Reconfigurable metamaterial for asymmetric and symmetric elastic wave absorption based on exceptional point in resonant bandgap

Elastic wave absorption at subwavelength scale is of significance in many engineering applications. Non-Hermitian metamaterials show the ability in high-efficiency wave absorption. However, the single functionality of metamaterials is an important limitation on their practical applications for lack of tunability and reconfigurability. Here, we propose a tunable and reconfigurable non-Hermitian piezoelectric metamaterial bar, in which piezoelectric bars connect with resonant circuits, to achieve asymmetric unidirectional perfect absorption (UPA) and symmetric bidirectional perfect absorption (PA) at low frequencies. The two functions can be arbitrarily switched by rearranging shunted circuits. Based on the reverberation-ray matrix (RRM) method, an approach is provided to achieve UPA by setting an exceptional point (EP) in the coupled resonant bandgap. By using the transfer matrix method (TMM) and the finite element method (FEM), it is observed that a non-Hermitian pseudo-band forms between two resonant bandgaps, and the EP appears at the bottom of the pseudo-band. In addition, the genetic algorithm is used to accurately and efficiently design the shunted circuits for desired low-frequency UPA and PA. The present work may provide new strategies for vibration suppression and guided waves manipulation in wide potential applications.


Introduction
Elastic wave absorption is of significance in many engineering applications. For example, perfect absorption (PA) enables accurate measurement, and improves safe reliability and ride comfort of train. However, wave absorption for noise reduction and vibration suppression on subwavelength scale remains a technological and scientific challenge [1][2][3][4] . Acoustic/elastic metamaterials with designed microstructures have been explored to control elastic wave propagation at low frequencies for two decades [1] , and have shown the unique properties and unprecedented functionalities, such as negative effective mass or stiffness [5][6] , negative refraction [7] , asymmetric transmission [8][9] , and cloaking [10] . Generally, locally resonant acoustic/elastic metamaterials have low-frequency bandgaps, in which wave cannot propagate, and can be used to isolate vibration [11][12] . However, matematerials cannot dissipate wave energy in lossless cases [13] . The existing energy is still a potential threat.
Many phenomena in non-Hermitian systems are associated with exceptional points (EPs) [20,22,[34][35][36] , which are the branch points or singularities, at which the eigenvalues of Hamiltonian operator or scattering matrix coalesce, and the corresponding eigenvectors degenerate, simultaneously. Particularly, EP-induced asymmetric wave propagation in non-Hermitian metamaterials has attracted growing interest, such as asymmetric wave reflection [18][19][37][38] and asymmetric mode switching [22] . When non-Hermitian metamaterials involve EPs for wave absorption, the unidirectional perfect absorption (UPA) shows functional applications in aeroacoustics [30,[39][40][41] . For unidirectional elastic wave absorption in solids, Norris and Packo [42] theoretically proposed that a pair of attachments on a beam is possible to achieve perfect one-way flexural absorption. Li et al. [3] designed a lossy metalayer to demonstrate unidirectional flexural wave absorption in a beam, in which the key is to reach a critical coupling.
Although there is increasing interest in elastic wave absorption via non-Hermitian metamaterials, the single functionality of these metamaterials is an important limitation on their practical applications for the lack of tunability and reconfigurability [43] . In this paper, we propose a reconfigurable non-Hermitian metamaterial by employing easily tunable electric circuits connected with piezoelectric elements to obtain UPA and bidirectional PA. Not only is the working frequency of the proposed metamaterial highly tunable at subwavelength scale due to drastic change of mechanical properties of shunted piezoelectric pitches [44][45][46][47][48] , but also the functions of the metamaterial can be arbitrarily switched between UPA and PA. It is also important that we use a new approach to achieving UPA, that is, accessing EP in non-Hermitian bandgap. Though EP in bandgap has been reported in aeroacoustics [39] , here, the mechanism is different. We induce EP in bandgap by constructing non-Hermitian pseudo-band. However, the physical fundamentals of wave propagation in solids and fluids are very different.
In this paper, we explore the reconfigurability and tunability of a piezoelectric metamaterial bar for UPA and PA. The rest of paper is arranged as follows. In Section 2, we illustrate the different characteristics of UPA and PA, and introduce the mathematical model of the proposed metamaterials. Section 3 provides a semi-analytical method, i.e., the reverberation-ray matrix (RRM) method, for calculating transmission and reflection of the metamaterial. Then, in Section 4, we demonstrate that EP can be set in electromechanical resonant bandgap by con-structing non-Hermitian pseudo-band, which gives rise to the UPA. The accurate manipulation of EP in bandgap is achieved by using the genetic algorithm, and reconfigurability and high tunability of the metamaterial are demonstrated in Section 5. Section 6 gives some conclusions.
2 Reconfigurable and tunable piezoelectric metamaterial bar

Configuration
Before we present the characteristics of UPA and PA, CPA is briefly discussed at first, since UPA appears after CPA in non-Hermitian metamaterials. The characteristics of CPA, UPA, and PA of a two-port system from scattering matrix perspective are illustrated in Fig. 1, in which S is the scattering matrix, A − and A + are amplitudes of incident waves, and D − and D + are amplitudes of outgoing waves. CPA needs two coherent incident waves. When the phases and amplitudes of two incident waves satisfy strict coherent conditions, the system will perfectly absorb the incident waves without outgoing waves, leading to CPA [31] . Two incident waves are needed to meet the coherent conditions, which is an obvious disadvantage of CPA for wave absorption. Thus, UPA has been developed [40] , which can absorb incident waves without this limitation. When UPA occurs in a reciprocal system, three elements of scattering matrix are zero, and one reflection coefficient remains non-zero. If one wants a system to absorb bidirectional incident waves but without coherent limit, all elements of the corresponding scattering matrix must be zero, as PA shown in Fig. 1. We propose a reconfigurable and tunable non-Hermitian piezoelectric metamaterial bar shown in Fig. 2 to achieve UPA and PA manipulating EP in resonant bandgap. The metamaterial bar with circular cross section is constructed with aluminum and piezoelectric material PZT-5H (lead zirconate titanate, PZT), and its unit cell with length l consists of aluminum bars and piezoelectric bars in alternate arrangement, in which the length of the aluminum bars at center is l b , the length of the aluminum bars at ends is l 0 , and the length of the piezoelectric bars is l p , as shown in Fig. 3(a). In one cell, two piezoelectric bars connect with two different shunt circuits with effective impedances Z 1 and Z 2 , respectively, and each circuit consists of an inductor and a resistor in series, as shown in Fig. 3(b). The ends of N -cell matematerial connect with semi-infinite aluminum bars, which are background media, as shown in Fig. 2. The polarization direction of PZT-5H is in the z-direction of global coordinate system Oxyz, whose origin is set at the center of symmetry of the whole structure, as shown in Fig. 2. The radius of the cross section of the composite bar is a constant r. Fig. 2 One-dimensional N -cell metamaterial bar with background media (color online)

Modeling, harmonic solution, and dispersion equation
The constitutive equations of piezoelectric materials are [49] ε = s E σ + dE, in which ε and σ represent the second-order strain tensor and the stress tensor, respectively, E and D represent the electric field intensity vector and the electric displacement vector, respectively, s E represents the elastic compliance tensor at constant electric field, d represents the piezoelectric coefficient tensor, and σ represents the dielectric constant at constant stress. Gauss's law in the differential form is D j,j = 0, and the dynamic equilibrium equation without body force is in which " · " represents time derivative, and ", j" represents spatial partial derivative in the j-direction. By neglecting transverse Poisson's effect and transverse piezoelectric effect, the reduced constitutive relations of one-dimensional piezoelectric bar along the z-direction are in which ρ p is the density of the piezoelectric bar.
In the local coordinate O x z , in which O is set at the end of a piezoelectric bar, as shown in Fig. 3(b), the piezoelectric bar is polarized in the z -direction and connected with the circuit with effective electrical impedance Z. Thus, the harmonic wave solutions with angular frequency ω for Eq. (3) are [50]  in which "i" is the unit of imaginary number, A 0 , A 1 , and A are amplitudes, and q p is the wavenumber in the piezoelectric bar. Substituting the solutions into Eq. (3), we obtain the wavenumber as Therefore, considering the superposition of the left-going wave and right-going wave, the solu- in which A b and A f represent the amplitudes of the left-and right-going waves, respectively. The electric boundary conditions for the piezoelectric bar shunted circuit with electrical impedance Z can be expressed as in which A p = πr 2 represents the cross-sectional area. Considering the electric boundary conditions in Eq. (6), we rewrite the solutions to Eq.
in which By substituting Eq. (7) into Eq. (2), the displacement and stress distributions for harmonic wave are expressed as in which For the boundaries z = 0 + and z = l − p , there are in which .
Hence, the transfer matrix of the piezoelectric bar connected with a circuit with electrical impedance Z is The shunted piezoelectric bar can be equivalent to non-piezoelectric bar with effective Young's modulus E e and effective density ρ e . The dynamic equation of the equivalent nonpiezoelectric bar is in which and q e = ρ e /(E e ω). Let T e = T p . Then, the effective E e and ρ e can be obtained.
The transfer matrices T 0 and T b of the aluminum bars with l 0 and l b , respectively, are in which E 0 is Young's modulus of aluminum. Consequently, the transfer matrix of one cell shown in Fig. 3 Using Bloch's theory, the dispersion equation of the metamaterial bar can be obtained as in which q represents the Bloch wavenumber.

Tunable electromechanical resonant bandgaps
It is known that the piezoelectric bar can be equivalent to a capacitor with effective capacitance C σ p = πr 2 ε σ 33 /l p . When the piezoelectric bar is shunted with an inductor with inductance L, the inductor and the piezoelectric bar can form a resonant circuit with resonant To show the high tunability of electromechanical-resonance-induced bandgap, three different electric boundary conditions are considered: The geometrical parameters and material properties of the metamaterial bar are listed in Table 1 and Table 2, respectively.
Solving Eq. (11), we obtain the band structures of the metamaterial for the three cases as shown in Fig. 4. When the piezoelectric bars are short circuit (Case A), there is only one bandgap below 150 kHz, which is induced by Bragg scattering resulting from the alternate arrangement of aluminum bars and piezoelectric bars, and the frequency 30 kHz is in passband, as shown in Fig. 4(a). When inductors are connected to the piezoelectric bars, the electromechanical resonance at f LC = 30 kHz can induce a resonant bandgap below the Bragg's bandgap at this frequency, as Case B shown in Fig. 4(b). Intuitively, when two resonances are created, such as Case C, there are two resonant bandgaps, as shown in Fig. 4(c). Definitely, the metamaterial is highly tunable based on changing the shunt circuits [44] . In these Hermitian cases without dissipation, passbands and bandgaps are clearly shown. However, it will be shown that the boundary between passband and bandgap is blurred in the non-Hermitian case with dissipation, and there is a formation of non-Hermitian pseudo-band between two resonant bandgaps. The pseudo-band is the key for setting EP in bandgap, which leads to UPA.

RRM method for calculating transmission and reflection
In this section, the calculation method for transmission and reflection coefficients of the metamaterial is given before we present the approach to achieving UPA in the next section. For a harmonic longitudinal wave propagation in the one-dimensional composite bar system shown in Fig. 2, the displacement distribution u 3 in the background media can be expressed as in which q is the wavenumber, A − and A + are the amplitudes of the incident waves, and D − and D + are the amplitudes of the outgoing waves from the metamaterial bar with length N l. Therefore, the scattering relations of the metamaterial are in which S is the scattering matrix of the metamaterial, t L (t R ) and r L (r R ) are the transmission and reflection coefficients, respectively, for incident wave from the negative (positive) z-direction, namely the left (right) side of the metamaterial bar.
Here, the RRM method [51] is used to calculate the transmission and reflection coefficients. When the N -cell metamaterial is divided into n homogeneous parts, the whole bar system shown in Fig. 2 including two semi-infinite background media, has n + 2 parts (n + 1 interfaces). The interfaces between aluminum bars and piezoelectric bars must belong to the n + 1 interfaces. The left background medium is labeled as zeroth part, and the first part is the leftmost part of n-part metamaterial. Thus, from left to right, the Jth pair of dual coordinates, (Oxz) J(J+1) and (Oxz) (J+1)J , for the Jth part, is shown in Fig. 5. In the coordinate (Oxz) J(J+1) , the displacement distribution of harmonic wave can be written as in which a J(J+1) and d J(J+1) represent the arriving wave amplitude and the departing wave amplitude, respectively. Therefore, the displacement and stress distributions, X J(J+1) = (u 3 , σ 33 ) T , at z J(J+1) = 0 are expressed as in which Substituting Eq. (14) into Eq. (16) yields the relations as follows: In the coordinates (Oxz) (J−1)J and (Oxz) J(J−1) , the displacement distribution leads to and the phase relations of wave amplitudes at the Jth interface are The continuous conditions at all interfaces lead to the scattering relation for the whole system,  There are 2 × 2(n + 1) unknown wave amplitudes in Eq. (21). In background media, the displacement is expressed in Eq. (12), and the wave amplitudes are marked as in which A − and D − also represent the arriving wave amplitude and the departing wave amplitude in the coordinate (Oxz) 10 , respectively, and A + and D + represent the arriving wave amplitude and departing wave amplitude in the coordinate (Oxz) (n+1)(n+2) , respectively. Thus, Eq. (21) can be rewritten as In terms of the phase relation (Eq. (20)), the phase relations for the whole system are a = P Hd, in which Thus, the scattering relation of the N -cell metamaterial is

Approaching to UPA
Based on the RRM method, the approach to achieving UPA is shown here by observing transmission and reflection. It has been shown that electromechanical resonance can induce resonant bandgap in passband at low frequencies. If two different electromechanical resonances are created, as shown in Fig. 4(c), two resonant bandgaps will appear. The corresponding transmittance and reflectance of 5-cell metamaterial with parameters in Fig. 4(c) are shown in Fig. 6(a). In this case without dissipation, i.e., resistance R = 0, the left reflection coefficient equals the right reflection coefficient. When the resistors have a non-zero resistance, e.g. R = 10 kΩ, the two reflection coefficients become obviously different in bandgaps and passbands, as shown in Fig. 6(b), resulting in asymmetric scattering. This asymmetric scattering is due to simultaneous asymmetric unit cell and intrinsic loss [37] , in which the former is caused by two different resonant circuits. Then, adjusting the resonant frequencies f LC1 and f LC2 close to each other, the passband between the two resonant bandgaps becomes extremely narrow, or disappears when the two resonant bandgaps are coupled to each other, as shown in Fig. 6(c). However, the prominent asymmetric reflections still remain, and are almost completely located in the coupled resonant bandgap. Subsequently, adjusting the loss, i.e., resistance, there is almost no change for the transmission in the bandgap, but remarkable change for the asymmetric reflections in the resonant bandgap, as shown in Fig. 6(d). It can be observed that one reflection coefficient remains to be large, and the other one tends to zero at a dip in the resonant bandgap. Definitely, this is UPA, which is indicated by red arrow in Fig. 6(d), at which three elements of scattering matrix are zero, and one reflection coefficient is finite. The absorption coefficients of the metamaterial for incident waves from the left side and the right side are defined as respectively. The absorption coefficients corresponding to the case shown in Fig. 6(d) are shown in Fig. 7. At UPA indicated by red arrow, the incident wave with frequency 29.53 kHz from the right side of the metamaterial can be absorbed perfectly, but part of the left incident wave is absorbed.

EP in bandgap
To confirm that this UPA is related to EP, we eximine the phases and eigenvalues of the scattering matrix. In respect of the scattering matrix, the EPs of non-Hermitian system indicate that the eigenvalues of the scattering matrix and their corresponding eigenvectors degenerate  simultaneously [18,[34][35][36]38] . The eigenvalues of the scattering matrix in Eq. (13) are in which t (= t R = t L ) results from the reciprocity of the system, and the corresponding eigenvectors are in which "T" represents transposition. It is obvious that if one reflection coefficient, either r L or r R , equals zero, the simultaneous degeneration of eigenvalues and eigenvectors occurs, and the EP is obtained. Figure 8 shows the phases of transmission and reflection coefficients and the eigenvalues of scattering matrix of the 5-cell metamaterial with UPA at 29.53 kHz. The right reflection coefficient has an abrupt phase change of π at 29.53 kHz, but not for the left reflection coefficient, as shown in Fig. 8(a). This is also a feature of EP. Figure 8(b) shows the eigenvalues in the complex plane coalesce at 29.53 kHz, corresponding to UPA. Thus, clearly, the UPA is related to the EP. To gain more insights into the UPA, we observe the band structure of the non-Hermitian metamaterial based on the transfer matrix method (TMM) and finite element method (FEM), as shown in Fig. 9. The results from the TMM and FEM are in different scenarios. The TMM results are obtained by solving Eq. (11) for wavenumber with given frequency, and the FEM results are obtained by solving Eq. (11) for frequency with given wavenumber using commercial software COMSOL Multiphysics. In FEM analysis, a two-dimensional axial symmetric model is used. The model involves solid mechanics module, electrostatics module, and electrical circuits module, and serendipity quadrilateral mesh with element size less than a tenth of a wavelength is adopted. It can be observed that there is a low-frequency resonant bandgap from the TMM shown in Fig. 9(b), but the FEM shows that there are two resonant bandgaps and a passband between them shown in Fig. 9(c). Considering the transmission coefficient shown in Fig. 6(d), we can conclude that the passband between the two resonant bandgaps is non-Hermitian pseudoband with extremely low transmittance, which is caused by dissipation of resistors, and thus the two resonant bandgaps are coupled together in fact. The frequency of EP is marked in Fig. 9(b) and indicated by red arrow, and the frequency of EP is also marked in Fig. 9(c) as horizontal dashed-dotted line. The interesting thing is that the frequency of EP or UPA is close to the bottom boundary of the non-Hermitian pseudo-band, as shown in Fig. 9(c). It is confirmed that EP is in resonant bandgap from band structure view. Our proposed metamaterial is highly tunable, so that UPA can be achieved at a subwavelength scale. The approach shown in Section 4 can be adopted to obtain a lower-frequency UPA. Here, we demonstrate an alternative optimization method based on the genetic algorithm for exact low-frequency UPA design. To achieve UPA, EP in bandgap, is equivalent to zero transmission and unidirectional zero reflection, i.e., r R = t L = r L = 0 or r L = t R = r R = 0. Thus, the optimization problem can be formed as LC2 , and f b2 LC2 are boundary values, and f 0 is the designed frequency.
The genetic algorithm based on MATLAB optimization toolbox is utilized to solve the optimization problem, in which transmission and reflection coefficients are calculated by the RRM method. The population size is 10, elite count is 2, and crossover fraction is 0.4. The algorithm stops if Case A, the average relative change in the best fitness function value over 50 generations is less than or equal to 1 × 10 −10 , or if Case B, the generation is over 5 000. Repeating 10 times, the optimization results of 5-cell metamaterial are shown in Table 3 for the UPA at f 0 = 10 kHz. All results show high absorption over 99.999% theoretically. The 6th set of parameters in Table 3 indicates the metamaterial with these parameters has the best absorption effect in the 10 cases. At the frequency f 0 = 10 kHz, the length of unit cell of the metamaterial approximates to one twelfth wavelength of background media, indicating that the metamaterial is at a subwavelength scale. The scattering coefficients of the 5-cell metamaterial with the optimized parameters for the best absorption are shown in Fig. 10(a) by the RRM method, and the corresponding FEM results are given in Fig. 10(b). For the FEM analysis in a frequency domian, perfectly matched layers (PMLs) are set outside the metamaterial and finite background aluminum bars. The frequency range with transmittance lower than 10 −2 can reasonably be recognized as bandgap.
Thus, the resonant bandgaps are marked by shadow areas in Fig. 10. Obviously, as shown in Figs. 10(c) and 10(d), the UPA occurs in coupled resonant bandgap, at which three elements of scattering coefficients tend to zero and only the left reflection coefficient is finite. Therefore, the metamaterial can perfectly absorb the right incident wave, and reflects the left incident wave. The UPA from the FEM with frequency 10.084 kHz, indicated by red arrow in Fig. 10(d), is little shifted to high frequency compared with the RRM method exactly at 10 kHz, indicated by red arrow in Fig. 10(c), but the relative error is less than 0.84%.

Reconfigurability
Our proposed metamaterial is multifunctional, and is reconfigurable by easily rearranging the electric circuits. To demonstrate the reconfigurability, the 5-cell metamaterial with the optimized parameters for the best UPA, i.e., parameters in the 6th group in Table 3, is denoted as configuration 1 shown in Fig. 11, in which the numbers "1" and "2" represent the corresponding piezoelectric bars connected with shunt circuits with effective impendence Z 1 and Z 2 , respectively. The configuration 1 is asymmetric. When UPA occurs, the wave field responses of the metamaterial in a frequency domain obtained by FEM analysis are shown in Fig. 11 for the left incident wave and the right incident wave with frequency 10.084 kHz. The line graphs show the normalized displacement along axisymmetric center line of the structure. The results show that there is no transmitted wave. When the incident wave is from the left, it is reflected by the metamaterial, and the reflected wave interferes with the incident wave so that an interference pattern forms. When the incident wave is from the right, the incident wave is perfectly absorbed, so that the reflected wave does not exist, giving rise to no interference pattern, and displacement response is constant on the right side of the metamaterial. Observing the wave pattern for the incident wave from the right, one can find that the wave is mainly confined in the area between the two piezoelectric bars in the rightmost cell. This means that the wave is mainly absorbed by the rightmost cell of the metamaterial. Changing the circuits of the metamaterial to symmetric configuration 2 is shown in Fig. 12, in which the two piezoelectric bars in the center cell connect with the circuit with effective impendence Z 2 , and PA is achieved. The metamaterial device in this case can absorb incident wave with any phase and any amplitude from arbitrary direction without such limitation of the CPA [31] .

Conclusions
In this work, we propose a tunable and reconfigurable non-Hermitian piezoelectric metamaterial bar for UPA and PA, which enables high-efficiency elastic wave absorption at a subwavelength scale and selectable performances for functional applications. The RRM method is used to analyze the transmission and reflection of the metamaterial. Based on the RRM method, the approach from the perspective of scattering coefficients is shown to achieve the UPA by coupling two resonant bandgaps to set EP in bandgap. Using the TMM and FEM to investigate the band structure of the metamaterial, we find that the coupling of the two resonant bandgaps gives rise to a non-Hermitian pseudo-band between them, and the EP appears at the bottom of the non-Hermitian pseudo-band. To design UPA at wanted frequency accurately and efficiently, we suggest an optimization method based on the genetic algorithm to design UPA. In addition, the reconfigurability of the metamaterial is demonstrated; for instance, the UPA and PA can be arbitrarily switched by easily rearranging shunt circuits of the metamaterial. Our approaches are efficient to design a multifunctional device for longitudinal wave absorption in a bar, and they can be easily expanded to other types of elastic waves, such as flexural wave in a beam and Lamb waves in a plate. Our work paves the way of designing non-Hermitian metamaterials in practical applications. For example, our metamaterial can be used to construct an experimental platform for accurate measurement by isolating vibration. In guided wave-based nondestructive testing, our method can be used to construct reflectionless boundaries to suppress disturbances from reflected waves on damage signals to improve accuracy of damage detection.
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