Topological rainbow trapping for elastic energy harvesting in graded SSH systems
Gregory J. Chaplain, Jacopo M. De Ponti, Giulia Aguzzi, Andrea Colombi, Richard V. Craster
TTopological rainbow trapping for elastic energyharvesting in graded SSH systems
Gregory J. Chaplain a , Jacopo M. De Ponti b,c , Giulia Aguzzi d , AndreaColombi d , Richard V. Craster a,e,f a Department of Mathematics, Imperial College London, 180 Queen’s Gate, SouthKensington, London SW7 2AZ b Dept. of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardoda Vinci, 32, 20133 Milano, Italy c Dept. of Mechanical Engineering, Politecnico di Milano, Via Giuseppe La Masa, 1,20156 Milano, Italy d Dept. of Civil, Environmental and Geomatic Engineering, ETH,Stefano-Franscini-Platz 5, 8093 Z¨urich, Switzerland e Department of Mechanical Engineering, Imperial College London, London SW7 2AZ,UK f UMI 2004 Abraham de Moivre-CNRS, Imperial College London, London SW7 2AZ, UK
Abstract
We amalgamate two fundamental designs from distinct areas of wave con-trol in physics, and place them in the setting of elasticity. Graded elasticmetasurfaces, so-called metawedges, are combined with the now classicalSu-Schrieffer-Heeger (SSH) model from the field of topological insulators.The resulting structures form one-dimensional graded-SSH-metawedges thatsupport multiple, simultaneous, topologically protected edge states. Theserobust, enhanced localised modes are leveraged for applications in elastic en-ergy harvesting using the piezoelectric effect. The designs we develop are firstmotivated by applying the SSH model to mass-loaded Kirchhoff-Love thinelastic plates. We then extend these ideas to using graded resonant rods, andcreate SSH models, coupled to elastic beams and full elastic half-spaces.
1. Introduction
Topological insulators are exotic materials in which protected edge or in-terfacial surface states exist between bulk band gaps, owing their existenceto broken symmetries within a periodic system. Despite their origins in
Preprint submitted to Elsevier June 12, 2020 a r X i v : . [ phy s i c s . a pp - ph ] J un uantum mechanical systems [1, 2, 3], there has been a flurry of intensiveresearch translating these effects into all flavours of classical wave propaga-tion, from electromagnetism and acoustics to mechanics and elasticity [4, 5];the protected edge modes can have attractive properties such as resilienceto backscattering from defects and impurities, and can exhibit unidirectionalpropagation. As such the physical phenomena surrounding topological in-sulators has naturally led to a concerted effort mapping such effects intometamaterial and photonic crystal (and their analogues) design [6, 7, 8].The nature of the symmetry breaking giving rise to a protected edge statedefines two classes of topological insulators. Inspired by the original quantummechanical systems exhibiting the quantum Hall effect (QHE), where timereversal symmetry (TRS) is broken through applied external fields [9, 10], hasled to so-called active topological materials [11, 12, 13, 14, 15]. Similarly thequantum spin Hall effect (QSHE), in which symmetry breaking is achievedthrough spin-orbit interactions (TRS is preserved)[16, 17], has engenderedpassive topological systems. Such systems have promulgated simpler topo-logical insulator motivated designs in continuum wave systems through thebreaking of geometric symmetries to induce topologically nontrivial bandgaps[18].Underpinning the topological nature of the Bloch bands defined by eachmaterial are associated invariants which characterise the geometric phase,that is the phase change associated with a continuous, adiabatic deformationof the system; most notably the Berry phase [19, 20], and its one dimensionalcounterpart, the Zak phase [21].Numerous translations of 2D topological insulators to wave physics havebeen realised, often based around honeycomb structures [22], which guaranteesymmetry-induced Dirac points that can be leveraged to induce edge statesat the interface between two topologically distinct media. These have beenreplicated for waveguiding applications for photonics [23], phononics [24] andplatonics [25, 26]. More nuanced interpretations have achieved beam splitterdesigns with square lattices [27, 28, 29]. Higher order topological effects, forhigher dimensional structures have also received much attention [30, 31].Despite their relative simplicity, 1D topological insulators serve not onlyas pedagogical examples, but posses important features for applications rang-ing from lasing [32] to mechanical transport [33]. Motivated by applicationsin elastic energy harvesting, we highlight a new modality of 1D topologicalinsulators, based on the well-established Su-Schrieffer-Heeger (SSH) model[34], via the amalgamation of this model with graded metawedge structures2 a Δ Δ = a - Δ S1 S1 S1 S1' S1' S1'S1 S1' S2' S2 S3 S3' a (a)(b)(c) Figure 1: Combining SSH interfaces with metawedge structures: (a) Shows an elasticversion of the SSH model, where an interface is encountered between structures S1 andS1 (cid:48) . The unit cells of each are highlighted in green, each shares the same periodicity a , buthave rod separation ∆ and ∆ respectively. (b) Shows conventional metawedge structureconsisting of periodically spaced rods of increasing height. (c) Shows the amalgamation ofthese geometries to produce several SSH interfaces for differing rods heights, marked bythe dashed red lines. The green material at the base of each rod represents a piezoelectricmaterial which is to be used for energy harvesting [39, 40]. [35, 36, 37, 38].Throughout this article we will examine the SSH model, due to its suc-cessful predictions and ease of its translation from topological wave physicsto mechanical systems [41], but now in the setting of elasticity. Figure 1highlights the overarching theme of this article, which combines the conven-tional SSH model with a graded system. Shown in Figure 1(a) is an elasticversion of the classical SSH interface, for resonant rods atop a beam. Theinterface at which a topological edge mode exists is highlighted by the reddashed line; this is where two related geometries meet. To the left of thisinterface are a set of periodic unit cells, of width a , each with two resonantrods set apart a distance ∆ from the centre of the cell: we call this structure31. To the right of the interface we consider structures, S1 (cid:48) , that consist ofunit cells of the same width, but this time with the rods placed a distance∆ = a − ∆ apart from the cell centre. In an infinite periodic array boththese structures are identical as there is merely a translation in the definitionof the unit cell. For that infinite array, then, taking advantage of periodicityto introduce Floquet-Bloch waves, both structures have the same dispersioncurves. We show in Section 2 how an edge mode arises at this interface,sometimes referred to as a domain wall.Shown in Figure 1(b) is the, now conventional, graded metawedge struc-ture [35], that has recently been utilised for energy harvesting purposes[39, 42]. This consists of periodically spaced rods, which increase in heightthrough an adiabatic grading. The utility of such devices is provided by theirability to manipulate and segregate frequency components by slowing downwaves which can reach effective local bandgaps at different spatial positions[40]. Our desire is to combine these two structures, as shown in Figure 1(c),to incorporate several, simultaneous, topologically protected edge modes forenergy harvesting applications. Such a structure is devised by alternatingbetween primed and un-primed pairs of structures for differing rod heights.To elucidate the design paradigm and conditions for existence of an edgemode, we firstly consider the simplified elastic model of a point mass loadedthin Kirchhoff-Love elastic plate. The expected existence of edge states ina one dimensional SSH chain is confirmed through calculation of the Zakphase via an efficient numerical scheme [43], corroborated via Fourier spectralanalysis of scattering simulations. The differences between the localised 1Dedge states and that of conventional band gap defect states are highlighted.This methodology is then extended to a topological system of resonatingrods atop an elastic beam. Recent experimental work has highlighted theexistence of such states in quasi-periodic resonant-loaded beams [44], whilstother works incorporate piezoelectric effects to tune the topological phasesof the bands [45]. Here we continue with the SSH model demonstratingthat, by the addition of piezoelectric materials, efficient energy harvestingfrom mechanical to electric energy is possible; this extends the applicationsof coupling piezoelectricity with topological insulators [46]. The motivationof coupling with the graded structures, as highlighted in Figure 1, is toextend the bandwidth from the single frequency at which the edge mode existthereby achieving broadband performance of the device with an attractivelycompact device. Finally, this model is extended to elastic halfspaces thatsupport Rayleigh waves, introducing the concept to broaden the scope of4opological groundborne vibration control.
2. SSH in Thin Elastic Plates
The equations governing flexural wave propagation in thin Kirchhoff-Love(KL) elastic plates [47] provide a flexible avenue for investigating a wide va-riety of wave manipulation effects; they efficiently predict wave behaviourin physical systems [48], with elegant solutions readily available for pointloaded scatterers [49]. Further to this, the Green’s function of the governingbiharmonic wave equation is non-singular and remains bounded, and as suchnumerical complications during the implementation of scattering simulationsare side-stepped, enabling efficient scattering calculations to be obtained byextending a method attributed to Foldy [50]. Recent advances for analysingone-dimensional, infinite, periodic structures, in such systems, [43] have gen-erated efficient methods for calculating their dispersion curves, enabling fastdesign and analysis. These features of the KL system, and the numericalease of its solution, motivate its use as a powerful toolbox for quickly char-acterising topological systems [51, 18, 52, 27, 53, 54].For a point mass loaded KL plate, loaded with J masses of value M ( j ) at positions x ( j ) , the restoring forces at the mass position are proportionalto the displacement of the mass at that point, resulting in the out-of-planeflexural wave displacement, w ( x ), being governed by the biharmonic waveequation, ( ∇ − Ω ) w ( x ) = Ω J (cid:88) j =1 M ( j ) w ( x ) δ ( x − x ( j ) ) . (1)We adopt a non-dimensionalised frequency such that Ω = ρhω /D , where ρ is the mass density of the plate and h is the plate thickness, with ω beingthe dimensional angular frequency. D is the flexural rigidity, which encodesthe Young’s modulus, E , and Poisson’s ratio, ν , of the plate through D = Eh / − ν ).Considering an infinite periodic line array of point masses, capable of sup-porting propagating Rayleigh-Bloch modes which exponentially decay per-pendicularly to the array, allows the governing equation to be formulatedas a generalised eigenvalue problem; we do so by partitioning the array andplate into periodic infinite strips and by formulating the wavefield, w ( x ) asa combination of a Fourier series and a decaying basis, as in [43]. Employ-ing Floquet-Bloch conditions, and invoking orthogonality, then characterises5he dispersion relation for an arbitrary periodic strip of width a . Adoptingthe nomenclature conventional with topological systems, the eigensolutions(wavefields) of this system of equations are then written as | w (cid:105) = (cid:88) n,m W nm exp[ i ( G n − κ ) x ] ψ m ( y ) , (2)where, for integer n , G = 2 nπ/a is a reciprocal lattice vector, κ is the Blochwavenumber and ψ m ( y ) is an exponentially decaying orthonormal Hermitefunction. The advantages of this approach allows a spectral Galerkin methodto accurately and rapidly characterise the dispersion relation, an example ofwhich is shown in Fig. 2.An advantage of having such explicit solutions for the eigenstates, satis-fying (1) with periodic modulation, is that they can be used to obtain keyinformation from the bulk bands in the form of topological invariants, specif-ically the Zak phase. To demonstrate the efficiency of this, and the existenceof 1D topological edge states we utilise the SSH model. This has been utilisedin many systems for transport [33, 55] and to identify the existence of edgemodes [56]. Here we wish to exploit this model to emphasise the features of1D topological defect states, and how they can be used for energy harvesting.To build the SSH model, in the setting of a 1D array of point loadedmasses on a KL elastic plate, we first consider an infinite, periodic 1D arrayconsisting of infinite unit strips of width a , with two masses of mass M placedsymmetrically about the strips origin a distance ∆ apart; this system hasthe dispersion relation highlighted in Fig. 2. As before, this cell configurationwill be labelled as structure S1. Due to the translational invariance presentin the infinite structure, the same periodic structure can be built by a trans-lation of the unit strip by a distance a/
2. In this new unit cell, the massesare separated symmetrically about the strip origin by ∆ = a − ∆ ; this con-figuration has an identical dispersion relation to S1, and we denote the unitcell of this structure S1 (cid:48) . Structures S1 and S1 (cid:48) can be seen in Fig. 2(a), withtheir calculated dispersion relation shown in Fig. 2(b) (as also confirmed bythe Fourier spectrum obtained through scattering simulations).We then form an SSH array by creating a 1D chain composed of repeatedcells of S1 and S1 (cid:48) which meet at an interface (Fig. 2(c)). To determinewhether a topological edge mode exists at this interface, we calculate the Zakphase [21] for each band defined by S1 and S1 (cid:48) ; each material composing theSSH array has a common band-gap and, provided each periodic structure hasa distinct Zak phase, the existence of an edge mode is guaranteed [57, 58, 59].6he Zak phase, ϕ Zakn , for the n th band is defined in terms of the Berryconnection A ( κ ) such that ϕ Zakn = (cid:90) BZ A ( κ ) d κ , (3)with A ( κ ) = i (cid:104) u κ | ∂ κ u κ (cid:105) , (4)where BZ denotes the Brillouin Zone; there are several efficient methods ca-pable of calculating such invariants [60, 61]. We opt to dove-tail the eigenso-lutions obtained from the spectral method (2) to calculate the Zak phase foreach band, by ensuring that | u n, κ (cid:105) is cell periodic such that | w (cid:105) = e − i κ · r | u n, κ (cid:105) .In doing so the required quantities are readily available from the obtainedeigensolutions. We evaluate (3) over the discretised BZ in κ -space such that (cid:90) BZ A ( κ ) d κ → (cid:88) κ j d κ (cid:104) u κ | ∂ κ u κ (cid:105) (cid:12)(cid:12) κ = κ j , (5)resulting in ϕ Zakn = − Im (cid:32) log J (cid:89) j =1 (cid:104) u n, κ j | u n, κ j +1 (cid:105) (cid:33) . (6)The periodic gauge condition is satisfied through | u n, κ J +1 (cid:105) = e − i G · r | u n, κ (cid:105) .Due to the intrinsic connection with Wannier charge centers [62, 63], pro-vided we have inversion symmetry with respect to the array axis, we areguaranteed a quantised Zak phase of 0 or π ; indeed this can be inferred fromthe symmetry properties of the band edge states [59]. In this setting, thesecorrespond to the flexural displacement fields being localised to the centre oredges of the strip respectively.As expected, we find distinct Zak phases as highlighted for the lowestband in S1 and S1 (cid:48) in Fig. 2(b). As such at the interface between S1 and S1 (cid:48) we have an analogue to an incomplete Wannier state: there exists an edgemode. This confirmed through the Fourier spectrum shown in Fig. 2(b).To visualise the edge mode we make use of the attractive Green’s functionapproach [49, 51] that can be employed to calculate the total wavefield, sub-ject to forcings F ( j ) from J masses. This can be evaluated quickly, obtaining w ( x ) = w i ( x ) + J (cid:88) j =1 F ( j ) g (cid:0) Ω , | x − x ( j ) | (cid:1) , (7)7 igure 2: (a) Schematics of the infinite strips, that periodically repeat, that characterisethe array and which are used for the dispersion curves for structures S1 and S1 (cid:48) , suchthat a = 1, M = 5, ∆ = 0 .
2, ∆ = a − ∆ . The dispersion curves, from the spectralmethod [43], for the normalised frequency, Ω( κ ) are shown in white, with the dotted whiteline showing the free space flexural ‘sound line’, which, for KL plates is not dispersionless.Plotted along the wavenumbers from κ = − X ≡ − π/a to κ = Γ ≡ κ = X ≡ π/a showing those for S1 (cid:48) . The two dispersion relations are clearlyidentical. Further to this, the Fourier spectrum is also shown in (b), through a FFT ofscattering simulations of the SSH geometry shown in (c). Corroborated by the calculationof the distinct Zak phases, which label the bands in (b), there is an edge mode within thebulk band gap, highlighted at Ω = 2 .
7. The topological nature of a bandgap is determinedby summation over the Zak phase of all the bands below this gap [64, 59], having nodependence on the bands above it. As such we only show the lowest dispersion branch ofthis system. where w i ( x ) is the incident field. Using the well-known Green’s function[49], g (Ω , ρ ) = ( i/ ) [ H (Ω ρ ) − H ( i Ω ρ )], the unknown reaction terms F ( j ) F ( k ) = M ( k ) Ω (cid:34) w i ( x ( k ) ) + J (cid:88) j =1 F ( j ) g (cid:0) Ω , | x ( k ) − x ( j ) | (cid:1)(cid:35) . (8)From this, fast Fourier transform (FFT) techniques can be utilised to obtainthe dispersion relation in κ -space, shown in Fig. 2(b). Using this method, wedemonstrate the characteristics of a 1D edge mode, by also evaluating thetime-averaged flux through [65] (cid:104) F (cid:105) = Ω2 Im (cid:0) w ( x ) ∇ w ∗ ( x ) − ∇ w ∗ ( x ) ∇ w ( x ) (cid:1) . (9) Figure 3: Scattered fields for (a) propagating, (c) edge mode and (e) conventional scatter-ing defect modes, excited at the interface of S1 and S1 (cid:48) , marked by the white stars. Thefield amplitudes are normalised with respect to the maximum amplitude of the scatteredfield of the topological edge mode, showing that it is approximately 17 times greater thanthe maximum amplitude of conventional defect modes. Panels (b,d,f) show a streamlineplot of the time averaged flux in the regions highlighted by the rectangular box in (a,c,e)for the propagating, edge and defect modes respectively. The chiral nature of the edgemode flux is markedly different from the other cases. igure 4: Testing disorder in the system, with (a-b) showing scattered fields and flux for aline-defect structure; a mass is removed in the first cell of S1 (cid:48) to the right of the interface(highlighted by the red point). (c-d) shows similar plots for an impurity-type defect; thered point represents an additional mass of M = 1 to the left of the source. In each case,the field is normalised with respect to the maximum amplitude of the perfect SSH case(Fig. 3(c)). The chirality of the flux is seen to be preserved. Shown in Fig. 3 are the scattered fields, for a monopolar point sourceplaced at the interface between structures S1 and S1 (cid:48) in the SSH model,for the parameters as defined in Fig. 2. Exciting at different frequenciesreveals three distinct modes present in the system: propagating, conventionaldefect and topologically protected edge modes. When exciting at Ω = 2,unsurprisingly a propagating Rayleigh-Bloch mode exists, transiting alongthe array in each direction. Increasing the frequency to lie within the bandgap, we see stark contrasts between the wavefields between the edge mode(Ω = 2 .
7, Fig. 3(c-d)) and a conventional localised defect state (Ω = 3Fig. 3(e-f)); the amplitude of the topologically protected edge state is nearly17 times that of the localised defect state, with its flux displaying chiral orbitswhich are indicative of edge modes, induced by the distinct topological phasesat the interface [52]. We further test the robustness of the topological edgestate, by introduced line and impurity defects, by the removal and addition ofextra masses respectively, demonstrated in Fig. 4: in each case the amplitudeand chirality of the fields are preserved.We have successfully shown that the SSH model can be implemented inthe setting of point mass loaded KL elastic plates. The existence of edgemodes is confirmed through a variety of numerical techniques. The purposeof exploring such features in this system is to motivate energy harvesting10pplications in elastic settings, particularly because the localised amplitudesof edge modes are so much greater than those for conventional defect modes.A key feature of such harvesting structures is the ability to recycle energyfrom a distance; until now we have only focused on source positions at theinterface between topologically distinct media. In order to assess the feasi-bility of harvesting devices, we explore the excitation of this mode from adistance.To do this, we consider a region, S0, consisting of the same geometricstructure as S1, but with a lower mass value ( M = 2 .
3) such that a prop-agating mode exists at the frequency of the edge mode in the SSH model.Then, at a given spatial position, we abruptly switch the mass value to beconsistent with S1 ( M = 5) - in this region an exponentially decaying modeis excited. The SSH interface is then encountered, by constructing a regionof S1 (cid:48) close to the interface between S0 and S1. A schematic of this is shown,along with the field and flux computations in Fig. 5, showing that it is possi-ble to excite this mode from a distance; the amplitude of the resulting modedepends on the decay length introduced in the transition region between S1and S1 (cid:48) , a feature which can be predicted from high frequency homogenisa-tion techniques [43]. Thus 1D edge modes can be excited by a source whichis external to the topological interface and this motivates energy harvestingapplications within such regions. Limitations of this simplistic system are,however, immediately apparent; there must be a propagating region beforethe interface, and the effect is extremely narrowband. To circumnavigatethese deficiencies, we turn to recent metawedge structures, and hybridise theSSH model with an adiabatic grading in a system of resonant rods atop anelastic beam.
3. The Graded SSH metawedge
The graded resonant metawedge [35] has proved an important source ofinspiration for the ‘trapping’ of energy, by a reduction in effective group ve-locity of propagating waves. The classical arrangement is that of resonantrods atop an elastic half-space, or elastic beam, as shown in Fig. 1(b). Inthis example, the rods adiabatically change in height from one unit cell tothe next, generating locally periodic cells; the global behaviour of the de-vice is inferred from the dispersion curves corresponding to an infinite arrayof each rod height [66]. As such, different frequency components encounterlocal bandgaps at different spatial positions. Similar to rainbow trapping11 igure 5: Exciting an edge mode using a step ‘grading’. (a) shows a schematic of the array,composed of structures S0 of M = 2 . (cid:48) , forming a step-SSH array. In region S0, the frequencyΩ = 2 . (cid:48) ; at thisfrequency an edge mode is excited, shown by the chiral fields in the inset. (b) shows thescattered field, normalised to the amplitude of the perfect SSH array (Fig. 3(c)). (c) showsthe absolute amplitude | w | along the array axis, indicating the increased amplitude at theinterface between S1 and S1 (cid:48) , demonstrating this edge mode can be externally excited. devices [67], the metawedge achieves local field enhancement which can beused for energy harvesting effects [39]. Despite the success, both in designand experimental verification, of a wide variety of effects exhibited by themetawedge and similar structures [36, 37, 38], this simplistic array has reflec-tions, due to Bragg scattering, at the ‘trapping’ positions. As such, energy isnot confined for prolonged periods due to intermodal coupling, and rainbowreflection phenomena is seen instead [40].Topological systems therefore seem attractive candidates for energy ex-traction, due to their resilience to back-scatter and strong confinement; thelonger energy is confined to a spatial position, the more energy that can beharvested [40]. This is more efficient for symmetry broken systems, where alack of coupling to reflected waves leads the energy to be more localised; anatural extension of this is to consider topological devices. Indeed, recent de-12igns for topological rainbow effects have been theorised for elasticity in per-forated elastic plates of varying thickness, based on topologically protectedzero-line-modes (ZLMs) between an interface of 2D square array structures[68, 27].Due to the low dimensionality of the 1D-SSH system, the SSH modelprovides an optimal arrangement for elastic energy harvesting as there isno propagating component of the edge mode. However, the caveat to thishas already been alluded to - this mode only exists for a very narrow rangeof frequencies. We therefore design a hybrid graded-SSH-metawedge which,based on a gentle adiabatic grading of alternating SSH structures, signifi-cantly increases the bandwidth of operation, serving as the perfect candidatefor topological rainbow trapping. Due to the strong interaction between thesymmetry broken structure and the edge mode, these devices offer an addi-tional benefit of being compact compared to classical metawedges. Figure 6: Graded SSH schematics: (a) shows the Sn-Sn (cid:48) -Sm (cid:48) -Sm altering cell structurefor heights h n , h m (b) shows graded-SSH-metawedge for 7 alternating SSH cells. Thegreen disks at the base of each rod represent the positioning of the piezoelectric materialdiscussed in Section 4. (cid:48) to be unit cells consisting of rods ofheight h arranged in the SSH configuration. The heights of the rods areadiabatically increased every two unit cells, with the arrangement being mir-rored: cells with rods of height h follow an S2 (cid:48) -S2 interface: this is repeatedalong the array. An example of the corresponding Sn-Sn (cid:48) -Sm (cid:48) -Sm geome-try (where n and m corresponding to heights h n , h m ) is shown in Fig.6(b).Throughout the following sections the existence of the edge modes is con-firmed for the now familiar S1-S1’ configuration, followed by an investigationinto the uses for elastic energy harvesting.
4. Topological rainbow trapping for elastic energy harvesting
Topological systems have been widely proposed as efficient solutions forelastic energy transport, guiding and localization [52, 27]. These concepts of-fer, amongst others, promising capabilities for energy harvesting, due to theenhancement of local vibrational energy present in the environment. Oneof the main challenges in elastic energy scavenging, is obtaining simultane-ously broadband and compact devices [69]. Broadband behaviour is usuallyachieved through nonlinear effects [70, 71] or multimodal response [72, 73],i.e. by exploiting multiple bending modes of continuous beams or arrays ofcantilevers. Whilst multimodal harvesting enhances the operational band-width, it is usually accompanied by an increase in the volume or weight ofthe device. This can affect the overall power density of the system as wellas the circuit interface, which becomes more complex with respect to singlemode harvesters. Conversely it is important to appreciate that multimodalschemes can be well integrated with metamaterial concepts, leading to trulymultifunctional designs [74] with enhanced energy harvesting capabilities.Here we propose a multimodal scheme, i.e. a broadband device, which issimultaneously compact due to the reduced number of required cells. Thedevice is similar to that in [39], but based on the excitation of local edgemodes through the graded-SSH-metawedge geometry (Fig.6). We recall thatthe physics of these arrays is primarily governed by the longitudinal (axial)resonances of the rods [35] which, along with the periodicity, determine band-gap positions through their resonance. The axial resonance frequency of therod is governed by the rod height [35]. By a simple variation of the length ofadjacent rods, an effective band-gap, that is both broad and sub-wavelength14 igure 7: Dispersion comparisons for (a) trivial and (b) SSH interfaces. (a-b) show thearrangement of rods atop a beam with 10 cells of width a consisting in (a) of structuresS1-S1 (grey rods), forming a trivial interface, with (b) showing an SSH interface between10 cells of structure S1 and 10 cells of S1’ (green rods). (c-d) shows the Fourier spectrumfor this arrangement from a scattering simulation where a source is placed at the inter-face, marked by the star and dashed black lines in (a-b). An edge mode appears insidethe bandgap defined by the longitudinal resonance of the rods, as expected in the SSHarrangement. Overlaid in (c-d) are the numerical dispersion curves for a perfectly peri-odic, infinite array of structures S1 (and simultaneously S1 (cid:48) ) represented by the colouredpoints in (c-d), with green points corresponding to vertical polarization of the rod (axialelongation), whilst blue refers to horizontal (flexural motion). The geometrical parametersare such that a = 30 mm with the SSH spacing ∆ = 10 mm. The rods have a heightof h = 82 mm and circular cross section of radius r = 3 mm. The beam has thickness t = 10 mm and width w = 30 mm, as is assumed to be infinitely long in the direction ofthe array. can be achieved. The addition of alternating SSH configurations introducesfrequency dependent positions of localized edge states. By the definition ofrainbow effects [40], this will hence define a true topological rainbow.To quantify the advantages of such designs for energy harvesting, wecompare its performance with a conventional rainbow reflection device [39]composed of equal number of rods, with identical grading angle and quantityof piezoelectric material.We firstly verify the existence of an edge mode, by considering two ar-rays, one only composed of equal rods with constant spacing, i.e. consisting15nly of structures S1 (Fig. 7(a)), and another with a transition between re-gions consisting of structures S1 and S1 (cid:48) , shown in Fig. 7(b), similar to theprevious examples (Figs. 1(a), 3). Both systems are made of aluminium( ρ = 2710 kgm − , E = 70 GPa and ν = 0 .
33) and composed of rods withlength 82 mm and circular cross section with 3 mm radius. The beam is de-fined by 10 mm thickness and 30 mm width, and is assumed to be infinitelylong in the direction of the wave propagation. The unit cell dimension is a = 30 mm, with the resonator separation inside the cell as ∆ = 10 mm(in structure S1) and ∆ = a − ∆ = 20 mm (in S1 (cid:48) ). We compute thedispersion curves for both configurations using Abaqus [75] with a user de-fined code able to impose Bloch-Floquet boundary conditions. As expected,the two dispersion relations are identical as can be seen from the directions − X − Γ and Γ − X in Fig. 7(b). To detect the presence of an edge mode,we excite both systems with a time domain frequency sweep in the range5 −
15 kHz, with a source inside the array and located at the interface be-tween S1-S1 (cid:48) . By inspection of the spatiotemporal Fourier transform of theresultant wavefield, an edge mode clearly appears inside the bandgap openedby an axial resonance. This can be seen through the colormap of the disper-sion curves where the polarisation is such that green points correspond tovertical (longitudinal) polarisation, whilst blue refers to horizontal (flexural)polarisation (see Fig. 7(c,d)).We consider graded line arrays of resonators, to simultaneously excite thearray from outside and to enlarge the bandwidth, based around the designsshown in Figures 6 and 8(a). To quantify the energy that can be stored,we insert PZT-5H piezoelectric disks ( ρ = 7800 kgm − , E = 61 GPa and ν = 0 .
31) of 2 mm thickness between the rods and the beam (shown as greendisks in Fig. 6). Due to the dominant axial elongation in the rod response, wemodel the piezoelectric coupling by means of the 33 mode piezoelectric coeffi-cient e = 19 . − , and constant-stress dielectric constant (cid:15) T /(cid:15) = 3500,with (cid:15) = 8 .
854 pFm − the free space permittivity. The device is composed of40 rods with height approximately from 5 mm to 100 mm and grading angle θ (cid:39) . ◦ . We compare the SSH rainbow system with a conventional rain-bow device, through a steady state dynamic direct analysis performed usingAbaqus with open circuit electric conditions. The infinite length of the beamis modeled using ALID boundaries at the edges [76]. We see rainbow effectsin both cases (Fig. 8(c,d)), i.e. spatial signal separation depending on fre-quency, but the voltage peaks are more localized and with higher amplitudein the SSH case. It is important to notice that this effect is more significant16 igure 8: (a)-(b) Schematics of graded-SSH-metawedge and conventional metawedge re-spectively. Open circuit voltage (c)-(d) and accumulated energy (e)-(f) for the graded SSHand conventional metawedges as a function of position along the array. in the steady state regime; a relatively long excitation is required in order toproperly activate the edge modes. Both systems are compared using a timedomain simulation with a frequency sweep in the range 10 −
40 kHz with asource duration of 40 ms. In order to quantify the amount of electric energystored in both cases, we attach each piezo disk to an electric load of 10 kΩby means of a user Fortran subroutine integrated with Abaqus implicit timedomain integration scheme. The accumulated energy as a function of time is17hown in Fig. 8(e,f). The excitation of the edge modes at discrete frequenciescan clearly be seen, with an approximate maximum value of stored energy of0 .
44 nJ. For the conventional metawedge, we see that energy is more evenlydistributed along space, with a maximum value of approximately 0 .
26 nJ.This implies that, once the edge modes have been efficiently excited in theSSH configuration, we obtain a local enhancement of approximately 40% ofthe trapped electric energy when compared to conventional reflective rainbowmetawedge configurations.Before finalising this design in 3D elastic half-spaces, we address the ques-tion of why altering the height of the resonators was chosen as the gradingparameter other than, say, the initial spacing ∆ in S1. Figure 9 showsthe rationale behind this, which is to ultimately achieve broadband perfor-mance. Figures 9(a,b) show the effects of the longitudinal resonance on theposition of the Bragg gap in which the edge mode lies; the taller the rod thelower in frequency the axial resonance, which pushes down in frequency theBragg gaps below it; the geometrically induced bandgaps are influenced bythe resonance frequencies of the rods, which permits a natural tunability ofthe devices. Exploiting this, as we do, by the adiabatic grading of the rodheights increases the range of frequencies at which the distinct edge modesexist and which can be therefore be exploited for harvesting as shown in Fig-ure 8. Contrary to this, if instead we chose to alter the spacing ∆ , a muchsmaller range of edge mode frequencies can be exploited. Given the previousdefinitions of S1, we see that ∆ < a/
2; for ∆ > a/ = a/ can take. This is highlighted in Figures 9(c,d), where thespacing ∆ i is marked with Roman subscripts to avoid confusion between ∆ and ∆ used in the definition of S1 and S1 (cid:48) . In each case the different ∆ i correspond to altering ∆ . We alter from ∆ b = 14 mm, which is close to thelargest separation where the geometries can be distinguished, in Fig. 9(c) to∆ c = 7 mm in Fig. 9(d). These show that the position of the bandgap, andhence edge mode frequency, is largely unaffected by the change in spacing.As such for the simplest designs, where there is only one grading parameter,the choice of grading the rod height leads to optimal performance.18 igure 9: Grading height and spacing: Polarised dispersion curves, with longitudinalmotion shown by green points, overlaid on the Fourier spectra of the SSH interfaces con-sisting of different parameters within S1 and S1 (cid:48) . The periodicity a = 30 mm and rodradius r = 3 mm remain constant throughout. Panels (a,b) show the effect of grading theheight of the resonators; the longitudinal resonances of the rods influence the position ofthe Bragg gap, allowing for increased bandwidth of the device. (a) has parameters suchthat ∆ = ∆ a = 10 mm, h = 100 mm whilst (b) has the same S1 spacing with h = 155mm. The edge modes are clearly visible. Note the different scaling on the frequency axis;the gap position has been decreased due to the longitudinal resonance of the rods. Panels(c,d) show the effect of grading the spacing ∆ in structures S1. There is a much smallerrange of values this can take as we are limited by the symmetry of the unit cell. Panel (c)shows the band gap opening for a large ∆ = ∆ b = 14 mm and (d) shows the edge modefor ∆ = ∆ c = 7 mm. . Graded-SSH-metawedge for Rayleigh waves Galvanised by the amplifications achieved by the 1D topological edgestates in elastic beams, we now turn our focus to full 3D (isotropic) elastichalf-spaces, patterned with arrays of resonant rods on the surface. Thisstructuration creates a so-called metawedge and these have been used toexhibit extraordinary control of surface Rayleigh waves in terms of rainbowdevices and tailored surface to body wave converters [35, 36, 38]; here weexplore graded SSH structures for the elastic half-space.Elastic half-spaces support a wider variety of waves than the motiva-tional KL plates or elastic beams. Surface Rayleigh waves propagate alongthe free surface, exponentially decaying with depth into the bulk, travel-ling at wavespeed c r , independently of any periodic structuring. Further tothis, two polarisations of body waves, namely compressional (P) and ver-tical/horizontal shear (SV and SH) waves exist, both travelling at differingwavespeeds c p and c s respectively such that c p > c s > c r [77]. Unlike recentlydesigned mode conversion devices [38], we focus here on exciting topologi-cally protected surface waves by utilising the now familiar SSH model of rods,but now placed atop such a half-space. Analysis of the dispersion curves oftwo structures S1 and S1 (cid:48) show the existence of an edge mode at the do-main boundary between the two geometries, shown in Fig. 10, and confirmedthrough scattering simulations in Fig. 11.We obtain these results from time domain simulations of an array consist-ing of 10 cells of S1 and 10 cells of S1 (cid:48) atop an aluminium halfspace, as shownin Fig. 10(c). These simulations are carried out with SPECFEM3D, an open-source code fully parallelised with MPI [78]. This software solves the full 3Delastic wave equation leveraging the spectral element method for space dis-cretization, with an explicit finite difference scheme for time integration. Inorder to reduce potential spurious reflections and emulate unbounded wavepropagation, we apply Stacey absorbing boundary conditions at the edges ofthe half-space. The top surface and the resonator boundaries are traction-freeto guarantee the propagation of surface waves.The system is initially excited with a broadband sweep source rangingfrom 10 to 50 kHz, polarised in the vertical direction ( z ). By double Fouriertransforming the output signal in both space and time we obtain the disper-sion curves as the Fourier spectrum in Fig. 10(a), which are then validated bymeans of an eigenvalue analysis in Comsol Multiphysics 5.4. We can readilycompute the band structure of an infinite (doubly) periodic array for the unit20 igure 10: Panel (a) Shows identical dispersion curves obtained using Comsol Multiphysicsfor S1 and S1 (cid:48) , using the geometries defined in (b) as white points, with Rayleigh, shear andcompressional sound lines shown in red, green and cyan respectively. Also shown, throughthe Fourier spectrum is the existence of an edge mode at the band gap centre. The sourceis placed at the interface between regions of structures S1 and S1 (cid:48) (each comprising 10cells), marked by the dashed red line in (c). The parameters of the unit cells and half-spaceare such that are a = 30 mm , h = 30 mm , t = 0 . , l = 1 m , w = 20 cm , d = 40 cm. cells designed in Fig. 10(b) with Bloch-Floquet boundary conditions appliedon the side boundaries, and a low-reflecting boundary condition on the bot-tom surface of the computational domain to avoid spurious reflections. Sincewe are only considering normally incident excitation we only display the dis-persion curves along Γ − X [38], allowing the one-dimensional behaviour tobe inferred.After a thorough analysis of the dispersion relations we select the inputfrequencies for the scattering simulations in SPECFEM3D. Here we excitethe array at the SSH interface with a sinusoidal source corresponding to prop-agating, scattering and edge mode frequencies, respectively at 35 kHz, 37 kHzand 37 . u z ) of the resonators at a frequency close to that of the edge mode; there isa large amplification (over 100 times) of the vertical displacement of the rodscompared to the maximum vertical surface displacement of the half-space,which further corroborates the rods as suitable candidates for harvesters onhalf-spaces.To confirm the characteristic chiral flux of the surface edge modes, wecalculate the time-averaged flux, only on the surface. To do this, we use asimplified asymptotic approximation for the surface Rayleigh wave, treatingthe governing equation as a simple scalar wave equation, with the wavespeedcorresponding to that of the Rayleigh wave, c r . This model has been exten-sively developed [79, 80] and adopted in the design of seismic lenses [81]. Itprovides a simplification for calculating the flux, (cid:104) F (cid:105) , which now takes theform (cid:104) F (cid:105) = Im ( φ ∗ ( ∇ φ )) , (10)where φ is the out of plane displacement of the surface and ∗ denotes com-plex conjugation. As such this approximation treats the Rayleigh wave as ascalar surface wave, and allows the nature of the edge modes to be seen, ashighlighted in Figs. 11, 12.As for the conventional graded metawedge devices, the design of graded-SSH-metawedge structures for energy harvesting depends on the desired op-erational frequencies. For conventional metawedges, the height and grad-ing profile is informed by the periodicity and resonances of the individualrods; for efficient harvesting these structures operate by slowing propagat-ing waves to encounter zero group velocity modes at some designed spatialposition. Such modes are always present at the band edge by virtue of theBragg condition. Alternatively, symmetry broken structures can obtain zerogroup velocity modes within the first BZ [40]. As for graded SSH systems,the edge mode appears at the centre of the band gap [82], and as such thisallows the tailored design of a stepwise SSH grading to operate over a rangeof frequencies, as highlighted in Section 4. There is a larger degree of free-dom when considering an elastic halfspace compared to, say the KL platemodel; surface Rayleigh waves exist independently of any structuring on thearray. As such, the broadband excitation of multiple individual edge modes ispossible. We highlight this through the design of a graded-SSH-metawedge,22 igure 11: (a) Schematic of SSH interface, as a cross section of that in Fig.10(c). Regionscomposed of S1 are shown as grey rods with S1 (cid:48) shown as green. A top view of the surfacedisplacement of the halfspace shows a propagating, edge and scattering mode in (b,c,d)respectively, normalised to the displacement of the edge mode. Their respective fluxes,within the dashed white rectangles, are shown in the insets. The arrays are forced atthe SSH interface (dashed black line). Again the difference in amplitude (with the edgemode having approximately 100 times the amplitude) and flux patterns demonstrate theexistence of the protected edge state. (e) Shows the vertical displacement of the rodsrelative to the maximum displacement of the surface in (c), for a frequency Ω = 37 . shown in Fig. 12(a). The array consists of an Sn-Sn (cid:48) -Sm (cid:48) -Sm configurationas introduced in the case of rods on an elastic beam. The dispersion curvesof each individual pair are computed in a similar manner to the examplearray in Fig. 10, and the heights selected so that there is an overlap betweenthe longitudinal dispersion curves of the shortest rods with the bandgapsof the tallest rods, which ensures the propagation of the total broadbandsignal through the array, with lower frequencies travelling furthest throughthe array. The rod heights range linearly from 20 mm to 50 mm, all withthe same cross sectional thickness defined in Fig. 10. A broadband Rayleighwave (10 −
60 kHz) of duration 38 ms excites the array, and by frequencydomain analysis we show the excitation of several edge modes at their pre-23icted frequencies and interfacial positions between the designed structures(Fig. 12(b-d)). To confirm these are independently excited edge modes weanalyse the flux through (10), which shows the chiral nature of this quantity,shown in Fig. 12(e-f). Despite there only being one cell of the correspond-ing SSH pairs for each edge mode (i.e. one cell of S1 and one cell of S1’),rather remarkably each predicted edge mode is excited as highlighted by theexample in Fig. 12(d) at the extremity of the array. As such, graded-SSH-metawedges on elastic halfspaces have much promise in the design of compacttopological rainbow trapping energy harvesters.To further emphasise the utility of these devices, and to better emu-late a true metawedge, we extend the device to include several rows of thegraded-SSH geometry, representing those considered in [36, 38]. This barrierconfiguration enables the excitation of more edgemodes in the perpendic-ular direction. We analyse this system as above, which corroborates withthe initial assumptions used to match the edge mode frequency and positionfrom the dispersion curves; the predictions were obtained from the Γ − X direction of the two-dimensional dispersion curves and applied to a single,one-dimensional array. In Figure 13, the edge modes exist at the same fre-quencies an spatial positions along the array for normally incident radiation.This arrangement further motivates energy harvesting devices and vibrationisolation effects due to the strong confinement of the topological edge modes.
6. Discussion
We have successfully implemented the SSH model in a variety of elasticwave regimes, extending the uses of one-dimensional topological modes. Thecoalescence between the, seemingly distinct, SSH topological insulator andthe graded metawedge provides a new avenue for topological rainbow devices;these provide significant broadening of the bandwidth over which these onedimensional edge modes can be utilised. Specifically we have shown elasticenergy harvesting applications and compared the graded-SSH-metawedge toconventional metawedges, showing a pronounced increase in extracted energy,of approximately 40%, due to the strong localisation of these modes.Further applications, be it in thin plates, elastic beams or 3D elastic half-spaces, include vibration isolation. These small scale models can be readilyscaled up to groundborne and mechanical vibration control [81], and as suchwe envisage new applications of established topological models by synthesiswith other metamaterial structures. 24 igure 12: (a) Schematic of Graded SSH metawedge composed of structures in the alter-nating primed un-primed configuration (shown by grey-green colors of the rods), similarto the beam structure. Rod heights range linearly from 20mm to 50mm, on an elastichalf space of dimensions defined in Fig. 10. The array is excited, from the left (shown bywhite arrow), with a broadband Rayleigh wave in the range 10 −
60 kHz with a duration of38 ms, with (b-d) showing a top view of the frequency domain response for three frequen-cies which correspond to the predicted edge modes obtained from the individual dispersioncurves for these heights (as in Fig. 10), with rod positions marked by white points. (e-g)Show the flux on the surface from the scalar approximation of the surface Rayleigh waveat the corresponding interface positions marked by the dashed white boxes in (b-d). Theenhanced amplitude at the predicted interface positions along with the chiral nature ofthe flux confirms the existence and separate excitation of several edge modes in the gradedsystem. igure 13: SSH-barrier-metawedge (a) shows a schematic of an SSH-barrier-metawedge,where five rows of the graded array condisered in Fig.12 are separated by the array pa-rameter a . The barrier array is excited, from the left, with a broadband Rayleigh wavein the range 10 −
60 kHz with a duration of 38 ms (b,d,f) show the multiple edge modesexcited in this configuration, matching the positions as in Fig. 12, with their correspondingfluxes shown in (c,e,g). This corroborates the use of the section of the dispersion curves(Γ − X ), which were calculated from a doubly periodic array (Fig.10) for the predictionof this effect for normally incident Rayleigh waves. Again, the edge modes are quicklyexcited, requiring only one cell of each pair of SSH geometries a a given height, furthermotivating compact SSH devices. cknowledgements The authors would like to thank Joseph Sykes, Matthew Proctor andSamuel Palmer for helpful discussions. A.C. was supported by the Am-bizione Fellowship PZ00P2-174009. The support of the UK EPSRC throughgrant EP/T002654/1 is acknowledged as is that of the ERC H2020 FETOpenproject BOHEME under grant agreement No. 863179.