Implementation of Hyperelastic Physics-Augmented Neural Networks in the Explicit Finite Element Codes Simcenter Radioss and OpenRadioss with Applications to Impact Events

Technical Summary: Implementation of Hyperelastic Physics-Augmented Neural Networks in Explicit Finite Element Codes

Problem Statement
Data-driven material modeling, particularly Physics-Augmented Neural Networks (PANNs), offers a promising avenue for capturing complex constitutive behaviors beyond classical models. However, a significant barrier to their adoption in industrial engineering simulations is the integration gap between machine learning frameworks (e.g., PyTorch, TensorFlow) and established explicit finite element solvers (e.g., SIMCENTER RADIOSS, OPENRADIOSS). Direct coupling introduces dependencies, reduces computational efficiency, and complicates standard workflows. Furthermore, while PANNs are theoretically robust, their practical implementation in high-performance explicit dynamics codes requires careful attention to computational cost, as these solvers rely on user-defined material routines written in compiled languages like Fortran where automatic differentiation is unavailable.

Methodology
The authors present a workflow to integrate PANN-based hyperelastic material models into SIMCENTER RADIOSS and OPENRADIOSS. The core methodology involves:

1. Training and Architecture: PANNs are trained in PyTorch to represent the scalar strain-energy density function ($\psi$) rather than predicting stresses directly. This ensures thermodynamic consistency. The network utilizes an Input-Convex Neural Network (ICNN) architecture with positive weights and convex, non-decreasing activation functions to enforce polyconvexity. The inputs are the isochoric invariants $I_1$ and a modified $I_2^* = I_2^{3/2}$.
2. Code Generation: A Python script automatically generates a standalone Fortran user material routine. This script exports the trained network parameters (weights and biases), defines the network architecture, and explicitly derives the stress formulation using the chain rule. This eliminates the need for external ML libraries during the simulation.
3. Stress Evaluation: The generated routine computes the Cauchy stress directly from the deformation gradient ($F$) using the left Cauchy-Green tensor ($b$). It avoids automatic differentiation by hard-coding the derivatives of the activation functions and invariants.
4. Activation Function Optimization: The study investigates replacing the standard SoftPlus activation function with the SQuarePlus (SQP) function. While SoftPlus is smooth and convex, it relies on logarithmic and exponential evaluations. SQP offers similar mathematical properties (smoothness, convexity, monotonicity) but uses algebraic operations involving square roots, which are computationally cheaper in Fortran.
5. Benchmarking: The implementation is validated using an explicit impact simulation of a "dragon" geometry struck by a rigid projectile. Three mesh resolutions are tested. The PANN results are compared against a classical Carroll hyperelastic model and evaluated for both qualitative stress distribution and computational runtime.

Key Contributions

• Automated Integration Framework: The authors provide a publicly available GitHub repository containing a Python script that converts trained PANN models into ready-to-use Fortran user material subroutines for RADIOSS. This allows users to integrate custom neural network models with minimal manual effort.
• Activation Function Analysis: The paper demonstrates that replacing SoftPlus with SQuarePlus significantly reduces the computational cost of the material routine. Benchmark tests show an isolated evaluation time reduction of approximately 87–88% for the activation function.
• Runtime Efficiency in Explicit Dynamics: The study quantifies the overhead of PANNs in explicit simulations. For a shallow network (one hidden layer, five neurons), the additional runtime compared to a classical Carroll model is minimal (3.6% for SoftPlus, 1.8% for SQuarePlus). Even for deeper networks, the overhead remains manageable, and the SQuarePlus variant consistently halves the additional cost compared to SoftPlus.
• Validation of Stability: The generated PANN routines are shown to produce numerically stable simulations under large deformations and complex contact conditions, yielding stress distributions qualitatively similar to classical constitutive laws.

Results

• Qualitative Performance: In the impact benchmark, PANN models (both SoftPlus and SQuarePlus variants) reproduced the nonlinear behavior of hyperelastic materials effectively. The von Mises stress distributions and transient displacement responses were comparable to the classical Carroll model, with no numerical instabilities observed.
• Quantitative Performance: The runtime analysis on the finest mesh (approx. 2.7 million elements) revealed that the SQuarePlus-based PANNs are more efficient than their SoftPlus counterparts.
  • Shallow Network (1x5): SoftPlus overhead was 3.6%; SQuarePlus overhead was 1.8%.
  • Deeper Network (2x5): SoftPlus overhead was 15.6%; SQuarePlus overhead was 7.8%.
• Parameter Efficiency: While the PANNs are slightly slower than the 3-parameter Carroll model, they utilize significantly more trainable parameters (20 for the shallow network, 50 for the deeper network), offering greater flexibility in representing complex material behaviors.

Significance and Claims
The paper claims to provide a practical route for incorporating machine-learning-based constitutive models into established explicit finite element workflows without requiring specialized solvers or external dependencies during runtime. The authors emphasize that their approach allows both commercial (SIMCENTER RADIOSS) and open-source (OPENRADIOSS) solvers to serve as testbeds for next-generation physics-based neural networks.

The work highlights that computational efficiency is a critical factor for the adoption of data-driven models in engineering. By demonstrating that SQuarePlus can reduce the computational overhead of PANN material routines by approximately half without sacrificing accuracy, the authors argue that compact PANN architectures are viable for routine engineering simulations. The presented framework is modest in its scope, focusing on isotropic hyperelasticity, but the authors note that the code-generation procedure could theoretically be extended to history-dependent materials and implicit solvers in future work.