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A compression driver is an electro-acoustic transducer with considerably higher efficiency than direct radiating loudspeakers, thanks to the increased radiation resistance caused by a large vibrating diaphragm placed in a compression chamber with small openings. The transition section between compression chamber and output waveguide, the phase plug, must be carefully designed to avoid irregularities in the output sound pressure level (SPL) as a function of frequency. Here we present a shape optimization method based on an implicit level-set description and adjoint sensitivity analysis, which enables a large number of design parameters and vast design freedom. The CutFEM approach, a fictitious domain finite element method, removes the need for mesh updates and makes the method robust and computationally inexpensive. Numerical experiments for a generic annular diaphragm compression driver are presented, with optimized designs showing only minor frequency irregularities. Two different objective functions are considered: one for maximum SPL and one where the SPL is fitted to that of a hypothetical ideal design; the latter approach is found to be more effective in reducing irregularities. Visco-thermal boundary-layer losses are included in a post-processing step, and, though the influence of losses is clearly noticeable, the overall performance is similar and the optimized designs still outperform the original design.

The linearized, compressible Navier-Stokes equations can be used to model acoustic wave propagation in the presence of viscous and thermal boundary layers. However, acoustic boundary layers are notorious for invoking prohibitively high resolution requirements on numerical solutions of the equations. We derive and present a strategy for how viscous and thermal boundary-layer effects can be represented as a boundary condition on the standard Helmholtz equation for the acoustic pressure. This boundary condition constitutes an O (delta) perturbation, where delta is the boundary-layer thickness, of the vanishing Neumann condition for the acoustic pressure associated with a lossless sound-hard wall. The approximate model is valid when the wavelength and the minimum radius of curvature of the wall is much larger than the boundary layer thickness. In the special case of sound propagation in a cylindrical duct, the model collapses to the classical Kirchhoff solution. We assess the model in the case of sound propagation through a compression driver, a kind of transducer that is commonly used to feed horn loudspeakers. Due to the presence of shallow chambers and thin slits in the device, it is crucial to include modeling of visco-thermal losses in the acoustic analysis. The transmitted power spectrum through the device calculated numerically using our model agrees well with computations using a hybrid model, where the full linearized, compressible Navier-Stokes equations are solved in the narrow regions of the device and the inviscid Helmholtz equations elsewhere. However, our model needs about two orders of magnitude less memory and computational time than the more complete model. 

Fictitious domain methods are attractive for shape optimization applications, since they do not require deformed or regenerated meshes. A recently developed such method is the CutFEM approach, which allows crisp boundary representations and for which uniformly well-conditioned system matrices can be guaranteed. Here, we investigate the use of the CutFEM approach for acoustic shape optimization, using as test problem the design of an acoustic horn for favorable impedance-matching properties. The CutFEM approach is used to solve the Helmholtz equation, and the geometry of the horn is implicitly described by a level-set function. To promote smooth algorithmic updates of the geometry, we propose to use the nodal values of the Laplacian of the level-set function as design variables. This strategy also improves the algorithm's convergence rate, counteracts mesh dependence, and, in combination with Tikhonov regularization, controls small details in the optimized designs. An advantage with the proposed method is that the exact derivatives of the discrete objective function can be expressed as boundary integrals, as opposed to when using a traditional method that uses mesh deformations. The resulting horns possess excellent impedance-matching properties and exhibit surprising subwavelength structures, not previously seen, which are possible to capture due to the fixed mesh approach.