Optimal Design of Broadband, Low-Directivity Graded Index Acoustic Lenses for Underwater Applications

TL;DR

This work tackles the challenge of designing underwater acoustic lenses that are broadband and exhibit low directivity to mitigate multipath effects. It formulates an optimal-control problem on the Helmholtz equation with density and bulk-modulus controls, solved via a Lagrangian gradient to transition smoothly from a gradient-index profile to a realizable hexagonal lattice of solid inclusions in water. Three 2D lenses are optimized for different central frequencies to achieve broadband, omnidirectional performance, with validation against a coupled acoustic-elastic FE model and analysis of microstructure dynamics through a neural-network inverse-design mapping. The results offer a framework for robust, broadband underwater sensing and communication components, with extensions to 3D designs and experimental validation identified as future directions.

Abstract

Manipulating underwater pressure waves is crucial for marine exploration, as electromagnetic signals are strongly absorbed in water. However, the multi-path phenomenon complicates the accurate capture of acoustic waves by receivers. Although graded index lenses, based on metamaterials with smoothly varying properties, successfully focus pressure waves, they tend to have high directivity, which hinders practical application. This work introduces three 2D acoustic lenses made from a metamaterial composed of solid inclusions in water. We propose an optimization scheme where the pressure dynamics is governed by Helmholtz's equation, with control parameters affecting each lens cell's density and bulk modulus. Through an appropriate cost function, the optimization encourages a broadband, low-directivity lens. The large-scale optimization is solved using the Lagrangian approach, which provides an analytical expression for the cost gradient. This scheme avoids the need for a separate discretization step, allowing the design to transition directly from the desired smooth refractive index to a practical lattice structure. As a result, the optimized lens closely aligns with real-world behavior. The homogenized numerical model is validated against finite elements, which considers acoustic-elastic coupling at the microstructure level. When homogenization holds, this approach proves to be an effective design tool for achieving broadband, low-directivity acoustic lenses.

Figures (7)

• Figure 1: Schematic representation of the computational domain $\Omega$. The lens occupies the control region $\Omega_c$, filled by inhomogeneous properties accounting for the grating. The background fluid occupies the remaining domains. $\Omega_f$ is the focal region where the sensor is placed.
• Figure 2: \ref{['fig:metamat-a']} schematic of the two parametric families of cells composing the grating; \ref{['fig:metamat-c']} the properties attainable by varying the parameters of this two cells. The union of the blue and the orange areas outline the constraint for the control space for the optimization problem. Please note that the star shaped cell has 12 tips even if the drawings show 6 tips for the sake of simplicity.
• Figure 3: Optimization results for the sizing procedure at the three frequencies. \ref{['fig:sizing costs-a']} comparison of the performance achieved by the lens with smooth varying properties and by the discrete devices. The dashed line shows the expected quadratic trend of the four devices. \ref{['fig:sizing costs-b']} values of the cost functional and \ref{['fig:sizing costs-c']} gain with respect to the radius of the discrete lenses.
• Figure 4: Properties and performance of the optimized lenses. The dashed lines highlight the incidence acceptance cone of $\pm\qty{50}{\degree}$. For the sake of clarity, the polar plots do not show values below -5. Note that, since the symmetry of the problem, the three lenses show a strong symmetry about the $x$ axis.
• Figure 5: From left to right. The first two lines shows amplitude and phase of the transfer function $TF$ of the three lenses; the bottom line shows the phase distortion.