A mesh-based neural energy method for the simulation of heterogeneous composites

The gist: This work introduces a mesh-based neural energy method (M-NEM) to simulate heterogeneous composites by extending the neural energy method (NEM) through mesh-based discretization, which addresses limitations of conventional collocation-based methods like geometric overfitting and automatic differentiation costs.

Methodology Overview

The M-NEM extends the NEM by introducing a mesh-based discretization of the displacement field via shape functions, which imposes a kinematic constraint to suppress oscillations and allows for algebraic computation of derivatives instead of automatic differentiation. Furthermore, it employs high-order Gaussian quadrature for accurate energy integration to eliminate aliasing errors. The domain is discretized into isoparametric elements where the neural network maps spatial coordinates of global nodes to their corresponding displacement vectors and uses shape functions Na(ξ) defined in the parent coordinates ξ for interpolation and then compute spatial derivatives algebraically as ∂u e/∂X=Xnen a=1 uˆe a ⊗∇XNa(ξ), where ∇XNa = J−T∇ξNa, and J is the geometric Jacobian of the mapping.

Addressing Conventional Limitations

The M-NEM specifically targets several challenges inherent in the conventional collocation-based neural energy method (C-NEM). By using shape function interpolation, the M-NEM prevents sub-sampling oscillations that arise in the C-NEM. The method also replaces automatic differentiation with algebraic shape function derivatives for strain computation, which avoids the cost of automatic differentiation and stabilizes training on heterogeneous materials. Finally, integration by Gaussian quadrature eliminates aliasing errors that occur when strain energy density varies abruptly across material interfaces.

Comparative Performance and Architecture Study

The M-NEM was validated through comparative case studies on (i) a rectangular coupon with a hard inclusion, (ii) a unit cell of the microstructure of a calcified hydrogel, and (iii) a volume element of a carbon fiber reinforced polymer (CFRP) microstructure. The quantitative performance comparison showed that the M-NEM yields stress errors around two orders of magnitude lower than the C-NEM while being faster across all architectures. For instance, in the coupon with stiff inclusion benchmark, the RBFNN demonstrated superior performance across all metrics.

Neural Architecture Analysis

A comparative study of neural architectures revealed that radial basis function neural networks (RBFNNs) yield optimal performance within the M-NEM, resolving sharp gradients at material interfaces with higher accuracy than multi-layer perceptrons (MLPs) with random Fourier feature (RFF) mapping and faster convergence than Kolmogorov-Arnold networks (KANs) The RBFNN architecture benefits most significantly from the transition to the M-NEM framework, exhibiting the largest relative reduction in runtime.

Conclusion and Robustness

The M-NEM demonstrated superior robustness regarding hyperparameter selection and domain discretization. Unlike its meshfree counterpart, the framework converged reliably with coarse discretizations and across a wide range of hyperparameters where the standard C-NEM failed. This inherent stability reduced the need for extensive hyperparameter tuning, making the method more accessible for practical applications. The M-NEM greatly outperformed the standard C-NEM, increasing solution accuracy by up to three orders of magnitude while decreasing computational effort by up to two orders of magnitude. The first case study showed that the M-NEM correctly captures the smooth stress distribution observed in the reference solution. The final conclusion is that this contribution proposed the M-NEM, addressing fundamental limitations of the C-NEM by introducing a mesh-based discretization. The method was tested on heterogeneous microstructures, including a coupon with a stiff inclusion, a bioinspired calcified hydrogel unit cell, and a CFRP microstructure.

How it works

The M-NEM suppresses oscillations through shape function interpolation. It also enables the algebraic computation of derivatives, which avoids automatic differentiation and stabilizes training on heterogeneous materials. The numerical integration is performed using Gaussian quadrature to evaluate the total potential energy over the element volume.

Constitutive Models Used

The simulations employed different constitutive formulations to characterize material behavior. Two distinct classes of material behavior were used: linear elasticity for small-deformation problems and hyperelasticity for applications involving large deformations and nonlinear material response. For hyperelasticity, the Neo-Hookean model was used with a compressible formulation including a logarithmic volumetric penalty term <ref:2610.108624, Neo-Hookean model.

1. No more physically inadmissible oscillations: The mesh-based neural energy method (M-NEM) imposes a kinematic constraint that suppresses oscillations by interpolating nodal displacements via shape functions, directly addressing the C-NEM's failure where a sufficiently expressive network may produce physically inadmissible displacement oscillations between collocation points.

2. Computational efficiency gain: The M-NEM replaces automatic differentiation with algebraic shape function derivatives for strain computation, which is noted as an improvement because it decouples the computation of spatial derivatives from the neural network’s automatic differentiation graph, which is now reserved solely for optimizing the parameters θ.

3. Improved integration accuracy: The method employs high-order Gaussian quadrature for accurate energy integration and replaces lower-order schemes that introduce errors, specifically addressing the challenge where stochastic or low-order integration schemes introduce significant aliasing errors in heterogeneous materials.

4. Enhanced gradient stability: The M-NEM allows for better optimization dynamics by using algebraic shape function derivatives, which is noted to be trivial for the linear output layer compared to automatic differentiation, leading to a reduction in runtime and alleviating memory bottlenecks.

5. Architecture selection optimization: The system can select the optimal neural architecture based on performance metrics; specifically, it shows that for complex problems like CFRP, the RBFNN produces a smooth stress field that correctly captures the concentrations, demonstrating superior accuracy over MLPs and KANs in high-stiffness regimes.

6. Robust convergence across discretizations: The M-NEM maintains stability across a wider range of settings and converge with coarser discretizations than those required by the C-NEM, preventing divergence where the C-NEM fails because the potential energy is evaluated only at the widely spaced collocation points.