Geometrical Reduced Order Modeling (ROM) by Proper Orthogonal Decomposition (POD) for the incompressible Navier Stokes equations

2018 ◽  
Author(s):  
Nissrine Akkari ◽  
Renaud Mercier ◽  
Vincent Moureau
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Proper Orthogonal

scholarly journals Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

2009 ◽  
Vol 629 ◽  
pp. 41-72 ◽  
Author(s):  
ALEXANDER HAY ◽  
JEFFREY T. BORGGAARD ◽  
DOMINIQUE PELLETIER
Keyword(s):  
Sensitivity Analysis ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Selection Step ◽  
Low Dimensional ◽  
Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

scholarly journals Investigation of the Effects of Length to Depth Ratio on Open Supersonic Cavities Using CFD and Proper Orthogonal Decomposition

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ibrahim Yilmaz ◽  
Ece Ayli ◽  
Selin Aradag
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Reduced Order ◽  
Modeling Approach ◽  
Depth Ratio ◽  
Minimum Number ◽  
Proper Orthogonal

Simulations of supersonic turbulent flow over an open rectangular cavity are performed to observe the effects of length to depth ratio (L/D) of the cavity on the flow structure. Two-dimensional compressible time-dependent Reynolds-averaged Navier-Stokes equations with k-ωturbulence model are solved. A reduced order modeling approach, Proper Orthogonal Decomposition (POD) method, is used to further analyze the flow. Results are obtained for cavities with severalL/Dratios at a Mach number of 1.5. Mostly, sound pressure levels (SPL) are used for comparison. After a reduced order modeling approach, the number of modes necessary to represent the systems is observed for each case. The necessary minimum number of modes to define the system increases as the flow becomes more complex with the increase in theL/Dratio. This study provides a basis for the control of flow over supersonic open cavities by providing a reduced order model for flow control, and it also gives an insight to cavity flow physics by comparing several simulation results with different length to depth ratios.

On Reduced Order Modeling of Transient Flows Using the Proper Orthogonal Decomposition

2012 ◽  
Author(s):  
Fariduddin Behzad ◽  
Brian T. Helenbrook ◽  
Goodarz Ahmadi
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Reduced Order Modeling ◽  
Orthogonal Decomposition ◽  
Basis Functions ◽  
Navier Stokes ◽  
Reduced Order ◽  
Split Method ◽  
Proper Orthogonal

Reduced-order modeling (ROM) of transient fluid flows using the proper orthogonal decomposition (POD) was studied. Particular attention was given to incompressible, unsteady flow over a two-dimensional NACA0015 airfoil in the laminar regime. When the airfoil sheds vortices, a transient blowing through a jet placed at the 10% chord location was imposed. POD modes were derived from the numerical solution of the flow obtained using an hp-finite element method. The ROM was obtained by a streamwise-upwind-Petrov-Galerkin (SUPG) projection of the incompressible Navier–Stokes equations onto the space spanned by the POD modes. The extraction of accurate POD-based reduced order model of this flow was explored using three different POD mode generation methods. The first approach was the split method, which superposes modes derived from simulations of the blowing jet with no flow and simulations of the baseline flow with no jet. The second method combined POD modes derived from simulations having both the jet and flow with modes obtained from simulation of only the flow. These modes were generated after the simulations reached the periodic state. The third and newly proposed approach was to generate a set of modes called “Generalized POD basis functions.” These modes were derived from simulations where the jet’s flow amplitude is varied slowly. For each method, the results were compared with detailed Finite Element solutions and the accuracy and efficiency of different methods were evaluated. The newly proposed “Generalized POD basis functions” approach predicted the transient response of the system most accurately.

scholarly journals A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flows

2019 ◽  
Vol 40 (2) ◽  
pp. 886-913 ◽  
Author(s):  
Max Gunzburger ◽  
Traian Iliescu ◽  
Michael Schneier
Keyword(s):  
Numerical Simulation ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Spatial Filter ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Numerical Discretization ◽  
Proper Orthogonal ◽  
Convection Dominated

Abstract Partial differential equations (PDEs) are often dependent on input quantities that are uncertain. To quantify this uncertainty PDEs must be solved over a large ensemble of parameters. Even for a single realization this can be a computationally intensive process. In the case of flows governed by the Navier–Stokes equations, an efficient method has been devised for computing an ensemble of solutions. To further reduce the computational cost of this method, an ensemble-proper orthogonal decomposition (POD) method was recently proposed. The main contribution of this work is the introduction of POD spatial filtering for ensemble-POD methods. The POD spatial filter makes possible the construction of the Leray ensemble-POD model, which is a regularized-reduced order model for the numerical simulation of convection-dominated flows of moderate Reynolds number. The Leray ensemble-POD model employs the POD spatial filter to smooth (regularize) the convection term in the Navier–Stokes equations, and diminishes the numerical inaccuracies produced by the ensemble-POD method in the numerical simulation of convection-dominated flows. Specifically, for the numerical simulation of a convection-dominated two-dimensional flow between two offset cylinders, we show that the Leray ensemble-POD method better reflects the dynamics of the benchmark results than the ensemble-POD scheme. The second contribution of this work is a new numerical discretization of the variable viscosity ensemble algorithm in which the average viscosity is replaced with the maximum viscosity. It is shown that this new numerical discretization is significantly more stable than those in current use. Furthermore, error estimates for the novel Leray ensemble-POD algorithm with this new numerical discretization are also proven.

scholarly journals A New Reduced Stabilized Mixed Finite-Element Method Based on Proper Orthogonal Decomposition for the Transient Navier-Stokes Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Aiwen Wang ◽  
Jian Li ◽  
Zhenhua Di ◽  
Xiangjun Tian ◽  
Dongxiu Xie
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Proper Orthogonal ◽  
Stabilized Mixed Finite Element

A reduced stabilized mixed finite-element (RSMFE) formulation based on proper orthogonal decomposition (POD) for the transient Navier-Stokes equations is presented. An ensemble of snapshots is compiled from the transient solutions derived from a stabilized mixed finite-element (SMFE) method based on two local Gauss integrations for the two-dimensional transient Navier-Stokes equations by using the lowest equal-order pair of finite elements. Then, the optimal orthogonal bases are reconstructed by implementing POD techniques for the ensemble snapshots. Combining POD with the SMFE formulation, a new low-dimensional and highly accurate SMFE method for the transient Navier-Stokes equations is obtained. The RSMFE formulation could not only greatly reduce its degrees of freedom but also circumvent the constraint of inf-sup stability condition. Error estimates between the SMFE solutions and the RSMFE solutions are derived. Numerical tests confirm that the errors between the RSMFE solutions and the SMFE solutions are consistent with the the theoretical results. Conclusion can be drawn that RSMFE method is feasible and efficient for solving the transient Navier-Stokes equations.

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