Liyun Cao

Yifan Zhu

Sheng Wan

Yi Zeng

Badreddine Assouar

On the Design of Non-Hermitian Elastic Metamaterial for Broadband Perfect Absorbers

Keywords: elastic metamaterials, non-Hermitian, broadband, low-frequency, perfect absorption

Elastic metamaterial, an engineered artificial material, has received much attention in physics and engineering communities due to its functional properties unavailable in natural materials. However, most elastic metamaterials, especially for their twodimensional structures, belong to the Hermitian category, making them difficult to adapt to real lossy structures and explore their loss modulation. In the present study, non-Hermitian loss-modulation beam and plate models based on complex wavenumber plane, structural dynamics, and mode-coupling scattering theory are established to design lossy elastic metamaterials (LEMs) for any solid material. Based on a unified closed-form solution for absorption, the subwavelength loss-modulation LEM can cope with the conventional challenge of achieving broadband and near-omnidirectional elastic wave perfect absorption. We reveal here the high-performance absorption of the LEM from greatly enhanced wave-energy dissipation by a combination of dissipationradiation-balance and multiple reflections in the non-Hermitian elastic wave system.

We numerically and experimentally verify the effectiveness of the theoretical model and demonstrate broadband and perfect absorption in a plate-like structure. Our work not only opens a new route to achieve broadband low-frequency vibration suppression in macro devices and microelectromechanical systems, more essentially, but it also provides an effective paradigm to wave engineering in non-Hermitian elastic wave systems.

Introduction

Ongoing development of elastic metamaterials [START_REF] Assouar | Acoustic metasurfaces[END_REF][START_REF] Chen | A reprogrammable mechanical metamaterial with stable memory[END_REF][START_REF] Dong | Achromatic metasurfaces by dispersion customization for ultra-broadband acoustic beam engineering[END_REF][START_REF] Zhu | Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial[END_REF] and metasurfaces [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF][START_REF] Lee | Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering[END_REF][START_REF] Liu | Source illusion devices for flexural lamb waves using elastic metasurfaces[END_REF][START_REF] Rong | Frequency-Coded Passive Multifunctional Elastic Metasurfaces[END_REF] has opened up new possibilities to control elastic waves scattering (solid object dynamic deform), which provides functional properties unavailable in natural materials. However, in most elastic metamaterial [START_REF] Dong | Customized broadband pentamode metamaterials by topology optimization[END_REF][START_REF] Muhammad | Lightweight architected lattice phononic crystals with broadband and multiband vibration mitigation characteristics[END_REF][START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Zhao | Broadband sub-diffraction and ultra-high energy density focusing of elastic waves in planar gradient-index lenses[END_REF] or metasurface (Cao et al., 2021d;[START_REF] Jin | Elastic Metasurfaces for Deep and Robust Subwavelength Focusing and Imaging[END_REF]Li et al., 2020a;[START_REF] Liu | Total reflection of flexural waves by circular meta-slab and its application in vibration isolation[END_REF][START_REF] Ruan | Reflective elastic metasurface for flexural wave based on surface impedance model[END_REF][START_REF] Shen | 3D-printed meta-slab for focusing flexural waves in broadband[END_REF][START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF][START_REF] Zhang | Vibration control of flexural waves in thin plates by 3D-printed metasurfaces[END_REF], especially for their twodimensional structures, the modulation of inherent loss is ignored because of the tiny damping of pure metal media. The corresponding systems are consequently treated as Hermitian, which are widely used to describe a closed physical system with respect to energy conservation. In a practical system, most solid structures are metal alloys and composites or introduce additional damping layers, which naturally have inevitable dissipation. Lately, considering loss modulation, the non-Hermitian description [START_REF] Graff | Controlling Sound in Non-Hermitian Acoustic Systems[END_REF]Hu et al., 2021b;[START_REF] Liu | Willis Metamaterial on a Structured Beam[END_REF]Wang et al., 2021a) has attracted growing attention in exploring new physics in the non-conservation system of optical waves [START_REF] Kawabata | Symmetry and topology in non-Hermitian physics[END_REF]Wang et al., 2021a) and acoustic waves [START_REF] Merkel | Control of acoustic absorption in one-dimensional scattering by resonant scatterers[END_REF]Romero-García et al., 2016;[START_REF] Zhou | Broadband impedance modulation via non-local acoustic metamaterials[END_REF], for example, unidirectional invisibility (Li et al., 2020b;[START_REF] Lin | Unidirectional Invisibility Induced by $\mathcal{P}\mathcal{T}$-Symmetric Periodic Structures[END_REF], enhanced sensing [START_REF] Chen | Exceptional points enhance sensing in an optical microcavity[END_REF][START_REF] Nair | Enhanced Sensing of Weak Anharmonicities through Coherences in Dissipatively Coupled Anti-PT Symmetric Systems[END_REF], topological characteristics (Hu et al., 2021a;Wang et al., 2021b) and asymmetric transmission [START_REF] Li | Tunable asymmetric transmission via Lossy acoustic metasurfaces[END_REF].

Mechanical vibration absorption of beam/plate-type structures is a conventional research topic. It has been widely investigated since the early 1950s, because of considerable significance in many engineering applications, for example, preventing vibration-induced fatigue damage, eliminating vibration interference to high-precision instruments, and fundamentally reducing noise sources. However, with the development of elastic-wave-based information technology in microelectromechanical devices and chip sensors in recent years, it is difficult for the conventional vibration absorption methods to meet the needs of lightweight and miniaturization. The conventional vibration absorption methods are represented by attaching many damping layers [START_REF] Kerwin | Damping of flexural waves by a constrained viscoelastic layer[END_REF][START_REF] Sun | Passive, adaptive and active tuned vibration absorbers-a survey[END_REF] and heavy dynamic vibration absorbers [START_REF] Hunt | The broadband dynamic vibration absorber[END_REF][START_REF] Shen | Parameters optimization for a novel dynamic vibration absorber[END_REF]. As a newly generated method of vibration absorption, the acoustic black hole [START_REF] Deng | Noise reduction via three types of acoustic black holes[END_REF][START_REF] Gao | Elastic wave modulation of double-leaf ABH beam embedded mass oscillator[END_REF][START_REF] Mccormick | Design optimization and performance comparison of three styles of one-dimensional acoustic black hole vibration absorbers[END_REF][START_REF] Pelat | Use of complex frequency plane to design broadband and sub-wavelength absorbers[END_REF][START_REF] Tang | Characterization of acoustic black hole effect using a onedimensional fully-coupled and wavelet-decomposed semi-analytical model[END_REF] is not useful for the low-frequency range (long wavelength) because its large structure dimension seriously hampers practical applications. Afterward, a sub-wavelength resonator [START_REF] Leng | Limits of flexural wave absorption by open lossy resonators: reflection and transmission problems[END_REF] with an attached damping layer in one-dimensional elastic beams was proposed to achieve perfect absorption of flexural waves in one specific frequency. A broadband active meta-layer (Li et al., 2021b) composed of piezoelectric sensors and actuators with complex feedback control loops in an elastic beam was also proposed for flexural wave absorption in a high-frequency range. However, narrowband or no ability to apply low frequencies is their weakness. To date, it is still a great challenge to realize subwavelength lightweight structures for broadband vibration absorption in the low-frequency regime. The loss modulation in non-Hermitian systems is a possible way to break through the challenge.

In the present research, we establish a unified non-Hermitian model for engineered elastic beam/plate-type structures and design subwavelength lightweight lossy elastic metamaterial (LEM) to break through the challenge of broadband vibration absorption in low frequency. Based on the theoretical model, we discuss the loss-modulationinduced perfect absorption physics and analytically derive the perfect absorption condition. Combined with weak coupling between multi-resonators, the absorption bandwidth of LEM is enlarged in a beam-type structure. Further, we achieve broadband and near-omnidirectional perfect absorption in a plate-type structure by the LEM with the coaction of dissipation-radiation-balance FP resonance and multiple reflections of the 0 th diffraction mode. Based on the theoretical model, we also show asymmetrical perfect absorption at the exceptional point. The theoretical models are validated by the corresponding experimental and simulation results. (1)

The model of the non-Hermitian elastic wave system

where is the propagation constant, . is the bending rigidity.

is flexural wavenumber and is the angular frequency. Along with the Bernoulli-Euler beam, Eq. ( 1) with is valid for a Kirchhoff plate of thickness d [25]. The in Eq. ( 1) differs only by the factor in the elastic beam and plate, where is Poisson's ratio. Therefore, theoretical models of the elastic beam are also applicable to the elastic plate for the vertically incident wave. We should point out that the fourth-order partial derivative equation Eq. ( 1), unlike the two-order one in optics/acoustics, has four solutions of wavenumber, i.e., two real ones and two extra pure imaginary ones. The imaginary ones are intrinsic non-propagation modes, i.e., flexural evanescent waves [START_REF] Graff | Controlling Sound in Non-Hermitian Acoustic Systems[END_REF], which hybridize scattering fields at all boundaries and interfaces of the elastic beam (plate) model. This makes the theoretical elastic wave model more complex and challenging than optic and acoustic ones. First, for an elastic wave system without loss, according to the continuities of displacement, slope, bending moment, shear force at the interface between the background beam and the waveguide resonator, we get the scattering equation of the interface ,

0 x -¥ < < 2 y b £ 2 z d £ ( ) , , x y z = x ( ) 0,0, w = u x L -¥ < < ( ) 4 4 , 0 w x t x b ¶ ae ö + = ç ÷ ¶ è ø b 4 d D b r = D k b w = w ( ) 4
where and .

is the scattering matrix.

The symbols t and r denote the transmission and reflection coefficients, respectively.

The subscripts 1 and 2 represent the background beam and the absorber, as shown in

1 l ! 1 s + 2 s + 1 s + ! 2 s + ! 1 s -2 s - 1 s - ! 2 s - ! ln s l 0.6, 0.1 к Ð = = 2 R rad g 1 l !
- + + ae ö ç ÷ ç ÷ = = × ç ÷ ç ÷ ç ÷ è ø S S MS ! ! ! ! ! ! !! !! ! ! !! !! [ ] 2 1 2 1 , , , T s s s s - - - - - = S ! ! [ ] 1 2 1 2
, , ,

where is the phase of the wave propagating a round-trip between the interfaces and . We should point out that Eq. ( 3) does not consider the near-field evanescent wave interference due to L being more than the half wavelength. Yet, it includes the contribution of the evanescent wave hybridization in all boundaries and interfaces. For example, in the above expression of is the phase shift of the

~1 2 к k k = 1 2 Ð D D = 1( 2) E 1( 2) r 1(2) u 1(2) d ( ) ( ) 1 2 1 2 k b w = w I 22 11 1 22 1 i N i i i N t t e R r
t t e r e r r e

f f f f - ¥ - - - - = = + × = + - å 2 2 3 /2 k L f p = - 1 l ! 2 l !
3 /2 p f flexural wave at the boundary , which is from the evanescent wave hybridization (see detailed derivation in Appendix G). However, optic/acoustic waves at a similar boundary have no phase shift.

The eigenvalue of the one-port elastic wave system equals the magnitude of the complex reflection coefficient, i.e., . After substituting a dimensionless complex wavenumber into Eq. ( 3

where

. ( 6
)
n=1 represents the first pair of Zero and Pole in the complex wavenumber plane. The n th pair will shift along the real wavenumber axis.

2 l ! s R l = ( ) 2 RE IM i k L k k L W = = + ! ! s R l = RE W IM W ln s l ( ) 0 s l W = s l ( ) ( ) 1 s s l l * * W = W * ( ) s l * W = ¥ Lossless Pole 11 ln i 2 r j W = - × Lossless Zero 11 ln i 2 r j W = + × ( ) 11 3 1 arg , 1 4 2 r n n p j p = - + ³ ( ) 1 n p -
The imaginary part of the Pole represents the radiative decay constant from the resonator to the background structure. The corresponding frequency of the Pole can be expressed as . It indicates that the radiative wave in the resonator will decay in the function of at the frequency of . Therefore, the quality factor of the resonance due to radiation can be expressed as ,

where is finite lifetime, . is radiative decay rate, which determines the resonance linewidth at its half-maximum [START_REF] Sauvan | Theory of the Spontaneous Optical Emission of Nanosize Photonic and Plasmon Resonators[END_REF].

We should point out that for the resonator described by the first pair of Zero and Pole in Fig. 1(d), its is about 8.4 according to Eq. ( 7). Its reflection coefficient is a fixed value of one for all real wavenumbers due to the lossless. However, its reflection phase spectrum has a rapid change at the resonance frequency, which is consistent with the positive and negative of rapidly changes on the real axis of the complex wavenumber plane.

In the lossy non-Hermitian elastic wave system, for the resonator with the dissipative decay constant , the complex reflection coefficient R in Eq. ( 3) can be replaced as ,

( )

Lossless rad Pole Im G = W ( ) ( )
w w t w é ù ê ú ë û = = = é ù × × ê ú ë W û W rad t ( ) Lossless rad rad Pole 1 1 Im t g w = = rad g rad Q R IM W RE 3 2 p W = V 12 21 11 22 1 i i t t e R r r e f f ¢ - ¢ - = + -
where is the phase constant with loss modulation . When the dissipative decay constant satisfies , the distribution of in Fig. 1(d) will move up .

It is the dissipation-radiation-balance, like the so-called critical coupling [START_REF] Cai | Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System[END_REF] in the optic system, however, here it is fulfilled in a different physical system with evanescent mode hybridization. Zero is located on the real wavenumber axis, as shown in Fig. 1

.

According to loss modulation of Eq. ( 10), the perfect absorption condition (PAC) in the non-Hermitian elastic wave system can be expressed as (see detailed derivation in

Appendix C) , (11)
where is the loss constant , and is the loss factor. Once the material parameters , geometric parameters , and loss factor meet PAC, the energy of the incident flexural wave will be perfectly absorbed. We point out that PAC applies to elastic beam/plate-type structures composed of any conventional material with internal loss or additional damping. To intuitively display the absorption, based on the resonator parameter described in Fig. 1(e), we plot the reflection energy Due to equals , the corresponding quality factor of the loss equals

f¢ i f f V ¢ = -× rad V = G In s l V W s R l = V 11 Los 11 s Pole i i 2 ln ln i r r j V j W = - × + × = - × Loss Zero j W = V ( ) ( ) 11 Re 2 l Im n r j à º à à 4 1 i h Ã= -× h 1(2) 1(2) 1(2) , , E r u 1(2)
. The total quality factor of the resonator with perfect absorption can be expressed as .

, where . Based on Eqs. ( 5), (7), and ( 12), Q varying with , for and , are shown in Fig. 1(g), respectively. Q of the perfect absorber decreases with the increase of .

Lossy elastic beam

As a typical example, we investigate the non-Hermitian LEM composed of a lightweight waveguide resonator with variable thickness d2, as shown in Fig. 2(a). For simplicity, the LEM has the same material of metal alloys with the background elastic beam of thickness d1. Increasing the thickness ratio can improve and of the model to get , according to Eq. (S5). Therefore, when is large enough to weaken the stiffness of the LEM (a large ), the real part of Pole can be rewritten as ,

where . Eq. ( 13) describes the so-called n th Fabry-Pérot (FP) resonance in the elastic wave system. Importantly, sufficiently weak stiffness of resonator (thickness ratio) to make the reflected wave have a modified phase is the key to the FP

2 R 2 1 R a = - V rad G loss Q rad Q rad 2 Q Q = rad loss 1 1 1 Q Q Q = + rad g 1 n = 2 n = rad g V = 1 2 d d d e = Ð к ( ) 11 arg 2 r p » d e Ð ( ) Lossless Pole Re j W = ( ) 2 Re , 1 2 k L n n p j p × = » + ³ 2 2 k b w = 11 r 2 p
resonance existence, which is different from that in optics and acoustics. Note that this modified phase can also be obtained by adjusting the material parameters of the resonator. The resonance frequency only depends on the length L. Here, we consider practical metal alloys or composite structures with small internal losses, described by the loss factor, which is often less than 0.1. The traditional idea is that it is impossible to achieve perfect absorption only through weak internal loss. However, based on the above-discussed loss-modulation non-Hermitian model, we break through this possibility. According to Eq. ( 11), the thickness ratio is surprisingly reduced to a constant regarding to ,

where is loss constant, , and . and are univariate functions of (see detailed derivation in Appendix D). Interestingly, it indicates that perfect absorption only depends on modulating the stiffness of the resonator (thickness ratio) and loss factor, and does not rely on the length L. The modulating method opens a new way of designing elastic wave absorbers and makes the loss factor become an essential design degree of freedom. Note that the effective loss factor of the resonator can be flexibly modulated by additional external damping [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF] for a wider design degree of freedom.

Based on Eqs. ( 13) and ( 14), once and satisfy Eq. ( 14), the LEMs with different lengths can achieve perfect absorptions whose different resonance frequencies are decided by Eq. ( 13), as shown in Fig. 2 Eq. ( 13), the LEM has a subwavelength size described by the ratio of the resonator length to the peak absorption wavelength

2 2 2 1 2 2 4 8 d J J y J J y J e J ae ö -- + + + ç ÷ = ç ÷ + è ø J 2e kh J - = (
. ( 15
)
The subwavelength size indicates that the compact and lightweight absorber can be applied to a lower frequency than the conventional ones [START_REF] Deng | Noise reduction via three types of acoustic black holes[END_REF][START_REF] Gao | Elastic wave modulation of double-leaf ABH beam embedded mass oscillator[END_REF][START_REF] Kerwin | Damping of flexural waves by a constrained viscoelastic layer[END_REF][START_REF] Mccormick | Design optimization and performance comparison of three styles of one-dimensional acoustic black hole vibration absorbers[END_REF][START_REF] Pelat | Use of complex frequency plane to design broadband and sub-wavelength absorbers[END_REF][START_REF] Sun | Passive, adaptive and active tuned vibration absorbers-a survey[END_REF][START_REF] Tang | Characterization of acoustic black hole effect using a onedimensional fully-coupled and wavelet-decomposed semi-analytical model[END_REF]. Based on Eqs. ( 7) and ( 13), the Q of the LEM based on n th resonance can be rewritten as .

( 16)

We plot the varying with L for and , respectively, as shown in Fig. 2(d).

These analytical results of Q are consistent with the simulated ones, which are independent of L. The reason is that in Eq. ( 16) quantifies the radiative decay at the interface without considering the resonator length L. In addition, the linewidth of LEM at its half-maximum can be expressed as .

According to Eq. ( 17), we plot for and , respectively, as shown in Fig.

2(e). The linewidth decreases with the increase of L.

To further reveal the absorption mechanism of the LEM, we extract the amplitude fields in the models from simulations, as shown in Fig. 2(f). The energy densities of wavefields (labeled P1 and P2 in Fig. 2(a)) are greatly strengthened in the LEM due to the resonance with loss modulation, which enhances energy absorption. For comparison, when the thickness d2 of the LEM is reduced to half to increase dissipative decay constant , but the loss-modulation PAC is not considered, the wavefields are not strengthened. The incident wave is hardly absorbed. It indicates that one key factor

( ) (
)

2 rad rad 2 2 16 4 2 n Q n p p p p + = + -G G Q 1 n = 2 n = rad G 1 l ! ( ) ( ) Loss 2 2 Pol ra 2 e d 4 2 2 Im ( ) R 2 2 e 1 D n d L f r p é ù × W ë û + G × D = = f D 1 n = 2 n = ( ) 2 Im k L
in obtaining the perfect absorption is significantly increasing the energy density of the resonator by judiciously harnessing the tailored losses.

Broadband absorption of LEM based on weak coupling

To realize broadband absorption, and inspired by the multi-resonance weak coupling theory for optic [START_REF] Cui | Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab[END_REF] and acoustic [START_REF] Huang | Compact broadband acoustic sink with coherently coupled weak resonances[END_REF]Yang et al., 2017)

where , , and are scattering coefficient vectors. is the coefficient vector of the incident wavefield and is the parameter matrix coupling all lossy resonators. Eq. ( 18) comes from the degradation of Eq. ( 24) (see detailed derivation in Appendix F).

First, we design every resonator capable of generating perfect absorption for different frequencies. These resonators and the background beam form a non-coaxial structure with an offset , as shown in the illustration of Fig. 3(b). We show that the absorption map is independent of , taking the center frequency of 1000 Hz as an example, as shown in Fig. 3(b). The independent proves that Eq. ( 14) based on the coaxial structure is still valid for these non-coaxial ones. The validity is due to the fact

( ) ( ) ( ) I II 1 L 2 III b b b b b ae ö ç ÷ × = ç ÷ ç ÷ ç ÷ è ø Τ G Τ G Τ ( ) I b T ( ) II b L T ( ) III b T 2 b G 1 b G s D s D
that the low-frequency flexural wave has the same deflection in the thickness direction of the thin plate. Based on Eq. ( 14), the absorption spectrum of each resonator is plotted by light grey curves in Fig. 3(c). All absorption peaks reach approximately the unity.

Further, as shown in Fig. 3(a), for example, the LEM is composed of ten resonators with different lengths L, which are 34.0δ, 33.6δ, 33.2δ, 32.8δ, 32.2δ, 31.7δ, 31.2δ, 30.4δ, 29.7δ, and 29.0δ mm, respectively, is a constant. We point out that extending the elastic beam model infinitely in the y-direction will form a plate. The plane strain model of this plate has the same absorption spectrum as this elastic beam model (Cao et al., 2021c) if its resonator lengths become 34.0, 33.6, 33.2, 32.8, 32.2, 31.7, 31.2, 30.4, 29.7 and 29.0 mm, i.e., .

Based on Eq. ( 18), we can get the analytical reflection coefficient of the LEM in the elastic beam, and the solid black line plots the absorption spectrum in Fig. 3(c). An average absorption coefficient higher than 0.8 in the frequency range from 851 Hz to 1196 Hz is obtained. Ten absorption peaks occur almost at the resonance frequencies of the resonators. It indicates that a weak coupling between resonators in the LEM induces broadband absorption. The simulated results, marked by circles in Fig. 3(c), are consistent with the analytical ones. We point out that all the results are not optimized, and the absorption effect of the LEM can be further improved by parameter optimization. On the other hand, increasing the number of resonators will increase the bandwidth. However, additional resonators will make the fabrication process of the specimen more difficult, which will limit the absorption bandwidth of the actual ( )

1 2 4 1 d u = - 1 d =
specimen. The non-local properties of the resonators [START_REF] Díaz-Rubio | Power flow-conformal metamirrors for engineering wave reflections[END_REF]Zhou et al., 2021) may be an effective way to break this limit.

To gain insights concerning the ten absorption peaks, the corresponding amplitude, In Fig. 4(a), the background plate, the gradient-length waveguides, and lossy multi-resonators are divided into region (I), region (II), and region (III) by the dotted lines and . In region (I), the governing equation for the flexural waves in the twodimension form is , where . The displacement field, including all diffraction modes in region (I), can be expressed as

3 M = 10 N = g 3 l ! 4 l ! ( ) (
)

2 I 2 2 1 11 2 + , , 0 D d w x yt t r ae ö ¶ Ñ Ñ = ç ÷ ¶ è ø 2 2 2 2 2 = x y ¶ ¶ Ñ + ¶ ¶ , ( 19
)
where is the Kronecker delta, and are the reflection amplitudes of j th order propagating and j th order evanescent diffraction modes, respectively. The number of propagation diffraction modes in the background structure can be decided by [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF] ,

where is the reciprocal lattice vector along the metasurface. In region (II) of one supercell, since the width of the subunit is much smaller than the operating wavelength, only the fundamental mode needs to be considered. The displacement of flexural waves in the m th waveguide can be expressed as

-× × -× × - × = = × + + å ! ,0 n d j  j  ! ( ) 0 =2 +1=2 roundup 2 / 1 J J k g × - 2 g g p = 2 2 1 = xj yj k k k - 2 2 1 ˆ= i xj yj k k k --- ( ) T I 1 , , j J é ù = ë û Τ e e e ! ! T , j j j é ù =   ë û e ! 1 M = 10 N = 1 M = 1 N = absorption coefficient
, ( 21
)
where , , , and are the amplitude coefficients of the fundamental mode at the left interface of the region (II). The coefficient vector can be defined as , where . In the region (III), considering the right free boundary of the multi-resonator, the displacement in the n th resonator on the edges of the m th waveguide can be expressed as

, ( 22
)
where and are the amplitude coefficients at the left interface of the resonator.

Ln is the length of the resonator. The coefficient vector can be defined as

, ( 23
)
According to the continuous boundary and orthogonality conditions, based on modified mode coupling methods, the scattering equation in the non-Hermitian elastic plate system can be derived as (see detailed derivation in Appendix E)

1 N = 3 M = 10 N = 3 M = 10 N = ( ) 1 1 1 1 II i i ( ) k x k x k x k x m m m m m w x a e b e c e d e -× -× × × = + + + m a m b m c m d ( ) [ ] T II 1 , , m M = Τ k k k ! ! [ ] T , , , m m m m m a b c d = k ( ) ( ) ( ) ( ) ( ) III i 1 i 2 i + i i 1 i 2 î 1 i ˆ i 1 i n n n n nn n n n n n nn n k L k x k x k L k x mn k L k x k x k L k x mn w e e e a
e e e b

-+ × + × -× - × × -+ × + × -× -× + × é ù = - + - ë û é ù + + + + ë û ˆmn a ˆmn b ( ) 1 III 1 = , , m mn m m n m n m mn mN M a b é ù é ù ê ú ê ú ê ú ê ú é ù ê ú ê ú = = ê ú ê ú ê ú ê ú ë û ê ú ê ú ê ú ê ú ë û ë û g G Τ G g g G g G ! ! ! ! , ( 24
)
where is a square matrix of size , and is a column vector of size . Eq. ( 24) can accurately calculate the response of all diffraction modes in the background structure.

When the reciprocal lattice vector of the metasurface is greater than 2k1, there is only 0 th propagation diffraction mode in the background plate, according to Eq. ( 20).

The mode has the maximum number E of multiple reflections due to phase shift matching at the interface. The maximum number E equals subunit number M in one supercell [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF]. The absorption coefficient of the LEM can be defined as

, where is the amplitudes of 0 th propagation diffraction mode in the background structure. When there is no gradient for the LEM, i.e., and in the model, the absorption spectrum of LEM in the plate model is almost consistent with that in the multi-resonators beam model, as shown in Fig. 4(b).

To quantitatively analyze the enhancement effect of multiple reflections on absorption, we replace the multi-resonator in Fig. 4 We have compared the corresponding simulation results with the analytical ones for three different frequencies, 900, 1000, and 1100 Hz (labeled P6, P7, and P8 in Fig.

I II L III ae ö ç ÷ × = ç ÷ ç ÷ ç ÷ è ø 1 2 Τ G Τ G Τ 1 G (2 4 2 ) (2 4 2 ) J M MN J M MN + + ´+ + 2 G (2 4 2 ) J M MN + + g 2 0 1 a = -Â 0 Â 1 M = 10 N = 1 N = 1 M = 1 N = ( ) ( ) ( )

4(e)

). An excellent agreement is obtained, as shown in Fig. 4(f). The nearomnidirectional perfect absorption stems from interface impedance matching in a wide

3 M = 3 M = ( ) ( ) 0 1 2 , 1 i f S i M L i M l = -× + Î ! " L ! 0 f l 0 1000 Hz f =
range of incident angles due to two points. The first is the high phase resolution of the LEM with a deep subwavelength unit width. The second is that flexural evanescent waves [START_REF] Liu | Source illusion devices for flexural lamb waves using elastic metasurfaces[END_REF] compensate for the impedance mismatch of the interface .

Asymmetrical perfect absorption at the exceptional point

To investigate the exceptional point (EP), we have designed the non-Hermitian elastic metamaterial described in section 5 as a two-port system, as shown in Fig. 5. The scattering waves in the left and right ports have complex displacement amplitudes , , and , , respectively. The subscripts 1 and 2 indicate the left and right ports, and the subscripts + and -represent incidence and reflection in the two ports, respectively. The scattering equation of the two-port system can be expressed as

, ( 25
)
where and are scattering vectors. The scattering matrix connects the incidences and reflections in the two ports. We should point out that the idea of this model comes from acoustics [START_REF] Wang | Extremely Asymmetrical Acoustic Metasurface Mirror at the Exceptional Point[END_REF], but it is implemented in challenging elastic waves with different loss systems. This section shows that our model has generalizability to the study of non-Hermitian elastic wave systems.

1 l ! 1 w - 1 w + 2 w -2 w + - + = × w ψ w [ ] 2
where and mean y-component wave vectors of the incident waves and j th order diffraction, respectively. According to Eq. ( 26), for the left incidence of , only the 0 th and -1 st order diffraction modes can propagate, corresponding to the specular reflection of and retroreflection of , respectively. For the right incidence of , only the 0 th and 1 st order diffraction modes are propagation ones, representing the specular reflection of 24), we can accurately obtain the reflection spectra of all diffraction modes. Owing to reciprocity, specular reflections from both sides are identical, i.e.,

. However, the extraordinary reflections and are different.

For simplicity, the resonator number in one subunit is set to 1, i.e., N = 1 in Eq. ( 24). All resonators are the same. We modulate the losses of the resonators simultaneously, which are characterized by an equivalent loss factor . The background plate and the gradient-length waveguides are considered small structural losses . Based on Eq. ( 24), we choose unit number M=20 to maximize the retroreflection efficiency by optimization. The operating frequency is 2340 Hz. The additional length constant , the resonator length, and the resonator thickness are modified to 31.14, 12.46, and 0.47 mm, respectively. Other geometric and material parameters remain the same as section 5. The eigenvalues of the scattering matrix are calculated by Eq. ( 24). Their trajectories vary with the loss , as shown in Fig. 5(b). At the EP where , the two eigenvalues coalesce together with the same eigenvectors . The same eigenvectors show that the

o r 30 q = - o r 30 q = ψ 0 1 1 0 L R L R r r r r - é ù = ê ú ë û ψ 0 0 L R r r = 1 L r - 1 R r eff h 0 h L ! 0 1 1 L L R r r r l ± - = ± × eff h eff 0.93 h = l ± [ ] 1, 0 T ± = v
initially orthogonal eigenvectors become parallel, which directly demonstrates the EP from the non-Hermiticity of the elastic wave system [START_REF] Wang | Extremely Asymmetrical Acoustic Metasurface Mirror at the Exceptional Point[END_REF].

We theoretically calculate the reflected wavefield at the EP by Eqs.

Experimental verification of theoretical models

Our proposed design method is universal and applicable to almost all solid beam/plate-type structures composed of any material (see absorption spectrums of r Re ( , )

w x y é ù ë û 1 0 L r -= 1 0.85 R r =
LEMs in elastic beams with different materials of concrete, cross-grain wood, plexiglass (PMMA), and Aluminum alloy, in Appendix J). We should point out that the proposed theoretical approach is valid for both the low and high-frequency ranges, verified by the corresponding numerical simulations. The size of the designed specimens for the high-frequency range is relatively small, which can facilitate specimen printing by the 3D printer and attaching the blue-tack layer in the boundary to design a non-reflection boundary (Cao et al., 2021c). Therefore, we have verified the proposed theory through experiments in the high-frequency range.

To verify the universality and facilitate specimen processing, we have chosen a 3D printer to fabricate three types of specimens with PLA material, which verify the design model in section 3, section 4, and section 5, as shown in the illustration of Figs.

6(e), 6(g), and 6(i), respectively. They are named as the single resonator specimen (an elastic beam with one resonator), the multi-resonators specimen (an elastic beam with multi-resonators), and the designed plate specimen, respectively. Fig. 6(a) shows the experiment setup. The absorption spectrum can be measured using the measurement mode "FFT" of Polytec Scanning Vibrometer 500 (PSV-500). To illustrate the absorption intuitively, we have realized the dynamic full wavefield in the single resonator specimen at the resonance frequency of 7.84 kHz by the "time" measurement mode (see Videos S1, Supplementary materials). The incident wave is not reflected when it enters the LEM, which means that it, indeed, perfectly absorbs flexural waves. In addition, the displacement deformations inside the LEM are greatly enhanced, which confirms that the loss modulation significantly increases energy density. For comparison, we have also measured the dynamic full wavefield at 9 kHz, away from the central frequency.

The measured result is presented in Videos S2, Supplementary materials, which shows f D that the wavefield is a standing wave, and the displacement deformation inside the resonator is not enhanced.

To verify the broadband absorption of LEM based on weak coupling, we print the multi-resonators specimen, i.e., an elastic beam with multi-resonators, as shown in Fig. 6(g). The LEM in the multi-resonators specimen consists of six resonators with different L. For experimental consistency, the six resonators are the same as those in six single resonator specimens in Fig. 6(b). The analytical absorption spectrum of the multi-resonators specimen, which is calculated by the theoretical model described by Eq. ( 18), is shown by the solid line in Fig. 6(f). The measured absorption spectrum is added by solid dots to Fig. 6(f). We can see some relatively large deviations, which is due to the amplification of the manufacturing error by the coupling between resonators. However, we can still find that the number of absorption peaks is consistent. More importantly, the frequencies of these peaks are consistent with the resonance frequencies of the six single resonators, described by six points in Fig. 6(b). To intuitively show their connection, we connect these peaks in Fig. 6(b) to the points in Fig. 6(f) by red dotted lines. They are one-to-one correspondence. The correspondence confirms that the broadband absorption of the LEM is from the coupling between the six resonators.

To verify the broadband absorption of the LEM in the plate-type structure, we print the designed plate specimen with PLA material, as shown in Fig. 6(i). The designed plate specimen is consistent with the model in Fig. 4(a), but it has fewer resonators to verify universality and simplify specimen processing, i.e., and 3 M = . Specific geometric parameters of the plate specimen can be found in Appendix I. For experimental consistency, the dimensions of the six resonators in the designed plate specimen are the same as the ones in the multi-resonators specimen in Fig. 6(g).

Solid dots show the measured absorption spectrum of the designed plate specimen in Fig. 6(h). It shows a high-efficiency average absorption ( ) in frequencies ranging from 6.72kHz to 8.42 kHz. The measured results are in good agreement with the theoretical ones based on Eq. ( 24). Some small observed deviations come mainly from the manufacturing error of the specimen and the imperfect excitation signal.

Compared with the absorption spectrum in Fig. 6(f), all absorption peaks reach one, and the curve is approximately a flat band. This confirms the broadband perfect absorption of the LEM in the plate-type structure.

Discussion and conclusion

We have established a theoretical model to design the subwavelength lightweight lossy elastic metamaterials in a non-Hermitian elastic beam or plate system, which can provide a general way of breaking through the challenges of broadband vibration absorption in low frequency. We have numerically and experimentally verified the effectiveness of the theoretical method in elastic beam/plate-type structures.

In acoustics, significant progress has been made in the low-frequency perfect absorption of sound waves [START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF]Mei et al., 2012). However, it is no longer adequate for low-frequency noise over 120 dB caused by mechanical vibrations (Li et al., 2021a). Our approach has the potential to solve this challenge by absorbing 6 N = 0.8 a > vibrational energy to suppress the sound emission. In addition, the lossy elastic metamaterial can contain no additional damping material. It could open real possibilities in the families of the elastic metamaterial to realize absorption in extreme and harsh environments of high (or low) temperature and corrosion (Cao et al., 2021b).

More than that, this work, in essence, paves the way for investigating loss-engineered wave scattering manipulation in elastic beam/plate-type structures, not limited to the dissipation-radiation-balance required by perfect absorption. from the interface to region 2, respectively. Therefore, the vectors and for both sides of the interface can be written as:

1 l ! 1 l ! ( ) ( ) i i i , j j j j k x k x k x k x t j
= + + + 1, 2 j = i j k x j A e - i j k x j B e j k x j C e - j k x j D e k b w = b 4 d D b r = 3 12 D d E = i 1 t e w × 1 l ! 1 l ! ( ) ( ) ( ) ( ) i L 1 1 11 i R 1 2
. (A3)

At the interfaces, the boundary conditions of displacement, slope, bending moment, and shear force must be satisfied (Cao et al., 2021a). According to these boundary conditions, we obtain the propagation equation

, ( A4
)
where N1 is the transfer matrix. From Eq. (A4), we can obtain = 1 0

, ( A5
= 0 0 r r t t ì é ù ï ë û í é ù ï ë û î k k ! ! R L 1 N = k k ( ) ( ) ( ) ( ) 2 2 2 11 2 2 2 2 1 i 1 2 1+ + 1+ Ðк к Ðк r Ðк к Ðк --- = ( ) 2 12 2 4 3 2 2( +1)( +1) +2 +2 +2 +1 к Ðк t к Ð к Ðк Ðк Ðк = 1 2 D Ð D = 1 2 k к k = i 1 t e w × 1 l ! L k R k T 21 21 T 22 22 L R = 0 0 = 1 0 t t r r ì é ù ï ë û í é ù ï ë û î k k ! ! L R 2 N = k k 1 s - 2 s - 1 s - ! 2 s - ! 1 l ! 12
ö ç ÷ ç ÷ = = × ç ÷ ç ÷ ç ÷ è ø S S MS ! ! ! ! ! ! !! !! ! ! !! !! [ ] 2 1 2 1 , , , T s s s s - - - - - = S ! ! [ ] 1 2 1 2
, , , T s s s s

+ + + + + = S ! ! I M b 4 d D b r = ( ) 4 2 1 d D b r u é ù = - ë û s R l = ( ) 2 RE IM i k L k k L W = = + ! ! ( ) ( ) IM IM 2 12 21 11 2 22 cos i sin 1 cos i sin L k s L k t t e r r e f f l f f × × -× = + - -× ! ! ! ! ! ! RE =2 3 2 k L f p - ! ! ( )
= + + + + - ! ! ! ! ! ! ( ) 11 Re a r = ( ) 11 Im b r = ( ) ( ) IM IM 1 4 2 2 2 + +2 c o s s i n + 1 L k L k m a b e a b e f f - × × é ù = - ë û ! ! ! ! s l IM IM
× ì + + = ï í + - = ï î ! ! tan = b a f - ! ( ) 11 3 1 = arg , 1 4 2 r n n p j p - + ³ RE L k j = × ! IM 11 2 In L k r × = ! Lossless Zero W Lossless Zero 11 In i 2 r j W º + × h . (C1)
According to , we can get .

(C2)

According to Eq. (C2), Eq. ( 9), and Eq. ( 12), we get .

(C3)

According to Eq. (C1) and Eq. (C3), we can get

, ( C4
)
where .

Appendix D. The thickness ratio is a constant regarding

The most practical metal alloys or composite structures have small internal losses, as described by the loss factor of less than 0.1. We can get where .

Solving the quartic equation of one unknown (D4) by the Ferrari method, we get the ( ) where .

( ) ( ) ( ) 1 1 1 4 4 2 2 1 4 4 2 2 2 1 1 i 1 i 1 1 s s s s s s d d k D D r w r w h h h h ae ö ae ö ae ö ç ÷ ç ÷ = - = - ç ÷ ç ÷ ç ÷ + + è ø è ø è ø ( ) 2 RE IM i k L k k L W = = + ! ! ( ) ( ) 2 2 Re Im i k k L W = + é ù ë û ( ) ( ) 11 2 2 Im R 2 e In r k L k L V j = -= = ( ) ( ) 11 Re 2 I Im n r j à º à 4 1 i h Ã= -× d e h 4 = 1 i 1 i 4 h h à -× » -×
= - = = -+ ( ) ( ) 2 2 1 2 2 2 4 2 2 3 3 3 3 3 1 2 1 2 4 0 2 6 0 2 8 0 3 4 0 3 3( 4 )/2 3 3( 4 )/2 9 3 ( ) ( ) 3 3 9 3 9 9 3 32 2 2 A B B AC A B B AC A A B C z z z z z z z z z z z z z z z z z z z z z z z z z J J J J J J J J J J J = - - - = - + + - = - - + - - = -+ = - = - = -× + - j ( ) ( ) I I x w x j ¶ = ¶ ( ) ( ) ( ) ( ) I I I 2 2 2 2 2 0 + 2 2 m x m w w D yy p x y M y y p l p u ì ae ö ¶ ¶ + - £ ï ç ÷ ç ÷ ¶ ¶ = í è ø ï - £ - î ( ) ( ) ( ) ( ) ( ) I I I 3 3 3 2 2 2 0 + 2 2 m x m w w D y y p x xy V y y p l p u ì é ù ¶ ¶ - + - -£ ï ê ú ¶ ¶ ¶ = í ë û ï - £ - î M M M M j J P P P ae ö ae ö ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ + = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø è ø H H H q Τ Τ H H H q H H H q ! ! " " " " # ! ! " " " " # ! ! ( ) ( ) ( ) i i 1 , sinc / 2 , sinc / 2 yj m yj m k y k y m j y j y j m j e k p e k p -× -× é ù = × × ë û H [ ] 1 = 1, 1, 1, 1 m q ( ) ( ) 0 i 1 0 sinc / 2 y m k y m y P m e k p - × = × 3 l ! ( ) ( ) 1 1 1 1 1 2 2 2 2 1 2 I I I L 2 2 2 2 1 2 2 2 2 2 1 2 j J m m m m m j J M M M M M j J P P P ae ö ae ö ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ + = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø è ø H H H q Τ Τ H H H q H H H q ! ! " " " " # ! ! " " " " # ! ! ( ) ( ) 0 i 2 0 0 i s i n c 2 y m k y m x y P m k e k p - × = - × × [ ] 2 1 1 1 1 = i , , i , m k k k k -- q ( ) ( ) ( ) i i 2 , i sinc 2 , sinc 2 yj m yj m k y k y m j x j y j x j y j m j k e k p k e k p - × - × é ù = × × × × ë û H 3 l ! 0 y g £ £ ( ) ( ) 1 1 3 1 3 3 1 3 1 3 1 I I I 1 3 L 3 3 3 3 1 3 3 3 3 3 m M j m M j j j j J m M J J J J P P P ae ö ae ö ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ + = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø è ø H q q q H Τ Τ q q q H q q q ! ! " # # # # ! ! " # # # # ! !
By applying the continuous boundary conditions of the x-component of displacement at the interface , and integrating to the z-direction in the region , we can get

( ) [ ] 3 1 1 1 1 , = , , , m j m j V V V V - - q ( ) ( ) 2 2 2 2 3 , j x j y j x j y j j g k k k k u u é ù = × - - - ë û H ! ( ) 2 1 1 = sin /2 yj m ik y yj pk c k p e V × × ( ) 2 2 3 0 0 j x y P g k k u = × - - 3 l ! ( ) ( ) 1 1 4 1 4 1 4 1 4 1 4 I I I 1 4 L 4 4 4 4 1 4 4 4 4 4 m M m M j j j j j m M J J J J J P P P ae ö ae ö ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ + = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø è ø H q q q H Τ Τ q q q H q q q ! ! " # # # # ! ! " # # # # ! ! ( ) [ ] 4 2 2 2 2 , = i , , i , m j m j V V V V × - -× q ( ) 3 2 4 0 0 0 i 2 i j x x y P g k k k u é ù = × - -- ë û ( ) ( ) ( ) { } 3 2 3 2 4 i 2 i , 2 j x j x j y j x j x j y j j g k k k k k k u u é ù é ù = × - -- -- ë û ë û H ! ! i 3 2 1 = sin ( /
= N N N N N N M M M M ´ǽ ö ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ × × ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø q p Τ Τ q p q p ! ! ! ! 1 1 1 1 1 1 1 1 6 1 1 1 1 4 i i i i i i N N k k k k k k k k k k k k - - é ù ê ú - - ê ú = ê ú ê ú - - ë û q ! ! ! ! 1 6 6 6 6 N n N ae ö ç ÷ ç ÷ ç ÷ = ç ÷ ç ÷ ç ÷ è ø p p p p ! ! ( ) ( ) ( ) ( ) ( ) 2 2 2 2 i 1 i 1 2i 2 6 2 2 2 2 2 2 i 1 i , i i 1 n n n n k L k L k L k L n n k k e k e k k e k e -+ × -+ × - × -× é ù = -+ + - -+ + - ë û p 4 l ! 1 2 z d £ , i 1 i n n n n k L k L k L k L n n D k k e k e k k e k e -+ × -+ × - × -× é ù = -+ + - + -+ ë û q 4 l ! 1 2 z d £ ( ) ( ) 8 8 8 R 8 8 8 II III = N N N M M M M ´ǽ ö ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ × × ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø è ø q p q Τ Τ p q p ! ! ! ! [ ] 8 1 1 1 1 1 = i , , i , D k k k k - - q [ ] 8 8 8 8 1
, , , ,

N N= p q q q ! ! ( ) ( ) ( ) ( ) ( )
O O O O O O O O ae ö - - - - ç ÷ = ç ÷ ç ÷ - - - - ç ÷ è ø 1 H H H H G Q Q Q Q Q N Q N Q N

Fig. 1

 1 Fig. 1(a) illustrates a typical lightweight absorber composed of a waveguide resonator

Fig. 1 .

 1 Fig. 1. Theoretical model. (a) A background elastic beam model (thickness d1 and width b) with a waveguide resonator (length L and thickness d2) absorbing flexural wave. (b) The interface between the background beam (region 1) and waveguide resonator (region 2) scatters incoming propagation waves with amplitudes , , incoming evanescent ones with amplitudes , , outgoing propagation ones with amplitudes , , and outgoing evanescent ones with amplitudes , , respectively. (c) Reflection of flexural waves in the waveguide resonator. (d) and (e) The contour plots illustrating the distribution of in the lossless and lossy scattering system, respectively [Here, ]. (f) The reflection energy ratio and absorption spectrum α. (g) Q varying with .

  Fig. 1(b). The superscript represents the evanescent mode. The scattering

  distribution of is shown in Fig. 1(d). In the complex wavenumber plane, Zero corresponds to the eigenvalue . satisfies , in which represents the conjugate. Pole corresponds to the eigenvalue . Therefore, a pair of Zero and Pole are symmetrically distributed about the real wavenumber axis, as shown in Fig. 1(d). The corresponding wavenumbers of Pole and Zero can be expressed as (see detailed derivation in Appendix B) ,

  , as shown in Fig. 1(f). The corresponding absorption coefficient is also added to Fig. 1(f), which shows a perfect absorption peak.

  (a).Fig. 2(b) shows the corresponding

  and 1000 Hz(labeled P1 and P2 in Fig. 2(a)) are shown in Fig.2(c). The corresponding simulated results are consistent with these analytical ones. According to

Fig. 2 .

 2 Fig. 2. Lossy elastic beam. (a) The map of absorption coefficient varying frequency and the length L of the resonator. The model in the illustration is an elastic beam of the thickness d1 4.5 mm with the waveguide resonator of the thickness d2 0.3 mm. The model composed of Mn-Cu alloy has a width b of 13 mm. (b) The map of the reflection phase. (c) Two absorption spectrums with α approximating one at 600 Hz and 1000 Hz (labeled P1 and P2 in Fig. (a)). (d)The quality factor Q varies with L for the absorber based on 1 st and 2 nd resonance, i.e., n=1 and 2 in Eq. (7). (e) The

  absorbers, we have established the elastic wave counterpart with unique evanescent wave hybridization. As shown in Fig. 3(a), the LEM is composed of parallel resonators with different lengths at the background elastic beam boundary. The scattering equation of the non-Hermitian elastic beam system is ,

  and energy flux fields (marked by the arrows in Figs. 3(d)-3(f)) in the model are obtained by simulations. Each amplitude field is highly concentrated to only one resonator when the excitation frequency approximates its resonance frequency. For example, the amplitude fields corresponding to three specific frequencies (labeled P3, P4, and P5 in Fig. 3(c)) are shown in Figs. 3(d)-3(f). These highly concentrated fields are a clear sign of the weak coupling between the resonators, leading to broadband absorption of the LEM.

Fig. 3 .

 3 Fig. 3. Broadband absorption in an elastic beam. (a) Ten waveguide resonators with different lengths are connected in parallel to the background beam in the theoretical model. The host beam and resonators have the same thickness as the model in Fig. 2(a). These resonators are numbered in their cross-section of the xy plane. (b) The resonator and the background beam form a non-coaxial structure in the illustration. The simulated absorption spectrum varies with frequency and for the resonator with the center frequency of 1000 Hz. (c) The light grey curve plots the absorption spectra of every single resonator. The analytical and simulated absorption spectra of the LEM are plotted. (d)-(f) The simulated amplitude and energy flux fields for three specific frequencies (labeled P3, P4, and P5 in Fig. 3(c)).

  Fig. 4(a) shows the theoretical model, M different units consisting of gradient-length

  and are x-component wave vectors of the propagation and evanescent flexural wave diffraction modes, respectively. The coefficient vector of the reflection diffraction field can be defined as , where .

Fig. 4 .

 4 Fig. 4. Broadband and near-omnidirectional absorption in an elastic plate. (a) Schematic diagram of the theoretical LEM model in the plate-type structure. The thicknesses of gradient-length waveguides are the same as that of the background plate, i.e., 4.5 mm. The widths of the supercell g and subunit b are 40.6 mm and 13 mm, respectively. (b) The absorption spectrum of the model without gradient waveguides ( and ). (c) The absorption spectrum of the single resonator ( and ) with the resonance frequency of 1000 Hz in the left subgraph. The

  of the theoretical model with varying with frequency and unit number M. (d) The absorption spectrum of the theoretical model with and . (e) The map of the absorption coefficient of the theoretical model with and varying with the frequency and the incident angle. (f) The corresponding simulated and analytical absorption coefficients for three frequencies of 900, 1000, and 1100 Hz.

  (a) with a single resonator, i.e., . The absorption spectrum of the single resonator ( , ) with the resonance frequency of 1000 Hz is shown in the left subgraph of Fig. 4(c). The corresponding absorption coefficient varying with frequency and M can be obtained by Eq. (24). The absorption spectrum is shown in the right subgraph of Fig. 4(c). When M increases, the frequency band with an absorption coefficient of approximately one (width of the magenta area in the y-direction) is gradually enlarged. For example, for

a

  the frequency marked by the red dotted line, the absorption coefficient of the single resonator is about 0.6 (marked by circle 1). The absorption coefficient of the LEM increases with M and reaches almost 0.95 when (marked by circle 2). Therefore, we choose to design the LEM, which can have a high absorption performance and increase the absorption coefficient of the single resonator from 0.6 to close to 1. As shown in Fig. 4(a), the lengths of three gradient-length waveguides can be calculated by . is the additional length constant. is the wavelength of flexural waves in the host plate in the frequency of . The absorption spectrum of LEM has broad absorption, as shown in Fig. 4(d). All absorption peaks in Fig. 4(b) are increased and reach almost one, i.e., broadband perfect absorption. The simulated results present an excellent agreement with the analytical ones. Further, based on Eqs. (24), we analytically solve the absorption coefficient varying with incident angle and frequency. The high absorption coefficient (exceeds 0.8) occurs in the frequency range from 836 to 1194 Hz, and for the angle ranging from -81° to 81°, as shown in Fig. 4(e). We define the highperformance absorption for the angle ranging from -90° to 90° as omnidirectional absorption. Therefore, the proposed LEM has high performance and achieves almost omnidirectional absorption, i.e., near-omnidirectional absorption.

Fig. 5

 5 Fig. 5 Asymmetrical perfect absorption. (a) a two-port system in the non-Hermitian elastic metamaterial. (b) The trajectories of the eigenvalues of the scattering matrix vary with the loss . (c) and (d) The reflected wavefield at the EP for the left and right incidences, respectively.

  the reflection coefficient of the corresponding diffraction mode. The subscripts of r indicate the diffraction order j, and the superscripts L and R indicate the left and right incidences, respectively. Based on the established analytical model described by Eq. (

(

  19) and (24). Figs. 5(c) and 5(d) illustrate the reflected wavefields for the left and right incidences, respectively. In the eight subgraphs from top to bottom, the first, second, third, and fourth rows of subgraphs show the analytical specular reflection fields from 0 th diffractions, the analytical retroreflection fields from 1 st or -1 st diffractions, the analytical total fields from all diffraction modes, and the simulated total fields, respectively. Due to reciprocity, specular reflections show a symmetrical pattern while strongly suppressed to enhance the overall asymmetric, as shown in the first row of subgraphs. The second row of subgraphs demonstrates the asymmetrical perfect absorption where no retroreflection occurs for the left incidence and a strong retroreflection is present for the right incidence. The third row of subgraphs manifests the EP of the system. Compared to the right incidence with strong reflection, the waves are almost perfectly absorbed for the left incidence. To validate the theory, we have carried out the corresponding numerical simulations, and show the results in the fourth row of subgraphs. Good agreement between the simulated and analytical results confirms the correctness of the theoretical model.

Fig. 6

 6 Fig. 6 Experimental verification of theoretical models. (a) The experiment setup. (b) For the single resonator specimen, the map of the analytical absorption coefficient varying with the frequency and the L. Six points corresponding to these measured absorption coefficients are added. (c) The map of Q and Δf of the absorption spectrum varying with L, calculated by Eqs. (16) and (17), respectively. The corresponding results obtained from these measured absorption spectrums are added. (d) The experimental absorption spectrum and the analytical one for the 2 nd single resonator specimen. (e) The printed single resonator specimen with PLA material. (f) For the multi-resonators specimen, the analytical and measured absorption spectrums are shown. (g) The printed multi-resonators specimen with PLA material. (h) The experimental absorption spectrum and the analytical one of the designed plate specimen. (i) The printed plate specimen with PLA material.

Fig. 6

 6 Fig. 6(d) also shows the agreement between the experimental absorption spectrum

  k are flexural stiffness and wavenumber of flexural waves.In the same way, we make an incident negative-going flexural wave of at the right side of the interface ,

  scattering matrix of the elastic plate systemIn the x-axis direction shown in Fig.4(a), the slope , bending moment M, and shear force V, in the two-dimension plate model are rewritten as ,

  x-components of shear force at the interface of x=0, (II) of one supercell, the transfer equation for waves propagating from the left to the right interface of the m th waveguide can be expressed as , (E8)

.

  By applying the continuous boundary conditions of the x-component of slop at the interface , and integrating it to the z-direction in the region , continuous boundary conditions of the x-component of bending moment at the interface , and integrating to the z-direction in the region ,

  vector of size . The corresponding amplitudes of diffraction modes can be calculated by solving Eq. (E14).Appendix F. The scattering matrix of the elastic beam systemWhen the unit number in the metasurface and the diffraction mode number are set to M=1 and J=1, respectively, the propagation constant changes from to , Eq. (E14) can be reduced to the scattering equation size . The reflection coefficient equals . The absorption coefficient of the LEM is .

2 z d£

Conflict of Interest

All authors declare that they have no conflict of interest.

Supplementary materials

Video S1: The dynamic full wavefield in the 2 nd single resonator specimen at the central frequency of 7.84 kHz measured by PSV-500.

Video S2: The dynamic full wavefield in the 2 nd single resonator specimen at 9 kHz (away from the central frequency) measured by PSV-500.

Appendix A. Analytical expression of the scattering matrix at the interface

As shown in Fig. 1 , where is the propagation constant, , and is the flexural rigidity. E is the Young modulus, ρ is density.

We make an incident positive-going flexural wave of at the left side of the interface , so the wave fields at the left and the right sides of the interface can be expressed as follows:

where the symbols t and r denote the transmission and reflection complex coefficients, respectively. The subscript ~ represents an evanescent flexural wave. 1 and 2 represent region 1 and region 2, respectively. The subscripts 11 and represent the propagating reflection wave and evanescent one from the interface to region 1, respectively. The subscripts 12 and represent the propagating transmission wave and evanescent one According to Eq. (A8), we can obtain (A9) .

(A10)

In the same way, we make an incident evanescent flexural wave of at the left and right sides of the interface , and obtain the coefficients of , , ,

The interface scatters incoming propagation waves with amplitudes , , incoming evanescent ones with amplitudes , , outgoing propagation ones with amplitudes , , and outgoing evanescent ones with amplitudes , .

respectively. Therefore, we can obtain the scattering equation at the interface :

Appendix B. The wavenumbers of Zero and Pole

The eigenvalue of the one-port elastic wave system equals the magnitude of the complex reflection coefficient, i.e., . Substituting a dimensionless complex wavenumber into Eq. ( 3), From Eq. ( B6) and (B7), we obtain the complex wavenumber , which corresponds to the Zero in the complex wavenumber plane, i.e., .

(B8)

Appendix C. PAC in the lossy elastic wave system

The wavenumber in the waveguide resonator with the loss fact is

Appendix G. Phase shift of at the free hard boundary

We set an incident flexural wave of at the left side of the hard boundary , so the scattering wave field with evanescent wave hybridization can be expressed From Eq. (G2), we get , i.e., the propagation reflection wave has the phase of . The incident flexural wave of has the phase of 0. Therefore, the propagation reflection wave has a phase shift of at the free hard boundary, due to the evanescent wave hybridization.

Appendix H. Experimental measurements

As shown in Fig. 4h, the PZT patches are driven by a signal generator (Tektronix AFG3022C). The blue-tack layer is bonded to the bottom of the plate specimen to absorb the downward propagating flexural wave excited by the PZT patch and the reflected flexural wave from the designed structure at the top. The length of blue-tack in the direction of propagation waves is 50 mm, which can be equivalent to a nonreflection boundary of flexural waves within the frequency range of the experiment.

The specimen surface is held perpendicular to the laser beam from PSV-500 scanning laser Doppler vibrometer through a C-clamp supporting the plate applied to the lower edge, as shown in Fig. 4h. The PSV-500 scanning head records the out-of-plane complex velocities of two measurement points in the far-field of the incident direction using the measurement mode "FFT". An ensemble average with 20 samples is used at every measurement point to ensure signal quality. The measured complex velocities in the two points are and , respectively. We can calculate the reflection coefficient as , where and are the distance between the two points, . is the measured wavenumber of the flexural wave in the host structure. The absorption coefficient of flexural waves can be calculated by .

Appendix I. The geometry of specimens in the experiments

For six single resonator specimens with different lengths, we name them as "1 st single resonator specimen", "2 nd single resonator specimen", "3 rd single resonator specimen", "4 th single resonator specimen", "5 th single resonator specimen", and "6 th single resonator specimen". Their resonator thicknesses are d2, and their resonator lengths are L1, L2, L3, L4, L5, and L6, respectively. Specific geometric parameters of the resonators are shown in Table S1.

Table S1 The geometric parameters of specimens in the experiments. All parameters are in units of mm.

3.0 0.2 6.6 6.8 7.0 7.1 7.2 7.3 2.5 9 1.5 7.5 13.5

Appendix J. Absorption spectrums of LEMs in elastic beams with different materials Our proposed design method is universal and applicable to almost all solid beams or plates composed of any material. Based on Eq. ( 6), we design LEMs in four elastic beams with different materials of concrete, cross-grain wood, plexiglass (PMMA), and Aluminum alloy. These LEMs are defined as the cases from C1 to C4. We show the absorption spectrums in Figure S1. The related material and geometric parameters of the cases from C1 to C4 are shown in Table S2. , and the loss factor (Cao et al., 2021c).

All simulations are performed by using the COMSOL Multiphysics 5.6 software (Solid Mechanics Module). All outer boundaries have any reflection by PMLs. We obtain the simulated absorption spectrum by post-processing in the same calculation method as the experiment.