Crystal plasticity finite element method (CPFEM) has been an integrated computational materials engineering (ICME) workhorse to study materials behaviors and structure-property relationships for the last few decades. These relations are mappings from the microstructure space to the materials properties space. Due to the stochastic and random nature of microstructures, there is always some uncertainty associated with materials properties, for example, in homogenized stress-strain curves. For critical applications with strong reliability needs, it is often desirable to quantify the microstructure-induced uncertainty in the context of structure-property relationships. However, this uncertainty quantification (UQ) problem often incurs a large computational cost because many statistically equivalent representative volume elements (SERVEs) are needed. In this article, we apply a multi-level Monte Carlo (MLMC) method to CPFEM to study the uncertainty in stress-strain curves, given an ensemble of SERVEs at multiple mesh resolutions. By using the information at coarse meshes, we show that it is possible to approximate the response at fine meshes with a much reduced computational cost. We focus on problems where the model output is multi-dimensional, which requires us to track multiple quantities of interest (QoIs) at the same time. Our numerical results show that MLMC can accelerate UQ tasks around 2.23×, compared to the classical Monte Carlo (MC) method, which is widely known as ensemble average in the CPFEM literature.
Quantifying uncertainty associated with the microstructure variation of a material can be a computationally daunting task, especially when dealing with advanced constitutive models and fine mesh resolutions in the crystal plasticity finite element method (CPFEM). Numerous studies have been conducted regarding the sensitivity of material properties and performance to the mesh resolution and choice of constitutive model. However, a unified approach that accounts for various fidelity parameters, such as mesh resolutions, integration time-steps and constitutive models simultaneously is currently lacking. This paper proposes a novel uncertainty quantification (UQ) approach for computing the properties and performance of homogenized materials using CPFEM, that exploits a hierarchy of approximations with different levels of fidelity. In particular, we illustrate how multi-level sampling methods, such as multi-level 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 homogenized materials properties. We show that by adaptively exploiting the fidelity hierarchy, we can significantly reduce the number of microstructures required to reach a certain prescribed accuracy. Finally, we show how our approach can be extended to a multi-fidelity framework, where we allow the underlying constitutive model to be chosen from either a phenomenological plasticity model or a dislocation-density-based model.
The structure-property linkage is one of the two most important relationships in materials science besides the process-structure linkage, especially for metals and polycrystalline alloys. The stochastic nature of microstructures begs for a robust approach to reliably address the linkage. As such, uncertainty quantification (UQ) plays an important role in this regard and cannot be ignored. To probe the structure-property linkage, many multi-scale integrated computational materials engineering (ICME) tools have been proposed and developed over the last decade to accelerate the material design process in the spirit of Material Genome Initiative (MGI), notably crystal plasticity finite element model (CPFEM) and phase-field simulations. Machine learning (ML) methods, including deep learning and physics-informed/-constrained approaches, can also be conveniently applied to approximate the computationally expensive ICME models, allowing one to efficiently navigate in both structure and property spaces effortlessly. Since UQ also plays a crucial role in verification and validation for both ICME and ML models, it is important to include UQ in the picture. In this paper, we summarize a few of our recent research efforts addressing UQ aspects of homogenized properties using CPFEM in a big picture context.
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.