# Stability of radiative heat transfer

Hi everyone. I have here a problem understanding the stability of heat transfer through radiation. I'll give you some background, and later on I'll describe the physical problem.
Background
I am simulating the unsteady radiative heat transfer between mutually visible surfaces of objects through FEM. I am using the method of panels, which considers constant temperature on each panel.
Depending on the material properties, an instability issue might arise while integrating in time. When this happens, surfaces begin to transfer heat from one region to another, with an increasing amplitude.
Problem
The mechanism of instability is as follows: one surface becomes slightly warmer than the other one, and as consequence, it emits more power; so in the next time step, it tends to become colder. The other surface, being slightly colder at the beginning, gets the energy and then becomes warmer. The cycle cannot be stopped unless there is some sort of damping.
Question
As far as I can think, this instability looks quite reasonable. It does happen here due to the discrete numerical integration; but does anybody knows why this doesn't happen in reality? Or it does happen?

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## Answers and Replies

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BvU
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Hi,
Your oscillations are well known in the simulation world, especially when the solution phase is block-oriented and the process has a strong feedback component. In reality things go simultaneously, so the effect you describe doesn't occur -- unless there are delays (that happens e.g. in control engineering).

But you already know what to do: dampen the effect until the feedback factor is small enough. But it slows down the calculations, unfortunately.

As far as I can think, this instability looks quite reasonable. It does happen here due to the discrete numerical integration; but does anybody knows why this doesn't happen in reality? Or it does happen?
I think you should just make the time step smaller, so the difference between the curent temperature and the equilibrium temperature isn't entirely corrected in a single timestep.I don't think this could happen in nature, since there are no discrete timesteps. It's also forbidden by the second law, unless there's an energy input.

Thanks for your replay!

While I do know where the problem is, I don't really see how I can damp this effect without introducing artificial things that might pollute the solution.
Reducing the time step is something I don't really want to do, because I have to keep an eye in performance.
I am using an implicit method (Euler backwards) for the time integration and I do have here a Newton iteration seeking for convergence of equilibrium at each time step. I have introduced there the derivative of the radiative heat with temperature hoping that that would help, but it didn't.
I think I should introduce a damper, something that holds the radiation when the variation of the temperature is too high.
Any idea?

Hi,
Your oscillations are well known in the simulation world, especially when the solution phase is block-oriented and the process has a strong feedback component. In reality things go simultaneously, so the effect you describe doesn't occur -- unless there are delays (that happens e.g. in control engineering).

But you already know what to do: dampen the effect until the feedback factor is small enough. But it slows down the calculations, unfortunately.
Any idea about how implementing an effective damper?

BvU
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2019 Award
One way would be to simultaneously solve, but that's a big change in approach.
The problems you experience remind me of the countercurrent chain of heat exchangers that caused problems for block-oriented solvers. (3.3. here ; check 4.2.3 too).

Bottom line: don't have a good and simple easy way out, sorry.

Basically you use successive substitution to solve a T4 problem so it diverges

One way would be to simultaneously solve, but that's a big change in approach.
The problems you experience remind me of the countercurrent chain of heat exchangers that caused problems for block-oriented solvers. (3.3. here ; check 4.2.3 too).

Bottom line: don't have a good and simple easy way out, sorry.

Basically you use successive substitution to solve a T4 problem so it diverges
Yest, it is a divergence problem. But it does not happen always. It commonly appears when one of the material properties (C or K) has an important discontinuity.
What do you mean with "block-oriented solver"?
I have also linearized all T4 terms, so that should not be an issue. Or do?

BvU
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block-oriented solvers use successive substitution. The term comes from chemical process simulation, which is what the article is about.

block-oriented solvers use successive substitution. The term comes from chemical process simulation, which is what the article is about.
Ok thanks :) I'll find more info about that.

After thinking about this and performing some tests, I am concluding that the source of this numerical instability is the discreteness of the FEM method.
I have now tried to stabilize the algorithm by taking the mean value between the last and former radiative heat vectors, and as far as I have tested it, it works.
I first calculate the starting radiation at the beginning of a time step, and for every Newton iteration, I take a new value that includes information from the previous iteration step. This new value is just the mean between the last and previous vectors.
I came to this solution because I realized that what should be somehow stopped, was the pulsation in the local exchange: when two regions begin a "coupled feedback exchange". So my hypothesis was that if you take the mean value, this pulsation is damped. At least if you think in the particular case when only the sense of the local exchanged heat is reversed between some regions, the reversion is then neutralized in one step.
At the end of the Newton iteration, the equilibrium state should be valid if the difference between the last and previous heat vectors is small, so that could be used as an extra convergence criteria.
I am not basing this in any theoretical analysis, and I have not analyzed the stability of the new algorithm, but it is a medicine that seems to work very well.