# Frequency domain harmonic vibroacoustics

Hello everyone,

I want to perform a vibroacoustic simulation. Below I describe the general setup, where an acoustic cavity \Omega_a interfaces a structural domain \Omega_s through one or multiple interfaces \partial \Omega_{int}. I am using FEniCSx for both domains.

The weak forms for such domains should be the following:

\text{Acoustic domain, time harmonic linear acoustics:}\\ \\ \int_{\Omega_a}\nabla p \cdot \nabla v d \Omega_a - \int_{\Omega_a}k_a^2p v d \Omega_a = \int_{\Omega_a} j\omega\rho_a q v d \Omega_a + \int_{\partial\Omega_{int}} \rho_a \omega^2 \left(\mathbf{u \cdot n_a}\right) v d\partial\Omega_{int}

\text{Structural domain, time harmonic linear elastodynamics:}\\ \\ \int_{\Omega_s} \sigma(\mathbf{u}) : \epsilon (\mathbf{w}) d\Omega_s - \int_{\Omega_s} \rho_s\omega^2 \mathbf{u\cdot w} d\Omega_s = \int_{\partial\Omega_{int}} -p\mathbf{n_s \cdot w} d\partial\Omega_{int}

*Note: I am using volumetric sources as acoustic input, so the structure is moving as a result of the fluid-structure interaction. There is no external force or load on the structure.

The equation are coupled through the Neumann BCs (-p\mathbf{n_s} is just the traction vector on the interface due to the acoustic pressure on it, while \frac{\partial p}{\partial n_a} = \rho_a \omega^2 u_n = \rho_a \omega^2 \mathbf{u \cdot n_a} is derived from the euler equation)

Usually this problem is solved with a monolithic approach, solving one algebraic system and getting both p and \mathbf{u} simultaneously.

The problem is that I need the interface to have non matching meshes, so I thought about solving it in with a partitioned approach, but I have not any experience in this topic.

Then my first question is: is preCICE for me? I see that almost everyone is solving time dependent problems here, but I am not (for now).
What happens in partitioned approach near “subsystem” eigenfrequencies, where the matrices become singular? Is there always a way to make the soultion converge?

I need to decide if I need to put my effort to build the monolithic matrix with non conforming meshes or building a partitioned algorithm. That’s why I need your advice!

I suppose that I would solve something like this:

1. Solve the acoustic domain for p with natural boundary condition (rigid walls) and get p
2. Use p for the Neuman BC on the structure (after interpolating it in the structure’s FunctionSpace) and get \mathbf{u}
3. Use \mathbf{u} for the Neuman BC on the acoustic domain (after interpolating it on the acoustic FunctionSpace) and get p
4. Iterate 2 and 3 until convergence.

Is this approach correct? I am sorry if the approach is wrong but, as I said, I’ve never studied partitioned approaches.
Can I implement something like this? Is it worth it? I need the results to be comparable to the one I would obtain with the monolithic approach.

Every advice that you give me would be precious, thank you very much.

Hello @bayswiss

Thank you for describing your problem in such detail. My first impression is that, yes, partitioned coupling with preCICE would be something worth pursuing in your case. Let me try to answer your questions:

• preCICE handles non-matching meshes well. We provide several data mapping methods which you can try out.
• It is true that most of our user base solves time dependent problems, but there have been users who have solved steady state or time independent problems too. This can be done, and we can discuss this further.

What happens in partitioned approach near “subsystem” eigenfrequencies, where the matrices become singular? Is there always a way to make the soultion converge?

This is hard to answer without trying things out. In preCICE, you can use acceleration techniques to ensure convergence of an implicitly coupled problem. This mainly relies on robust Quasi-Newton algorithms. How would this look for your time independent case is something that would warrant a wider discussion. One possible strategy would be in doing fake time steps until you get convergence.

Overall I believe preCICE is definitely worth exploring for your case. Please let us know if we can answer any more questions.

1 Like

Thank you very much for the detailed answer.
I will definitely give it a try and I will give you updates on the outcome!

About this, I think that I am going to ask question in the future for sure 2 Likes

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