A better compression driver? CutFEM 3D shape optimization taking viscothermal losses into account

TL;DR

The paper tackles the challenge of designing a radial phase plug for a compression driver while accounting for viscothermal losses. It introduces a boundary-layer viscothermal model expressed as a generalized impedance (Wentzell) boundary condition and integrates it into a 3D CutFEM framework with level-set geometry to enable gradient-based shape optimization. The authors develop a fully discrete shape calculus to compute derivatives of a frequency-range objective, and compare two strategies: maximizing output power and tracking an ideal, lossless response. Numerical results show that the tracking approach, when losses are included during optimization, can produce designs whose frequency responses closely match the ideal, suggesting a revival of radial phase-plug designs in practical drivers. Overall, the work advances viscothermal acoustics optimization in 3D and demonstrates the practical impact of combining boundary-layer theory with CutFEM and discrete shape calculus for engineering design.

Abstract

The compression driver, the standard sound source for midrange acoustic horns, contains a cylindrical compression chamber connected to the horn throat through a system of channels known as a phase plug. The main challenge in the design of the phase plug is to avoid resonance and interference phenomena. The complexity of these phenomena makes it difficult to carry out this design task manually, particularly when the phase-plug channels are radially oriented. Therefore, we employ an algorithmic technique that combines numerical solutions of the governing equations with a gradient-based optimization algorithm that can deform the walls of the phase plug. A particular modeling challenge here is that viscothermal losses cannot be ignored, due to narrow chambers and slits in the device. Fortunately, a recently developed, accurate, but computationally inexpensive boundary-layer model is applicable. We use this model, a level-set geometry description, and the Cut Finite Element technique to avoid mesh changes when the geometry is modified by the optimization algorithm. Moreover, the shape calculus needed to compute derivatives for the optimization algorithm is carried out in the fully discrete case. Applying these techniques, the algorithm was able to successfully design the shape of a set of radially-directed phase plugs so that the final frequency response surprisingly closely matches the ideal response, derived by a lumped circuit model where wave interference effects are not accounted for. This result may serve to resuscitate the radial phase plug design, rarely used in today's commercial compression drivers.

Figures (14)

• Figure 1: A conceptual cylindrical compression driver. Cross-section view through axis (left) and orthogonal to axis (right).
• Figure 2: The frequency response, given a fixed diaphragm acceleration, of the lumped model for various compression ratios $\kappa$ and chamber depths $d$. (Plots scaled by the maximum sound pressure of all graphs.)
• Figure 3: Left: a conceptual cylindrical compression driver with annular openings, placed according to Smith's guidelines, to suppress radial modes in compression chamber. Right: the function of the phase plug is to guide the sound from the slits into the throat of a horn. Cross-section view through axis.
• Figure 4: Cutaway drawing of a compression driver with radial slits.
• Figure 5: The computational domain with the initial shape of the phase-plug wall $\Gamma_\text{p}$. Symmetry conditions hold at $\Gamma_\text{sym}$, consisting of the full back side together with the surfaces indicated with tinted color in the figure.

Theorems & Definitions (8)

• Remark 2.2
• Remark 4.1
• Remark 6.1
• Remark 7.1
• Remark 8.1
• Theorem B.1
• Remark B.2
• Theorem B.3