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

[Fande Kong]

Barry,

Thanks. I see.


On Tue, Feb 18, 2014 at 6:56 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
> 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,
>
>
>
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