Data-Driven Multi-Scale Modeling and Optimization for Elastic Properties of Cubic Microstructures

The present work addresses gradient-based and machine learning (ML)-driven design optimization methods to enhance homogenized linear and nonlinear properties of cubic microstructures. The study computes the homogenized properties as a function of underlying microstructures by linking atomistic-scale and meso-scale models. Here, the microstructure is represented by the orientation distribution function that determines the volume densities of crystallographic orientations. The homogenized property matrix in meso-scale is computed using the single-crystal property values that are obtained by density functional theory calculations. The optimum microstructure designs are validated with the available data in the literature. The single-crystal designs, as expected, are found to provide the extreme values of the linear properties, while the optimum values of the nonlinear properties could be provided by single or polycrystalline microstructures. However, polycrystalline designs are advantageous over single crystals in terms of better manufacturability. With this in mind, an ML-based sampling algorithm is presented to identify top optimum polycrystal solutions for both linear and nonlinear properties without compromising the optimum property values. Moreover, an inverse optimization strategy is presented to design microstructures for prescribed values of homogenized properties, such as the stiffness constant (C11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{11}$$\end{document}) and in-plane Young’s modulus (E11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{11}$$\end{document}). The applications are presented for aluminum (Al), nickel (Ni), and silicon (Si) microstructures.