If we are using a very coarse value of time step, it will not recover a very good numerical solution for problem in hand. One parameter that I can think about is the heat diffusion coefficient ''LAMBDA''. If LAMBDA is high it mean the heat will dissipate very quickly from one part to the other and we need a small value of time step to capture the result. If our selected valued of time step is very coarse it will not be able to capture the solution correctly. Also I can think off the boundary conditions that will affect the transfer of heat. Are there other parameter need to be considered and is there any relation from which we can select the minimum value of time step for an FEM solution with specified mesh density.
The relation between mesh density and time step is Courant/CFL number. For diffusion problem you need to have a low number, typically much less than 1.
I suggest to play around with some CFL numbers for your given problem, and see whether the solution converge at that numbers. In practice, a solution can be considered converged when the solution doesn't change much if you further reduce the CFL)
As such there is no rule for this. You have to do mesh sensitivity / convergence analysis for a particular time stepping or vice-versa..
Hi Firdaus and Pankaj, thanks for the reply. As suggested I am doing the same convergence analysis, but I am working on 3D problems and it takes ages for an analysis to complete and converge.
Hi Deepak
Thanks for reply. I have downloaded the mentioned paper and checked the Equation (6) i.e. Δt ≥ ΔI /6D
The relation impose a minimum value of Δt to be used, which means that Δt should be greater than or equal to this value. It will not work if Δt is chosen less than that suggested by this relation. In contrast for my case I need to choose the max value of Δt that I can use to get an accurate solution. It will not work if I use greater value of Δt. ? Did u get my point?
Also in the paragraph just before relation (6), it is mentioned that "A convergence study is carried out to determine the optimum mesh size and time step". This is somehow confusing for me. If we have a relation to select the optimum value of Δt then whats the need to do the convergence study to select the optimum time step????
Waiting for any guidance from your side.
All my experience with transient heat transfer analysis is with using the NX NASTRAN solver with plate elements. So hopefully this is applicable.
In the NX NASTRAN Thermal Analysis User's Guide they give an equation to give guidance on selecting a time step size. It is delta t = (1/n)*((delta x)^2)*((rho*cp)/k) "where n is the modification number of the time scale, delta x is the mesh size (smallest element dimension), rho is the material density, cp is the specific heat, and k is the thermal conductivity. A suggest value of n is 10. For highly nonlinear problems, a small step size is recommended".
It seems to have worked well for my model.
I would prefer the equation based on the thermal difusibility which is conductivity/(rho x cp) since the higher this parameter the "faster" heat penetrates (or leaves) the body. Only conductivity is not enough.
Hi tyler,
thanks for the help. In my case I am solving the general trainsient heat transfer problem defined by equation
∂u/∂t+λ∆u=f
where u is our unknown temperature and f is the heat source/sink.
We define initial condition
u=u_0
and Robin type boundary condition
∂u/∂n+hu=g
In my case I am not interested in any specific material, I am using general value of λ=0.1.
Now Considering the equation you suggest for selecting the delta t i.e.
delta t = (1/n)*((delta x)^2)*((rho*cp)/k)
For my case can i take a general value of ((rho*cp)/k)=λ=0.1 ?
For my case delta x=0.06 and taking n=10 as suggested , the delta t comes out to be
delta t=0.00004444
is that right?
Waiting for you guidance
Hi Tyler Lemonds
Please see if you have time and can guide me regarding selection of Dt for my case.
Also Nick hamburger if you can add some useful info.
thanks in advance
Regards
In your equation you NEGLECT an aspect of transient heat transfer: the fact that the stuff increases its temperature and so absorb part of the heat flow. You consider only the conduction and this is not enough for a transient it is OK ONLY for steady stade transfer. This was the reason I mentioned the difusibility which being related to the specific heat, specific mass and conduction is more representative for transient analysis.
Hi Nick
Thanks a lot for your help and guidance. Considering the value of 'rho' and 'cp', we will get a certain value for the term conductivity/(rho x cp). Now suppose we have a value for this term conductivity/(rho x cp), how can we then select the value of DT?
Regards
Iqbal
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