scholarly journals Construction of reduced order models for the non-linear Navier-Stokes equations using the proper orthogonal fecomposition (POD)/Galerkin method.

2013 ◽  
Author(s):  
Jeffrey Fike
Keyword(s):  
Galerkin Method ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Reduced Order Models ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Non Linear ◽  
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 A POD-BASED REDUCED-ORDER STABILIZED CRANK–NICOLSON MFE FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER–STOKES EQUATIONS

2015 ◽  
Vol 20 (3) ◽  
pp. 346-368 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Algorithm Implementation ◽  
Proper Orthogonal ◽  
Crank Nicolson

We firstly employ a proper orthogonal decomposition (POD) method, Crank–Nicolson (CN) technique, and two local Gaussian integrals to establish a PODbased reduced-order stabilized CN mixed finite element (SCNMFE) formulation with very few degrees of freedom for non-stationary parabolized Navier–Stokes equations. Then, the error estimates of the reduced-order SCNMFE solutions, which are acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the algorithm implementation for the POD-based reduced-order SCNMFE formulation are provided, respectively. Finally, some numerical experiments are presented to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order SCNMFE formulation is feasible and efficient for finding numerical solutions of the non-stationary parabolized Navier–Stokes equations.

Improved extrapolation of steady turbulent aerodynamics using a non-linear POD-based reduced order model

2012 ◽  
Vol 116 (1184) ◽  
pp. 1079-1100 ◽  
Author(s):  
R. Zimmermann ◽  
S. Görtz
Keyword(s):  
Fluid Dynamics ◽  
Stokes Equations ◽  
Solution Space ◽  
Order Reduction ◽  
Navier Stokes ◽  
Space Representation ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Non Linear ◽  
The Cost

AbstractA reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.

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