# Broadband Frequency and Spatial On-Demand Tailoring of Topological Wave Propagation Harnessing Piezoelectric Metamaterials

- Structural Dynamics and Controls Laboratory, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, United States

Many engineering applications leverage metamaterials to achieve elastic wave control. To enhance the performance and expand the functionalities of elastic waveguides, the concepts of electronic transport in topological insulators have been applied to elastic metamaterials. Initial studies showed that topologically protected elastic wave transmission in mechanical metamaterials could be realized that is immune to backscattering and undesired localization in the presence of defects or disorder. Recent studies have developed tunable topological elastic metamaterials to maximize performance in the presence of varying external conditions, adapt to changing operating requirements, and enable new functionalities such as a programmable wave path. However, a challenge remains to achieve a tunable topological metamaterial that is comprehensively adaptable in both the frequency and spatial domains and is effective over a broad frequency bandwidth that includes a subwavelength regime. To advance the state of the art, this research presents a piezoelectric metamaterial with the capability to concurrently tailor the frequency, path, and mode shape of topological waves using resonant circuitry. In the research presented in this manuscript, the plane wave expansion method is used to detect a frequency tunable subwavelength Dirac point in the band structure of the periodic unit cell and discover an operating region over which topological wave propagation can exist. Dispersion analyses for a finite strip illuminate how circuit parameters can be utilized to adjust mode shapes corresponding to topological edge states. A further evaluation provides insight into how increased electromechanical coupling and lattice reconfiguration can be exploited to enhance the frequency range for topological wave propagation, increase achievable mode localization, and attain additional edge states. Topological guided wave propagation that is subwavelength in nature and adaptive in path, localization, and frequency is illustrated in numerical simulations of thin plate structures. Outcomes from the presented work indicate that the easily integrable and comprehensively tunable proposed metamaterial could be employed in applications requiring a multitude of functions over a broad frequency bandwidth.

## Introduction

In recent years, it has been recognized that wave control utilizing elastic metamaterials can enhance performance and expand functionalities in many applications, such as energy harvesting, noise isolation, sensing, communications, filtering, and cloaking (Hussein et al., 2014; Fang et al., 2018; Wang et al., 2020). One objective of elastic metamaterials that has received significant attention is the confinement of elastic waves to a specified path or location through the formation of a waveguide. Conventional elastic waveguides obtain wave confinement by creating an inclusion where the wave can localize within a periodic lattice (Kafesaki et al., 2000; Khelif et al., 2004; Benchabane et al., 2005; Oudich et al., 2010; Casadei et al., 2012; Liu et al., 2020). These conventional waveguides can suffer from performance degradation when disorder (e.g., a sharp turn in the waveguide) or defects (e.g., a manufacturing imperfection) exist within the periodic lattice (Sun and Wu, 2006; Pal and Ruzzene, 2017). To avoid this performance degradation and expand the functionalities of elastic waveguides, the concepts that underly topologically protected conducting states in electronic materials (Hasan and Kane, 2010) have been employed in elastic metamaterials (Huber, 2016; Ma et al., 2019). Topological metamaterials are immune to various classes of disorder and defects that are oftentimes found in practical engineering applications, thus enabling lossless transmission along any desired wave path. To achieve topologically protected wave propagation that is localized to waveguides in two-dimensional systems, the elastic analogs of the quantum Hall effect (QHE), quantum spin Hall effect (QSHE), and quantum valley Hall effect (QVHE) have been extensively investigated.

Protected wave propagation that is confined to a waveguide is obtained through the activation of localized topological edge states that arise from the QHE, QSHE, and QVHE. Due to the active/moving components that are generally required to break time-reversal symmetry (TRS) for the QHE (von Klitzing, 1986; Nash et al., 2015; Wang P. et al., 2015; Wang Y. T. et al., 2015) and the complex geometries necessary to achieve a double Dirac cone (a degeneracy where four cones meet) in the dispersion relation for the QSHE (Kane and Mele, 2005; Mousavi et al., 2015; Süsstrunk and Huber, 2015; Wu and Hu, 2015; Chaunsali et al., 2018; Miniaci et al., 2018), the relatively simpler QVHE has garnered significant attention. The QVHE requires the formation of a single Dirac cone (a degeneracy where two cones meet) in the dispersion relation of the unit cell. This single Dirac cone is opened with a lattice perturbation that breaks space inversion symmetry (SIS), which produces a bandgap that can support topological edge states due to valley-dependent topological properties (Nakada et al., 1996; Rycerz et al., 2007; Xiao et al., 2007; Peres, 2010). Topologically protected wave transmission according to the elastic analog of the QVHE has been obtained in reticular structures (Vila et al., 2017; Liu and Semperlotti, 2019; Miniaci et al., 2019) and continuous thin plates with periodically placed masses (Chen et al., 2017; Yu et al., 2018; Zhu et al., 2018) or inclusions (Du et al., 2020). These structures emulate the elastic QVHE by exploiting a periodic impedance mismatch (Bragg scattering mechanism), and thus the resulting topological edge states exist at frequencies corresponding to wavelengths that are dependent on the lattice constant. The addition of local mass resonators (Torrent et al., 2013; Pal and Ruzzene, 2017; Lera et al., 2019; Wang et al., 2019; Zhang et al., 2020) and acoustic black holes (Ganti et al., 2020) to continuous plates has facilitated the achievement of topological edge states at frequencies that are determined by the characteristics of the local element (mass resonator or acoustic black hole). Topological wave propagation at low frequencies corresponding to wavelengths that are larger than the lattice constant has been demonstrated by carefully selecting the properties of these local elements (Pal and Ruzzene, 2017; Chaunsali et al., 2018; Wang et al., 2019; Zhang et al., 2020). Subwavelength topological metamaterials such as these could be highly valuable for engineering applications that require wave control at low frequencies in a size-constrained environment.

To extend beyond the functionalities available in fixed structures and enable adaptivity to varying operating requirements and external conditions, many recent investigations have focused on introducing tunability to topological metamaterials. The spatial path of topological wave propagation has been demonstrated to be adjustable by applying an external magnetic field (Zhang et al., 2019), modifying mechanical boundary conditions (Darabi and Leamy, 2019; Tang et al., 2019), adding an elastic foundation (Al Ba’ba’a et al., 2020), connecting negative capacitance piezoelectric circuitry (Riva et al., 2018; Darabi et al., 2020), or switching stable states in bistable elements (Wu et al., 2018). The shape and localization of topological edge states have been tuned by exploiting an applied strain field (Liu and Semperlotti, 2018). Initial studies involving real-time frequency tuning of topological edge states have shown that an applied temperature (Liu et al., 2019), strain field (Nguyen et al., 2019), or electrical field (Zhou et al., 2020) can increase the frequency range over a limited region that is related to the Bragg mechanism or the magnitude of lattice perturbation.

The tunable topological metamaterials that have been investigated thus far have each concentrated on tailoring an individual characteristic of the topological wave to unlock novel functions and enhance performance in devices exhibiting elastic wave control. However, to fully realize the potential of topological metamaterials in wave control platforms and enable robust performance over a broad set of functionalities, an elastic metamaterial with multiple tunable topological characteristics must be developed. To date, concurrent tunability of the frequency and spatial characteristics of topological edge states has yet to be fully explored in a singular elastic metamaterial. Such a metamaterial would be of great benefit for devices such as wave filters, multiplexers, and energy harvesters that must route energy over a large frequency bandwidth. Besides primarily focusing on one adaptive characteristic, the currently established tunable topological metamaterials generally rely on the Bragg mechanism to generate a Dirac cone. Due to the interaction of platform-specific tuning parameters with the underlying physics of the Bragg mechanism, these metamaterials are either incapable of on-line frequency tuning (e.g., they are path tunable only) or can only practically do so over a limited range. Moreover, structures fabricated from these metamaterials would need to be large to control energy at low frequencies that correspond with fundamental system modes, which can be critical in structural applications. As a result, an easily integrable topological metamaterial that is capable of subwavelength elastic wave control and programmable over a broad frequency bandwidth has yet to be developed.

To advance the state of the art, the presented research proposes a piezoelectric topological metamaterial harnessing integrated resonant circuitry for comprehensively tunable subwavelength wave control. The goal of this investigation is to uncover insights into critical adaptive parameters and synthesize a framework for the attainment of programmable topological wave propagation using resonant electromechanical metamaterials. In contrast to previous works, the proposed methodology and approach has achieved clear advancements: (a) the rich tailorable characteristics of the piezoelectric metamaterial enable the concurrent adaptation of the frequency range, path, and mode shape of topological edge states and (b) this on-demand tunability of topological properties is achieved for the first time in a subwavelength (i.e., compact) and load-bearing thin plate structure.

To accomplish the research objective, this manuscript presents the evaluation of the dispersion relation for the unit cell from the plane wave expansion (PWE) method. The PWE method is selected because of its concise nature and computational efficiency when compared to the often-used finite element (FE) method, since there can be a low (e.g., 50–100) number of degrees of freedom required to achieve an accurate result from PWE calculations for thin plate structures (Xiao et al., 2012; Pal and Ruzzene, 2017). This method is utilized to establish the working principle for the achievement of topological edge states from the QVHE. Beyond establishing a working principle for the attainment of topological edge states for fixed system parameters, the advantages of the PWE method facilitate a comprehensive parameter study through rapid calculations of the metamaterial dispersion relation. A parametric analysis is performed to identify and explore an achievable operating region for tunable topological wave propagation. Numerical computations of the dispersion relation for a finite strip of connected unit cells uncover how circuit parameters can enable the adjustment of topological edge states. Further analysis is conducted to investigate how topological edge state adaptivity can be augmented by the connection of negative capacitance to enhance electromechanical coupling and lattice reconfiguration via shorting circuits to obtain an additional Dirac cone. Finally, the activation of topological edge states for the achievement of guided topological wave propagation at lattice interfaces and boundaries is revealed by numerical simulations of a thin plate structure.

## Concept and Theoretical Model

### System Description

As shown in Figure 1, the proposed piezoelectric metamaterial is comprised of a bimorph thin plate with substrate (gray) and piezoelectric (yellow) layers. The substrate layer has thickness

**FIGURE 1**. **(A)** Isometric view of piezoelectric metamaterial, with unit cell enclosed in dashed lines. **(B)** Cross-section and **(C)** top view of metamaterial unit cell. Blue indicates electrode geometry connected to circuit 1, red indicates electrode geometry connected to circuit 2.

### Governing Equations

In the derivation of the theoretical model for the metamaterial, small deflections and a thin structure (in the

where

where

where

To generalize the results, a nondimensionalization scheme is adopted and the resulting equations are:

where the non-dimensional flexural displacement, voltage, time, and length scales are defined as:

### Dispersion Relation

To attain the dispersion diagram (i.e., band structure) of the metamaterial, the dispersion relation of the unit cell is analyzed using the PWE method. For this study, the number of electrode pairs (capacitors) in the unit cell is set to

where

**FIGURE 2**.

Eq. 10 is obtained. Similarly, by substituting Eq. 8 into Eq. 7, Eq. 11 is obtained. The following equations define the dispersion relation of the metamaterial unit cell:

where

To evaluate the dispersion relation for the system, Eqs 10, 11 are arranged in the form of the classical eigenvalue problem:

where the eigenvalues

### Negative Capacitance Circuitry

Per Eq. 12,

This term, which is present in Eq. 10, measures the level of effective electromechanical coupling when accounting for the connected negative capacitor. Thus, by careful selection of the negative capacitance ratio (e.g.,

Since

There is a physical explanation for the derived stability criterion. The parallel-connected negative capacitor reduces the effect of the inherent capacitance of an electrode pair, causing a reduction in the total capacitance present in the corresponding circuit (

In addition to changing the effective electromechanical coupling, the inclusion of negative capacitance influences the effective nondimensional tuning frequency of the resonant circuits. Through observation of Eq. 11, this influence can be measured as:

where

## Working Principle– Obtainment of Tunable Topological Wave Propagation

In this section, the working principle is outlined for the attainment of subwavelength topological wave propagation using the proposed metamaterial. A unit cell dispersion analysis is conducted to identify lattices with nontrivial topological characteristics and define metrics that are indicators of waveguide performance. A finite strip analysis is undertaken to investigate how localized topological edge states can be obtained through the connection of topologically distinct lattices. Numerical simulations are performed to illustrate how these edge states can be exploited to achieve guided topological wave propagation that is tunable in both the frequency and spatial domains.

### Unit Cell Dispersion Analysis

The band structure of the proposed metamaterial is obtained through a unit cell dispersion analysis. For this study, PZT-5H (a commonly utilized piezoceramic) is selected as the material for the piezoelectric layers and aluminum is selected as the material for the substrate layer. The material properties associated with PZT-5H and aluminum that are used in dispersion calculations are listed in Table 1. The geometric dimensions specified for the analysis are also included in Table 1. The layer thicknesses

Due to the presence of

To obtain topological edge states per the QVHE, different inductance values are selected for each of the two circuits in the unit cell (

**FIGURE 3**. **(A)** Dispersion curves (specifically, the first and second bands) for **(B)** Schematic of unit cells for Type A (**(C)** Mode shapes evaluated at the

Where

where

The topological interface state must be excited in a way that does not activate the bulk modes of a given structure to obtain localized topological wave propagation. To achieve this goal, the interface state is excited at a frequency within the common topological bandgap that is found in the dispersion diagrams of the Type A and Type B lattices (

Based on this discussion, the performance criteria defined for this investigation that are obtainable from the unit cell dispersion analysis are 1)

### Topological Interface States

A dispersion analysis is conducted for a finite strip of unit cells containing an interface between Type A and Type B lattices to demonstrate the emergence of topologically protected interface states. For this finite strip analysis, all parameters are defined identically to the unit cell analysis discussed in the previous section (

where

**FIGURE 4**. **(A)** Band structure for a finite strip (|**(B)** Band structure and schematic for a finite strip with a Type II interface. At the bottom, an antisymmetric mode shape that is calculated from the interface state at

An additional localized state exists near

Two observations are gained from the finite strip analysis. First, the proposed metamaterial enables the obtainment of symmetric and antisymmetric topological interface states, which aligns with previous investigations into the QVHE (Zhu et al., 2018). Due to the tunability of the circuit parameters in the proposed metamaterial, switching between these interface state types (symmetric and antisymmetric) could easily be achieved in practice. Second, the finite strip dispersion analysis supports the performance requirement for a nontrivial topological bandgap (

### Path Tunable Topological Wave Propagation

The ability to achieve topological wave propagation from the interface states is investigated with FE simulations of a plate constructed from the proposed metamaterial. The proposed metamaterial enables concurrent tunability of the topological wave path, mode shape, and frequency, which advances upon previously developed platforms that are generally narrowband in nature and only focus on one tailorable characteristic. Each of the proposed metamaterial’s tailorable properties are investigated separately in this manuscript to simplify the analysis. The path tunability of the topologically protected waveguide is examined in this section, while comprehensive analyses on frequency and mode shape tailoring are contained in section *Parametric Study– Frequency and Mode Shape Tunability*.

The plate used for FE simulations consists of a 16 by 20 lattice of unit cells, with low reflection conditions applied along all outer boundaries to suppress reflections (see left column of Figure 5 for plate schematics). The lattice contains an interface between Type A and Type B unit cells, which is enclosed by the black lines in Figures 5A–D. A resistance (

**FIGURE 5**. Schematics (left column) and steady-state response (right column) for guided wave propagation along **(A)** straight, **(B)** sharp corner, **(C)** shallow corner, and **(D)** “Z-shape with defect” interfaces. In the schematics, R1, R2, and R3 represent output signal receivers and black lines enclose the interface between Type A and Type B lattices. Circuit parameters are defined as:

## Parametric Study– Frequency and Mode Shape Tunability

In this section, a parametric study is undertaken to extensively explore the adaptive characteristics of the proposed metamaterial and develop a detailed understanding of system parameters that govern performance. A framework is developed to assess the frequency and interface mode shape tunability of the metamaterial and uncover insights into the impact of electromechanical coupling on topological elastic wave control.

### Frequency Tunability of the Dirac Point

Since topological wave propagation occurs at a frequency near the Dirac frequency (

**FIGURE 6**. **(A)** Dirac frequency **(B)** Evolution of the first and second bands (all bands are shown as black lines) with increasing **(C)** Evolution of third band for same set of parameters. **(D)** Band structure comparison for proposed metamaterial with high tuning frequency (

According to Figure 6A, the achievable frequency range for

### Achievable Operating Region for Topological Interface States

A lattice perturbation must be applied to open a topological bandgap from the Dirac point and achieve topological edge states per the QVHE. While the previously presented analysis (see section *Working Principle– Obtainment of Tunable Topological Wave Propagation*) demonstrates this working principle under a fixed set of parameters, further exploration is required to fully evaluate the tunability of the proposed metamaterial. Thus, a parametric study involving the inductance perturbation parameter

**FIGURE 7**. **(A)** Valley Chern magnitude |**(B)** Relative bandgap **(C)** Schematic for finite strip with Type I interface that is used to generate interface modes. Interface mode shapes for finite strip with unit cell parameters indicated by orange circle, square, triangle, star, and pentagon markings in **(B)**.

Aside from enabling an adjustable frequency range for topological interface states, the proposed metamaterial also facilitates the tailoring of interface mode shape and localization. The ability to adjust interface mode localization is investigated by varying the inductance perturbation parameter

### Influence of Electromechanical Coupling on Topological Wave Tunability

The effect of electromechanical coupling on the vibration attenuation performance of piezoelectric metamaterials is well documented (Hagood and von Flotow, 1991; Tang and Wang, 2001; Sugino et al., 2017). In this investigation, the influence of the effective electromechanical coupling factor

**FIGURE 8**. **(A)** Relative bandgap **(C)** Schematic for finite strip with Type I interface that is used to generate interface modes. Interface mode shapes for finite strip with unit cell parameters indicated by orange circle, star, and pentagon markings in **(A)**. **(B**, **D)** report the same results as **(A, C)**, respectively, for PZT-5H with negative capacitance circuitry (

The expansion of the achievable operating region signifies an increased level of tunability. In terms of frequency range, the achievable operating region is extended to lower frequencies (while maintaining an upper bound of approximately

These findings indicate that electromechanical coupling plays a crucial role in determining the extent of the tunability of the topological interface state. Thus, care must be taken to maximize the effective electromechanical coupling through material selection and geometric design. In addition, negative capacitance circuitry could be utilized to artificially enhance the coupling and achieve topological interface states with a broader frequency range and increased displacement localization.

### Finite Element Evaluation of Frequency and Mode Shape Tunable Topological Waves

FE simulations of plates constructed from the proposed metamaterial are conducted to verify that tunable topological wave propagation can be realized by activation of the topological interface states contained within the achievable operating region. Plates with straight line and Z-shaped Type I interfaces are harmonically excited at the locations indicated by the arrows in Figure 9A. Geometric, material, and negative capacitance (*Working Principle–Obtainment of Tunable Topological Wave Propagation*, where wave propagation for |

**FIGURE 9**. **(A)** Schematics for thin plate metastructures with straight and Z-shaped Type I lattice interfaces (enclosed in black lines). A harmonic out-of-plane point excitation is applied where indicated by the arrow. Negative capacitance circuity is connected such that **(B)** Steady-state displacement fields for **(C)** Steady-state displacement fields for

Notably, for the low-frequency case (

## Lattice Reconfiguration

The architecture of the proposed metamaterial can also be exploited to enhance adaptivity through lattice reconfiguration, which has been used in previous investigations to achieve tailorable bandgaps and waveguides in topologically trivial structures (Thota et al., 2017; Thota and Wang, 2018). In this section, lattice reconfiguration is explored as a mechanism to enhance the frequency range of topological waves and obtain additional topological edge states.

### Formation of Additional Dirac Point Through Lattice Reconfiguration

Figure 10A contains schematics for three different lattice configurations that are attainable with the proposed metamaterial. Figure 10B contains the corresponding dispersion diagrams for each lattice configuration with circuit conditions specified such that

**FIGURE 10**. **(A)** Schematics for three lattice configurations attainable through circuit tailoring in the proposed metamaterial. Circuits are connected to electrodes that are represented by blue and red circles. For Configuration 3, electrodes pertaining to shorted circuits are omitted for clarity. Unit cell (enclosed in black dashed lines) circuit parameter details and lattice symmetries are listed below each schematic. **(B)** Dispersion diagrams corresponding to lattice Configurations 1-3 calculated using the PWE method with

### High-Frequency Interface State From Dirac 2

The same process described in previous sections for Dirac 1 is followed to construct interface states from Dirac 2. Figure 10A includes the schematic for a lattice (defined as Configuration 2: Perturbed Lattice) with the inductance perturbation parameter

**FIGURE 11**. **(A)** Band structure for a finite strip (|**(B)** A symmetric mode shape that is calculated from the interface state at **(C)** (top row) Schematics for thin plate metastructures with straight and Z-shaped Type I lattice interfaces (enclosed in black lines). A harmonic (

To demonstrate guided wave propagation, plates are constructed with straight and Z-shaped Type I interfaces and excited harmonically (

### Boundary States From Dirac 2

Results from the finite strip dispersion analysis for a Type I interface (Figure 11) suggest that boundary states are readily achievable from Dirac 2. To further investigate the formation of boundary states from Dirac 2, a dispersion analysis is conducted for a finite strip with 18 Type B unit cells (no interface is present, see schematic in Figure 12A). The left boundary is specified as fixed, the right boundary is specified as free, and periodic boundary conditions are applied in the

where

**FIGURE 12**. **(A)** Band structure for a finite strip (|**(B)****(top)** Schematic for thin plate metastructure with fixed and free boundaries selected to attain boundary states. A harmonic (**(bottom)** Steady-state displacement field illustrating flexural displacement confinement along plate boundaries.

A plate is constructed with boundary conditions (free and fixed) specified such that the outlined boundary states are supported at all four boundaries. The plate is excited harmonically (

## Conclusion

This research proposes and develops a topological metamaterial that harnesses resonant piezoelectric circuitry to enable comprehensively tunable elastic wave control. Overall, this investigation advances the state of the art by enabling and exploring the adaptation of topological wave path, frequency, and edge mode shape in a single mechanical platform for the first time. The proposed metamaterial operates over a broad frequency bandwidth and can be integrated in a compact fashion for applications that require control of large-wavelength (i.e., low-frequency) waves due to its subwavelength characteristic, which constitutes a breakthrough in the field of tunable topological elastic waves.

In this manuscript, the tunability of wave path, frequency range, and edge states is explored through a systematic analysis of the dispersion properties and dynamic response characteristics of the proposed metamaterial. A subwavelength and frequency tunable Dirac point (Dirac 1) is uncovered, and it is revealed that symmetric and antisymmetric topological interface states can be obtained from it. FE simulations illuminate how these interface states can be activated to achieve guided elastic wave transmission that is robust to disorder and defects and can be manipulated on-demand into a myriad of desired directions. A deeper understanding of the adaptive characteristics of the metamaterial is formed through parametric studies. A finite tunable frequency range and its underlying physical basis are discovered for Dirac 1. An achievable operating region is defined for topological interface states derived from Dirac 1, where it is learned that interface states that meet specified performance metrics are achievable over a wide frequency bandwidth that comprises a subset of the Dirac 1 frequencies. These findings offer new insights into frequency tunability that may be leveraged in future studies concerning Dirac dispersions and the QVHE in locally resonant elastic metamaterials. Further exploration of the achievable operating region illuminates how circuit parameters can be utilized to tune the displacement field of a waveguide by tailoring the localization and shape of the interface state. The operating region is used as a framework to study the role of electromechanical coupling in topological wave propagation for the first time. Results indicate that increased electromechanical coupling enhances the frequency range and achievable interface mode localization of the interface states. Lattice reconfiguration is also investigated as a method to tailor the topological properties of the proposed metamaterial. Analysis of a triangular lattice obtained through shorting circuits reveals that a second Dirac point (Dirac 2) can be achieved in a high-frequency range that extends beyond the operating region for Dirac 1 interface states. Boundary states and additional interface states are shown to be obtainable from Dirac 2, and numerical simulations illustrate how these states can be exploited to achieve exceptional guided wave phenomena.

The outcomes from this investigation provide fundamental insights into the influence of locally resonant elements, electromechanical coupling, and lattice reconfiguration in adaptive topological metamaterials exhibiting the QVHE, presenting a basis for further exploration. The proposed topological metamaterial may be employed to achieve subwavelength elastic wave control that is robust to practical considerations (eg., sharp corners or lattice imperfections) and adaptive in real-time to shifting operating requirements and external conditions. These beneficial features could be harnessed to improve performance and expand functionalities in a range of applications requiring adaptive and robust elastic wave control, such as vibration isolators, wave filters/multiplexers, and energy harvesters.

## Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

## Author Contributions

PD and KW formulated the presented idea. PD developed the mathematical framework, performed analytical and numerical computations, and drafted the manuscript. KW discussed results with PD, provided research direction/advisement, and edited the manuscript.

## Funding

This material is based upon work supported by the National Science Foundation under Award No. 1661568. PD also acknowledges financial support from the Rackham Merit Fellowship at the University of Michigan.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

The authors thank Xiang Liu and Megan Hathcock at the University of Michigan Structural Dynamics and Controls Lab for their useful suggestions and participation in discussions regarding topological metamaterials and FE modeling using COMSOL Multiphysics.

## Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmats.2020.602996/full#supplementary-material.

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Keywords: metamaterial, piezoelectric, topological, tunable, elastic waveguide, quantum valley Hall effect, subwavelength, lattice reconfiguration

Citation: Dorin P and Wang KW (2021) Broadband Frequency and Spatial On-Demand Tailoring of Topological Wave Propagation Harnessing Piezoelectric Metamaterials. *Front. Mater.* 7:602996. doi: 10.3389/fmats.2020.602996

Received: 04 September 2020; Accepted: 28 October 2020;

Published: 12 January 2021.

Edited by:

Maria Laura De Bellis, University of Studies G.d’Annunzio Chieti and Pescara, ItalyReviewed by:

Emanuele Reccia, University of Cagliari, ItalyMarco Pingaro, Sapienza University of Rome, Italy

Copyright © 2021 Dorin and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Patrick Dorin, pdorin@umich.edu