Michele Romanelli, Samir
Beneddine, Ivan Mary, Héloïse
Beaugendre, Michel Bergmann, Denis Sipp
International Journal of Heat
and Fluid Flow 99 (2023) 109097
Abstract: This article
presents a data-based methodology to build
Reynolds-Averaged Navier–Stokes (RANS)
wall models for aerodynamic simulations at
low Mach numbers. Like classical
approaches, the model is based on
nondimensional local quantities derived
from the wall friction velocity 𝑢𝜏, the
wall viscosity 𝜇𝑤, and the wall density
𝜌𝑤 . A fully-connected neural network
approximates the relation 𝑢+ = 𝑓 (𝑦+ ,
𝑝+ ). We consider reference data
(obtained with RANS simulations based on
fine meshes up to the wall) of attached
turbulent flows at various Reynolds
numbers over different geometries of
bumps, Raelicovering a range of wall
pressure gradients. After training the
neural networks on a subset of the
reference data, the paper assesses their
ability to accurately recover data for
unseen conditions on meshes that have been
trimmed from the wall up to an interface
height where the learned wall law is
applied. The network’s interpolation and
extrapolation capabilities are quantified
and carefully examined. Overall, when
tested within its interpolation and
extrapolation capabilities, the neural
network model shows good robustness and
accuracy. The global error on the skin
friction coefficient is a few percent and
behaves consistently over all the
considered test cases.
Download: [pdf] |
Michel Bergmann, Antoine
Fondanèche, Angelo Iollo
Journal of Computational Physics,
471 (2022)
Abstract: A
quadtree-based fully Eulerian finite volume
approach for the simulation of
fluid-structure interaction problems is
presented. Both fluid and structure phases,
which are assumed to be incompressible and
viscous, are solved monolithically on the
whole computational domain. The
discretization stencils are limited to the
first layer of neighbors thus enhancing the
efficiency of the parallel computations
while limiting the numerical order of the
finite volume discretizations that can be
reached. The behavior of hyperelastic
structures is described with the non-linear
Mooney-Rivlin model. The simulation of
several two dimensional test cases is
performed on uniform and quadtree grids and
results are compared with the literature. To
illustrate the versatility of the numerical
model presented, a biomedical application,
the axisymmetric simulation of a blood flow
in a cardiac pump, is presented.
Download: [pdf] |
Michel Bergmann
Bioinspir. Biomim. 17
065010, 2022
Abstract: A computational model is developed to investigate the jump of a self-propelled dolphin out of water. This model relies on the Navier–Stokes equations, where a fictitious domain approach with the volume penalization method is used for fluid-structure coupling, and the continuous surface force approach is used to model the water–air interface, the latter being tracked in a level-set framework. The dolphin’s geometry is based on freely available data from the literature. While body deformation is imposed, the leading linear and angular displacements are computed from Newton’s laws. Numerical simulations show that it is necessary to generate large propulsives forces to allow the jump out of water. When the dolphin is out of water, its trajectory follows a purely ballistic one. Download: [pdf] |
Inferring
characteristics of bacterial swimming in
biofilm matrix from time-lapse confocal
laser scanning microscopy
Guillaume Ravel, Michel
Bergmann, Alain Trubuil, Julien Deschamps,
Romain Briandet, Simon Labarthe
eLife 11:e76513 (2022). Abstract: Biofilms are
spatially organized microorganism colonies
embedded in a self-produced matrix,
conferring to the microbial community
resistance to environmentalstresses. Motile
bacteria have been observed swimming in the
matrix of pathogenic exogeneous host
biofilms. This observation opened new
promising routes for deleterious biofilms
biocontrol: these bacterial swimmers enhance
biofilm vascularization 10 for chemical
treatment or could deliver biocontrol agent
by microbial hitchhiking orlocal synthesis
Hence, characterizing swimmer trajectories
in the biofilm matrix is of particular
interest to understand and optimize its
biocontrol. In this study, a new methodology
is developed to analyze time-lapse confocal
laser scanning images to describe and
compare the swimming trajectories of bac- 15
terial swimmers populations and their
adaptations to the biofilm structure. The
method is based on the inference of a
kinetic model of swimmer population including mechanistic
interactions with the host biofilm. After
validation on synthetic data, the
methodology is implemented on images of
three different motile Bacillus species
swimming in a Staphylococcus aureus biofilm.
The fitted model allows to 20 stratify the
swimmer populations by their swimming
behavior and provides insights into the
mechanisms deployed by the micro-swimmers to
adapt their swimming traits to the biofilm
matrix.
Download: [pdf] |
ADER scheme for
incompressible Navier-Stokes equations on
overset grids with a compact transmission
condition
Michel Bergmann, Michele
Giuliano Carlino, Angelo Iollo, Haysam
Telib
Journal of Computational Physics 467 (2022) 111414 Abstract: A space-time
Finite Volume method is devised to simulate
incompressible viscous flows in an evolving
domain. Inspired by the ADER method (based
on a Finite-Element-
prediction/Finite-Volume-correction
approach), the Navier-Stokes equations are
discretized onto a space-time overset grid
which is able to take into account both the
shape of a possibly moving object and the
evolution of the domain. A compact
transmission condition is employed in order
to mutually exchange information from one
mesh to the other. The resulting method is
second order accurate in space and time for
both velocity and pressure. The accuracy and
efficiency of the method are tested through
reference simulations.
Download: [pdf] |
SECOND ORDER ADER
SCHEME FOR UNSTEADY ADVECTION-DIFFUSION ON
MOVING OVERSET GRIDS WITH A
COMPACT TRANSMISSION CONDITION MICHEL BERGMANN , MICHELE GIULIANO CARLINO, ANGELO IOLLO SIAM J. SCI. COMPUT. 44, No. 1, pp. A524-A553 Abstract: We propose a
space-time finite volume scheme on moving
Chimera grids for a general
advection-diffusion problem. Special care
is devoted to grid overlapping zones in
order to devise a compact and accurate
discretization stencil to exchange
information between different mesh
patches. Like in the arbitrary high order
derivatives method, the equations are
discretized on a space-time slab. Thus,
instead of time-dependent spatial
transmission conditions between relatively
moving grid blocks, we define
interpolation polynomials on arbitrarily
intersecting space-time cells at the block
boundaries. Through this scheme, a
mesh-free FEM-predictor/FVM-corrector
approach is employed for representing the
solution. In this discretization
framework, a new space-time local Lax-
Friederichs stabilization speed is defined
by considering both the advective and the
diffusive nature of the equation. The
numerical illustrations for linear and
nonlinear systems show that background and
foreground moving meshes do not introduce
spurious perturbation to the solution,
uniformly reaching second order accuracy
in space and time. Finally, it is shown
that several foreground meshes, possibly
overlapping and with independent
displacements, can be employed thanks to
this approach.
Download: [pdf] |
The DGDD Method for
Reduced-Order Modeling of Conservation
Laws
Riffaud S., Bergmann M., Farhat C., Grimberg S., Iollo A. J. Comput. Physics, 2021. Abstract: The
discontinuous Galerkin domain
decomposition (DGDD) method couples
subdomains of high-fidelity polynomial
approximation to regions of
low-dimensional resolution for the
numerical solution of systems of
conservation laws. In the low- fidelity
regions, the solution is approximated by
empirical modes constructed by Proper
Orthogonal Decomposition and a
reduced-order model is used to predict the
solution. The high-dimensional model
instead solves the system of conserva-
tion laws only in regions where the
solution is not amenable to a
low-dimensional representation. The
coupling between the high-dimensional and
the reduced-order models is then performed
in a straightforward manner through
numerical fluxes at discrete cell
boundaries. We show results from
application of the proposed method to
parametric problems governed by the
quasi-1D and 2D compressible Euler
equations. In particular, we investigate
the prediction of unsteady flows in a
converging-diverging nozzle and over a
NACA0012 airfoil in presence of shocks.
The results demonstrate the stability and
the accuracy of the proposed method and
the significant reduction of the
computational cost with respect to the
high- dimensional model.
Download: [pdf] https://doi.org/10.1016/j.jcp.2021.110196 |
Calibration of
projection-based reduced-order models for
unsteady compressible flows
Zucatti V., Wolf W. & Bergmann M. J. Comput. Physics, 433, 2021. Abstract: An analysis
of calibration for reduced-order models
(ROMs) is presented in this work. The
Galerkin and least-squares Petrov-Galerkin
(LSPG) methods are tested on compressible
flows involving a disparity of temporal
scales. A novel calibration strategy is
proposed for the LSPG method and two test
cases are analyzed. The first consists of a
subsonic airfoil flow where boundary layer
instabilities are responsible for
trailing-edge noise generation and the
second comprises a supersonic airfoil flow
with a transient period where a detached
shock wave propagates upstream at the same
time that shock-vortex interaction occurs at
the trailing edge. Results show that
calibration produces stable and long-time
accurate Galerkin and LSPG ROMs for both
cases investigated. The impact of hyper-
reduction is tested on LSPG models via an
accelerated greedy missing point estimation
(MPE) algorithm. For the first case
investigated, LSPG solutions obtained with
hyper- reduction show good comparison with
those obtained by the full order model.
However, for the supersonic case the
transient features of the flow need to be
properly captured by the sampled points of
the accelerated greedy MPE method.
Otherwise, the dynamics of the moving shock
wave are not fully recovered. The impact of
different time-marching schemes is also
assessed and, differently than reported in
literature, Galerkin models are shown to be
more accurate than those computed by LSPG
when the non-conservative form of the
Navier-Stokes equations is solved. For the
supersonic case, the Galerkin and LSPG
models (without hyper-reduction) capture the
overall dynamics of the detached and oblique
shock waves along the airfoil. However, when
shock-vortex interaction occurs at the
trailing-edge, the Galerkin ROM is able to
capture the high-frequency fluctuations from
vortex shedding while the LSPG presents a
more dissipative solution, not being able to
recover the flow dynamics.
Download: [pdf] https://doi.org/10.1016/j.jcp.2021.110196 |
Fluid-solid Floquet
stability analysis of self-propelled
heaving foils
Benetti Ramos, L., Marquet, O., Bergmann, M. and Iollo, A. Journal of Fluid Mechanics, 910, 2021. Abstract: We
investigate the role of linear mechanisms in
the emergence of nonlinear horizontal
self-propelled states of a heaving foil in a
quiescent fluid. Two states are analysed: a
periodic state of unidirectional motion and
a quasi-periodic state of slow back and
forth motion around a mean horizontal
position. The states emergence is explained
through a fluid–solid Floquet stability
analysis of the non-propulsive symmetric
base solution. Unlike a purely hydrodynamic
analysis, our analysis accurately determines
the locomotion states onset. An unstable
synchronous mode is found when the
unidirectional propulsive solution is
observed. The obtained mode has a propulsive
character, featuring a mean horizontal
velocity and an asymmetric flow that
generates a horizontal force accelerating
the foil. An unstable asynchronous mode,
also featuring flow asymmetry and a non-zero
velocity, is found when the back and forth
state is observed. Its associated complex
multiplier introduces a slow modulation of
the flapping period, agreeing with the
quasi-periodic nature of the back and forth
regime. The temporal evolution of this
perturbation shows how the horizontal force
exerted by the flow is alternatively
propulsive or resistive over a slow period.
For both modes, an analysis of the velocity
and force perturbation time-averaged over
the flapping period is used to establish
physical instability criteria. The behaviour
for a large solid-to-fluid density ratio of
the modes is thus analysed. The asynchronous
fluid–solid mode converges towards the
purely hydrodynamic one, whereas the
synchronous mode becomes marginally unstable
in our analysis not converging to the purely
hydrodynamic analysis where it is never
destabilised.
Download: [pdf] https://doi:10.1017/jfm.2020.1021 |
Methodology for Numerical
Simulations of Ellipsoidal Particle-Laden
flow.
Lambert B., Weynans L. & Bergmann M. International Journal for Numerical Methods in Fluids, 2020. Abstract: Despite
being relevant in many natural and
industrial processes, suspensionsof
nonspherical particles have been largely
underinvestigated compared with the
extensive analyses made on the
gravity-driven motions of spherical
particles. One of the main reasons for this
disparity is the difficulty of accurately
correcting the short-range hydrodynamic
forces and torques acting on complex
particles. These effects, also known as
lubrication, are essential to the suspension
of the particles and are usually poorly
captured by direct numerical simulation of
particle-laden flows. In this article, we
propose a partitioned volume
penalization-discrete element method solver,
which estimates the unresolved hydrodynamic
forces and torques. Corrections are made
locally on the surface of the interacting
particles without any assumption on the
particle global geometry. Numerical
validations have been made using ellipsoidal
particles immersed in an incompressible
Navier-Stokes flow.
Download: [pdf] https://doi.org/10.1002/fld.4809 |
Local Lubrication model
for spherical particles within an
incompressible Navier-Stokes flow.
Lambert B., Weynans L. & Bergmann M. Phys. Rev. E. 97, 033313, 2018. Abstract: The
lubrication forces are short-range
hydrodynamic interactions essential to
describe suspension of the particles.
Usually, they are underestimated in direct
numerical simulations of particle-laden
flows. In this paper, we propose a
lubrication model for a coupled volume
penalization method and discrete element
method solver that estimates the unresolved
hydrodynamic forces and torques in an
incompressible Navier-Stokes flow.
Corrections are made locally on the surface
of the interacting particles without any
assumption on the global particle shape. The
numerical model has been validated against
experimental data and performs as well as
existing numerical models that are limited
to spherical particles.
Download: [pdf] https://doi.org/10.1103/PhysRevE.97.033313 |
A zonal Galerkin-free POD
model for incompressible flows
Bergmann M., Ferrero A., Iollo A., Lombardi E., Scardigli A & Telib H. J. Comput. Physics, 352, 301-325, 2018. Abstract:
A domain decomposition method which
couples a high and a low-fidelity model is
proposed to reduce the computational cost
of a flow simulation. This approach
requires to solve the
high-fidelity model in a small portion of
the computational domain while
the external field is described by a
Galerkin-free Proper Orthogonal
Decomposition (POD) model.
We propose an error indicator to determine
the extent of the interior domain and to
perform an optimal coupling between the
two models. This zonal approach can be
used to study multi-body
configurations or to perform detailed
local analyses in the framework of
shape optimisation problems. The
efficiency of the method to perform
predictive low-cost
simulations is investigated for an
unsteady flow and for an aerodynamic shape
optimisation problem.
Download: [pdf] https://doi.org/10.1016/j.jcp.2017.10.001 |
A finite-difference
method for the variable coefficient
Poisson equation on hierarchical Catresian
meshes
Raeli A., Bergmann M & Iollo A. J. Comput. Physics, 325, 59-77, 2018. Abstract:
We consider problems governed by a linear
elliptic equation with varying
coefficients across internal interfaces.
The solution and its normal derivative can
undergo significant variations through
these internal boundaries. We present a
compact finite-difference scheme on a
tree-based adaptive grid that can be
efficiently solved using a natively
parallel data structure. The main idea is
to optimize the truncation error of the
discretization scheme as a function of the
local grid configuration to achieve
second-order accuracy. Numerical
illustrations are presented in two and
three-dimensional configurations.
Download: [pdf] https://doi.org/10.1016/j.jcp.2017.11.007 |
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.
Download: [pdf] https://doi.org/10.1016/j.jcp.2016.07.022 |
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 consistent 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 developed 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 number 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.
Download: |
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.
Download: [pdf] http://dx.doi.org/10.1088/1748-3182/9/4/046001 |
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.
Download: [pdf] https://doi.org/10.4208/cicp.220313.111013a |
Reduced
Order Models at Work in Aeronautics and
Medicine
Bergmann M., Colin T., Iollo A., Lombardi D., Saut O. and Telib H.. Modeling, Simulation and Applications, 9, 2013. Abstract: We review a few applications of reduced-order modeling in aeronautics and medicine. The common idea is to determine an empirical approximation space for a model described by partial differential equations. The empirical approximation space is usually spanned by a small number of global modes. In case of time-periodic or mainly diffusive phenomena it is shown that this approach can lead to accurate fast simulations of complex problems. In other cases, models based on definition of transport modes significantly improve the accuracy of the reduced model. [pdf] https://doi.org/10.1007/978-3-319-02090-7_11 |
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.
Download: [pdf] https://doi.org/10.1016/j.jcp.2010.09.017 |
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.
Download: [pdf] http://dx.doi.org/10.3166/jesa.45.575-593 |
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.
Download: [pdf] https://doi.org/10.1002/fld.2025 |
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.
Download: [pdf] https://doi.org/10.1016/j.jcp.2008.09.024 |
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).
Download: [pdf] https://doi.org/10.1016/j.jcp.2008.04.034 |
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.
Download: [pdf] https://doi.org/ 10.1002/cnm.990 |
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. Download: [pdf] https://doi.org/10.1051/meca:2007028 |
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). Download: [pdf] https://doi.org/10.1007/978-3-540-71439-2_19 |
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. Download: [pdf] |
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. Download: [pdf] |
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.
Download: [pdf] https://doi.org/10.1063/1.2033624 |
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] |
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] |
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.
Download: [pdf] |
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.
Download: [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.
Download: [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. Download: [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.
Download: [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://] |
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.
Download: |
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.
Download: |
65rd
Annual
APS/DFD Meeting
San Diego, California, USA, november 18-20, 2012. 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.
Download: |
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.
Download: |
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.
Download: |
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.
Download: |
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. Download: [Slides pdf (2.2Mo)] |
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. Download: [Slides pdf (3.4Mo)] |
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. Download: [Slides pdf (1.8Mo)] [Extended abstract pdf(174k)] |
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. Download: [Slides pdf (1.4Mo)] |
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. Download: [Acte pdf(570k)] [Slides pdf(4.0Mo)] |
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. Download: [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. Download: [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. Download: [Acte pdf(800k)] [Slides pdf(1.6Mo)] |
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. Download: [Acte pdf(127k)] [Slides pdf(1.3Mo)] |