COMSOL 6.3 - Optimizing the Shape of a Horn

This example shows how to apply boundary shape optimization to a simple axisymmetric horn. For the sake of simplicity, the on-axis far-field sound pressure level is maximized for a single frequency and in a single direction. The focus is on the optimization procedure and optimization solver settings. The objective function is computed using the built-in exterior field optimization function. The deformation of the geometry is introduced using the Shape Optimization functionality of COMSOL Multiphysics.

The model was inspired by the work of Erik Bängtsson, Daniel Noreland, and Martin Berggren (Ref. 1).

This application requires the Acoustics Module and the Optimization Module.

Figure 1: The initial configuration is a simple cone (the z < 0 part of the gray area) in an infinite baffle.

Model Definition

A plane-wave mode feeds an axisymmetric horn radiating from an infinite baffle toward an open half space; see Figure 1. The radius of the waveguide attached to the horn throat is assumed to be fixed, as well as the depth of the horn and the size of the horn mouth attached to the baffle. By varying the curvature of the initially conical neck of the horn, its directivity and impedance can be changed.

The surface of the horn neck is parameterized assuming that the radius and the z position of the horn deviate from the simple cone by a set of 8th-order Bernstein polynomials. The maximum displacement (in the two directions) is given by the dmax parameter. The number of optimization variables is determined by the order of the polynomial. Using a higher order gives more freedom and potentially a better final value of the objective function, but it also makes the optimization process more sensitive and can generate a shape that is less suitable for production.

The dedicated built-in optimization function for the exterior-field sound pressure level is evaluated on axis and used as objective function (the expression Lp_pext_opt(0, Lfar), where

is the far-field evaluation distance). The optimization problem is solved with the MMA optimization solver with an iteration limit of 50 and no move limit.

Results and Discussion

By changing the shape of the horn within the limits of the selected parameterization, the on-axis sound pressure level can be raised by about 1 dB compared to the simple cone in Figure 1.

Figure 2: The final shape of the horn, optimized for on-axis SPL at 5000 Hz.

The improvement is rather small, because the initial design also shows a marked directivity, as can be seen from Figure 3. Obviously, the optimal shape with respect to on-axis SPL leads to deep undesirable minima in other directions.

Figure 3: Radiation plot of the original (dashed blue) and final (solid black) designs.

Optimizing with respect to a slight off-axis direction can give you a more uniform far-field pattern, but may also result in a deep minimum on the axis. Try for example to set the off-axis angle to 22°.

Varying the frequency reveals that the on-axis improvement is robust over a wide frequency band; see Figure 4.

Figure 4: The objective function is plotted as a function of the frequency for the initial as well as the optimized design.

To search for a stable and practically useful horn design, you might instead create a composite objective function as a weighted sum of transmission values evaluated for a number of discrete directions, or choose to minimize the deviation from the mean SPL over a range of angles. In addition, you would also want to optimize with respect to more than one frequency, and experiment with different parameterizations.

Notes About the COMSOL Implementation

COMSOL Multiphysics implements the parameterization using a feature.The mesh is allowed to move freely in the conical part of the horn, but otherwise kept fix.

The amplitude of the boundary displacement is restricted by the maximum displacement and the polynomial order. These constraints are intended to keep the mesh element volumes positive at all times.

A time-harmonic Pressure Acoustics, Frequency Domain interface solves for the pressure field inside the horn and in a small spherical domain surrounding its opening.

Set up the optimization problem in the study node.

Reference

1. E. Bängtsson, D. Noreland, and M. Berggren, “Shape Optimization of an Acoustic Horn,” Technical Report 2002-019, Department of Information Technology, Uppsala University, May 2002.

Acoustics_Module/Optimization/horn_shape_optimization

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

Geometry 1

Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.025.

Locate the section. In the text field, type -0.2.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.2.

In the text field, type 90.

Click

Click the

button in the toolbar.

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type 0 0.1 0.1 0.025 0.025 0.

In the text field, type 0 0 0 -0.175 -0.175 -0.175.

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
f0	5[kHz]	5000 Hz	Optimization frequency
df	50[Hz]	50 Hz	Optimization frequency bandwidth
Lfar	1[m]	1 m	Far-field evaluation distance

Component 1 (comp1)

Free Shape Domain 1

In the toolbar, click

and choose .

Select Domain 2 only.

Polynomial Boundary 1

In the toolbar, click

Select Boundary 8 only.

In the window for , locate the section.

From the dmax list, choose .

In the text field, type 0.03.

Locate the section. In the n text field, type 8.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Pressure Acoustics, Frequency Domain (acpr)

Port 1

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

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

This gives you a plane wave with the amplitude 1 Pa propagating in the positive z direction.

Exterior Field Calculation 1

In the toolbar, click

and choose .

Select Boundary 6 only.

In the window for , locate the section.

From the 0 list, choose .

Perfectly Matched Boundary 1

In the toolbar, click

and choose .

Select Boundary 10 only.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the fmax text field, type f0.

From the list, choose .

In the text field, type 10.

Click

Study 1 - Reference Solution

In the window, click .

In the window for , type Study 1 - Reference Solution in the text field.

Before starting the actual optimization, it is a good practice to check the model setup by solving the one with the default parameters. This way, you can also study the reference state which you intend to improve.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(3500,50,6500).

Locate the section. In the column of the table, under , clear the checkbox for .

In the toolbar, click

Results

Sound Pressure Level, 3D (acpr)

Click the

button in the toolbar.

In the window, under click .

Radiation Pattern 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type Lfar.

Radiation Pattern 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type Lfar.

Shape Optimization

In the window, under right-click and choose .

Acoustic Pressure (acpr), Acoustic Pressure, 3D (acpr), Exterior-Field Pressure (acpr), Exterior-Field Sound Pressure Level (acpr), Sound Pressure Level (acpr), Sound Pressure Level, 3D (acpr)

In the window, under , Ctrl-click to select , , , , , and .

Right-click and choose .

Reference solution

In the window for , type Reference solution in the text field.

Root

Next, add a new study for the optimization.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 2

Step 1: Frequency Domain

In the window for , locate the section.

In the text field, type f0-df/2 f0 f0+df/2.

In the window, click .

In the window for , type Study 2 - Optimized Solution in the text field.

Shape Optimization

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 50.

Clear the checkbox.

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


Expression	Description
comp1.Lp_pext_opt(0, Lfar)

Set up a MaxMin problems such that frequency associated with the worst objective function is prioritized.

From the list, choose .

In the toolbar, click

By making the nonlinear tolerance stricter than that of the optimization solver, you ensure that the optimization does not fail because each solution is not sufficiently converged. An optimality tolerance of 1e-4 is still stricter than the accuracy of this low-resolution finite element model.

Step 1: Frequency Domain

In the window, expand the > node, then click > .

In the window for , locate the section.

From the list, choose .

In the text field, type 1e-6.

Solution 2 (sol2)

In the window, expand the node.

In the window, expand the > > > node, then click .

In the window for , locate the section.

From the list, choose .

In the window, expand the > > > > node, then click .

In the window for , click to expand the section.

In the text field, type 1e-4.

Shape Optimization

In the window, under click .

In the window for , locate the section.

Select the checkbox.

From the list, choose .

In the toolbar, click

Results

Acoustic Pressure (acpr) 1

The first default plot is the acoustic pressure. It should look like the image below.

Sound Pressure Level (acpr) 1

In the window, click .

In the toolbar, click

Your plot of the sound pressure level should now look like Figure 2.

To see a direct comparison of the exterior-field polar pattern before and after optimization, modify the exterior-field plot of the sound pressure level in the rz-plane. Modify it to only plot the results in the positive half plane (z > 0), increase the resolution, and change some Coloring and Style options. The resulting plot of the exterior field should look like Figure 3. Note that 0 deg on the polar graph corresponds to the vertical z-axis.

Exterior-Field Sound Pressure Level (acpr) 1

In the window, click .

In the window for , locate the section.

From the list, choose .

In the text field, type 2.

Radiation Pattern 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. From the list, choose .

In the ϕ text field, type -90.

In the ϕ text field, type 180.

Find the subsection. In the text field, type Lfar.