Preprint for "Physics-Informed Neural Networks for Multiscale Large Deformation Analysis of Metamaterials"

Available online at www.sciencedirect.com Structural Integrity Procedia 00 (2023) 000–000 www.elsevier.com/locate/procedia Fracture, Damage and Structural Health Monitoring Physics-Informed Neural Networks for Multiscale Large Deformation Analysis of Metamaterials Haolin Lia,∗, Zahra Sharif Khodaeia, Michal Kotoulb, M.H. Aliabadia aDepartment of Aeronautics, Imperial College London, London, SW7 2AZ, UK bInstitute of Solid Mechanics, Mechatronics and Biomechanics, Brno University of Technology, Faculty of Mechanical Engineering, Brno, Czech Republic Abstract Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving solid mechanics problems. In this work, we propose a novel PINN architecture designed for homogenisation problems of metamaterials under large deformation. The architecture incorporates periodic functions to ensure exactly imposed boundary conditions and employs an energy-based loss for efficient training. Three representative metamaterial structures—octet truss, gyroid, and spindoid—are selected as case studies. The results demonstrate that the proposed PINN achieves accuracy comparable to finite element analysis (FEA), while offering improved computational efficiency for high-volume-fraction structures. Beyond accuracy and speed, the meshfree nature and flexibility of PINNs provide clear advantages, highlighting their potential as a scalable tool for modelling complex materials. ©2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of Professor Ferri Aliabadi. Keywords: Physics informed neural network; Homogenisation; Metamaterials; Multiscale Analysis 1. Introduction Physics-informed neural networks (PINNs) have seen rapid development in recent years Raissi et al. [2019], Karniadakis et al. [2021]. Their application in solid mechanics has also begun to emerge. As a new meshfree method, PINNs differ not only from mesh-based approaches such as the finite element method (FEM), but also from other meshfree approaches like radial basis function (RBF) methods Li and Liu [2002] or the boundary element method Aliabadi [2002], owing to their global implementation and approximation nature. Compared with these methods, PINNs as PDE solvers for solid mechanics offer several advantages: a) they do not require prescribed meshes, which are often expensive and sensitive to generate in FEM; b) they provide inherently smooth and differentiable approximations of the solution field, unlike most traditional numerical methods that allow only limited derivative orders, restricting their ∗Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000. E-mail address: [email protected] 2210-7843 ©2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of Professor Ferri Aliabadi. PREPRINT 2Author name /Structural Integrity Procedia 00 (2023) 000–000 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 systematic 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 efficiency and reduced accuracy compared with traditional numerical methods. To address these challenges, researchers have proposed different 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 equivalent 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]. Another 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 effective way to evaluate material properties from complex microstructures and serves as a bridge between microand 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 suffers from several drawbacks, including the difficulty 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 homogenisation 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. 2. Methodology 2.1. Homogenisation of metamaterial structures 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 (1) where Pis the first Piola–Kirchhoffstress tensor, Cis the elastic constitutive tensor, and Fis the deformation gradient: F(x)=∇(x+u(x)) (2) where uis the displacement field. In the cell problem, xis periodic over the domain with period L, as denoted by Eq. (1). A schematic of the cell problem is shown in Fig. 1. The effective domain/geometry of the metamaterial structure is denoted by Ω. This work employs a single-phase metamaterial as the study object, so Cis constant in Eq. (1). To solve the PDE system in Eq. (1), the following boundary conditions are required:            1 VZΩ F(x)dV =¯ F,∀x∈Ω n·P(x)=0,∀x∈∂Ω (3) PREPRINT Author name /Structural Integrity Procedia 00 (2023) 000–000 3 Fig. 1. Periodic domain in homogenisation problems. where ¯ Fdenotes the applied average strain and ∂Ωrepresents the domain boundary, Vis the volume of the domain. The first relation in Eq. (4) prescribes the macro strain applied to the cell, and the second imposes homogeneous Neumann boundary conditions. The objective of the cell problem is to obtain the homogenised effective first Piola–Kirchhoffstress: ¯ P=1 VZΩ P(x)dV,∀x∈Ω(4) 2.2. Hyperelasticity model This work implements the large-deformation formulation via the Complete Lagrangian formulation, by introducing hyperelasticity. The functional associated with Eq. 1defining the potential energy is: H(X)=ZΩ Ψ(X(x)) dV (5) where Xdenotes the deformed configuration: X(x)=x+u(x)(6) and Ψrepresents the strain energy. Minimising Eq. (5) is equivalent to solving the strong-form PDE in Eq. (1). In this work, the strain energy Ψis based on the Neo-Hookean hyperelastic model: Ψ=C10 2¯ I1−3+1 D1 (J−1)2,(7) where •C10 is a material constant that governs the deviatoric (shear) response of the material, related to the shear modulus by µ=2C10. •D1is a material constant that governs the volumetric response (compressibility), related to the bulk modulus by K=2 D1. •¯ I1=J−2/3I1is the first deviatoric invariant of the right Cauchy–Green tensor, •I1=tr(C), with C=FTFthe right Cauchy–Green tensor, •J=det(F) is the Jacobian of the deformation gradient F. PREPRINT 4Author name /Structural Integrity Procedia 00 (2023) 000–000 2.3. PINN model for Homogenisation PINN approximates the solution field Xby 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·2π L!,cos x·2π L!;θ!,(8) where θdenotes 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. 5is exactly satisfied once the macroscopic strain ¯ Fis 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)=ZΩ Ψ(x;θ)dV,(9) The proposed neural network architecture inherently enforces the boundary conditions in Eq. 2. Therefore, no additional boundary-condition loss term is required. 3. Case study We employ three typical metamaterial structures for our case studies: octet-truss metamaterials, gyroid metamaterials, and spindoid metamaterials. The octet truss is a classic lattice of interconnected struts; it is stretch-dominated, offering high stiffness-to-weight efficiency 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 stiffness 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 (surfacebased), and the spindoid (irregular patterns). They also span different volume fractions and have strikingly different appearances, making them ideal representatives for comparative study. The geometries of the three metamaterials are shown in Fig. 2. 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 different 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 PREPRINT Author name /Structural Integrity Procedia 00 (2023) 000–000 5 the Neo-Hookean hyperelasticity model used in the PINN (Eq. 7) is consistent with the one used in Abaqus, which facilitates validation. Fig. 3. Finite-element models of metamaterial structures: (a) octet-truss metamaterial; (b) gyroid metamaterial; (c) spindoid metamaterial. 3.1. Single-axis strain deformation First, we apply a macroscopic tensile strain purely in the z(vertical) direction as an example of single-axis strain. The resulting homogenised stresses are given in Table.1. It is seen from the table that The PINN achieves accuracy comparable to the FEM results. The deformed configurations are illustrated in Fig. 4as a showcase. Pxx Pyy Pzz Pxy Pxz Pyz Octet Truss FEM 0.087 0.087 1.344 0.0137 0.223 0.223 PINN 0.089 0.089 1.344 0.0101 0.223 0.224 Gyroid FEM 0.327 0.327 4.712 0.134 0.720 0.720 PINN 0.328 0.328 4.724 0.137 0.721 0.723 Spindoid FEM 1.153 1.153 17.740 0.292 0.846 0.846 PINN 1.152 1.157 17.788 0.292 0.845 0.850 Table 1. Placeholder caption for a table with three rows and seven columns. 3.2. Multiple-axis strain deformation Second, we apply random macroscopic strains in all three axes, including both normal and shear components. The homogenised stresses are reported in Table.2. Again, the PINN yields results comparable with the FEM reference. The deformed configurations are shown in Fig. 5for illustration. Pxx Pyy Pzz Pxy Pxz Pyz Octet Truss FEM 1.170 1.210 0.722 0.124 0.183 0.162 PINN 1.175 1.217 0.743 0.125 0.184 0.163 Gyroid FEM 1.845 2.753 3.385 2.133 2.153 2.905 PINN 1.890 2.742 3.399 2.142 2.154 2.911 Spindoid FEM 4.728 20.134 12.134 7.747 7.602 6.112 PINN 4.738 20.277 12.135 7.749 7.609 6.116 Table 2. Placeholder caption for a table with three rows and seven columns. PREPRINT 6Author name /Structural Integrity Procedia 00 (2023) 000–000 Fig. 4. Deformed configurations under single-axis strain: (a) octet-truss metamaterial; (b) gyroid metamaterial; (c) spindoid metamaterial. The color represents the displacement distributions. Fig. 5. Deformed configurations under random multi-axial strain: (a) octet-truss metamaterial; (b) gyroid metamaterial; (c) spindoid metamaterial. The color represents the displacement distributions. 4. Discussion The results demonstrate that the proposed PINN method achieves accuracy comparable to that of FEM across both single-axis and multiple-axis strain cases. For all three metamaterial structures, the homogenised stresses predicted by PINN closely match those obtained from FEM, validating the correctness of the formulation and the implementation of the energy-based approach. In terms of efficiency, the computing performance of PINN and FEM shows different behaviour depending on the structural volume fraction (ϕ). For the low-volume-fraction octet truss (Vf=0.029), the computational times are comparable (PINN ≈8 min; FEM ≈10 min). For the gyroid (Vf=0.140), the gap widens (PINN ≈15 min; FEM ≈ 47 min). For the high-volume-fraction spindoid (Vf=0.307), the PINN method becomes significantly faster (PINN ≈23 min; FEM ≈296 min). This difference arises because FEM’s computational cost grows non-linearly with the number of mesh elements, whereas PINN exhibits only minor sensitivity to the increase in collocation points. The effectiveness of the proposed PINN model can be attributed to two key design choices. First, the adoption of an energy-based loss term enables faster convergence compared to the conventional strong-form PDE loss. Second, the PREPRINT Author name /Structural Integrity Procedia 00 (2023) 000–000 7 network architecture enforces boundary conditions exactly, so the optimisation focuses solely on minimising strain energy. This eliminates the balance problem among multiple competing loss terms that often complicates standard PINN training. Beyond accuracy and speed, the PINN method provides additional advantages. It naturally supports parallel computing, which is particularly useful in engineering applications. It also avoids the need for mesh generation, thereby simplifying preprocessing and improving scalability. Moreover, the framework can be readily extended to other problem classes, such as history-dependent material models, by leveraging transfer learning. In contrast, FEM computations must be restarted from scratch when problem settings are modified. Overall, these results suggest that the proposed PINN approach is not only a viable alternative to FEM but also offers unique advantages in flexibility and scalability, making it a promising tool for future studies of complex metamaterial systems. 5. Conclusion This work has presented a physics-informed neural network framework designed for homogenisation of metamaterials subject to large deformation. By embedding periodicity directly into the neural network architecture and adopting an energy-based loss, the method enforces boundary conditions exactly and avoids balancing multiple loss terms. Comparative studies with finite element analysis show that the proposed PINN achieves similar accuracy, while demonstrating faster performance in complex, high-volume-fraction structures such as spindoids. Additional benefits of the PINN approach include its meshfree formulation, ease of parallelisation, and adaptability to extended problem classes through transfer learning. Taken together, these results suggest that PINNs provide an effective and versatile alternative to conventional solvers, opening new possibilities for computational mechanics and metamaterial design. Data availability Regarding the computational procedures see Li, H.. (2025). Data for “Physics-Informed Neural Networks for Multiscale Large Deformation Analysis of Metamaterials” (1.0.0). 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