Nonlocal elastic metasurfaces: enabling broadband wave control via intentional nonlocality
Hongfei Zhu, Timothy F. Walsh, Bradley H. Jared, Fabio Semperlotti
aa r X i v : . [ phy s i c s . a pp - ph ] F e b Nonlocal elastic metasurfaces: enabling broadbandwave control via intentional nonlocality
Hongfei Zhu , Timothy F. Walsh , Bradley H. Jared , and Fabio Semperlotti Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana47907, USA Sandia National Laboratories, Albuquerque, New Mexico 87185, USA * e-mail: [email protected] † Sandia National Laboratories is a multimission laboratory managed and operated by National Technology andEngineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S.Department of Energy’s National Nuclear Security Administration. With main facilities in Albuquerque, N.M., andLivermore, C.A., Sandia has major R&D responsibilities in national security, energy and environmentaltechnologies, and economic competitiveness.
ABSTRACT
While elastic metasurfaces offer a remarkable and very effective approach to the subwalength control of stress waves, theiruse in practical applications is severely hindered by intrinsically narrow band performance. This work introduces the conceptof intentional nonlocality as a fundamental mechanism to design passive elastic metasurfaces capable of an exceptionallybroadband operating range. The nonlocal behavior is achieved by exploiting nonlocal forces, conceptually akin to long-rangeinteractions in nonlocal material microstructures, between subsets of resonant unit cells forming the metasurface. These long-range forces are obtained via carefully crafted flexible elements whose specific geometry and local dynamics are designed tocreate remarkably complex transfer functions between multiple units. The resulting nonlocal coupling forces enable achievingphase gradient profiles that are function of the wavenumber of the incident wave.The identification of relevant design param-eters and the assessment of their impact on performance are explored via a combination of semi-analytical and numericalmodels. The nonlocal metasurface concept is tested, both numerically and experimentally, by embedding a total-internal-reflection design in a thin plate waveguide. Results confirm the feasibility of the intentionally nonlocal design concept and itsability to achieve a fully passive and broadband wave control.
The concept of metasurface was originally pioneered in optics and, shortly afterwards, extended to acoustics . Metasur-faces quickly gained popularity thanks to their ability to achieve wave front control via deep subwavelength artificial interfaces.This characteristic stands in stark contrast with traditional metamaterials whose ability to control incoming wave fronts is lim-ited by the strict dependence between the size and number of unit cells, and the wavelength of the incoming wave. The mostdirect consequence of this physical constraint is the existence of a minimum dimension of a metamaterial slab (typically onthe order of several wavelengths) that is necessary to control incoming waves. On the contrary, metasurfaces are very thinlayers (typically a fraction of the dominant wavelength) that allow an abrupt control of the wave fronts, therefore offering apowerful alternative to circumvent a key practical limitation of metamaterials.The response of a metasurface subject to an incoming wave is described by the Generalized Snell’s Law (GSL) whichallows predicting directions of anomalous reflection and refraction across an interface characterized by a spatial phase gradient.It is indeed this capability to encode the most diverse phase gradient profiles within the metasurface that allows achievingunconventional effects including, but not limited to, wave front shaping , surface modes generation , and ultra-thin lenses .More recently, the concept of metasurface was also extended to elastodynamics in an effort to achieve abrupt controlof elastic waves in solid media. Initially, Zhu et al. explored the behavior of an elastic metasurface in transmission modeand investigated the possibility of controlling the direction and the shape of a refracted wave front generated by an incomingplanar wave. Later, the study was extended to show that metasurfaces could act as subwavelength barriers, therefore com-pletely preventing the transmission of elastic waves across the interface . Despite the great success in the application ofmetasurfaces to wavefront control and shaping, their intrinsic narrowband performance has greatly limited their use in real-world applications. The limited operating frequency range is a direct consequence of the underlying design based on locallyresonant unit cells. In passive metasurfaces, local resonances are used to achieve abrupt phase changes as propagating wavescross the interface. As the frequency of the incoming wave is detuned with respect to the operating frequency of the individual1nit cells, the metasurface looses quickly its efficiency and ability to control the wave.In order to address this important shortcoming of passive metasurfaces, researchers have explored the possibility to useactive designs, involving piezoactuators and control logics, to extend the operating range. Chen et al introduced a pro-grammable active elastic metasurface with sensing and actuating units, that achieved wave control functionalities over abroadband range of harmonic signals. Despite the good success of the active approach, the overall increased complexity andthe significant barriers in scaling the design to either larger structures or high intensity waves makes this approach not alwaysviable. It follows that the ability to synthesize fully passive broadband designs still represents a major research endeavor andholds the potential to revolutionize metasurface-based devices.To address this major challenge, we introduce the concept of intentional non-locality . The study and modeling of non-local system properties have been a long standing challenge in many areas of engineering and physics. While the generalconcept of non-locality holds a practically universal meaning, strictly connected to the idea of action at a distance, technicalnuances can arise depending on the specific area of application. In classical mechanics of solids, the concept of non-localityimplies that the local response of the medium at a target location (typically expressed in terms of a stress field) does not onlydepend on the state of the medium (typically expressed in terms of a strain field) at that same location but also from the stateof other distant locations. The ensemble of these distant points, whose state affects the response at the target location, forms asurface (in 2D) or a volume (in 3D) known as the area, or volume, of influence. In elastodynamics, the concept of nonlocalityresults in a dependence of the stress (or, equivalently, the strain) field on the wavenumber . Note that the typical nonlocalcharacter of many natural materials is very subtle and, in most applications, it is considered as a higher order effect.The concept of nonlocality has also found applications in electromagnetism. Compared to the mechanical properties,nonlocality has a much more pronounced effect on electromagnetic properties . This is also the reason why nonlocal de-signs have found natural applications on electromagnetic metasurfaces , metagratings and metamaterials . Morerecently, this concept was also extended to acoustic metasurfaces .To-date, no attempts were made to develop nonlocal elastic metasurfaces. In part, this technology gap follows from theweak nonlocal mechanical effect in most natural materials, as previously mentioned. In this study, we propose the concept ofnonlocality that is intentionally introduced into the design of an elastic metasurface in order to extend its operating range byleveraging wavenumber-dependent mechanical properties. More specifically, starting from a classical locally resonant elasticmetasurface design , we exploit the concept of action at a distance by coupling selected unit cells via carefully designedconnecting elements. These elements are conceived explicitly to create wavenumber-dependent coupling forces that mimic,at the macroscale, the nonlocal effect typical of nonlocal forces in elastic continuum microstructures.In this study, the concept of intentional nonlocality will be illustrated by applying it to a total-internal-reflection typeof metasurface (TIR-MS) , which was chosen as basic benchmark system. Note that, despite this choice, the proposedconcept is extremely general and applicable to any type of metasurface. A specific semi-analytical transfer-matrix model wasdeveloped to study the effect of the nonlocality on the dynamic behavior of the metasurface and to facilitate the synthesis ofpractical physical designs for numerical and experimental testing. Then, the nonlocal TIR-MS concept was embedded in athin plate waveguide and its performance assessed via a combination of numerical and experimental tests. In order to facilitate the understanding of the nonlocal design, this section provides a brief overview of the basic designprinciples of a classical passive elastic metasurface , which in the following will be referred to as the local design. In theclassical problem of wave transmission across an interface between dissimilar materials, the angle of refraction is entirelydetermined by the impedance mismatch between the two materials and it is carefully described by the Snell’s law. For a givenwavelength, the angle of refraction is fixed once the two materials are chosen. However, when in presence of a metasurfacethat encodes a prescribed spatial phase gradient (along the metasurface direction), the refracted angle is no longer controlledonly by the impedance mismatch between materials but also by the phase gradient itself. In fact, metasurfaces are typicallyemployed to control the reflection and refraction of waves between halves spaces made of the same material (hence withoutan intrinsic impedance mismatch).When in presence of an interface with a spatial gradient profile (Fig. 1a), the direction of the refracted wave front isobtained from the Generalized Snell’s Law (GSL) :sin ( θ t ) λ t − sin ( θ i ) λ i = π d φ d y (1)where d φ / d y is the phase gradient, while θ i and θ t indicate the directions of the incident and refracted waves, respectively.In passive elastic metasurfaces, the phase gradient is often realized by using a periodic array of locally resonant cellsproviding a linearly-varying, step-like phase modulation covering the range [0,2 π ]. Figure 1a shows a conceptual view of a ypical local elastic metasurface as well as a zoomed-in view of the locally resonant unit cell. The latter consists of a hollowframe with an internal resonator realized via a small mass attached to two slender beams connected to the frame. Previousstudies showed that the dynamics of the resonator can be easily controlled in order to match specific phase profiles anddesign constraints while limiting the fabrication complexity. Figure 1. (a) Schematic view of an elastic waveguide with an embedded local metasurface. Some of the fundamentalparameters such as the angle of incidence θ i , the angle of anomalous refraction θ t , and the phase gradient d φ d y are shown. Theinset also shows a schematic view of a typical locally resonant unit from which the metasurface is assembled. (b) Schematicillustration introducing the concept of intentional nonlocality realized via flexible elements connecting different unit cells.While, in the local design, the desired phase gradient profiles can be efficiently achieved at a single target frequency, itis not possible to maintain the target phase gradient over a wide frequency range. This limitation is a direct consequenceof the fundamental operating mechanism of the metasurface based on local resonances. A possible approach to overcomethis limitation could include the design of a multi-resonant unit cells (e.g. using internal multi-degree-of-freedom resonators)within the basic local metasurface design. The increased number of design parameters would improve the ability to tune thephase gradient profile versus frequency. Although conceptually possible, this approach would likely result in a very complexdesign of the unit cells and in an operating range still limited by the number of local resonances available per unit cell. The concept of nonlocal metasurface presented in this study is based on the fundamental idea of a nonlocal supercell. Anonlocal supercell consists of a prescribed number of locally resonant units whose internal resonators are coupled to eachother via specifically designed flexible links. The role of these links is to provide, at macroscopic level, an action at a distancethat is conceptually equivalent to the role that nonlocal forces play in nonlocal microstructures. The main objective of theselong-range connections is to enrich the dynamics of each individual unit cell which is now influenced also by the dynamics ofdistant cells. The most immediate effect of these coupling forces is the generation of a wavenumber-dependent dynamics, foreach individual cell, that can be made as complex as needed to achieve a target phase profile over a broad range of operatingfrequencies. Once the nonlocal supercell design is available, the nonlocal metasurface can be assembled by realizing a periodicdistribution of identical nonlocal supercells.For elastic metasufaces, a possible implementation of the nonlocal design is shown in Fig. 1b. The design includes flexiblebeam-like links that connect the local resonators of multiple individual units. As explained in detail in the following, in orderto obtain complex phase profiles over an extended frequency range, the connecting elements must exhibit an elaborate localdynamics that can be obtained in a variety of ways, including proper choices of the cross-sectional properties, materials, andlocal resonances. Within this general design framework, the synthesis of the links’ final configuration requires the formulationand solution of an optimization problem. The objective of the optimization procedure is to select the links’ physical parameterscapable of producing the desired nonlocal forces that ultimately enable maintaining the target phase profile.Dedicated analysis tools are needed in order to simulate the response of these nonlocal systems, specifically, to understandthe effect of the design parameters on the physical behavior of the supercell and to allow the synthesis of physically realizabledesigns. The overall modeling approach follows the following strategy. The nonlocal supercells are initially modeled using implified lumped parameter (e.g. mass-spring) systems that allows formulating the system dynamics via transfer matrix (TM)method. In the TM approach, the resonant unit is represented via a lumped parameter system, conceptually shown in Fig. 2a,which allows capturing the relevant dynamics while requiring a reduced set of system parameters. This simplified approachis possible because the metasurface operates within a deep-subwavelength regime (i.e. the dominant wavelength is muchlarger than the characteristic unit cell dimension). In addition, the long-range coupling forces between local unit cells can besimulated by means of mass-spring connections having frequency-dependent properties. Once the unit cells are coupled toform the nonlocal supercell, the links’ parameters can be tuned to achieve a frequency-dependent phase profile.
Figure 2.
Schematics illustrating the conversion of the unit cell model from a continuum to a discrete representation. Thisconversion is at the basis of the development of the transfer matrix model. (a) (top) Schematic of a typical locally resonantunit cell, and (bottom) its lumped parameter counterpart represented by a mass-in-mass-spring model. (b) (top) Schematicshowing the integration of the lumped unit cell model within the elastic waveguide as well as the amplitude of the incident( A ), reflected ( B ), and transmitted ( A ) waves. (bottom) Schematic showing the integration of local resonant units to form anonlocal supercell. The connecting links are represented by lumped elements having stiffness k c and mass m c . Thesuperscript ( (cid:3) ) in the amplitude coefficients represents the units number. In this section, we present the main highlights of the mathematical model developed to design and simulate the dynamics ofthe nonlocal metasurface. Complete details can be found in the Supplementary Information (SI ). As previously mentioned,the dynamic behavior of a unit cell is estimated by means of a discrete (mass-in-mass)-spring model via a transfer matrix(TM) approach. Figure 2a shows both the continuum and the discrete representations of the unit cell and it clarifies the role ofthe lumped parameters. In the discrete model, the unit cell is made of a mass M that captures the cell’s frame, a mass m thatrepresents the internal resonator, and a spring k in that connects the resonator to the frame. The resulting model is a classicalmass-in-mass system . The unit cell is connected to the exterior boundaries (representing the waveguide) by effectivesprings having stiffness K e f f .Assuming time-harmonic response at an angular frequency ω , the governing equations of the (mass-in-mass)-spring units re given by, M ∂ w M ∂ t = K e f f ( w + w L ) − K e f f w M + k in ( w m − w M ) (2a) m ∂ w m ∂ t = k in ( w M − w m ) (2b) F = K e f f ( w M − w ) (2c) F L = K e f f ( w L − w M ) (2d)where w and F represent the boundary displacement and shear force, while the subscript 0 and L indicate that the variable istaken at either the input or the output boundary, respectively and the superscript M and m of w indicate the displacement ofthe frame mass M and the internal resonator m . L is the length of the unit cell. Solving the above system of equations, thequantities at the output boundary can be expressed as a function of the same quantities at the input boundary, (cid:20) w L F L (cid:21) = T (cid:20) w F (cid:21) (3)where the transfer matrix T is given by, T = T T T T = K ef f − ω ( M + kinmkin − m ω ) K ef f K ef f − ω ( M + kinmkin − m ω ) K ef f − ω ( M + k in mk in − m ω ) K ef f − ω ( M + kinmkin − m ω ) K ef f (4)The TM model can then be extended to simulate the nonlocal supercell in which multiple units are coupled via connectingelements, as shown in Fig. 2b. While the physical design of the connecting beams will require additional steps, at this stagetheir dynamics is captured via discrete spring-mass elements that model the effective stiffness k c and mass m c of the link. Asan examples, considering the case where two units are coupled together by a link, the equations of motion for the unit I or IIare given by: M ( (cid:3) ) ∂ w M ( (cid:3) ) ∂ t = K ( (cid:3) ) e f f ( w ( (cid:3) ) + w ( (cid:3) ) L ) − K ( (cid:3) ) e f f w M ( (cid:3) ) + k ( (cid:3) ) in ( w m ( (cid:3) ) − w M ( (cid:3) ) ) (5a) m ( (cid:3) ) ∂ w m ( (cid:3) ) ∂ t = k ( (cid:3) ) in ( w M ( (cid:3) ) − w m ( (cid:3) ) ) + k c ( w mc − w m ( (cid:3) ) ) (5b) F ( (cid:3) ) = K ( (cid:3) ) e f f ( w m ( (cid:3) ) − w ( (cid:3) ) ) (5c) F ( (cid:3) ) L = K ( (cid:3) ) e f f ( w ( (cid:3) ) L − w m ( (cid:3) ) ) (5d)where the superscript ( (cid:3) ) can be either I or II and it refers to the different unit cells in the nonlocal supercell, and the basicunits are coupled by the connecting beam whose governing equation is given by, m c ∂ w mc ∂ t = k c ( w m ( I ) + w m ( II ) ) − k c w mc (6) m c and k c are the effective parameters characterizing the link in the form of a spring with mass (as shown in Fig. 2b), and w mc is the displacement of the lumped link.By solving the above set of differential equations, the response at the outer boundary of the coupled supercell can beexpressed as a function of the quantities at the input boundary using a nonlocal transfer matrix ˜ T (4 × w ( I ) L f ( I ) L w ( II ) L f ( II ) L = ˜ T w ( I ) f ( I ) w ( II ) f ( II ) (7)Once the nonlocal transfer matrix ˜ T is determined, the boundary response of each unit can be further related to the incident,reflected, and refracted waves in order to extract both the phase and the transmission coefficients of the non-local supercell. ore details on the methodology as well as a practical example of the simulation of a nonlocal supercell can be found in SI .Results also include a direct comparison with the numerical predictions obtained from a three-dimensional finite elementmodel. Despite the several simplifications introduced in the TM approach, the two models are in good agreement henceindicating that the simplified approach is perfectly capable of capturing the behavior of the unit cells, the connecting links,and the metasurface as a whole. In this section, we apply the proposed methodology to design a nonlocal total-internal-reflection metasurface (NL-TIR-MS)embedded in a plate waveguide. We recall that a local TIR-MS consists in a metasurface that is specifically designedto exhibit a phase gradient profile exceeding the requirements for total internal reflection. Under these conditions, incomingwaves with any arbitrary angle of incidence (but in the neighborhood of the target operating frequency, i.e. the resonancefrequency of the local unit cells) will not be able to cross the interface and will be reflected back. Note that the selection of theTIR-MS as benchmark system to test the nonlocal concept does not limit the generality of the results. Similar performancecan be envisioned also for transparent metasurfaces.Both the metasurface and the host waveguide have thickness t = .
35 mm. The target operating bandwidth is set to ∆ f = [ . , . ] kHz. Note that this selection of the operating range is somewhat arbitrary and it was chosen exclusively forconvenience of the experimental measurement. The nonlocal approach per se is not limited to a specific frequency range. Theoverall dimensions of the basic unit are L (= mm ) × W (= mm ) × t (= . mm ) , as shown in Fig. 2a. Recall that, inorder to ensure TIR conditions , the phase shift gradient should always exceed the critical value d φ / d y = π / λ , and thetransmission coefficients of each unit should have a comparable value. Assuming that the supercell contains n basic units tocover the 2 π range, then it must be π nW ≥ π / λ which results in a condition on the number of units n ≤ λ W . Over the selectedfrequency range, the wavelength λ approximately varies between 12 . W and 10 . W , hence a nonlocal supercell of n = Figure 3. (a) Schematics of the three selected locally resonant unit cells that will form the final nonlocal TIR-MS. (b) and(c) shows both the transmission phase and the transmission amplitude versus frequency for the three units.
Step 1: the first step consists in selecting the physical design of the locally resonant unit cells. Note that, contrarily to thelocal design which requires setting the fundamental resonance of the unit cells at the target frequency, in the nonlocal designthe individual unit cells do not need to be designed for a specific frequency. In fact, it is even preferable not to start with unitsoperating at the same frequency because this would bias the dynamic behavior of the nonlocal metasurface in favor of a singlefrequency. The most direct effect of this bias would be the need for extreme values of the effective properties (e.g. either zeroor negative mass and stiffness values) required from the links in order to guarantee broadband performance. On the other side, n appropriate selection of the basic units can facilitate achieving a broadband design. In general, it was found to be a gooddesign rule for the nonlocal metasurface to select units whose phase contribution (compared with the adjacent units) is closeto 2 π / n (where n=3 in the present case) over the selected frequency range. This first step of the design can be performed byacting on the basic geometric parameters of the unit cells in order to tune their local resonances and, consequently, the phaseprofile. For the example presented in this study, the three selected units are shown in Fig. 3a. Details on the geometric designcan be found in SI . Their amplitude and phase responses were extracted using the discrete TM model and the results aresummarized in Fig. 3b and c. It can be seen that, although the phase profiles of the three units are slowly varying over theselected frequency range 0, 2 π / − π /
3, the corresponding phase jumps between adjacent units (Fig. 3b) are always inthe neighborhood of the target value.
Step 2: once the fundamental resonant units forming the supercell are selected, the next step requires the identification ofthe nonlocal (coupling) interaction forces needed to achieve the target performance of the TIR-MS over the selected operatingrange. Equivalently, the former objective can be stated by saying that the frequency-dependent effective properties (i.e. m c ( ω ) and k c ( ω ) ) that identify the dynamics of each connecting element must be determined such that the phase jump betweenadjacent units is maintained at the constant value 2 π /
3. Note that, although the amplitude of the response of each unit shouldalso be maintained to comparable levels to each other (a condition that guarantees the transmission coefficients of each unit tobe equivalent), for the TIR-MS design the ability to guarantee a constant phase gradient poses a more stringent requirement. Inother terms, the ability to yield the desired phase gradient is more important than matching the response amplitude at the unitlevel. The phase response of the local and nonlocal three-unit-supercell are presented in Fig. 4a and b, respectively. Resultsshow that the nonlocal design is capable of producing a reasonably constant phase at the desired value 2 π / ).The corresponding effective properties characterizing the three coupling elements are shown in Fig. 5a, b, and c, respectively.A visual analysis of these wavenumber-dependent functions allows making an important consideration. The dynamic massprofile of unit cell Figure 4. (a) The phase gradient of the local supercell without coupling elements. (b) The predicted phase gradient from thelumped nonlocal supercell model after the application of the optimized coupling elements.
Step 3: once the necessary dynamic mass and stiffness profiles of the connecting elements are identified, the third andlast step consists in converting these synthetic long-range transfer functions into physically realizable designs of the threecoupling elements. To simplify the design process and avoid the need for topology optimization, we assume that the basicgeometry of the connecting elements is fixed to be a thin rectangular beam having piece-wise constant thickness. Whilechanges in thickness can help controlling the flexibility and the global dynamics of the link, it was already pointed out thatsharp variations in the coupling forces requires local resonances. Hence, the beam connectors will also include attachedresonators tuned at selected frequencies. Also in this case, in order to avoid the use of topology optimization, we assumedthese resonator to be in the form of a cantilever dumbbell. Based on the above assumptions, the topology of the links is entirely igure 5.
The required dynamic stiffness and dynamic mass profiles necessary to achieve the broadband TIR effect. Resultsare presented for the three coupling beams: (a) beam I (connecting unit 1 to 2), (b) beam II (connecting unit 2 to 3), and (c)beam III (connecting unit 1 to 3).defined, so that the profiles of the effective properties (or, equivalently, of the nonlocal coupling forces) can be matched byperforming a simple parameter optimization on the geometric design variables. This step was performed using a commercialfinite-element-based optimizer (COMSOL Multiphysics). The resulting physical configurations of the three coupling beamsare shown in Fig. 6 and their dynamic responses are evaluated in Fig. 6a, b and c (dashed lines), and compared with the targetprofiles (solid lines). Overall, the dynamic properties of the three beams are in good agreement with the target profiles. Notethat this study focuses on presenting and validating the concept of intentionally nonlocal metasurfaces, not on the evaluationof optimal designs. Hence, it is merely highlighted here that a concurrent optimization approach could provide better matchof the nonlocal coupling terms.
Figure 6.
Comparison of the dynamic stiffness and dynamic mass profiles provided by the synthetic (dashed lines) and thephysical (solid lines) designs. The insets in each figure show the corresponding physical design of the beam connectors andlocal resonators: (a) beam I, (b) beam II, and (c) beam III.
The previous section presented the physical design of the connecting elements capable of guaranteeing to the metasurface abroadband TIR effect. Based on this physical design, a complete 3D model of the nonlocal metasurface (Fig. 7a) embeddedin a flat plate was assembled. Note that the nonlocal metasurface was simply obtained by periodically repeating the basicnonlocal supercell (see inset I in Fig. 7a) along the interface direction. The side view of the non-local supercell ( y − z plane) isalso provided (see inset II in Fig. 7a) in order to clarify the connection strategy between units. Further details on the connectionscheme can be found in SI .The waveguide with the embedded nonlocal metasurface was modeled via finite element analysis using a commericalsoftware (Comsol Multiphysics). The metasurface was excited by a point load source located at a distance of 0.12m from the igure 7. Numerical simulations obtained from a full 3D model and showing the performance of the nonlocal TIRmetasurface. (a) Schematics showing the 3D model of the assembled nonlocal TIR-MS embedded in a thin waveguide. Theinsets show the isometric (I) and the side (II) view of the nonlocal connectors mounted on the resonating cores. Differentcomponents are marked by colors: basic unit cells (yellow) and connecting links (dark grey). (b) Spatially-averagedamplitude ratio of the out-of-plane component of the velocity obtained from the response of the waveguide before and afterthe MS interface. Both local and nonlocal designs are compared. The wide dip observed in the nonlocal MS response(highlighted by the dashed black box) indicates the broadband wave blocking effect. (c) full field simulations showing theout-of-plane particle displacement at five selected frequencies within the nonlocal MS operating range. The white starmarker shows the location of the point load source excitation. lane of the metasurface (see the white star marker in Fig. 7c), and acting in the out-of-plane z − direction, hence generatingpredominantly A guided modes. Perfectly matched layers (PML) were applied at all the boundaries to eliminate the effectof unwanted reflections. In order to assess the effect of the nonlocal metasurface on the propagating wave, the model of theequivalent local metasurface design was developed and used as a reference. This model consisted in the same set of resonantunit cells as those used in the nonlocal TIR-MS design but without the nonlocal connections and the large masses (drawingsprovided in SI ).Figure 7b shows the comparison of the spatially-averaged amplitude ratios calculated from the transverse particle dis-placement ( z − direction) before and after the MS. Both the reference (local) MS and the NL-TIR-MS are shown. The largereduction in the amplitude ratio of the NL-TIR-MS metasurface (marked by the dashed black box in Fig. 7b) compared to thereference waveguide is a clear indicator of the broad operating range of the proposed nonlocal design. Full field numericalresults are shown in Fig. 7c for five selected frequencies inside the operating bandwith. The results are still presented in termsof out-of-plane particle displacement of the A wave field. The direct comparison of these wave patterns shows that, althougha small fraction of the incident wavefront can still be transmitted through the metasurface, the nonlocal design is capable ofproducing a drastic reduction of the transmitted wave intensity over a broad frequency range. In order to put these resultsin perspective, note that the numerically calculated bandwidth for the nonlocal design (approximately ∆ f = [ . , . ] kHz)corresponds to a 37 .
2% relative bandwidth with respect to its center frequency f = . where available data showed approximately a 250 Hz bandwidth (only 4 .
54% of the center frequency f =5.5 kHz), the nonlocal design provides a range that is approximately eight times larger. In order to validate the concept of intentional nonlocality as a viable approach to the synthesis of passive broadband metasur-faces and to confirm the performance of the NL-TIR-MS, we built an experimental testbed following the configuration thatwas numerically studied in Fig. 7a. The plate waveguide and the supporting frames of each resonant unit cell were made outof aluminum and fabricated via computer controlled machining (CNC). The nonlocal connecting links (including the threeresonating masses) were made out of 316L stainless steel and fabricated via additive manufacturing techniques (i.e. metalprinting). The links were later glued on the corresponding unit cells slots realized in the waveguide. Further details on thefabrication process are discussed in SI .The test setup is shown in Fig. 8. Figure 8a provides a front view of the setup consisting in a plate waveguide withthe embedded nonlocal MS, while Fig. 8b shows a closeup view of the metasurface. In Fig. 8c, from top to bottom, thethree panels show different views (Iso-, Front- and Top-) of the nonlocal links assembled on the three resonant masses. Theplate waveguide was mounted in an aluminum frame while a single Micro-Fiber-Composite (MFC) actuator was glued on thesurface of the plate in order to apply the external excitation (see the red star marker in Fig. 8a). Viscoelastic tapes were appliedall around the boundary of the plate so to reduce back-scattering. The response of the plate was measured under white noiseexcitation (bandwidth 3-6 kHz) by using a scanning laser Doppler vibrometer. Both the local (reference) and the nonlocalconfigurations were tested.Figure 8d presents the comparison of the spatially-averaged amplitude ratios collected from the measured out-of-planeparticle velocity in the regions before and after the MS. As already discussed for the numerical results, the large reduction inthe amplitude ratio in the frequency band 3 . − .
55 kHz (see the black dashed box in Fig. 8d) identifies the region in whichthe nonlocal metasurface is effective and provides broadband wave blocking effect. Note that, although the operating rangeis slightly smaller than the one predicted via numerical simulations, it still covers a range of 1 kHz, that is approximatelya 24 .
69% relative bandwidth with respect to the center frequency f =4.05 kHz. Compared to the numerical results, weobserve a reduction in the operating range and a global shift towards lower frequencies. These differences are due to structuralmodifications applied to the original design during the fabrication phase, to some variability in the properties of the 3D printedconnecting beams, to the coarseness of the printing process, and to inaccuracies during the assembly procedure of the nonlocallinks (which where glued on location). Despite these differences, the experimental results still show a marked increase in theoperating range that is approximately six times that observed in local MS.Figure 9 summarizes the results of the experimental measurements in terms of full field data. In particular, Fig. 9a andb show the response of both the reference and the NL-TIR-MS waveguides at four selected frequencies ( f = . f = . f = .
3, and f = .
75 kHz, see the four yellow dashed line in Fig. 8b as well) within the operating range of the nonlocalmetasurface. The response was measured in terms of out-of-plane ( z − direction) velocity V z amplitude field. As expected, inthe nonlocal TIR-MS waveguide the vibrational energy is mostly confined in the area preceding the MS, while only a smallfraction propagates through the MS interface. On the other hand, in the reference (local) waveguide, the wave could penetratethe MS practically unaffected. The direct comparison of these wave patterns further support the validity of the intentionallynonlocal concept to achieve passive broadband metasurfaces. igure 8. Experimental setup consisting of a thin plate waveguide with an embedded nonlocal TIR-MS. The waveguide wasinserted into an aluminum frame providing clamped boundary condition on two sides of the waveguide. The excitation wasprovided by a MFC patch glued on the surface (see red star marker). (a) The front view of the experimental test bed. (b)Zoomed-in view of the nonlocal TIR-MS. (c) 3D metal printed nonlocal beams mounted on the the resonating masses. Threeviews are shown (Iso, Front, Top) moving from the top to the bottom panel. The connectors are later glued in place on thecorresponding unit cells realized in the supporting waveguide. (d) The spatially-averaged amplitude ratio obtained from themeasured out-of-plane velocity field before and after the MS. The comparison between the local and the nonlocal designclearly shows the broadband nature of the latter highlighted by the large drop in the amplitude ratio (see dashed black box).
We have presented and experimentally validated the concept of intentional nonlocality in order to design passive elastic meta-surfaces capable of broadband performance. Contrarily to the classical metasurface design, where wave control is achievedby engineering the phase gradient via local resonances, in the nonlocal design the response at a point of the metasurface de-pends simultaneously on the response of multiple distant points along the interface. This unique behavior was possible due tothe use of specifically designed connecting elements that provided carefully tuned effective dynamic properties and coupledtogether multiple distant units. In essence, these coupling elements represent a macroscopic implementation of the concept ofaction-at-a-distance typical of nonlocal microscopic strictures.Dedicated modeling tools were also developed in order account for the nonlocal coupling forces between unit cells and tocharacterize the dynamic behavior of nonlocal supercells. As a practical example, the methodology was applied to the designof a broadband total internal reflection metasurface. The performance of this design was validated via both full field numericalsimulations and experimental testing. Both results confirmed the validity of the nonlocal concept and the ability to achievebroadband performance via a fully passive approach.While the nonlocal design was tested on a TIR type metasurface, the concept is very general and it is expected to beapplicable to any kind of metasurface. The ability to overcome the intrinsic narrowband nature of traditional metasurface igure 9.
Full field experimental data showing the performance of the nonlocal TIR-MS in comparison to the local design.The images show the out-of-plane particle velocity field at four selected frequencies within the operating bandwith of thenonlocal MS. (a) the out-of-plane particle velocity amplitude for the reference (local) MS configuration. (b) the out-of-planeparticle velocity amplitude for the nonlocal TIR-MS. The comparison of the wave patterns clearly indicates the broadbandwave blocking effect of the non-local TIR-MS compared to a local design.designs is expected to eliminate a significant barrier to the use of these passive wave-control devices and to pave the way to awide range of practical applications.
Acknowledgements
The authors gratefully acknowledge the financial support of the Sandia National Laboratory under the Academic Allianceprogram, grant
Supplementary information
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