Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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On the universality of the frequency spectrum and band-gap optimization of quasicrystalline-generated structured rods

Lorenzo Morini

Lorenzo Morini

School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, Wales, UK

[email protected]

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Zafar Gokay Tetik

Zafar Gokay Tetik

School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, Wales, UK

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Gal Shmuel

Gal Shmuel

Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

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Massimiliano Gei

Massimiliano Gei

School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, Wales, UK

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    The dynamical properties of periodic two-component phononic rods, whose elementary cells are generated adopting the Fibonacci substitution rules, are studied through the recently introduced method of the toroidal manifold. The method allows all band gaps and pass bands featuring the frequency spectrum to be represented in a compact form with a frequency-dependent flow line on the surface describing their ordered sequence. The flow lines on the torus can be either closed or open: in the former case, (i) the frequency spectrum is periodic and the elementary cell corresponds to a canonical configuration, (ii) the band gap density depends on the lengths of the two phases; in the latter, the flow lines cover ergodically the torus and the band gap density is independent of those lengths. It is then shown how the proposed compact description of the spectrum can be exploited (i) to find the widest band gap for a given configuration and (ii) to optimize the layout of the elementary cell in order to maximize the low-frequency band gap. The scaling property of the frequency spectrum, that is a distinctive feature of quasicrystalline-generated phononic media, is also confirmed by inspecting band-gap/pass-band regions on the torus for the elementary cells of different Fibonacci orders.

    This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

    1. Introduction

    In the last 50 years, the investigation of wave propagation in structured media and their applications in different areas of engineering have attracted significant interest from the scientific community. In this context, the contribution of Prof. Slepyan and his collaborators was essential for understanding and predicting several phenomena, in particular, transition waves in periodic and bistable structures [14], interaction between surface modes and fractures [5,6], dissipation and phase transition in lattice materials [79] and solitary nonlinear waves [10,11]. These fundamental studies, together with the results obtained by other authors [1215], have inspired a very active field of research, concerning the design of phononic structures with the aim of achieving and controlling non-standard wave propagation phenomena, such as wave focusing [16], frequency filtering [17], cloaking [18,19] and negative refraction [20,21]. Recently, the intriguing dynamical properties of a class of two-phase periodic structured solids, whose unit cells are generated according to the Fibonacci substitution rule, have been presented [22,23]. This particular family of composite structures belongs to the subset of quasicrystalline media [24] and the portion of Floquet–Bloch frequency spectra of its members are characterized by a self-similar pattern which scales according to the factors linked to the Kohmoto's invariant of the family itself [25].

    This work provides new insights on the relationship between the geometrical and constitutive properties of the elementary cells and the layout of pass bands/band gaps for the same type of quasicrystalline-generated phononic rods. By considering harmonic axial waves, we show that the corresponding frequency spectrum can be represented on a two-dimensional toroidal manifold similar to that introduced in [26,27] to study Floquet–Bloch waves in periodic laminates. This manifold is universal for all two-phase configurations and the dispersion properties of the concerned rod can be inferred from the features of the frequency-parametrized flow lines lying on the toroidal domain, which is composed of band gap and pass-band regions. We identify a particular subclass of rods whose flow lines on the torus are closed, thus describing a periodicity in the spectrum at an increasing frequency, and show that the subclass coincides with that of the so-called canonical structures introduced by Morini and Gei [23]. The local scaling governing the pass band/band gap layout about certain relevant frequencies (i.e. the canonical frequencies) is confirmed and highlighted through the analysis of the flow lines on the torus.

    The universal representation of the spectrum on the toroidal surface allows us to rigorously estimate the band gap density for rods of any arbitrary Fibonacci elementary cell. We find that for canonical configurations, this quantity varies with the ratio between the lengths of the phases, corresponding to the slope of the flow lines. Conversely, for generic non-canonical rods, the band gap density is independent of the lengths of the cells and is defined by the ratio between the area of the band gap subdomain and the total surface of the torus [2830]. The provided examples show that this ratio can be easily evaluated numerically.

    We further demonstrate how the compact representation of the spectrum on the two-dimensional torus can be exploited to either optimize the design of the elementary cells to achieve the widest low-frequency band gap or to determine rigorously where the maximal band gap is located in the spectrum for a given configuration. In the examples that we report, we have based this investigation on analytical expressions of the boundaries of band gap regions that can be easily obtained for low-order elementary cells. Unlike the standard procedure based on partial evaluation of the spectrum [3133], the proposed optimization strategy provides exact rigorous results, and it can be easily generalized to Fibonacci cells of higher order.

    2. Waves in quasicrystalline-generated phononic rods

    We introduce a particular class of infinite, one-dimensional, two-component phononic rods consisting of a repeated elementary cell where two distinct elements, say L and S, are arranged in series according to the Fibonacci sequence [24]. The repetition of such a cell implies periodicity along the axis and then the possibility of applying the Floquet–Bloch technique in order to study harmonic wave propagation. The two-component Fibonacci sequence is based on the following substitution rule [34]

    Expression (2.1) implies that the ith (i = 0, 1, 2, …) element of the Fibonacci sequence, here denoted by Fi, obeys the recursive rule Fi=Fi1Fi2, where the initial conditions are F0=S and F1=L (in figure 1, elementary cells designed according to sequences F2, F3 and F4 are displayed).1 The total number of elements of Fi corresponds to the Fibonacci number n~i given by the recurrence relation n~i=n~i1+n~i2, with i≥2, and n~0=n~1=1. The limit of n~i+1/n~i for i corresponds to the so-called golden mean ratio (1+5)/2.
    Figure 1.

    Figure 1. Elementary cells for infinite Fibonacci rods based on F2=LS, F3=LSL and F4=LSLLS. Symbols r and l denote right-hand and left-hand boundaries of the cell, respectively.

    Further in the text, we will refer to those structured rods as Fibonacci structures. According to the general criterion for the classification of the one-dimensional quasiperiodic patterns proposed in [35], these structures are quasicrystalline. Quasicrystalline media possess characteristic features that make them an intermediate class between periodic ordered crystals and random media [36,37]. An example of these interesting and intriguing properties is the self-similarity of the frequency spectrum [23]. The focus of this paper is on the analysis of the universal structure of this spectrum and on its application to predict, modulate and optimize the corresponding stop/pass band layout. We will show that the universality of the spectrum is closely related to the properties of the Floquet–Bloch dispersion relation exploited in [22] and summarized in this Section.

    Let us introduce the mechanical and geometric parameters of elements L and S. The lengths of the two phases are indicated with lL and lS, while AX, EX and ϱX (X∈{L, S}) denote cross-section area of each bar, Young's modulus and mass density per unit volume of the two adopted materials. For both elements, we define the displacement function and the axial force along the rod as u(z) and N(z) = EAu′(z), respectively, where z is the coordinate describing the longitudinal axis. The governing equation of harmonic waves in each section assumes the form

    where ωR+ is the circular frequency (simply the ‘frequency’ in the following) and the term ϱX/EX corresponds to the reciprocal of the square of the speed of propagation of longitudinal waves in material X. The solution of (2.2) is given by
    where CX1 and CX2 are integration constants, to be determined by the boundary conditions.

    To obtain the dispersion diagram of the periodic rod, displacement ur and axial force Nr at the right-hand boundary of the elementary cell have to be given in terms of those at the left-hand boundary, namely ul and Nl (figure 1), as

    where Uj = [uj Nj]T (j = r, l) and Ti is a transfer (or transmission) matrix of the cell Fi. This matrix is the result of the product Ti=p=1n~iTX, where TX (X∈{L, S}) is the transfer matrix, which relates quantities across a single element, given by

    Transfer matrices Ti have some important properties that can be exploited: (i) they are unimodular, i.e. detTi=1, and (ii) follow the recursion rule

    with T0 = TS and T1 = TL.

    The Floquet–Bloch theorem implies that Ur=exp(ikLi)Ul, where Li is the total length of the fundamental cell Fi and the imaginary unit appearing in the argument of the exponential function should not be confused with the index i. By combining this condition with (2.4), we obtain the dispersion equation

    The solution of equation (2.7) provides the complete Floquet–Bloch spectrum and allows us to obtain the location of band gaps and pass bands associated with the infinite rods here considered.

    Equivalently, we can study the dispersion properties of these structures by evaluating the eigenvalues of the transfer matrix. As Ti is unimodular, it turns out that the characteristic equation of the waveguide is given by

    By substituting eikLi = λ in equation (2.8) and multiplying it by e−ikLi, the condition eikLi + e−ikLi − trTi = 0 is achieved, leading to
    where ηi = trTi/2.

    By observing equation (2.9), we can easily deduce that all the information concerning harmonic axial wave propagation in a Fibonacci structure is contained in the half trace ηi of the corresponding transfer matrix. Waves propagate when |ηi| < 1 (kLiR{x:x=hπ,hZ}), band gaps correspond to the ranges of frequencies where |ηi| > 1 (k is a complex number with a non-vanishing imaginary part), whereas |ηi| = 1 characterizes standing waves (kLi{x:x=hπ,hZ}).

    We note that both the transfer matrix (2.5) and the dispersion relation (2.9) possess a form identical to that derived in [38,39] and used in [26,27,40] to study antiplane shear waves in periodic two-phase, multiphase and quasicrystalline laminates, respectively. Further in the paper, we will exploit this mathematical analogy generalizing the approach proposed in [26] to study the universal structure of the frequency spectrum of Fibonacci phononic rods.

    3. Universal structure of the frequency spectrum

    The analysis of the universal structure of the frequency spectrum will take advantage of the introduction of the following variables [26,2830]:

    The unimodularity property of Ti, together with the relationship (2.6), implies the following recursive rule for the half trace ηi+1 [23]:
    where the initial conditions are

    The quantity

    quantifies the impedance mismatch between the phases L and S, and it depends on their constitutive parameters but not on lengths of the single elements L and S. When γ = 1 there is no contrast between phases and the waveguide behaves as a homogeneous one. Expressions (3.3) show that for any given value of γ, η0, η1 and η2 are 2π-periodic functions of ζS and ζL. The generic half trace ηi can be derived by means of successive iterations of the recursive formula (3.2) by assuming (3.3) as initial conditions. Therefore, at any order i, ηi is also a 2π-periodic function, separately, of ζS and ζL as it is defined through sums and products of functions with the same period. This implies that we can consider the half trace ηi as a function of a two-dimensional torus of edge length 2π, whose toroidal and poloidal coordinates are ζS and ζL, respectively. This function is independent of the lengths of the two phases L and S. The toroidal domain is composed of two complementary subspaces that are associated with the two inequalities introduced earlier in the discussion after equation (2.9), namely |ηi(ζS, ζL)| < 1 identifies a pass-band subdomain, whereas |ηi(ζS, ζL)| > 1 corresponds to a band-gap one. The two regions might not be simply connected and the collection of lines of separation between the two subdomains, in which |ηi(ζS, ζL)| = 1, denotes a standing wave solution. The measures of the two regions are univocally determined by the value of the parameter γ.

    A sketch of the toroidal domains for cells F2 and F3 is displayed in figure 2a,b where the set of physical properties tabled in table 1 have been assumed (for that choice, γ ≈ 2.125). In both plots, the pink zone corresponds to the pass-band region, whereas the band-gap one is painted in grey.

    Figure 2.

    Figure 2. (a,b) Toroidal domains of edge length 2π for Fibonacci cells F2 (a) and F3 (b) with γ ≈ 2.125. The pass-band regions where |η2| < 1 and |η3| < 1 are depicted in pink. The band-gap ones (|η2| > 1 and |η3| > 1) are highlighted in grey. An example of a periodic, closed flow line is reported in blue in each panel. (c,d) Square identification of the π-periodic torus for cells F2 and F3; light blue, light red and light brown regions correspond to the subdomains D2(γ) and D3(γ) defined for γ ≈ 8.031 (AS/AL = 0.0625), 2.125 (AS/AL = 0.25) and 1.170 (AS/AL = 0.5625), respectively. Red dots denote the intersection of the flow lines with the boundary of Di for the case γ ≈ 2.125. (e,f ) Dispersion diagrams for Fibonacci cells F2 (e) and F3 (f ) with γ ≈  2.125 (AS/AL = 0.25) and values of other the mechanical and geometrical parameters reported in table 1. (Online version in colour.)

    Table 1. Mechanical and geometrical parameters adopted in the numerical calculations.

    ES = EL = 3.3 GPa ϱS = ϱL = 1140 kg m−3 AL = 4AS = 1.963 × 10−3 m2 lL = 0.07 m

    Equation (2.9) shows that |ηi(ζS, ζL)| is invariant under the transformation

    so that, as pointed out in [26], the map on the torus can be equivalently represented on a reduced π-periodic torus. The latter can be conveniently represented through the so-called square identification [41], in which the curved domain is flattened and transformed to a square whose edges are still described by coordinates ζS and ζL, both ranging now between 0 and π. In the new square representation, the band-gap subdomain (|ηi(ζS, ζL)| > 1) is denoted by Di(γ). In the following, the square equivalent π-periodic torus with the domain Di(γ) will be indicated with Ti. At times, we will also refer to it as the ‘reduced torus’ for the cell Fi.

    In figure 2c,d, the reduced tori T2 and T3 are reported. The light blue, light red and light brown regions in both plots denote the subdomains D2(γ) and D3(γ) determined for γ ≈ 8.031,  2.125 and 1.170, respectively. In particular, the light red ones are the representation of the band gap domains depicted in grey on the original 2π-periodic tori reported just above in the same figure 2a,b, respectively.

    The spectrum for a Fibonacci rod of any arbitrary order can, therefore, be studied by analysing the dynamic flow parametrized ζ(ω) = (ζS(ω), ζL(ω)) on the corresponding reduced torus, where the frequency ω plays the role of a time-like parameter. This flow is the image on Ti of the trajectories described by the angles ζS and ζL on the original torus. Two examples of the latter are the blue lines reported in figure 2a,b. In order to represent these flow lines on Ti, we interpret expression (3.1) as the equation of a rectilinear trajectory lying on the square. Now, for any arbitrary Fibonacci cell Fi for which a specific indication for lengths lL and lS is provided, we can depict the trajectory (3.1) on Ti as those illustrated for F2 and F3 in the two plots of figure 2c,d. For this purpose, if we consider values of the frequency such that ϱX/EXωlX>π, by recalling the invariance of Ti and of its subdomain Di(γ) with respect to transformations (3.5), expression (3.1) can be written in the transformed form as

    Consequently, the trajectory (3.1) reported on Ti appears as a set of parallel segments as those reported in blue in figure 2 c,d, and the flow ζ(ω) can be expressed as
    The segments shown in figure 2c,d are the images of the flow lines illustrated in figure 2a,b, respectively. By examining these lines, we can easily observe that they trace a closed trajectory on the torus. In the next section, the class of structures, whose spectra are described by this particular type of flow lines, is defined and characterized in detail.

    The values of ω, for which the lines of the flow (3.7) intersect the boundary of the subdomain Di(γ), coincide with the extremes of the band gaps. These intersections are highlighted with red points in figure 2c,d for waveguides generated by F2 and F3 for γ ≈ 2.125. The same band gaps are illustrated in the classical dispersion diagrams of figure 2e,f .

    A parametric equation for the flow lines on Ti is easily derived from equation (3.6)

    and the angular coefficient
    defines the direction of the flow (i.e. the slope of the blue segments shown in figure 2c,d). In particular, the segment emerging from the origin for ω = 0+ (i.e. m = n = 0) has equation ζL(ω) = βζS(ω). In the next section, we discuss how rational and irrational values of ratio (3.10) are associated with Fibonacci rods possessing periodic and non-periodic spectra, respectively, corresponding to closed and open trajectories on the 2π-periodic torus, respectively. Both these two different behaviours are studied by analysing the flow lines on Ti. Relevant indications concerning the band gap density and the different properties of rods with periodic and non-periodic spectra are obtained by using this universal approach.

    4. Analysis of the flow lines on the reduced torus

    Let us analyse the different types of trajectories (lines) that can describe the flow ζ(ω) on the torus. The condition for closed periodic lines is the existence of a frequency interval Ω such that [41]

    By combining expressions (4.1) with equation (3.1), we derive the relationships
    and then the ratio

    We can deduce from expression (4.3) that the trajectories on the torus are periodic if the ratio β is a rational number. This condition is exactly the same as that introduced in [23] and necessary to realize Fibonacci structures with a periodic spectrum, which are called in that article canonical structures. Therefore, canonical configurations correspond to closed flow trajectories on the torus. Considering the original 2π-periodic torus, these are closed helicoidal orbits on the surface as those reported in figure 2a,b. The two whole numbers j and q represent the number of cycles, namely 2π rotations, about the toroidal and poloidal axes. As an example, both blue trajectories of figure 2a,b correspond to j = 1 and q = 2 and then to β = 2. On Ti, the closed flow lines associated with canonical structures become a finite number of parallel segments. The periodicity of the dispersion diagram is verified in figure 2e,f where the band gap limits already highlighted in the companion graphs plotted above (i.e. (c) and (d), respectively) are marked with red points.

    In figures 3, 4 and 5, examples of periodic flow lines for canonical structures generated by the repetition of cells F2, F3 and F4 are reported. For the calculations, we considered two phases S and L of the same material (ES = EL and ϱS = ϱL, see table 1) so that parameters γ and β become

    As a consequence, the areas of subdomains Di2 depend only on ratio AS/AL, while the direction of flow is defined by lS/lL. Moreover, according to the classification provided in [23], the analysed rods belong to the second family of canonical configurations.
    Figure 3.

    Figure 3. Half-trace function (a) and flow lines on diagram T2 (b) for a F2 canonical Fibonacci rod characterized by the parameters listed in table 1 and lS/lL = 1/2 (γ ≈ 2.125, β = 2). Coloured dots in both panels mark the extremes of the band gaps. (Online version in colour.)

    Figure 4.

    Figure 4. Half-trace function (a) and flow lines on diagram T2 (b) for a F3 canonical Fibonacci rod characterized by the parameters listed in table 1 and lS/lL = 1/2 (γ ≈ 2.125, β = 2). Coloured dots in both panels mark the extremes of the band gaps. (Online version in colour.)

    Figure 5.

    Figure 5. Half-trace function (a) and flow lines on diagram T2 (b) for a F4 canonical Fibonacci rod characterized by the parameters listed in table 1 and lS/lL = 1/2 (γ ≈ 2.125, β = 2). Coloured dots in both panels mark the extremes of the band gaps. (Online version in colour.)

    In the plots on the left-hand side of each of figures 35, diagrams are presented of the half traces η2, η3, η4 reported as functions of ω for an interval of frequencies which coincides with the half period of the spectrum. We use coloured dots to earmark the extremes of the intervals where |η2|, |η3|, |η4| > 1, defining the band gaps. The flow lines on T2, T3 and T4 are reported on the right-hand side of each figure. Their intersection with the boundaries of D2, D3 and D4, which identify the extreme of the band gaps, are indicated with the same coloured dots. We used the same colour cod in both diagrams of traces and Ti in order to associate the corresponding band gap in the two different representations. We note that the flow diagrams in Ti highlight all the band gaps contained in the half period of the canonical structures, and then the successive band gaps can be visualized using the same finite number of segments on Ti and applying the transformation (3.6). Therefore, for canonical structures generated by any arbitrary cell Fi, the band gap density φi is given by the ratio between the measure of the intersections between the flow lines and the subdomain Di, and the total length of the flow lines. The latter is given by the sum of all the parallel segments reported in figures 35 and corresponding to j2+q2π. This ratio depends on both the area of Di and the direction of the flow lines, and then on both γ and β parameters.

    The values of the band gap density for three different examples of canonical structures with elementary cells from F2 to F8 are reported in figure 6. We assumed the same constitutive properties used for the results shown in figures 35 (table 1) and three different ratios lS/lL which, in this particular case, correspond to three values of β, namely 1, 2 and 7/2 (see equation (4.4)2). According to the definition provided in [23], those three ratios are associated with canonical structures which belong to the first, the second and the third family. The three families are distinguished by different stop and pass band layouts, but they all possess periodic spectra with properties depending exclusively on β. Figure 6 shows that the value of the band gap density is different for cells of the same order i, but with distinct values of the parameter β. This confirms, as we have already mentioned, that the band gap density of canonical rods depends on the ratio lS/lL. As a consequence, if we assume given constitutive properties of the phases S and L (i.e. ES, EL, ϱS and ϱL) and given cross-sections AL and AS, and then we determine univocally the domain Ti and the area of the subspace Di, we can modulate the band gap density by simply varying the ratio lS/lL. Indeed, by changing this parameter, we assign a different direction to the flow lines on the torus or equivalently to the slope of the segments on the square identification of Ti, determining the band gap intervals which coincide with intersections of the flow trajectories with the subdomain Di.

    Figure 6.

    Figure 6. Band-gap density reported for Fibonacci canonical rods designed according to elementary cells F2 to F8 whose constitutive properties are listed in table 1 (γ ≈ 2.125). Three different values of the ratio lS/lL are assumed: 1, 1/2, 2/7, corresponding to β = 1 (Family no. 1), β = 2 (Family no. 2) and β = 7/2 (Family no. 3), respectively. (Online version in colour.)

    By observing figure 6, we note that, for all the three types of canonical rods here analysed, the band gap density increases with the index i following a logarithmic trend. This is in agreement with the results presented in [42,43] for electronic and optic systems subjected to quasiperiodic Fibonacci potentials.

    In addition to the canonical ones, we can define a different class of waveguides whose ratio β is irrational. In this case, the spectrum is not periodic and the corresponding flow lines are open and cover ergodically the whole torus with uniform measure [44]. In this situation, it is commonly said that the orbits are dense on the torus [45]. Consequently, the flow trajectories on Ti consist of an infinite number of parallel segments which, in turn, cover ergodically the whole square domain. Therefore, the band gap density is given by the ratio between the area of the subdomain Di and the area π2 of the square. Since the measure of Di is determined only by the parameter γ, which is independent of the ratio lS/lL, for non-canonical rods the band gap density does not depend on that ratio.

    The fundamental differences between the flow lines of a canonical waveguide and those of non-canonical one are pointed out in figure 7. Figure 7a,b display the variation of the half trace η2 with the frequency and the trajectories on the reduced torus T2 for a canonical structure with parameters listed in table 1 and lS/lL = 1/2, the same considered in figure 3. The variation of η2 is plotted for a frequency range equal to its period (0 < ω≲305 krad s−1). The corresponding extremes of band gaps both in the half-trace diagram and in T2 are marked using points with the same colours. As anticipated, due to the periodicity of the flow lines, all band gaps and pass bands in the frequency spectrum can be represented through the two parallel segments reported in figure 7b. Indeed, by observing this figure, the first and the third band gap, whose extremes are denoted by red and green points, respectively, overlap as well as the second and the fourth ones whose extremes are marked with magenta and yellow points, respectively.

    Figure 7.

    Figure 7. Half-trace diagrams and flow lines on T2 associated with cells F2 characterized by the parameters listed in table 1 and (a,b) lS/lL = 1/2, (c,d) lS/lL=1/2+1/500, (e,f ) lS/lL=1/2+31/500, (g,h) lS/lL=1/2+101/500. In coloured dots, mark the extremes of homologous band gaps. (Online version in colour.)

    The three pairs of figures 7c,d, 7e,f , 7g,h illustrate the diagrams of the half trace η2 and the flow lines on T2 for cells F2 with lS/lL=1/2+1/500, lS/lL=1/2+31/500 and lS/lL=1/2+101/500, respectively. We assumed three different perturbations of the length ratio in order to have three irrational values of β and then three examples of non-canonical configurations. Their spectra are studied in the same range of frequencies of the canonical waveguide in figure 7a,b. We observe that for the three irrational ratios the half trace η2 is no longer periodic, and the number of band gaps in the same frequency range increases with respect to the canonical case. Due to the lack of periodicity, band gaps are characterized by widths and relative distances that are all different from each other. This implies that the representation of each of them on T2 is associated with a different parallel segment, as shown in figure 7d,f ,h. These segments are the images on the reduced torus of the three flow lines, which, in this case, are infinite. At an increase of the frequency range for the half traces in figure 7c,e,g, more and more segments are needed in order to depict the set of band gaps on the right-hand counterparts (figure 7d,f ,h, respectively), up to cover the whole domain of T2. Therefore, for all the three non-canonical rods analysed, it is confirmed that the band gap density φi is given by the ratio between the area of D2 and π2. In general,

    Unlike canonical structures, this value is univocally determined by the parameter γ and is independent of lS/lL.

    We can now generalize the analysis provided for waveguides generated by F2 to any arbitrary Fibonacci cell Fi. In analogy with the previous examples, we consider two phases with the same properties (table 1) and lS/lL=3/10 and lS/lL=1/2, corresponding to β=10/3 and β=2. We solve numerically the dispersion relation (2.9) over increasing intervals of frequencies, and at each iteration we estimate the ratio between the total length of the band gaps and the whole length of the frequency range. Calculations are carried out for structures designed according to cells F2, F3, F4 and F5; the results are shown in figures 8 and 9. Red lines with circle marks and blue lines with square marks map the band gap density for lS/lL=3/10 and lS/lL=1/2, respectively. For both cases, and in each panel, we note the convergence of the data to the black horizontal line that corresponds to φi in (4.5). These ratios can be estimated numerically or analytically for cell F2 (see explicit expression derived in [26]), and in this case they are 0.5090 for F2, 0.5098 for F3, 0.5938 for F3 and 0.6334 for F5. The convergence observed for all panels in figures 8 and 9 demonstrates that for non-canonical structures the band gap density at a given value of γ is independent of the lengths of the phases S and L. Therefore, we can state that the band gap density is a universal property of classes of non-canonical waveguides characterized by a prescribed γ and an elementary cell Fi. This is in agreement with the results reached in [26], where it is shown that for irrational values of a parameter analogous to our β the band gap density of two phase laminates is independent of the thicknesses of the layers.

    Figure 8.

    Figure 8. Numerical study of convergence of the band-gap density for non-canonical structures with elementary cells F2 and F3 whose properties are listed in table 1 (γ ≈ 2.125). We assumed two irrational values for the length ratio, i.e. lS/lL=3/10 (red circle markers) and lS/lL=1/2 (blue square markers).

    Figure 9.

    Figure 9. Numerical study of convergence of band-gap density for non-canonical structures with elementary cells F4 and F5 whose properties are listed in table 1 (γ ≈ 2.125). We assumed two irrational values for the length ratio, i.e. lS/lL=3/10 (red circle markers) and lS/lL=1/2 (blue square markers). (Online version in colour.)

    5. Band gap optimization using universality properties

    The compact representation of the frequency spectrum on Ti is now used to formulate rigorously and solve two types of optimization problems in periodic quasicrystalline-generated rods. We focus on the case of F3 for which analytical representations of the boundaries of the band gaps are available, but the same approach can be easily applied to higher-order cells with the aid of implicit expressions similar to those obtained in [27].

    The band gap subdomain D3 is composed of two identical regions for any values of the parameter γ: one, namely D3, lies on the portion of T3 delimited by the intervals 0 ≤ ζS ≤ π and 0 ≤ ζL ≤ π/2; the other, D3+, occupies the portion delimited by the intervals 0 ≤ ζS ≤ π and π/2 ≤ ζL ≤ π (figures 2d and 4b). The former is considered for the maximization of gap width, but the same methodology can be applied to D3+. All points of the boundary of D3 satisfy the condition η3(ζS, ζL) = − 1 and define the curves Cl and Cu whose analytical expressions are

    where the upper curve Cu (lower one Cl) corresponds to the plus (minus) sign in the numerator. The width of the generic band gap {ωB − ωA} is related to the length of the associated interval along the flow line, whose endpoints A(ζAS, ζAL) and B(ζBS, ζBL) lie on Cu and Cl, respectively, through the relationship
    where νS=ES/(ϱSlS). An equation analogous to (5.2) is obtained in [26], where it is used to derive exact expressions for the bounds of the band-gap widths in two-phase laminates as functions of the geometrical and physical properties of the unit cells. Since points A and B belong to the flow lines on T3, their coordinates satisfy the relationships
    where ζX = ζX(ω). Equations (5.2) and (5.3), together with expressions (5.1) for the curves Cl and Cu, enable us to maximize the width of {ωB − ωA} through the flow lines defined on the basis of the physical and geometrical properties of the elementary cells.

    (a) Identification of the widest band gap for a prescribed structure

    We first consider a given cell F3 with prescribed physical and geometrical properties. Our purpose is to determine the interval {ωB − ωA} defining the widest band gap in the frequency spectrum of the structure. As β = (ζBL − ζAL)/(ζBS − ζAS), expression (5.2) can be written in the following form:

    In this case, νS and β are known and the goal is achieved by finding the value of the translation coefficient α associated with the largest ΔζS. By imposing that both the points A and B lie on the flow line (5.3) and that ACl and BCu, the following equations for the coordinates ζBS and ζAS are established
    and then ΔζS = ζBS − ζAS becomes
    By eliminating α between (5.5) and (5.6), it turns out that
    The aim is now to determine the values of ζAS and ζBS that maximize the quantity (5.7) and are also a solution of equation (5.8). Then, the corresponding α can be evaluated by means of (5.5) and (5.6). The problem can be solved graphically for any cell F3 through the two diagrams reported in figure 10. For the calculations, we considered a non-canonical configuration with the parameters listed in table 1 and lS/lL=1/2+31/500.
    Figure 10.

    Figure 10. Widest band gap for a non-canonical cell F3 designed assuming the parameters listed in table 1 and lS/lL=1/2+31/500. (a) Contour plot of the function ΔζS(ζAS, ζBS). Red, blue and black lines are associated with equations Δα(ζAS, ζBS) = 0, ΔζS(ζAS, ζBS) = 0.589 and ζBS + ζAS = π, respectively. (b) Graphic solution of system including equations (3.9) and (5.13); red, blue and black lines correspond to equations (5.13)1, (5.13)2 and (3.9), respectively; the green dot is placed at n = 6, m = 10. (Online version in colour.)

    The contour plot in figure 10a shows the variation of the function (5.7) on the whole two-dimensional domain 0 ≤ {ζAS, ζBS} ≤ π, while the red line reported in the same figure is determined by the values of ζAS and ζBS satisfying equation (5.8). Point P, whose coordinates are the solution to (5.8) and maximize ΔζS, is denoted by the yellow dot. It corresponds to the intersection between the red line and the blue curve, defined in this case through the equation ΔζS = 0.589. We note that this point also coincides with the intersection between the curve (5.8) and the line ζBS = π − ζAS. Consequently, the coordinates ζAS and ζBS can be derived as the solution of the system

    By substituting (5.9)2 into (5.9)1, we obtain
    For the set of physical and geometrical properties assumed in the example, the solution of (5.10) is ζAS = 1.278 and ζBS = 1.863. Using these values in equation (5.5) (or (5.6)), α = 1.692 is determined. Remembering that in this case lS/lL=1/2+31/500=1/β, equation (5.3) provide ζAL = 0.324 and ζBL = 1.246.

    We determined the translation coefficient of the flow segment corresponding to the widest band gap among all those detected in the spectrum of the structure, as well as the coordinates on T3 of the points A(ζAS, ζAL) and B(ζBS, ζBL), associated with ωA and ωB. A and B are denoted by red dots in figure 11b, and the width ωB − ωA = 22.066 krad s−1 can be calculated through equation (5.4). On the basis of the definition (3.6), ωA and ωB are given by

    or, alternatively,
    where n and m are two whole numbers satisfying condition (3.9).
    Figure 11.

    Figure 11. Widest band gap for a non-canonical cell F3 designed assuming the parameters listed in table 1 and lS/lL=1/2+31/500, highlighted both on the diagram of the half trace η3 (a) and on the reduced torus T3 (b). The extremes of the gap, namely ωA = 759.69 krad s−1 and ωB = 781.76 krad s−1 (a) and A(ζAS, ζAL) and B(ζBS, ζBL) on T3 (b), are marked with red points. (Online version in colour.)

    The invariance of T3 and D3 with respect to the transformations (3.5), together with the conditions ACl and BCl+, provides the following system of implicit equations

    The values of n and m corresponding to the extremes ωA and ωB of the maximal band gap are given by a pair of integer solutions of system (5.13) that satisfies the relationship (3.9). They can be found through a diagram like the one reported in figure 10b, where the solutions of equation (5.13)1 and (5.13)2 correspond to the red and blue contours, respectively, and the black line is defined by equation (3.9). The green dot denotes the intersection of the three curves at n = 6 and m = 10, which are the required numbers in this case. By substituting them together with the previously calculated ζAS, ζBS, ζAL, ζBL and the physical properties of the cell in expressions (5.11) and (5.12), we finally determine ωA = 759.69 krad s−1 and ωB = 781.76 krad s−1. These extremal values are highlighted using the red dots in the diagram of the half trace η3 reported in figure 11a.

    The illustrated method can be easily applied to cells of higher order through the general approach developed in [27], where analytical expressions for the boundaries of the band gap subregions of periodic laminates with an arbitrary number of phases are derived. This original procedure provides several fundamental advantages with respect to the standard optimization methods based on the numerical evaluation of the frequency spectrum. This is obvious especially in the case of non-canonical structures as this is the case where the spectrum is not periodic, and then, in principle, calculations over an infinite frequency domain should be performed to determine the widest band gap. Since, in practice, calculations must be truncated, such an approach yields only an approximate solution. Moreover, there is currently any rigorous way to predict how the considered truncated subdomain allows an accurate estimation compared to the real infinite case. Contrarily, through the formulation over the torus T3, the problem is solved in closed form, without any approximation, avoiding the numerical calculations required by the evaluation of large portions of the frequency spectrum. It is also worth remarking that the solutions m and n can be relatively high, at a frequency for which, due to the effects of lateral inertia, the simple one-dimensional axial model might be no longer valid.

    (b) Optimization of the lowest band gap through variation of the geometrical properties

    The second example of optimization, which can be formulated rigorously and solved by exploiting the representation of the spectrum on T3, is here illustrated. Let us consider a cell F3 with parameters listed in table 1, such that γ ≈ 2.125 and the slope of the flow lines becomes β = lL/lS. Our aim is now to find the value of β that maximize the lowest band gap of the spectrum.

    This one, i.e. {ωB − ωA}, is detected by the intersection between the region D3 and the flow segment starting from the origin of the plane SζL. Similarly to the case studied in §a, A(ζSA,ζLA)Cl and B(ζSB,ζLB)Cl+, and

    since α = 0 in this problem. Equation (5.14)1, together with the condition ACl, provides the following expression for β:
    By substituting (5.15) into (5.14)2 and imposing BCu, we obtain
    Assuming that lS, and then νS, is known, the expression for the width of the band gap (5.2) can be written in the normalized form
    where β is given by (5.15), ζAL and ζAL can be expressed as functions of ζAS and ζAS using (5.1). We now have to determine the values of ζAS and ζBS that maximize Δω¯ and are solution of equation (5.16). The problem is solved graphically using the diagram reported in figure 12a. The contour plot herein shows the variation of Δω¯ on the whole two-dimensional domain 0 ≤ {ζAS, ζBS} ≤ π, while the red line reported in the same figure is the plot of equation (5.16). Point Q, whose coordinates are the solution of (5.16) and maximize Δω¯, is denoted by the yellow dot. It corresponds to the intersection between the red line and the blue contour, the latter defined through equation Δω¯=1.46. For the set of constitutive and geometrical parameters here considered, we have ζAS = 1.215 and ζBS = 2.675. By employing these values in equation (5.1), we get ζAL = 0.345 and ζBL = 0.769, and then, eventually, β = (ζBL − ζAL)/(ζBS − ζAS) = 0.284. Therefore, this solution provides the slope of the flow segment corresponding to the widest lowest band gap, and its extreme A(ζAS, ζAL) and B(ζAS, ζAL) on the reduced torus T3 are marked with the red dots in figure 12b. This result is valid for any given value of lL≠0 which is assumed to be known for the calculations, and then the optimization procedure does not depend separately on lengths lS and lL, but only on their ratio β.
    Figure 12.

    Figure 12. Optimization of the lowest band gap for a waveguide designed according to the parameters listed in table 1. (a) Contour plot of the function Δω¯(ζSA,ζSB). Red and blue lines are associated with equation (5.16) and Δω¯=1.46, respectively. Point Q, marked in yellow, corresponds to ζAS = 1.215 and ζBS = 2.675, i.e. the solutions of equation (5.16), and maximize Δω¯. (b) Identification on T3 of the maximal lowest band gap obtained for β = 0.284 whose extremes are points A and B. (Online version in colour.)

    The illustrated method provides an exact solution to the problem of the maximization of the lowest band gap, which is of practical importance in several operative scenarios involving different types of phononic structures (see, e.g. [3133]). The formulation over the reduced torus can be easily extended to the case of an arbitrary cell Fi and represents a promising alternative to the direct approach based on partial evaluation of the frequency spectrum evaluation for all possible ratios lS/lL.

    6. Scaling of the band gaps observed on the reduced torus

    The universal representation of the spectrum on the reduced torus Ti can be exploited to check the local scaling occurring between band gaps at determined frequencies, as shown earlier in [22,23,40] for different types of quasicrystalline phononic structures. Following their approach, let us identify with Ri = (xi, yi, zi) a point whose coordinates correspond to xi = ηi+2, yi = ηi+1 and zi = ηi. On the basis of the recursive relation (3.2), the change of point Ri to Ri+1 can be described as the evolution of the nonlinear discrete map

    We can easily demonstrate (e.g. [22]) that the invariant
    is a constant, independent of index i. It is worth noting that J = I/4, where I is the Kohmoto's invariant defined in [23].

    For any given value of the frequency, and then of the flux variables ζS and ζL, equation (6.2) defines a manifold whose equation in the continuous three-dimensional space Oxyz is x2 + y2 + z2 − 2xyz − 1 = J(ω), the so-called Kohmoto's surface [25]. The points obtained by iterating map (6.1) are all confined on this surface and describe an open, discrete trajectory. Each Kohmoto's surface possesses six saddle points, say ± Pk (k = 1, 2, 3), whose coordinates are ±P1=(±21+J(ω),0,0), ±P2=(0,±21+J(ω),0), ±P3=(0,0,±21+J(ω)). They are connected through a closed (periodic) orbit generated by the six-cycle transformation obtained by applying six times map (6.1), in other words, T6(Pk)=Pk. Moreover, it can be also verified that T3(Pk)=Pk. The frequencies ωc at which a generic Ri coincides with one of these saddle points are called canonical frequencies and are exactly midway of the semi-period of the spectrum of canonical structures [23]. For instance, in the cases addressed in figures 35 and 7, the period is approximately 305 krad s−1 and canonical frequencies are approximately 305/4 krad s−1 and (3/4) 305 krad s−1.

    In the neighbourhood of ωc, the corresponding point Ri locates in the vicinity of a saddle point, therefore the discrete trajectory traced by transformation of the point Ri itself at an increasing index on the Kohmoto's surface is then studied as a small perturbation of the periodic orbit with map (6.1) linearized about the six saddle points. The derived linearized transformation has an eigenvalue that is equal to one and an additional pair of them given by

    In both [22,23], it is shown that the quantity κ+(ω) governs the local scaling occurring between localized ranges of the spectrum of cell Fi and that of Fi+6, while λκ+ is the scaling factor between Fi and Fi+3. In particular, across a canonical frequency, the width of a band gap in the diagram of cell Fi+6 centred at frequency ωc, say {ωVi+6 − ωUi+6}, is related to that of {ωVi − ωUi} in the diagram of cell Fi centred at the same frequency by the following scaling law:
    where κ = κ+(ωc). Similarly, the following relationship can be established between the widths of {ωVi+3 − ωUi+3} and {ωVi − ωUi}:

    As an example, let us consider Fibonacci canonical cells Fi whose parameters are those in table 1 (γ ≈ 2.125) and lS/lL = 1 (β = 1). For this class of structures, local scaling governed by (6.4) and (6.5) at ωc = 37.596 krad s−1 is analysed. The numerical results are illustrated in figure 13, where the band gap associated with F5 is compared with that corresponding to F8 using close-up views of both the diagrams η5 and η8 (respectively, (a) and (c)) and the flow lines on the reduced tori T5 and T8 (respectively, (b) and (d)). The canonical frequency ωc is indicated with green points and the magenta dot-dashed vertical lines on the left-hand sides, while the extremes of the band gaps U and V are denoted by the red points on the right. We note also in this case the perfect correspondence between the band gaps detected through the trace diagrams and the intersections of the flow lines with the subdomains D5 and D8. Concerning the band gap reported in figure 13a,b, numerical calculations yield ωV5 − ωU5 = 3.407 krad s−1 and λ = 18.12. By using the relationship (6.5), the value ωV8 − ωU8 ≈ (ωV5 − ωU5)/λ = 0.188 krad s−1 is obtained, which is in very good agreement with the value provided by direct estimation of the band gap highlighted in figure 13c,d (i.e. ωV8 − ωU8 = 0.186 krad s−1). We record the same scaling behaviour by comparing ωV5 − ωU5 with ωV11 − ωU11 centred at the same ωc. In this case, the scaling factor is κ = κ+(ωc) = 328.25, the actual range ωV11 − ωU11 measures 0.0103 krad s−1, whereas relationship (6.4) provides ωV11 − ωU11 ≈ (ωV5 − ωU5)/κ = 0.0104 krad s−1.

    Figure 13.

    Figure 13. Half-trace diagrams (left) and flow lines on the domain Ti (right) corresponding to Fibonacci rods whose properties are listed in table 1 with lS/lL = 1. Cells F5 (a,b) and F8 (c,d) are considered. The extremes of the band gap centred at the canonical frequency ωc = 37.596 krad s−1, that is indicated with a green point, are marked with red points. (Online version in colour.)

    The proposed example demonstrates how, in addition to the standard representation of the dispersion diagram, the typical scaling properties can be also pinpointed and estimated through the universal representation of the torus.

    7. Concluding remarks

    The characteristic features of the frequency spectrum for elastic waves propagating in a two-phase periodic medium can be revealed through its universal representation on a two-dimensional toroidal surface composed of pass band and band gap subregions. Frequency-dependent flow lines belonging to the surface can be defined for each configuration of the waveguide. In this paper, we exploited this possibility to investigate axial waves for a class of periodic rods whose elementary cell is generated through the Fibonacci substitution rule, an example of quasicrystalline sequence.

    First, we have established the mechanical and geometrical conditions for which an elementary cell of the Fibonacci sequence may display closed flow lines on the torus, a circumstance that corresponds to the periodicity of the frequency spectrum and of the layout of pass bands and band gaps. We concluded that the required combination of parameters corresponds to that leading to the concept of canonical structures introduced by Morini & Gei [23]. For these types of arrangements, it turned out that the band gap density depends on the lengths of the two phases. Conversely, for non-canonical rods, the flow lines cover ergodically the torus and their band gap density is independent of the lengths of the constituents.

    Second, we addressed analytically two illustrative band gap optimization problems, based on element F3 of the Fibonacci sequence. Analytical expressions of the boundaries of band gap regions on the torus were exploited, on the one hand, to guide the design of the elementary cell to achieve the widest low-frequency band gap, on the other, to detect the maximal band gap in the spectrum for a given configuration. Thanks to the availability of the expressions of the boundaries of the band-gap regions on the torus, the proposed optimization technique is considerably more robust in comparison with the standard procedure based on the partial evaluation of the frequency spectrum [3133], which necessarily relies on numerical algorithms.

    In the final section, the local scaling governing the spectrum of quasicrystalline-generated phononic rods about certain relevant frequencies, as revealed in [23], was investigated and confirmed through the analysis of the flow lines on the torus.

    The presented approach, based on the representation of pass band and band gap sub-regions on the toroidal manifold, can be easily extended to study other wave phenomena governed by an equation similar to (2.2) in different periodic systems, i.e. prestressed laminates, photonic crystals and composite nanostructures. Moreover, through the definition of an appropriate set of invariants that fully characterize the pass band/band gap layout, similar universality properties can be detected in spectra associated with different types of equations, such as, for example, those related to flexural systems [46,47], thin soft dielectric films [48] and plane strain laminates [49].

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    Author's contributions

    L.M. and Z.G.T. carried out the numerical calculations. L.M. and M.G. performed the data analysis. L.M., G.S. and M.G. drafted the manuscript. All authors read and approved the manuscript.

    Competing interests

    The authors declare that they have no competing interests.


    L.M. and M.G. were funded by the European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie Actions COFUND grant SIRCIW, agreement no. 663830. G.S. was supported from ISF (grant no. 1912/15) and BSF (grant no. 2014358).


    The authors gratefully acknowledge Dr Ram Band, Department of Mathematics, Technion-Israel Institute of Technology, for their insightful discussions on the research topic of the paper.


    1 Henceforth, the notation Fi will indicate both the sequence and the elementary cell of the structured rod.

    2 From now on the dependency on γ of Di will be dropped.

    One contribution of 12 to a theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

    Published by the Royal Society. All rights reserved.