Automatic Differentiation

Automatic Differentiation:

• Materials with Automatic Differentiation (tensortrax)
• Materials with Automatic Differentiation (JAX)

FElupe supports multiple backends for constitutive material formulations with automatic differentiation. The default backend is based on tensortrax which ships with FElupe. For more computationally expensive material formulations, jax may be the preferred option.

It is straightforward to switch between these backends.

tensortrax (default)

import felupe as fem
import felupe.constitution.tensortrax as mat
import tensortrax.math as tm

def neo_hooke(C, mu):
    "Strain energy function of the Neo-Hookean material formulation."
    return mu / 2 * (tm.linalg.det(C) ** (-1/3) * tm.trace(C) - 3)

umat = mat.Hyperelastic(neo_hooke, mu=1.0)

JAX

import felupe as fem
import felupe.constitution.jax as mat
import jax.numpy as jnp

def neo_hooke(C, mu):
    "Strain energy function of the Neo-Hookean material formulation."
    return mu / 2 * (jnp.linalg.det(C) ** (-1/3) * jnp.trace(C) - 3)

umat = mat.Hyperelastic(neo_hooke, mu=1.0)

Note

The default backend is available in the top-level package namespace, this includes all models from felupe.constitution.tensortrax.models.hyperelastic and felupe.constitution.tensortrax.models.lagrange as well as the material classes felupe.constitution.tensortrax.Material and felupe.constitution.tensortrax.Hyperelastic.