Steering of Acoustic Reﬂection from Metasurfaces through Numerical Optimization

Space–coiling metamaterials that exploit the so–called generalized Snell’s law are suitable for the subwavelength manipulation of acoustic waves, leading to extraordinary refracting and reﬂecting metasurfaces. Typically, space–coiling metasurfaces are characterized by a narrow operating range in frequency due to the intrinsic connection between the design wavelength and the characteristic dimensions of the metasurface. The main goal of the present work is to design a broadband and modular space–coiling based metasurface, extending the range of frequency where the eﬀective metabehaviour is achieved. The metasurface building set is composed by eight diﬀerent unit cells, each introducing a tailored phase shift in the ﬁeld, that can be combined side by side to produce the desired acoustic eﬀect. The design parameters of each cell are selected as the result of an optimization process that minimizes the dependency on the operating frequency of the metabehaviour, keeping the overall thickness below a quarter of the design wavelength. Signiﬁcant results are obtained for benchmark problems.

, nevertheless it is worth mentioning a specific class of acoustic metamaterials, so called "space-coiling metamaterials". While the majority of acoustic metamaterials exploit local resonances to produce their effects, the space-coiling approach is based on labyrinthine structures that can induce several peculiar behaviours.
When a metamaterial device is built with a thickness (deeply) below the working wavelength, it is called a metasurface. Space-coiling metamaterials are inherently suitable for designing metasurfaces due to their feature of coiling up space. This reduced thickness is often desirable for a metamaterial as it allows its applicability in a broader range of real applications, hence there is a great interest in research and development of metasurfaces. The most interesting designs that can be found in literature present metasurfaces engineered to show peculiar behaviours like arbitrarily-shaped virtual surfaces to modify the local reflection angle [16][17][18], or to obtain anomalous transmission for lens-like behaviors or even to transform the propagation pattern from spherical to plane or surface waves [19,20].
Typically, the design of a metasurface for anomalous reflection/refraction is based on the choice of an operating wavelength λ = λ 0 , that allows to define the constructive parameters of the surface itself, and the resulting effective bandwidth is typically narrow. Some authors investigated so far the off-design performance of their metadevices in the attempt to obtain broadband effectiveness [21][22][23] and explored also different design strategies, such as the use of pentamode metamaterials for modifying the local speed of sound [24], even with remarkable results. However, their advancements are not transferable to the design of a space-coiling broadband acoustic metasurface.
An interesting observation can be found in the work by [25], where it is pointed out that a non-dispersive relation is required between the frequency and the phase shift gradient provided by the metasurface in order to achieve a broadband behaviour. This linear relation (or inverse with wavelength), in the cited work, is obtained using elementary cells with straight channels as building blocks. Even though this geometrical choice allows the designed phase shift gradient to remain unaltered for frequencies different from the design one, not using a space-coiling strategy necessarily yields to a higher required thickness of the surface, at least half of the wavelength of the lowest frequency of interest.
The aim of the present work is to improve the broadband behaviour of a metasurface for anomalous reflection, based on space-coiling metamaterials. The adoption of the space-coiling technology, joint with optimization, empowers us to overcome the issue of the limited bandwidth of the device and, at the same time, exploit the advantages ensured by its reduced thickness.
The design of the metasurface is described in Sec. II, followed by the formalization of the optimization process in Sec. III. Results from the optimization are reported in Sec. IV along with the numerical simulations of the optimized metasurface.

II. Metasurface
The design of a metasurface for anomalous acoustic reflection usually starts with the design of the elementary cells to be used as 'bricks' of the device, each one introducing a different phase shift ∆φ = φ r − φ i in the acoustic reflected or transmitted field, in the full range 0-2π. This approach is based on the generalized Snell's law for acoustic reflection and refraction [26], that stems out from its analogous in optics [27]: where λ is the wavelength of the incident wave, θ i is its angle with respect to the metasurface, θ r is the reflection angle, θ t is the refraction angle, n is the refraction index of the incidence region (n equals to 1 in air). These equations link the extraordinary reflection/refraction behaviour with the presence of a gradient of induced phase delay in the reflected/refracted wave on the metasurface; when there is no induced phase gradient on the surface the relation between incident and reflected wave is, as expected, described by the traditional Snell's law. Combining the elementary units, the designer would be able to reproduce a phase gradient profile at will and hence obtain the desired local extraordinary reflection angle by discretizing in space the continuous function φ(x, λ). The unit cell adopted in the present work is illustrated in Figure 1. a y = λ 0 /8 is the width of the cell.
w = 1 mm is the thickness of the internal walls.
d = (a x − w n w )/n w is the channel width.
l = a y − d − 2t is the width of the internal walls.

Fig. 1 Geometry of the reference unit cell (n
The acoustic response of the elementary cells is obtained from direct linear lossless acoustic simulations using the commercial Finite Element Method (FEM) solver Comsol Multiphysics®, in terms of the argument of the complex reflection coefficient of the cell, ∆φ = arg(R). In order to obtain an adequate resolution for the design of space delay gradients, the range 0-2π is chosen to be divided in intervals of π/4, therefore it will be covered using eight cells, each giving a phase delay spaced by π/4. The resulting metasurface is in consequence modular: the set of eight elementary cells represents the unitary element of the metasurface, that can be replicated to obtain a surface with the desired extension. This modularity is not achievable following the approach described by Zhu et al. in [25]. Despite the remarkable results obtained at off-design frequencies for a small metasurface, a repeatable elementary block of cells is not identifiable, as at off-design frequencies an undesired jump in the phase delay gradient would arise at any conjunction of two metasurface elements, thus leading to an undesirably growing thickness together with the extension of the device. The phase delay gradient continuity between the last and the first cell of a metasurface unit will be a fundamental requirement during the optimization, allowing to keep the modularity of the metasurface also at off-design conditions.

III. Optimization problem definition
The problem of the phase manipulation of acoustic waves by a metasurface is addressed through an optimization process, including in the merit function the desired metabehaviour of the elementary cells, in order to obtain their constructive parameters as solution. As previously stated, one of the major goals to be reached with this work through the optimization approach, is the improvement of the off-design behaviour, i.e. the broadband effectiveness, of a space-coiling metasurface.
A generic optimization problem consists in the research of the set of variables x that yields to a minimum of the N J objective functions J k (x, y) while satisfying the N g + N h constraints g(x, y) and h(x, y), and can be formalized as follows: where y is the vector of the parameters, x is the vector of the N x design variables bounded by x L n and x U n in the design space D, g i (x) are the N g inequality constraints and h j (x) are the N h are equality constraints.
The mathematical formalization of the objective function plays a key role in the formulation of an optimization problem. In the present study, the objective includes the phase shift provided by each elementary cell within the considered frequency range; these phase delays change accordingly to the desired phase shift target dynamics. A fundamental requirement for the optimized set of cells is the capability to give a phase delay covering the full 0-2π range, in order to be effectively combined to design the desired ∂∆φ/∂ x. To reach the required phase delay ∆φ dynamics, it is necessary to correctly include the phase shift provided by the metasurface in the objective function. The distance between the current global phase delay ∆φ(x, f ) and the target phase delay ∆φ(x, f ) has to be quantified and included into the optimization process. To this end, the distance δ p between ∆φ(x, f ) and ∆φ(x, f ) in the L p space can be defined in the space-frequency domain D = [x min , x max ] × [ f min , f max ] as: being ∆x = x max − x min the length of the metasurface and ∆ f = f max − f min the analyzed frequency interval. The distance δ p defined above represents the objective of the optimization problem, and the L 2 norm (euclidean distance) appears to be a good candidate to represent the difference between the actual phase shift and the target one in the domain D. It is interesting to note that the spatial variable x can be expressed as function of the number of cells N c and of the cell size a y (here supposed constant). The objective function of the problem can be numerically evaluated as T is the design variables vector, collecting all the cells' geometrical variables, and y is the vector of the parameters (ambient temperature and pressure, medium density, flow velocity, etc.). Table 1 Design variables of each cell: reference value, lower bound and upper bound.

Design Variable Reference Lower Bound Upper Bound
The value of t max is dependent on a x and n w to guarantee the consistence of the constructive parameters and its value can be assessed evaluating an inequality constraint as it follows: It is worth noting that in this work, instead of solving the original constrained problem, a pseudo-objective function (including the constraint) has been defined using the quadratic penalty function method. The minimization is provided by a Particle Swarm Optimization (PSO) algorithm, originally introduced by Kennedy and Eberhart [28]. PSO is based on the social-behavior metaphor of a flock of birds or a swarm of bees searching for food, and belongs to the class of heuristic algorithms for evolutionary derivative-free global optimization. The used PSO is an original deterministic implementation (DPSO) of the CNR-INSEAN Resistance & Optimization team [29,30]. Systematic studies on both the formulation and the performance of DPSO can be found in [31,32], with the definition of suitable guidelines for SBO (Simulation-Based Optimization) applied on ship design. Moreover, an initial efficiency comparison between the multiobjective DPSO and the genetic algorithm has been performed in [33] for aeronautical problems.

IV. Numerical results
As benchmark case, the array of eight cells is used to build a metasurface able to steer the reflected field acting like an inclined wall. The ∂∆φ(x, λ, χ)/∂ x is designed to obtain a reflection angle θ r = π/6 + θ i with λ 0 = 0.1 m ( f 0 = 3430 Hz) according to Eq.1. For the case of a plane wave impinging the surface, two sets of results are presented: the first is of a metasurface built with non-optimized cells able to reproduce the desired behaviour at the design frequency, for the second the range of operating frequencies is extended through the single-objective optimization previously described.
The objective function in Eq.(4) was built using N f = 9 frequencies equally spaced in the range 3230 − 3630 Hz and, for each frequency, the target phase delay of the i-th cell was defined such that ∆φ( The latter condition is introduced to drive the optimization towards a set of cells able to guarantee the modularity of a metasurface built using them. The optimization analysis has been performed within the framework FRIDA (FRamework for Integrated Design of Aircraft). The minimization problem was solved making use of 1000 iterations with 72 particles, i.e. 72.000 objective function evaluations. Less than 10.000 iterations are needed for the optimization algorithm to reach a substantial convergence (Fig. 2), with a reduction of the objective function value of the 70% of its reference value. The non-optimized metasurface is able to guarantee the desired extraordinary reflection just in a neighborhood (a) f = 3080 Hz.

Fig. 3 Scattered fields from the non-optimized metasurface impinged by a plane wave incoming from the upper side at different frequencies.
of the design frequency. As can be seen from Fig.3, at slightly lower or higher frequencies, the performance of the metasurface rapidly deteriorates.
The reflected pressure fields from the optimized metasurface when impinged by a plane wave are shown in Fig.4 for the design frequency f 0 and two off-design frequencies, exhibiting an effective steering even though with slightly different angles. This change in the reflection angle is expected from Eq.1 as the metasurface is optimized aiming at a ∂∆φ/∂ x constant in frequency.
To better compare the performance of the reference and the optimized solutions, the absolute value of the scattered pressure is evaluated over a reference line L parallel to the metasurface at x = 1 m in an octave band centered in the design frequency. Figure 6 clearly shows that the optimization widened the effective frequency band: when the extraordinary reflection is happening, the scattered acoustic pressure is focused in the lower part of the y-axis, that corresponds to the right hand side of Figs 3 and 4, while ineffective bands, shadowed in pictures, are characterized by a non organized reflection pattern.

V. Conclusions
The exploitation of the generalized Snell's law allows to design metasurfaces for extraordinary reflection. As shown in literature, the operative range of such deeply subwavelength devices is narrow in frequency. In this article, a new approach to improve the off-design performance of space-coiling cells and metasurfaces is developed by means of numerical optimization. The minimized objective function deals with the L 2 norm, i.e. the euclidean distance, between the required phase delay and the one induced by each cell, aiming at the phase matching, not only for the design frequency, but over a desired range. The optimization approach demonstrated to be effective in the design of this extraordinary reflecting surface, broadening the effective frequency range of the device. The algorithm was able to provide a set of eight cells that guarantee the desired phase delay dynamic while respecting all the imposed geometric constraints. The process can be interpreted as a robust design with respect to the actual working frequency of the device, and it is interesting to note how the resulting effective range is closely related to the frequency range considered in the objective function with f min and f max .