Helmholtz solver and eigen values

Numerical methods and mathematical models of Elmer
supreet
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Helmholtz solver and eigen values

Post by supreet »

The wave number (k) in the Helmholtz equation is the eigen value of the equation. Thus, the boundary conditions and the eigen values are not independent. For example, in an open-open ideal pipe resonator, imposition of zero pressure fluctuations at the open ends (BCs) results in wave numbers equal to integral multiples of pi. An open-closed resonator has different wave numbers.

The Helmholtz solver asks for both angular frequency and the boundary conditions. Why is that? Before I checked out the solver tutorial, I was hoping I could just input the BCs and the solver would output the first few frequencies (eigen value). What am I missing?
kevinarden
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Re: Helmholtz solver and eigen values

Post by kevinarden »

Are you trying to do something like this
https://github.com/ElmerCSC/elmerfem/tr ... zStructure
supreet
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Re: Helmholtz solver and eigen values

Post by supreet »

I've a annular cylindrical combustor for which I've to compute resonant modes and their frequencies, which can be accomplished by the Helmholtz equation (HEq).

My question is rather about the solver. If the Helmholtz solver is indeed solving the HEq, why is it that we are required to mention frequency too. The BCs will determine the mode shape and frequency.
raback
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Re: Helmholtz solver and eigen values

Post by raback »

Hi

HelmholtzSolver has been written as the harmonic version of the solver. Hence you need to give frequency. This solver cannot be used to compute eigenmodes.

There is also a WaveSolver that can be solved in transient, harmonic or eigenmode mode. See test case "WaveEquEigen". Basically this solver creates global matrices for 0th, 1st and 2nd order in time, and based on the users choice treats them as transient, harmonic or eigenmode. This is a more generic approach but the specialized harmonic version was created 1st.

-Peter
supreet
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Re: Helmholtz solver and eigen values

Post by supreet »

Hi Peter,

Thanks for the information.

But can you please correct my understanding of what the Helmholtz solver is doing. It's solving the equation d^2(f) + k^2 f = 0 (where d = nabla and k is related to the angular frequency). The discretized form of the equation would be A F + k^2 F = 0, where A is the discretization matrix and F the solution vector. Matrix A also encompasses information from the boundaries. Therefore, values of k for which the equation is satisfied must be related to the eigenvalues of the matrix A, which is dependent on BCs. Why then, should we provide both BCs and the angular frequency? Given the BCs, only certain solutions are possible associated with certain eigenvalues (k).

I am certain there is a lack of understanding on my part. It'd be very helpful if someone corrects me.

Thanks.
kevinarden
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Re: Helmholtz solver and eigen values

Post by kevinarden »

supreet
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Re: Helmholtz solver and eigen values

Post by supreet »

Hi Kevin,

I did check out the manual before, but it wasn't helpful.

Do you see any fault in my understanding as I've stated in the post above?
kevinarden
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Re: Helmholtz solver and eigen values

Post by kevinarden »

The wave solver Chapter 14 of the manual solves the equation the way you want, it is still the Helmholtz equation. It is the general version of the Helmholtz equation.
Chapter 12 titled Helmholtz Solver is a specific form of the equation using harmonic solution so it needs a frequency to solve in the harmonic domain.
An example of chapter 14 general Helmholtz is here
https://github.com/ElmerCSC/elmerfem/tr ... veEquEigen
supreet
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Re: Helmholtz solver and eigen values

Post by supreet »

Thank you. I'll follow the tutorial you and Peter have specified.

But that still doesn't clear my doubts as to what the Helmholtz solver is doing. If there are faults in my understanding, of which I am certain there are, I'd be pleased if anyone points them out and corrects me.
raback
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Re: Helmholtz solver and eigen values

Post by raback »

Hi

The solution of the linear system for (d^2 + k^2)f with only zero BCs has trivial solution f=0. You know that you have some frequency, and want to wake it up you need to provide some source (which is in the very same frequency).

If you don't know your system frequency then you're solving a different problem - an eigenvalue problem - which uses totally different machinery where you need to find not only the eigenmodes f_i but the eigenvalues k^2_i. That problem does not need any BCs because all cavities have resonant frequencies.

-Peter
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