In the Rigid Walled Room episode we seen how to model a rectangular room with rigid walls. We driven the room at the modal frequencies and compared the solution field with the theoretical modal shapes, finding that the results matched single modal shapes real well until, at a frequency high enough, the contribution of multiple modes (in addition to that related to the driving modal frequency) became important. In this episode we will look at making the model more realistic. To do so, we will investigate the low frequency response of a home studio.

Getting less Step by Step

So far all episodes involved detailed step-by-step instructions. This makes sense as I want this series to act as a tutorial, and some explanation is needed as these tools are not very straightforward to learn. However, the previous episodes cover the basics in great detail, so we can avoid repeating the procedure to setup geometries, meshes and solver and cut to the chase. From this episode step by step guidance will be given only when dealing with something not already covered in the previous episodes.

Project Files

All the files used for this project are available at the repositories below:

• From 10 Hz to 100 Hz.
• From 125 Hz to 400 Hz.

How to add Complexity

The best way to add complexity to a model is through incremental steps, going from simple to complex. If we start off a model with a high complexity, and look at the solution, how do we understand that it makes sense? Could we have made an error in setting one of the many parameters? And what is the impact of the many parameters and features of the model in isolation? These, and many more questions, become tricky to answer when everything is entangled in a final field.

So, we should do something different. We already seen how to model the Rigid Walled Room, one of the simplest systems in acoustics (and already a pretty complex one). Then we should proceed with baby steps. For example, something like this would be nice:

1. Make the shape more realistic, putting an omni-directional uniform velocity source in the room but keeping the walls rigid.
2. Modify the above and make the impedance of the surfaces (walls, floor, ceiling, doors and windows) more realistic.
3. Modify the above and simulate the behaviour of furniture.
4. Modify the above to simulate the behaviour of acoustic treatment for the room.
5. Modify the above to include more realistic sound radiators.

The list could go on, and actually many steps can be broken in many substeps, but this should give you the idea.

A Note for Arch Linux Users

Looks like Salome Platform 9.3.0 does not like the latest mesa version. In order to work with it, I had to downgrade to mesa-19.3.4-2. So, if you get this kind of error when opening the Geometry module:

OpenGL_Window::CreateWindow: glxCreateContext failed.

you can try to downgrade your mesa. For guidance about downgrading packages, refer to the Arch Linux wiki.

Wait, Low Frequency Response?

Yes, for the time being. The reason is that we are solving for the acoustic field inside a large room. Now, remember our rule for the maximum mesh size $s$ given the frequency $f$ at which we are running the simulation ($\lambda$ is the wavelength and $c_{0}$ is the speed of sound):

$$s<\frac{\lambda}{10}=\frac{c_{0}}{10f}$$

We can plot it as a function of frequency (assuming $c_{0}$ to be $343$ $\frac{\text{m}}{\text{s}}$):

Note that the scale of both axes is logarithmic. As you can see, the maximum mesh size rapidly drops with frequency. This means that more elements will be needed for higher frequency. At $1$ $\text{kHz}$ the maximum mesh size is already as small as $3.4$ $\text{cm}$. This is a very small number, and solving for a realistic room size will be very hard. For the room we will be considering, this amounts to a total number of elements in the order of $6$ millions! Solving for such a system is pretty computationally intensive. For this reason, when it comes to acoustics, Finite Element Analysis (FEA) for large volumes (such as rooms, for example) is rarely employed at frequencies above $1$ $\text{kHz}$. Ray tracing methods are preferred at high frequencies instead.

So, why not to just use ray tracing? Well, ray tracing works in the so called optical approximation of acoustics, that holds at high frequency but not at low frequency. Wave methods, such as FEA, are then preferred ones at low frequency.

Similar considerations go with other kinds of modelling techniques. Normally, certain modelling techniques are best applied when a certain number of constrain hold, which guarantee maximal accuracy. Hence, to have a full picture of a system, it is not unusual to having to resort to multiple modelling strategies, each to maximise the accuracy in a certain region of the space of parameters describing the system.

Frequencies Under Study

We will be solving for the steady state acoustic field inside our room at a number of different frequencies. Hence, our mesh size should allow for good accuracy also at the highest frequency of the study. However, the plot above shows to us that a fine mesh will be overkill for frequencies that are significantly lower than the maximum frequency for which the mesh is tuned. Hence, it will be best to split the study in frequency ranges.

In the following study we will sweep across the third octave nominal centre frequencies up to $400$ $\text{Hz}$. We will do so by using two meshes in two separate studies as follows:

Note how we will not use one mesh up to the maximum third octave centre frequency it can work with, but the one just below. We do this as a sort of “safety margin” for accuracy. The meshes above will be prepared in the usual way (NETGEN 1D-2D-3D algorithm and Second Order elements) but we will see that we will face convergence issues. But let’s not get ahead of ourselves.

Geometry

This time I will not cover the geometry of the system in a step by step fashion. If you are new to FreeCAD you can refer to the previous episodes. If you need help to model your room in FreeCAD, refer to the online FreeCAD resources (such as the forum and documentation). You can also refer to the FreeCAD file included in the GitLab repositories mentioned above for guidance.

In the previous episodes we made a CAD model of the air enclosed within the room, and only that. In this episode we will model the air of the room but we will put another 3D object inside it, to act as our source. In the model on the repositories the room volume is generated by tracing a 2D plant of a room. Then, two surfaces, flush with the walls, are used to act as doors. You can model your room as you like, and once you have done you can put a sphere anywhere inside it. This sphere will act as our omni-directional radiator. Once you have done, select both the room and the sphere object (as shown in Figure 2 and export this selection as BREP.

Meshing

As always, you can import your BREP file, that now contains more than one solid, inside the Geometry module in Salome. You can now proceed to explode the imported geometry into Solid entities, as always. However, there is now an important thing to consider.

We have two 3D objects in our geometry. We must make sure that they are meshed properly, with “continuity”, that is, the mesh of one fades with continuity into the mesh of the other, without gaps. Also, the surface between the two 3D objects must “belong” to both objects, so that it can be used to apply boundary conditions to the both of them or, in multi-physics problems, to act as a Structure Interface. This is a so called conformal mesh, and the way we do this is through domain partition.

To do a partition select, among the solids we just exploded from the original geometry, the 3D object representing the room (Solid_2 in my case) and proceed as following:

• From the top menu, choose Operations and then Partition. This will open a new window.
• In the new window, click the arrow on the left of Tool Objects and select the sphere source solid (Solid_1 in my case), as shown below.

This will create a partition object, which you can then explode in two solid sub-entitites. You can now proceed and explode each one of these sub-entities in faces. The hierarchy of results is shown below:

So, to sum up:

• We imported the file geometry.brep, thus creating the geometry.brep_1 object.
• We exploded geometry.brep_1 into two solids, Solid_1 and Solid_2 (see the previous episodes for instructions on how to explode things).
• We created a partition from Solid_2 (the room volume) by using Solid_1 (the source model) as a tool, resulting in the partition Partition_1.
• We exploded the partition Partition_1 into solids.
• We exploded each of the solids in Partition_1 in faces. You will note that the spherical surface enclosing the sphere source appears under Solid_3 (which is the sphere) and Solid_4 (which is the room). This is the crucial bit.

Good, so now we only have to do the mesh(es). Switch to the Mesh module, and make sure you select the partition object from the Object Browser (as shown in the figure above). Then define your mesh in the usual way (see to the previous episodes for guidance). We will use our old trustworthy NETGEN 1D-2D-3D algorithm with any of the maximum sizes defined above, a minimum size of $0.01$ $\text{mm}$, Fine fineness and make sure it is Second order (see below for an example). Remember that typically Salome wants the sizes to be specified in millimetres for geometry created as in these tutorials.

Now, make sure that once the mesh is setup, you create Create Groups from Geometry in the usual way. But we will need to include in the groups all the sub-entities of our partition, as shown below.

The solid entities will become Groups of Volumes and the face entities will become Groups of Faces, as shown below.

We then finally compute and export our mesh as UNV. When we load the UNV file into Elmer then Elmer will treat the Groups of Volumes as bodies and the Groups of Faces as boundaries. This means that we will be able to apply different governing equations to different bodies, and couple equations, but we will not do this today. We will leave the sphere itself ungoverned (which will generate various warnings when we solve), and just apply an uniform velocity boundary condition, to its surface, in the usual way.

Below is a picture of the resulting mesh as visualised with Salome. You can see that it looks pretty conformal.

Solving

This largely goes like the Rigid Walled Room episode, so I will just list the model setup briefly, using the study from 16 Hz to 100 Hz as an example.

Model Setup

• Put the following in the Free text:

$ f = 16 20 25 31.5 40 50 63 80 100
$ p = 1.205
$ U = 10

f is the vector of third octave centre frequencies we will be solving for, p is the density of air at room temperature, and U is the surface velocity of our sphere source.

• Set the Simulation type to Scanning.
• Set the Timestep intervals to the number of elements in f (9 in this case).
• Set the coordinate scaling to 0.001, as the coordinates in our mesh are millimetres instead of meters.

Equation

Add an Helmholtz equation in the usual way (see the previous episodes if you need guidance). Ensure that it is applied to the correct body, i.e. the room volume. One way to do so is to choose Model > Set body properties and double click on any surface of the room. This will allow to assign the equation to it.

Also, since we will be solving at many frequencies, I suggest you tick Abort if the solution did not converge in the Linear system solver settings, as shown below. This will stop the solver as soon as it cannot converge to a solution, so that we do not risk to go great lengths analysing a solution that is, most likely, not that good.

For low frequency a value of Max. iterations of $500$ will be OK, but after $125$ $\text{Hz}$ you will see the study having an harder time converging to a stable solution. With $2000$ iterations (as shown below) you should be able to solve up to $315$ $\text{Hz}$ (see the Results section for more observations).

Material

Add Air (room temperature) in the usual way. Remember to put this MATC expression for Density:

Real MATC "p"

Boundary Condition

Add $2$ boundary conditions, a rigid wall and an uniform velocity radiator.

The rigid one has $0$ flux, for both real and imaginary parts.

The radiator has this MATC expression for the imaginary part of the flux:

Variable time; Real MATC "2 * pi * f(tx - 1) * p * U"

To apply them, I suggest you choose Model > Set boundary properties. Then, double click all the external boundaries of the room (including the doors) and set them to rigid. Finally, open the boundary condition editor again for the radiator condition. There should be only one boundary left to which you can apply it.

Sif Files

You can see my sif files here ($16$ $\text{Hz}$ to $100$ $\text{Hz}$) and here ($125$ $\text{Hz}$ to $400$ $\text{Hz}$) to check that you setup the model the same way I did.

Results

You will see that when we deal with frequencies lower than $125$ $\text{Hz}$ Elmer can converge to a solution pretty easily (less than $30$ iterations). However, things get quite harder to deal with at higher frequency. In fact, I could not get the solution at $400$ $\text{Hz}$ to converge at all!

But first, let’s have a look.

The animation above, prepared with ParaView (see here for an introduction and few tricks) shows the steady state Sound Pressure Level (SPL) in the room for each of our study frequencies, with the exception of $400$ $\text{Hz}$, since the solver could not converge at that frequency. The room is shown transparent, and two slices are operated in the domain, one vertical and one horizontal, passing through the source. The curves inside the room are curves of constant SPL.

Well, clearly a surface velocity of $10$ $\frac{\text{m}}{\text{s}}$ is pretty crazy, and yields pressures up to $130$ $\text{dBSPL}$. But, since our equation is linear, this high level does not impact the shape of the field: had we chosen an lower surface velocity we would have had the same solution, just lower peak SPL. So, we can study the resulting shape with some sense of generality.

We can see that the steady state field grows in complexity the higher the frequency. Few features are seen at $16$ $\text{Hz}$. In fact, all the SPL variation is concentrated close to the source, and the rest of the air has an pretty uniform SPL value. But at high frequencies we start seeing many nodal surfaces of very complex shape, cutting between zones of high pressures. This complexity of the field is one of the causes of the convergence issues at high frequency, as we will see in the next episodes.

Conclusion

We seen in this episode how to create models with more than one body. We also seen how computationally intensive it is to solve for high frequency problems, due to the mesh size needing to be smaller, and we seen that the field becomes more complex the higher the frequency. Finally, we seen that convergence at high frequency is more complicated to achieve. So, before we dig into the study of the results some more, and go ahead with adding additional complexity, we should stop for a moment and investigate these two issues:

• What can we do to aid convergence?
• What can we do to shorten computation times?

So, in the next episode we will look at the problem of convergence.

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License.

• Physics
• Elmer
• FEM
• Linear System
• Single Physics
• Steady State
• Modal
• Series: Home Studio