Sean,
Thanks for the wonderfully clear answer. For my function f, I can
basically treat the line integral like a "normal" volume integral,
setting the direction using the delta functions. This would be the right
approach for a distributed adaptive setting as well, no?
Yes. You'd write the integral over the entire volume as a sum over
integrals over the locally owned set of cells -- each processor computes
on term in this sum, and you then just sum up a number over all processors.
It makes no
assumptions about cell layout and is defined at the functional level.
Correct.
If I may ask a follow up with an added wrinkle - it concerns the
interplay of material_id (cells) and functions:
What if I don't have the form of f?
In my case, I have been using the material_id marker at the cell level
to indicate different materials, that in the assembly loops, then
provide different coefficient values to the local sums.
Suppose that the distribution of these cell identities(material_id) is
updated based on the results of some FE field calculations. Then given
the configuration of the cells at that time step, I would like to do the
line integral (essentially compute the weight of the material above the
point), and go on to another FE solve. However, I do not have a
functional form to put in my line integral.
I don't think I quite understand. The function f(x,y,z) in your case
will have the form
if (some_condition) // depends on the material_id.
return something;
else
return something_else;
I don't know how to describe the weight function globally, but locally
(at the cell level) it is defined.
That's all you need. You compute these integrals as sums over all
(locally owned) cells, where you use quadrature. So at each quadrature
point, you're going to evaluate what value f(x,y,z) has. But for this
evaluation, you know what cell you're on, etc.
By way of an example, take a look at this postprocessor that computes
the root-mean-square of the velocity over the entire domain in a flow
solver (the only really important parts are lines 56-76):
https://github.com/geodynamics/aspect/blob/master/source/postprocess/velocity_statistics.cc
"The material of a cell may be used during matrix generation in order to
implement different coefficients in different parts of the domain. It is
not used by functions of the grid and dof handling libraries." [1]
Now see also this: https://github.com/dealii/dealii/pull/4492/files
Generally, I am hoping to better understand the relationship of the
mesh( things like looping over cells, querying for the local properties,
...) and the Function object class which returns values based on the
spatial position.
There really isn't very much of one. Function objects are functions that
are dependent on x,y,z. Querying local properties would correspond to
functions that depend on K (the set of cells).
Best
W.
--
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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