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

The gist: This work presents an approach to integrate physics-augmented neural networks into explicit finite element solvers SIMCENTER RADIOSS and OPENRADIOSS by developing a framework for transferring pretrained network architectures and parameters to a standalone user material routine.

How it works

The core idea involves embedding physical constraints directly into the neural network architecture, creating Physics-Augmented Neural Networks (PANNs) which combine machine learning flexibility with engineering reliability. A central idea is to use the network to represent the scalar potential ψ instead of predicting stresses directly, allowing stresses and tangent stiffness matrices to be obtained by differentiating this potential using automatic differentiation approaches embedded in machine learning frameworks. This construction helps ensure thermodynamic consistency, and additional constraints like monotonicity or convexity can be enforced through the choice of network structure, weights, and activation functions.

Integration into Solvers

The paper focuses on integrating these PANN-based hyperelastic material models into the explicit finite element solver OPENRADIOSS. A framework for transferring pretrained network architectures and their corresponding parameters to a standalone user material routine is developed. The networks are trained using the PYTORCH framework, but the procedure can be adapted to other frameworks like TENSORFLOW. This enables the use of PANNs within the finite element technology available in SIMCENTER RADIOSS and OPENRADIOSS without requiring specialized solvers. The Python code that generates the material model directly in Fortran is provided in a public GitHub repository.

Computational Efficiency and Implementation

The paper places particular emphasis on computational efficiency, demonstrating how replacing the commonly used SoftPlus activation function by the SQuarePlus function can reduce the computational cost of the PANN material routine. The SQuarePlus function satisfies required smoothness, monotonicity, and convexity properties while being less expensive to evaluate in the generated material routine. For the one-hidden-layer network considered, the normalized element-force runtime increased by only 3.6 % for the SoftPlus activation and by 1.8 % for the SQuarePlus activation compared to the Carroll model.

Material Model Formulation and Comparison

The material behavior is restricted to isotropic hyperelasticity, formulated in terms of a scalar-valued strain-energy density function ψ that depends only on the deformation state. The formulation decomposes the strain-energy density into an isochoric and a volumetric contribution: ψ = ψisoI1, I2 + ψvol(J). The isochoric part describes distortional deformation using models such as the Neo-Hooke or Mooney-Rivlin [21, 22], and the volumetric part depends on the volume measure J. A classical reference model, the Carroll material model, is used for comparison.

Validation and Results

An example impact simulation demonstrates that the generated PANN user material can reproduce the nonlinear behavior characteristic of hyperelastic materials under large strains. The results show that the generated PANN user materials can be applied to an explicit impact simulation without numerical instabilities and yield physically plausible stress fields. The PANN models and the Carroll model do not yield identical fitted material responses and also differ slightly in their initial shear moduli. The results show that the generated PANN user materials can be applied to a complex explicit impact simulation involving large deformations, contact, and a large number of constitutive evaluations.

Conclusion

The presented implementation concept is not restricted to the simple hyperelastic PANN architecture considered in this work. The proposed code-generation approach provides a practical step toward the use of machine-learning-based constitutive models in engineering simulations. The results indicate that physics-augmented neural-network constitutive models can be incorporated into established explicit finite element workflows with manageable computational overhead. The SQuarePlus activation therefore represents a promising alternative for efficient PANN-based constitutive models in explicit finite element simulations. The presented implementation concept is not restricted to the simple hyperelastic PANN architecture considered in this work. The framework can in principle be extended to material models with internal variables and history-dependent evolution equations. Future work should therefore focus on generalizing the code-generation procedure to broader PANN architectures and more advanced constitutive formulations. The computation of consistent tangent stiffness matrices should be incorporated into the code-generation framework. The proposed code-generation approach therefore provides a practical step toward the use of machine-learning-based constitutive models in engineering simulations. The presented results indicate that physics-augmented neural-network constitutive models can be incorporated into established explicit finite element workflows with manageable computational overhead.

How it works

The stress evaluation is formulated in terms of the left Cauchy-Green deformation tensor. The network derivative is obtained by a forward pass through the hidden layers followed by a backpropagation step to the invariant inputs. For one hidden layer, only the derivative of the activation function is required in the final stress expression. The resulting PANN formulation is written as ψPANN = ψPANN iso I1, I∗2 + ψvol(J).

How it works

The stress evaluation involves computing the invariants and the invariant derivatives multiplied by b according to Eqs. (47)–(49). The complete stress evaluation can be written in a compact form and translated into the generated Fortran material module. The final Cauchy stress is then evaluated as τ = Jσ = 2∂ψ/∂b b.

How it works

The paper investigates the replacement of the SoftPlus function with the SQuarePlus function to reduce computational cost, showing a net time saving with SQuarePlus. The observed improvement in the complete finite element simulation is smaller than the speedup obtained in the isolated activation function benchmark. The SQuarePlus activation therefore represents a promising alternative for efficient PANN-based constitutive models in explicit finite element simulations.

How it works

The generated material routine starts with the definition of the network size, the trained weights and biases, and the material parameters. The routine evaluates Cauchy stress σ from the deformation gradient F. The script is provided in a public GitHub repository and can be adapted to different network architectures.

1. The improved AI system can perform material modeling within explicit finite element workflows by generating user material subroutines automatically from pretrained physics-augmented neural networks (PANNs). This enables users to integrate their own neural-network models with limited manual effort, requiring merely the specification of the network architecture and the corresponding trained parameters.

2. The system can utilize SQuarePlus activation functions to reduce computational overhead in material routines. The paper demonstrates that this replacement can lead to a reduction in the additional runtime of the PANN material routine by approximately one half compared to SoftPlus, particularly for deeper architectures.

3. The AI system can achieve thermodynamic consistency and ensure physical constraints are embedded directly into the network architecture by using PANNs where key principles of classical constitutive modeling are embedded directly into the network architecture.

Data-driven material modeling techniques have gained significant attention due to their ability to capture complex constitutive behaviors beyond the limitations of classical material models. Physics-augmented neural networks (PANNs), which embed physical constraints directly into their architecture, combine the flexibility of machine learning with the reliability required for engineering simulations. This work presents an approach to integrate such network architectures into the explicit finite element solvers Simcenter Radioss and OpenRadioss (Siemens). A framework for transferring pretrained network architectures and their parameters to a standalone user material routine is developed. Networks are trained using PyTorch, though the procedure can be adapted to other frameworks such as TensorFlow, enabling the use of PANNs within existing finite element technology without requiring specialized solvers. Particular emphasis is placed on computational efficiency. The influence of network architecture on simulation performance is investigated, and strategies for reducing evaluation costs while preserving accuracy are discussed. Specifically, replacing the SoftPlus activation function with SQuarePlus is shown to reduce computational cost. A publicly available GitHub repository automates the generation of Fortran user material routines, requiring only the specification of the network architecture and trained parameters. An example impact simulation demonstrates that the generated PANN user material reproduces the nonlinear behavior characteristic of hyperelastic materials under large strains, providing a practical route toward machine-learning-based constitutive models in explicit finite element simulations.