Acoustic-Structure Interaction

This model provides a tutorial on modeling the interaction of sound with elastic, solid structures by using a specific example of an elastic cylinder in water. The sound will cause movement of the solid cylinder, which in turn will induce new sound waves in the air; thus, full two-way coupling between the acoustic medium (water) and the cylinder is required to realistically simulate this situation.

This model is similar to the interaction of acoustic signals (sound) with most everyday objects: Liquid or gas acoustics coupled to structural objects have application in many engineering fields, for example, loudspeakers, acoustic sensors, nondestructive testing, or medical ultrasound diagnostics of the human body.

The intent with this tutorial model is to illustrate the modeling process rather than to provide an exhaustive illustration of the acoustic-solid interaction capabilities of COMSOL.

Model Definition

This model simulates the behavior of a solid cylinder in an water domain with an incident acoustic wave in the water. The object’s walls are impacted by the acoustic pressure. The model calculates the frequency response from the solid and then feeds this information back to the acoustics domain so that it can analyze the resulting wave pattern. As such, the model is a good example of a scattering acoustic-solid interaction problem.

Figure 1: Geometric setup of an aluminum cylinder immersed in water.

To set up the model, use the Acoustic-Solid Interaction, Frequency Domain multiphysics interface. This interface involves two physics interfaces: Solid Mechanics and Pressure Acoustics, Frequency Domain. It also sets up the Acoustic-Structure Boundary multiphysics coupling.

Figure 1 illustrates the aluminum cylinder immersed in water. The incident wave is 60 kHz, in the ultrasound region. The cylinder is 2 cm in height and has a diameter of 1 cm. The water acoustic domain is truncated as a sphere with a reasonably large diameter. What drives the system is an incident plane wave from the surroundings into the spherical boundary. The harmonic acoustic pressure in the water on the surface of the cylinder acts as a boundary load in the 3D solid to ensure continuity in pressure. The model calculates harmonic displacements and stresses in the solid cylinder, and it then uses the normal acceleration of the solid surface in the acoustics domain boundary to ensure continuity in acceleration.

Domain Equations

The default feature models harmonic sound waves in the water domain by means of the Helmholtz equation for sound pressure:

Where p is the pressure (SI unit: N/m2), ρc is the density (kg/m3), ω is the angular frequency (SI unit: rad/s), and cc is the speed of sound (SI unit: m/s). Note that both the density and the speed of sound can be complex valued (hence the subscript “c”) in order model fluids with dissipating properties. In this model they are real valued as no damping is modeled.

Table 1: Acoustics Domain Data.
Quantity	value	description
ρc	997 kg/m3	Density
cc	1500 m/s	Speed of sound
f = ω/2π	60 kHz	Frequency

To calculate the harmonic stresses and strains in the solid cylinder for a frequency-response analysis, use the default model feature under the Solid Mechanics interface. The material data comes from the built-in database for Aluminum 3003-H18.

Boundary Conditions

Outer Perimeter

On the outer spherical perimeter of the water domain (Figure 1) specify an incident plane wave to represent an incoming sound wave. A superimposed spherical wave is allowed to travel out of the system as a response from the cylinder. In the Pressure Acoustics, Frequency Domain interface you implement this scenario by using the prepared boundary condition. Such boundary condition is useful when the surroundings are only a continuation of the domain.

Table 2: Radiation Boundary condition settings.
Quantity	value	Description
k		Incident wave direction vector
p0	1 Pa	Pressure amplitude

The incident wave direction k is controlled by the two angles 0 < θ < π and 0 <

< 2π. The incident field is defined in the subfeature.

Interface Cylinder-Water

The coupling between the fluid domain (pressure waves) and the solid is automatically done via the multiphysics coupling. The automatic boundary condition sets the boundary load F (force/unit area) on the solid cylinder to

where ns is the outward-pointing unit normal vector seen from inside the solid domain. While on the fluid side the normal acceleration experienced by the fluid is set equal to the normal acceleration of the solid. Mathematically this means that

where na is the outward-pointing unit normal vector seen from inside the acoustics domain, and the normal acceleration an is equal to (na · u) ω2, where u is the calculated harmonic-displacement vector of the solid structure.

Hard-Wall Comparison

As a reference you can also study a simpler model where the solid interface is regarded as a hard wall. This implies that the cylinder will not be affected by sound, but its presence will nonetheless affect how the sound is distributed. In the model this is achieved by setting a fixed constraint on all the solid boundaries, that is, u = 0. This reduces the above condition (an = 0) to the sound hard boundary condition

Results and Discussion

Figure 2 displays the sound pressure as a slice plot. It is clear from which direction the sound wave propagates into the domain. The values of the deformation are very small, but the acceleration is large enough to have an impact on the sound waves.

Figure 2: Displacement of the cylinder and the sound-pressure plot (dB) of the acoustic waves in the coupled problem. The arrow lengths are proportional to the surface acceleration, which is a direct measure of the sound-pressure interaction between the water and the cylinder.

Figure 3 shows a comparison between the hard-wall example and the full aluminum solid model. Near the cylinder wall the plot shows that the sound pressure level (SPL) is higher on the upstream side for the hard-wall case than for the aluminum model. This is also the case on the downstream side, where the SPL for the hard-wall model is higher than for the aluminum model. This shows that the hard wall reflects more than the aluminum case. The conclusion is that the mechanical properties of the metal object have an impact on the acoustic signature.

Figure 3: Sound pressure level on impact and on the shadow side of the cylinder.

Acoustics_Module/Tutorials,_Pressure_Acoustics/acoustic_structure

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

You may either add the parameters manually or load them from a text file.

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file acoustic_structure_parameters.txt.

To add parameters manually, you can do as follows.

In the table, enter the following settings:


Name	Expression	Value	Description
f	60[kHz]	60000 Hz	Frequency
phi	(-pi/6)[rad]	-0.5236 rad	Wave direction angle, phi
theta	(4*pi/6)[rad]	2.0944 rad	Wave direction angle, theta
k1	sin(theta)*cos(phi)	0.75	Incident wave direction vector, X component
k2	sin(theta)*sin(phi)	-0.43301	Incident wave direction vector, Y component
k3	cos(theta)	-0.5	Incident wave direction vector, Z component
R	30[mm]	0.03 m	Model domain radius

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 5.

In the text field, type 20.

Locate the section. In the text field, type -10.

Click

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R.

Click

Click the

button in the toolbar.

To see the interior:

Click the

button in the toolbar.

Definitions

Next, define a number of selections as sets of geometric entities to use when setting up the model.

Fluid Domain

In the toolbar, click

In the window for , type Fluid Domain in the text field.

Select Domain 1 only.

Solid Domain

In the toolbar, click

In the window for , type Solid Domain in the text field.

Select Domain 2 only.

Radiation Boundaries

In the toolbar, click

In the window for , type Radiation Boundaries in the text field.

Locate the section. Select the check box.

Locate the section. From the list, choose .

Solid Boundaries

In the toolbar, click

In the window for , type Solid Boundaries in the text field.

Select Domain 2 only.

Locate the section. From the list, choose .

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Water, liquid (mat1)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Aluminum 3003-H18 (mat2)

In the window, click .

In the window for , locate the section.

From the list, choose .

Now, set up the physics of the problem by defining the domain physics conditions and the boundary conditions.

Pressure Acoustics, Frequency Domain (acpr)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Spherical Wave Radiation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Incident Pressure Field 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the p0 text field, type 1.

From the c list, choose .

From the list, choose .

Specify the ek vector as


k1	x
k2	y
k3	z

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Mesh 1

Free Tetrahedral 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type 1500[m/s]/f/6.

This setting is appropriate for the fluid domain and corresponds to 6 elements per wavelength.

Add another Size node to specify a different maximum element size for the solid domain.

Size 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type 3.

Click

Click the

button in the toolbar.

Click the

button in the toolbar.

Click the

button in the toolbar to return to the default state.

Study 1 - Sound Hard Cylinder

In the window, click .

In the window for , type Study 1 - Sound Hard Cylinder in the text field.

Locate the section. Clear the check box.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type f.

Disable the Solid Mechanics interface, which corresponds to the hard cylinder case.

Locate the section. In the table, clear the check box for .

In the table, clear the check box for .

In the toolbar, click

Before visualizing this solution, include the structural analysis of the cylinder and compute the corresponding solution. You can do this by adding one more study.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 2 - Aluminum Cylinder

In the window, click .

In the window for , type Study 2 - Aluminum Cylinder in the text field.

Locate the section. Clear the check box.

Step 1: Frequency Domain

In the window, click .

In the window for , locate the section.

In the text field, type f.

In the toolbar, click

Results

To reproduce the plot in Figure 3, which compares the sound pressure levels along a diameter in the propagation direction for the two cases, begin by defining datasets as follows.

Cut Line 3D 1

In the toolbar, click