We investigate the effect of hierarchical designs on the bandgap structure of periodic lattice systems with inner resonators. A detailed parameter study reveals various interesting features of structures with two levels of hierarchy as compared with one level systems with identical static mass. In particular: (i) their overall bandwidth is approximately equal, yet bounded above by the bandwidth of the single-resonator system; (ii) the number of bandgaps increases with the level of hierarchy; and (iii) the spectrum of bandgap frequencies is also enlarged. Taking advantage of these features, we propose graded hierarchical structures with ultra-broadband properties. These designs are validated over analogous continuum models via finite element simulations, demonstrating their capability to overcome the bandwidth narrowness that is typical of resonant metamaterials.

## I. INTRODUCTION

Acoustic/elastic metamaterials are receiving increasing attention due to their extraordinary dynamic properties^{1–3} and novel applications, such as invisibility cloaks,^{4–6} active and non-reciprocal wave control,^{7,8} superlens and super-resolution imaging,^{9–11} and seismic protection.^{12} One of their most impressive features is their ability to prohibit the propagation of elastic waves at certain frequencies: the so-called bandgaps or stopbands. In conventional phononic crystals, these bandgaps are generated via Bragg scattering when the propagating wavelength is comparable to the characteristic size of the microstructure. In contrast, locally resonant metamaterials rely on the vibration of internal resonators, and generate bandgaps in the vicinity of the resonant frequency.^{13} As a result, resonant metamaterials can be constructed to produce bandgaps at very low frequencies, which is particularly promising for sound and vibration isolation applications. However, they typically suffer from very narrow bandwidths, which severely limits their applicability. Many efforts have then been devoted to tailor the bandgaps of composite structures, and, in particular, to achieve lower and wider bandgaps. Some of the strategies employed include topology optimization,^{14–16} damping engineering,^{17,18} and electromechanical coupling.^{19,20}

Recently, the design of general composite structures has been influenced by the remarkable mechanical properties of natural materials (e.g., wood, nacre, or bone). These often have a multi-level hierarchical design that delivers effective properties, such as combined static strength and toughness, that far exceed the performance of the individual phases.^{21–24} The impact of hierarchical architectures on wave propagation is much less studied. Yet, emerging results in this area reveal their capability to broaden the overall bandwidth of the structure. For instance, Zhang and To^{25} demonstrated an increase of one or two orders of magnitude for the total bandwidth induced by Bragg scattering in hierarchical layered composites.

The calculation of the bandgap structure of periodic materials, whether hierarchical or not, often necessitates a numerical treatment. Only simple models such as mass-spring systems are amenable to analytic solutions. These discrete systems have thus long been used to reveal the exotic effective properties of acoustic metamaterials,^{26,27} and are here considered as a blueprint for understanding the effect of hierarchy on the wave filtering properties of resonant systems. In particular, we use lattice models with mass-in-mass resonators to generate unit cells with multiple levels or hierarchy (see Fig. 1). Models of this sort have been previously considered and their effective dynamic mass has been rigorously analyzed.^{28–30} Yet, our interest in the bandgap structures requires consideration of both the effective mass and stiffness.^{31} Furthermore, in light of bio-inspired materials, a rigorous analysis of hierarchical structures with different number of hierarchies but with consistent control variables, e.g., identical static mass, is in need.

In this study, we perform a systematic analysis of periodic lattice systems with resonant unit cells 1 and 2 (single- and dual-resonators, respectively), as shown in Fig. 1, as a function of the parameters involved in the hierarchical design. We will show in Sec. II that, when the total mass of both unit cells coincide, the overall bandwidth of Unit cell 2 is bounded above by that of Unit cell 1. Yet, more bandgaps are created and the frequency spectrum of the stopbands can be greatly enlarged.^{32} These results motivate the design of graded hierarchical structures as promising architectures to create ultra-broadband systems, in contrast with the usually narrow bandgaps associated with resonant structures. The theoretical predictions based on the band diagram of the individual unit cells are then verified in Sec. III by finite element simulations of continuum models that mimic the lattice designs. The results indicate an increase in the overall bandwidth, thanks to the hierarchical design, that can go well beyond 100%.

## II. THEORETICAL MODEL

### A. Lattice model

We begin by describing the resonant periodic lattice models that will be used to understand the effect of hierarchy on the bandgap structure. We construct these by starting from a simple one-dimensional mass-spring chain, which will be the 0th level of hierarchy. Next, each of the following levels is constructed by replacing the most inner mass with a resonant unit (mass-in-mass system). The resulting unit cells for hierarchy levels zero, one and two, are depicted in Fig. 1, where the labels used for the material parameters are also indicated.

The simplicity of these structures enables an analytical characterization of the dispersion relations. For Unit cell 0, a straightforward application of Floquet-Bloch theorem to the equations of motion leads to the following well-known result:

where *ω* is the angular frequency, *q* is the wave number, and *L* is the length of the unit cell.

A similar strategy to the above described may be used to deliver the dispersion relations associated with Unit cells 1 and 2. Alternatively, and in complete equivalence, these may be obtained via a recurrence strategy. More precisely, the dispersion relation of Unit cell *n* can be obtained by replacing the value of the most inner static mass in the expression for level *n* − 1, with the dynamic mass of the resonant unit it replaces (recall the construction of the hierarchical structures). For Unit cells 1 and 2, we thus have, respectively,

where $\omega 0=k2/m2$ and $\omega 1=k3/m4$. As may have been observed, this recursive strategy is highly effective, and it does not necessitate the construction of the system of equations describing the motion of all degrees of freedom of the multiple-resonator system, and subsequent condensation.

The dispersion relations given by Eqs. (2) and (3) are depicted in Fig. 2 within the first Brillouin zone for some specific values of the parameters (these may be found in the caption). Note that no units are assigned to these parameters for the mass-spring models unless real materials are taken into consideration (Sec. III B), and all the units are consistent in the following analyses. Qualitatively, it can be observed that the number of bandgaps increases by one for each new level of hierarchy; and that the lower edge of the first bandgap may be reduced. This therefore raises important questions: (i) can the total width of the bandgaps be increased with the help of hierarchical designs? (ii) May multiple unit cells be combined to deliver a single broad bandgap? To answer the first question, we perform next a detailed parameter study of the resonant structures characterized by Unit cells 1 and 2. The second question will be the scope of Sec. III.

### B. Effect of relevant parameters

The periodic hierarchical structure given by Unit cell 2 is constructed by splitting the mass *m*_{2} of Unit cell 1 into two parts (*m*_{3} and *m*_{4}), which are connected by the spring *k*_{3}; cf. Fig. 1. With the goal of obtaining a better understanding of the effect of hierarchy in the design, we keep the total mass constant in this splitting process. In particular, we consider a non-dimensional parameter *θ* ∈ [0, 1] and define the individual masses as *m*_{3} = *θm*_{2} and *m*_{4} = (1 − *θ*)*m*_{2}. We further define the ratio *η* = *k*_{3}/*k*_{2} to quantify the strength of the new spring *k*_{3}.

The dispersion relation for Unit cell 2, given by Eq. (3), may be rewritten as a function of *θ* and *η*, giving

with $\omega 0=k2/m2$ and $\omega 1=\eta k2m2(1\u2212\theta )$, while we note that Unit cell 1 is recovered for *θ* = 1 or *η* → *∞*. The corresponding band diagram is depicted in Fig. 3, where the values of the lower and upper edges of the three bandgaps are shown as a function of *θ* and *η* in subfigures (a) and (b), respectively. For the former, a fixed value of *η* = 1 is chosen, whereas, for the latter, *θ* is considered to have the intermediate value of 0.5. The remainder of the parameters are specified in the caption. In these figures, each bandgap is color-coded, with blue, red and black denoting the first, second and third stopband, respectively. We note that for lattice systems, the upper most bandgap extends to infinity theoretically, and, as such, only the lower bound of the third stopband is represented in the figure.

The limiting cases *θ* ∈ {0, 1} and *η* ∈ {0, +*∞*}, all correspond to single-resonator systems, and the three bandgaps degenerate into two stopbands. As previously remarked, both *θ* = 1 and *η* → *∞* correspond to Unit cell 1. For *θ* = 0, the intermediate mass disappears and the springs *k*_{2} and *k*_{3} are connected in series, whereas for *η* = 0, the most inner mass is disconnected from the remainder of the unit cell, resulting in a single-resonator with a lighter internal mass. All these limiting structures are shown in Fig. 3 for completeness.

For intermediate values of *θ* and *η,* it can be observed from Fig. 3 that the lower edge of the last stopband is approximately constant, with a slight increase at very low values of *θ*. Similarly, the lower edge of the first bandgap does not exhibit much variability, although a monotonic decrease is observed with decreasing values for *θ* or *η*, the latter being more pronounced near *η* = 0. Such behaviors contrast with the remainder of the bandgap edges, which depict a significant sensitivity with respect to the control variables. Additionally, and more importantly, the upper edge of the second bandgap can be much larger than the upper bound of the first bandgap for Unit cell 1 (*θ* = 1 or *η* → *∞*).

We are now ready to examine the effect of the parameters *θ* and *η* on the overall bandwidth, here denoted as *ω*_{tot}. We define such quantity as the sum of the three bandgaps up to a cut-off frequency that is higher than the lower edge of the upper most bandgap (we recall that the last stopband extends to infinity). The results are shown in Fig. 4, where the dependency of *ω*_{tot} on *m*_{2} and *k*_{2} is also presented for the case of the single-resonator system in subfigure (c).

Examining first the dependence on *θ*, it can be observed from Fig. 4(a) that the overall bandwidth increases smoothly from *θ* = 0 to *θ* = 1. Note that *θ* = 0 and *θ* = 1 correspond to single-resonator systems with the same mass for the inner resonator but different stiffness for the inner spring (*k*_{2}*k*_{3}/(*k*_{2} + *k*_{3}) < *k*_{2} for the former and *k*_{2} for the latter). For such single mass-in-mass systems, the behavior is shown in Fig. 4(c), where it is observed that a higher inner stiffness increases the bandwidth. This thus explains why *ω*_{tot} is lower at *θ* = 0 than *θ* = 1, and that at *θ* = 0, *ω*_{tot} increases with an increasing value of *η*.

A similar analysis can be performed to Fig. 4(b), where the dependence of the overall bandwidth, *ω*_{tot}, on *η* is depicted. We also observe here a smooth increase of *ω*_{tot} between the two limiting cases, *η* = 0 and *η* → *∞*. These two extremes also correspond to single-resonator systems, in this case, with identical inner stiffness and different mass of the resonator (*θm*_{2} for *η* = 0 and *m*_{2} for *η* → *∞*). According to Fig. 4(c), a higher inner mass leads to a larger *ω*_{tot} and, consequently, the overall bandwidth is larger for increasing values of *η*.

Summarizing, Fig. 4 indicates that, for specific values of *θ* and *η*, the total bandwidth of a resonant system with hierarchy level 2 is bounded above by the bandwidth of the structure with level 1 hierarchy. A more detailed analysis of the derivatives in the (*θ*, *η*)-space, see Fig. 5, shows that this behavior actually extends over the full parameter space. This implies that a higher bandwidth is not obtained by increasing the level of hierarchy. However, the sensitivity on the parameters is rather small, and thus, the multiple bandgaps generated by the hierarchical structure can be seen as a split of the stopbands corresponding to the single-resonator unit.

Even though the introduction of a hierarchical design cannot be used *per se* to access a wider bandwidth, it is important to emphasize that multiple bandgaps arise from the multiple-resonator system. Furthermore, as illustrated in Fig. 3, the lower edge of the first bandgap for Unit cell 2 can be lower than that of Unit cell 1, and the upper edge of the second bandgap for Unit cell 2 may also be higher than the upper edge of the first bandgap for Unit cell 1. In other words, the range of the bandgap frequencies is enlarged. This motivates the following section (Sec. III), where different structural units are combined to deliver a large broadband system.

## III. GRADED DESIGN

The combination of different unit cells to form graded designs is a promising avenue for tuning the bandgap structure of acoustic/elastic metamaterials, and their fabrication is becoming much more accessible, thanks to recent advances in additive manufacturing. Some interesting applications of graded designs include, without being comprehensive, rainbow trapping devices that spatially separate and filter the frequency components of broadband acoustic waves;^{33} low-frequency bending waveguides;^{34} graded grating phononic crystal slabs for asymmetric Lamb wave transmission;^{35–37} or omnidirectional broadband filters capable of bending incoming waves toward an absorbing inner core. Additional designs that are closer to the ones of interest in this study include graded resonant phononic crystals with multicoaxial cylindrical inclusions,^{38} which has potential applications in seismic protection.^{12} Finally, we note that graded designs have also been investigated in other contexts, such as shock attenuation^{39} or thermal transport.^{40}

We explore here hierarchical graded systems in view of their potential to generate structures with an enhanced bandwidth. We first use lattice systems to describe the design process under the assumption that each unit cell is infinitely periodic (Sec. III A). Finite continuous models are then generated to validate the approach in a more realistic setting (Sec. III B).

### A. Lattice model

We first propose two designs with graded values of *θ* or *η* while maintaining the other variable fixed. These are constructed by combining various unit cells with overlapping bandgaps. It is then expected that the overall stopband of the resulting structures, assumed to contain various identical unit cells of each type, coincides with the union of the individual bandgaps. Under these assumptions, graded structures lying in the intervals *θ* ∈ [0.12, 1.0] [Fig. 3(a)] and *η* ∈ [0.4, 2.0] [Fig. 3(b)] would have an overall bandwidth of 3.738 and 2.513, respectively, while the bandwidth for the corresponding single-resonator system is 1.472. This represents an increase of 153.9% and 70.7% for the *θ*- and *η*-graded metastructures. Additionally, we note that the lower edge of the resulting stopband is also lower than that of the non-graded structure, which is often a desired feature.

The anticipated increase in the bandwidth for the previous designs is quite remarkable. Yet, it may be argued, at least qualitatively, that an improvement with respect to a non-graded single-resonator structure is to be expected, thanks to the additional control variable (*θ* or *η*). To clarify this point and further highlight the effect of hierarchy on graded designs, we also perform a comparison between identical graded structures with Unit cells 1 and 2 (for a fixed value of *θ* and *η*), obtained by changing the value of *m*_{2}; see Fig. 6(a) for the bandgap structures of Unit cells 1 and 2 as a function of *m*_{2}. In this case, if we consider mass values of *ξm*_{2}, with *ξ* ∈ [1,4] the expected bandwidth of the system with hierarchical design is 1.875, whereas for the single-resonator system, it is 0.933. This represents an increase of 100.9%, which is also quite remarkable. Additionally, and similarly to the previous designs, the lower edge of the bandgap is also reduced, which provides an additional attractive feature to the hierarchical metastructures. Figure 6(b) shows for completeness the bandgap structure of both unit cells as a function of *k*_{2}. In this case, the bandgap spectrum for Unit cell 2 is also larger than that of Unit cell 1. However, there is a gap in the frequency space between the first and the second bandgap which precludes the design of a graded architectures based on Unit cell 2 and different values of *k*_{2}, to deliver a single broad stopband.

### B. Continuum model and finite element simulation

The results obtained in Sec. III A pertain to infinite periodic structures, which is, of course, unrealistic for practical applications; and it is actually incompatible with the notion of graded designs, where distinct unit cells are combined. It is therefore important to test the approach of Sec. III A, with the transmission loss (TL) in finite graded structures as a function of frequency. These studies are precisely the focus of this section, for which we further consider more realistic unit cells. More precisely, we use plane strain continuum models that mimic lattice designs, following the approach used by Liu *et al.*^{31} for single-resonator systems; see Figs. 7(a) and 7(b) for a schematic representation of the continuum version of Unit cells 1 and 2.

We briefly summarize some important considerations needed to construct the continuum models so that they are equivalent to their lattice analogue. The elastic material used to represent the masses in the discrete system shall have a high density and large Young's moduli. For their part, the springs shall be mimicked by light and soft materials. In addition, the wavelength in the “mass” part in the range of frequency of interest should be much longer than the size of the unit cell. More precisely, Liu *et al.*^{31} showed that *λ*/*L* should be larger than 10, where *λ* is the wavelength corresponding to the highest frequency of the passing bands and *L* is the length of the unit cell. Following these principles, we select the material properties for the linear elastic materials corresponding to *m*_{1}, *m*_{2}, and the springs, and these are listed in Table I. The precise values of *k*_{1} and *k*_{2} are obtained by changing the geometry of the connecting unit between two masses, and these are estimated by uniaxial type of tests in the actual geometry via finite element simulations.^{31} Finally, Unit cell 2 for different values of *θ* and *η* is constructed from the geometry shown in Fig. 7(a), by varying the density and Young's modulus of the elements corresponding to *m*_{3}, *m*_{4}, and *k*_{3}, respectively. To assess the capability of the suggested continuum designs to mimic the mass-in-mass discrete systems, we compare in Fig. 8 their dispersion relations for various values of *θ*, obtaining an excellent agreement. The curves depicted for the continuum model are obtained using the finite element software COMSOL Multiphysics,^{41} by applying Bloch boundary condition on the boundaries of the unit cells, and imposing additional constraints to only obtain the locally resonant modes in the longitudinal direction of the irreducible Brillouin zone.

. | Density . | Young's modulus . | Poisson's ratio . |
---|---|---|---|

. | (kg/m^{3})
. | (GPa) . | . |

m_{1} | 2000 | 290 | 0.30 |

m_{2} | 7000 | 400 | 0.30 |

springs | 930 | 1 | 0.46 |

. | Density . | Young's modulus . | Poisson's ratio . |
---|---|---|---|

. | (kg/m^{3})
. | (GPa) . | . |

m_{1} | 2000 | 290 | 0.30 |

m_{2} | 7000 | 400 | 0.30 |

springs | 930 | 1 | 0.46 |

The transmission loss for finite structures is calculated in the frequency domain using COMSOL Multiphysics.^{41} This is achieved by applying an input excitation on one end of the sample, while the other end remains free. For computational purposes, we will attach perfectly matched layers (PML) at the two ends of the computational domain to suppress undesired reflections; and further add additional matrix material between the sample and the PML, where the excitation and receiving signals are located, cf. Fig. 7(b). Periodic boundary conditions are also applied on top and bottom boundaries to avoid boundary effects. The transmission loss is then obtained by measuring the ratio of the magnitudes of the output (*U*_{out}) and input (*U*_{in}) displacements; more precisely, TL = 10 log(*U*_{out}/*U*_{in}). If the transmission loss is far below 0 in a certain range of frequencies, the input wave propagation will be attenuated by the structure, and hence a bandgap can be identified.

Before we present the transmission loss for the graded designs, we first show the results for non-graded structures composed of 100 Unit cell 2 with uniform *θ* along the continuum model. Figures 9(a)–9(d) present the TL of the non-graded structures with *θ* = 0.25, 0.50, 0.75, and 1.0, respectively, where the case *θ* = 1.0 corresponds to Unit cell 1. It can be seen that the transmission loss is in excellent agreement with the bandgaps predicted by the mass-spring model, indicated by the shaded regions.

The full structure for the graded design is shown in Fig. 7(b). The central part of the bar is composed of uniform sections, denoted as S_{1}, S_{2}, and S_{N}, each of which contains identical unit cells. Geometrical and/or material parameters vary between sections, leading to a graded structure.

The transmission loss for the graded designs based on Unit cells 2 with varying values of *θ* or *η* is illustrated in Figs. 10(b) and 11(b), respectively, where the results are compared with the transmission loss over an ungraded structure composed of Unit cells 1. Recall that the total static mass of the resonators is equal in both unit cells, and thus this comparison showcases the advantages of the additional degrees of freedom induced by the hierarchical designs. In these graded structures, the domain is composed of 10 (*θ*) or 5 (*η*) sections, each of which contains 10 and 20 unit cells, respectively, while the ungraded structures also contain a total of 100 unit cells. The precise values of the parameters in each of the sections are *θ* = {0.15, 0.17, 0.20, 0.24, 0.30, 0.40, 0.50, 0.60, 1.0, 1.0} and *η* = {0.4, 1.2, 1.6, 1.8, 2.0}. As may be observed in the figures, the resulting overall bandgap is very well approximated by the union of the bandgaps of the different sections composing the graded design, obtained as if they were infinitely periodic lattice systems [see darken region in Figs. 10(a) and 11(a)]. More precisely, for the *θ*-graded structure, a 124.2% gain in the overall bandwidth is achieved, while for the *η*-graded structure, such a gain is of 66.7%. Both of these values are close to the lattice model predictions, which amount to 126.5% and 70.4%, respectively.

Next, Fig. 12(b) depicts the transmission loss of metastructures with Unit cells 1 and 2 and varying values of *m*_{2}. This is achieved by means of a factor *ξ* applied in the 5 sections of the domain, each containing 20 unit cells, i.e., we consider *ξm*_{2}, where *ξ *= {0.55, 0.65, 0.75, 1, 1.5}. In these simulations, the values of *θ* and *η* are kept constant in the sample to illustrate the effect of hierarchy on graded metastructures. The results depict the superiority of the hierarchical design, with an overall bandwidth that is 54.3% broader than the structure with Unit cell 1. Again, the obtained result matches well the anticipated 50.0% gain from the lattice model, cf. Fig. 12(a). A similar analysis could be done for sections with varying *k*_{2}, while maintaining *m*_{2} fixed. In this case, a broad bandgap spectrum is also expected. Yet, analogously to the material system considered in Sec. III A, a gap exists between the first and second bandgap of Unit cell 2, making it a less favorable design.

### C. Discussion

It is important to mention that the proposed designs with graded ratio *θ* (and similarly for *η*-graded structures) rely on the existence of overlapping bandgaps; that is, the maximum frequency of the first stopband shall be larger than the minimum value of the second bandgap, see Fig. 3(a)—here, maximum and minimum are understood as a function of *θ* and fixed value of the other parameters. This condition is satisfied for large enough values of *m*_{2} as can be inferred from Fig. 6(a) for the material system of Sec. III A. In particular, it is there observed for a fixed value of *θ* = 0.5 that the passing band between the first and second bandgap of Unit cell 2 becomes narrower with increasing *m*_{2}, and after a critical value, it is fully contained in the bandgap of Unit cell 1, or equivalently, of Unit cell 2 with *θ* = 1. Considering the full range of *θ* ∈ [0, 1], the critical value of *m*_{2} can be computed and is shown in Figs. 13(a) and 13(b) as a function of *k*_{2} and *k*_{3}, respectively. The values of the remainder of the parameters are indicated in the caption, and correspond to the system analyzed in Sec. III A.

The trends of Figs. 13(a) and 13(b) can be explained as follows. With increasing value of *k*_{2}, the upper edge of the first bandgap of the single-resonator (equivalent to dual-resonator with *θ* = 1) increases faster than the lower edge of the second bandgap of the dual-resonator structure as shown in Fig. 6(b). As a result, the possibility of getting overlapped bandgaps increases and the critical value of *m*_{2} decreases. As for the effect of *k*_{3}, Fig. 3(b) shows that the passband between the first and the second stopbands enlarges with increasing *η*, and thus the value of *m*_{2} required to close the gap is also increased.

Additionally, it is important to remark that the computed critical values for *m*_{2} are also optimal for achieving the maximum bandwidth for the hierarchical architectures, cf. Fig. 13(c). In other words, for *m*_{2} larger than the critical value, a hierarchical design can be constructed with a continuous broadband filter. Yet, a further increase of *m*_{2} beyond such a critical point will not provide further benefits, but, on the contrary, it will reduce the total bandwidth.

## IV. CONCLUSION

This article presents a detailed analysis of the acoustic bandgap of resonant hierarchical lattice structures. The results indicate that the overall bandwidth of double-resonator systems is approximately equal, albeit bounded by that of the single-resonator unit with the same static mass; and this holds true for all possible values of the parameters controlling the hierarchical design (*θ* and *η*). Yet, the lower (upper) edges of the first bandgaps can decrease (increase) significantly, and these can strongly vary with the hierarchical parameters. Based on these observations, we combine multiple hierarchical units to deliver graded metastructures with ultra-broadband filtering capabilities. This design strategy is validated over finite size continuum systems, where the transmission loss is calculated by means of finite element simulations. The results are in excellent agreement with the predictions made by the band diagram of the individual cells computed for analogous lattice models.

## ACKNOWLEDGMENTS

The authors acknowledge the support from NSF Grant No. CMMI-1401537.

## References

Here, the bandgap *spectrum* refers to the frequency content of the finite stopbands, which spans between the lower edge of the lowest bandgap and the upper edge of the upper most finite bandgap.

COMSOL Multiphysics™ v. 5.3. COMSOL AB, Stockholm, Sweden (2017).