Re: PDE of vector variable

Mon, 10 Dec 2012 15:06:40 -0800

Ah, I see this is more difficult than I realized. Luckily this part of the problem is not a high priority for me, I was mainly interested in the temperature calculation, but I was curious how to deal with vector equations like this one. Hopefully I won't run into anything else like this.

Thanks,
Adam


On 12/10/2012 05:22 PM, Daniel Wheeler wrote:


On Sat, Dec 8, 2012 at 4:26 PM, Adam Stone <[email protected]> wrote:
Sorry to be asking so many questions, but I've hit yet another hurdle. This is sort of a continuation of my laser heating problem. Thanks to the mailing list I was able to obtain the correct temperature distribution, but the next part of the problem is to use that temperature distribution to calculate a thermo-elastic expansion given by



Adam,

Good that you are making some progress. There is a problem here. FiPy doesn't support creating equations that are both anisotropic and coupled (or vectorized). To solve the above you need a representation of an equation as follows,

   $ \partial_t q^i = \partial_k \left[ \Gamma^{kli}_j \partial_l \q^j \right] $

where $ \Gamma^{kli}_j$ is a diffusion tensor and $i$ and $j$ run over the number of equations and $k$ and $l$ run over the number of dimensions. Currently, in FiPy you can have either

   $ \partial_t q^i = \partial_k \left[ \Gamma^i_j \partial_k \q^j \right] $

 or

   $ \partial_t q = \partial_k \left[ \Gamma^{kl} \partial_l \q \right] $

but not both. I can't see a way to make it work at present other than just using an explicit representation.

On an alternative branch, I have a working code for coupled hyperbolic equations of the form

   $ \partial_t q^i = A^{ki}_j \partial_k q^j $

which is similar to what you want. I am sure it is straight forward getting the $\Gamma^{kli}_j$ tensor notation working given that we have the sub parts working. I'll file a ticket and hopefully do something on this for the next release.

Sorry about that.


--
Daniel Wheeler


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