PSI - Issue 80

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

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use in high-order problems; c) they retain all the strengths of neural networks used in learning tasks, giving PINNs strong potential for large-scale simulations and inverse problems. Thanks to these advantages, the use of PINNs to solve solid mechanics problems has become increasingly popular in both academia and engineering Haghighat et al. [2021], Bai et al. [2023], Hu et al. [2024], Wang et al. [2024]. A sys tematic investigation in Haghighat et al. [2021] has demonstrated the feasibility of applying PINNs to such problems using the original PINN formulation. However, their results also highlight several limitations, including low e ffi ciency and reduced accuracy compared with traditional numerical methods. To address these challenges, researchers have proposed di ff erent strategies. One approach is to reformulate the loss function from the strong form PDE to a weak form, often through energy-based loss minimisation. Although the energy functional is mathematically equiv alent to the strong form PDE, this weak formulation is easier to implement and shows faster convergence. Notable examples include the Deep Energy Approach Samaniego et al. [2020] and the Deep Ritz Method Yu et al. [2018]. An other approach exploits the specific features of solid mechanics PDEs, which often involve complex geometries. Here, geometric information is integrated into the PINN architecture or solution process, leading to geometry-aware deep learning methods. Examples include XPINN Jagtap and Karniadakis [2020], PINNs with exact boundary condition enforcement Wang et al. [2023], and Finite-PINN Li et al. [2024]. Among typical solid mechanics problems, the so-called cell problem, or homogenisation problem, plays a central role Charalambakis [2010], Li et al. [2023a]. It provides an e ff ective way to evaluate material properties from complex microstructures and serves as a bridge between micro- and macro-scale modelling. With the development of advanced materials such as composites and metamaterials, homogenisation has become increasingly important in meeting both academic and engineering demands. FEM remains the most widely used solver for homogenisation, but it also su ff ers from several drawbacks, including the di ffi culty of meshing highly complex geometries and the high computational cost of 3D analyses Li et al. [2023b, 2022, 2025]. In this context, we employ the emerging PINN approach to overcome the challenges faced by FEM in homogeni sation problems. We propose a novel neural network architecture tailored to the requirements of homogenisation. Our study focuses on metamaterials, which typically exhibit complex microstructures, and addresses homogenisation under large deformations, a critical factor in predicting their mechanical properties. The proposed method and case studies are presented in the following sections. The governing cell (homogenisation) problem for large deformations is defined as:    ∇ · P ( x ) = 0 , ∀ x ∈ Ω P ( x ) = C : F ( x ) , ∀ x ∈ Ω x ∼ x + L where P is the first Piola–Kirchho ff stress tensor, C is the elastic constitutive tensor, and F is the deformation gradient: F ( x ) = ∇ ( x + u ( x )) (2) where u is the displacement field. In the cell problem, x is periodic over the domain with period L , as denoted by Eq. (1). A schematic of the cell problem is shown in Fig. 1. The e ff ective domain / geometry of the metamaterial structure is denoted by Ω . This work employs a single-phase metamaterial as the study object, so C is constant in Eq. (1). To solve the PDE system in Eq. (1), the following boundary conditions are required: (1) 2. Methodology 2.1. Homogenisation of metamaterial structures

V  Ω

   1

F ( x ) dV = ¯ F , ∀ x ∈ Ω n · P ( x ) = 0 , ∀ x ∈ ∂ Ω

(3)