Computer simulations often involve both qualitative and numerical inputs. Existing Gaussian process (GP) methods for handling this mainly assume a different response surface for each combination of levels of the qualitative factors and relate them via a multiresponse cross-covariance matrix. We introduce a substantially different approach that maps each qualitative factor to an underlying numerical latent variable (LV), with the mapped value for each level estimated similarly to the correlation parameters. This provides a parsimonious GP parameterization that treats qualitative factors the same as numerical variables and views them as effecting the response via similar physical mechanisms. This has strong physical justification, as the effects of a qualitative factor in any physics-based simulation model must always be due to some underlying numerical variables. Even when the underlying variables are many, sufficient dimension reduction arguments imply that their effects can be represented by a low-dimensional LV. This conjecture is supported by the superior predictive performance observed across a variety of examples. Moreover, the mapped LVs provide substantial insight into the nature and effects of the qualitative factors.

Mechanical degradation and resultant capacity fade in high-capacity electrode materials critically hinder their use in high-performance rechargeable batteries. Despite tremendous efforts devoted to the study of the electro–chemo–mechanical behaviours of high-capacity electrode materials, their fracture properties and mechanisms remain largely unknown. Here we report a nanomechanical study on the damage tolerance of electrochemically lithiated silicon. Our in situ transmission electron microscopy experiments reveal a striking contrast of brittle fracture in pristine silicon versus ductile tensile deformation in fully lithiated silicon. Quantitative fracture toughness measurements by nanoindentation show a rapid brittle-to-ductile transition of fracture as the lithium-to-silicon molar ratio is increased to above 1.5. Molecular dynamics simulations elucidate the mechanistic underpinnings of the brittle-to-ductile transition governed by atomic bonding and lithiation-induced toughening. Our results reveal the high damage tolerance in amorphous lithium-rich silicon alloys and have important implications for the development of durable rechargeable batteries.

The data-driven approach is emerging as a promising method for the topological design of multiscale structures with greater efficiency. However, existing data-driven methods mostly focus on a single class of microstructures without considering multiple classes to accommodate spatially varying desired properties. The key challenge is the lack of an inherent ordering or “distance” measure between different classes of microstructures in meeting a range of properties. To overcome this hurdle, we extend the newly developed latent-variable Gaussian process (LVGP) models to create multi-response LVGP (MR-LVGP) models for the microstructure libraries of metamaterials, taking both qualitative microstructure concepts and quantitative microstructure design variables as mixed-variable inputs. The MR-LVGP model embeds the mixed variables into a continuous design space based on their collective effects on the responses, providing substantial insights into the interplay between different geometrical classes and material parameters of microstructures. With this model, we can easily obtain a continuous and differentiable transition between different microstructure concepts that can render gradient information for multiscale topology optimization. We demonstrate its benefits through multiscale topology optimization with aperiodic microstructures. Design examples reveal that considering multiclass microstructures can lead to improved performance due to the consistent load-transfer paths for micro- and macro-structures.

Summary Gaussian process (GP) metamodels have been widely used as surrogates for computer simulations or physical experiments. The heart of GP modeling lies in optimizing the log‐likelihood function with respect to the hyperparameters to fit the model to a set of observations. The complexity of the log‐likelihood function, computational expense, and numerical instabilities challenge this process. These issues limit the applicability of GP models more when the size of the training data set and/or problem dimensionality increase. To address these issues, we develop a novel approach for fitting GP models that significantly improves computational expense and prediction accuracy. Our approach leverages the smoothing effect of the nugget parameter on the log‐likelihood profile to track the evolution of the optimal hyperparameter estimates as the nugget parameter is adaptively varied. The new approach is implemented in the R package GPM and compared to a popular GP modeling R package (GPfit) for a set of benchmark problems. The effectiveness of the approach is also demonstrated using an engineering problem to learn the constitutive law of a hyperelastic composite where the required level of accuracy in estimating the response gradient necessitates a large training data set.

Materials design can be cast as an optimization problem with the goal of achieving desired properties, by varying material composition, microstructure morphology, and processing conditions. Existence of both qualitative and quantitative material design variables leads to disjointed regions in property space, making the search for optimal design challenging. Limited availability of experimental data and the high cost of simulations magnify the challenge. This situation calls for design methodologies that can extract useful information from existing data and guide the search for optimal designs efficiently. To this end, we present a data-centric, mixedvariable Bayesian Optimization framework that integrates data from literature, experiments, and simulations for knowledge discovery and computational materials design. Our framework pivots around the Latent Variable Gaussian Process (LVGP), a novel Gaussian Process technique which projects qualitative variables on a continuous latent space for covariance formulation, as the surrogate model to quantify "lack of data" uncertainty. Expected improvement, an acquisition criterion that balances exploration and exploitation, helps navigate a complex, nonlinear design space to locate the optimum design. The proposed framework is tested through a case study which seeks to concurrently identify the optimal composition and morphology for insulating polymer nanocomposites. We also present an extension of mixed-variable Bayesian Optimization for multiple objectives to identify the Pareto Frontier within tens of iterations. These findings project Bayesian Optimization as a powerful tool for design of engineered material systems.