====== TRPO2 ======
===== Description =====

For the axisymmetric element, the symmetry axis is Y.  \\ \\
{{  :elements:plani.png?250|}}
The element is defined by 4 nodes specified in NODES following the order indicated in the figure. \\ \\
The element must be used with law [[laws:ptrns|PTRNS]] \\
\\
The resolution method is the UPWIND FUPG method (with second order derivatives of the interpolation functions). The time discretization must therefore be small enough (Courant number less than unity). \\ \\

When the problem is mostly convective (Peclet number $P_e>2$), oscillations may persist. 
\[C_r=\frac{|\underline{v}|\Delta t}{dx}\]
\[P_e=\frac{|\underline{v}|dx}{a_L|\underline{v}|+D_m}\]
With:
  * $|\underline{v}|$ = Norm of the mean rate of convection on the element
  * $\Delta t$ = Time step
  * $dx$ = length of the element in the direction of the mean rate of convection
  * $a_L$ = longitudinal dispersivity
  * $D_m$ = Molecular diffusion
\\
Type: 172 \\
Implemented by: J-P. Radu, 1996

==== Files ====
Prepro: TRPO2A.F \\
Lagamine: TRPO2B.F

===== Input file =====
^Title (A5)^^
|TITLE|"TRPO2" in the first 5 columns|
^Control data (I5)^^
|NELEM|Number of elements|
^Definition of the elements (3I5/4I5)^^
|NINTE|Number of integration points (4)|
|LMATE|Material law|
|IPRIN|Index for the printing during the execution: \\ 0 = no printing \\ 1 = printing in the *.out file of $C_r$ and $P_e$ of the element for each time step \\ 2 = printing via the *.BEM file in the *.DES1 file of the maximum $C_{r}$ and $P_{e}$ of the elements, at each time step \\ 3 = 1 + 2|
|NODES(4)|List of nodes|
===== Results =====
SIGMA(8):
  * $f_{x,tot}$ the total pollutant flow in the x direction
  * $f_{y,tot}$ the total pollutant flow in the y direction
  * $f_{x,disp}$ the dispersive pollutant flow in the x direction
  * $f_{y,disp}$ the dispersive pollutant flow in the y direction
  * $f_{e,trans}$ the pollutant flow stored by temporal variation of concentration
  * $f_{e,conv}$ the pollutant flow stored by convection and unloaded by degradation
  * $f_{e,m-im}$ the pollutant flow transfered from mobile water to still water
  * $f_{e,degr}$ quantity of degraded pollutant

These flows are given with respect to the fluid volume except for total flows that are given with respect to the porous medium volume. \\
\\
6 state variable are also given: 
  * $v_x$ Darcy's flow rate in the x direction
  * $v_y$ Darcy's flow rate in the y direction
  * divv the divergence of Darcy's rate
  * $c_{im}$ pollutant concentration in the still water
  * $c_m$ the pollutant concentration in the moving water (if semi-coupled analysis, INDN = 3)
  * $S_r$ saturation degree