Chaos–Galerkin solution of stochastic Timoshenko bending problems

Silva, Cláudio R. Ávila da; Beck, André Teófilo

doi:10.1016/j.compstruc.2011.01.002

Cited by 12 publications

(8 citation statements)

References 28 publications

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“…is a Hilbert space. Because V ¼ L 2 X; F ; P; H 1 0 0; l ð Þ À Á , L 2 (X, F , P) and H 1 0 0; l ð Þ are separable spaces it follows that V is a separable space too, (Ávila da Silva and Beck, 2011). Besides that V ' L 2 X; F ; P ð Þ H 1 0 0; l ð Þ and {c i } i[N ,{f i } i[N are total orthonormal sets in L (X, F , P) and H 1 0 0; l ð Þ, then by the tensor product theorem, (Reed and Simon, 1980, p. 51), in Hilbert spaces with measure, {(c f ) i } i[N is an enumerable total set inV.…”

Section: Preliminariesmentioning

confidence: 99%

A counterexample to FORM and SORM

Júnior

Damázio

Matioli

et al. 2020

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Purpose This paper aims to presents a counterexample that points to an inconsistency generated by the first- and second-order approximation methods, FORM and SORM, respectively, procedures for elliptical problems. Design/methodology/approach The classical results of theory measure and functional analysis were used. Findings The FORM and SORM are known to find solutions in a Gaussian space. This procedure does not satisfy the conditions of the Lax–Milgram theorem and does not assure the existence and uniqueness of the solution. Research limitations/implications This paper alerts the engineering research community that uses these methods, initiating discussion and improvement of FORM and SORM procedures. Practical implications This paper puts in check the feasibility of using FORM and SORM in engineering problems. Originality/value From the moment they were introduced to the engineering and scientific communities, the FORM and SORM were taken as the bases for solving various problems found in the literature and indifferent documents scattered throughout the world over the past 50 years, for FORM, and 40 years, for SORM. Even though it was a very serious fault, at least for elliptical problems, pointed out in this work, it went unnoticed all those years by the research community. Therefore, the contribution of this paper is to present the engineering community that uses FORM and SORM in elliptical problems an unnoticed failure since the introduction of these methods.

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Section: Preliminariesmentioning

confidence: 99%

A counterexample to FORM and SORM

Júnior

Damázio

Matioli

et al. 2020

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“…Let and be such that (H1)-(H3) ( (2) and (3)) are satisfied. Then, a solution to the problem defined in (12) exists, and it is unique in .…”

Section: Existence and Uniqueness Of The Solutionmentioning

confidence: 99%

“…Theoretical solutions to the abstract variational problem in (1), associated with the problem in (12), are found in ≃ 2 (Ω, F, ) ⊗ . Numerical solutions are obtained in less abstract spaces: continuous functions of class 2 , sequentially dense in , and a family of generalized chaos polynomials, belonging to the Askey-Wiener scheme, are used.…”

Section: Existence and Uniqueness Of The Solutionmentioning

confidence: 99%

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Application of Galerkin Method to Kirchhoff Plates Stochastic Bending Problem

Júnior

Kist

Santos

2014

ISRN Applied Mathematics

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In this paper, the Galerkin method is used to obtain approximate solutions for Kirchhoff plates stochastic bending problem with uncertainty over plates flexural rigidity coefficient. The uncertainty in the rigidity coefficient is represented by means of parameterized stochastic processes. A theorem of Lax-Milgram type, about existence and uniqueness of the theoretical solutions, is presented and used in selection of the approximate solution space. The Wiener-Askey scheme of generalized polynomials chaos (gPC) is used to model the stochastic behavior of the displacement solutions. The performance of the approximate Galerkin solution scheme developed herein is evaluated by comparing first and second order moments of the approximate solution with the same moments evaluated from Monte Carlo simulation. Rapid convergence of approximate Galerkin's solution to the first and second order moments is observed, for the problems studied herein. Results also show that using the developed Galerkin's scheme one gets adequate estimates for accrued probability function to a random variable generated by the stochastic process of displacement.

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“…Recently, Sepahvand et al [11] and Ernst et al [12] investigated the use of polynomial chaos in various problems and studied its convergence. The method of generalized polynomial chaos with the stochastic Galerkin method has been used to analyze several stochastic problems in applied mechanics [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Influence of Physical and Geometrical Uncertainties in the Parametric Instability Load of an Axially Excited Cylindrical Shell

Silva

Brazão

Gonçalves

2015

Mathematical Problems in Engineering

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This work investigates the influence of Young’s modulus, shells thickness, and geometrical imperfection uncertainties on the parametric instability loads of simply supported axially excited cylindrical shells. The Donnell nonlinear shallow shell theory is used for the displacement field of the cylindrical shell and the parameters under investigation are considered as uncertain parameters with a known probability density function in the equilibrium equation. The uncertainties are discretized as Hermite-Chaos polynomials together with the Galerkin stochastic procedure that discretizes the stochastic equation in a set of deterministic equations of motion. Then, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all nonlinear modes that couple with the linear modes. So, a particular solution is selected which ensures the convergence of the response up to very large deflections. Applying the standard Galerkin method, a discrete system in time domain that considers the uncertainties is obtained and solved by fourth-order Runge-Kutta method. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the parameters. Special attention is given to the influence of the uncertainties on the parametric instability and time response, showing that the Hermite-Chaos polynomial is a good numerical tool.

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Chaos–Galerkin solution of stochastic Timoshenko bending problems

Cited by 12 publications

References 28 publications

A counterexample to FORM and SORM

A counterexample to FORM and SORM

Application of Galerkin Method to Kirchhoff Plates Stochastic Bending Problem

Influence of Physical and Geometrical Uncertainties in the Parametric Instability Load of an Axially Excited Cylindrical Shell

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