On 06/04/2012 07:14 PM, steve wrote:
OK. So it seems that for the above, you could use directly dw_laplace and dw_volume_dot terms, with material arguments equal to 2\pi r and 2\pi k^2 r, respectively. Or just r, and put the known constants directly into the equations string.
Hmm.. OK. I'll try to wrap my head around that. ;-)
I will try to support you, don't worry.
Where \Omega is now the 2-D interior of the 'r-z' plane. \Gamma is reduced to a 1D boundary, but d\Gamma is going to get a factor of 2\pi r as well to take into account the 'distance' around in the theta direction.
Since the weak form only depends on the gradient, and '1/r' only shows up in the theta part of the gradient, it doesn't appear at this point. The only 'r' factor is in the volume element itself.
I thought (googled) that the 'r' part of the cylindrical Laplacian was (d
\partial):
\frac{1}{r} \frac{d}{dr} (k r \frac{du}{dr}), which is
k \frac{d^2 u}{dr^2} + \frac{k}{r} \frac{du}{dr}.
so in weak form it comes to a laplace-like term with different coefficients by the 'u' and 'z' parts, and a gradient-like term w.r.t. 'r' (the second one in the line above). Just curious...
So let's see if I can work this out. To convert to the weak formulation you'd start with the laplacian and integrate against an arbitrary function. If we just focus on the 'r' part it would go like so:
\int s \frac{1}{r} \frac{d}{dr} (k r \frac{du}{dr}) dV
but the r part of dV is r dr (forgetting about the 2\pi for now) so that's
\int s \frac{1}{r} \frac{d}{dr} (k r \frac{du}{dr}) r dr
which looks like:
k \int s \frac{d}{dr} (r \frac{du}{dr}) dr
But if that gets integrated by parts it's works out to be
k (s r \frac{du}{dr})\big\vert_{\Gamma} - k \int \frac{ds}{dr} \frac{du}{dr} r dr
Which is I think more or less what I had before. Have I got that right?
Probably :) (would have to compile the LaTeX) The difference is, that I thought about first making the d/dr derivative by parts, and then doing the weak form from the resulting two terms, while you did the opposite. Not sure which way is better...
r.