Models

This module contains premade jax-transformable partial differential equation (PDE) formulations, weak forms, user-defined elements, and time integration procedures.

The models-functions return callables with appropriate arguments for the use required in the other modules of AutoPDEx. These functions can then be passed e.g. to the solver via static_settings[‘model’].

For instance, a typical weak form function has the following arguments:

x (jnp.ndarray): Spatial coordinates at the integration point.
ansatz (function): Ansatz function representing the field variable (e.g., displacement or temperature).
test_ansatz (function): Test ansatz function representing the virtual displacement or virtual temperature.
settings (dict): Settings for the computation.
static_settings (dict): Static settings for the computation.
int_point_number (int): Integration point number.
set: Number of domain.

Note: Some of the models work for degrees of freedoms (dofs) as a jnp.ndarray and some work for dicts of jnp.ndarrays. It is specified in each docstring.

User potentials/elements

`mixed_reference_domain_potential`(...)	Constructs a multi-field 'user potential' for integration of a potential in the reference configuration of finite elements.
`mixed_reference_domain_residual`(...)	Constructs a multi-field 'user residual' for integration of a weak form in the reference configuration of finite elements.
`mixed_reference_surface_potential`(...)	Constructs a multi-field 'user potential' for integration of a potential in the reference configuration of surface elements.
`mixed_reference_surface_residual`(...)	Constructs a multi-field 'user residual' for integration of a weak form in the reference configuration of surface elements.
`isoparametric_domain_integrate_potential`(...)	Constructs a local integrand fun (user potential) for integration of functions in the reference configuration of isoparametric elements.
`isoparametric_domain_element_galerkin`(...[, ...])	Constructs an isoparametric domain element for Galerkin methods using a given weak form function and DOFs as jnp.ndarray.
`isoparametric_surface_element_galerkin`(...)	Constructs an isoparametric surface element for Galerkin methods using a given weak form function and DOFs as jnp.ndarray.

Functions for coupling with DAE solver

`mixed_reference_domain_residual_time`(...)	Constructs a multi-field 'user residual' for time-dependent weak forms.
`mixed_reference_domain_int_var_updates`(...)	Similar as mixed_reference_domain_residual_time, but generates a function that doesn't compute the residual, but the internal variables based on the local_int_var_updates_fun.

Convenience functions for modelling

`aug_lag_potential`(lag, constr, eps)	Augmented Lagrangian potential function for an inequality constraint (constr >= 0).
`kelvin_mandel_extract`(a)	Converts a 6-component Kelvin-Mandel representation vector into a 3x3 symmetric matrix.

Linear equations

`transport_equation`(c)	Transport equation: du/dt + c * du/dx = 0
`poisson`([coefficient_fun, source_fun])	Poisson equation in n dimensions: coefficient * Laplace(Theta) + source_term = 0
`poisson_weak`([coefficient_fun, source_fun])	Poisson equation in n dimensions as weak form.
`poisson_fos`(spacing[, coefficient_fun, ...])	Poisson equation as set of first order models augmented by curl(v)=0
`heat_equation`(diffusivity_fun)	Space-time heat equation with thermal diffusivity alpha.
`heat_equation_fos`(diffusivity_fun)	Space-time heat equation with thermal diffusivity alpha as a first order system with curl-augmentation for more than 2 spatial dimensions
`d_alembert`(wave_number_fun)	Space-time d'Alembert operator 1 temporal [-1] and (n-1) spatial [:n-1] dimension with wave number c.
`d_alembert_fos`(wave_number_fun, spacing)	Constructs a first-order system for the space-time d'Alembert operator with wave speed c.
`linear_elasticity`(youngs_mod_fun, ...[, ...])	Constructs the strong form of linear elasticity in Voigt notation, displacement based.
`linear_elasticity_weak`(youngs_mod_fun, ...)	Constructs the weak form of linear elasticity in Voigt notation, displacement based.
`linear_elasticity_fos`(youngs_mod_fun, ...[, ...])	Constructs the first order system (FOS) of linear elasticity in Voigt notation, displacement based.
`neumann_weak`(neumann_fun)	Constructs the weak form of Neumann boundary conditions for the virtual work of surface tractions or heat inflow.

Nonlinear equations

`burgers_equation_inviscid`()	Constructs the inviscid Burgers' equation: du/dt + u * du/dx = 0.
`hyperelastic_steady_state_fos`(...[, ...])	Constructs a first-order PDE system for given strain energy function
`hyperelastic_steady_state_weak`(...[, ...])	Constructs the weak form of hyperelasticity for given strain energy function
`navier_stokes_incompressible_steady`(...[, ...])	Constructs the steady-state incompressible Navier-Stokes equations.
`navier_stokes_incompressible`(dynamic_viscosity)	Constructs the transient incompressible Navier-Stokes equations.

Strain energy functions

`neo_hooke`(F, param)	Computes the strain energy for a neo-Hookean material model of Ciarlet type given the deformation gradient.
`isochoric_neo_hooke`(F, mu)	Computes the strain energy for an isochoric neo-Hookean material model.
`linear_elastic_strain_energy`(F, param)	Computes the strain energy for a linear elastic material for small deformations.