Please if anybody can help me by looking at the information below and tell me if I am on the right track;
Total simulation time = the number of time steps* time step size
for example:
time step size= 5e-6 sec ;
number of time steps= 1000 ;
Then: ∆t=Total simulation time= 5e-3 sec.
>>The Nyquist criterion states that a periodic wave can be correctly reconstructed when the sampling frequency is greater than double the highest frequency component in the phenomenon. >>
Considering the maximum desired frequency for acoustic simulation to be 2000 Hz, we would have:
the minimum wavelength would be λ_min= c⁄f_max =343⁄2000=0.1715 m and accordingly, for good simulation, we need at least 10 elements for the minimum wavelengths, so, the maximum element size would be:
∆x=Minimum element lenght=λ_min⁄10= 0.01715 m or 17.15 mm
Now considering CFL condition for convergence:
C=∆t∑u/∆x ≤1 or ∆t≤16.6 μsec
So, if we set the maximum mesh size equal to 10 mm and the time sep size equal to 5 μsec, then we should not worry about the simulation.
But the questions here are:
1- What are the relationship between Nyquist criteria (in signal processing) and appropriate total simulation time for acoustic numerical simulation?
2- Can we say that the lowest valid frequency after the solution would be two times of 1/∆t? which in this example would be 2*(1/5e-3)= 400 Hz.
So, the simulation would be valid for the frequency range of 400 to 2000 Hz. is this conclusion correct?
I'll try to give you a brief resume
- The total time for the simulation T=n*dt must cover the largest wavelenght (smallest frequency) of your problem, that is should be at least equal to the fundamental period of time of your acoustic problem.
- The time step dt must be so small to resolve the highest frequency fmax of your physical problem, that is dt=O(1/fmax).
- The Nyquist wavenumber is by definition kc=T/(2*dt) =n/2. Since the frequency is f=k*2*pi/T, you get the Nyquist frequency fc= pi/dt.
In conclusion, you need to have an estimation of the highest possible frequency of your problem.
Thank you professor Denaro,
your response was really helpful,
So, if the highest and lowest frequency components were 2000 Hz and 20 Hz respectively, we could say that dt= pi/2000=0.00157 sec, and total time would be 2*(1/20)=0.1 sec.
Is this correct?
Dear Hossein Ahmadian
the lowest frequency corresponding the the wavenumber k=1 leads to estimate the total time for the period T according to
f=2*pi*k/T evaluated for k=1 -> 20=2*pi/T -> T=pi/10=0.314 s
Be careful that this is a physical period of time after that the solution has correlated. If you have some numerical transient from initial arbitrary condition you need to wait some time (the solution must forget the initial condition) before starting to compute T.
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