Spectral identification of nonlinear multi-degree-of-freedom structural systems with fractional derivative terms based on incomplete non-stationary data

Santos, Ketson Roberto Maximiano dos; Brudastova, Olga; Kougioumtzoglou, Ioannis A.

doi:10.1016/j.strusafe.2020.101975

Cited by 17 publications

(3 citation statements)

References 58 publications

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“…Further discussion of Grassmannian kernels and their specific use for GDMaps can be found in dos Santos et al 59…”

Section: Grassmannian Kernelsmentioning

confidence: 99%

“…It is constructed based on the projection embedding

Π : 𝒢 (p, n) \to ℝ^{n \times n}

such that

\Pi \left(\boldsymbol{\Psi} \right)=\boldsymbol{\Psi} {\boldsymbol{\Psi}}^T

. This kernel is defined as

k_{p r} (𝒳, 𝒴) = ‖ Ψ_{x}^{T} Ψ_{y} ‖_{F}^{2},

or equivalently in terms of principal angles 51,58

k_{p r} (𝒳, 𝒴) = \sum_{i = 1}^{p} \cos^{2} (θ_{i}) .

Further discussion of Grassmannian kernels and their specific use for GDMaps can be found in dos Santos et al 59 …”

Section: Grassmann Manifoldmentioning

confidence: 99%

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Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Santos

Giovanis

Kontolati

et al. 2022

Numerical Meth Engineering

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A novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics (GH) is developed for predicting the response of complex physical phenomena. The method utilizes GDMaps to obtain a low-dimensional representation of the underlying behavior of physical/ mathematical systems with respect to uncertain input parameters. Using this representation, GH, an out-of-sample extension technique, is employed to create a global map from the input parameter space to a Grassmannian diffusion manifold. GH is further employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate model is verified with three examples. The first problem is a toy example used to illustrate the technique. In the second example, errors associated with the various mappings are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone theory of plasticity.

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“…Further discussion of Grassmannian kernels and their specific use for GDMaps can be found in dos Santos et al 59…”

Section: Grassmannian Kernelsmentioning

confidence: 99%

“…It is constructed based on the projection embedding

Π : 𝒢 (p, n) \to ℝ^{n \times n}

such that

\Pi \left(\boldsymbol{\Psi} \right)=\boldsymbol{\Psi} {\boldsymbol{\Psi}}^T

. This kernel is defined as

k_{p r} (𝒳, 𝒴) = ‖ Ψ_{x}^{T} Ψ_{y} ‖_{F}^{2},

or equivalently in terms of principal angles 51,58

k_{p r} (𝒳, 𝒴) = \sum_{i = 1}^{p} \cos^{2} (θ_{i}) .

Further discussion of Grassmannian kernels and their specific use for GDMaps can be found in dos Santos et al 59 …”

Section: Grassmann Manifoldmentioning

confidence: 99%

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Santos

Giovanis

Kontolati

et al. 2022

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…Further discussion of Grassmannian kernels and their specific use for Grassmannian diffusion maps can be found in dos Santos et al [17].…”

mentioning

confidence: 99%

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Santos,

Giovanis,

Kontolati

et al. 2021

Preprint

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In this paper, a novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics is developed for predicting the response of engineering systems and complex physical phenomena. The method utilizes the GDMaps to obtain a low-dimensional representation of the underlying behavior of physical/mathematical systems with respect to uncertainties in the input parameters. Using this representation, geometric harmonics, an out-of-sample function extension technique, is employed to create a global map from the space of input parameters to a Grassmannian diffusion manifold. Geometric harmonics is also employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate modeling is verified with three examples. The first problem is a toy example used to illustrate the development of the technique. In the second example, errors associated with the various mappings employed in the technique are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone (STZ) theory of plasticity. In all examples, accurate predictions are obtained, showing that the present technique is a strong candidate for the application of uncertainty quantification in large-scale models.

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Stochastic bifurcation and dynamic reliability analyses of nonlinear MDOF vehicle system with generalized fractional damping via DPIM

Chen,

Meng

et al. 2024

Nonlinear Dyn

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Spectral identification of nonlinear multi-degree-of-freedom structural systems with fractional derivative terms based on incomplete non-stationary data

Cited by 17 publications

References 58 publications

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Stochastic bifurcation and dynamic reliability analyses of nonlinear MDOF vehicle system with generalized fractional damping via DPIM

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