[petsc-users] Dirichlet boundary condition for a nonlinear system

[Barry Smith]

On Feb 18, 2014, at 5:39 PM, Fande Kong <fd.kong at siat.ac.cn> wrote:

> Hi all,
> 
> I am just trying to solve a nonlinear system resulted from discretizating a hyperelasticity problem by finite element method. When I solve a linear PDE, I never put boundary solution either in a solution vector or a matrix, but instead, I put boundary condition to the right hand size (load).

    You adjust the right hand side to have zero as the boundary conditions. This can be written as 

      (A_II   A_IB ) ( X_I )       (F_I)
      (A_BI  A_BB)(X_B)   =   (F_B)

    Which is equivalent to 

      (A_I  A_B) (X_I)         (F_I) - (A_B)*(X_B) 
                        (0)       = 

      A_I X_I  = F_I - A_B*X_B

    In the nonlinear case you have

      F_I(X_I,X_B)    = ( 0 )
      F_B(X_I,X_B)      ( 0) 

     where you know X_B  with Jacobian

       (J_II  J_IB)   
       (J_BI J_BB) 
   
    Newtons’ method on all variables gives

      (X_I)^{n+1}     =  (X_I)^{n}     +  (Y_I)
      (X_B)                  (X_B)              (Y_B)

    where   JY = F which written out in terms of I and B is 

        (J_II  J_IB)   (Y_I)      =   F_I( X_I,X_B)
       (J_BI J_BB)   (Y_B)         F_B(X_I,X_B)

    Now since X_B is the solution on the boundary the updates on the boundary at zero so Y_B is zero so this system reduces to

        J_II   Y_I     = F_I(X_I,X_B) so Newton reduces to just the interior with 

    (X_I)^{n+1}     =  (X_I)^{n}     +  J_II^{-1} F_I(X_I,X_B)

    Another way to look at it is you are simply solving F_I(X_I,X_B) = 0 with given X_B so Newton’s method only uses the Jacobian of F_I with respect to X_I

   Barry



     


   


> How can I do a similar thing when solving a nonlinear system using a newton method?
> 
> Thanks,
> 
> Fande,