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The mixed deep energy method for resolving concentration features in finite strain hyperelasticity
Jan N. Fuhg, Nikolaos Bouklas
TL;DR
PINN and DEM formulations have difficulty resolving fine stress and displacement concentrations in finite-strain hyperelasticity. The paper proposes mDEM, which adds stress outputs and Delaunay integration, and reports concentration fields comparable to FEM in challenging experiments.
Problem
PINN and DEM formulations struggle to resolve fine stress and displacement features such as concentration fields in solid-mechanics applications.
Method
mDEM extends DEM by adding first Piola-Kirchhoff stress components as network outputs and uses Delaunay-based integration for non-grid-like training points.
Results
mDEM resolves stress and displacement concentrations near clamps, localized tractions, and holes at levels comparable to FEM, while DEM and PINN miss some fine features.
Takeaways & Limitations
The mixed formulation improves concentration-feature resolution and traction-boundary treatment for the investigated finite-strain hyperelasticity problems.
Takeaways & Limitations
The constitutive framework assumes a hyperelastic strain-energy function, with the paper using a compressible Neo-Hookean-type model.
Abstract
from arXiv · showhide
The introduction of Physics-informed Neural Networks (PINNs) has led to an increased interest in deep neural networks as universal approximators of PDEs in the solid mechanics community. Recently, the Deep Energy Method (DEM) has been proposed. DEM is based on energy minimization principles, contrary to PINN which is based on the residual of the PDEs. A significant advantage of DEM, is that it requires the approximation of lower order derivatives compared to formulations that are based on strong form residuals. However both DEM and classical PINN formulations struggle to resolve fine features of the stress and displacement fields, for example concentration features in solid mechanics applications. We propose an extension to the Deep Energy Method (DEM) to resolve these features for finite strain hyperelasticity. The developed framework termed mixed Deep Energy Method (mDEM) introduces stress measures as an additional output of the NN to the recently introduced pure displacement formulation. Using this approach, Neumann boundary conditions are approximated more accurately and the accuracy around spatial features which are typically responsible for high concentrations is increased. In order to make the proposed approach more versatile, we introduce a numerical integration scheme based on Delaunay integration, which enables the mDEM framework to be used for random training point position sets commonly needed for computational domains with stress concentrations. We highlight the advantages of the proposed approach while showing the shortcomings of classical PINN and DEM formulations. The method is offering comparable results to Finite-Element Method (FEM) on the forward calculation of challenging computational experiments involving domains with fine geometric features and concentrated loads.
1 Introduction
Neural networks have become a prominent meshfree approach for solving solid-mechanics boundary-value problems, including PINNs based on PDE residuals and DEM based on energy minimization. The section situates these methods within broader data-driven mechanics research and outlines the paper’s mDEM development and evaluation.
- Neural-network methods: Neural networks are used as universal approximators to solve solid-mechanics boundary-value problems from scratch.Recent advances in GPU hardware and neural-network technologies have contributed to their growing adoption.
- PINNs: PINNs solve PDE problems through automatic differentiation and strong-form residuals over meshfree collocation points.They can incorporate constraints from experiments and support inverse parameter-estimation problems.
- Deep Energy Method: The Deep Energy Method instead defines the neural-network loss through potential-energy minimization rather than governing-equation residuals.The method uses a neural network as a global displacement shape function over the body.
- Paper scope: The paper introduces an mDEM formulation and Delaunay-based integration scheme, then compares it with PINN and DEM implementations in computational experiments.The planned experiments address the proposed method’s performance in finite-strain hyperelasticity.
2 Finite-strain hyperelasticity
Finite-strain hyperelasticity describes motion, deformation, equilibrium, boundary conditions, and constitutive response through the deformation gradient, stress, and strain energy. For conservative quasi-static loading, minimizing potential energy yields a deformation satisfying static equilibrium.
- Kinematics and boundaries: An elastic body occupies a bounded domain whose boundary is partitioned into displacement and traction regions.The motion maps referential positions X to current positions x through ϕ(X,t), with displacement field u.
- Kinematics: The deformation gradient is obtained from the gradient of the motion with respect to the initial position.It is written as F = Gradϕ(X).
- Equilibrium and boundary conditions: Quasi-static solid mechanics is governed by an equilibrium equation involving the first Piola-Kirchhoff stress tensor and body force.The boundary conditions prescribe displacement on one boundary portion and surface traction on the other.
- Constitutive model: Hyperelasticity postulates a strain-energy function from which the first Piola-Kirchhoff stress tensor is computed.The paper specifies a compressible Neo-Hookean-type strain energy using Lamé constants and deformation measures.
- Energy formulation: For conservative loads without inertia, minimizing the body’s potential-energy functional produces a deformation that fulfills static equilibrium.The energy functional is minimized with respect to the motion ϕ.
3 Deep neural networks, Physics-informed Neural Networks and the Deep Energy Method
Neural networks approximate displacement fields over computational domains, while PINNs enforce strong-form PDE residuals and DEM minimizes potential energy. DEM reduces derivative requirements but depends on effective integration over collocation points.
- Neural networks: Neural networks use trainable weights and biases to approximate input-output mappings through optimization of a loss function.
- Physics-informed Neural Networks: PINNs approximate displacement fields and require automatically differentiated stresses to satisfy equilibrium and boundary-condition residuals.
- Physics-informed Neural Networks: Collocation points provide neural-network inputs whose outputs represent the approximated fields over the computational domain.
- Deep Energy Method: DEM minimizes potential energy over collocation points instead of using the governing PDE residual as its loss function.
- Deep Energy Method: DEM requires only first-order automatic differentiation rather than the second-order derivatives required by PINNs, but it needs effective domain integration.
4 Mixed Deep Energy Method (MDEM)
The mixed Deep Energy Method adds stress outputs to DEM to address the averaging of stress and displacement concentrations. It also introduces Delaunay-based integration for training-point sets containing fine geometric features.
- Motivation: DEM and PINN formulations can average out stress and displacement concentrations in solid-mechanics and multiphysics applications.
- Mixed Deep Energy Method: mDEM adds stress components as neural-network outputs that must satisfy traction boundary conditions alongside displacement outputs.
- Mixed Deep Energy Method: mDEM connects displacement and stress outputs by enforcing constitutive behavior derived from the displacement output and hyperelastic material model.
- Delaunay integration: Delaunay integration addresses the difficulty of integrating over random sample points representing continua with fine geometric features and stress concentrations.
- Delaunay integration: A constraint Voronoi mesh can represent a plate containing a C-shaped hole for numerical integration.
5 Numerical examples
The numerical experiments compare mDEM with DEM, PINN, and FEM on problems containing stress or displacement concentrations. mDEM more reliably resolves localized features, while DEM and PINN often fail to recover concentrated stresses or fine displacement fields.
- Experimental setup: The experiments compare mDEM, DEM, and PINN against FEM, using FEM as the ground truth.The methods are evaluated on uniaxial loading, localized traction, and a beam with a circular hole.
- Localized traction boundary: For localized traction, only mDEM accurately resolves the displacement concentration around the applied load and produces proficient stress fields compared with FEM.PINN loss convergence does not ensure recovery of the fine FEM features.
- Beam with a circular hole: For the beam with a circular hole, DEM and mDEM closely agree with FEM for displacement, whereas PINN does not find the correct solution.The PINN loss quickly saturates, suggesting stagnation in local minima.
6 Conclusion and outlook
The paper concludes that mDEM improves concentration-feature resolution in finite-strain hyperelasticity and performs comparably to FEM across the investigated problems. It also introduces Delaunay integration for non-grid training points and identifies three-dimensional and time-dependent applications as future work.
- Conclusion: mDEM adds first Piola-Kirchhoff stress components as neural-network outputs to improve DEM’s resolution of concentration features.The formulation also introduces Delaunay-based integration for non-grid-like training point positions.
- Conclusion: Across three computational experiments, mDEM yields results comparable to FEM, whereas DEM and PINN do not reliably resolve local stress and displacement features.The experiments involve stress concentrations and compare mDEM, DEM, PINN, and FEM solutions.
- Outlook: Future work will apply the formulation to three-dimensional and time-dependent computations.