A random field-based method to estimate convergence of apparent properties in computational homogenization

This work addresses the challenge of efficiently determining the representative volume size in computational homogenization by exploiting the requirement that mechanical response quantities are statistically homogeneous and ergodic random fields. The proposed computational homogenization approach focuses on empirically determining the autocorrelation functions of output response quantities through post processing finite element analyses. Once the autocorrelation functions are known, only trivial computations are needed to determine the variance of apparent properties as a function of domain size without any additional finite element computation by utilizing a simple formula relating the autocorrelation function to the variance of spatially averaged quantities. This approach improves upon the current established approach by circumventing the need to analyze numerous successively larger domains in order to determine convergence of apparent properties, which can be computationally prohibitive. Furthermore, similar previous analytical expressions for the variance of apparent properties are asymptotic with respect to domain size while the expression in the proposed approach is exact. After presenting the formulation, the method is first demonstrated for a one-dimensional bar model where analytical expressions can be derived. Then the approach is demonstrated on two numerical examples: (1) a plane strain, linear elastic domain with a lognormal random field describing its compliance, and (2) a stochastic polycrystalline microstructure of a Nickel super alloy having a crystal plasticity model for its constitutive law.