=====================
Behaviour integrators
=====================

A Behaviour integrator computes residual and jacobian terms for a
material. Behaviour integrators are associated with:

- a physical phenomenon (mechanics, heat transfer, etc.),
- a modeling hypothesis (plane strain, plane stress, etc.),
- a material symmetry (isotropy, geometry).

.. contents::
    :depth: 3
    :local:

.. note::

   Behaviour integrators are generally declared using 
   the :code:`addBehaviourIntegrator` method of the
   :code:`NonLinearEvolutionProblem` class, as in the
   following example:

   .. code:: c++

      mechanics.addBehaviourIntegrator("Mechanics", "beam",
                                       "src/libBehaviour.so",
                                       "MicromorphicDamageI_SpectralSplit");

   Based on the generic category :code:`Mechanics`, this declaration
   automatically handles the selection of
   a behaviour integrator consistent with the modeling hypothesis
   declared by the nonlinear evolution problem and the material
   symmetry declared by the behaviour.

Behaviour integrators and :code:`MFront` behaviours
===================================================

Behaviour integrators rely on external behaviours that shall be
generated by :code:`MFront`. In practice, those external behaviours are compiled by :code:`MFront`
into shared libraries that are loaded dynamically when an behaviour
integrator is loaded.

Mechanical behaviour integrator
===============================

A mechanical behaviour integrator computes the residual and jacobian
associated with one of the following variational operators:

.. math::
   
   \begin{aligned}
   \int_{\Omega_r} \underline{\sigma}\,\colon\,\underline{\varepsilon}^{\star}\,\mathrm{d}\,V & \quad\quad &\text{(small strain)} \\
   \int_{\Omega_r} \underline{P}\,\colon\,\underline{F}^{\star}\,\mathrm{d}\,V & \quad\quad & \text{(finite strain)}
   \end{aligned}

where:

- :math:`\Omega_r` is the domain occupied by the material of interest in the
  reference configuration,
- :math:`\underline{\sigma}` is the `Cauchy` stress (expressed in the
  reference configuration for a small strain analysis,
- :math:`\underline{\varepsilon}^{\star}` is the virtual deformation,
- :math:`\underline{P}` is the first Piola-Kirchhoff stress,
- :math:`\underline{F}^{\star}` is the virtual deformation gradient.

The mechanical behaviour integrators available in :code:`MFEM/MGIS`
support:

- small and finite strain behaviours,
- isotropic and orthotropic behaviours,
- plane strain, plane stress and tridimensional modeling hypotheses.


Heat transfer behaviour integrators
===================================

Nonlinear heat transfer behaviour integrators compute the residual and jacobian
associated with the following variational operator:

.. math::
   
   -\int_{\Omega_r} \vec{j}\,\cdot\,\vec{\nabla} T^{\star}\,\mathrm{d}\,V

where:

- :math:`\Omega_r` is the domain occupied by the material of interest in the
  reference configuration,
- :math:`\vec{j}` is the heat flux,
- :math:`\vec{\nabla} T^{\star}` is the virtual temperature.

The heat transfer behaviour integrators available in :code:`MFEM/MGIS`
support:

- small and finite strain behaviours,
- isotropic and orthotropic behaviours,
- plane strain, plane stress and tridimensional modeling hypotheses.

Micromorphic damage behaviour integrators
=========================================

Micromorphic damage behaviour integrators  compute the residual and jacobian
associated with the following variational operator:

.. math::
   
   \int_{\Omega_r} \vec{b}_{\chi}\,\cdot\,\vec{\nabla} d_{\chi}^{\star}+a_{\chi}\,d_{\chi}^{\star}\,\mathrm{d}\,V

where:

- :math:`\Omega_r` is the domain occupied by the material of interest in the
  reference configuration,
- :math:`b_{\chi}` is the thermodynamic force associated with the
  gradient of the micromorphic damage :math:`\vec{\nabla} d_{\chi}`,
- :math:`a_{\chi}` is the thermodynamic force associated with the
  micromorphic damage :math:`d_{\chi}`,
- :math:`d_{\chi}^{\star}` is the virtual micromorphic damage.

The micromorphic damage integrators available in :code:`MFEM/MGIS` support:

- isotropic behaviour in :math:`2D` (plane strain and plain stress) and :math:`3D` computations,
- orhotropic behaviours in :math:`2D`.