[petsc-users] Dirichlet boundary condition for a nonlinear system
Sanjay Govindjee
s_g at berkeley.edu
Tue Feb 18 22:31:58 CST 2014
One alternate to this is that on the first step the initial value of X_B
is not
taken as the know value but rather some prior known value. Then for the
first
iteration Y_B turns out to be the known increment; in subsequent
iterations it
is take as 0. This can sometimes be helpful with convergence in tough
problems.
-sg
On 2/18/14 5:56 PM, Barry Smith 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,
More information about the petsc-users
mailing list