On 06/02/2017 07:37 AM, Sean McGovern wrote:
What's a good way to integrate a quantity in the vertical direction? For
example, how do I find the weight of water above a point (assuming variable
density, so that the integration is necessary)?
In a structured regular mesh, I would fix the x and y coordinates of the grid
and iterate over the z coordinate adding up the different weight contributions
per cell.
What would the way to emulate this in dealii be (in a globally refined
context)?
What you want to do (a line integral) is, in general, a difficult operation on
non-uniform meshes. As with all integrals, you will want to approximate it
with some kind of quadrature rule, which would require you to evaluate the
solution at quadrature points that are located *somewhere*, as far as the mesh
is concerned. You can do this evaluation with VectorTools::point_value(), and
this will work if you use a relatively small number of quadrature points
located along your vertical line. But the function is expensive if you need a
lot of quadrature points.
An alternative is to rewrite the line integral as a volume integral. For
example, your integral
\int_a^b f(x0,y0,z) dz
(where 'f' is a function of your finite element solution, and x0,y0 are the
coordinates of the vertical line you're trying to integrate along) is equal to
\int\int\int f(x,y,z) \delta(x-x0) \delta(y-y0) dz dy dz
If you replace the delta functions by an appropriate approximation (e.g., a
Gaussian with a width equal to several times the mesh size), then the integral
is approximated by the volume integral
\int\int\int f(x,y,z) K(x,y,z) dz dy dz
with some kernel K(x,y,z). This is now just any other ordinary volume integral
where K(x,y,z) is explicitly given. It is easy to integrate on a finite
element mesh.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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