[petsc-users] solving stokes-like equation in 3D staggered grid in irregular domain with petscsection

[Bishesh Khanal]

On Wed, Nov 13, 2013 at 10:06 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> Bishesh Khanal <bisheshkh at gmail.com> writes:
>
> > Within A, for now, I can consider mu to be constant, although later if
> > possible it can be a variable even a tensor to describe anisotropy. But
> to
> > start with I want put this a constant.
> > The original equations start with mu (grad(u) + grad(u)^T) but then
> > simplifications occur due to div(u) = f2
>
> Rework that step in case of variable mu.
>

> > I'm mostly interested in the phenomenon in A with my model, here B is the
> > extension of the very irregular domain of A to get a cuboid. Here, in B I
> > release the div(u) = f2 constraint and just put a regularisation to
> > penalize large deformation. What is of importance here is to compensate
> the
> > net volume expansion in domain A by corresponding contraction in domain B
> > so that the boundaries of the cuboid do not move. It does not
> particularly
> > represent any physics except probably that it gives me a velocity field
> > having a certain divergence field that penalizes big deformations.
>
> Okay, sounds like it's already an artificial equation, so you should be
> able to leave in a normal equation for p, with a big mass matrix on the
> diagonal,
>
> div(mu(grad(u))) - grad(p) = f1
> div(u)           - c(x) p  = f2
>
> c(x) = 0 in domain A and c(x) is large (the inverse of the second Lamé
> parameter) in domain B.
>
> Thanks, this looks quite reasonable. I'll try to experiment with it.


> > I do not know much about FEM. But some of the reasons why I have avoided
> it
> > in this particular problem are:  (Please correct me on any of the
> following
> > points if they are wrong)
> > 1. The inputs f1 and f2 are 3D images (in average of size 200^3) that
> come
> > from other image processing pipeline; it's important that I constrain u
> at
> > each voxel for div(u) = f2 in domain A. I am trying to avoid having to
> get
> > the meshing from the 3D image(with very detailed structures), then go
> back
> > to the image from the obtained u again because I have to use the
> obtained u
> > to warp the image, transport other parameters again with u in the image
> > space and again obtain new f1 and f2 images. Then iterate this few times.
>
> Okay, there's nothing wrong with that.
>
Thanks for the confirmation.
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