration algorithm. The turbulence model is based on a large-eddy simulation model initially fitted on experimental data. This numerical procedure allows massive parallelization by using existing distributed linear-algebra libraries. The test case under consideration is the two blades NREL ametest wind turbine that has been intensively studied in wind tunnel.

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[Extended abstract pdf(0.5Mo)] [Acte pdf(1.7Mo)] [Slides pdf(9.1Mo)] [http://]
A Simplified Design Tool For Large Wind Turbine Blades

Jin X., Gaillardon B., Bergmann M. & Iollo A.

The European Wind Energy Association conference
Copenhagen, Danemark, April 16-19, 2012.


Abstract: The development of a design tool for large wind turbine blades was presented in this article.
The approch is composed of three main steps: Modeling and numerical simulation, Calibration and Optimization. This tool uses a model composed of Blade Element Momentum (BEM) theory combined with a Generalized Actuator Disc model. This model relies on the solution of a system of Partial Differential Equations (PDEs) and respects a reasonable compromise between model complexity and computational reliability. The performance of the power extraction is evaluated when a steady solution is obtained after pseudo temporal iterations. The aerodynamic forces are the main source terms in the Navier-Stokes equations. To recompensate the neglected 3D effects in this model, an automatic calibration mechanism was implemented and some good results were obtained and presented in this article. With the same Nelder-Mead Simplex algorithm for calibration, the twist optimization was also carried on. Plus, a graphical user interface is developped. Finally, a real optimited twist distribution was deduced with the optimized twist and the calibrating one. Validation of this conception will be carried on in the next step of this research. To conclude, this simplified model is low time-costly, user-friendly and efficient.


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[Extended abstract pdf(0.5Mo)] [Acte pdf(1.7Mo)] [Slides pdf(9.1Mo)] [http://]
Modeling and simulation of fish like swimming

Bergmann M. & Iollo A.

The sixth International Conference on Computational Fluid Dynamics
Saint Petersburg, Russia, July 12-16, 2010.


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.

Download:
[Extended abstract pdf(0.5Mo)] [Acte pdf(1.7Mo)] [Slides pdf(9.1Mo)] [http://]
Improvement of Reduced Order Modeling based on Proper Orthogonal Decomposition

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

The Fifth International Conference on Computational Fluid Dynamics
Seoul, Korea, July 7-11, 2008.


Abstract: This study focuses on stabilizing Reduced Order Model based on Proper Orthogonal Decomposition (POD) and on improving the POD functional subspace. A modified reduced order model (ROM) that incorporates directly the pressure term is proposed. The ROM is obtained by seeking a solution that lives in the POD subspace and at the same time minimizes the Navier-Stokes residuals. Both ROM stabilization and POD subspace adaptation make use of methods based on the fine scale equation that is approximated using the residuals of the Navier-Sokes equations.  Results are shown for the 2D confined cylinder wake flow.

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[Extended abstract pdf(0.5Mo)] [Acte pdf(1.7Mo)] [Slides pdf(9.1Mo)] [http://]
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.

44th IEEE Conference on Decision and Control and European Control Conference ECC 2005
Sevilla, Sapin, december 12-15, 2005.


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 mean drag minimization of the wake 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 OrderModel (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 a high fidelity model. A lot of computational work is indeed saved because the optimization process is now based only on low-fidelity models. When the TRPOD is applied to the wake flow configuration, this approach leads to a relative mean drag reduction of 30% for reduced numerical costs.

Download:
[Acte pdf(1.4Mo)] [Slides pdf(3.9Mo)] [http://]
Optimal Rotary Control of The Cylinder Wake Using POD Reduced Order Model

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

First European Forum on Flow Control ( LEA-CEAT, Univ. Poitiers)
Poitiers, France, October 11-14, 2004.


Abstract: This communication investigates the optimal control approach for the active control and drag optimization of incompressible viscous flow past cylinders. The control function is the time harmonic 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. Since the POD basis represents only velocities, we minimize a drag-related cost function 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 the optimal control function determined with the POD ROM. A cost reduction factor of respectively one hundred and six hundred is obtained for respectively the CPU time and the memory.

Download:
[Acte pdf(170k)] [Slides pdf(2.7Mo)]
Optimal rotary control of the cylinder wake using POD reduced order model

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

2nd AIAA Flow Control Conference
Portland, Oregon, USA, june 28 - july 1, 2004.
AIAA paper 2004-2323.

Abstract: This article investigates the optimal control approach for the active control and drag optimization of incompressible viscous flow past cylinders. The control function is the time harmonic 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. Since the POD basis represents only velocities, we minimize a drag-related cost function 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 the optimal control function determined with the POD ROM. A cost reduction factor of respectively one hundred and six hundred is obtained for respectively the CPU time and the memory. Finally, limits of the performance of our approach are discussed.

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Influence of the caudal fin rigidity on swimmer propulsion efficiency

Bergmann M., Iollo A. and Mittal R.

EUROMECH Colloquium 549 on Immersed Boundary Methods
Leiden, The Netherlands, June 17-19 2013.

Abstract: The aim of this study is to estimate the influence of caudal fin elasticity on swimmer propulsion. The swimmer paradigm is a simplified fish model where the fins are limited to a caudal one. This caudal fin can be either solid or elastic. The fin spine elasticity is modeled by lumped spring and dampers. The effect of the elasticity will be shown on 2D or 3D self propelled fishes.

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On the efficiency of fish like swimming

Bergmann M. & Iollo A.

65rd Annual APS/DFD Meeting
  San Diego, California, USA, november 18-20, 201
2.

Abstract: The aim of this work is to investigate numerically the efficiency of fish like swimming. We are interested in the Gray paradox stating that the power required for a fish to swim can be about seven times the muscular power available for propulsion. Even if this paradox has been contested in the literature it was confirmed experimentally at MIT. Our goal is then to perform numerical simulations of flow around a self propelled fish for several values of the Reynolds number. The Navier-Stokes equations are discretized onto a cartesian mesh and the interface between the fluid and the fish is computed using immersed boundary like method. The motion of the fish is computed in a Lagrangian way from the Newton's laws.

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

Bergmann M. & Iollo A.

63srd 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.

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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.

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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.

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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.

Abstract: In this communication we investigate the optimal control approach for the drag minimization of the circular cylinder wake flow in the laminar regime (Re = 200). The control function is the time harmonic angular velocity of the rotating cylinder. The resolution of the discretized optimality system, built from the Navier-Stokes equations as state equation, leads to tremendous computational costs. With the aim of making computationally effective the optimization process, a Proper Orthogonal Decomposition (POD) Reduced Order Model (ROM) is then derived to be used as state equation. The range of validity of the POD ROM is generally limited to a vicinity of the design parameters in the control parameter space. Therefore, to overcome this difficulty, 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 prove that the iterates produced by the TRPOD algorithm will converge to the solution of the high fidelity optimization problem. Due to the use of reduced order models, the computational work involved by the TRPOD is then greatly reduced. Finally, the application of the TRPOD to the cylinder wake flow configuration leads to a relative mean drag reduction greater than 30% for reduced numerical costs.

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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.


Abstract: The optimal control approach for the active control of the circular cylinder wake flow considered in the laminar regime (Re = 200) is investigated. The objective is the mean drag minimization of the wake where the control function is the time harmonic angular velocity of the rotating cylinder. In order to reduce the computational costs, the optimization process is not based on the Navier-Stokes equations as state equations but rather on low-fidelity models derived with the Proper Orthogonal Decomposition (POD). Since the range of validity of this POD Reduced Order Model (ROM) is generally restricted to the vicinity of the design parameters in the control parameter space, the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), is used to update the ROMs 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 a high fidelity model. When the TRPOD is applied to the wake flow configuration, this approach leads to a relative mean drag reduction of 30% for reduced numerical costs.

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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.


Abstract: This communication investigates the optimal control approach for the active control and drag optimization of incompressible viscous flow past cylinders. The control function is the time harmonic 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. Since the POD basis represents only velocities, we minimize a drag-related cost function 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 the optimal control function determined with the POD ROM. A cost reduction factor of respectively one hundred and six hundred is obtained for respectively the CPU time and the memory.

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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.

Abstract: An active control strategy dedicated to separated flows is presented. The methodology is applied to the wake flow behind a circular cylinder at a Reynold's number of 200. This configuration is studied numerically through the use of the Proper Orthogonal Decomposition (POD). Our objective is to minimize the drag by unsteady rotation of the cylinder (Tokumaru and Dimotakis, 1991). The application of the POD to flow realizations yields an optimal, in an energetic sense, basis set. Unfortunately, the basis functions derived for the uncontrolled flow were incapable of successfully capturing the flow due to a change in the cylinder rotation. To solve this problem, we first generate "generalized basis functions" by rotating the cylinder with a chirp excitation. Then, a low-order dynamical system was obtained by Galerkin projection of the Navier-Stokes equations onto the generalized POD functions. Next, the control surface motion was incorporated into the POD model using the control function method introduced by Graham et al. (1999). Finally, following the method introduced by Ravindran (2000), this reduced order model is used as the state equations in the optimality system derived to estimate the flow control parameters. Results obtained with this reduced order adaptive controller based on POD will be presented.

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[Slides pdf (1.4Mo)]



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.

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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.


Abstract: L'objectif de cette étude est de minimiser, par rotation sinusoïdale, le coefficient de traînée moyen d'un cylindre circulaire en régime laminaire. Une procédure d'optimisation couplant modêle 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 and trust region method (TRPOD). Finally, 30% of relative mean drag reduction is found for reduced numerical costs.

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[Acte pdf(570k)] [Slides pdf(4.0Mo)]
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.


Abstract: Cette communication présente une synthèse des travaux réalisés par notre groupe sur le contrôle de l'écoulement de sillage laminaire en aval d'un cylindre circulaire. L'objectif est de démontrer, qu'il est possible, moyennant certaines précautions qui seront décrites, de résoudre un problème de contrôle d'écoulement par une procédure couplant contrôle optimal ou sous-optimal et un modèle réduit de dynamique construit par POD. Cette approche conduit à  une réduction relative du coefficient de trainée moyen de l'ordre de 25 à  30% pour des coûts de calcul limités.

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[Acte pdf (1.0Mo)] [Slides pdf(3.4Mo)] [http://]
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.

Abstract: L'objectif de cette étude est de minimiser, par rotation sinusoïdale, le coefficient de trainée 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 trainée 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 and trust region method (TRPOD). Finally, 30% of relative mean drag reduction is found for reduced numerical costs.

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[Acte pdf(570k)] [Slides pdf(9.4Mo)]
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.

Abstract: Cette communication aborde le contrôle d'écoulement par une méthode couplant contrôle optimal et réduction de dynamique. L'objectif est de démontrer que cette approche conduit à  une amélioration importante des performances aérodynamiques et qu'elle s'accompagne d'une réduction drastique du coût de synthèse de la loi de contrôle, permettant en cela de s'intéresser à  des configurations industrielles. Pour des raisons de facilité de mise en oeuvre, la pertinence de la méthode est évaluée sur une configuration décollée générique constituée par le sillage laminaire bidimensionnel d'un cylindre circulaire. Dans cette étude, la loi de contrôle est l'évolution temporelle de la vitesse tangentielle du cylindre. La fonction objectif à  minimiser par contrôle optimal est la trainée aérodynamique. L'écoulement est simulé numériquement pour un nombre de Reynolds égal à  200 par une méthode d'éléments finis. En raison des coûts numériques importants (temps CPU et encombrement mémoire) liés à  l'approche par contrôle optimal, un modèle d'ordre réduit basé sur la Décomposition Orthogonale aux Valeurs Propres (Proper Orthogonal Decomposition, POD) est utilisé comme équations d'état pour résoudre le problème d'optimisation par la méthode des multiplicateurs de Lagrange. Lorsque les équations de Navier-Stokes sont résolues à  nouveau en utilisant la loi de contrôle déterminée à  l'aide du système réduit POD, une réduction de 25% du coefficient de trainée moyen est obtenue. Comparé au cas oû les équations de Navier-Stokes sont utilisées comme équations d'état pour résoudre le problème de contrôle optimal, notre approche nécessite un temps de calcul et un stockage mémoire respectivement 100 fois et 600 fois inférieurs. Finalement, les limites de l'approche couplant modèle réduit POD et contrôle optimal sont abordées.

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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.

Abstract: Dans cette étude, nous considérons pour problème modèle d'une configuration décollée, l'écoulement autour d'un cylindre de section circulaire pour un nombre de Reynolds de 200. Notre objectif est de minimiser l'instationnarité de sillage par rotation sinusoïdale du cylindre autour de son axe principal. La résolution de problèmes d'optimisation de grande taille restant d'un coût prohibitif, il existe une vraie demande de modèles d'ordre faible de dynamique permettant de représenter pour un coût de calcul limité l'essentiel de la dynamique non linéaire du système. L'approche spécifique suivie dans ces travaux est de construire un modèle réduit de dynamique basé sur la POD (Proper Orthogonal Decomposition). Par la suite, par utilisation d'une méthode adaptative proposée par Ravindran (2000), ce modèle est utilisé comme équation d'état dans le système optimal développé pour déterminer les paramètres de contrôle de l'écoulement. Les premiers résultats issus de la boucle d'optimisation sont finalement présentés.

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[Acte pdf(127k)] [Slides pdf(1.3Mo)]









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

Bergmann M.

Smai 2009
La Colle sur Loup, may 25-29, 2009.

Abstract: Dans cette communication nous nous intéressons à  la simulation numérique de la nage de poissons dans un fluide réel. Mathématiquement, l'écoulement est gouverné par les équations de Navier Stokes incompressibles. Numériquement ces équations sont discrétisées en espace sur une grille cartésienne par différences finies centrées d'ordre deux pour les termes diffusifs et décentrées d'ordre trois pour les termes convectifs et par une méthode de pro jection de type prédicteur correcteur en temps (Chorin) . Les obstacles (poissons) autour duquel est résolu l'écoulement sont modélisés avec une technique de pénalisation des équations dans laquelle les obstacles sont considérés comme des milieux poreux avec une très faible perméabilité [1]. Le système total (fluide+obstacles) peut alors être considéré comme un seul  écoulement. Des fonctions level set sont utilisées pour capturer les interfaces entre le fluide et les obstacles [3]. Enfin, les interactions entre le fluide et les obstacles sont prises en compte par un calcul des forces exercée par le fluide sur l'obstacle. Apès avoir validé l'outil numérique, nous présenterons quelques résultats de simulations 2D pour la nage de poissons et de méduses. Nous montrerons qu'au cours de leur nage les poissons géèrent une allée de Von Karman inversée [2] qui est l'origine d'une force de poussée. Nous montrerons que la forme des poissons ainsi que la loi de nage (amplitude du mouvement de la queue) sont grandement influents sur le déplacement. Nous nous intéresserons également l'efficacité de la nage en groupe.