Origami inspired phononic structure with metamaterial inclusions for tunable angular wave steering

Autonomous systems have seen tremendous growth in the last decade due to sensory components becoming less expensive and computational power of processors increasing.¹ Without human operators, these machines are entirely dependent on sensing technology to map their environment and make intelligent decisions in real time. Autonomous systems typically use remote sensing technology where the distance of objects is determined by transmitting waves and interpreting the reflected signals,² such as LiDAR. While LiDAR is the most ubiquitous remote sensing technology, it requires complex, costly hardware. Conversely, sonic systems have the potential to be cheaper, smaller, and less complex, which makes them very attractive to be integrated into a variety of autonomous systems.³ As shown in medical,^4–6 nondestructive evaluation,^7,8 and underwater^9,10 sensing technologies, the ability to generate and manipulate an acoustic beam is fundamental to acoustic sensing. However, acoustic beams are not typically generated using a single transducer, but arrays of transducers, known as acoustic phased arrays.^11,12 Their complexities detract from the cost, power, and space saving advantages of acoustics sensing for autonomous system mapping.

As an alternative to acoustic phased arrays, researchers have explored gradient index metamaterials to produce controllable collimated beams with limited space and a single transducer.^13–18 To achieve planar wavefront modulation, researchers developed gradient based metastructures composed of labyrinthine unit cells,¹³ Helmholtz resonators,^14,15 and periodic structures with varying inclusion shapes.^16–18 While these studies were able to control collimated beams, each individual structure could only output waves at a single angle or frequency; a different structure would be required to change the output wave angle or frequency. In an effort to create a programmable structure capable of acoustic wave steering, researchers explored an acoustic metastructure composed of gradient based Helmholtz resonators with an advanced fluid pumping system.¹⁹ With the fluid pumping system, they were able to dynamically control the height of the Helmholtz resonator cavity and thus the angle of transmission of output waves. Although this study can produce on-demand acoustic wave steering, the structure requires a large number of synchronized actuators, which greatly increases complexity and makes control of the acoustic lens difficult.

In addition to gradient index metamaterials, phononic crystals have been shown to possess acoustic wave manipulation capabilities. Researchers inspired by the presence of Dirac cones in the band structure of graphene have used phononic crystals to study wave transport phenomena near Bloch mode degeneracies.^20–25 Dirac points can be found at the intersection of modes, and at these degeneracies, acoustic waves propagate freely without backscattering, a phenomenon that can be used to realize structures that create collimated acoustic beams.^26–32 The ability to tune the frequency of these collimated beams would be very beneficial to acoustic sensing systems in that it would allow the system to focus on a variety of distances. To shift Dirac cone frequency, researchers have explored expandable inclusions,^34,35 rotatable inclusions,³⁶ and adjustable resonant inclusions.^37,38 While these complex structures are tunable to some degree, the tunable range was limited and difficult to implement physically.

Breaking away from Dirac cones at high points of symmetry, Zhang et al.³³ explored the production of oblique waves with Dirac points around the irreducible Brillouin zone boundary with phononic structures. They showed that it is possible to produce oblique waves when the Dirac point of a phononic structure is located between X and M. At a single frequency, the angle of the incident wave could vary from 0° to 25° by modifying the inclusion shape and thus bulk material properties of the structure. However, no physical mechanism was proposed to produce these inclusion changes. Additionally, like the majority of phononic Dirac studies cited above, the analysis was limited to inclusions with a slower speed of sound than the surrounding medium, e.g., rubber inclusions in water. Therefore, while this approach produces varied angle oblique waves, it cannot be directly utilized in air. In reality, the ability to produce varied angle oblique waves in air is essential to acoustic sensing for scanning in multiple directions.

In summary, the current literature demonstrates that the Dirac physics inherently possessed by some phononic structures can be used to produce controllable oblique plane waves with limited space and a single transducer. However, the interesting phenomenon produced by these structures is highly dependent on lattice parameters, which results in acoustic structures that can only produce a plane wave at a single frequency and single incident angle. Additionally, the majority of these studies are very difficult to achieve in an air medium. Therefore, for angular wave control with phononic structures to be realistic and useful for acoustic sensing, an easily tunable phononic structure that can achieve a broad range of Dirac point manipulation in an air medium still needs to be developed, which is a considerable research challenge.

To shift the Dirac cone frequency and location around the Brillouin zone in a phononic lattice, the material properties or geometry of the structure have to be able to change dramatically. Origami structures have been used for adaptive wave control for both elastic and acoustic waves.^39–42 Recently, Thota et al.^43–45 investigated a method to extend the tunable range of phononic bandgaps using origami-based reconfigurable systems. Specifically, a folded Miura-origami sheet was used as a base to create a phononic lattice that could transition from a hexagonal to a square to a second hexagonal Bravais lattice as a single degree of freedom system. The reconfigurable model was able to dramatically alter the phononic structure's bulk material properties and thus alter its band structure. While interesting and promising, none of these previous studies have focused on Dirac point manipulation.

Building upon the origami concept,^43–45 our new investigation is to expand the scope to tune both the frequency and location of Dirac cones in phononic lattices. Our goal is to advance the state of the art and explore a new idea of modulated origami phononic structures with metamaterial inclusions for tunable angular wave control for acoustic remote sensing in air.

In this article, we first discuss the design of the origami structure to understand what lattice geometries are available through the tuning mechanism (Sec. II). We then develop insight into the material properties of the lattice scatterers required to produce Dirac points for the available lattices (Sec. III) and apply that insight to design suitable metamaterial-based scatterers (Sec. IV). With our proposed origami structure defined, we numerically investigate angular wave control with the structure (Sec. V). We then conclude with a summary of our advances (Sec. VI).

As shown in Fig. 1(a), the proposed structure consists of a folded Miura-origami sheet as a base, light gray, with magnetic cylindrical rods, dark gray, attached to the vertices. Each rod is magnetically connected to a metallic inclusion, blue and yellow, placed inside an acoustic waveguide composed of two rigid reflecting plates. For this work, the acoustic waves propagate only inside the waveguide in between the plates as illustrated in Fig. 1(b), and the magnetic rods provide the mechanism through which the Bravais lattice is reconfigured. Since the Miura-origami sheet is a single degree of freedom system, applying a force to one point at the corner of the structure will modulate the magnetic rods to form the Bravais lattices described in Secs. II–V.

The location of the vertices of a Miura-origami sheet can be defined with the following four parameters: two crease lengths (a, b), a sector angle (γ), and a folding angle (⁠$θ$⁠). As shown in Fig. 1(c), the folding angle is defined by the dihedral angle between facets and the XY plane. We then use the four parameters in the kinematic relationships given by Eq. (1)⁴³ to calculate the inclusion locations at the vertices of our sheet as shown in Fig. 1(c). A folding angle of 0° would create a completely flat plate in the XY plane, while a folding angle of 90° would create a completely compressed vertical structure in the YZ plane. Folding of the structure moves the vertices of the Miura-origami sheet and thus changes the phononic lattice symmetry:

The crease lengths and sector angle determine what lattice configurations a Miura-origami design can realize, and by selecting these two parameters, the Miura origami is able to achieve four of the five Bravais lattices: rectangle, center-rectangle, square, and hexagon.⁴⁴ To achieve angular control, we need to shift the location of Dirac cones around the Brillouin zone border. Although center-rectangular and hexagonal Bravais lattices have been shown to possess Dirac points, they do not possess them in locations suitable for angular control. Square and rectangular lattices have tunable Dirac cones along the XM border, so we chose a sector angle and crease lengths that can achieve square and rectangular lattices. To achieve a large range of operational frequencies, we design the origami structure such that we can transition between multiple square and rectangular lattices. Miura-origami parameters that produce square and rectangular Bravais lattices must follow Eqs. (2) and (3), where n is an integer.

Square Bravais lattice

Rectangular Bravais lattice

To determine the Miura-origami parameters that would produce three rectangular Bravais lattices, where one of the rectangular lattices is classified as square, we perform a parametric study. Solving Eqs. (2) and (3) with n = 1 to 3 produces the parametric design space in Fig. 2. In yellow, blue, and gray, we see the γ and a/b values that will produce rectangular Bravais lattices for n values of 1, 2, and 3. In black, we see the γ and a/b values that will produce square Bravais lattices. The square Bravais lattice is considered a special case of the rectangular Bravais lattices, so the black curves are highlighting designs that are also in the rectangular design space. Overlapping regions represent multiple lattice configurations achievable with the same design parameters. Ultimately, the Miura-origami design with a crease length ratio (a/b) of 1.03 and a sector angle (γ) of 73° is chosen with lattices achievable in all three rectangular design regions. This design produces a structure that is able to transition to two rectangular Bravais lattices with folding angles of 62.2° and 75.5° and a square lattice with a folding angle of 84.2°, as shown in Fig. 3.

With the geometry of the Miura-origami lattice determined, the next step is to design the inclusions. Phononic structures with useful degeneracies are most often designed with inclusions that have a slower speed of sound than the surrounding medium. For example, researchers have used rubber inclusions surrounded by epoxy,²⁶ theoretical air inclusions surrounded by water,²⁸ and rubber inclusions surrounded by water.^31,32 Achieving a square or rectangular phononic structure with useful degeneracies in air is challenging since air has such a slow speed of sound. Additionally, there is very little insight into exactly what combination of material parameters will produce these useful degeneracies.

In order for this structure to be used in autonomous remote sensing technology, it has to be able to produce directional collimated beams in air. To determine exactly what material properties would produce tunable Dirac points, we perform a parametric study where we vary the relative density (⁠$ρ∗$⁠) and the relative speed of sound (⁠$c∗$⁠) of a 22 cm square inclusion for our two rectangular and square lattices. The relative density and speed of sound can be defined by Eq. (4) where $ρmedia$ and $cmedia$ are 1.225 kg/m³ and 343 m/s, respectively. Using the plane wave expansion method,⁴⁶ the band structure for each lattice with the varied material properties is calculated. Each band structure is then reviewed for a Dirac point on the XM boundary:

The realizable design space is shown in Fig. 4(a). The yellow regions represent the relative speed of sound and relative density combinations that produce Dirac points, while the blue regions represent material combinations that do not produce Dirac points. From this design space, we can see that structures with tunable Dirac points have inclusions with significantly higher relative densities and lower speeds of sound than the surrounding medium, air. While previous research has shown it is possible to achieve a Dirac point with the speed of sound of the inclusions less than the speed of sound of the medium, we have shown that for all three lattices, the speed of sound must be less than 70% of the speed of sound of the surrounding medium to achieve a Dirac point. While not shown in these plots, achievable Dirac points disappear as the relative density becomes impractically large. Furthermore, if we calculate the relative bulk moduli of these structures using $B∗=c∗2ρ∗$⁠, as shown in Fig. 4(b), we determine that the inclusions must be both dense and very compressible, which is a difficult material combination to achieve.

Based on the discussions in Sec. III, we make advances to realize inclusions attaining the unusual material properties needed to obtain Dirac points using metamaterial techniques. Specifically, each inclusion is composed of arrays of highly subwavelength unit cells consisting of Helmholtz cavities operated off resonance. The Helmholtz cavity is composed of two parts, shown in Fig. 5 in blue and yellow. The blue piece slides over the yellow piece as displayed in the cross-sectional diagrams in parts (b) and (c) of Fig. 5. Springs are connected to all four corners inside of the cavity, so increasing the waveguide height will allow the structure to expand due to the force of the springs and decreasing the waveguide height will compress the structure. The Helmholtz cavities can be completely defined by the following six properties: post width, waveguide height, opening width, wall thickness, cavity height, and cavity width.

Using the parametric study from above, we aim to create a metamaterial unit cell to achieve a relative density of 13.7 and a relative bulk modulus of 0.67, material properties that produce a Dirac point on the XM boundary of the Brillouin zone boundary at 550 Hz. We vary the Helmholtz cavity dimensions to determine their effects on the relative density and bulk modulus curves. Using an iterative process, we increase or decrease these dimensions individually until we reach the desired relative density and bulk modulus at the desired frequency. Ultimately, we tune the post width (1.3 mm), waveguide height (3 mm), opening width (10 mm), wall thickness (0.7 mm), cavity height (103 mm), and cavity width (14.8 mm) to achieve these parameters. Using the effective medium theory described in Refs. 47–50, we map the Helmholtz cavity to an effective density and bulk modulus, the results of which can be seen in parts (a) and (b) of Fig. 6. The desired relative density of 13.7 and the relative bulk modulus of 0.67 are shown by yellow points in Figs. 6(b) and 6(c).

These yellow points are on relatively flat parts of the curve, away from the resonant peak. Avoiding the resonant peak is necessary to make sure the desired Dirac point is in a sufficiently large partial bandgap, as highly dispersive material properties result in smaller partial bandgap sizes. The flatness of these curves, and therefore the metamaterial dispersion, is directly related to the density of metamaterial unit cells in the inclusion. As the number of cavities increases, lower effective bulk moduli can be obtained in lower dispersion metamaterials. To achieve the results shown in Fig. 6, we use a 15 by 15 array of Helmholtz cavities, which are illustrated in Fig. 5(d). To verify that our effective material properties realize a Dirac cone on the XM boundary, we calculate the band structure of a 32 cm square inclusion with our 22 cm square array of Helmholtz cavities using COMSOL, the results of which are shown in Fig. 6(b). As expected, the structure produces a Dirac point at 550 Hz, as highlighted by the yellow points.

While modulating the origami structure between lattice configurations modulates the frequency of the collimated beam, we still need a mechanism to modulate the angle of the oblique wave and thus the location of the Dirac point on the Brillouin zone boundary. As described at the beginning of this section, the Helmholtz cavities are designed such that the height is tunable. By adjusting the waveguide height, we are able to control the inclusion of bulk material properties. In the square lattice described above, an inclusion height of 6 cm produces a relative density of 11.3 and a relative bulk modulus of 0.95 at 600 Hz, while an inclusion height of 10.3 cm produces a relative density of 13.85 and a relative bulk modulus of 0.92 at 400 Hz. The effects of these material properties on the Dirac point location will be explored in Sec. V.

To demonstrate angular wave control with the proposed Miura-origami phononic structure, the band structure and full structure excitation is studied for the origami lattice folding at angles 62.2°, 75.5°, and 84.2°, which correspond to rectangular, rectangular, and square lattice configurations, respectively, with varied inclusion heights. The inclusions in this work are composed of steel cavities embedded in air. The density of air is 1.225 kg/m³, while the longitudinal wave velocity is 343 m/s. For this study, steel is treated as rigid. All dimensions of the Helmholtz cavities, except for the height of the cavities, can be found in Fig. 5. The heights of the inclusions can be found in Fig. 7. The band structures of the phononic crystals are calculated by performing an eigenvalue analysis of the two rectangular and square lattice unit cells. Floquet Bloch periodic boundary conditions are applied to all sides of the unit cells. The wave vectors are swept along the irreducible Brillouin zone, which spans from Γ to X to M to Y to Γ for rectangular lattices and from Γ to X to M to Γ for square lattices. The resulting band structures can be seen in parts (a) and (c) of Fig. 7.

To determine the full structure excitation, a frequency domain simulation is performed at each of the Dirac point frequencies determined in the unit cell analysis. A 2D simulation is performed with the calculated effective material properties applied to the inclusion domains. Floquet Bloch periodic boundary conditions are applied to the top and bottom boundaries of the structures, and perfectly matched layers are used on the left and right boundaries to simulate non-reflective domains to absorb all outgoing waves. A line source is used to excite the structures as highlighted by the black circles in parts (b) and (d) of Fig. 7 on the resulting pressure field distributions. All mesh elements are smaller than one tenth of the shortest wavelength.

While only four folding angle and inclusion height combinations are shown, the proposed origami structure is able to produce collimated beams at angles from 0° to 64° in the frequency range of 294–599 Hz, meaning the structure can scan its environment in multiple directions and focus on a variety of distances. As shown in Fig. 7, modulation of the folding angle produces dramatic changes in both the frequency and location of the Dirac points. Adjusting the inclusion height allows for fine tuning of both the frequency and location of the Dirac points. Modulation of inclusion height has a significant impact on both the frequency and angle of plane waves for square lattices, while it mostly changes the frequency of plane waves for rectangular lattices, characteristics that could be very useful in acoustic sensing. Furthermore, this structure has shown the ability to convert a line source into a plane wave, meaning the structure has the ability to convert the output from a single transducer into a directional plane wave for acoustic sensing.

We advance the state of the art and explore a new idea of origami phononic structure with metamaterial inclusions to achieve broad range tunable angular wave control for acoustic remote sensing in air. In this study, we gained insight into the required metamaterial properties to produce Dirac cones in air and designed an origami-based reconfigurable phononic structure to realize tunable Dirac points. Through an analysis of the origami lattice geometry, we developed an adaptive phononic structure capable of angular wave control from 0° to 66° in the frequency range of 294–606 Hz, a range much larger than the state of the art. In addition to dramatically changing the frequency and angle of the useful degeneracies, this phononic structure is physically easy to realize and is a simple two degree of freedom system that requires minimal effort to reconfigure.

Future work includes fabricating and experimental demonstration of the origami inspired phononic structure with metamaterial inclusions. The origami facets can be made out of aluminum sheet metal connected with polythene adhesive sheet, and the metamaterial inclusions can be 3D printed. While metal inclusions were proposed in this study, other materials such as plastic could be used if a thin magnet is placed at the bottom of the inclusion. With the analysis tools developed in this study, the system can be designed to meet desired requirements for a variety of sensing applications. The proposed mechanism can form and steer collimated, highly directional beams in desired directions, which can enable a new generation of a beam forming sensor for sonar-like applications in air that require only one transducer and thus very low power.

This research is partially supported by the University of Michigan Collegiate Professorship and by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1256260. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

The data that support the findings of this study are available from the corresponding author upon reasonable request.