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

[Matthew Knepley]

On Thu, Jun 12, 2025 at 4:58 AM Mark Adams <mfadams at lbl.gov> wrote:

> 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)
>

One other trick you can use is to have

  -fieldsplit_0_mg_coarse_pc_type svd

This will use SVD on the coarse grid of GAMG, which can handle the null
space in A00 as long as the prolongation does not put it back in. I have
used this for the Laplacian with Neumann conditions and for freely floating
elastic problems.

  Thanks,

     Matt


> 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
>>
>>

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

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