Manual pages | Cast3M

$$$$ KBBT     NOTICE  CHAT      11/09/12    21:16:41     7124           
                                             DATE     11/09/12
                                                                                
    Operateur KBBT                           Voir aussi : NAVI KMAB
    --------------                                        KMBT  DUDW   
                                                          EQEX   
    SYNTAXE ( EQEX ) :   Cf operateur EQEX                 
    __________________


      'OPER'  'KBBT' coef      'INCO' 'UN' 'PRES'


                                                                                
                                                                                
     Description :
     _____________

 The KBBT operator discretizes the terms Div U and Grad P  by a finite
element method, in order to keep symetric the linear system.
                            -------------

     Contents :
     __________

     coef    coefficient
             FLOTTANT
             or CHPOINT SCAL SOMMET  (volumic porosity)
             or CHPOINT VECT SOMMET  (directional porosity)
             or MOT

     beta    stabilization parameter
             FLOTTANT

     UN      Velocity field
             CHPOINT VECT SOMMET or MOT
     PRES    Pressure field
             CHPOINT SCAL CENTRE   or MOT
             CHPOINT SCAL CENTREP1 or MOT
             CHPOINT SCAL CENTREP0 or MOT
             the type has to be given with the option key word INCOD

  When coefficients are of type MOT, the operator looks for data in INCO
  table at the index corresponding to the given name.


    Comments
    ________

 Considering the Stokes or Navier-Stokes equations for an incompressible
 viscous flow.

 A U + Grad P = F       : momentum equation

 -Div U       = 0       : continuity equation


 where U and P are respectively the velocity and the pressure, A an
 invertible operator (usually diffusion/advection operator).


 In the variational formulation the term  'Grad P' is integrated by part
which leads to a symetric system the continuity equation being written :

   - Div U = 0  .

        t
 | A  -B |(U)  (F)
 |       |( ) =( )
 |-B   0 |(P)  (0)

        t                 /
 where B is the matrix of | P Div W dv
                          |v
                                       (W test function for the velocity)

                          /
    B is the matrix of    | q Div V dv
                          |v
                                      (q test function for the pressure)

The KBBT operator constructs the matrices corresponding to the operators
B and Bt (only B is stored)

 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
 REMARK : Considering the sign of the second equation an eventual source
        term must be negative.
 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


  1/ Associated boundary conditions :

 Integrating by parts and using the divergence theorem we get :

   /                /          /
   | W*Grad P dv =  | P n ds - | P Div W dv
   |v               |s         |v

 The surface integral is ommited which leads to the natural induced
 boundary condition:

   /
   | P n ds = 0   (n outward normal)
   |s

 Additional induced boundary conditions can appear whith other operator
 like LAPN.
 See operator TOIM in order to impose a value to the surface integral.

  ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
  2/ If all normal velocities are prescribed (zero or not) all around the
     boundaries, it is NECESSARY to impose the pressure at a point.
     It ist the case for all incompressible flows in a closed cavity.
  :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

  3/ If the coefficient is of type CHPOINT SCAL SOMMET
    (volumic porosity H) the operator computes :

   /                  /            /                 /
   | W*H*Grad P dv =  | WHP n ds - | P H Div W dv  - | P W Grad H  dv
   |v                 |s           |v                |v


  4/ If the coefficient is of type CHPOINT VECT SOMMET
    (directional Porosity Hi ) the operator computes :

   /                   /              /                  /
   | W*Hi*Grad P dv =  | WHi P n ds - | P Hi Div W dv  - | P W Grad Hi dv
   |v                  |s             |v                 |v



    Options : (EQEX)
    _________

    OPTI INCOD CENTRE
               CENTREP1
               CENTREP0