Computational Discovery of Microstructured Composites with Optimal Stiffness-Toughness Trade-Offs

TL;DR

This approach implements a nested-loop proposal-validation workflow to bridge the simulation-to-reality gap and find microstructured composites that are stiff and tough with high sample efficiency and provides a blueprint for computational design in various research areas beyond solid mechanics.

Abstract

The conflict between stiffness and toughness is a fundamental problem in engineering materials design. However, the systematic discovery of microstructured composites with optimal stiffness-toughness trade-offs has never been demonstrated, hindered by the discrepancies between simulation and reality and the lack of data-efficient exploration of the entire Pareto front. We introduce a generalizable pipeline that integrates physical experiments, numerical simulations, and artificial neural networks to address both challenges. Without any prescribed expert knowledge of material design, our approach implements a nested-loop proposal-validation workflow to bridge the simulation-to-reality gap and discover microstructured composites that are stiff and tough with high sample efficiency. Further analysis of Pareto-optimal designs allows us to automatically identify existing toughness enhancement mechanisms, which were previously discovered through trial-and-error or biomimicry. On a broader scale, our method provides a blueprint for computational design in various research areas beyond solid mechanics, such as polymer chemistry, fluid dynamics, meteorology, and robotics.

Figures (4)

• Figure 1: Schematic of our approach. ( A) Workflow of the proposed nested-loop pipeline. Our system integrates three distinctive evaluators: a mechanical tester, an FEM-based simulator, and a CNN-based predictor, which vary in evaluation speed and accuracy. Data flow directions are indicated by colored arrows. Arrows from slower, more accurate evaluators to faster, less accurate evaluators illustrate the propagation of microstructure performance data for improving the latter's accuracy. Arrows pointing in the opposite direction represent proposed Pareto-optimal designs from faster evaluators to slower ones for validation. The self-loop at the predictor depicts an evolutionary strategy for finding its Pareto front. ( B) Illustration of a microstructure and the corresponding microstructured composites. The microstructure is defined by a 2D pattern that demonstrates the spatial arrangement of two base materials with contrasting properties. ( C) Sample photographs of manufactured microstructured composites, annotated by the source microstructure patterns.
• Figure 2: The inner loop of the proposed workflow with neural network-accelerated multi-objective optimization (NMO). ( A) Workflow of NMO, illustrated by a zoomed-in snapshot of Fig. \ref{['fig:pipeline']}A. The simulator has 10 exposed parameters, including material model parameters of rigid, soft, and interface base materials plus a global damping coefficient. ( B) Average prediction errors of Young's modulus and toughness in NMO over 500 iterations, calculated for all discovered microstructures and specifically those on the simulation Pareto front. Shaded regions indicate standard deviations estimated from adjacent data points. ( C) Evolution of design proposal quality, as characterized by the Pareto hypervolume of 10 proposed designs in each iteration, over 500 iterations. NMO is compared with NSGA-II and a random sampling strategy. ( D), Comparison of the final Pareto hypervolumes from NMO and its simplified alternative (NMO one iteration) that only trains the predictor with 5,000 random designs and proposes designs back to the simulator once. ( E) Comparison between NMO and other multi-objective optimization algorithms in Pareto hypervolume growth within a budget of 5,000 simulation evaluations. The baselines comprise our modified NSGA-II algorithm, topology optimization (TO) (e.g., BESO querin1998evolutionaryhuang2008topology and SIMP bendsoe1989optimal), and multi-objective Bayesian optimization (MOBO) (e.g., DGEMO lukovic2020diversity and TSEMO bradford2018efficient). MOBO algorithms are stopped at 2,000 simulations due to exceeding a time limit of 24 hours. Each solid curve is an average of repeats using five random seeds and the colored region around each curve indicates standard deviation. ( F) Number of evaluations required for NMO and other baseline algorithms to reach a target hypervolume, marked by the dashed black line in ( E).
• Figure 3: The outer loop of the proposed workflow. ( A) Experiment data of discovered microstructures and the simulation gamut at the end of Round 0. The sim-to-real gap is defined as the symmetric difference between the Pareto hypervolumes of the experimental Pareto front and the simulation Pareto front. ( B) Workflow illustration of the outer loop as simplified from Fig. \ref{['fig:pipeline']}A. ( C) Evolution of experiment data and the simulation gamut within four rounds of the outer loop, where the sim-to-real gap shrinks substantially. ( D) Pareto hypervolume of experiment data and the simulation gamut, and the area of the sim-to-real gap over four rounds of the outer loop. ( E) Representative microstructures discovered by the pipeline with optimal trade-offs. Numbers in parentheses indicate the improvement of a design in Young’s modulus and toughness compared to homogeneous composites with various volume fractions of the rigid material. ( F) Average relative simulation error on Young's modulus and toughness in each round of the outer loop. The error is calculated for all 50 discovered microstructures to showcase the improvement in simulation accuracy. ( G) Quality of microstructure designs proposed by the simulator over the entire outer loop. Proposal quality is evaluated by the experimental Pareto hypervolume of 8 proposed designs in each round. The quality of the 10 randomly chosen microstructures before Round 1 (Rand.) is used as a reference.
• Figure 4: Analysis of families and intrinsic toughening mechanisms using discovered Pareto-optimal microstructures. ( A) Schematic of the analysis workflow. Microstructures near the Pareto front are grouped into families and further split into subfamilies (see Note S4 and Fig. S17). The mechanical performance of each family is verified, while the pattern variation in each subfamily is studied in a low-dimensional embedding space. ( B) Seed microstructures of four families and their mechanical performance compared with homogeneous composites. ( C) Representative stress-strain curves of microstructures in each family collected from mechanical testing. ( D and E) The Isomap embedding space of an example subfamily before ( D) and after ( E) interpolation. Some representative patterns are illustrated. Microstructure properties are encoded in colors (Young’s modulus: the blue channel; toughness: the red channel). Seed patterns are highlighted in orange boxes. ( F- H) Toughening mechanisms observed in experiment and simulation: bridging, deflection, and branching. Snapshots are taken from validation microstructures in Family 1 ( F), 2 ( G), and 3 ( H).