HyperElBase

Short description

GAMM AG DATA collaborative repository for hyperelastic strain energy functions to build benchmarks

Scope and background

At the 13th GAMM AG Data workshop in Darmstadt (Feb 11/12, 2025), the creation of a set of baseline models for the use of benchmarks and simulations was decided on. This repository collects different models for implementation in pytorch or TensorFlow. A provisional interface is teased to allow the use from TensorFlow as well as from pytorch, independent of the implementation (not all features will be accessible though).

A first demonstration for how to use the models is presented in the base class torch_HypEl for models written for pytorch. It contains also a unified demonstration sub-routine and some utilities, e.g., to generate random deformation tensors or relative errors. The demo sub-routine generalizes to all child classes of torch_HypEl, as long as they provide an analytical expression of the first Piola stress through stress(self, x).

Notation and code design

The notation for the inputs/outputs is as follows:

• An orthonormal coordinate system is asserted throughout for simplicity.
• The input is x[i, j, k] = F[i][j, k], i.e., x[i] is the $i$th deformation input, and the trailing 2 indices denote the component of the tensor according to
  $\boldsymbol{F}_{(i)} = \left( \begin{array}{ccc} F_{11} & F_{12} & F_{13} \\ F_{21} & F_{22} & F_{23} \\ F_{31} & F_{32} & F_{33} \end{array} \right)_{(i)}$
• The forward(self, x) function provides the strain energy w[i]=$W(\boldsymbol{F}_{(i)})$.
• The [optional] stress(self, x) function is used to validate the stresses $\boldsymbol{P}_{(i)}=\partial_{\boldsymbol{F}} W(\boldsymbol{F}_{(i)})$; this function can contain a closed-form implementation of the Piola stress.

Documentation of the models

The documentation will be placed in a LaTeX document for now. This document (including a compiled version thereof) will be added soon.

Utility routines

Utilities are provided to facilitate testing etc.

• rel_err(a, a_ref, eps) computes relative errors and is safe w.r.t. division by zero errors, as long as $\epsilon>0$
• RandomDeformations(n, amp) defines a dataset of $n$ deformations defined via
  $ \boldsymbol{F}_{(i)} = \left( \begin{array}{ccc} 1 + X & X & X \\ X & 1+X & X \\ X & X & 1+X \end{array} \right)_{(i)},$
  where $X \sim $ amp * $\mathcal{U}([-1, 1])$ (i.e. uniformly random entries of the displacement gradient).
• demo_HypEl(HypElModel, n, amp) uses the model HypElModel and computes the stresses using auto-differentiation (via autograd_stress(self, x) from the base class torch_HypEl). They are compared to the analytical stress from stress(self, x) and the relative errors are printed for each sample.
• demo_NeoHooke() illustrates the use of demo_HypEl for a Neo Hooke model.