Discussion Forum

I assume that you mean that the imaginary parts are not only of the order of numerical noise.

Can you upload an example of this? Usually, the cause is that there are some contributions in the model that you would not immediately recognize as damping.

I just ran a case like this and the imaginary part is 8-9 orders of magnitude lower than the real part. Your setup is probably fine.

Thanks Henrik and René.

At first, I'm sorry for that I did not notice that the text "When using a search for eigenfrequencies, the nonsymmetric eigenvalue solver is always used, even when the problem to solve is symmetric" is written in the Reference Manual.

When using the nonsymmetric eigenvalue solver, is complex eigenvector always calculated regardless of the characterization of matrices and real eigenvalues (eigenfrequencies)?

I uploaded two simple model files. It was created on Japanese language environment but using English label in Studies and Results.

In most case, the imaginary part is always much smaller than the real part.

In the acoustic case, the eigenvector normalized by the maximum value should be ±1 real value at the corner location. Please confirm this.

In the solid case, please check on the imaginary part of the results computed by the nonsymmetric solver.

I actually want to solve the acoustic problem with damping, but I would like to understand a better settings for these problems because I may be terribly misunderstanding about basic problems.

Best regards.

The acoustics model shows large imaginary parts in one case: where the shift frequency is zero. You can trigger the same behavior in Solid Mechanics by disabling the Fixed Constraint.

What happens is that when there are too few Dirichlet conditions, you get eigenvalues that are almost zero (theoretically zero). For solid mechanics, that means rigid body motions and for acoustics, a constant arbitrary pressure. Such boundary conditions are not necessarily physically wrong. In particular for acoustics, having sound hard walls everywhere is reasonable. For structural mechanics, a rocket in space would be without constraints.

Such problems are ill-conditioned if you set Search for eigenvalues around to be zero. A sloppy explanation is that the solver at some point will work with the ‘dynamic stiffness matrix’ -omega^2*M + K, where omega is the given frequency, M is the mass matrix, and K is the stiffness matrix. With insufficient boundary conditions, K in itself is singular, and with omega = 0 there is no contribution from the mass matrix.

For such rank deficient problems, you should always use a Search for eigenvalues around frequency that is non-zero. The best is if it is close the first non-trivial eigenfrequency. Then the imaginary contamination will be negligible.

I'm not sure- but it's possible that when OP is talking about the eigenvector he means the values of displacement u,v,w that correspond to the eigenVALUE (or eigenfrequency).

The displacements u,v,w (and pressure) are phasors so of course they can be complex.

On the matter of normalization of eigenmodes- there are at least two different settings for this. I don't know if either of them result in a maximum eigenmode magnitude of exactly one.

Thanks you very much for your very valuable advice.

In the simple rectangular acoustic with sound hard wall case, I think that the rigid body mode its eigenfrequency is nearly 0Hz is physically important for the comparison with the theoretical solution and the direct solution at low frequency response. That's why I have tried Search for eigenvalues around zero.

But in general case, I could understand Search for eigenvalues around frequency should not be zero. Is it correct that each eigenmode possibly have a complex eigenfrequency if I set Search for eigenvalues around to be zero numerically?

And about eigenvectors I am most interested in, is it possible for a real symmetric matrix to have complex value regardless the settings about Search for eigenvalues? Study1 and Study3 for Acoustic Case and Study2 for Solid Case are examples. Is it difficult to obtain perfect real eigenvectors if an nonsymmetric eigenvalue solver is selected? I could get real eigenvectors and eigenfrequency if I solved these matrices calculated by COMSOL with MATLAB. I used LiveLink and eigs function it may be solving with real symmetric solver.

For normalization of eigenvectors, I tried to compare the amplitudes of simple acoustic theoretical real modes (such as cos(Ax)cos(By)cos(Cz)) since they are always 1 at the corners. However, they may not match since the complex amplitudes are normalized.

Best regards.

ill-conditioned if you set Search for eigenvalues around to be zero. A sloppy explanation is that the solver at some point will work with the ‘dynamic stiffness

I'm not sure- but it's possible that when OP is talking about the eigenvector he means the values of displacement u,v,w that correspond to the eigenVALUE (or eigenfrequency).

The displacements u,v,w (and pressure) are phasors so of course they can be complex.

On the matter of normalization of eigenmodes- there are at least two different settings for this. I don't know if either of them result in a maximum eigenmode magnitude of exactly one.

That is how I read it. The pressure phasor should be real here though, when there is no damping, and they almost are.

ill-conditioned if you set Search for eigenvalues around to be zero. A sloppy explanation is that the solver at some point will work with the ‘dynamic stiffness

I'm not sure- but it's possible that when OP is talking about the eigenvector he means the values of displacement u,v,w that correspond to the eigenVALUE (or eigenfrequency).

The displacements u,v,w (and pressure) are phasors so of course they can be complex.

On the matter of normalization of eigenmodes- there are at least two different settings for this. I don't know if either of them result in a maximum eigenmode magnitude of exactly one.

That is how I read it. The pressure phasor should be real here though, when there is no damping, and they almost are.

Thanks René

What I understood on this discussion are...

In the case of solving with nonsymmetric solver, the sound pressure eigenvector should be real when there is no damping, and they almost are but not zero perfectly. I believe there exist cases that's imaginary part are not small, so I must check each case.

I wanted to compare with theoretical modal superposition or numerical case calculated by other solver, but I'll pass on that for now. Why use an nonsymmetric solver for a real symmetric problem? Does it depend on the search method?

Thank you for all.