Variational Approach to Probabilistic Finite Elements

Liu, W. K.; Besterfield, Glen; Belytschko, Ted

doi:10.1061/(asce)0733-9399(1988)114:12(2115)

Cited by 69 publications

(10 citation statements)

References 14 publications

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“…The uncertainty of the constitutive model, represented by the PDF and covariance matrix of the calibration parameters, together with uncertainties from other sources, should be appropriately propagated for the prediction of material/structure performance. A variety of methods can be chosen for UP [72,74,[79][80][81][82][83] and in this paper, we propose to use the method of stochastic projection with PCE. It allows high-order representation and the UP with arbitrary probability measure with fast convergence.…”

Section: Uncertainty Propagation From Multiresolution Constitutive Modelmentioning

confidence: 99%

Complexity science of multiscale materials via stochastic computations

Liu

Siad

Tian

et al. 2009

Numerical Meth Engineering

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SUMMARYNew technological advances today allow for a range of advanced composite materials, including multilayer materials and nanofiber-matrix composites. In this context, the key to developing advanced materials is the understanding of the interplay between the various physical scales present, from the atomic level interactions to the microstructural composition and the macroscale behavior. Using the developing 'multiresolution data sets mechanics', the 'predictive science-based governing laws of the materials microstructure evolutions' are derived and melted into a 'stochastic multiresolution design framework.' Under such a framework, the governing laws of the materials microstructure evolution will be essential to assess, across multiple scales, the impact of multiscale material design, geometry design of a structure, and the manufacturing process conditions, by following the cause-effect relationships from structure to property and then to performance.The future integrated multiscale analysis system will be constructed based on a probabilistic complexity science-based mathematical framework. Its verification, validation and uncertainty quantification are done through carefully designed experiments, and the life-cycled materials design for products design and manufacturing is performed through the use of petascale computing. The various techniques of microstructure reconstruction result in the generation of model microstructures that, at some level, has the same statistical properties as the real heterogeneous media. Having reconstructed the heterogeneous medium, one can then evaluate its effective properties via direct numerical simulation and compare these values with experimentally measured properties of the actual medium. The proposed analysis will be dynamic in nature to capture the multi-stage historical evolvement of material/structure performance over the life span of a product. In addition to providing more accurate assessment of structure performance with stochastic multiscale analysis, our development will provide the capability of predicting structure failures and system reliability to enable more reliable design and manufacturing decisions in product development.

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Section: Uncertainty Propagation From Multiresolution Constitutive Modelmentioning

confidence: 99%

Complexity science of multiscale materials via stochastic computations

Liu

Siad

Tian

et al. 2009

Numerical Meth Engineering

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“…The analysis of nonlinear structural systems with parameter uncertainties is very limited to special cases such as static problems and numerical simulations. Static problems involving system nonlinearities and stochasticity were studied by Liu et al (1986Liu et al ( , 1988. Liu et al (1987) considered the dynamic response to a step input of an elastoplastic beam with isotropic hardening whose plastic modulus was modeled as a Gaussian random field.…”

Section: Nonlinear Systemsmentioning

confidence: 99%

Progress in Structural Dynamics With Stochastic Parameter Variations: 1987-1998

Manohar

Ibrahim

1999

Applied Mechanics Reviews

131

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This article is an update of an earlier paper by Ibrahim (1987) and is aimed at reviewing the work published during the last decade in the area of vibration of structures with parameter uncertainties. Different types of uncertainty modeling are described in terms of material and geometric properties. These models are considered in terms of Gaussian or non-Gaussian distributions. Computational stochastic algorithms including stochastic finite element methods and Monte Carlo simulation are dominating a major part of current activities. Recent analytical developments of the random eigenvalue problem are reviewed with reference to typical structural elements. These developments include the implementation of statistical energy analysis, stochastic boundary element methods, and interval algebra. Other topics include forced vibration of single-and multi-degree-offreedom systems including nonlinear systems, localization in disordered periodic structures, and experimental results. Computational stochastic mechanics has found several industrial applications including aerospace, automotive and composite structural elements. The review also covers developments in the areas of statistical modeling of high frequency vibrations. There are 183 references.

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“…Current stochastic ®nite element methods mainly include the Monte-Carlo FEM (Shinozuka and Wen 1972;Shinozuka and Astill 1972), the perturbation SFEM (Nakagiri and Hisada 1982; Liu et al 1986Liu et al , 1988aLiu et al , 1988b, the Neumann expansion approach (Yamazaki et al 1988) and the Karhunen-Loeve expansion method Spanos 1990, 1991). Besides, the Galerkin's method was also applied to develop SFEM (Lawrence 1987).…”

Section: Introductionmentioning

confidence: 99%

A stochastic finite element method for fatigue reliability analysis of gear teeth subjected to bending

Peng

Liu

et al. 1998

Computational Mechanics

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A stochastic ®nite element method (SFEM) is developed for accurate structural reliability analysis. Using the second-order three-moment reliability analytical model, this method takes into account such random factors as load, material parameters and especially geometry randomness. The calculation of the bending fatigue strength reliability of a cantilever beam is carried out as a numerical example to verify the present method. MonteCarlo FEM and SFEM based on the ®rst-order secondmoment model are used in the example to compare with the proposed method. By incorporating the fatigue theory of gears, the present method is then used to analyze the bending fatigue strength reliability of a spur gear. The effects of random variables' coef®cient of variation and skewness and the gear's correction factor (not random variable) on the gear's reliability are also investigated.

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Variational Approach to Probabilistic Finite Elements

Cited by 69 publications

References 14 publications

Complexity science of multiscale materials via stochastic computations

Complexity science of multiscale materials via stochastic computations

Progress in Structural Dynamics With Stochastic Parameter Variations: 1987-1998

A stochastic finite element method for fatigue reliability analysis of gear teeth subjected to bending

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