The multifaceted nature of uncertainty in structure-property linkage with crystal plasticity finite element model

Abstract

Uncertainty quantification (UQ) plays a critical role in verifying and validating forward integrated computational materials engineering (ICME) models. Among numerous ICME models, the crystal plasticity finite element method (CPFEM) is a powerful tool that enables one to assess microstructure-sensitive behaviors and thus, bridge material structure to performance. Nevertheless, given its nature of constitutive model form and the randomness of microstructures, CPFEM is exposed to both aleatory uncertainty (microstructural variability), as well as epistemic uncertainty (parametric and model-form error). Therefore, the observations are often corrupted by the microstructure-induced uncertainty, as well as the ICME approximation and numerical errors. In this work, we highlight several ongoing research topics in UQ, optimization, and machine learning applications for CPFEM to efficiently solve forward and inverse problems. The first aspect of this work addresses the UQ of constitutive models for epistemic uncertainty, including both phenomenological and dislocation-density-based constitutive models, where the quantities of interest (QoIs) are related to the initial yield behaviors. We apply a stochastic collocation (SC) method to quantify the uncertainty of the three most commonly used constitutive models in CPFEM, namely phenomenological models (with and without twinning), and dislocation-density-based constitutive models, for three different types of crystal structures, namely face-centered cubic (fcc) copper (Cu), body-centered cubic (bcc) tungsten (W), and hexagonal close packing (hcp) magnesium (Mg). The second aspect of this work addresses the aleatory and epistemic uncertainty with multiple mesh resolutions and multiple constitutive models by the multi-index Monte Carlo method, where the QoI is also related to homogenized materials properties. We present a unified approach that accounts for various fidelity parameters, such as mesh resolutions, integration time-steps, and constitutive models simultaneously. We illustrate how multilevel sampling methods, such as multilevel Monte Carlo (MLMC) and multi-index Monte Carlo (MIMC), can be applied to assess the impact of variations in the microstructure of polycrystalline materials on the predictions of macroscopic mechanical properties. The third aspect of this work addresses the crystallographic texture study of a single void in a cube. Using a parametric reduced-order model (also known as parametric proper orthogonal decomposition) with a global orthonormal basis as a model reduction technique, we demonstrate that the localized dynamic stress and strain fields can be predicted as a spatiotemporal problem. The fourth aspect of this work highlights the constitutive model calibration using an optimization under microstructure-induced uncertainty with Bayesian optimization. To account for natural variability or the aleatory uncertainty of microstructure, we average the loss function over an ensemble of microstructures and couple the Monte Carlo estimator with an asynchronous parallel Bayesian optimization to calibrate a phenomenological constitutive model. The framework is demonstrated for 304L stainless steel. The fifth aspect of this work solves a stochastic inverse problem in the structure-property relationship. In this aspect, we seek to consistently learn a distribution of microstructure features, in the sense that the forward propagation of this microstructure feature distribution through CPFEM matches a target distribution of homogenized materials properties.