Uncertainty analysis in solid mechanics with uniform and triangular distributions using stochastic perturbation-based Finite Element Method

In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the gene...

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Vydáno v:	Finite elements in analysis and design Ročník 200; s. 103648
Hlavní autor:	Kamiński, Marcin
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.03.2022 Elsevier BV
Témata:	Finite element analysis Finite element method Iterative stochastic perturbation technique Least squares method Mechanical systems Mechanics Monte-Carlo simulation Parameter uncertainty Perturbation Polynomials Probability theory Semi-analytical method Solid mechanics Statistical analysis Stochastic finite element method Structural response Triangular probability distribution Uncertainty analysis Uniform probability distribution Stochastic finite element method Triangular probability distribution Uniform probability distribution Iterative stochastic perturbation technique Monte-Carlo simulation Semi-analytical method
ISSN:	0168-874X, 1872-6925
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Abstract	In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions. •New formulation of the iterative generalized stochastic perturbation technique.•Implementation of the Stochastic Finite Element Method for the triangular PDF.•Implementation of the Stochastic Finite Element Method for the input uniform PDF.•Comparative numerical analysis with Monte-Carlo simulation.•Comparative numerical analysis with the semi-analytical probabilistic method.
AbstractList	In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions. In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions. •New formulation of the iterative generalized stochastic perturbation technique.•Implementation of the Stochastic Finite Element Method for the triangular PDF.•Implementation of the Stochastic Finite Element Method for the input uniform PDF.•Comparative numerical analysis with Monte-Carlo simulation.•Comparative numerical analysis with the semi-analytical probabilistic method.
ArticleNumber	103648
Author	Kamiński, Marcin
Author_xml	– sequence: 1 givenname: Marcin surname: Kamiński fullname: Kamiński, Marcin email: Marcin.Kaminski@p.lodz.pl organization: Department of Structural Mechanics, Faculty of Civil Engineering, Architecture and Environmental Engineering, Łódź University of Technology, Al. Politechniki 6, 90-924, Łódź, Poland
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Keywords	Stochastic finite element method Triangular probability distribution Uniform probability distribution Iterative stochastic perturbation technique Monte-Carlo simulation Semi-analytical method
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Snippet	In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty...
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SubjectTerms	Finite element analysis Finite element method Iterative stochastic perturbation technique Least squares method Mechanical systems Mechanics Monte-Carlo simulation Parameter uncertainty Perturbation Polynomials Probability theory Semi-analytical method Solid mechanics Statistical analysis Stochastic finite element method Structural response Triangular probability distribution Uncertainty analysis Uniform probability distribution
Title	Uncertainty analysis in solid mechanics with uniform and triangular distributions using stochastic perturbation-based Finite Element Method
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