[Dune] gridcheck.cc : Integral over outer normals is not always zero.

[Martin Nolte]

Hi Aleksejs,

the divergence theorem only holds for volumes, not for surfaces. On a surface, 
the surface integral over all normals equals the integral over the mean 
curvature. And this is exactly what the message states: In case of a surface 
grid, the integral might be non-zero if the mean curvature of is nonzero.

If you encounter this message for a volume grid (i.e., a grid with dimgrid = 
dimworld), then something went wrong. In this case, the divergence theorem holds 
and, as you stated, the surface integral over all normals should be zero. Apart 
from a bug in your normal implementation, this might also result from an 
insufficient quadrature order. Would you consider filing a bug report in the 
latter case?

Best,

Martin

PS: I am merely the author of the warning, not of the test. Originally, the test 
simply failed for surface grids with element geometries of nonzero mean 
curvature (which is plain wrong).

On 03/18/2015 04:23 PM, Aleksejs Fomins wrote:
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> Dear Dune,
>
> When running the gridcheck.cc I notice I encounter the following warning:
>
> - -- Checking Intersection Iterator
> Warning: Integral over outer normals is not always zero.
>           This behaviour may be correct for entities with nonzero curvature.
> Warning: Integral over outer normals is not always zero.
>           This behaviour may be correct for entities with nonzero curvature.
>
> Could the person who wrote this test please explain what this means.
>
> I assume that this refers to the integral Int(vec{n} dS) over the surface of an element.
> If this is indeed the case, then this integral should be zero by divergence theorem.
> In particular, could you explain the case in which the integral would not be zero
>
> Greetings,
> Aleksejs
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-- 
Dr. Martin Nolte <nolte at mathematik.uni-freiburg.de>

Universität Freiburg                                   phone: +49-761-203-5630
Abteilung für angewandte Mathematik                    fax:   +49-761-203-5632
Hermann-Herder-Straße 10
79104 Freiburg, Germany