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 ado October 10, 2000 10:27

numerical scheme

Dear friends, i am trying to use numerical methods in order to determine thermal diffusivity and thermal conductivity of unknown material ranged from highly diffused to lower one. The method used is well known and called "parameter estimation" which use the comparison between experimental and numerical temperature-time histories of the material under investigation. I am using a fully implicit numerical scheme. I got very good result in steady state (no variation of temperature with time). This allows to get the thermal conductivity of any material. However, I found something strange. A difference exists between the numerical and the experimental curves in transient state. This difference become more and more important with the "decrease " of thermal diffusivity value. Because of this, it is not possible at the present stage to get a satisfactory results for thermal diffusivity. Personaly, I guess this is due to the numerical scheme i am using which is well known to be unconditionaly stable but not very accurate..But on the other hand, how can this happen only for lower diffuse materials and not for the highers like metal liquids??? waiting for your answers and advise. ado

 Sebastien Perron October 10, 2000 20:59

Re: numerical scheme

Have you tried a second order schemes such as Gear?

For this scheme, The approximation of the time derivative is

(3u^(n+1)-4u^n+u^(n-1))/2dt

for a constant time step dt. The other terms are evaluted at time t=t^(n+1)

This scheme is also unconditialy stable and might gives better results.

 Jim Park October 11, 2000 11:09

Re: numerical scheme

Your accuracy will likely depend on the non-dimensional time step, which is

delt(non) = delt(dim)*(alpha)/L^2,

where L is a physical dimension, alpha is the thermal diffusivity, and delt(dim) is the dimensional time step.

If you keep a constant dimensional time step and mesh, changing the material (diffusivity) changes the accuracy.

You should experiment with various (and likely much smaller) time steps until you have a better 'feel' for the relationships between your numerical parameters. This will require running a lot of cases and carefully analyzing the differences in the solutions.

Good luck!

 K.S.Ravichandran October 12, 2000 08:20

Re: numerical scheme

Ado,

1. I assume that your spatial discretization is 2nd order and time is 1st order. The steady state comparisons then will perhaps produce good results assuming that your linear model is a good approximation of the materails in question, and this is independent of the time step. To make sure that this is the case you must examine the modified equation actually approximated by the differential equation and verify that at steady state all delta_t terms vanish.

As Perron has said, a second order in time computation may be a guide to the role played by time step in the accuracy of the computation in the transient stage.

As for the error being more pronounced at lower diffusivity values, again the leading order truncation term must be examined to determine if for lower diffusivity, truncationerror magnitude can be held constant by a smaller mesh and time step.

I must again empahasise that the mathematiacl model is linear and is good enough to approximate material behavior.

Ravi

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