Finite Element Method

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2010/04/25
443 words
by Fredrik Olsen

One of the specializations I’m considering the most for my masters is “Scientific calculations and simulations”. It’s a mix between programming and physics which I like. It’s about creating a model of the physical problem then letting a computer solve it. Anyway, one of the courses I’m taking right now is a sort of primer for that specialization, and so far I like it, even though I haven’t done much yet. The course is on the finite element method, which is a specific method of letting computers solve problems.

I did the first exercise yesterday and the result looked kind of neat so I want to show you. The exercise was about calculating the temperature distribution in a rectangular disc with a hole in it. Inside the hole the temperature is 1000 degrees and outside the disc the temperature is 100 degrees, so the question is what does the temperature look like inside the disc? Another thing that you “automatically” calculate is the flow of heat from the inside to the outside of the disc.

This is what the disc looks like divided up into 336 element on which we’ll calculate the temperature.

Disc with 336 elements

And when I let the computer solve it, it’ll look like the following, which shows the temperature distribution.

Disc with 336 elements solved for temperature distribution

This is all pretty well and good. To give you an idea of what use this is, this thing could be a model of what the temperature would look like inside a large square metal bar with a pipe inside it, inside could flow something very warm while the entire thing would be dipped in a coolant. This solution lets us know what the temperature will look like inside the metal bar, as well as how much heat will flow to the outside. With this information we could learn how to optimize the cooling process. A very simplified model of course, but it’s just an introductory course. Now, being the engineer that I am, of course I could be happy with just 336 elements, I naturally have to over-do this thing. For no reason what so ever.

Disc with around 1500 elements

Disc with around 1500 elements solved

This last one took around 3-4 hours to solve with my computer :P I’m happy I don’t have to pay for computer time!

What I’m worried about in this field however is that as things get more complicated we need to use more sophisticated software, and I think there will be less and less focus on doing algorithms and more and more focus on just using commercial software. I don’t want to be a user, I want to be a creator.

2 comments

tetris

  • DeXimE

    Pretty cool & nice graphics!
    This is pure theory right? IRL you would need to know the thicknes of the disc? What if it's less thick on the outside than on the inside, it would be harder to calculate? Or would it still be the same? Probably the type of material will infuence the flow of the heat too, right? Just wondering :)

    • http://tetrisrockstar.com/ Fredrik

      In this case it is a very physically valid model for a cross-section of a beam (i.e. how the distribution would look like inside a metal beam that's “very” long, it wouldn't be accurate for the ends of the beam).

      If I were to model it for a flat disc then yes I would have to account for the thickness, and I would have to give boundary conditions for the big flat surfaces as well, and it wouldn't be very interesting if it was a thin disc (it would just be the surrounding temperature, heat would dissipate too fast for it to be interesting).

      Varying thickness wouldn't actually be that much more difficult to calculate, if you know a function for the thickness or have some sort of fairly accurate data for how it looks like, you would just need to put that into a computer and the software would “mesh” (divide up into elements) the object and calculate the temp. distribution automatically.

      That's what sort of bothers me, that when things get more advanced, things get more “automatically”. Unless of course you start to work for the people creating the software that does the things “automatically”.

      Oh, and material is accounted for in this model with material-specific constants found through testing. You can even have non-isotropic materials, i.e. materials where heat flows easier and one direction and such, that's very very simple to account for. The material in this example would be an isotropic material with relatively good heat-conductivity, like metal.