Coupled equations: Electricity x Thermics
=========================================

Unknows are the electrical potential :math:`U` and the temperature :math:`T` or :math:`\theta`.

.. figure:: Legend-derivatives.png
    :scale: 50%

Model
-----
.. figure:: conductivity-x-thermics.png
    :align: center

    The coupled system potential x temperature

A tension is applied by setting :math:`U=0` on the left side and :math:`U=1` on the right side. The temperature is set to 0 at these same boundaries, while a convective exchange applies elsewhere at the boundary, the external convective temperature being :math:`T_{\rm ext}`.

The other parameters of the problem are the material properties:

* the electric conductivity :math:`\sigma = \sigma_0+\sigma_1 \theta`,
* the thermal conductivity :math:`\kappa`,
* the volumic thermal capacity :math:`c_p` (that enables deducing temperature increase from heat production),
* the exchange coefficient :math:`\beta`.

Here is the pseudosystem with only the volumic contributions. In red, the test functions of :math:`\theta`; in blue, those of :math:`u`.

.. figure:: coupling-UxT-system.png

That is a fully coupled multiphysics system of equations. We'll see how to define the matrix and the right-hand side.

Sensitivities
-------------
Furthermore, some additional results will be provided: the derivatives of the solution with respect to some coefficients
(electrical conductivity, thermal exchange coefficient). For this, the only additional need is the formulation of the right-hand side derivatives
as the solver will perform the corresponding line-solves with the matrix obtained at the last Newton iteration to give the proper result
(there is a negative sign in the box).

Code
----
.. literalinclude:: ./nvi_Conductivity_x_Thermics_0.cpp

Results
-------

The results are shown for the electrical potential and its sensitivities:

.. figure:: VisuU1.png
    :align: center
    
    'U, ∂U/∂σ1, ∂U/∂ß'

and for the temperature and its sensitivities:

.. figure:: VisuT1.png
    :align: center
    
    'T, ∂T/∂σ1, ∂T/∂ß'

Log file
--------
.. literalinclude:: Conductivity_x_Thermics_0_gmsh2.log
    :language: text