Uncertainty quantification for multiscale disk forging of polycrystal materials using probabilistic graphical model techniques

The mechanical properties of a deformed workpiece are sensitive to the initial microstructure before processing commences. Generally, the initial microstructure is random in nature and location specific. The location-dependence of the microstructure dramatically increases the dimensionality of the stochastic input and thus leads to the “curse of dimensionality” in stochastic deformation problems. In this work, a graph-theoretic approach is used to address the stochastic multiscale deformation problem and compute the propagation of the initial microstructure uncertainty to the forged disk properties. Following the finite element representation of the multiscale deformation problem, a graphical representation is introduced with nodes in the graph representing the macro- and meso- scale random variables and links between nodes modeling the dependence relations between variables at each scale and across scales. Model reduction techniques are employed locally in the graph to represent the initial random microstructure. Then the conditional distribution of the multi-output mechanical responses on the low-dimensional representation of the initial microstructure is factorized into a product of local potential functions. An expectation-maximization algorithm is used to learn the non-parametric representation of these potentials using a set of training data. A non-parametric loopy belief propagation method is applied to perform uncertainty quantification tasks. The non-parametric nature of the model is able to capture non-Gaussian features of the responses. The developed framework can be used as a surrogate model to predict the mechanical response fields for any input microstructure realization as well as our confidence on such predictions. A multiscale disk forging example of FCC nickel is presented to demonstrate the accuracy and efficiency of the constructed framework for addressing uncertainty quantification problems in multiscale deformation processes.