Dynamic Mode Decomposition for Construction of Reduced-Order Models of Hyperbolic Problems with Shocks

In this paper, a frequently employed technique named the sparsity-promoting dynamic mode decomposition (SPDMD) is proposed to analyze the velocity fields of atmospheric motion. The dynamic mode decomposition method (DMD) is an effective technique to extract dynamic information from flow fields that is generated from direct experiment measurements or numerical simulation and has been broadly employed to study the dynamics of the flow, to achieve a reduced-order model (ROM) of the complex high dimensional flow field, and even to predict the evolution of the flow in a short time in the future. However, for standard DMD, it is hard to determine which modes are the most physically relevant, unlike the proper orthogonal decomposition (POD) method which ranks the decomposed modes according to their energy content. The advanced modal decomposition method SPDMD is a variant of the standard DMD, which is capable of determining the modes that can be used to achieve a high-quality approximation of the given field. It is novel to introduce the SPDMD to analyze the atmospheric flow field. In this study, SPDMD is applied to extract essential dynamic information from the 200 hPa jet flow, and the decomposed results are compared with the POD method. To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, the POD/SPDMD reduced-order models are constructed, respectively. The results show that both modal decomposition methods successfully extract the underlying coherent structures from the 200 hPa jet flow. And the DMD method provides additional information on the modal properties, such as temporal frequency and growth rate of each mode which can be used to identify the stability of the modes. It is also found that a fewer order of modes determined by the SPDMD method can capture nearly the same dynamic information of the jet flow as the POD method. Furthermore, from the quantitative comparisons between the POD and SPDMD reduced-order models, the latter provides a higher precision than the former, especially when the number of modes is small.

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On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach

Energies ◽

10.3390/en13246513 ◽

2020 ◽

Vol 13 (24) ◽

pp. 6513

Author(s):

Nassir Cassamo ◽

Jan-Willem van Wingerden

Keyword(s):

Wind Farm ◽

Dynamic Mode Decomposition ◽

Dynamic Mode ◽

Reduced Order Models ◽

Decomposition Approach ◽

Reduced Order ◽

Mode Decomposition ◽

Non Linear ◽

Farm Systems ◽

Turbine Wake

The high dimensions and governing non-linear dynamics in wind farm systems make the design of numerical optimal controllers computationally expensive. A possible pathway to circumvent this challenge lies in finding reduced order models which can accurately embed the existing non-linearities. The work presented here applies the ideas motivated by non-linear dynamical systems theory—the Koopman Operator—to an innovative algorithm in the context of wind farm systems—Input Output Dynamic Mode Decomposition (IODMD)—to improve on the ability to model the aerodynamic interaction between wind turbines in a wind farm and uncover insights into the existing dynamics. It is shown that a reduced order linear state space model can reproduce the downstream turbine generator power dynamics and reconstruct the upstream turbine wake. It is further shown that the fit can be improved by judiciously choosing the Koopman observables used in the IODMD algorithm without jeopardizing the models ability to rebuild the turbine wake. The extensions to the IODMD algorithm provide an important step towards the design of linear reduced order models which can accurately reproduce the dynamics in a wind farm.

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Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena

Journal of Computational Physics ◽

10.1016/j.jcp.2020.109229 ◽

2020 ◽

Vol 407 ◽

pp. 109229 ◽

Cited By ~ 3

Author(s):

Hannah Lu ◽

Daniel M. Tartakovsky

Keyword(s):

Dynamic Mode Decomposition ◽

Dynamic Mode ◽

Reduced Order Models ◽

Reduced Order ◽

Mode Decomposition ◽

Lagrangian Dynamic

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Analyzing Nonlinear Dynamics via Data-Driven Dynamic Mode Decomposition-Like Methods

Complexity ◽

10.1155/2018/6920783 ◽

2018 ◽

Vol 2018 ◽

pp. 1-21 ◽

Cited By ~ 8

Author(s):

Soledad Le Clainche ◽

José M. Vega

Keyword(s):

Nonlinear Dynamics ◽

Growth Rates ◽

Transient Behavior ◽

Dynamic Mode Decomposition ◽

Data Driven ◽

Dynamic Mode ◽

Reduced Order Models ◽

Reduced Order ◽

Mode Decomposition ◽

The Given

This article presents a review on two methods based on dynamic mode decomposition and its multiple applications, focusing on higher order dynamic mode decomposition (which provides a purely temporal Fourier-like decomposition) and spatiotemporal Koopman decomposition (which gives a spatiotemporal Fourier-like decomposition). These methods are purely data-driven, using either numerical or experimental data, and permit reconstructing the given data and identifying the temporal growth rates and frequencies involved in the dynamics and the spatial growth rates and wavenumbers in the case of the spatiotemporal Koopman decomposition. Thus, they may be used to either identify and extrapolate the dynamics from transient behavior to permanent dynamics or construct efficient, purely data-driven reduced order models.

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On the potential of reduced order models for wind farm control: a Koopman dynamic mode decomposition approach

10.20944/preprints202010.0352.v1 ◽

2020 ◽

Author(s):

Nassir Cassamo ◽

Jan-Willem van Wingerden

Keyword(s):

Wind Farm ◽

Dynamic Mode Decomposition ◽

Dynamic Mode ◽

Reduced Order Models ◽

Decomposition Approach ◽

Reduced Order ◽

Mode Decomposition ◽

Non Linear ◽

Farm Systems ◽

Turbine Wake

The high dimensions and governing non linear dynamics in wind farm systems make the design of numerical optimal controllers computationally expensive. A possible pathway to circumvent this challenge lies in finding reduced order models which can accurately embed the existing non linearities. The work here presented applies the ideas motivated by non linear dynamical systems theory - the Koopman Operator - to an innovative algorithm in the context of wind farm systems - Input Output Dynamic Mode Decomposition - to improve on the ability to model the aerodynamic interaction between wind turbines in a wind farm and uncover insights into the existing dynamics. It is shown that a reduced order linear state space model can reproduce the downstream turbine generator power dynamics and reconstruct the upstream turbine wake. It is further shown that the fit can be improved by judiciously choosing the Koopman observables used in the IODMD algorithm without jeopardizing the models ability to rebuild the turbine wake. The extensions to the IODMD algorithm provide an important step towards the design of linear reduced order models which can accurately reproduce the dynamics in a wind farm.

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Dynamic Mode Decomposition Reduced-Order Models for Multiscale Kinetic Plasma Analysis

2021 United States National Committee of URSI National Radio Science Meeting (USNC-URSI NRSM) ◽

10.23919/usnc-ursinrsm51531.2021.9336498 ◽

2021 ◽

Author(s):

Indranil Nayak ◽

Fernando L. Teixeira

Keyword(s):

Dynamic Mode Decomposition ◽

Dynamic Mode ◽

Reduced Order Models ◽

Plasma Analysis ◽

Reduced Order ◽

Mode Decomposition

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Modeling the Turbulent Wake Behind a Wall-Mounted Square Cylinder

Logic Journal of IGPL ◽

10.1093/jigpal/jzaa060 ◽

2020 ◽

Author(s):

Christian Amor ◽

José M Pérez ◽

Philipp Schlatter ◽

Ricardo Vinuesa ◽

Soledad Le Clainche

Keyword(s):

Proper Orthogonal Decomposition ◽

Turbulent Wake ◽

Orthogonal Decomposition ◽

Dynamic Mode Decomposition ◽

Dynamic Mode ◽

Square Cylinder ◽

Complex Flow ◽

Reduced Order ◽

Mode Decomposition ◽

Proper Orthogonal

Abstract This article introduces some soft computing methods generally used for data analysis and flow pattern detection in fluid dynamics. These techniques decompose the original flow field as an expansion of modes, which can be either orthogonal in time (variants of dynamic mode decomposition), or in space (variants of proper orthogonal decomposition) or in time and space (spectral proper orthogonal decomposition), or they can simply be selected using some sophisticated statistical techniques (empirical mode decomposition). The performance of these methods is tested in the turbulent wake of a wall-mounted square cylinder. This highly complex flow is suitable to show the ability of the aforementioned methods to reduce the degrees of freedom of the original data by only retaining the large scales in the flow. The main result is a reduced-order model of the original flow case, based on a low number of modes. A deep discussion is carried out about how to choose the most computationally efficient method to obtain suitable reduced-order models of the flow. The techniques introduced in this article are data-driven methods that could be applied to model any type of non-linear dynamical system, including numerical and experimental databases.

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