Design of a specimen to train path-dependent deep learning material models from a single uniaxial test: eliciting strain diversity via automatically differentiable elastoplastic topology optimization

Technical Summary: Design of a Specimen to Train Path-Dependent Deep Learning Material Models from a Single Uniaxial Test

Problem Statement
Artificial neural networks (ANNs) have demonstrated the ability to accurately learn nonlinear, path-dependent material behavior, often outperforming traditional phenomenological constitutive laws. However, training these data-driven models typically requires large, diverse datasets spanning a wide range of stress-strain trajectories. Currently, such datasets are predominantly generated via synthetic unit cell simulations rather than physical experiments. Standard experimental specimens, such as dogbone geometries, fail to generate sufficiently diverse stress-strain paths under simple uniaxial loading. Consequently, training complex neural network models (e.g., those with millions of parameters) from physical experiments would require an unreasonably large number of tests or complex multi-axial loading setups. This paper addresses the gap between the data requirements of deep learning material models and the limited information content of standard mechanical tests.

Methodology
The authors propose a framework that integrates automatically differentiable elastoplastic topology optimization with data-driven material model updating. The workflow consists of three distinct stages:

1. Topology Optimization (Data Generator Design):
   The core innovation is the design of a specimen geometry that maximizes strain diversity under simple monotonic uniaxial loading. The authors formulate an optimization problem where the objective is to maximize the Shannon entropy of the strain distribution within the specimen.

   • Differentiability: To enable gradient-based optimization, the authors utilize the JAX framework for automatic differentiation (AD). This allows for the exact computation of gradients through the time-stepping integration of the elastoplastic constitutive model, avoiding the laborious manual derivation of adjoint formulations required by conventional methods.
   • Objective Function: Strain diversity is quantified by discretizing the strain space into cells and constructing a differentiable histogram using Gaussian kernels (soft assignment) to assign strain samples to cells. The Shannon entropy of this histogram serves as the objective function to be maximized.
   • Constraints: The optimization employs a density-based approach (SIMP scheme) with a cone filter and threshold projection to ensure manufacturable, binary (black-and-white) designs.

2. Dataset Generation:
   Once the optimized geometry is determined, it is subjected to cyclic uniaxial loading (tension-compression) via Finite Element Analysis (FEA). Local stress-strain paths from all solid elements within the specimen are collected to form a comprehensive dataset. This dataset is intended to span a wide range of deformation states, including plastic deformation.

3. Material Model Updating (Training):
   The generated dataset is used to train a Recurrent Neural Network (RNN) material model, specifically utilizing Gated Recurrent Units (GRUs). The authors employ the Automatically Differentiable Model Updating (ADiMU) framework (implemented in HookeAI) to train the network. To rigorously test the data efficiency of the proposed specimen, an overparameterized model (two GRU layers with 500 hidden units each, totaling >2 million parameters) is trained. The model is evaluated on independent, randomly generated polynomial stress-strain paths.

Key Contributions

• Novel Specimen Design Strategy: This work presents the first integration of elastoplastic topology optimization with the training of deep learning surrogate models. It demonstrates that a single, complex specimen geometry can be designed to generate sufficient strain diversity for training path-dependent models.
• Automatically Differentiable Optimization: The paper leverages AD to solve the elastoplastic topology optimization problem, enabling the use of complex, non-standard objective functions (entropy-based strain diversity) that are difficult to formulate with traditional adjoint methods.
• Data Efficiency: The study shows that a single optimized specimen under simple loading can generate datasets capable of training large neural networks, potentially reducing the experimental burden associated with data-driven material modeling.

Results

• Superior Performance vs. Standard Specimens: GRU models trained on data from the topology-optimized specimen achieved significantly lower Normalized Root Mean Squared Error (NRMSE) compared to models trained on standard dogbone or notched specimens. For 2D stress components, the optimized specimen yielded NRMSE values around 9–11%, whereas dogbone specimens resulted in errors exceeding 55–166%.
• Strain Diversity and Redundancy: Analysis of the datasets revealed that the optimized specimen produces a much broader coverage of the strain space. A redundancy study (pruning analysis) showed that the dogbone dataset contained high redundancy (requiring nearly the full dataset to maintain accuracy), whereas the optimized dataset had lower redundancy, with 65% of the data sufficient to achieve near-optimal accuracy.
• Generalization: The framework demonstrated robustness when the underlying material model was changed. An optimized specimen designed using a von Mises plasticity model was used to generate data for a Drucker–Prager material model. The resulting GRU model successfully captured the Drucker–Prager behavior, indicating the specimen's ability to generate informative data even when the constitutive assumptions differ from the design phase.
• 3D Extension: The methodology was extended to 3D, showing that optimizing for in-plane strain diversity also yields informative out-of-plane stress responses, although shear stress prediction remained less accurate due to limited exposure in the training data.

Significance and Claims
The paper claims to establish the feasibility of "Material Testing 2.0," a paradigm where a single, carefully designed experiment replaces standardized tests for calibrating complex material models. The authors assert that this is the first work to integrate elastoplastic topology optimization with the training of recurrent neural network surrogates from a single optimized specimen.

The significance lies in the potential to substantially reduce the experimental burden associated with data-driven material modeling. By replacing the need for extensive multi-axial testing campaigns with a single, information-rich uniaxial test, the proposed method aims to enable the practical application of large, path-dependent neural network models to real-world materials. The authors note that while the current study is fully computational and relies on FEA-generated datasets, the optimized geometries are feasible for manufacturing. Future work plans to validate these designs experimentally using full-field measurement techniques like Digital Image Correlation (DIC) to capture global displacement and force data, which would then be used to train or calibrate models within the ADiMU framework. The work does not claim global optimality of the specimen geometry but demonstrates that maximizing strain-state diversity through topology optimization is a viable strategy for training high-capacity data-driven models.