[petsc-users] FEM Implementation of NS with SUPG Stabilization

[Brandon Denton]

By natural coordinates, I am referring to the reference element coordinates. Usually these are represented as (xi, eta, zeta) in the literature.

Yes. I would like to have the Jacobian and the derivatives of the map available within PetscDSSetResidual() f0 and f1 functions.  I believe DMPlexComputeCellGeometryFEM() function provides this information. Is there a way to get the cell, shape functions as well? It not, can we talk about this more? I would like to understand how the shape functions are addressed within PETSc. Dr. Kirk's approach uses the shape function gradients in its SUPG parameter. I'd love to talk with you about this is more detail.






________________________________
From: Matthew Knepley <knepley at gmail.com>
Sent: Wednesday, October 11, 2023 3:13 PM
To: Brandon Denton <bldenton at buffalo.edu>
Cc: Jed Brown <jed at jedbrown.org>; petsc-users <petsc-users at mcs.anl.gov>
Subject: Re: [petsc-users] FEM Implementation of NS with SUPG Stabilization

On Wed, Oct 11, 2023 at 2:09 PM Brandon Denton <bldenton at buffalo.edu<mailto:bldenton at buffalo.edu>> wrote:
Thank you for the discussion.

Are we agreed then that the derivatives of the natural coordinates are required for the described approach? If so, is this something PETSc can currently do within the point-wise residual functions?

I am not sure what natural coordinates are. Do we just mean the Jacobian, derivatives of the map between reference and real coordinates? If so, yes the Jacobian is available. Right now I do not pass it
directly, but passing it is easy.

  Thanks,

     Matt

Matt - Thank you for the command line option for the 2nd derivatives. Those will be needed to implement the discussed approach. Specifically in the stabilization and shock capture parameters. (Ref.: B. Kirk's Thesis). What is a good reference for the usual SUPG method you are referencing? I've been looking through my textbooks but haven't found a good reference.

Jed - Thank you for the link. I will review the information on it.

Sorry about the attachment. I will upload it to this thread later (I'm at work right now and I can't do it from here).
________________________________
From: Jed Brown <jed at jedbrown.org<mailto:jed at jedbrown.org>>
Sent: Wednesday, October 11, 2023 1:38 PM
To: Matthew Knepley <knepley at gmail.com<mailto:knepley at gmail.com>>
Cc: Brandon Denton <bldenton at buffalo.edu<mailto:bldenton at buffalo.edu>>; petsc-users <petsc-users at mcs.anl.gov<mailto:petsc-users at mcs.anl.gov>>
Subject: Re: [petsc-users] FEM Implementation of NS with SUPG Stabilization

Matthew Knepley <knepley at gmail.com<mailto:knepley at gmail.com>> writes:

> On Wed, Oct 11, 2023 at 1:03 PM Jed Brown <jed at jedbrown.org<mailto:jed at jedbrown.org>> wrote:
>
>> I don't see an attachment, but his thesis used conservative variables and
>> defined an effective length scale in a way that seemed to assume constant
>> shape function gradients. I'm not aware of systematic literature comparing
>> the covariant and contravariant length measures on anisotropic meshes, but
>> I believe most people working in the Shakib/Hughes approach use the
>> covariant measure. Our docs have a brief discussion of this choice.
>>
>> https://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Flibceed.org%2Fen%2Flatest%2Fexamples%2Ffluids%2F%23equation-eq-peclet&data=05%7C01%7Cbldenton%40buffalo.edu%7Cd9372f934b26455371a708dbca80dc8e%7C96464a8af8ed40b199e25f6b50a20250%7C0%7C0%7C638326427028053956%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=skMsKDmpBxiaXtBSqhsyckvVpTOkGqDsNJIYo22Ywps%3D&reserved=0<https://libceed.org/en/latest/examples/fluids/#equation-eq-peclet>
>>
>> Matt, I don't understand how the second derivative comes into play as a
>> length measure on anistropic meshes -- the second derivatives can be
>> uniformly zero and yet you still need a length measure.
>>
>
> I was talking about the usual SUPG where we just penalize the true residual.

I think you're focused on computing the strong diffusive flux (which can be done using second derivatives or by a projection; the latter produces somewhat better results). But you still need a length scale and that's most naturally computed using the derivative of reference coordinates with respect to physical (or equivalently, the associated metric tensor).


--
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

https://www.cse.buffalo.edu/~knepley/<http://www.cse.buffalo.edu/~knepley/>
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