Uncertainty quantification for random fields estimated from effective moduli of elasticity (original) (raw)
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Computational Materials Science, 1998
This paper deals with ®nite element analysis of mechanical response of simple structural members with random nonuniformity of material parameters under deterministic loads. Procedures for a stochastic response analysis are presented, such as parameter identi®cation techniques and algorithms for description of spatial variability of material properties in a ®nite element structure. Direct Monte±Carlo simulation was used in conjunction with a standard ®nite element code to simulate the random mechanical response for a large number of specimens. An elastic±viscoplastic constitutive model was used to predict the deformation behaviour of these structures under constant strain rate conditions. A statistical evaluation of the results shows a signi®cant in¯uence of geometrical irregularity on the degree of randomness of the local and overall mechanical response.
Computers & Structures, 1995
In this paper, a stochastic finite element method is presented for the analysis of structures having statistical uncertainities in both material properties and externally applied loads, which are modelled as a homogeneous Gaussian stochastic process. The Neumann expansion technique has been used for the inversion of a stochastic stiffness matrix. The digital simulation technique was adopted to generate the deviatoric part of the stiffness matrix and load vector. A beam problem was taken up for comparison of results obtained by Neumann expansion and the direct Monte Carlo simulation technique.
Random homogenization analysis in linear elasticity based on analytical bounds and estimates
International Journal of Solids and Structures, 2011
In this work, random homogenization analysis of heterogeneous materials is addressed in the context of elasticity, where the randomness and correlation of components' properties are fully considered and random effective properties together with their correlation for the two-phase heterogeneous material are then sought. Based on the analytical results of homogenization in linear elasticity, when the randomness of bulk and shear moduli, the volume fraction of each constituent material and correlation among random variables are considered simultaneously, formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from Random Factor Method. Results from the Random Factor Method and the Monte-Carlo Method are compared with each other through numerical examples, and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods. Moreover, the correlation coefficients of random effective properties are obtained by the Monte-Carlo Method. The Random Factor Method is found to deliver rapid results with comparable accuracy to the Monte-Carlo approach.
Strain and stress computations in stochastic finite element methods
International Journal for Numerical Methods in Engineering, 2008
This paper focuses on the computation of statistical moments of strains and stresses in a random system model where uncertainty is modeled by a stochastic finite element method based on the polynomial chaos expansion. It identifies the cases where this objective can be achieved by analytical means using the orthogonality property of the chaos polynomials and those where it requires a numerical integration technique. To this effect, the applicability and efficiency of several numerical integration schemes are considered. These include the Gauss-Hermite quadrature with the direct tensor product-also known as the Kronecker product-Smolyak's approximation of such a tensor product, Monte Carlo sampling, and the Latin Hypercube sampling method. An algorithm for reducing the dimensionality of integration under a direct tensor product is also explored for optimizing the computational cost and complexity. The convergence rate and algorithmic complexity of all of these methods are discussed and illustrated with the non-deterministic linear stress analysis of a plate. uncertainties can be modeled and their effects can be analyzed using a computational framework based on probability theory.
International Journal of Mechanical Sciences, 1996
Statistical uncertainty is bound to occur due to the randomness in material and geometric properties, support conditions, soil variability, etc. in structural engineering problems. An attempt has been made to study the stochastic structural responses, in particular, their mean and variances under such uncertain system parameters. The random parameters are modeled as homogeneous Gaussian stochastic processes and discretized by efficient local averaging method. The discretized Gaussian field is simulated by Cholesky decomposition of the respective covariance matrix. The present paper takes the advantage of Neumann expansion technique in deriving the finite element solution of response variability within the framework of Monte Carlo simulation. Neumann expansion technique needs inversion of only the deterministic part of the stiffness matrix for all sample structures, and thus increases the computational efficiency. Numerical examples are presented to study the advantage of Neumann expansion based simulation method with respect to accuracy and efficiency. The comparison of the results shows that the values approach towards that obtained by direct Monte Carlo simulation as the order of expansion in Neumann series is increased.
An Introduction to Stochastic Finite Element Method Analysis of Hyperelastic Structures
Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)
The main idea of this work is to demonstrate an application of the generalized iterative stochastic perturbation technique to numerical analysis of the hyperelastic materials and structures with Gaussian random parameter, where the input random variable is a magnitude of the vertical uniformly distributed load. Theoretical apparatus is connected with the general order Taylor expansion of both input and state parameters with random coefficients and analytical derivation of their first four probabilistic moments and coefficients. Our computational implementation is released with the Response Function Method having polynomial basis of the order minimizing variance and maximizing correlation of the least squares fitting to the series of numerical experiments. Computational experiment concerns the hyperelastic rubber-like prismatic beam under three-point bending discretized in the FEM system ABAQUS with the use of various 3D brick finite elements. Large deformations in the vertical symmetry plane of this structure are analyzed in the stochastic context-by determination of their expectations, coefficients of variations, skewness and kurtosis for different increments of the external load. It enables also to recover the basic probabilistic characteristics of the stress-strain curve of such a material, whose further comparison with the experiments will allow a full validation of such a probabilistic model. The entire probabilistic algorithm together with statistically optimized Weighted Least Squares Method fitting are implemented in the symbolic algebra package MAPLE. The proposed scheme of the Stochastic Finite Element Method is contrasted with the crude Monte-Carlo scheme and also with the semi-analytical calculations of the same probabilistic characteristics by direct integration of the response functions.
Structural reliability and stochastic finite element methods
Engineering Computations, 2018
Purpose This paper aims to provide a comprehensive review of uncertainty quantification methods supported by evidence-based comparison studies. Uncertainties are widely encountered in engineering practice, arising from such diverse sources as heterogeneity of materials, variability in measurement, lack of data and ambiguity in knowledge. Academia and industries have long been researching for uncertainty quantification (UQ) methods to quantitatively account for the effects of various input uncertainties on the system response. Despite the rich literature of relevant research, UQ is not an easy subject for novice researchers/practitioners, where many different methods and techniques coexist with inconsistent input/output requirements and analysis schemes. Design/methodology/approach This confusing status significantly hampers the research progress and practical application of UQ methods in engineering. In the context of engineering analysis, the research efforts of UQ are most focused...
Communications in Statistics - Simulation and Computation, 2020
This article presents error estimates for the expected value and variance of the stochastic process of transverse displacement of an Euler-Bernoulli beam, with uncertainty in the mechanical properties of the materials. Estimators are obtained using the Neumann-Monte Carlo, Yamazaki et al. (1988) methodology. The theoretical results are unprecedented and use the properties of the Neumann series. The error rates are shown to have exponential decay. To evaluate the performance of the error estimates, numerical experiments are presented for the problem of stochastic bending of beams.
Non-parametric structural reliability analysis using random fields and robustness evaluation
Proceedings Weimarer Optimierungs-und …, 2006
In reliability and robustness analysis, imperfections of a mechanical or structural system, such as material properties or geometrical deviations, are modelled as random fields in order to account for their fluctuations over space. A random field normally comprises a huge number of random variables. The present paper proposes a method to reduce the random variables set. This reduction is performed on the basis on a robustness analysis. In this way, numerical difficulties can be avoided and the efficiency of the subsequent reliability analysis is enhanced. As an example, the reliability of a cylindricral shell structure with random imperfections is studied. Within this example, the imperfections are discretized by Stochastic Finite Element methods. it is demonstrated, how robustness analysis is employed in order to identify the most relevant random variables. The probability of failure is computed by Monte Carlo simulation involving Latin Hypercube sampling. The failure criterion is derived from a comparison of the linear buckling loads of the perfect and the imperfect structures. This so-called non-parametric structural reliability analysis is a new method to estimate the safety and reliability of finite element structures in such cases where a CAD-based parametrization is not possible or not meaningful. The probabilistic and structural analysis tasks are performed with the optiSLang, SoS and SL ang software packages.
Stochastic finite element analysis of composite structures based on material microstructure
Composite Structures, 2015
The linking of microstructure uncertainty with the random variation of material properties at the macroscale is particularly needed in the framework of the stochastic finite element method (SFEM) where arbitrary assumptions are usually made regarding the probability distribution and correlation structure of the macroscopic mechanical properties. This linking can be accomplished in an efficient manner by exploiting the excellent synergy of the extended finite element method (XFEM) and Monte Carlo simulation (MCS) for the computation of the effective properties of random two-phase composites. The homogenization is based on Hill's energy condition and involves the generation of a large number of random realizations of the microstructure geometry based on a given volume fraction of the inclusions and other parameters (shape, number, spatial distribution and orientation). In this paper, the mean value, coefficient of variation and probability distribution of the effective elastic modulus and Poisson ratio are computed taking into account the material microstructure. The effective properties are used in the framework of SFEM to obtain the response of a composite structure and it is shown that the response variability can be significantly affected by the random microstructure.