How to solve a one dimensional diffusion problem in each node?

[pengfejeac]

Hi there!

I'm working on a three dimensional problem, where it requires solving a one dimensional diffusion problem in each node. I checked some of the previous discussion about nodal kernel and ODE kernel, but none of those seems to be applicable to my case. On the other hand, I tried to customize a nodal kernel for my case, but nodal kernel doesn't have "_grad_phi" and "_phi", I have no idea on how to construct a diffusion problem without "_phi", so I'm stuck.

Any help will be much appreciated!

[GiudGiud]

Hello

Which finite element family are you planning to use for the main variable? And is there only one variable?

Nodal kernels being at nodes, a lot of the families don't have a continuous gradients there and would not work for that purpose.

Do you have to solve at nodes? Or could you solve three 1D diffusion problem volumetrically instead?

Guillaume

[pengfejeac]

Wow, that sounds really advanced. Let me clear things up a little bit.

We can also have transfers between multiapps. This would however mean you need to create multiple multiapps and not just one multiapp with many child app.

So what you are suggesting is create a "branch" like structure with one Main app, a couple of intermediate app and a ton of sub app, is that correct? And that could be done by grouping?

Communicate with neighbors sounds like finite difference, but that should work if I can implement this branch.

[GiudGiud]

You're right it might be too complicated. I dont think anyone has done this before.

When you do these 1D solves, are you able to code up the entire solve in an object? If so you can use NodalKernels to perform the 1D solves on every node.
NodalKernels can even contribute to the residual and Jacobian of the main solve.

[pengfejeac]

Sorry about the late reply.

I'm capable of coding diffusion with certain boundary conditions and calculate entries in Jacobian matrix explicitly. But I'm having trouble understanding the difference between Nodal Kernel, Kernel and Array Kernels in terms of "JxW" term. Will that value be different if I apply Nodal kernel instead of Array kernel? Is there a way that I can avoid that?

As far as I understand, JxW means Jacobian times weight of Gauss quadrature rules, but I can't see the meaning of it if we are interpreting a node.

[GiudGiud]

A nodal kernel is supposed to be a local equation only active on that node. The variable is often very local, with completely independent solves on each node (but you can still pull in information from other nodes if you want)

An array kernel is for a field variable that extends over the entire mesh (or a few subdomains) but also has multiple components. For exmaple, the neutron flux with several energy groups is often an array variable. Or an array of multiple species being advected in a fluid.

[pengfejeac]

This is probably the last update. So I managed to code diffusion on each node without using any sub-app (Basically hand-coded residual and Jacobian entries as you suggested) and it takes a huge amount of time to debug. But it eventually worked. I stick with array variable instead of nodal variable, but I guess nodal variable would also work.

Thanks for your help! :D