PDF
Bessel Panel
Introduction
The Bessel panel (patented by Philips, see Ref. 1) is a way to arrange a number of loudspeakers so that the angular sound distribution resembles that of a single speaker. This benchmark model is a study of the near and far sound fields created by 25 loudspeakers arranged as an array. The solution is compared with analytical results.
The sound field outside the computational domain is determined by using the built-in BEM-FEM Multiphysics coupling between Pressure Acoustics, Frequency Domain and Pressure Acoustics, Boundary Elements. The FEM interface is used to model the acoustics in the vicinity of the point sources and the BEM interface models the rest of the radiation problem.
Model Definition
A Bessel panel consists of a number of loudspeakers placed equidistantly in a row. The speakers are driven with different signals, some of them in counterphase. For a system of five speakers, the input (voltage and current) is weighted by the factors 1, 2, 2, −2, and 1. This results in an approximately homogeneous polar far-field distribution.
This model combines five Bessel panels in the same pattern to approximate a purely radial sound field. Figure 1 is a sketch of this assembly and the input to each speaker.
Figure 1: The Bessel panel combination used in the model. The circles represent the speakers and the numbers represent their input. Each row and each column is a Bessel panel in itself.
Model Equation
The Pressure Acoustics, Frequency Domain interface is used to model the pressure field near the point sources in this model. The interface solves the Helmholtz equation
(1)
where p is the pressure, ρ is the density of the medium, ω = 2π  f is the angular frequency, c is the speed of sound, and QL (SI unit: 1/s2) is a monopole source representing a loudspeaker. In this model, a frequency of  f = 200 Hz is used. Each loudspeaker L is represented by a monopole point source emitting a flow of strength SL = 10−2 m3/s nL , where nL is the weight factor shown in Figure 1. The Helmholtz equation is also solved in the Pressure Acoustics, Boundary Elements interface but with all right-hand-side sources equal to zero.
Analytical Solution
Each monopole point source is described mathematically as
where δ refers to the 3D Dirac delta function, xL is the location of the speaker L, and x is the spatial coordinate. Let G be the Green’s function for the Helmholtz equation, that is, the solution to the equation
where k is the wave number given by ω/c. The equation has an outgoing spherical wave solution, given by
The corresponding analytical solution to Equation 1 (for a constant density) is given by the convolution of ρG and , that is
(2).
This analytical solution is set up as a COMSOL Multiphysics variable and compared with the simulation result (see Figure 6).
Model Geometry and setup
The model geometry is shown in Figure 2. The distance between two neighboring loudspeakers is 0.5 m. A box extends 0.2 m around the array of points. The void domain exterior of the solid box is modeled with the BEM interface.
Figure 2: The model geometry.
Results and Discussion
Figure 3 shows the pressure distribution in a slice at x = 0.2 m close to the loudspeaker plane (located at x = 0 m). In this immediate vicinity of the sources, the sound field is still very inhomogeneous. The sound pressure level is shown in a slice at x = 0 m in Figure 4. In both plots the small white region can be removed by increasing the spatial resolution in the Grid 3D dataset used for the BEM evaluation.
Figure 3: Slice plot of the pressure distribution at 200 Hz. The slice is parallel with the yz-plane and situated at x = 0.2 m.
Figure 4: Slice plot of the sound pressure level distribution at 200 Hz. The slice is parallel with the yz-plane and situated at x = 0 m.
Figure 5 shows the exterior-field sound distribution at a distance of 100 m from the speakers. Note that the scale limits are equal to the global extremes of the sound pressure level. Hence the sound pressure level in any two given directions does not differ by more than 3 dB.
Figure 5: Sound pressure level (dB) at a distance of 100 m from the loudspeakers represented as a 3D radiation pattern plot.
Figure 6 plots the computed exterior-field pressure at a radial distance of 100 m versus polar angle in the positive xz-plane and compares it to the analytical solution. As the plot shows, the computed solution is very close to the analytical solution. The accuracy could probably be increased slightly by refining the mesh.
Figure 6: Sound pressure level (dB) at a radial distance of 100 m in the xz-plane (zero azimuthal angle) as a function of the polar angle from the xy-plane. The blue line represents the computed solution and the green line the analytical solution.
Reference
1. “Bessel Panels — High-power Speaker Systems with Radial Sound Distribution,” Technical Publication 091, Philips Export BV, 1983.
Application Library path: Acoustics_Module/Tutorials,_Pressure_Acoustics/bessel_panel
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Acoustics>Pressure Acoustics>Pressure Acoustics, Frequency Domain (acpr).
3
Click Add.
4
In the Select Physics tree, select Acoustics>Pressure Acoustics>Pressure Acoustics, Boundary Elements (pabe).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select General Studies>Frequency Domain.
8
Click  Done.
Geometry 1
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 0.4.
4
In the Depth text field, type 2.4.
5
In the Height text field, type 2.4.
6
Locate the Position section. From the Base list, choose Center.
7
Click  Build Selected.
Point 1 (pt1)
1
In the Geometry toolbar, click  More Primitives and choose Point.
2
In the Settings window for Point, locate the Point section.
3
In the y text field, type -1.
4
In the z text field, type -1.
5
Click  Build Selected.
Array 1 (arr1)
1
In the Geometry toolbar, click  Transforms and choose Array.
2
Click the  Wireframe Rendering button in the Graphics toolbar.
3
Select the object pt1 only.
4
In the Settings window for Array, locate the Size section.
5
In the y size text field, type 5.
6
In the z size text field, type 5.
7
Locate the Displacement section. In the y text field, type 0.5.
8
In the z text field, type 0.5.
9
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
10
From the Show in physics list, choose Point selection.
This makes the object Array 1 available as a predefined point selection, for example when you later specify flow point sources.
11
Click  Build All Objects.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
S
0.01[m^3/s]
0.01 m³/s
Flow source
f0
200[Hz]
200 Hz
Frequency
c0
343[m/s]
343 m/s
Speed of sound
Definitions
Variables 1
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Qs
-S*i
m³/s
Source strength
4
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Point.
5
Click  Paste Selection.
6
In the Paste Selection dialog box, type 5 9 25 29 in the Selection text field.
7
Click OK.
Variables 2
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Qs
2*S*i
m³/s
Source strength
4
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Point.
5
Click  Paste Selection.
6
In the Paste Selection dialog box, type 6 10 14 26 in the Selection text field.
7
Click OK.
Variables 3
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Qs
-2*S*i
m³/s
Source strength
4
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Point.
5
Click  Paste Selection.
6
In the Paste Selection dialog box, type 7 8 15 19 20 24 27 28 in the Selection text field.
7
Click OK.
Variables 4
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Qs
-4*S*i
m³/s
Source strength
4
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Point.
5
Click  Paste Selection.
6
In the Paste Selection dialog box, type 11 17 18 22 23 in the Selection text field.
7
Click OK.
Variables 5
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Qs
4*S*i
m³/s
Source strength
4
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Point.
5
Click  Paste Selection.
6
In the Paste Selection dialog box, type 12 13 16 21 in the Selection text field.
7
Click OK.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Air.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Air (mat1)
Add the material to all domains and the infinite void selection. This is all of the surroundings, where the BEM problem is solved.
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose All domains and voids.
Pressure Acoustics, Frequency Domain (acpr)
Monopole Point Source 1
1
In the Model Builder window, under Component 1 (comp1) right-click Pressure Acoustics, Frequency Domain (acpr) and choose Points>Monopole Point Source.
2
In the Settings window for Monopole Point Source, locate the Point Selection section.
3
From the Selection list, choose Array 1.
4
Locate the Monopole Point Source section. In the QS text field, type Qs.
Pressure Acoustics, Boundary Elements (pabe)
Apply the BEM interface to the surroundings by only selecting the voids, that is, not the solid where the FEM interface is used.
1
In the Model Builder window, under Component 1 (comp1) click Pressure Acoustics, Boundary Elements (pabe).
2
In the Settings window for Pressure Acoustics, Boundary Elements, locate the Domain Selection section.
3
From the Selection list, choose All voids.
Multiphysics
Now, set up the multiphysics coupling between the BEM and the FEM pressure acoustics models.
Acoustic BEM-FEM Boundary 1 (aab1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Boundary>Acoustic BEM-FEM Boundary.
2
In the Settings window for Acoustic BEM-FEM Boundary, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
Definitions
Now set up the analytical expression for the far field (see Equation 2). Call this variable p_an. You will use it later for comparison with the numerical results.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Point.
4
From the Selection list, choose Array 1.
Variables 6
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
r
sqrt((dest(x)-x)^2+(dest(y)-y)^2 +(dest(z)-z)^2)
m
Distance between source and observation point
p_an
intop1(Qs*acpr.omega*acpr.rho*exp(-i*acpr.k*r)/(4*pi*r))
Pa
Analytic pressure at observation point
The operator dest() evaluates its argument on the destination side, independently of the integration. In other words, p_an is a variable defined on the modeling domain. When evaluated in the far field, it gives the analytical acoustic pressure in the point (x,y,z) to use in the radiation pattern plot. The variables are defined globally (entire model).
Mesh
Proceed and generate the mesh using the Physics-controlled mesh functionality. The frequency controlling the maximum element size is per default taken From study. Set the desired Frequencies in the study step. In general, 5 to 6 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. In this model, we use 6 elements per wavelength; the default Automatic is to have 5.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
In the Frequencies text field, type f0.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarse.
4
Locate the Pressure Acoustics, Frequency Domain (acpr) section. From the Number of mesh elements per wavelength list, choose User defined.
5
In the text field, type 6.
6
Click  Build All.
7
Click the  Go to Default View button in the Graphics toolbar.
Study 1
In the Home toolbar, click  Compute.
Results
Acoustic Pressure (acpr)
After solving the model, six default plots have been created. The first three stem from the FEM model (Pressure Acoustics, Frequency Domain) and the last three from the BEM model (Pressure Acoustics, Boundary Elements).
The FEM interface creates surface plots of the pressure and the sound pressure level, as well as an isosurface plot.
The BEM interface automatically creates a Grid 3D dataset where the BEM solution can be evaluated (using kernel evaluation). The first of the three BEM plots shows the pressure on the BEM boundary, in this case this is the boundary in common with the FEM interface. The last two plots depict the pressure and sound pressure levels on the BEM surface as well as in cross sections through the grid dataset.
Take a look at the default plots and then modify them as follows.
Grid 3D 1
1
In the Model Builder window, expand the Results>Datasets node, then click Grid 3D 1.
2
In the Settings window for Grid 3D, locate the Parameter Bounds section.
3
Find the First parameter subsection. In the Minimum text field, type -2.
4
In the Maximum text field, type 2.
5
Find the Second parameter subsection. In the Minimum text field, type -6.
6
In the Maximum text field, type 6.
7
Find the Third parameter subsection. In the Minimum text field, type -6.
8
In the Maximum text field, type 6.
9
Click to expand the Grid section. In the y resolution text field, type 60.
10
In the z resolution text field, type 60.
Multislice 1
1
In the Model Builder window, expand the Acoustic Pressure (pabe) node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Multiplane Data section.
3
Find the x-planes subsection. From the Entry method list, choose Coordinates.
4
In the Coordinates text field, type 0.2.
5
Find the y-planes subsection. In the Planes text field, type 0.
6
Find the z-planes subsection. In the Planes text field, type 0.
Surface 1
In the Model Builder window, right-click Surface 1 and choose Disable.
Multislice 2
1
In the Model Builder window, under Results>Acoustic Pressure (pabe) right-click Multislice 1 and choose Duplicate.
2
In the Settings window for Multislice, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (sol1).
4
Locate the Expression section. In the Expression text field, type acpr.p_t.
5
Click to expand the Title section. From the Title type list, choose None.
6
Click to expand the Inherit Style section. From the Plot list, choose Multislice 1.
7
In the Acoustic Pressure (pabe) toolbar, click  Plot.
8
Click the  Go to YZ View button in the Graphics toolbar.
Compare the result with that in Figure 3.
Sound Pressure Level (pabe)
Compare the result with that in Figure 4.
Go back to the default 3D view and then create plots of the spatial response of the Bessel panel.
1
Click the  Go to Default View button in the Graphics toolbar.
3D Spatial Response
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type 3D Spatial Response in the Label text field.
3
Locate the Color Legend section. Select the Show units check box.
Radiation Pattern 1
1
In the 3D Spatial Response toolbar, click  More Plots and choose Radiation Pattern.
2
In the Settings window for Radiation Pattern, locate the Expression section.
3
In the Expression text field, type pabe.Lp_t-74.4.
4
Locate the Evaluation section. Find the Angles subsection. In the Number of elevation angles text field, type 50.
5
In the Number of azimuth angles text field, type 100.
6
Find the Sphere subsection. From the Sphere list, choose Manual.
7
In the Radius text field, type 100.
8
Locate the Coloring and Style section. From the Grid list, choose Finer.
9
In the 3D Spatial Response toolbar, click  Plot.
10
Click the  Zoom Extents button in the Graphics toolbar.
Compare the result with the plot in Figure 5. Note that you may need to click Zoom Extents when looking at other figures again.
Spatial Response in xz-plane
1
In the Home toolbar, click  Add Plot Group and choose Polar Plot Group.
2
In the Settings window for Polar Plot Group, type Spatial Response in xz-plane in the Label text field.
Radiation Pattern 1
1
In the Spatial Response in xz-plane toolbar, click  More Plots and choose Radiation Pattern.
2
In the Settings window for Radiation Pattern, locate the Expression section.
3
In the Expression text field, type pabe.Lp_t.
4
Locate the Evaluation section. Find the Angles subsection. In the Number of angles text field, type 360.
5
Find the Evaluation distance subsection. In the Radius text field, type 100.
6
Find the Normal vector subsection. In the y text field, type 1.
7
In the z text field, type 0.
The default reference direction (1,0,0) means that 0° in the polar plot corresponds to the positive x-axis direction.
8
Click to expand the Legends section. Select the Show legends check box.
9
From the Legends list, choose Manual.
10
In the table, enter the following settings:
Legends
Computed
Radiation Pattern 2
1
Right-click Radiation Pattern 1 and choose Duplicate.
2
In the Settings window for Radiation Pattern, locate the Expression section.
3
In the Expression text field, type 10*log10(0.5*abs(p_an)^2/abs(acpr.pref_SPL)^2).
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
5
Find the Line markers subsection. From the Marker list, choose Point.
6
In the Number text field, type 90.
7
Locate the Legends section. In the table, enter the following settings:
Legends
Analytical
8
In the Spatial Response in xz-plane toolbar, click  Plot.
What you see now is the sound pressure level at a distance of 100 m from the panel as a function of the polar angle at zero azimuthal angle. This plot should resemble the one in Figure 6.