Low frequency acoustic stop bands in cubic arrays of thick spherical shells with holes

We analyse the propagation of pressure waves within a fluid filled with a three-dimensional array of rigid coated spheres (shells). We first draw band diagrams for corresponding Floquet-Bloch waves. We then dig a channel terminated by a cavity within each rigid shell and observe the appearance of a low frequency stop band. The underlying mechanism is that each holey shell now acts as a Helmholtz resonator supporting a low frequency localized mode: Upon resonance, pressure waves propagate with fast oscillations in the thin water channel drilled in each shell and are localized in each fluid filled inner cavity. The array of fluid filled shells is approximated by a simple mechanical model of springs and masses allowing for asymptotic estimates of the low frequency stop band. We finally propose a realistic design of periodic macrocell with a large defect surrounded by 26 resonators connected by thin straight rigid wires, which supports a localized mode in the low frequency stop band.

In the tracks of photonic crystals, phononic crystals [1] have provided a fillip for research in acoustic stop band structures [2, 3] within which light or sound is prohibited to propagate due to multiple scattering between periodically spaced inclusions. In 2000, Liu et al. provided the first numerical and experimental evidence of locally resonant structures for elastic waves in 3D arrays of thin coated spheres [4] wherein low frequency stop bands occur. This seminal work paved the way towards acoustic analogues of electromagnetic meta-materials, such as fluid-solid composites [5]. Inspired by the research monograph on multi-structures [6], Movchan and Guenneau subsequently proposed to use arrays of cylinders with a split ring cross section as building blocks for 2D localised resonant acoustic structures displaying negative refraction [7,8]. Such split ring resonators, introduced by John Pendry in the context of electromagnetic waves [9], also work for in-plane elastic waves [10]. Milton, Briane and Willis provided a thorough mathematical frame for such effects including cloaking for certain types of elastodynamic waves in structural mechanics [11]. For instance, coupled in-plane pressure and shear waves were numerically shown to be detoured around a finite size obstacle by a specially designed cloak with an anisotropic heterogeneous elasticity tensor (without the minor symmetries) [12]. Acoustic metamaterials via geometric transform can thus in theory achieve unprecented control of elastic and pressure waves [13,14].
Li and Chan independently proposed a similar type of negative acoustic metamaterial [15]. In a recent work, Fang et al. experimentally demonstrated a dynamic effective negative stiffness in a chain of water filled Helmholtz's resonators for ultrasonic waves [16]. It has been also shown using homogenisation theory that surface water waves propagating within an array of fluid filled Helmholtz's resonators display a negative effective density [17]. A focussing effect through a finite array of such resonators was numerically achieved, with a resolution of a third of the wavelength. Similar effects have been experimentally demonstrated for a doubly periodic array of Helmholtz's resonators shaped as soda cans [18]. In the present paper, we would like to extend these concepts to pressure waves propagating in a three-dimensional array of resonators.

II.
MOTIVATION: SPECTRAL PROPERTIES OF A PERIODIC ARRAY OF RIGID SPHERES As a preamble, let us start with an illustrative numerical result for a spectral problem for the Helmholtz operator within a periodic cubic array of rigid spheres: the unknown is a pressure wave field, here sound in water (wave speed c = 1483 m.s −1 ). Neumann boundary conditions are prescribed on the contour of each defect and standard Floquet-Bloch conditions are set on an elementary cell of the periodic structure. The finite element formulation was implemented in the COMSOL Multiphysics Package to compute the eigenvalues and to generate the corresponding eigenfields. We present in figure 1 the structure and in figure 2 the corresponding dispersion diagram for eigenfrequencies ω as a function of the Floquet-Bloch parameter k: along the horizontal axis we have the values of modulus of k, where k stands for the position vector of a point on the contour ΓXM U within the irreducible Brillouin zone. We note the absence of bandgaps with the presence of rigid spheres. This lack of intervals of forbidden frequencies motivates the present study: how can one create a stop band without further increasing the arXiv:1802.02043v1 [physics.class-ph] 29 Dec 2017 size of the rigid spheres?  1 ρ(x, y, z) ∇u(x, y, z) + ω 2 λ(x, y, z) u(x, y, z) = 0 (1) where ρ, λ and u are the density, the bulk modulus and the pressure field.
Due to the periodicity of the lattice, it is customary to require that the eigenfunctions be of the Floquet-Bloch type. So, for a cubic array of unit cells Y , where k x , k y and k z are components of the Bloch vector k within the Brillouin zone The implementation in the finite element package is fairly straightforward. We first multiply equation (1) by a smooth function V and using the Green's formula, we obtain the so-called weak form of the time-harmonic acoutic equation where ∂f /∂n = ∇f ·n with n the unit outward normal to the boundary ∂Y of Y , and ds the infinitesimal surface element on ∂Y . We note that the weak formulation holds for heterogeneous fluids as ρ and λ can be spatially varying. In particular, this model works for perforated domains such as a homogeneous fluid filled with a periodic array of rigid osbtacles. For the finite element implementation, (3) is discretised using test functions taking values on nodes of a tetrahedral mesh of the basic cell (first order tetrahedral elements), see e.g. [19] for further details. From (3), we note that setting rigid boundary conditions conditions on an inclusion amounts to assuming Neumann (natural) homogeneous data, whereas transmission conditions at the interface between various fluid phases mean that the quantity ρ −1 ∂u/∂n is preserved across the interface.
Let us now consider a periodic array of defects Ω 1 , ..., Ω N embedded in an elementary cell Y =]0; 1[ 3 . Let u(x, y, z) satisfy the Helmholtz equation in Y \ N j=1 Ω j . We also assume that u satisfies Neumann boundary condiditons on the contours of defects, where n denotes the unit outward normal to the boundary ∂Ω j of a defect Ω j : We would like to consider a particular case when the defects Ω 1 , ..., Ω N are spherical shells with thin water channels connecting a fluid-filled interior cavity to the exterior surrounding fluid. These defects can be modelled as multistructures [6] in the following way, where a (N ) and b (N ) are given constants and Π (j) ε(N ) is the thin channel.

IV. ASYMPTOTIC APPROXIMATION: A DISCRETE SPRING-MASS MODEL
In this section, we derive an asymptotic approximation of the field within thin channels Π where l j is the length of the j th bridge, εh j (t) the radius of its varying cross-section D ε (parametrized by t).
Here, ε is a small positive non-dimensionnal parameter.
Without loss of generality, and for the sake of simplicity, we drop the superscript j. In Π ε , the time-harmonic wave equation takes the rescaled form with the Neumann boundary conditions The field u is approximated in the form To leading order, we obtain Hence, . Assuming that u (0) is given, we derive that the function u (1) satisfies the following model problem on the scaled cross-section The condition of solvability for the problem has the form: Hence, we have shown that to the leading order we can approximate the field u within the thin channel Π ε by the function u (0) which satisfies the Helmholtz's equation in one-space dimension. We now assume that the field is periodic over the cell since it is localized. This shows that the average of the eigenfield over the macro-cell vanishes. Indeed, let χ 1 denotes the value of the field in the large body Σ of the multi-structure Ω and let χ 2 (which we normalize to 1) denotes the value of the field within the complementary area of the macro-cell Y \ Ω excluding the thin channels. Taking V = 1 in (3), we deduce that This shows that the average of the field u over Y vanishes, hence by neglecting the volume of the thin channels, we obtain where meas Σ and meas Y \Ω denote respectively the areas of Σ and Y \ Ω. We now consider two cases. The first one is the study of an array of simple spherical shells with either one or six thin channels, and the other one is the study of an array of double spherical shells with one thin channel in each shell. Since we have p thin channels, we have p separate eigensolutions V j , (j = 1, ..., p), corresponding to the vibrations of thin domains Π where All the channels are connected to Ξ, hence, V 1 (l 1 ) = ... = V p (l p ) = V . We note that V j (0) is equal to a non-zero constant.
The solution of the problem (15) − (17) has the form where c = λ/ρ and the frequency ω is given as the solution of the following equation: where we invoked Newton's second law. Looking at a first low frequency, we deduce an explicit asymptotic approximation This estimate actually holds for the frequency ω 2 of the upper edge of the phononic band gap. We note that if we take V (0) = 0 instead of V (0) = meas(Ξ)/meas(Y \ Ω), we estimate the frequency of the lower-edge of the phononic band gap.
A. Eigenfrequency estimate in the case of a single spherical shell with one or six thin channels We report in figures (4) and (7) finite element computations for a periodic cell of sidelength d with a simple spherical shell with thin channels. The interior and exterior radii of the shell are respectively 0.3m and 0.4m, the thin channels have the same length 0.1m and radii 0.01m. Therefore, the frequency estimates are (in Hertz): ω 1 ∼ 227.301 , ω 2 ∼ 235.889 (22) for one thin channel, which are in good agreement with the finite element values for six thin channels. This demonstrates that the discrete model provides accurate estimates for the lower and upper edges of the ultra-low frequency stop band. This is therefore a useful tool which can be used in the design of acoustic metamaterials. The eigenfunction corresponding to the eigenfrequency ω * 1 = 224.6155Hz for one thin channel. Blue color corresponds to nearly vanishing amplitude of the eigenmode u, while red color corresponds to it maximum value. The pressure field u is constant inside the inner cavity and outside the shell, but it varies rapidly inside the thin channel: it is a localised eigenmode responsible for the stop band in figure 4, which is well approximated by a spring mass model.

B. Eigenfrequency estimate in the case of a double spherical shell with one thin channel
In the numerical example we now have ε 2 h 2 2 = 3.14.10 −4 m 2 , ε 2 h 2 1 = 7.85.10 −5 m 2 , l 2 = 0.1m, l 1 = 0.05m, and the masses (in kilogram) ) of spherical rigid shells (inner radius R = 0.3m and outer radius R = 0.4m) with six thin channels (length 0.1m and radius 0.01m) representing the frequency ω (Hz) of pressure waves in water versus the wavenumber |k| (m −1 ), projection of the Bloch vector k along the edges of ΓXM U . We note the appearance of a frequency stop band for ω ∈ [548, 588]Hz which is wider and at higher frequencies than the stop band in figure 4: the more identical thin channels, the higher the resonant frequency of the localised mode. FIG. 8: The eigenfunction corresponding to the eigenfrequency ω * 1 = 548.3515Hz for six thin channels responsible for the stop band in figure 7. This frequency is well approximate by the sping mass discrete model which provides us with the frequency estimate ω1 = 556.772Hz. m 1 = ρV 1 = 4000 3 πr 3 1 m 2 = ρV 2 = 4000 3 π r 3 1 + (b 3 2 − a 3 2 ) + 10 3 ε 2 h 2 1 l 1 (26) where ρ is the density of water (∼ 10 3 kg.m −3 ), V 1 and V 2 the volumes water occupies in Ξ 1 and Ξ 2 , r 1 is the interior radius for the domain Ξ 1 and a 2 , b 2 are respectively the interior and exterior radii for the domain Ξ 2 . In our case, r 1 = 0.1m, a 2 = 0.15m and b 2 = 0.2m. The formula (21) gives the following values for the first eigenfrequencies (in Hertz) of the multistructures Π ε(1) Ξ (1) et Π ε(2) Ξ (2) : The corresponding frequencies (in Hertz) associated with the standing waves in the periodic structure were obtained numerically, and they are ω * 1 = 979.0268 , ω * 2 = 470.286 (28) Formula (21) gives a good estimate for the eigenfrequency ω * 1 , but we observe a discrepancy in the approximation of ω * 2 . The estimate for the eigenfrequency ω * 2 can be improved if the domain Π ε(2) Ξ (2) is replaced by the domain Π ε(2) Ω (2) Π ε(1) Ω (1) . In this case, the eigenfrequency ω 2 is approximated by the first positive eigenvalue of the problem where V 1 (x), V 2 (x) are the eigenfunctions defined on (0, l 1 ) and (0, l 2 ), respectively, and the masses m 1 , Taking into account that ω 2 = O(ε), we deduce that it can be approximated as the first positive solution of the following algebraic equation: m 1 m 2 l 1 l 2 ω 4 − λω 2 (l 2 I 1 m 1 + I 1 l 2 m 2 + I 2 m 1 l 1 ) + λ 2 I 1 I 2 = 0 (36) so that ω 2 ∼ 520.121Hz, which provides a more accurate approximation of ω * 2 = 470.286H.

V. CONCLUSION
In this paper, we have seen that it is possible to sculpt the Bloch spectrum of three-dimensional phononic crystals ad libitum simply by digging some holes and adding cavities in rigid spheres periodically arranged along a cubic lattice. One of the main achievements of the numerical study is the appearance of ultra-low frequency stop bands at frequencies predicted quantitalively by an asymptotic model. We also conducted sozme basic shape optimization (by varying the size, diameter and number of channels in a rigid sphere of constant radius) in order to enhance the control of the location and the number of low frequency stop bands, thanks to our asymptotic estimates. Importantly, a cubic array of rigid spheres does not support any complete stop band, even in the densely packed configuration. Our findings are thus twofold: Multistructures open not only an original route toward ultra-low frequency stop bands (associated with very flat dispersion curves i.e. localized eigenmodes) but also offer the first paradigm of a complete stop band for threedimensional pressure waves in a fluid. Finally, we illustrate in figure 12 a possible application of the ultra-low frequency stop band in order to localize a mode of a wavelength much larger than the pitch of the array of resonators. Similarly, one could envisage to reflect, detour, or focus, pressure waves using the low frequency stop band within which effective parameters are expected to take negative values as it is now well-established for the two-dimensional counterpart of such kind of acoustic metamaterials.