3D shape optimization of loudspeaker cabinets for uniform directivity

Abstract

This paper presents a method to perform gradient-based shape optimization to minimize the root mean square deviation of the exterior acoustic sound pressure level distribution in front of an initially spherically shaped loudspeaker. The work includes several examples of how different multi-frequency optimization strategies can affect the final optimized design performance. This includes testing, averaging, and weighting of multi-frequency cost functions or using a minimax formulation. The shape optimization technique is based on an acoustic Boundary Element Method coupled to a Lumped Parameter loudspeaker model. To control and alter the deformation of the loudspeaker cabinet the optimization method adapts a spherical free-form deformation approach based on Bernstein polynomials. For the particular optimization problems presented, it is shown that improvements in the root mean square deviation of the sound pressure level in front of the loudspeaker can be achieved between 1 and 5 kHz. In the best-case scenario, less than a 1 dB sound pressure level (SPL) variation is observed between on-axis and a 70° off-axis response in the range 2 to 5 kHz. The widest frequency bandwidth and smoothest response of the root mean square deviation is found by utilizing the minimax formulation.

Appendices

Appendix 1: Renderings of the optimized designs

Photo-realistic renderings of the optimized designs are shown in the Fig. 20.

Fig. 20

Photo-realistic renderings of the optimized designs In the Figure, a is the design optimized with $\phi _{A,1}$ and $w_k=1$, b is the design optimized with $\phi _{A,1}$ and $w_k=w_{fd}$, c is the design optimized with $\phi _{B,1}$, d is the design optimized with $\phi _{A,2}$ and $w_k=1$, and e is the design optimized with $\phi _{B,2}$. Remark that all renderings are on the same scale to illustrate the difference in size between optimized designs

Appendix 2: Note on semi-analytic sensitivities and step size

The design sensitivities are obtained from the semi-analytical adjoint approach described in Sect. 7 meaning that matrix design derivatives are calculated with finite differences. Therefore, the sensitivities can be dependent on the step length. To show the sensitivity to step length ratio, four randomly chosen design sensitivity calculations for different step sizes are shown in Fig. 21. Additionally, the semi-analytical adjoint sensitivities are compared to a pure forward finite difference sensitivity at different step sizes. As is seen, the sensitivities are stable within the range of $10^{-5}$ to $10^{-8}$ and should therefore be chosen in this range.

Fig. 21

A set of four randomly chosen design sensitivities based on the frequency averaged cost function calculated for different step lengths. The sensitivities are both calculated with the semi-analytical adjoint approach (blue) and a forward finite difference scheme (red). The sensitivities are calculated for the initial design. (Color figure online)

Appendix 3: Mesh dependency of optimized design

When performing shape optimization without any re-meshing, there is the risk that the optimization becomes mesh dependent to an unacceptable degree. In such cases, the optimizer can optimize on numerical error rather than the physical effects. To study this potential mesh sensitivity, the design based on $\phi _{A,1}$ and equal weighting (the design in Fig. 9) is re-meshed with a denser mesh consisting of 24,002 nodes and 12,000 elements. The SPL frequency response for the $0^{\circ }$ and $70^{\circ }$ field point is shown in Fig. 22a and b, respectively. The Figures show close to no changes in the SPL response between the mesh used during optimization and the mesh that is re-meshed with the denser mesh. Hence, it is expected that the mesh utilized during optimization can be safely used.

Fig. 22

The SPL frequency response based on the optimized design in Figure 9 using the cost function $\phi _{A,1}$ with equal weighting. In the figure, a is calculated at $\psi = 0^{\circ }$ and b is calculated at $\psi = 70^{\circ }$. The blue curves are calculations based on the mesh used during optimization and the red curves are re-meshed with a denser mesh using 24,002 nodes and 12,000 elements. (Color figure online)

Appendix 4: Visualization of control points for optimized designs

The location of the control points for the five optimized designs is visualized in Fig. 23.

Fig. 23

The geometry of the optimized designs including the control points. In the figure, a is the optimized design using $\phi _{A,1}$ and $w_k=1$, b is the design using $\phi _{A,1}$ and $w_k=w_{fd}$, c is design using $\phi _{B,1}$, d is the design using $\phi _{A,2}$ and $w_k=1$, and e is the design using $\phi _{B,2}$