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Physics-Informed Neural Networks for Shell Structures

Jan-Hendrik Bastek, Dennis M. Kochmann

arXiv:2207.14291v1cs.CEcs.LG

TL;DR

Thin-shell modeling is challenging because shell equations are solved on curved midsurfaces and conventional finite-element approaches face implementation complexity and locking concerns. This paper applies a PINN with Naghdi shell theory to arbitrarily curved shells; weak-form solutions match FEM results across three benchmarks, while strong-form training can converge poorly.

  • Problem

    Shell equations on curved midsurfaces pose numerical challenges, while shell finite-element implementations are complex and require careful choices to avoid locking.

    Source: §1 · pp. 1–2

  • Method

    The paper formulates a PINN for small-strain response of arbitrarily curved shells using linear Naghdi shell theory and evaluates it against FEM solutions.

    Source: §1 · p. 2

  • Results

    Across three classical benchmarks, PINN predictions compare well with FEM, and the weak form consistently outperforms the strong form, which can converge poorly in some settings.

    Source: §5 · p. 16

  • Takeaways & Limitations

    The results indicate that PINNs can represent shell solution fields even in thin-thickness cases where classical methods are prone to locking, without numerical adaptations.

    Source: §5 · p. 16

    Convergence worsens as shell thickness decreases, and membrane–bending energy scaling leaves an ill-conditioned loss in the thin-thickness limit.

    Source: §5 · p. 16, §4.4 · p. 16

Abstract

from arXiv · show

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.

1. Introduction

Thin-shell equations must be solved on a curved midsurface, making their numerical treatment distinct from problems in Euclidean space. The paper presents a PINN for arbitrarily curved shells and evaluates it against finite-element solutions.

1. Introduction

  • Shell equations are posed on a two-dimensional manifold embedded in 3D, and generally require numerical solutions in engineering practice.Analytical solutions exist only for a small class of special cases.Source: §1 · p. 1
  • Shell finite-element implementations are technically complex, and avoiding locking still requires experienced shape-function and discretization choices.Locking arises when the selected finite-element space cannot represent certain deformation modes.Source: §1 · p. 2
  • PINNs incorporate governing equations into the loss function, allowing physical-system responses to be obtained without response data.This differs from data-driven learning based on correlations in experimental or simulation datasets.Source: §1 · p. 2
  • Prior PINN work addressed plates and, with reduced accuracy especially without external data, composite plates and shallow cylindrical shells coupled to Extreme Learning Machines.Source: §1 · p. 2
  • The proposed framework uses linear Naghdi shell theory for small-strain response, including shear deformation relevant to comparably thick shells.It is evaluated against FEM solutions for arbitrarily curved continuous shells.Source: §1 · p. 2
  • The study compares the framework with reference FEM solutions across three shell structures with distinct curvatures and boundary conditions.Source: §1 · p. 2

2. Naghdi shell model

The Naghdi shell model describes a curved shell through its midsurface chart and local geometric measures, then expresses deformation and equilibrium using displacement and rotation fields. Its variational formulation supplies the weak form used for equilibrium.

2.1. Geometrical preliminaries

  • The chart maps reference-domain coordinates to the physical midsurface, from which local covariant basis vectors and geometric measures are constructed.The local tangent vectors span the tangent plane, while the third basis vector is normal to the midsurface.Source: §2.1 · p. 3
  • Surface derivatives account for changes in the local basis, while the Christoffel symbol and second fundamental form describe in-plane and out-of-plane curvature.Source: §2.1 · pp. 3–4
  • The formulation relates covariant and convected bases and uses the metric determinant to express physical surface area in reference coordinates.Source: §2.1 · p. 4

2.2. Shell model

  • Assuming constant thickness and straight, inextensible normal fibers, the model represents displacement as translation plus rotation-dependent displacement through the thickness.The Naghdi strain contributions comprise membrane, bending, and shear strains.Source: §2.2 · pp. 4–5
  • For homogeneous isotropic material, the elasticity tensors use plane-stress conditions through the thickness, and the shear correction factor is κ = 5/6.Source: §2.2 · p. 5
  • Equilibrium follows from minimizing total potential energy subject to essential boundary conditions, forming the weak formulation; the strong form is an equivalent alternative.The formulation also specifies natural boundary conditions on the free boundary and defines shell stress resultants and moments.Source: §2.2 · p. 5
  • The five-parameter model uses three displacement components and two tangential rotation components, neglecting rotations of material fibers about their own axis.Displacements are represented in the global Cartesian frame and transformed to the local frame.Source: §2.2 · p. 6

3. Physics-informed neural network framework

The framework trains a neural network to approximate solutions of shell equations from collocation points, using either their strong form or a variational weak form. For shell applications, it predicts displacements and rotations from curvilinear coordinates, with trial functions enforcing essential boundary conditions and the weak form simplifying the loss.

3.1. General PINN setting

  • In the general PINN setting, a feed-forward network approximates the PDE solution, and training minimizes sampled residuals computed using automatic differentiation.The residuals are evaluated at collocation points and optimized with first- or second-order methods.Source: §3.1 · p. 6

3.2. Application to shells

  • The shell PINN predicts global displacements and local rotations from curvilinear coordinates, using the chart to assemble the shell equations and the trial function to impose Dirichlet conditions.The equations can define the training loss in either strong or weak form.Source: §3.2 · p. 6, Figure 2 · p. 7
  • For the strong form, the loss combines equilibrium equations with natural- and essential-boundary residuals, evaluated using collocation points sampled in the domain and on its boundaries.The residual counts are denoted Nc, NN, and ND; Sobol sampling is used unless stated otherwise.Source: §3.2 · pp. 7–8
  • The weak-form loss minimizes the potential energy and avoids extra natural-boundary equations, reducing the highest derivative order and simplifying the loss to a single term.This also avoids potential ambiguity in weighting multiple loss terms.Source: §3.1 · p. 6, §3.2 · p. 8

3.3. Implementation details

  • The implementation tests three shell case studies and uses a three-hidden-layer, 50-neuron-per-layer GELU network optimized with PyTorch’s default L-BFGS solver.The cases include a hyperbolic paraboloid, the Scordelis-Lo roof, and a hemisphere.Source: Figure 3 · p. 9, §3.3 · p. 9

4. Results

The study compares PINN and converged FE solutions across three shell benchmarks with distinct shapes, boundary conditions, and loads. Weak-form PINNs generally match FE solutions, including for Scordelis-Lo, while training slows in the thin-thickness limit.

4.1. Partially-clamped hyperbolic paraboloid

  • For the partly clamped paraboloid, strong-form training failed to converge to FEM while the weak form matched it across membrane- and bending-dominated zones.The weak form embeds natural boundary conditions and reduces derivative order; training took approximately 70s/epoch for the strong form versus 3.8s/epoch for the weak form (4.5s/epoch for Nc = 16,384).Source: §4.1 · pp. 10–11
  • For the fully clamped paraboloid, the strong-form PINN approximated FEM but remained notably less accurate than the weak-form PINN.The strong form was viable in this simpler setting, while the weak form still gave a much better approximation for the same training epochs.Source: §4.1 · p. 11

4.2. Scordelis-Lo roof

  • ˆu3,FEM = −0.302 and ˆu3,PINN = −0.297; the weak-form PINN approximated Scordelis-Lo FEM solution fields to high accuracy.Both values approach the benchmark reference value of ˆu3 = −0.3024 after consistent rescaling.Source: §4.2 · p. 14
  • As thickness decreases, membrane and shear energies scale with t but bending energy with t^3, stiffening loss gradients and extending training to as many as 10,000 epochs.The authors leave further convergence improvements in this setting outside the scope of this work.Source: §4.2 · pp. 14–15

4.3. Hemisphere under concentrated load

  • For the fully clamped hemisphere under a concentrated load, the weak-form PINN compared well with FEM, with Table B.1 reporting high agreement.Source: §4.3 · p. 15

4.4. Discussion

  • Across three benchmarks, weak-form PINNs agreed well with FEM and showed no fundamental failure to approximate solution fields, including in evaluated thin-shell scenarios.The authors report that PINNs numerically confirmed the absence of locking in these scenarios, but may not match traditional solvers' efficiency at small thicknesses.Source: §4.4 · p. 15

5. Conclusion

The paper applies a PINN with Naghdi shell theory to small-strain response on arbitrary shell manifolds and compares it with FEM. Across three benchmarks, the weak form performs consistently better than the strong form and can represent solutions in thin-thickness settings, though training convergence and computational efficiency remain challenges.

5. Conclusion

  • PINNs solve the Naghdi shell equations on arbitrary manifolds and compare well with FEM across three classical benchmarks.The formulation uses three translational and two rotational degrees of freedom in a non-Euclidean domain.Source: §5 · p. 16
  • The weak form consistently outperforms the strong form, whose competing loss terms lead to poor convergence in some settings.The paper attributes worse convergence with decreasing thickness to stronger solution-field changes and increasingly imbalanced loss terms.Source: §5 · p. 16
  • PINNs showed no fundamental failure to approximate physical solution fields in the thin-thickness limit and passed the Scordelis-Lo roof benchmark.The authors present this as numerical confirmation of no locking in the evaluated scenarios, while noting that computational efficiency may lag traditional solvers.Source: §5 · p. 16
  • Further work should study thin-limit convergence and reduce computational effort before PINNs can compete with established FEM techniques.The authors also identify inverse problems and topology optimization as promising extensions, with autodifferentiation providing analytical sensitivities.Source: §5 · pp. 16–17

Appendix A. Derivation of the strain measures

Appendix A gives derivation steps for the Green-Lagrange strain tensor components, accounting for changing midsurface base vectors.

Appendix A. Derivation of the strain measures

  • The derivation begins from equation (14) and computes the required derivatives, omitting dependence on ξ_i for conciseness.The supplied passages identify the starting relation and subsequent derivative step, but do not show their full expressions.Source: Appendix A · p. 20
  • Because derivatives follow curves on the midsurface with changing base vectors, their evaluation includes the variation of those vectors.This is the geometric consideration stated for the derivative step.Source: Appendix A · p. 20
  • Replacing the 3D basis vectors in (A.1) with those from (9) yields the Green-Lagrange strain tensor components.Source: Appendix A · p. 20

Appendix B. Solution details for the presented benchmarks

Appendix B collects solution fields omitted from the main article and reports L2-errors across the three benchmarks.

Appendix B. Solution details for the presented benchmarks

  • Appendix B presents the remaining solution fields for all three benchmarks alongside L2-errors for each case and field.Source: Appendix B · p. 20

Appendix B.1. Solution fields for the partly clamped hyperbolic paraboloid

Appendix B.1 compares weak-form PINN and FE solution fields for the gravity-loaded, partly clamped hyperbolic paraboloid.

Appendix B.1. Solution fields for the partly clamped hyperbolic paraboloid

  • Figure B.13 compares four weak-form PINN and FE fields for the partly clamped, gravity-loaded hyperbolic paraboloid using Nc = 16,384.The reference-domain displacements are scaled by 0.005 for consistency with Figure 7.Source: Figure B.13 · p. 21

Appendix B.2. Solution fields for the Scordelis-Lo roof

Figure B.14 compares FE and weak-form PINN predictions for the Scordelis-Lo roof solution fields in the reference domain.

Appendix B.2. Solution fields for the Scordelis-Lo roof

  • The comparison covers the Scordelis-Lo roof benchmark in the reference domain.Source: Figure B.14 · p. 22
  • Figure B.14 compares FE and weak-form PINN predictions for ˆu1, ˆu2, θ1, and θ2.Source: Figure B.14 · p. 22
  • Displacements are scaled by 0.001 for consistency with Figure 8.Source: Figure B.14 · p. 22

Appendix B.3. Solution fields for the fully clamped hemisphere

Figure B.15 compares FE and weak-form PINN solution fields for the fully clamped hemisphere under a concentrated load. The accompanying text identifies boundary artifacts in the FEM θ1,2 fields due to mesh irregularities.

Appendix B.3. Solution fields for the fully clamped hemisphere

  • Figure B.15 compares FE and weak-form PINN predictions for ˆu2, ˆu3, θ1, and θ2 under a concentrated load.Source: Figure B.15 · p. 23
  • Figure B.15 limits the displayed radius to 0.96 because FE-mesh artifacts otherwise dominate the error plots.Source: Figure B.15 · p. 23
  • FEM θ1,2 solution fields show boundary artifacts attributed to mesh irregularities, including in θ1 as shown in Figure B.16.The effect persists with increased mesh resolution; a fully regular mesh may mitigate it only for regular geometries.Source: Appendix B.3 · p. 24

Appendix B.4. Errors of the three reported benchmarks

Table B.1 reports relative L2 errors of weak-form PINN solution fields against FEM solutions for three case studies.

Appendix B.4. Errors of the three reported benchmarks

  • Table B.1 gives relative L2 errors of weak-form PINN solution fields compared with FEM solutions across three case studies.Source: Table B.1 · p. 24
  • Parenthesized errors in Table B.1(a) correspond to increased collocation-point sampling with Nc = 16,384.Source: Table B.1 · p. 24
  • The table caption identifies the comparison scope but does not provide error values in the supplied passage.Source: Table B.1 · p. 24

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