***************
**surf_charge**
***************

::

	EQ =surf_charge {Galerkin_wt} QS {Interpol_fnc} <float1> <float2> <float3>
	<float4> <float5>

-----------------------
**Description / Usage**
-----------------------

This card provides information for solving a conservation equation for the total surface
charge in a 2-dimensional bar (or shell) element.. Note that this equation is not yet
available in three dimensions and is in fact untested at this time. The card entries are as
follows:

+-----------------+----------------------------------------------------------+
|**surf_charge**  |Name of the equation to be solved.                        |
+-----------------+----------------------------------------------------------+
|{Galerkin_wt}    |Two- or four-character value that defines the type of     |
|                 |weighting function for this equation, where:              |
|                 |                                                          |
|                 | * **Q1**-Linear                                          |
|                 | * **Q2**-Quadratic                                       |
+-----------------+----------------------------------------------------------+
|**QS**           |Name of the variable associated with the shell curvature  |
|                 |equation.                                                 |
+-----------------+----------------------------------------------------------+
|{Interpol_fnc}   |Two- or four-character value that defines the             |
|                 |interpolation function used to represent the variable     |
|                 |**QS** where:                                             |
|                 |                                                          |
|                 | * **Q1**-Linear Continuous                               |
|                 | * **Q2**-Quadratic Continuous                            |
+-----------------+----------------------------------------------------------+
|<float1>         |Multiplier for mass terms. Set to 1.0.                    |
+-----------------+----------------------------------------------------------+
|<float2>         |Multiple for advection terms. Set to 1.0.                 |
+-----------------+----------------------------------------------------------+
|<float3>         |Multiplier for boundary terms. Set to 1.0.                |
+-----------------+----------------------------------------------------------+
|<float4>         |Multiplier for diffusion terms - required but not         |
|                 |currently implemented.                                    |
+-----------------+----------------------------------------------------------+
|<float5>         |Multiplier for source terms - required but not currently  |
|                 |implemented.                                              |
+-----------------+----------------------------------------------------------+

------------
**Examples**
------------

The following is a sample card that uses bilinear surface charge interpolation and
weight function:
::

   EQ = surf_charge Q2 QS Q2 1.0 1.0 1.0 0.0 0.0

-------------------------
**Technical Discussion**
-------------------------

The surface charge conservation equation implemented is:

.. figure:: /figures/302_goma_physics.png
	:align: center
	:width: 90%

where :math:`\sigma` is the surface charge unknown, :math:`D_s` is the surface diffusion coefficient, 
e is the electrical permittivity, :math:`\underline{n}` is the unit normal vector to the surface, and 
:math:`\underline{E}` = –:math:`\Delta` V is the electric field vector. Here, advection contributions
are not considered.

This is a special type of shell equation which depends on the gradient of a bulk variable
(here, electric potential V). Since values of these variables away from the surface are
normally not accessible during assembly of shell equations, this term has to be applied
as a special type of boundary condition (WEAK_SHELL_GRAD) which is set up to
evaluate sensitivities to interior bulk variable degrees of freedom . This term, though
physically an integral part of the surface charge equation, is applied through the
SURFACE_ELECTRIC_FIELD_BC boundary condition.



--------------
**References**
--------------

Notz, Patrick K. Ph.D. thesis. Purdue University, 2000.

..
	TODO - Line 67 contains a photo that needs to be written as an equation.