Eigenfrequency in free vibration of a solid body

[yuchen liu]

Hello dealii mates,

A short description of the problem here:
I am solving the eigenvalues for a linear elastic solid body, where free boundaries are applied (zero Neumann). This project follows the tutorial step-36 and step-17 using Lame parameters for isotropic materials. 

Unfortunately, or I deserve it as a newbie, got a petsc error about zero pivot, which I presume is due to the six rigid body modes. The website below mentions creating nullspace, but I have no idea how to do it. Additionally, I want to see how many rigid modes I have to ensure my boundary conditions are applied correctly. The reason is that I previously encountered three rigid modes, which is so wrong for free vibration, due to improperly indexed Jacobian (not dealii). Can anyone help on this or generally offer some suggestions? I would greatly appreciate that in advance. 

[0]PETSC ERROR: Zero pivot in LU factorization: https://petsc.org/release/faq/#zeropivot
[0]PETSC ERROR: Zero pivot row 1 value 4.23516e-22 tolerance 2.22045e-14

Please forgive me for being greedy here, this wee project will be the basement of my next electromechanical problem, which highly depends on static condensation with an inhomogeneous rhs vector. If anyone has found something similar or might be helpful in tutorials, please let me know, thanks!

Cheers,

Yuchen Liu

[Wolfgang Bangerth]

Yuchen,
a typical eigenproblem is of the form
A x = lambda M x
and the requirement is that M is a definite matrix that can be inverted,
whereas A can have a nullspace (which then corresponds to zero eigenvalues).
My *expectation* (which may not correspond to the truth) is that the
eigensolver should then only apply an LU decomposition to the M matrix, but
not to the A matrix. But the A matrix is the one that has the rigid body
motions as nullspace. I would try and ascertain first that M is in fact
invertible.

Best
W.
--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[yuchen liu]

Dear Prof Bangerth,

Thanks for your reply. Aye, you are absolutely correct! 

I checked my global mass matrix and confirmed that it is indeed singular. Then, I found out that my local mass matrices are also singular, which explains the issue. After printing out the local matrices, I realised I had made a dumb mistake in the assembly process. The pattern for this 3D problem I obtained was: 

AAABBB...

AAABBB

AAABBB

BBBCCC

BBBCCC

BBBCCC

which is clearly incorrect and is of course singular... The pattern is supposed to be:

A00B00...

0A00B0

00A00B

B00C00

0B00C0

00B00C

The reason is that I mistakenly used the same approach in step-36 to assemble my mass matrix,

cell_mass_matrix(i, j) += rho * fe_values.shape_value(i, q_point) * fe_values.shape_value(j, q_point) * fe_values.JxW(q_point);

and this needs to be modified in the 3D dynamic problem as follows: 

cell_mass_matrix(i, j) += ((i % dim == j % dim)?
                                            rho * fe_values.shape_value(i, q_point) * fe_values.shape_value(j, q_point) * fe_values.JxW(q_point):
                                            0); 

Now my code is running well, and all 6 rigid body modes correctly identified. Tbh, my way is not wise, but hope it is helpful to others encountering the same issue. 

Best regards,

Yuchen Liu

[Wolfgang Bangerth]

On 7/17/25 00:36, yuchen liu wrote:
>
> I checked my global mass matrix and confirmed that it is indeed singular.
> Then, I found out that my local mass matrices are also singular, which
> explains the issue. After printing out the local matrices, I realised I had
> made a dumb mistake in the assembly process.

We all make dumb mistakes all the time. That's how we learn. I'm glad I could
help you point in the right direction and that you figured it out!