Model Reduction by Proper Orthogonal Decomposition Coupled With Centroidal Voronoi Tessellations (Keynote)

Volume 1: Fora, Parts A and B ◽

10.1115/fedsm2002-31051 ◽

2002 ◽

Cited By ~ 9

Author(s):

Qiang Du ◽

Max Gunzburger

Keyword(s):

Complex Systems ◽

Model Reduction ◽

Turbulent Flows ◽

Implementation Strategies ◽

Practical Implementation ◽

Voronoi Tessellations ◽

Centroidal Voronoi Tessellations ◽

Low Dimensional Approximations ◽

Low Dimensional ◽

Proper Orthogonal

Proper orthogonal decompositions (POD) have been used to define reduced bases for low-dimensional approximations of complex systems, including turbulent flows. Centroidal Voronoi tessellations (CVT) have been used in a variety of data compression and clustering settings. We review both strategies in the context of model reduction for complex systems and propose combining the ideas of CVT and POD into a hybrid method that inherets favorable characteristics from both its parents. The usefulness of such an approach and various practical implementation strategies are discussed.

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Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder

Journal of Fluid Mechanics ◽

10.1017/s0022112008001821 ◽

2008 ◽

Vol 606 ◽

pp. 339-367 ◽

Cited By ~ 40

Author(s):

DANIELE VENTURI ◽

XIAOLIANG WAN ◽

GEORGE EM KARNIADAKIS

Keyword(s):

Circular Cylinder ◽

Turbulent Flows ◽

Stokes Equations ◽

Stochastic Simulations ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Spatial Modes ◽

Laminar Wake ◽

Low Dimensional ◽

Proper Orthogonal

We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.

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Hierarchical Structure of Slow Manifolds of Reacting Flows

Zeitschrift für Physikalische Chemie ◽

10.1515/zpch-2014-0599 ◽

2015 ◽

Vol 229 (6) ◽

Cited By ~ 4

Author(s):

Viatcheslav Bykov ◽

Alexander Neagos ◽

Alexander Klimenko ◽

Ulrich Maas

Keyword(s):

Model Reduction ◽

Turbulent Flows ◽

Mathematical Description ◽

Hierarchical Modeling ◽

Molecular Transport ◽

Reacting Flows ◽

Transport Processes ◽

Elementary Reactions ◽

Hierarchical Nature ◽

Low Dimensional

AbstractNowadays the mathematical description of chemically reacting flows uses very often reaction mechanisms with far above hundred or even thousand chemical species (and, therefore, a large number of partial differential equations must be solved), which possibly react within more than a thousand of elementary reactions. These chemical kinetic processes cover time scales from nanoseconds to seconds. An analogous scaling problem arises for the length scales. Due to these scaling problems the detailed simulation of three-dimensional turbulent flows in practical systems is beyond the capacity of even today's super-computers. Using simplified sub-models is a way out of this problem. The question arising in mathematical modeling of reacting flows is then: How detailed, or down to which scale has each process to be resolved (chemical reaction, chemistry-turbulence-interaction, molecular transport processes) in order to allow a reliable description of the entire process. Both the chemical source term and the transport term have one important property, namely, they cause the existence of low-dimensional attractors in composition space. When these manifolds can be constructed (described) and parametrized by a small number of variables, it can be used to reformulate and reduce the mathematical description for modeling reacting flows. In this work the hierarchical nature of these low-dimensional manifolds of slow motions is discussed. It is demonstrated how this important feature of reacting flows is accounted for by the standard model reduction methods (like e.g. PEA and QSSA methods) as well as by recently developed concepts of model reduction. The use of the hierarchical nature for identification of the low-dimensional manifolds to devise hierarchical modeling concepts (e.g. for turbulent reacting flows) is additionally discussed.

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Technical Note: Practical implementation strategies of cycloidal computed tomography

Medical Physics ◽

10.1002/mp.14821 ◽

2021 ◽

Author(s):

Oriol Roche i Morgó ◽

Fabio Vittoria ◽

Marco Endrizzi ◽

Alessandro Olivo ◽

Charlotte K. Hagen

Keyword(s):

Computed Tomography ◽

Implementation Strategies ◽

Technical Note ◽

Practical Implementation

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Snapshot selection for groundwater model reduction using proper orthogonal decomposition

Water Resources Research ◽

10.1029/2009wr008792 ◽

2010 ◽

Vol 46 (8) ◽

Cited By ~ 45

Author(s):

Adam J. Siade ◽

Mario Putti ◽

William W.-G. Yeh

Keyword(s):

Proper Orthogonal Decomposition ◽

Model Reduction ◽

Orthogonal Decomposition ◽

Groundwater Model ◽

Selection For ◽

Proper Orthogonal

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Global Stability Analysis of the Wake behind a Circular Cylinder Based on the POD-Chiba Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.137.72 ◽

2011 ◽

Vol 137 ◽

pp. 72-76

Author(s):

Wei Zhang ◽

Xian Wen ◽

Yan Qun Jiang

Keyword(s):

Stability Analysis ◽

Circular Cylinder ◽

Global Stability ◽

Dimensional Model ◽

Global Stability Analysis ◽

The Stability ◽

Low Dimensional ◽

Proper Orthogonal ◽

Calculated Amount ◽

Good Agreement

A proper orthogonal decomposition (POD) method is applied to study the global stability analysis for flow past a stationary circular cylinder. The flow database at Re=100 is obtained by CFD software, i.e. FLUENT, with which POD bases are constructed by a snapshot method. Based on the POD bases, a low-dimensional model is established for solving the two-dimensional incompressible NS equations. The stability of the flow solution is evaluated by a POD-Chiba method in the way of the eigensystem analysis for the velocity disturbance. The linear stability analysis shows that the first Hopf bifurcation takes place at Re=46.9, which is in good agreement with available results by other high-order accurate stability analysis methods. However, the calculated amount of POD is little, which shows the availability and advantage of the POD method.

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Boundary Feedback Control of the Burgers Equations by a Reduced-Order Approach Using Centroidal Voronoi Tessellations

Journal of Scientific Computing ◽

10.1007/s10915-009-9310-4 ◽

2009 ◽

Vol 43 (3) ◽

pp. 369-387 ◽

Cited By ~ 4

Author(s):

Hyung-Chun Lee ◽

Guang-Ri Piao

Keyword(s):

Feedback Control ◽

Voronoi Tessellations ◽

Boundary Feedback Control ◽

Reduced Order ◽

Burgers Equations ◽

Boundary Feedback ◽

Centroidal Voronoi Tessellations

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

Journal of Fluid Mechanics ◽

10.1017/s0022112009006363 ◽

2009 ◽

Vol 629 ◽

pp. 41-72 ◽

Cited By ~ 66

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.

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