Finite Difference Method + Radiating Boundary = Screwed?

Akilae729

I'm working on a small MATLAB heat transfer code that is going to use the finite difference method to solve for the temperature distribution in a solid part made of multiple materials.

For everything that only "sees" conduction, this is super easy. Set up the nodes, set up a GIANT matrix with the algebraic equations, keep track of local material properties, and solve. Done. Easy.

The problem here is that there is a radiation boundary condition on the outer surface. Since this is a function of T^4, I can't solve this with linear algebra.

Is there a clever way to handle a radiation boundary condition????

I could always randomly assign temperatures to the outer nodes and let MATLAB's solver (fminsearch probably) mess with them until I get an appropriate energy balance. But I'm not terribly confident in MATLAB's ability to do this with a large number of nodes.

Kipling

It's steady state? The PDE toolbox I believe can handle steady state. It shouldn't be able to handle it in a time dependent solving method. It's going to take an insane amount of time if you guess wrong though.

The MATLAB forums would probably help a bit more - they are pretty good guys over there.

Akilae729

It's all steady state. Fortunately, I only have a couple hundred nodes, so MATLAB ok for solving. Basically, I've written the conduction only nodes into a linear algebra solver and I use the MATLAB solver to grab the remaining nodes.

After talking to some professors, it looks like there isn't a clever way to fix the boundary condition issue.

I've had a mixed bag with the MATLAB forums. And there are a few engineers floating around here that typically have good ideas.

Fuzzy Cumulonimbus Cloud

Can you express it as an ODE and use one of the hojillion solvers available for ODEs?