fol.responses

Sensitivity analysis and response evaluation provided by FoLax.

This module contains response objects that can be evaluated on top of a finite element (FE) state and a corresponding control field. A response typically represents a scalar objective or constraint functional of the form

\[J(d, u) = \sum_{e=1}^{N_e} \int_{\Omega_e} \phi(d(x), u(x)) \, \mathrm{d}\Omega ,\]

where d denotes the control field, u denotes the FE solution (DOFs), and the integrals are evaluated numerically using Gauss quadrature.

In addition to evaluating the scalar response value, FoLax responses support sensitivity analysis with respect to:

• State variables (DOFs), via automatic differentiation.

• Control variables, via automatic differentiation.

• Shape variables (nodal coordinates), via automatic differentiation.

• Adjoint-based gradients for large-scale problems where direct differentiation through the solver is expensive.

Finite element response class

Notes

• Element-wise response evaluation is performed using the FE element available in the associated fol.loss_functions.fe_loss.FiniteElementLoss instance. The response integrand is evaluated at Gauss points and summed over all elements.

• Adjoint sensitivities are computed by building the adjoint right-hand side from the response derivative with respect to the state, and by using the transpose of the FE Jacobian provided by the loss function.

• Finite-difference utilities are provided for verification of both control and shape sensitivities. Forward difference (FWD) and central difference (CD) are supported.

• The response formula is provided as a string and compiled into a JAX-jitted function during initialization. The formula is evaluated at Gauss points using the interpolated control value and interpolated DOF values.