PSI - Issue 80

Haolin Li et al. / Procedia Structural Integrity 80 (2026) 23–30 Author name / Structural Integrity Procedia 00 (2023) 000–000

26

4

2.3. PINN model for Homogenisation

PINN approximates the solution field X by a fully connected neural network. In this work, we propose a neural network architecture tailored to homogenisation problems: X ( x ) = ¯ F : L + NN sin x · 2 π L , cos x · 2 π L ; θ , (8) where θ denotes all trainable parameters in NN . This architecture projects the neural network approximation into a periodic space, consistent with the physical setting. As a result, the boundary condition in Eq. 5 is exactly satisfied once the macroscopic strain ¯ F is prescribed. The energy-based loss, defined from Eq. 6, is then employed to solve the PDE system. The optimisation problem is expressed as: min θ L ( x ) = Ω Ψ ( x ; θ ) dV , (9) The proposed neural network architecture inherently enforces the boundary conditions in Eq. 2. Therefore, no addi tional boundary-condition loss term is required. We employ three typical metamaterial structures for our case studies: octet-truss metamaterials, gyroid metamate rials, and spindoid metamaterials. The octet truss is a classic lattice of interconnected struts; it is stretch-dominated, o ff ering high sti ff ness-to-weight e ffi ciency and long serving as a benchmark for lightweight structural design. The gyroid is a triply periodic minimal surface with a smooth, labyrinth-like geometry; it provides isotropic sti ff ness and a high surface-to-volume ratio, making it a representative surface-based metamaterial. Spindoids are a newer class with complex, irregular patterns; they enable unusual anisotropy and non-linear deformation, exhibiting properties not found in traditional lattices or TPMS structures. We selected these three because they are typical yet distinct: the octet truss (strut-based), the gyroid (surface based), and the spindoid (irregular patterns). They also span di ff erent volume fractions and have strikingly di ff erent appearances, making them ideal representatives for comparative study. The geometries of the three metamaterials are shown in Fig. 2. 3. Case study

Fig. 2. Metamaterial structures: (a) octet-truss metamaterial; (b) gyroid metamaterial; (c) spindoid metamaterial.

For the implementation, we use the collocation method to evaluate the integral in the loss function, Eq. 9. The collocation points are uniformly distributed in the 3D unit cell with a resolution of 256 × 256 × 256. Because the three structures have di ff erent volume fractions, the numbers of collocation points are 492,616 for the octet truss, 2,261,719 for the gyroid, and 5,163,526 for the spindoid, respectively. To validate the results, we use finite element analysis (FEA) as the reference; the finite element models of the three structures are also shown in Fig. 3. Note that