Proper Orthogonal Decomposition (Pod) of a Class of Vibroimpact Oscillations

Azeez, Mohammed F. Abdul; Vakakis, Alexander F.

doi:10.1006/jsvi.2000.3264

Cited by 173 publications

(92 citation statements)

References 17 publications

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“…POD is known in other fields as Principal Components Analysis (PCA) [84] (with the mean subtracted out), the Hotelling transform [85], Empirical Orthogonal Functions (EOF) [86,87], and/or the Karhunen-Loéve (KL) decomposition [88]. The method has been utilized in a large and diverse set of scientific and engineering applications [2][3][4][5][6][7][8][9][10][11][12][13].…”

Section: The Proper Orthogonal Decomposition and Galerkin Projectionmentioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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Abstract. The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without physics-based governing equations. We review standard model reduction techniques such as Proper Orthogonal Decomposition (POD) with Galerkin projection and Balanced POD (BPOD). Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory. These data-driven methods can mitigate the challenge of not having a well-characterized set of governing equations. We illustrate that this equation-free approach that is being applied to measurement data from complex systems can be extended to include inputs and control. Three specific research examples are presented that extend current equation-free architectures toward the characterization and control of complex systems. These examples motivate a potentially revolutionary shift in the characterization of complex systems and subsequent design of objective-based controllers for data-driven models.

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Section: The Proper Orthogonal Decomposition and Galerkin Projectionmentioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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“…The above nonlinear reference solution is then used for determining a projection basis with the Proper Orthogonal Decomposition method (PODmethod), which is known to be particularly relevant for nonlinear problems [23,25,42,43,24,44]. Let u(s) be the Ê n -displacement vector corresponding to the finite element discretization of the displacement field {u(x, s) , x ∈ Ω}.…”

Section: Construction Of the Mean Nonlinear Reduced Static Computatiomentioning

confidence: 99%

“…This basis can be obtained by one of several techniques such as the Proper Orthogonal Decomposition method (POD method) [23,24,25], which is known to be particularly efficient for nonlinear static cases. One can also rely on linear elastic modes of vibration [26,27], or selected linear elastic modes appropriately enriched, e.g.…”

Section: Introductionmentioning

confidence: 99%

Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation

Capiez‐Lernout

Soize

Mignolet

2014

Computer Methods in Applied Mechanics and Engineering

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The paper presents a complete experimental validation of an advanced computational methodology adapted to the nonlinear post-buckling analysis of geometrically nonlinear structures in presence of uncertainty. A mean nonlinear reduced-order computational model is first obtained using an adapted projection basis. The stochastic nonlinear computational model is then constructed as a function of a scalar dispersion parameter, which has to be identified with respect to the nonlinear static experimental response of a very thin cylindrical shell submitted to a static shear load. The identified stochastic computational model is finally used for predicting the nonlinear dynamical post-buckling behavior of the structure submitted to a stochastic ground motion.

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“…The POD has been used to obtain approximate, lowdimensional descriptions of turbulent fluid flows [22,37], structural vibrations and chaotic dynamical systems [7]. Many recent investigations have examined impacting systems [1,2] and thermics [47].…”

Section: Model Reduction Using the Proper Orthogonal Decompositionmentioning

confidence: 99%

The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains

Yvonnet¹,

He²

2007

Journal of Computational Physics

266

164

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To cite this version:Julien Yvonnet, Qi-Chang He. The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. AbstractThis paper presents a new multi-scale method for the homogenization analysis of hyperelastic solids undergoing finite strains. The key contribution is to use an incremental nonlinear homogenisation technique in tandem with a model reduction method, in order to alleviate the complexity of multiscale procedures, which usually involve a large number of nonlinear nested problems to be solved. The problem associated with the Representative Volume Element (RVE) is solved via a model reduction method (Proper Orthogonal Decomposition). The reduced basis is obtained through pre-computations on the RVE. The technique, coined as Reduced Model Multiscale Method (R3M), allows reducing significantly the computation times, as no large matrix needs to be inverted, and as the convergence of both macro and micro problems is enhanced. Furthermore, the R3M drastically reduces the size of the data base describing the history of the micro problems. In order to validate the technique in the context of porous elastomers at finite strains, a comparison between a full and a reduced multiscale analysis is performed through numerical examples, involving different micro and macro structures, as well as different nonlinear models (Neo-Hookean, Mooney-Rivlin). It is shown that the R3M gives good agreement with the full simulations, at lower computational and data storage requirements.

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Proper Orthogonal Decomposition (Pod) of a Class of Vibroimpact Oscillations

Cited by 173 publications

References 17 publications

Including inputs and control within equation-free architectures for complex systems

Including inputs and control within equation-free architectures for complex systems

Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation

The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains

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