underworld.function.analytic module
===================================

This module provides a suite of models which satisfy the
Stokes system of equations.

All models are considered across a unit square (or cube)
domain, and utilise (unless otherwise stated) free-slip
conditions on all boundaries.

Each model object provides a set of Underworld Functions
for description of physical quantities such as velocity,
pressure and viscosity.

For numerical validation in Underworld, we construct a
Stokes system with appropriate domain and boundary
conditions. Viscosity and body forces are set directly
using corresponding Function objects provided by the solution
class. Generated numerical solution for velocity and
pressure (or derived quantities) may then be compared
with exact solutions provided by solution objects.

Where appropriate, solution classes provide Latex descriptions
of the body force and viscosity functions via the `eqn_bodyforce`
and `eqn_viscosity` attributes. Note the following definitions for
rectangular and step functions:

.. math:: 

    \operatorname{rect}(c, w; x) = \left\{\begin{array}{rl}
    1 & \text{if } |x - c| \le \frac{w}{2} \\
    0 & \text{if } |x - c| >    \frac{w}{2},
    \end{array}\right.

and

.. math:: 

    \operatorname{step}(\alpha,\beta,c; x) = \left\{\begin{array}{rl}
    \alpha & \text{if } x \le c \\
    \beta & \text{if } x > c.
    \end{array}\right.




.. module:: underworld.function.analytic

Classes
~~~~~~~

.. autosummary::
    :nosignatures:

    underworld.function.analytic.SolA
    underworld.function.analytic.SolB
    underworld.function.analytic.SolC
    underworld.function.analytic.SolCx
    underworld.function.analytic.SolDA
    underworld.function.analytic.SolDB2d
    underworld.function.analytic.SolDB3d
    underworld.function.analytic.SolH
    underworld.function.analytic.SolKx
    underworld.function.analytic.SolKz
    underworld.function.analytic.SolM
    underworld.function.analytic.SolNL

.. autoclass:: underworld.function.analytic.SolA
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolB
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolC
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolCx
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolDA
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolDB2d
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolDB3d
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolH
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolKx
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolKz
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolM
    :members:
    :show-inheritance:
.. autoclass:: underworld.function.analytic.SolNL
    :members:
    :show-inheritance: