Acoustic Voxels: Computational Optimization of Modular Acoustic Filters

Acoustic filters are used to “produce a desired sound pitch or to attenuate undesired noise”, and find many applications in the real world, from wind instruments (e.g., flutes, trumpets, clarinets, etc.) and mufflers (e.g., engine noise muffler, acoustic earmuffs) to hearing aids. However, designing customized acoustic filters with specific properties remains a challenge, with the current approach(s) being “complicated and unintuitive” due to their iterative trail-and-error based process. While the design space for these acoustic filters has traditionally been limited to simple geometric shapes, the recent strides in additive manufacturing have enabled the fabrication of complex geometries, thus opening up new possibilities for the expanding the design space. Motivated by this, this paper presents Acoustic Voxels, a computational method for the design of acoustic filters, which relies on modular design based on a simple shape primitive (a hollow cube with circular holes on some or all of its faces) to build a complex assembly. Moreover, the modular design enables a “fast and accurate” estimation of a given assembly’s acoustic performance, which in turn allows for optimization of its structure to achieve desired filtering properties. While the paper provides detailed mathematical formulations of the proposed method, in the interest of brevity, the key details are presented in this summary.

The authors use a hollow cube with six cylinders, each extruding from a cube face, as the primitive acoustic resonator. The hollow cube has a variable length, and the extruding cylinders all have the same radius and length, permitting the assembly of complex structures by connecting an inlet and an outlet. Using a simple primitive allows:

• easy filling of an interior volume of arbitrary shape,

• fast and accurate computation of the transmission matrix for an arbitrary assembly, and

• a one-dimensional shape space (hollow cube with variable size) making it easy to sample from.

Maintaining the assumption that the (complex-valued) acoustic pressure and the velocity in the frequency domain are uniformly distributed across the cross sections of all ports, the authors extend the acoustic transmission matrix to six ports. Then, the multi-port transmission matrices are pre-computed for a set of frequency values (upto 4500 Hz) and cube sizes and stored in a database, and are later used to interpolate matrices for any given frequency and cube size, this speeding up the optimization step. The task of computing the structure of an assembly of primitives and their parameters given the desired acoustical properties is an inverse problem, and the authors tackle it using combinatorial and continuous optimization. The combinatorial aspect comes from the problem of deciding how to connect the primitive shapes, whereas determining their geometric parameters (cube size) is a continuous optimization task. The objective function is defined as mean squared error between the acoustic filtering quantity and th target quantity summer over a discretized range of frequencies. To constrain the assembly to a fixed shape, the 3D mesh of the desired shape is voxelized to form a lattice whose each grid cell represents a potential primitive resonator. A string of binary bits is used to indicate the lattice grid connectivity, and a vector is used to denote the cube sizes. The combinatorial optimization problem is solved by using an efficient version of sequential Markov Chain Monte Carlo sampling, known as Sequential Monte Carlo (SMC). SMC uses random connectivity sampling (achieved using rejection sampling of random bit strings) and connectivity perturbation (achieved by flipping a randomly selected bit in the bit string). For the continuous optimization, the authors compute the gradient of the objective function w.r.t. the sizes of the primitives and use limited-memoryu BFGS bounded (L-BFGS-B) method to minimize the objective. To summarize, the optimization is done using a hybrid method comprised of stochastic optimization using SMC and a gradient-based quasi-Newton optimization done using L-BFGS-B, and the authors note that the local gradient descent step of the continuous optimization “complements the combinatorial sampling” done using SMC. For a validation of their method, the authors compare their results against those obtained by a finite-element solver for mechanics and with independent, third party tests conducted by the world’s largest manufacturer of acoustic equipment. They demonstrated strong agreement of their results with those obtained from both the finite-element simulation and the laboratory experiments, while also being 77,000 times faster than the finite element method. Next, they present results for 3 applications:

• muffler design for engine noise and acoustic earmuffs, where they show that their design can selectively filter out certain noise frequencies and their harmonics,

• wind instrument design, where their design of 3 different trumpets can produce notes which are in tone with playable notes, and

• acoustic signatures, where they demonstrate that they can tag 3D shapes by their acoustic signatures as well as encode 4-bit strings in the transmission loss curves of 3D shapes.

This paper proposes Acoustic Voxels, a computational method based on combinatorial and continuous optimization for the modular design of acoustic filters using a simple primitive that achieve desired acoustical properties. The proposed method fabricates shapes based on additive manufacturing, and produces filter designs that are accurate when compared against finite-element simulations and in laboratory tests, while being much faster. The paper is written with great clarity, and the authors have taken special care to introduce appropriate background about potentially unfamiliar terms, making it quite easy to read and follow. One of the potential limitations of this work, however, is their assumption of unidirectional sound flow (“as the sound waves flow along the same direction”; Page 5; Section 4.2). This assumption discounts the possibility of reverberations, and incorporating them would make the outputs more robust and accurate, although it might be difficult to model them in the current multi-port transmission matrix.