  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.
