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See detailA Bayesian framework to identify random parameter fields based on the copula theorem and Gaussian fields: Application to polycrystalline materials
Rappel, Hussein UL; Wu, Ling; Noels, Ludovic et al

in Journal of Applied Mechanics (in press)

For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application ... [more ▼]

For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application approaches the length scale of the micro-structure however, spatially fluctuating parameter fi elds (which vary from one realization of the fi eld to another) can be incorporated to make the model capture the stochasticity of the underlying micro-structure. Randomly fluctuating parameter fields are often described as Gaussian fields. Gaussian fi elds however assume that the probability density function of a material parameter at a given location is a univariate Gaussian distribution. This entails for instance that negative parameter values can be realized, whereas most material parameters have physical bounds (e.g. the Young's modulus cannot be negative). In this contribution, randomly fluctuating parameter fi elds are therefore described using the copula theorem and Gaussian fi elds, which allow di fferent types of univariate marginal distributions to be incorporated, but with the same correlation structure as Gaussian fields. It is convenient to keep the Gaussian correlation structure, as it allows us to draw samples from Gaussian fi elds and transform them into the new random fields. The bene fit of this approach is that any type of univariate marginal distribution can be incorporated. If the selected univariate marginal distribution has bounds, unphysical material parameter values will never be realized. We then use Bayesian inference to identify the distribution parameters (which govern the random fi eld). Bayesian inference regards the parameters that are to be identi fied as random variables and requires a user-defi ned prior distribution of the parameters to which the observations are inferred. For the homogenized Young's modulus of a columnar polycrystalline material of interest in this study, the results show that with a relatively wide prior (i.e. a prior distribution without strong assumptions), a single specimen is su ciffient to accurately recover the distribution parameter values. [less ▲]

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See detailBayesian Identification of Mean-Field Homogenization model parameters and uncertain matrix behavior in non-aligned short fiber composites
Mahamedou, Mohamed; Zulueta Uriondo, Kepa; Chung, Chi Nghia et al

in Composite Structures (2019), 220

We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other ... [more ▼]

We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other hand, the composite material constituents behaviors, including their variability. In particular, we characterize the model parameters of a Mean-Field Homogenization (MFH) model and the elastic matrix behavior, including the inherent dispersion in its Young's modulus, of non-aligned Short Fibers Reinforced Polymer (SFRP) composites. The inference is achieved by considering as observations experimental tests conducted at the SFRP composite coupons level. The inferred model and material law parameters can in turn be used in Mean-Field Homogenization (MFH)-based multi-scale simulations and can predict the confidence range of the composite material responses. [less ▲]

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