The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview

Kerschen, Gaëtan; Golinval, Jean‐Claude; Vakakis, Alexander F.; Bergman, Lawrence A.

doi:10.1007/s11071-005-2803-2

Cited by 777 publications

(464 citation statements)

References 76 publications

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“…Many different methods have been proposed in the past, one of the most popular being the Proper Orthogonal Decomposition (POD) method [3][4][5]29]. In this paper, the NNMs of the unforced structure are tested as basis functions for reducing the dynamics for shell models with harmonic forcing.…”

Section: Discussionmentioning

confidence: 99%

“…This is realized by imposing the invariance of sub-spaces as the key property that must be conserved when non-linearities come into play, and leads to the definition of NNMs as invariant manifolds of the phase space [1]. As the method used for computing the NNMs is purely non-linear, it is expected to give better results than using the LNMs, or modes obtained via the proper orthogonal decomposition (POD) method [2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

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Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Touzé

Amabili

Thomas

2008

Computer Methods in Applied Mechanics and Engineering

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. Reduced-order models for large-amplitude vibrations of shells including in-plane inertia. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 197 (21-24) AbstractNon-linear normal modes (NNMs) are used in order to derive reduced-order models for large amplitude, geometrically non-linear vibrations of thin shells. The main objective of the paper is to compare the accuracy of different truncations, using linear and non-linear modes, in order to predict the response of shells structures subjected to harmonic excitation. For an exhaustive comparison, three different shell problems have been selected: (i) a doubly curved shallow shell, simply supported on a rectangular base; (ii) a circular cylindrical panel with simply supported, in-plane free edges; and (iii) a simply supported, closed circular cylindrical shell. In each case, the models are derived by using refined shell theories for expressing the strain-displacement relationship. As a consequence, in-plane inertia is retained in the formulation. Reduction to one or two NNMs shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases, and this limitation is extended to two times the thickness for two of the selected models.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Touzé

Amabili

Thomas

2008

Computer Methods in Applied Mechanics and Engineering

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“…The proper orthogonal decomposition, also known as Karhunen-Loève decomposition, may serve to ensure the correlation transformation in case of N < n within two steps, namely the order reduction of the design matrix by projecting high-dimensional data into a lower-dimensional space and orthogonal transformation of the sample covariance or correlation matrix to the basis of the largest eigenvalues. A bibliographical review and applications of the proper orthogonal decomposition is given in [11]. The discrete modelling of the proper orthogonal decomposition is achieved by the singular value decomposition.…”

Section: Orthogonal Transformation Using Proper Orthogonal Decompositionmentioning

confidence: 99%

Latin Hypercube Sampling Based on Adaptive Orthogonal Decomposition

Roos¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. In the field of real world or simulation experiments, latin hypercube designs are very useful to approximate implicit given functions based on experimental designs of rectangular grid of points used e.g. for an efficient optimization of black-box experiments. The so called standard latin hypercube sampling is based on an idea of [8] to simulate correlated multivariate relationships based on uncorrelated variables. Hereby, the linear or rank correlations between the random variables can be reduced using a Cholesky decomposition of the covariance matrix. However, this algorithm requires that the number of samples is higher than the number of input variables to ensure a positive definite and non-singular covariance matrix. Furthermore, a rearrangement of the latin hypercubes so that they will have the same ordering to increase the multidimensional uniformity introduces an unwanted pseudo linear correlation of the simulated variables.In this paper, the new method of proper orthogonal decomposition methods in adaptive combination with re-sorting the rank-order of the original marginal distributions not only eliminates the drawbacks of the standard latin hypercube sampling, but also reduces the number of experimental designs necessary. In contrast to the standard latin hypercube sampling, the adaptive approach results in a further improvement of the independency and the multidimensional uniformity of the variables and shows an improved efficiency in comparison to optimization-based methods. Furthermore, a lower-rank approximation of the design matrix based on singular value decomposition ensures a positive definite and non-singular correlation matrix to reduce the correlation error in a stepwise manner during the adaptive procedure.3333

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“…The most diffused ones are probably the nonlinear normal modes (under this name are classified the techniques based on the centre manifold theorem, the normal form theory and the inertial manifold) [1][2][3][4][5][6][7][8][9], including the most diffused version with asymptotic approach, the discretization of the equations of motion by using global (i.e. defined on the whole structure) admissible functions [10][11][12][13], the proper orthogonal decomposition method [14][15][16][17][18] and the natural mode discretization [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

Amabili

2013

Phil. Trans. R. Soc. A.

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Reduced-order models are essential to study nonlinear vibrations of structures and structural components. The natural mode discretization is based on a twostep analysis. In the first step, the natural modes of the structure are obtained. Because this is a linear analysis, the structure can be discretized with a very large number of degrees of freedom. Then, in the second step, a small number of these natural modes are used to discretize the nonlinear vibration problem with a huge reduction in the number of degrees of freedom. This study finds a recipe to select the natural modes that must be retained to study nonlinear vibrations of an angle-ply laminated circular cylindrical shell that the author has previously studied by using admissible functions defined on the whole structure, so that an accuracy analysis is performed. The higher-order shear deformation theory developed by Amabili and Reddy is used to model the shell.

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The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview

Cited by 777 publications

References 76 publications

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Latin Hypercube Sampling Based on Adaptive Orthogonal Decomposition

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

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