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Bridging the gaps among various categories of stochastic microstructures remains a challenge in the design representation of microstructural materials. Each microstructure category requires certain unique mathematical and statistical methods to define the design space (design representation). The design representation methods are usually incompatible between two different categories of stochastic microstructures. The common practice of pre-selecting the microstructure category and the associated design representation method before conducting rigorous computational design restricts the design freedom and hinders the discovery of innovative microstructure designs. To overcome this issue, this paper proposes and compares two novel methods, the deep generative modeling-based method and the curvature functional-based method, to understand their pros and cons in designing mixed-category stochastic microstructures for desired properties. For the deep generative modeling-based method, the Variational Autoencoder is employed to generate an unstructured latent space as the design space. For the curvature functional-based method, the microstructure geometry is represented by curvature functionals, of which the functional parameters are employed as the microstructure design variables. Regressors of the microstructure design variables-property relationship are trained for microstructure design optimization. A comparative study is conducted to understand the relative merits of these two methods in terms of computational cost, continuous transition, design scalability, design diversity, dimensionality of the design space, interpretability of the statistical equivalency, and design performance.

 

Quantification and propagation of aleatoric uncertainties distributed in complex topological structures remain a challenge. Existing uncertainty quantification and propagation approaches can only handle parametric uncertainties or high dimensional random quantities distributed in a simply connected spatial domain. There lacks a systematic method that captures the topological characteristics of the structural domain in uncertainty analysis. Therefore, this paper presents a new methodology that quantifies and propagates aleatoric uncertainties, such as the spatially varying local material properties and defects, distributed in a topological spatial domain. We propose a new random field-based uncertainty representation approach that captures the topological characteristics using the shortest interior path distance. Parameterization methods like PPCA and β-Variational Autoencoder (βVAE) are employed to convert the random field representation of uncertainty to a small set of independent random variables. Then non-intrusive uncertainties propagation methods such as polynomial chaos expansion and univariate dimension reduction are employed to propagate the parametric uncertainties to the output of the problem. The effectiveness of the proposed methodology is demonstrated by engineering case studies. The accuracy and computational efficiency of the proposed method is confirmed by comparing with the reference values of Monte Carlo simulations with a sufficiently large number of samples.