From Hyperelastic Physics‐Augmented Neural Networks to Fortran User Materials: Standalone Implementation and Computational Performance in Explicit Dynamics

ABSTRACT

Data‐driven material modeling techniques have gained significant attention due to their ability to capture the complex constitutive behavior beyond the limitations of classical material models. Physics‐augmented neural networks (PANNs), which incorporate physical restrictions into their architecture, combine the flexibility of machine learning with properties required for reliable engineering simulations. As a result, they have become increasingly relevant in computational material modeling. This work presents an approach to integrate pretrained isotropic, nearly incompressible, invariant‐based hyperelastic PANNs into the explicit finite element solvers

and (Siemens). To this end, a Python framework transfers the trained parameters and the required constitutive expressions of the supported feed‐forward architecture to a standalone Fortran user material routine. The analytically expressed first derivatives of the strain‐energy density are included in the generated code. The finite element simulation therefore does not require an external machine‐learning library or a specialized solver. In addition to the integration strategy, particular emphasis is placed on the computational efficiency of the resulting user material subroutine. The influence of the network architecture on the overall simulation performance is investigated, and approaches for reducing evaluation costs while preserving the quality of the fitted material response are discussed. In this context, replacing the commonly used SoftPlus activation function by the SQuarePlus function reduces the additional constitutive evaluation costs for the investigated networks. To facilitate practical use, a publicly available GitHub repository ( https://github.com/OVGU‐CoMe/PANN_Radioss ) provides the code generator and the implemented material routines. Within the supported model class, the generator accounts for different numbers and widths of fully connected hidden layers. Additional invariant inputs can be incorporated through localized changes to the input and derivative definitions, whereas different constitutive formulations or solver interfaces require more comprehensive adaptations. As a complex example, an impact simulation demonstrates that the generated PANN user materials can be applied to explicit analyses involving large strains and (self‐)contact. The results illustrate the feasibility and computational performance of the investigated hyperelastic PANNs in and . They also show how a commercial and an open‐source finite element solver can be used to study this specified class of machine‐learning‐based constitutive models.