Calculating an integral over the boundary

[Kyle Schwiebert]

I'm trying to replicate the simulation described in this paper. How could I compute the integrals in equations 4 and 5? I'm sure it's covered in the tutorial, but I couldn't find it after searching in several different steps in the tutorial. Could someone please point me to an example in the tutorials where this type of computation with a computed solution? The only examples I could find seem to only discuss computing L2, or H1 errors with a special function that avoid manually computing the integral. Is the easiest thing to do as is done in the matrix assembly?

Finally, how would one initialize the vectors v_l and v_d? Could it be done easily with an appropriate AffineConstraints object?

Thanks in advance to anyone who can help.

[Wolfgang Bangerth]

On 9/27/21 1:43 PM, Kyle Schwiebert wrote:
> I'm trying to replicate the simulation described in this paper

> <https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwias-berlin.de%2Fpeople%2Fjohn%2FELECTRONIC_PAPERS%2FJoh04.IJNMF.pdf&data=04%7C01%7CWolfgang.Bangerth%40colostate.edu%7Ce97881b432a742a2b53008d981ef1a37%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C637683686653742835%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=dDqBEMcMDHgOSMr4G2McKiWUmifKo5sSZpc3ReFfUQ8%3D&reserved=0>.

> How could I compute the integrals in equations 4 and 5? I'm sure it's
> covered in the tutorial, but I couldn't find it after searching in
> several different steps in the tutorial. Could someone please point me
> to an example in the tutorials where this type of computation with a
> computed solution? The only examples I could find seem to only discuss
> computing L2, or H1 errors with a special function that avoid manually
> computing the integral. Is the easiest thing to do as is done in the
> matrix assembly?

Kyle,
in essence you just have to loop over all cells and integrate up the
quantity you have in the formula for a given choice of v_l and v_d. This
works in a similar way to this example here where we compute the angular
momentum:

https://github.com/geodynamics/aspect/blob/master/source/simulator/nullspace.cc#L357-L459

> Finally, how would one initialize the vectors v_l and v_d? Could it be
> done easily with an appropriate AffineConstraints object?

As long as you satisfy the boundary conditions, it's your choice. So you
could, for example, start with a zero vector, evaluate boundary values
as indicated on the boundary in question, and then just go through the
constraints you get as a result and apply them to your zero vector (via
AffineConstraints::distribute(), applied to the zero vector). That would
probably be what I'd do.

Or you just take v_l/v_d as functions that are analytically described
(i.e., that are not finite element functions to begin with).

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Kyle Schwiebert]

Thank you for the help. I think that has the pretty much sorted out for me. One additional question: On this test problem people also like to know the pressure drop across the obstacle. This basically involves calculating the pressure at two points in the mesh and finding their difference. I see in step 13 this can be done by hoping that the points you need to evaluate are vertices of the mesh. Is there a more general solution--or a way to ensure a certain point is in the mesh? FYI I'm using the very helpful builtin meshing scheme for this problem: GridGenerator::channel_with_cylinder()

Thanks,

Kyle

[Daniel Arndt]

Kyle,

Have a look at VectorTools::point_value(https://www.dealii.org/current/doxygen/deal.II/namespaceVectorTools.html#a7be5c7eed52308898dfaad91c4cff204).

Best,

Daniel

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