Stochastic isogeometric analysis of free vibration of functionally graded plates considering material randomness

Hien, Ta Duy; Noh, Hyuk-Chun

doi:10.1016/j.cma.2017.02.007

Cited by 62 publications

(7 citation statements)

References 53 publications

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“…More recently, spline based isogeometric analysis has found its way into the stochastic community. Stochastic methods have been proposed to quantify uncertainty due to material randomness in linear elasticity [41,31], static analysis of plates [69], vibrational analysis of shells [44], static and dynamic structural analysis of random composite structures [15] and functionally graded plates [26,42,43]. In [71] a method is proposed to quantify the effect due to uncertainty in shape.…”

Section: Challenges In Numerical Solution Of the Kle By Means Of The ...mentioning

confidence: 99%

A matrix-free isogeometric Galerkin method for Karhunen-Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

Mika¹,

Hughes²,

Schillinger³

et al. 2020

Preprint

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The Karhunen-Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L 2 -orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrixfree and parallel matrix-vector product for iterative eigenvalue solvers. Two higher-order three-dimensional benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

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Section: Challenges In Numerical Solution Of the Kle By Means Of The ...mentioning

confidence: 99%

A matrix-free isogeometric Galerkin method for Karhunen-Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

Mika¹,

Hughes²,

Schillinger³

et al. 2020

Preprint

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“…In the previous study, Hien and Noh (2017) dealt with the response variability of eigenvalues for the vibration problem of functionally graded plates with uncertain material properties. In this work, we focus on developing the SIGA for the static bending problem of functionally graded plates with uncertain elastic modulus.…”

Section: Introductionmentioning

confidence: 99%

Perturbation based stochastic isogeometric analysis for bending of functionally graded plates with the randomness of elastic modulus

Nguyen

2020

Lat. Am. j. solids struct.

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We propose, in this paper, stochastic isogeometric analysis (SIGA) is a type of non-statistic approach in which combines the perturbation technique with the standard isogeometric analysis, in particular for static behavior of functionally graded plates with the uncertain elastic modulus. We assume that the spatial random variation of elastic modulus can be modeled as a two dimensional Gaussian random field in the plane of the plate. The random field is discretized to set of random variables using the integration point method. The system equations of SIGA are created using the NURBS functions for approximation displacement fields in conjunction with the first-order and second-order perturbation expansions of random fields, stiffness matrix, displacement fields. Besides the non-statistic approach, Monte Carlo simulation is presented for validation. The accuracy and appropriateness of the non-statistic approach are demonstrated via comparisons of the present results with those given by the stochastic finite element method in the literature and by the Monte Carlo analysis as well. The numerical examples are employed to investigate the effect of the randomness of elastic modulus and system parameters on the first and second statistical moments of displacement.

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“…Stochastic analysis, on the other hand, may use one or more random variables as input, resulting in outputs that are also random. As it involves the sampling of random variables, stochastic analysis is the method that most faithfully reproduces reality (Hien & Noh, 2017). Values are randomized using models that effectively describe the probability distribution of the analyzed parameters (Stefanou, 2009).…”

Section: Introductionmentioning

confidence: 99%

Dynamic behavior of the macauba palm (Acrocomia aculeata) fruit-rachilla system using the stochastic finite element method

et al. 2020

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The search for alternative energy sources has fomented the study of several crops. The macauba palm crop, for instance, has been highlighted because of its particular relevance in Brazil due to its wide distribution across Brazilian territory and its potential for yielding high amounts of oil per cultivated hectare. However, the species is still most commonly harvested via extractivism, which results in low yields. Therefore, we aimed to analyze the dynamic behavior of the fruit-rachilla system when subjected to mechanical vibration to gather baseline information for the subsequent development of macauba harvesting machines. The fruit-rachilla system of the species was modeled for different fruit maturation stages and plant accessions. Natural frequencies and modes of vibration were determined by the stochastic finite element method (FEM), adopting the specific mass and the modulus of elasticity of the system as random variables, which enabled us to compile a dataset of natural frequencies based on the variability of the system properties. The mean values of the natural frequencies obtained in the vibration assays were 26.02 Hz at the green maturation stage and 21.22 Hz at the ripe maturation stage. The mean values of natural frequencies found in the simulation by stochastic FEM, referring to the third mode of vibration, were 26.05 Hz at the green maturation stage and 21.23 Hz at the ripe maturation stage. We concluded that the natural frequencies of the macauba fruit-rachilla system on the basis of different plant accessions showed a decreasing behavior during fruit maturation. The modes of vibration characterized by pendulum displacement did not differ among plant accessions or between fruit maturation stages.

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Stochastic isogeometric analysis of free vibration of functionally graded plates considering material randomness

Cited by 62 publications

References 53 publications

A matrix-free isogeometric Galerkin method for Karhunen-Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

A matrix-free isogeometric Galerkin method for Karhunen-Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

Perturbation based stochastic isogeometric analysis for bending of functionally graded plates with the randomness of elastic modulus

Dynamic behavior of the macauba palm (Acrocomia aculeata) fruit-rachilla system using the stochastic finite element method

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