Acoustic Cloaking

In the tree, select .

The geometry of this model is simple but a bit repetitive to create, due to the large number of circles involved. You will therefore import a geometry sequence from a file. The instructions for building the geometry can be found in the appendix at the end of this document.

Geometry 1

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file acoustic_cloaking_geom_sequence.mph.

In the toolbar, click

The geometry should look like that in Figure 1.

button in the toolbar.

The file you have just imported contains a few parameters that are used to generate the geometry. Now proceed to add some additional parameters that will be used in the analysis.

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
rhob	1.25[kg/m^3]	1.25 kg/m³	Density, background material
cb	343[m/s]	343 m/s	Speed of sound, background material
f0	300[Hz]	300 Hz	Frequency of the analysis
lam0	cb/f0	1.1433 m	Wavelength

The material properties in the cloaking involve dependencies on the inner and outer radius, the local distance from the center, and the properties of the surrounding background fluid. It is convenient to define these expressions as variables.

Definitions

Cylindrical System 2 (sys2)

In the toolbar, click

and choose .

In the window for , locate the section.

Find the subsection. In the table, enter the following settings:


x (m)	y (m)
x1	y1

Radial Coordinate: Homogenized Cloak

In the window, right-click and choose .

In the window for , type Radial Coordinate: Homogenized Cloak in the text field.

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
r	sys2.r	m	Radial coordinate

Radial Coordinate: 50 Layer Cloak

Right-click and choose .

In the window for , type Radial Coordinate: 50 Layer Cloak in the text field.

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
r	sqrt((x-x3)^2+(y-y3)^2)	m	Radial coordinate

Radial Coordinate: 20 Layer Cloak

Right-click and choose .

In the window for , type Radial Coordinate: 20 Layer Cloak in the text field.

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
r	sqrt((x-x4)^2+(y-y4)^2)	m	Radial coordinate

Variables: Acoustic Cloak Data

In the toolbar, click

In the window for , type Variables: Acoustic Cloak Data in the text field.

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
rho1	rhob(r+sqrt(2r*R1-R1^2))/(r-R1)	kg/m³	Density, Material 1
c1	cb(R2-R1)/R2r/(r-R1)	m/s	Speed of sound, Material 1
rho2	rhob^2/rho1	kg/m³	Density, Material 2
c2	c1	m/s	Speed of sound, Material 2
K1	rho1*c1^2	Pa	Bulk modulus, Material 1
K2	rho2*c2^2	Pa	Bulk modulus, Material 2
K	2K1K2/(K1+K2)	Pa	Effective bulk modulus
rho_tangential	2rho1rho2/(rho1+rho2)	kg/m³	Density along layers
rho_normal	(rho1+rho2)/2	kg/m³	Density perpendicular to the layers

Now, proceed to add the materials. Material 1 and Material 2 are the two materials with varying properties that will be used in the layers to create an acoustic cloak. The homogenized material is an anisotropic material which properties also vary with thickness and are equivalent to a model with a large number of layers.

Materials

Air

In the window, under right-click and choose .

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

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rhob	kg/m³	Basic
Speed of sound	c	cb	m/s	Basic

Material 1

Right-click and choose .

In the window for , type Material 1 in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rho1	kg/m³	Basic
Speed of sound	c	c1	m/s	Basic

Material 2

Right-click and choose .

In the window for , type Material 2 in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rho2	kg/m³	Basic
Speed of sound	c	c2	m/s	Basic

Homogenized Material

Right-click and choose .

In the window for , type Homogenized Material in the text field.

Locate the section. From the list, choose .

Click to expand the section. In the tree, select .

The homogenized material has a diagonal density matrix with terms that change value with the radius. The bulk modulus also varies with the radius.

In the tree, select .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Effective density	{rho_eff11, rho_eff22, rho_eff33} ; rho_effij = 0	{rho_normal, rho_tangential, rho_tangential}	kg/m³	Anisotropic acoustics model
Effective bulk modulus	K_eff	K	Pa	Anisotropic acoustics model
Density	rho		kg/m³	Basic
Speed of sound	c		m/s	Basic

Pressure Acoustics, Frequency Domain (acpr)

Homogenized Model

In the window, under right-click and choose .

In the window for , type Homogenized Model in the text field.

The outer boundaries need a radiation condition to make sure that no waves are reflected at this interface.

Cylindrical Wave Radiation 1

Right-click and choose .

Select Boundary 154 only.

In the window for , locate the section.

Specify the r0 vector as


x1	x
y1	y

Homogenized Model

Now, proceed to add a for each of the geometries. The field is a plane wave propagating in the direction. The phase of this field is updated on each of the domains to make sure that the incident acoustic field is the same in all the domains.

Background Pressure Field 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the p0 text field, type 1.

From the c list, choose .

From the list, choose .

In the φ text field, type acpr.bpf1.k*x1.

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Anisotropic Acoustics 1

In the toolbar, click

and choose .

Select Domain 4 only.

In the window for , locate the section.

From the list, choose .

No Cloak Model

In the window, right-click and choose .

In the window for , type No Cloak Model in the text field.

Cylindrical Wave Radiation 2

In the toolbar, click

and choose .

Select Boundary 209 only.

In the window for , locate the section.

Specify the r0 vector as


x2	x
y2	y

Background Pressure Field 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the p0 text field, type 1.

From the c list, choose .

In the φ text field, type acpr.bpf2.k*x2.

Symmetry 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

50 Layer Cloak Model

Right-click and choose .

In the window for , type 50 Layer Cloak Model in the text field.

Cylindrical Wave Radiation 3

In the toolbar, click

and choose .

Select Boundary 153 only.

In the window for , locate the section.

Specify the r0 vector as


x3	x
y3	y

Background Pressure Field 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the p0 text field, type 1.

From the c list, choose .

In the φ text field, type acpr.bpf3.k*x3.

Symmetry 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

20 Layer Cloak Model

Right-click and choose .

In the window for , type 20 Layer Cloak Model in the text field.

Cylindrical Wave Radiation 4

In the toolbar, click

and choose .

Select Boundary 208 only.

In the window for , locate the section.

Specify the r0 vector as


x4	x
y4	y

Background Pressure Field 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the p0 text field, type 1.

From the c list, choose .

In the φ text field, type acpr.bpf4.k*x4.

Symmetry 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Mesh 1

In this model, the mesh is set up manually. Proceed by directly adding the desired mesh component.

Free Triangular 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 lam0/6.

In general, five to six second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. This model uses six elements per wavelength.

We will add a mapped mesh in the acoustic cloak domains. As the domains have properties that change with the radius, it is a good idea to keep a structured mesh with elements along the varying direction.

Mapped 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Since the variation of properties is more pronounced toward the inner center of the acoustic cloak, add a skewed distribution to make sure that the mesh is sufficiently fine throughout the domain.

Distribution 1

Right-click and choose .

Select Boundary 4 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 20.

In the text field, type 4.

The same distribution should be applied to the other edge of the domain, but with the reverse direction.

Distribution 2

In the window, right-click and choose .

Select Boundary 55 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 20.

In the text field, type 4.

Select the check box.

Distribution 3

Right-click and choose .

Select Boundary 156 only.

In the window for , locate the section.

In the text field, type 40.

In the window, right-click and choose .

The mesh should look like this.

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type f0.

In the toolbar, click

The following steps will update the default plots to include a cleaner view of the model with annotations of the different cloaks.

Results

Total Acoustic Pressure (acpr)

In the window for , type Total Acoustic Pressure (acpr) in the text field.

Locate the section. Clear the check box.

Line 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.

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

Locate the section. From the list, choose .

From the list, choose .

Selection 1

Right-click and choose .

Select Boundaries 1, 2, 105–108, 151–156, and 208–211 only.

In the toolbar, click

Total Acoustic Pressure (acpr)

In the window, under click .

Table Annotation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the table, enter the following settings:


x-coordinate	y-coordinate	Annotation
x1	y1+4.3[m]	Homogenized cloak
x2	y2+4.3[m]	No cloak
x3	y3+4.3[m]	50 Layer cloak
x4	y4+4.3[m]	20 Layer cloak

Locate the section. Clear the check box.

From the list, choose .

In the toolbar, click

Total Acoustic Pressure (acpr)

button in the toolbar.

In the window, click .

In the toolbar, click

The plot should look like that in Figure 2.

Sound Pressure Level (acpr)

In the window, under right-click and choose .

Remove the default Sound Pressure Level plot and duplicate the Total Acoustic Pressure plot created previously. Doing this, you make sure that the annotations will be carried over to the new plots.

Total Sound Pressure Level (acpr)

In the window, right-click and choose .

In the window for , type Total Sound Pressure Level (acpr) in the text field.

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type acpr.Lp_t.

Locate the section. Click

In the dialog box, select in the tree.

The plot should look like that in Figure 3.

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Scattered Sound Pressure Level (acpr)

In the window, right-click and choose .

In the window for , type Scattered Sound Pressure Level (acpr) in the text field.

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type acpr.Lp_s.

Selection 1

Right-click and choose .

Select Domains 1, 2, 54, and 55 only.

In the toolbar, click

The plot should look like that in Figure 4.

Effective Speed of Sound in Principal Directions

In the toolbar, click

and choose .

In the window for , type Effective Speed of Sound in Principal Directions in the text field.

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

Select Domain 4 only.

Locate the section. From the list, choose to create a dedicated view for this plot.

Locate the section. Select the check box.

Principal Stress Surface 1

button in the toolbar.

In the toolbar, click

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

In the toolbar, click

and choose .

In the window for , locate the section.

Find the subsection. In row , set to 0.

In row , set to 0.5.

Find the subsection. In row , set to acpr.c_eff1x+acpr.c_eff2x.

In row , set to acpr.c_eff1y+acpr.c_eff2y.

In row , set to 0.

In row , set to acpr.c_eff3x.

In row , set to acpr.c_eff3y.

In row , set to 0.

The first and second effective speeds of sound are identical in magnitude, but one of them is perpendicular to the plane. Both terms are added to the first direction so that the arrows are consistent.

In the toolbar, click

The plot should look like this.

Total Acoustic Pressure Along Cloak Boundary

In the toolbar, click

and choose .

In the window for , type Total Acoustic Pressure Along Cloak Boundary in the text field.

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

Locate the section.

Select the check box. In the associated text field, type Pressure (Pa).

Line Graph 1

Right-click and choose .

Select Boundary 156 only.

In the window for , locate the section.

In the text field, type acpr.p_b.

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

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Background pressure field

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

From the list, choose .

In the text field, type 100.

In the toolbar, click

Line Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type acpr.p_t.

Locate the section. Find the subsection. From the list, choose .

Locate the section. In the table, enter the following settings:


Legends
Homogenized cloak model

Line Graph 3

Right-click and choose .

In the window for , locate the section.

Select Boundary 155 only.

Locate the section. In the table, enter the following settings:


Legends
50 layer cloak model

Line Graph 4

Right-click and choose .

In the window for , locate the section.

Select Boundary 210 only.

Locate the section. In the table, enter the following settings:


Legends
20 layer cloak model

In the toolbar, click

The plot should look like that in Figure 5.

Speed of Sound in Homogenized Material

In the toolbar, click

and choose .

In the window for , type Speed of Sound in Homogenized Material in the text field.

Locate the section. From the list, choose .

Locate the section.

Select the check box. In the associated text field, type Speed of sound (m/s).

Line Graph 1

Right-click and choose .

Select Boundary 55 only.

In the window for , locate the section.

In the text field, type acpr.c_eff1.

Locate the section. From the list, choose .

In the text field, type x.

Locate the section. Select the check box.

Find the subsection. Clear the check box.

Select the check box.

Line Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type acpr.c_eff3.

The plot should look like that in Figure 6.

button in the toolbar.

In the toolbar, click

Appendix: Geometry Sequence Instructions

Add Component

In the toolbar, click

and choose .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

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

Geometry 1

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 4.

In the text field, type 180.

Locate the section. In the text field, type x1.

In the text field, type y1.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	R2
Layer 2	R1

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2*R2.

In the text field, type 180.

Locate the section. In the text field, type x2.

In the text field, type y2.

Locate the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	R2
Layer 2	R1

Circle 3 (c3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2*R2.

In the text field, type 180.

Locate the section. In the text field, type x3.

In the text field, type y3.

Locate the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	R2
Layer 2	(R2-R1)/50
Layer 3	(R2-R1)/50
Layer 4	(R2-R1)/50
Layer 5	(R2-R1)/50
Layer 6	(R2-R1)/50
Layer 7	(R2-R1)/50
Layer 8	(R2-R1)/50
Layer 9	(R2-R1)/50
Layer 10	(R2-R1)/50
Layer 11	(R2-R1)/50
Layer 12	(R2-R1)/50
Layer 13	(R2-R1)/50
Layer 14	(R2-R1)/50
Layer 15	(R2-R1)/50
Layer 16	(R2-R1)/50
Layer 17	(R2-R1)/50
Layer 18	(R2-R1)/50
Layer 19	(R2-R1)/50
Layer 20	(R2-R1)/50
Layer 21	(R2-R1)/50
Layer 22	(R2-R1)/50
Layer 23	(R2-R1)/50
Layer 24	(R2-R1)/50
Layer 25	(R2-R1)/50
Layer 26	(R2-R1)/50
Layer 27	(R2-R1)/50
Layer 28	(R2-R1)/50
Layer 29	(R2-R1)/50
Layer 30	(R2-R1)/50
Layer 31	(R2-R1)/50
Layer 32	(R2-R1)/50
Layer 33	(R2-R1)/50
Layer 34	(R2-R1)/50
Layer 35	(R2-R1)/50
Layer 36	(R2-R1)/50
Layer 37	(R2-R1)/50
Layer 38	(R2-R1)/50
Layer 39	(R2-R1)/50
Layer 40	(R2-R1)/50
Layer 41	(R2-R1)/50
Layer 42	(R2-R1)/50
Layer 43	(R2-R1)/50
Layer 44	(R2-R1)/50
Layer 45	(R2-R1)/50
Layer 46	(R2-R1)/50
Layer 47	(R2-R1)/50
Layer 48	(R2-R1)/50
Layer 49	(R2-R1)/50
Layer 50	(R2-R1)/50
Layer 51	(R2-R1)/50

Circle 4 (c4)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2*R2.

In the text field, type 180.

Locate the section. In the text field, type x4.

In the text field, type y4.

Locate the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	R2
Layer 2	(R2-R1)/20
Layer 3	(R2-R1)/20
Layer 4	(R2-R1)/20
Layer 5	(R2-R1)/20
Layer 6	(R2-R1)/20
Layer 7	(R2-R1)/20
Layer 8	(R2-R1)/20
Layer 9	(R2-R1)/20
Layer 10	(R2-R1)/20
Layer 11	(R2-R1)/20
Layer 12	(R2-R1)/20
Layer 13	(R2-R1)/20
Layer 14	(R2-R1)/20
Layer 15	(R2-R1)/20
Layer 16	(R2-R1)/20
Layer 17	(R2-R1)/20
Layer 18	(R2-R1)/20
Layer 19	(R2-R1)/20
Layer 20	(R2-R1)/20
Layer 21	(R2-R1)/20

button in the toolbar.

Delete Entities 1 (del1)

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Domain 3 only.

On the object , select Domain 52 only.

On the object , select Domain 22 only.

The geometry should look like this.

Now that the geometry has been generated, you can proceed to simplify some of its features.

Ignore Edges 1 (ige1)

In the toolbar, click

and choose .

In the window for , locate the section.

Clear the check box.

In the dialog box, type 54-106 183-205 in the text field.

In the window for , click

Ignore Vertices 1 (igv1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the dialog box, type 56-110 191-215 in the text field.

With this clean geometry, you can proceed to generate a few selections that will be used in the model.

In the window for , click

Geometry 1

Selection: Material 1

In the toolbar, click

and choose .

In the window for , type Selection: Material 1 in the text field.

Locate the section. Click

In the dialog box, type igv1: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76 in the text field.

Selection: Material 2

In the toolbar, click

and choose .

In the window for , type Selection: Material 2 in the text field.

Locate the section. Click

In the dialog box, type igv1: 3, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 56, 59, 61, 63, 65, 67, 69, 71, 73, 75 in the text field.