[petsc-users] Questions Regarding PETSc and Solving Constrained Structural Mechanics Problems

[Mark Adams]

Adding this to the PETSc mailing list,

On Thu, Jun 12, 2025 at 3:43 AM hexioafeng <hexiaofeng at buaa.edu.cn> wrote:

>
> Dear Professor,
>
> I hope this message finds you well.
>
> I am an employee at a CAE company and a heavy user of the PETSc library. I
> would like to thank you for your contributions to PETSc and express my deep
> appreciation for your work.
>
> Recently, I encountered some difficulties when using PETSc to solve
> structural mechanics problems with Lagrange multiplier constraints. After
> searching extensively online and reviewing several papers, I found your
> previous paper titled "*Algebraic multigrid methods for constrained
> linear systems with applications to contact problems in solid mechanics*"
> seems to be the most relevant and helpful.
>
> The stiffness matrix I'm working with, *K*, is a block saddle-point
> matrix of the form (A00 A01; A10 0), where *A00 is singular*—just as
> described in your paper, and different from many other articles . I have a
> few questions regarding your work and would greatly appreciate your
> insights:
>
> 1. Is the *AMG/KKT* method presented in your paper available in PETSc? I
> tried using *CG+GAMG* directly but received a *KSP_DIVERGED_PC_FAILED*
> error. I also attempted to use *CG+PCFIELDSPLIT* with the following
> options:
>

No


>
>     -pc_type fieldsplit -pc_fieldsplit_detect_saddle_point
> -pc_fieldsplit_type schur -pc_fieldsplit_schur_precondition selfp
> -pc_fieldsplit_schur_fact_type full -fieldsplit_0_ksp_type preonly
> -fieldsplit_0_pc_type gamg -fieldsplit_1_ksp_type preonly
> -fieldsplit_1_pc_type bjacobi
>
>    Unfortunately, this also resulted in a *KSP_DIVERGED_PC_FAILED* error.
> Do you have any suggestions?
>
> 2. In your paper, you compare the method with *Uzawa*-type approaches. To
> my understanding, Uzawa methods typically require A00 to be invertible. How
> did you handle the singularity of A00 to construct an M-matrix that is
> invertible?
>
>
You add a regularization term like A01 * A10 (like springs). See the paper
or any reference to augmented lagrange or Uzawa


3. Can i implement the AMG/KKT method in your paper using existing *AMG
> APIs*? Implementing a production-level AMG solver from scratch would be
> quite challenging for me, so I’m hoping to utilize existing AMG interfaces
> within PETSc or other packages.
>
>
You can do Uzawa and make the regularization matrix with matrix-matrix
products. Just use AMG for the A00 block.



> 4. For saddle-point systems where A00 is singular, can you recommend any
> more robust or efficient solutions? Alternatively, are you aware of any
> open-source software packages that can handle such cases out-of-the-box?
>
>
No, and I don't think PETSc can do this out-of-the-box, but others may be
able to give you a better idea of what PETSc can do.
I think PETSc can do Uzawa or other similar algorithms but it will not do
the regularization automatically (it is a bit more complicated than just
A01 * A10)

Thanks,
Mark

>
> Thank you very much for taking the time to read my email. Looking forward
> to hearing from you.
>
>
>
> Sincerely,
>
> Xiaofeng He
> -----------------------------------------------------
>
> Research Engineer
>
> Internet Based Engineering, Beijing, China
>
>
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