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

[Mark Adams]

On Thu, Jun 12, 2025 at 8:44 AM Matthew Knepley <knepley at gmail.com> wrote:

> 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.
>
>
Good point.
You can also use -pc_gamg_parallel_coarse_grid_solver to get GAMG to use a
on level iterative solver for the coarse grid.


>   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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