Reference problem
=====================


Geometry
---------

In 2D, we consider a square with side :math:`\mathrm{1m}`. In 3D, we consider a cube with side :math:`\mathrm{1m}`.


Material properties
----------------------

Intrinsic permeability: :math:`\kappa \mathrm{=}0.05{m}^{2}\mathrm{.}{\mathit{Pa}}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1}`

Biot coefficient: :math:`b\mathrm{=}1.`

Young's module: :math:`E\mathrm{=}2.5\mathit{Pa}`

Poisson's ratio: :math:`\nu \mathrm{=}0.25`

So we have the following Lamé coefficients: :math:`\lambda \mathrm{=}\mu \mathrm{=}1.0\text{Pa}`


Boundary conditions and loads
-------------------------------------

The boundary conditions are the Dirichlet conditions corresponding to the available analytical solution.

In 2D, we define :math:`A\mathrm{=}2\mathrm{\times }{\pi }^{2}\mathrm{\times }\kappa` and we define the mechanical volume load:

:math:`f(t,x,y)\mathrm{=}\left[\begin{array}{c}\mathrm{cos}(\pi x)\mathrm{sin}(\pi y)\\ \mathrm{sin}(\pi x)\mathrm{cos}(\pi y)\end{array}\right]\pi {e}^{\mathrm{-}\mathit{At}}\left[\mathrm{-}(\lambda +2\mu )+b\right]`

In 3D, we define :math:`A\mathrm{=}3\mathrm{\times }{\pi }^{2}\mathrm{\times }\kappa` and we define the mechanical volume load:

:math:`f(t,x,y,z)\mathrm{=}\left[\begin{array}{c}\mathrm{cos}(\pi x)\mathrm{sin}(\pi y)\mathrm{sin}(\pi z)\\ \mathrm{sin}(\pi x)\mathrm{cos}(\pi y)\mathrm{sin}(\pi z)\\ \mathrm{sin}(\pi x)\mathrm{sin}(\pi y)\mathrm{cos}(\pi z)\end{array}\right]\pi {e}^{\mathrm{-}\mathit{At}}\left[\mathrm{-}(\lambda +2\mu )+b\right]`


Initial conditions
--------------------

The initial conditions correspond to the analytical solution that is available.