Advanced material modeling in FEniCSx

FEniCSx is a powerful code-generation software package for automating complex PDE systems. While hyperelastic materials can be formulated using UFL operators and automatic differentiation (AD) of a free energy potential, more complex materials often require solving for the evolution of internal state variables at the material point level.

However, this two-level procedure is not easily enabled by FEniCSx, which is not designed for it. This work aims to provide a simple way of defining such implicit constitutive equations, albeit with a slight computational and memory overhead.

The library can handle various implementation types of constitutive equations, starting with user-defined constitutive updates using NumPy and JAX, which are ideal for educational purposes [1]. For better computational performance and behavior complexity, the library also supports material models defined by the MFront code generator.

Additionally, advanced material modeling techniques such as multiscale material models using Thermodynamics-based Artificial Neural Networks (TANN) can also be used very easily.

• Github repository: https://github.com/bleyerj/dolfinx_materials
• Online documentation: https://bleyerj.github.io/dolfinx_materials/

References

[1] Andrey Latyshev, Jérémy Bleyer, Corrado Maurini, Jack S Hale. Expressing general constitutive models in FEniCSx using external operators and algorithmic automatic differentiation. 2024.