[docs]
class TorchModel:
r"""Isotropic hyperelastic material formulation based on a given strain energy
density function ``fun(I1, I2, I3, **kwargs)`` per unit undeformed volume. The
gradients are carried out by automatic differentiation using PyTorch.
.. math::
\psi = \psi(I_1, I_2, I_3)
.. note::
PyTorch uses single-precision by default. This must be considered in numeric
simulations, i.e. the error tolerance should not exceed
``np.sqrt(torch.finfo(torch.float).eps)`` (approx. ``tol=5e-4``). For double-
precision, enable ``torch.float64`` as default.
.. code-block:: python
import torch
torch.set_default_dtype(torch.float64)
Examples
--------
>>> import hyperelastic
>>> def yeoh(I1, I2, I3, C10, C20, C30):
>>> "Yeoh isotropic hyperelastic material formulation."
>>> return C10 * (I1 - 3) + C20 * (I1 - 3) ** 2 + C30 * (I1 - 3) ** 3
>>> model = hyperelastic.models.invariants.TorchModel(
>>> yeoh, C10=0.5, C20=-0.05, C30=0.02
>>> )
>>> framework = hyperelastic.InvariantsFramework(model)
>>> umat = hyperelastic.DistortionalSpace(framework)
"""
def __init__(self, fun, **kwargs):
self.fun = fun
self.kwargs = kwargs
def _grad(self, fun, x, numpy=False, allow_unused=True, **kwargs):
"Return the gradient of a sum of a tensor ``fun`` w.r.t. ``x``."
import torch
out = None
if hasattr(fun, "grad_fn"):
out = torch.autograd.grad(
fun.sum(), x, allow_unused=allow_unused, **kwargs
)[0]
if numpy:
if out is not None:
return out.detach().numpy()
return out
def _astensors(self, *args):
"Convert a list/tuple of arrays to torch-tensors."
import torch
tensors = [torch.Tensor(arg) for arg in args]
[tensor.requires_grad_(True) for tensor in tensors]
return tensors
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def gradient(
self,
I1,
I2,
I3,
statevars,
tensor=False,
numpy=True,
create_graph=False,
retain_graph=False,
):
"""The gradient as the partial derivative of the strain energy function w.r.t.
the invariants."""
if not tensor:
I1, I2, I3 = self._astensors(I1, I2, I3)
f = self.fun(I1, I2, I3, **self.kwargs)
kwargs = dict(numpy=numpy, create_graph=create_graph, retain_graph=retain_graph)
return (*[self._grad(f, x, **kwargs) for x in [I1, I2, I3]], statevars)
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def hessian(self, I1, I2, I3, statevars, numpy=True):
"""The hessian as the second partial derivatives of the strain energy function
w.r.t. the invariants."""
I1, I2, I3 = self._astensors(I1, I2, I3)
dWdI1, dWdI2, dWdI3, statevars = self.gradient(
I1, I2, I3, statevars, tensor=True, create_graph=True, numpy=False
)
return [
self._grad(f, x, numpy=numpy)
for f, x in zip(
[dWdI1, dWdI2, dWdI3, dWdI1, dWdI2, dWdI1], [I1, I2, I3, I2, I3, I3]
)
]
