#include <darcy.hpp>

Collaboration diagram for darcy_nonlinear:

Data Structures
struct	Private
	Private Members. More...

Private Attributes
std::shared_ptr< Private >	Members

	darcy_nonlinear (int argc, char **argv)
	Constructor. More...

	~darcy_nonlinear ()
	Destructor. More...

LifeV::Real	run ()
	To lunch the simulation. More...

Detailed Description

Includes.

LifeV non-linear and transient Darcy test case

Author: A. Fumagalli aless.nosp@m.io.f.nosp@m.umaga.nosp@m.lli@.nosp@m.mail..nosp@m.poli.nosp@m.mi.it

A 2D Darcy transient and non-linear test with Dirichlet boundary condition. Let us set $\Omega = (0,1)^3$ and $I_T = \Omega \times [0,4)$ , solve the problem in dual-mixed form

$\left\{ \begin{array}{l l l } \Lambda^{-1} \left( p \right) \sigma + \nabla p = f_v & \mathrm{in} & I_T, \vspace{0.2cm} \\ \displaystyle \phi \frac{\partial p}{\partial t} + \nabla \cdot \sigma + \pi p - f = 0 & \mathrm{in} & I_T, \vspace{0.2cm} \\ p = g_D & \mathrm{on} & \partial \Omega \times [0,4),\vspace{0.2cm} \\ p = p_0 & \mathrm{in} & \Omega. \end{array} \right.$

Furthermore the data are

$\begin{array}{l l l} f_v(x,y) = \left( ty - tx, -t^2 y^2 \right)^T & \pi(x,y) = 1 & f(x,y) = tx^2 - 2t^2 - t - (6y + 2 t^2 y)(1 + t^4 x^4 + y^6 + 2 t^2 x^2 y^3) - (3y^2 + t^2 y^2)(6y^5 + 6t^2 x^2 y^2) + t^2 x^2 + y^3, \vspace{0.2cm} \\ g_D(x,y,z) = t^2 x^2 + y^3, & \phi(x,y) = 0.5, & \Lambda (x,y,p) = \left( \begin{array}{c c} 1 & 0 \\ 0 & 1+p^2 \end{array} \right), \\ p_0(x,y) = y^3. & & \end{array}$

The analytical solutions are

$p(x,y,z) = t^2 x^2 + y^3, \quad \sigma(x,y,z) = \left( \begin{array}{c} - 2t^2 x + t (y - x) \\ - (3y^2 + t^2 y^2)( 1 + t^4 x^4 + y^6 +2 t^2 x^2 y^3) \end{array} \right).$