Hollow Cylinder

Fluid acoustics coupled to structural objects (vibroacoustics), such as membranes or plates, represents an important application area in many engineering fields. Some examples are loudspeakers, acoustic sensors, nondestructive impedance testing, and medical ultrasound diagnostics of the human body. Through the use of the Acoustic - Solid Interaction, Frequenccy Domain multiphysics interface, this model demonstrates how fluid acoustics can be easily coupled to structures.

Model Definition

This model provides a general demonstration of an acoustic fluid phenomenon in 3D coupled to a solid object. In this study, the solid object is a capped, hollow aluminum cylinder filled with and immersed in water. The acoustic waves created by a source inside the cylinder impact the cylinder walls.

Figure 1: A hollow aluminum cylinder is immersed in water. The white line inside the cylinder indicates the line source, and the tiny sphere next to the line shows the position of the point source. The simulation domain is bounded by a large sphere.

Figure 1 illustrates the aluminum cylinder immersed in water. The cylinder is 2 cm in height and has an outer diameter of 1 cm. The thickness of its walls is 0.5 mm.

The water-filled acoustic domain outside the cylinder is truncated to a sphere with a reasonably large diameter. In two different versions of the model, the system is driven either by a line source coinciding with the axis of the cylinder and located entirely within the cylinder or by a point source in the interior of the cylinder. The frequency studied is 60 kHz, that is, in the ultrasound region. The fluid and solid domains are coupled using the built-in multiphysics coupling. The coupling ensures a fully two-way coupled analysis of the problem.

Domain and Boundary Conditions

Water Domain

For harmonic sound waves, in the frequency domain, the model solves the Helmholtz equation defined in the Pressure Acoustics, Frequency Domain interface.

Solid Domain

You calculate the harmonic stresses and strains inside the solid cylinder walls using the same equation formulation as a frequency response analysis in the 3D Solid Mechanics interface. The material data comes from the built-in database for Aluminum 3003-H18.

Outer Perimeter

On the outer spherical perimeter of the water domain (Figure 1), use the predefined condition. This boundary condition allows a spherical wave to travel out of the system, giving only slight reflections for the non-spherical components of the wave. For mathematical details on the radiation boundary condition, see the Acoustics Module User’s Guide.

Cylinder-Water Interface

The model uses the multiphysics interface, in which the multiphysics coupling condition automatically apply to all acoustic-structure boundaries. The condition ensures that the acoustic pressure is added as a load on the solid and that the solid normal acceleration is added as a source on the acoustic domain.

edge and point settings

In the two cases considered, the sound waves are generated by either a point monopole source or a line source. A line source along the z-axis defines the following right-hand-side source term to the Helmholtz equation:

Here Prms is the free space reference rms power per unit length of a line source of length Ledge placed in a homogeneous medium extending to infinity. Furthermore, δ(2)(r) is the Dirac delta function in two dimensions, r denoting the projection of the position vector onto the xy-plane.

A monopole point source of rms power Prms located at the point R = R0 in an infinite homogeneous space defined the following source term

where δ(3)(R) is the Dirac delta function in three dimensions. Any type of confinement results in higher power usage.

Results

Figure 2 shows the resulting deformations and local sound pressure level from a computation with a line source.

Figure 2: Sound-pressure level plot (dB) of the acoustic waves in the coupled problem, using a line source inside the cylinder. The surface of the cylinder shows its deformation (mm).

In Figure 3, you see the solution from an off-centered point. Note that the deformation color scale is different, implying that this source gives deformations that at least locally are greater than those from the line source.

Figure 3: Sound-pressure level (dB) and structural deformations (mm) with a point source inside the cylinder. Some of the surfaces are hidden to reveal the pressure distribution inside the cylinder.

Acoustics_Module/Vibrations_and_FSI/hollow_cylinder

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

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
edgeL	1.7[cm]	0.017 m	Length of line source
cylh	20[mm]	0.02 m	Height of the cylinder
cylr	5[mm]	0.005 m	Radius of the cylinder
cylth	0.5[mm]	5E-4 m	Thickness of the cylinder
f0	60[kHz]	60000 Hz	Frequency of the analysis
c0	1482[m/s]	1482 m/s	Speed of sound in water
lam0	c0/f0	0.0247 m	Wavelength at f0

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 cylr.

In the text field, type cylh.

Locate the section. In the text field, type -cylh/2.

Click to expand the section. Select the check box.

Select the check box.

In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	cylth

Click

Click the

button in the toolbar.

With wireframe rendering, you now see the contours of both cylinders.

Line Segment 1 (ls1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the text field, type (cylr-cylth)/3.

In the text field, type -edgeL/2.

Locate the section. In the text field, type (cylr-cylth)/3.

In the text field, type edgeL/2.

Point 1 (pt1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type (cylr-cylth)/3.

In the text field, type (cylr-cylth)*2/3.

In the text field, type (cylh/2-cylth)/3.

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1.5*cylh.

Click

Click the

button in the toolbar.

The geometry should look like this.

Definitions

Solid

In the toolbar, click

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

Select Domains 2–8 and 10–16 only.

Water

In the toolbar, click

In the window for , type Water in the text field.

Select Domains 1 and 9 only.

Line Source

In the toolbar, click

In the window for , type Line Source in the text field.

Locate the section. From the list, choose .

Select Edge 56 only.

This is the line that will act as a source. The selection should look like this.

Point Source

In the toolbar, click

In the window for , type Point Source in the text field.

Locate the section. From the list, choose .

Select Point 32 only.

Point 32 lies inside the cylinder and is the only point that is not connected to any edges. The selection should look like this.

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 .

Pressure Acoustics, Frequency Domain (acpr)

By default, all domains are using the Pressure Acoustics model, which holds for fluids. Use the Water selection to specify where Pressure Acoustics is applied.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

The water reference pressure is 1 µPa. This does not affect the physics, but only serves as a reference in the definition of the sound pressure level.

Spherical Wave Radiation 1

In the toolbar, click

and choose .

Select Boundaries 1–4, 32, 33, 47, and 48 only.

These are the boundaries that will have a radiation condition on them. The selection should look like this.

Line Source 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the Prms text field, type 0.5.

Monopole Point Source 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the Prms text field, type 0.5.

You will select either the line source or the point source under the study by enabling .

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Multiphysics

Now is a good time to inspect the node. Notice that the Acoustic-Structure Boundary feature is automatically added to all boundaries, but only applied to the boundaries that have water on one side and the solid on the other.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Size

In the window, under click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type lam0/5.

This value, combined with the default choice of second-order elements, makes sure that the rule-of-thumb minimum of 10-12 degrees of freedom per wavelength for the solution to be reliable is satisfied.

In the text field, type lam0/10.

Click

Free Quad 1

In the toolbar, click

and choose .

Select Boundaries 25–27, 30, 45, and 53 only.

The model uses some free quad mesh to control what will be the sweep direction in the solid.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type cylth*3.

Select the check box.

In the associated text field, type cylth.

Click

Swept 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Distribution 1

Right-click and choose .

With the sweep operation, the 2D mesh previously created will be sweep through the domains. In general it is recommended to have at least two quadratic elements through the thickness.

In the window for , locate the section.

In the text field, type 2.

Click

Study 1 - Line Source

In the window, click .

In the window for , type Study 1 - Line Source in the text field.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type f0.

Now, disable the point source by modifying the physics tree.

Locate the section. Select the check box.

In the tree, select .

Click

In the toolbar, click

The default plots include all the surfaces of the model, obscuring the cylinder. A selection added to the dataset will make sure that the plots shows the requested variables on the selection.

Results

Study 1 - Line Source/Solution 1 (sol1)

In the window, expand the node, then click .

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 5–10, 12–24, 26–31, 34–46, and 49–64 only.

The selection should look like this.

Four default plots are created after running this model. In the following steps a additional plots will be added to show the space distribution of the acoustic field.

Slice 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 1.

Click to expand the section. From the list, choose .

In the toolbar, click

The plot should look like this.

Slice 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type acpr.Lp_t.

Locate the section. From the list, choose .

In the text field, type 1.

Locate the section. From the list, choose .

In the toolbar, click

The plot should look like this.

Filter 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type x>0.

Surface 1

In the window, right-click and choose .

In the window for , click to expand the section.

From the list, choose .

In the toolbar, click

The plot should look like this.

Create a new plot of the deformation and sound pressure level. Then proceed to solving the system again with a monopole point source.

Displacement/SPL - Line Source

In the toolbar, click

and choose .

In the window for , type Displacement/SPL - Line Source in the text field.

Click to expand the section. From the list, choose .

In the text area, type SPL and Displacement.

Locate the section. Select the check box.

Multislice 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type acpr.Lp_t.

Add a filter to only show the SPL in the water domain outside of the hollow cylinder.

Filter 1

Right-click and choose .

In the window for , locate the section.

In the text field, type y>-(cylr-cylth).

Surface 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type solid.disp.

Locate the section. From the list, choose .

Deformation 1

Right-click and choose .

In the toolbar, click

Your plot should now look like Figure 2. The remaining instructions show you how to set up a second study and replace the line source with a point source, while disabling the line source.

Create a group with all plots to facilitate the navigation through the different results.

Acoustic Pressure (acpr), Acoustic Pressure, Isosurfaces (acpr), Displacement/SPL - Line Source, Sound Pressure Level (acpr), Stress (solid)