# A Finite difference formulation ideas (journal verification)

1. May 15, 2017

### maistral

Hi. I am trying to simulate this paper since apparently I have a lot of time.

Scrolling down to the last page, he simulated a transient 2D heat conduction plate with composite slabs on it. Darkest one is copper, lighter one is steel, lightest one is glass.

If you look closely, the authors said they maintained two sides at 100OC and the other remaining sides at 50OC. Note that the plate was initially at 0OC.

My question is, judging the diagrams they made - how did they do this? My hypothesis is that the boundary is varying with time, yes? So does that mean that the boundary conditions are functions with respect to time? Or am I missing something here; that my finite differencing is wrong? Currently my finite difference is setting the boundaries at 100 and 50, so at any time t the boundaries are 100 and 50.

2. May 15, 2017

### scottdave

I have seen situations like this described. If I remember correctly, you model it as it is touching something which maintains the temperature constant over time (100 for two boundaries and 50 for the other two). The material itself starts to absorb heat from the touching surface, and starts to rise in temperature (which material near it absorbs heat and starts to rise, etc). Yes you will see the (copper?) material very near the boundary rise with time. Near the corners notice how it approaches a very sharp change in temperature (see how it looks as time goes toward infinity). Hopefully that helps you.

3. May 15, 2017

### maistral

Then I guess my hypothesis is correct. You model the boundaries as something that varies with time.

Newton's law of cooling on the boundaries as functions of time is it? Or are there other models that you can suggest? Thanks!

4. May 15, 2017

### phyzguy

No, I don't think so. The boundaries are at 100C on the front and back edges and 50C on the left and right edges, independent of time. As scottdave says, the material heats up with time and approaches the steady state condition shown in the bottom left. The plots aren't plotting the actual boundary points, but are starting at the first grid point inside the boundary - that is why the plots are changing with time.

5. May 15, 2017

### maistral

That was my initial assumption too. But I couldn't explain the final diagram where it did end up at 100 and 50

Unless they used a ridiculously small step, or they chose to not plot the boundaries in the first parts

6. May 15, 2017

### phyzguy

I'm not sure I understand you . I don't think they are plotting the actual boundaries in any of the plots. But the first grid point, which is very near the boundary, gets very close to the boundary temperature as time goes on. Does that make sense?

7. May 15, 2017

### maistral

Yes I do understand what you mean. But check their final diagram.

If they did not plot the boundaries how could it have ended up at 100 or 50? Shouldn't it be a bit lesser?

8. May 15, 2017

### phyzguy

It probably is a little bit less than 100 and greater than 50. How can you tell?

9. May 15, 2017

### maistral

... googly eyes. Lol.

Thanks for answering. I think my issue is resolved.