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Neural network solvers for parametrized elasticity problems that conserve linear and angular momentum

Wietse M. Boon, Nicola R. Franco, Alessio Fumagalli

TL;DR

This work tackles the challenge of creating data-driven surrogates for parametrized elasticity that preserve fundamental physical laws. It employs a mixed finite element formulation with stress, displacement, and rotation, and enforces linear and angular momentum via a constraint operator, decomposing the stress into a force-balancing part computed by a spanning-tree based solver and a learnable homogeneous correction. Two learnable solvers, Split and Corrected, integrate neural surrogates with the constraint structure to guarantee exact momentum conservation while delivering accurate stress fields and fast online evaluations. Across linear and nonlinear 2D/3D test cases, the Corrected approach achieves state-of-the-art accuracy and exact conservation, while the Split method preserves momentum but can introduce numerical artifacts. The results demonstrate that physics-consistent ROMs can provide orders-of-magnitude speedups with rigorous adherence to conservation laws, offering practical benefits for structural analysis and geomechanics.

Abstract

We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce conservation of linear and angular momentum. Due to the saddle-point structure, the resulting system is computationally demanding to solve directly, and we therefore propose an efficient solution strategy based on a decomposition of the stress variable. First, a triangular system is solved to obtain a stress field that balances the body and boundary forces. Second, a trained neural network is employed to provide a correction without affecting the conservation equations. The displacement and rotation can be obtained by post-processing, if necessary. The potential of the approach is highlighted by three numerical test cases, including a non-linear model.

Neural network solvers for parametrized elasticity problems that conserve linear and angular momentum

TL;DR

This work tackles the challenge of creating data-driven surrogates for parametrized elasticity that preserve fundamental physical laws. It employs a mixed finite element formulation with stress, displacement, and rotation, and enforces linear and angular momentum via a constraint operator, decomposing the stress into a force-balancing part computed by a spanning-tree based solver and a learnable homogeneous correction. Two learnable solvers, Split and Corrected, integrate neural surrogates with the constraint structure to guarantee exact momentum conservation while delivering accurate stress fields and fast online evaluations. Across linear and nonlinear 2D/3D test cases, the Corrected approach achieves state-of-the-art accuracy and exact conservation, while the Split method preserves momentum but can introduce numerical artifacts. The results demonstrate that physics-consistent ROMs can provide orders-of-magnitude speedups with rigorous adherence to conservation laws, offering practical benefits for structural analysis and geomechanics.

Abstract

We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce conservation of linear and angular momentum. Due to the saddle-point structure, the resulting system is computationally demanding to solve directly, and we therefore propose an efficient solution strategy based on a decomposition of the stress variable. First, a triangular system is solved to obtain a stress field that balances the body and boundary forces. Second, a trained neural network is employed to provide a correction without affecting the conservation equations. The displacement and rotation can be obtained by post-processing, if necessary. The potential of the approach is highlighted by three numerical test cases, including a non-linear model.

Paper Structure

This paper contains 17 sections, 2 theorems, 62 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Model problems
  3. A mixed finite element discretization
  4. A spanning tree solver for the stress
  5. Preliminaries on neural network based non-linear regression
  6. Black-box models: definition and training
  7. The POD-NN strategy
  8. Learnable solvers that preserve linear and angular momentum
  9. The Split approach
  10. The Corrected approach
  11. Numerical experiments
  12. Linearized elasticity in 2D
  13. Linearized elasticity in 3D
  14. Non-linear elasticity in 2D
  15. Conclusions
  16. ...and 2 more sections

Key Result

Lemma 3.1

The discrete spaces given by eq: discrete spaces satisfy the following condition: with $C$ independent of the mesh size $h$.

Figures (7)

  • Figure 1: Example of spanning trees for a two-dimensional grid, on the left with single root (green dot) and on the right with multiple roots. Each root is associated to a facet on the boundary on which a displacement boundary condition is imposed.
  • Figure 2: Errors distribution in the stress variable for the case study of \ref{['subsec:ex1']}. $\Sigma_h$-relative errors (left) are obtained by the summands in \ref{['eq: l2mre']}. Similarly, measurements for the constraint violation (right) refer to \ref{['eq: constraint violation']}.
  • Figure 3: Comparison between FOM and conservative solvers for the footing problem of Section \ref{['subsec:ex1']}. Parameter values have been selected at random from the test set: $g_y=1.167e-03$, $f_y=7.800e-03$, $\mu=1.055$, $\lambda=1.593$. Top: quiver plots of the displacement field $u$. Bottom: original domain (grid) and exaggerated, deformed domain according to the computed displacement (grey; exaggeration factor: 150).
  • Figure 4: $\Sigma_h$-relative errors in the stress and constraint violation for the second test case, \ref{['subsec:ex2']}. Panels read as in \ref{['fig:boxplot-case-1']}.
  • Figure 5: Comparison between FOM and conservative solvers for the cantilever problem of Section \ref{['subsec:ex2']}. Parameter values have been selected at random from the test set: $\mu= 1.826$, $\lambda= 0.590$, $f_z= 0.0182$. Top: quiver plots of the displacement field $u$. Bottom: original domain (grid) and corresponding deformed domain (grey). The quiver plots are directly generated in PyVista sullivan2019pyvista using the displacement as cell data. The generation of warp plots, on the other hand, relies on an interpolation procedure (intrinsic to PyVista) which maps cell data to point data. This may generate visual artifacts when the mesh is coarse, as apparent at the right end of the domain.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Example 2.1: Hooke's law
  • Example 2.2: Hencky-von Mises
  • Remark 2.3: Hyperelasticity
  • Example 2.4: Iterative solver for the Hencky-von Mises model
  • Lemma 3.1
  • proof
  • Remark 3.2: Multiple roots
  • Remark 3.3
  • Definition 4.1: Layer
  • Definition 4.2: Feed forward neural network
  • ...and 6 more