[petsc-users] Dirichlet boundary condition for a nonlinear system
Barry Smith
bsmith at mcs.anl.gov
Tue Feb 18 19:56:00 CST 2014
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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