COMSOL 6.4 - Shape Optimization of a Rectangular Loudspeaker Horn in 3D

Shape Optimization of a
Rectangular Loudspeaker Horn in 3D

This model shows how shape optimization can be used to design an acoustic loudspeaker horn. The horn is rectangular and this property can be maintained throughout the optimization by scaling the horn differently for different values of the z-coordinate. The scaling is performed isotropically in the x and y directions using a Bernstein polynomial. The objective function is computed using the built-in exterior field optimization function evaluated on-axis. The resulting design is exported to a second component for optimization with an objective function defined as the exterior field pressure evaluated off-axis. For this second optimization, the boundary of the horn is allowed to vary freely for, while only translation and rotation is allowed for the speaker boundary.

Problem Definition

The initial geometry of the horn is a pyramid, as shown in Figure 1. The horn and speaker are placed in an infinite baffle configuration. Symmetry is exploited, so only half of the horn is modeled.

Figure 1: The initial geometry consists of a sphere outside the horn. The horn is placed in an infinite baffle and has the shape of a pyramid. A circle at the bottom of the horn is used for the idealized loudspeaker speaker driver.

Acoustic Problem

The horn and baffle boundaries are modeled as sound hard. The loudspeaker driver is represented with a simplified circular piston and a lumped model; see Lumped Loudspeaker Driver. The lumped model is coupled to the acoustic domain using the feature. The feature is used to compute the radiated field at a distance of 1.2 m in front of the horn. The problem is only computed for a single frequency.

First Shape Optimization

The feature is used to scale the horn based on a Bernstein polynomial. The z-coordinate is used as argument for the function, and boundary condition are applied, so that no scaling is performed for the minimum and maximum value of the argument. The function is used in a feature to scale the horn isotropically in the x and y directions.

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(0,0,R0)). To limit the effects of impedance matching a constraint is imposed on the power of the speaker (the expression acpr.lsb1.P_front). The optimization problem is solved with the MMA optimization solver with an iteration limit of 20 and a move limit of 0.1 (to limit the design change between any two iterations)

The resulting design is exported to a new component using a dataset.

Second Shape Optimization

The design is now allow to change more freely in the sense that a feature is used for the sides of the horn, while a feature is used on the speaker boundary. The latter is set up to allow translation and rotation around the y-axis.

The solver settings and objective function are recycled from the first optimization, but the objective function is changed such that the exterior field SPL is evaluated off-axis using the expression comp2.Lp_pext_opt(R0*sin(theta0),0,R0*cos(theta0)). The evaluation point is rotated by an angle theta0 in the xz-plane. There is still a constraint on the speaker power, similar to the first on-axis optimization.

Results and Discussion

Figure 2 compares the radiation pattern for the initial design with the two optimized designs. The design optimized for off axis performance has the highest pressure both on- and off-axis, which can probably be attribute to the higher degree of design freedom.

Figure 2: The exterior-field sound pressure level is evaluated at different angles for the three designs.

Looking at the actual designs one can see that the last optimization essentially rotates the horn, as shown in Figure 3.

Figure 3: The total acoustic pressure is evaluated in a slice for the three designs. The intensity is plotted with streamlines, while half the speaker boundary is shown in gray.

Acoustics_Module/Optimization/rectangular_horn_shape_optimization

Modeling Instructions

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Global Definitions

Parameters 1

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

Geometry 1

Create the geometry. To simplify this step, insert a prepared geometry sequence.

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

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The geometry should now look like that in Figure 1. Note that the inserted geometry is parameterized and that the parameters used are automatically added to the list of global parameters in the model.

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