Stochastic Design Optimization of Microstructures with Utilization of a Linear Solver

Microstructure design can have a substantial effect on the performance of critical components in numerous aerospace applications. However, the stochastic nature of metallic microstructures leads to deviations in material properties from the design point, and it alters the performance of these critical components. In this work, an inverse stochastic design approach is introduced such that the material is optimized while accounting for the inherent variations in the microstructure. The highlight is an analytical uncertainty quantification model via a Gaussian distribution to model propagation of microstructural uncertainties to the properties. A metallic microstructure is represented using a finite element discretized form of the orientation distribution function. A stochastic optimization approach is proposed that employs the analytical model for uncertainty quantification, to maximize the yield strength of Galfenol microstructure in a compliant beam when constrained by uncertainties in the designed natural frequency of vibration. The results of the stochastic optimization approach are validated using a Monte Carlo simulation. It is also shown that multiple microstructure solutions can be identified using the null space of the linear systems involved in the optimization.