Data-driven Design of Isotropic and High-Stiffness TPMS-based Amorphousness-Induced Architected Material (TAAM)

Technical Summary: Data-driven Design of Isotropic and High-Stiffness TPMS-based Amorphousness-Induced Architected Material (TAAM)

Problem Statement
Triply periodic minimal surfaces (TPMS) are widely utilized in architected materials due to their high stiffness-to-weight ratios and smooth, continuous morphologies that mitigate stress concentrations. However, their inherent geometric regularity and symmetry frequently result in elastic anisotropy. This direction-dependent stiffness limits their reliability under complex or uncertain loading conditions, which is a critical constraint for applications such as biomedical implants, protective systems, and aerospace components. While previous strategies have attempted to mitigate this anisotropy through unit cell hybridization, symmetry-constrained assemblies, or variable thickness modulation, these approaches often compromise surface continuity, suffer from poor intercell connectivity, or operate within narrow structural subspaces that limit the exploration of the stiffness–isotropy trade-off.

Methodology
The authors propose a novel framework for TPMS-based amorphousness-induced architected materials (TAAMs). This approach introduces "designable amorphousness"—a controllable geometric disorder—as a tunable design variable to break the symmetry of periodic TPMS structures without relying on uncontrolled stochastic disorder.

1. Design Space Construction:

   • The base architecture is Schoen's I-Wrapped Package (IWP) minimal surface.
   • Geometric disorder is introduced via two mechanisms:
     • Amorphousness Constants: Parameters ($\alpha, \beta, \gamma$) modulate the implicit equation to create geometric irregularity.
     • Random Rotation: Each unit cell within the design domain is rotated using a random rotation matrix ($A$) defined by angles ($\theta_x, \theta_y, \theta_z$) sampled from a uniform distribution.
   • The design space is defined by the relative density (controlled by an iso-value) and the six parameters governing amorphousness and rotation.

2. Computational Homogenization:

   • Effective stiffness tensors ($\mathbb{C}$) are computed using Finite Element Method (FEM) under affine boundary conditions.
   • Anisotropy is quantified using the ratio of minimum to maximum directional elastic moduli ($E_{min}/E_{max}$), serving as a proxy for the universal anisotropy index.
   • Stiffness performance is benchmarked against the Hashin-Shtrikman (HS) upper bound for isotropic cellular structures.

3. Data-Driven Optimization:

   • A Multi-Objective Bayesian Optimization (MBO) framework is employed to navigate the trade-off between maximizing stiffness and minimizing anisotropy.
   • A Gaussian Process Regression (GPR) surrogate model, utilizing a Matern 5/2 kernel, approximates the mapping from design parameters to mechanical metrics.
   • Two acquisition functions are tested independently: Expected Hypervolume Improvement (EHVI) and Probability of Hypervolume Improvement (PHVI). The optimization process involves an initial Latin Hypercube Sampling (LHS) of 20 designs followed by 100 iterative updates guided by FEM simulations.

4. Experimental Validation:

   • Selected optimal designs (one from PHVI, one from EHVI) were fabricated using Fused Deposition Modeling (FDM) with PLA.
   • Specimens were tested via quasi-static uniaxial compression along three principal axes to validate numerical predictions regarding stiffness, crushing strength, plateau strength, and energy absorption.

Key Contributions

• Concept of Designable Amorphousness: The paper introduces a systematic method to incorporate tunable geometric disorder into periodic TPMS architectures, expanding the design space beyond traditional symmetric or hybridized unit cells.
• Optimization Framework: It demonstrates the efficacy of a data-driven MBO framework using GPR and specific acquisition functions (EHVI/PHVI) to discover Pareto-optimal solutions that balance stiffness and isotropy.
• Performance Validation: The study provides experimental evidence that TAAMs can achieve significantly improved elastic isotropy while maintaining high stiffness across various relative densities, outperforming conventional TPMS and existing benchmark designs (such as HTAM and CFCC) in the stiffness–isotropy trade-off.

Results

• Numerical Findings: The MBO framework successfully identified design configurations that occupy previously unexplored regions of the design space. The PHVI-guided design exhibited slightly higher stiffness, while the EHVI-guided design demonstrated superior isotropic performance. Both outperformed standard IWP and other benchmark structures.
• Experimental Findings: FDM-fabricated TAAM specimens showed strong agreement with numerical predictions. Uniaxial compression tests confirmed that the optimized structures maintained high stiffness and exhibited improved isotropy compared to regular TPMS structures. The manufacturing process (FDM) proved capable of producing the complex, support-free geometries with high fidelity.

Significance
The paper claims that this approach offers a robust and scalable pathway for developing architected materials tailored for applications requiring isotropic performance. By integrating controllable geometric disorder with data-driven optimization, the proposed TAAM framework addresses the fundamental limitation of elastic anisotropy in TPMS structures. The authors suggest that this method enables the simultaneous optimization of stiffness and isotropy, providing a general solution for designing materials suitable for biomedical implants, protective systems, and aerospace components where loading conditions are complex and unpredictable. The study underscores the potential of "designable amorphousness" as a fundamental design principle for next-generation mechanical metamaterials.