Physics-Informed Neural Networks for Shell Structures

TL;DR

The paper develops a Physics-Informed Neural Network (PINN) framework to predict the small-strain response of arbitrarily curved shells using the Naghdi shell model, formulating the problem on the shell midsurface which resides in a non-Euclidean domain. It compares strong-form and weak-form (energy-based) PINN losses, showing that the weak form yields accurate field predictions across three benchmarks while the strong form can fail to converge, particularly under partial boundary conditions. The approach demonstrates that PINNs can avoid locking effects that plague classical finite-element formulations, and remains robust across different shell geometries, boundary conditions, and curvature, albeit with training-time considerations in the thin-thickness limit. The work highlights the potential of PINNs as a mesh-free, calculus-enabled tool for validating locking-free behavior and as a platform for future inverse problems and topology optimization in shell mechanics.

Abstract

The numerical modeling of thin shell structures is a challenge, which has been met by a variety of finite element (FE) and other formulations -- many of which give rise to new challenges, from complex implementations to artificial locking. As a potential alternative, we use machine learning and present a Physics-Informed Neural Network (PINN) to predict the small-strain response of arbitrarily curved shells. To this end, the shell midsurface is described by a chart, from which the mechanical fields are derived in a curvilinear coordinate frame by adopting Naghdi's shell theory. Unlike in typical PINN applications, the corresponding strong or weak form must therefore be solved in a non-Euclidean domain. We investigate the performance of the proposed PINN in three distinct scenarios, including the well-known Scordelis-Lo roof setting widely used to test FE shell elements against locking. Results show that the PINN can accurately identify the solution field in all three benchmarks if the equations are presented in their weak form, while it may fail to do so when using the strong form. In the thin-thickness limit, where classical methods are susceptible to locking, training time notably increases as the differences in scaling of the membrane, shear, and bending energies lead to adverse numerical stiffness in the gradient flow dynamics. Nevertheless, the PINN can accurately match the ground truth and performs well in the Scordelis-Lo roof benchmark, highlighting its potential for a drastically simplified alternative to designing locking-free shell FE formulations.

Figures (16)

• Figure 1: Definition of the shell midsurface based on the chart $\bm{\phi}$, which maps from the reference domain $\omega$ to the physical domain $\Omega$. Besides the global frame with basis $\{\bm{e}_1,\boldsymbol{e}_2,\bf_3\}$, we construct from $\bm{\phi}$ a local covariant basis $\{\bm a_1,\boldsymbol{a}_2,\boldsymbol{a}_3\}$ at any point $P$, with $\boldsymbol{a}_1$ and $\boldsymbol{a}_2$ spanning the local tangent plane and $\boldsymbol{a}_3$ being normal to the midsurface.
• Figure 2: PINN architecture and corresponding loss. The PINN predicts the (scaled) global deformations $\hat{\boldsymbol{u}}^*$ and rotations $\bm{\theta}^*$ at a collocation point $\bm\xi_i$, which are multiplied by the trial function $\varphi$ to impose the Dirichlet BCs. Based on these five parameters and the given chart $\phi$, the shell equations can be assembled equivalently in their strong or weak forms, which define the loss function for the training of the network.
• Figure 3: Considered case studies. a) A hyperbolic paraboloid fully clamped on one edge and subject to gravity loading. b) The Scordelis-Lo Roof benchmark Belytschko1985, consisting of a partly clamped cylindrical shape subject to gravity loading ($\theta_0=40^{\circ}$). c) A fully clamped hemisphere subject to a vertical load at its center, modeled by a Gaussian kernel.
• Figure 4: Comparison of the predicted $\hat{u}_3$-displacement field as obtained from the FE and PINN frameworks (based on both the strong and weak forms with $N_{\text{c}}=2{,}048$) for the partly-clamped hyperbolic paraboloid subject to gravity loading, shown in the reference domain. Displacements are scaled by a factor of $0.005$ for consistency with Figure \ref{['fig:3d_sol_hyperb_parab']}.
• Figure 5: Average relative $L_2$-error of the five solution fields, computed as $\frac{1}{5}\left(\sum_{i=1}^3\sqrt{ (u_{i,\text{FEM}}-u_{i,\text{PINN}})^2/u_{i,\text{FEM}}^2}+\sum_{j=1}^2\sqrt{ (\theta_{j,\text{FEM}}-\theta_{j,\text{PINN}})^2/\theta_{j,\text{FEM}}^2}\right)$, with respect to the FEM solution over 100 training epochs based on the strong and weak forms. We first consider the original problem, i.e., a partly clamped shell, and in addition investigate the performance of both the strong and weak forms for the simpler problem of a fully clamped shell. $^*$Trained using $N_{\text{c}}=16{,}384$ (other results for $N_{\text{c}}=2{,}048$).