A generalized uncertainty propagation criterion from benchmark studies of microstructured material systems (original) (raw)
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Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM
International Journal for Numerical Methods in Engineering, 2011
An extended finite element method (XFEM) coupled with a Monte Carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials. The methodology allows for an arbitrary number, aspect ratio, location and orientation of elliptic inclusions within a matrix, without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure. Moreover, the number of degrees of freedom in the enriched elements is dynamically reallocated for each Monte Carlo sample run based on the given volume fraction. The main advantage of the proposed XFEM-based methodology is a major reduction in the computational effort in extensive Monte Carlo simulations compared with the standard FEM approach. Monte Carlo and XFEM appear to work extremely efficiently together. The Monte Carlo approach allows for the modeling of the size, aspect ratios, orientations, and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions. Numerical results are presented and the uncertainty of the homogenized elastic properties is discussed.
2012
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Uncertainty propagation in a multiscale CALPHAD-reinforced elastochemical phase-field model
Acta Materialia, 2019
ICME approaches provide decision support for materials design by establishing quantitative process-structure-property relations. Confidence in the decision support, however, must be achieved by establishing uncertainty bounds in ICME model chains. The quantification and propagation of uncertainty in computational materials science, however, remains a rather unexplored aspect of computational materials science approaches. Moreover, traditional uncertainty propagation frameworks tend to be limited in cases with computationally expensive simulations. A rather common and important model chain is that of CALPHAD-based thermodynamic models of phase stability coupled to phase field models for microstructure evolution. Propagation of uncertainty in these cases is challenging not only due to the sheer computational cost of the simulations but also because of the high dimensionality of the input space. In this work, we present a framework for the quantification and propagation of uncertainty in a CALPHADbased elasto-chemical phase field model. We motivate our work by investigating the microstructure evolution in Mg 2 (Si x Sn 1−x) thermoelectric materials. We first carry out a Markov Chain Monte Carlo-based inference of the CALPHAD model parameters for this pseudobinary system and then use advanced sampling schemes to propagate uncertainties across a high-dimensional simulation input space. Through high-throughput phase field simulations we generate 200,000 time series of synthetic microstructures and use machine learning approaches to understand the effects of propagated uncertainties on the microstructure landscape of the system under study. The microstructure dataset has been curated in the Open Phase-field Microstructure Database (OPMD), available at http://microstructures.net.
A Multiscale Design Approach with Random Field Representation of Material Uncertainty
2008
An integrated design framework that employs multiscale analysis to facilitate concurrent product, material, and manufacturing process design is presented in this work. To account for uncertainties associated with material structures and their impact on product performance across multiple scales, efficient computational techniques are developed for propagating material uncertainty with random field representation. Random field is employed to realistically model the uncertainty existing in material microstructure, which spatially varies in a product inherited from the manufacturing process. To reduce the dimensionality of random field representation, a reduced order Karhunen-Loeve expansion is used with a discretization scheme applied to finite-element meshes. The univariate dimension reduction method and the Gaussian quadrature formula are used to efficiently quantify the uncertainties in product performance in terms of its statistical moments, which are critical information for design under uncertainty. A control arm example is used to demonstrate the proposed approach. The impact of the initial microscale porosity random field produced during a casting process on the product damage is studied and a reliability-based design of the control arm is performed.
Micromechanically based stochastic finite elements: length scales and anisotropy
Probabilistic Engineering Mechanics, 1996
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Uncertainty Quantification of Microstructural Properties due to Experimental Variations
19th AIAA Non-Deterministic Approaches Conference, 2017
Electron backscatter diffraction scans are an important experimental input for microstructure generation and homogenization. Multiple electron backscatter diffraction scans can be used to sample the uncertainty in orientation distribution function: both point to point within a specimen as well as across multiple specimens that originate from the same manufacturing process. However, microstructure analysis methods typically employ only the mean values of the orientation distribution function to predict properties, and the stochastic information is lost. In this work, analytical methods are developed to account for the uncertainty in the electron backscatter diffraction data during property analysis. To this end, a linear smoothing scheme is developed in the Rodrigues fundamental region to compute the orientation distribution function from the electron backscatter diffraction data. The joint multivariate probability distributions of the orientation distribution function are then modeled using a Gaussian assumption. The uncertainty in engineering properties that are obtained by homogenization are also computed. It is shown that the uncertainty in nonlinear properties can be analytically obtained using direct transformation of random variables in the homogenization approach.
A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty
International Journal of Plasticity, 2021
Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of uncertainty in the computational predictions delivered by them. In this work, a sequential multiscale model has been developed, incorporating discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model to predict the size effect in plastic deformations of metallic micro-pillars. The DDD simulations include uniaxial compression of micro-pillars with different sizes and over a wide range of initial dislocation densities and spatial distributions of dislocations. An SGP model is employed at the continuum level that accounts for the size-dependency of flow stress and hardening rate. Sequences of uncertainty analyses have been performed to assess the predictive capability of the multiscale model. The variance-based global sensitivity analysis determines the effect of parameter uncertainty on the SGP model prediction. The multiscale model is then constructed by calibrating the continuum model using the data furnished by the DDD simulations. A Bayesian calibration method is implemented to quantify the uncertainty due to microstructural randomness in discrete dislocation simulations (density and spatial distributions of dislocations) on the macroscopic continuum model prediction (size effect in plastic deformation). The outcomes of this study indicate that the discrete-continuum multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the DDD results. Additionally, depending on the macroscopic features represented by the DDD simulations, the SGP model can reliably predict the size effect in plasticity responses of the micropillars with below 10% of error.
On two micromechanics theories for determining micro–macro relations in heterogeneous solids
Mechanics of Materials, 1999
The average-®eld theory and the homogenization theory are brie¯y reviewed and compared. These theories are often used to determine the eective moduli of heterogeneous materials from their microscopic structure in such a manner that boundary-value problems for the macroscopic response can be formulated. While these two theories are based on dierent modeling concepts, it is shown that they can yield essentially the same eective moduli and boundary-value problems. A hybrid micromechanics theory is proposed in view of this correspondence. This theory leads to a more accurate computation of the eective moduli, and applies to a broader class of microstructural models. Hence, the resulting macroscopic boundary-value problem gives better estimates of the macroscopic response of the material. In particular, the hybrid theory can account for the eects of the macrostrain gradient on the macrostress in a natural manner. Ó