Data-driven methods for computational mechanics: A fair comparison between neural networks based and model-free approaches

TL;DR

This work conducts a fair, data-consistent comparison between data-driven, model-free DDCM and neural-network based constitutive surrogates for hyperelastic materials under finite strains. By using identical datasets and FE solvers across Cook membrane and punch benchmarks, it isolates the strengths of each approach: DDCM achieves high accuracy when data densely covers the relevant states, while NN surrogates generalize better under noise and limited data, particularly for out-of-sample states. The study introduces isotropy-enriched DDCM and locally convex data embeddings to improve robustness, and demonstrates that both methods are viable with complementary regimes of applicability. Practically, the findings guide data collection and method selection for computational mechanics problems, highlighting trade-offs in accuracy, robustness, and online cost.

Abstract

We present a comparison between two approaches to modelling hyperelastic material behaviour using data. The first approach is a novel approach based on Data-driven Computational Mechanics (DDCM) that completely bypasses the definition of a material model by using only data from simulations or real-life experiments to perform computations. The second is a neural network (NN) based approach, where a neural network is used as a constitutive model. It is trained on data to learn the underlying material behaviour and is implemented in the same way as conventional models. The DDCM approach has been extended to include strategies for recovering isotropic behaviour and local smoothing of data. These have proven to be critical in certain cases and increase accuracy in most cases. The NN approach contains certain elements to enforce principles such as material symmetry, thermodynamic consistency, and convexity. In order to provide a fair comparison between the approaches, they use the same data and solve the same numerical problems with a selection of problems highlighting the advantages and disadvantages of each approach. Both the DDCM and the NNs have shown acceptable performance. The DDCM performed better when applied to cases similar to those from which the data is gathered from, albeit at the expense of generality, whereas NN models were more advantageous when applied to wider range of applications.

Figures (30)

• Figure 1: Architecture of the neural network used for modelling compressible hyperelasticity.
• Figure 2: Geometry, boundary and loading conditions of the Cook membrane. The mesh with an overlay of the source and changed mesh is shown as well. Source mesh is solid blue, changed mesh is solid white.
• Figure 3: Relative $L^2$ norm of displacements for Cook membrane with Ciarlet law for DDCM procedures. The labels on top represent the size of the dataset used in a simulation.
• Figure 4: Relative $L^2$ norm of stress for Cook membrane with Ciarlet law for DDCM procedures.
• Figure 5: Relative $L^2$ norm of displacements for neural networks trained on energy, results for Cook membrane with Ciarlet law. The results shown for the average displacement errors are averaged for all training runs of the neural networks. Standard deviations of the averaged errors are shown in parentheses.