A multiscale approach for model reduction of random microstructures

The mechanical properties of a deformed workpiece are sensitive to the initial microstructure. Often, the initial microstructure is random in nature and location specific. To model the variability of properties of the workpiece induced by variability in the initial microstructure, one needs to develop a reduced order stochastic input model for the initial microstructure. The location-dependence of microstructures dramatically increases the dimensionality of the stochastic input and causes the “curse of dimensionality” in a stochastic deformation simulation. To quantify and capture the propagation of uncertainty in multiscale deformation processes, a novel data-driven bi-orthogonal Karhunen–Loève Expansion (KLE) strategy is introduced. The multiscale random field representing random microstructures over the workpiece is decomposed simultaneously into a few modes in the macroscale and mesoscale. The macro modes are further expanded through a second-level KLE to separate the random and spatial coordinates. The few resulting random variables are mapped to the uniform distribution via a polynomial chaos (PC) expansion. As a result, the stochastic input complexity is remarkably reduced. Sampling from the reduced random space, new microstructure realizations are reconstructed. By collecting the properties of workpieces with randomly sampled microstructures, the property statistics are computed. A high-dimensional multiscale disk forging example of FCC nickel is presented to show the merit of this methodology, and the effect of random initial crystallographic texture on the macroscopic properties.