Abstract

We study nonreciprocity in a passive linear waveguide augmented with a local asymmetric, dissipative, and strongly nonlinear gate. Strong coupling between the constituent oscillators of the waveguide is assumed, resulting in broadband capacity for wave transmission. The local nonlinearity and asymmetry at the gate can yield strong global nonreciprocal acoustics, in the sense of drastically different acoustical responses depending on which side of the waveguide a harmonic excitation is applied. Two types of highly nonreciprocal responses are observed: (i) Monochromatic responses without frequency distortion compared to the applied harmonic excitation, and (ii) strongly modulated responses (SMRs) with strong frequency distortion. The complexification averaging (CX-A) method is applied to analytically predict the monochromatic solutions of this strongly nonlinear problem, and a stability analysis is performed to study the governing bifurcations. In addition, we build a machine learning framework where neural net (NN) simulators are trained to predict the performance measures of the gated waveguide in terms of certain transmissibility and nonreciprocity measures. The NN drastically reduces the required simulation time, enabling the determination of parameter ranges for desired performance in a high-dimensional parameter space. In the predicted desirable parameter space for nonreciprocity, the maximum transmissibility reaches 40%, and the transmitted energy varies by up to three orders of magnitude depending on the direction of wave transmission. The machine learning tools along with the analytical methods of this work can inform predictive designs of practical nonreciprocal waveguides and acoustic metamaterials that incorporate local nonlinear gates.

1 Introduction

Reciprocity is a basic feature in linear time-invariant elastodynamics and acoustics. Following the Maxwell-Betti reciprocity theorem, the response of a linear time-invariant (LTI) elastodynamic system is invariant when switching the points of excitation and measurement [1,2]. A break of acoustic reciprocity enables, e.g., wave propagation only in a preferred direction paving the way for various types of novel acoustic elements such as acoustic diodes [36], rectifiers [7], frequency convertors [8], topological insulators [9], and resonators with reconfigurable bandwidth properties [10]. Specifically, nonreciprocal acoustics have been widely studied in kinetic [11,12], active [13], and nonlinear media [1418], where break of nonreciprocity is achieved by applying external biases, introducing time-variant properties and imposing nonlinearity coupled with asymmetry [19], respectively. Passive nonreciprocal acoustics through application of intentional nonlinearity has been achieved by imposing local asymmetric acoustic elements such as vibro-impactors [3], bilinear oscillators [8], or a hierarchical series of nonlinear oscillators [4]. These local nonlinear elements integrated into an otherwise LTI elastodynamic waveguide can result in global nonreciprocal acoustics without requiring any external source of energy, i.e., in a passive way.

However, a typical feature of nonlinear acoustics is its dependence on energy (or the specific type of the applied excitation), whereas additional features are instabilities and bifurcations that often occur in the acoustics. Hence, it is much more challenging to solve nonlinear problems compared to linear ones. Moreover, nonreciprocal acoustic waveguides can only be predictively designed by analytically or numerically solving the governing nonlinear governing equations, which is often a formidable task, especially when in spatially extended systems that involve many subcomponents. Some examples of investigating nonlinear nonreciprocal acoustics involve asymptotic analysis [20], and the implementation of nonlinear normal modes [21,22], reduced order models [23] or Poincaré maps [24]. However, these methods are only applicable to systems with simplifications such as homogeneous structure (i.e., with no disorder), weak nonlinearity, or possessing local nonlinear elements with a few degrees-of-freedom. In addition, since the acoustics of nonlinear waveguides is dependent on energy, the computational load associated with direct numerical simulations is often prohibitive for predictive design. These limitations of traditional analytical and numerical approaches motivate for an alternative and more efficient way for predictive design of nonlinear nonreciprocal waveguides.

Despite the traditional approaches of acoustical analysis and simulations for ordinary differential equations, machine learning based simulators have attracted the attention of researchers as an alternative way for cost-efficient engineering simulation. Machine learning algorithms build models that can be “trained” (or that they “learn”) by feeding them appropriate data sets. Compared with direct numerical simulations, a trained machine learning simulator is much more efficient at the cost of acceptable errors; hence, machine learning methods have been widely applied to various fields of engineering computation, e.g., in finite elements [25], computational fluid dynamics [26] and particle-based simulations [27]. However, the applications of machine learning techniques in field of nonreciprocal acoustics are still limited. Recently, Wang et al. [28] developed a machine learning model that is applicable to a class of weakly coupled LTI waveguides that were excited by harmonic excitations and were locally augmented by nonlinear “gates.” With local nonlinearity and asymmetry, these weakly coupled waveguides exhibited strong acoustic nonreciprocity with mainly two types of responses, that is, monochromatic and strongly modulated responses (SMRs). The machine learning model correctly predicted the acoustics in a parameter space, enabling the parametric study and optimized design for nonreciprocity and transmissibility. However, the study in Ref. [28] was restricted only to waveguides with weak coupling and weak nonlinearity, so that the nonreciprocity was realized only in a narrow frequency band. Moreover, the group velocity of propagating waves was slow due to the weak coupling, and therefore, the acoustics was sensitive to structural damping. This motivates us to extend the study in Ref. [28] to the case of strong coupled LTI waveguides augmented by local nonlinear and asymmetric gates, supporting broad-band wave transmission and expanding the design space of the system parameters. This is the aim of the present work.

We start with a numerical study to illustrate that the strongly coupled waveguide, when augmented by a local nonlinear asymmetric gate features monochromatic responses and SMRs—as in the weakly coupled case [28]. Then, the bifurcations that generate the nonreciprocal acoustics in this system are discussed. These are based on analytical solutions by employing complexification averaging (CX-A) analysis, which yields closed form solutions for the monochromatic responses and enables stability analysis that reveals the bifurcations governing the transition from nonreciprocal monochromatic responses to SMRs. Extending the study to SMRs, machine learning is employed. To this end, the system and excitation parameters are considered as the inputs of the machine learning model, whereas the objective performance measures (in terms of nonreciprocity and transmissibility) are considered as outputs. An artificial neural net (NN) is trained based on numerical simulation results on a randomly-sampled dataset. The trained NN simulator drastically decreases the computational effort with acceptable errors of prediction. The NN and analytical predictions are then verified by direct numerical simulations. Finally, we combine the NN predicted results and the analytical results to investigate the robust range of system and excitation parameters for strongly nonreciprocal acoustics of the nonlinear gated waveguide.

2 Numerical Investigation of the Non-Reciprocal Acoustics

The considered one-dimensional (1D) nonlinear waveguide is composed of two semi-infinite 1D linear subwaveguides coupled with a nonlinear gate as is shown in Fig. 1. Each semi-infinite subwaveguide is composed of identical grounded damped linear oscillators coupled to their neighbors with linear coupling springs. Based on experimental experience with similar waveguides [29,30], the dissipation effects of the coupling springs are much smaller than the grounding ones and, therefore, are neglected for convenience. In contrast to Ref. [28], the linear coupling stiffnesses in the two subwaveguides are assumed to be strong. The nonlinear gate consists of two linear oscillators coupled through an essential cubic nonlinear spring, with identical masses, damping coefficients but dissimilar grounding stiffnesses, and, hence, different linearized natural frequencies. The nonlinearity and asymmetry in the overall waveguide are exclusively induced through the nonlinear gate, and so, the resulting nonlinear acoustics is expected to be nonreciprocal [20]. A harmonic excitation is applied on one of the oscillators in either one of the linear semi-infinite subwaveguides. Accordingly, the semi-infinite subwaveguide subject to the excitation is designated as the upstream one, while the nondirectly excited subwaveguide (which is initially at rest) is named as the downstream one. The oscillators are ordered sequentially from the gate to the far field, with the oscillators named as En(n=1,2,3,) in the upstream subwaveguide and An(n=1,2,3,) in the downstream subwaveguide. Moreover, the oscillators in the nonlinear gate are named as E0 and A0, connected to the upstream and downstream linear subwaveguides, respectively. The equations of motion are expressed as follows, with all initial conditions assumed to be zero
mẍ0+cẋ0+k1x0+kc(x0y0)3+kL(x0x1)=0mÿ0+cẏ0+k2y0+kc(y0x0)3+kL(y0y1)=0mẍn+cẋn+kxn+kL(xnxn1)+kL(xnxn+1)=2Fpcosωtδ(np),n>0mÿn+cẏn+kyn+kL(ynyn1)+kL(ynyn+1)=0,n>0
(1)

where xn and yn denote the displacements of oscillators En and An with n=1,2,3,. Also, overdot denotes differentiation with respect to the time t, and m, k, c and kl denote the mass, linear grounding stiffness, viscous damping coefficient, and linear coupling stiffness of the oscillators in the linear semi-infinite waveguides, respectively. The grounding stiffnesses of the gate oscillators E0 and A0 are detuned from k and denoted as k1 and k2, respectively, whereas the coupled stiffness of these oscillators is purely cubic (without a linear term) with coefficient kc. The harmonic excitation is applied at oscillator Ep of the upstream subwaveguide, with frequency ω and amplitude 2Fp, and δ denotes the Dirac function indicating that all other oscillators are not excited by an external force.

Fig. 1
Schematic of the strongly coupled subwaveguides with a nonlinear gate
Fig. 1
Schematic of the strongly coupled subwaveguides with a nonlinear gate
Close modal
As mentioned, in this work, the case with strong coupling in the linear subwaveguides is considered, where the linear and nonlinear stiffness coefficients, kL and kc, respectively, are of the same order of magnitude with the grounding stiffness coefficients k, k1, and k2. Therefore, system (1) represents a strongly nonlinear dynamical problem. First, the following normalized system is introduced
x0+2ξx0+(1+dσ1)x0+dα(x0y0)3+d(x0x1)=0y0+2ξy0+(1+dσ2)y0+dα(y0x0)3+d(y0y1)=0xn+2ξxn+xn+d(xnxn1)+d(xnxn+1)=[dApeiω̂τ+cc]δ(np),n>0yn+2ξyn+yn+d(ynyn1)+d(ynyn+1)=0,n>0
(2)
where ω0=k/m denotes the natural frequency of the oscillators in the linear subwaveguides, τ=ω0t is the normalized time, cc is the complex conjugate, prime differentiation with respect to τ, and i=(1)1/2. In addition, d=kL/k denotes the coupling stiffness normalized by the grounding stiffness, where d=O(1) for strong coupling, ξ=c/2mk is the normalized critical viscous damping ratio, ω̂=ω/ω0 is the normalized excitation frequency, Ap=Fp/(dk) is the normalized excitation amplitude, α is the normalized nonlinear stiffness, and σ1,2 is the normalized detuning parameters of the grounding stiffnesses of the gate oscillators E0 and A0, respectively. With the dissipation terms in the subwaveguides ignored, the normalized dispersion relation is given by
Ωω0=1+4dsin2θ2,0θπ
(3)
where Ω is the frequency, θ is the wavenumber, and the limiting values of the frequency define the passband of each linear subwaveguide. By selecting the excitation frequency ω̂ within the normalized passband (3), the semi-infinite subwaveguides can transmit harmonic waves in the broad band range 1ω̂1+4d. Since the dispersion relation (3) is monotonic, for each excitation frequency in the passband, there exists a unique wavenumber θ[0,π] that is excited in the subwaveguides. The nonlinear acoustics of the gated waveguide is then governed by the scattering—transmission and reflection—of these harmonic waveguides at the local gate, which, in turn, depends on the damping ratio ξ, the normalized detuning parameters σ1,σ2, the wavenumber θ corresponding to the excitation frequency, the normalized coupling d, the normalized nonlinear stiffness α and the normalized excitation amplitude Ap; moreover, we assume that σ1σ2 in order to induce asymmetry at the gate. Also, for convenience, we introduce the normalized damping ratio ζ=ξ/d.

As shown in Ref. [28] where weak coupling in the subwaveguides of Fig. 1 was studied, nonreciprocal acoustic responses were realized in the corresponding narrow pass band, with energy transfer allowed in one direction and prevented from the other. Moreover, the nonreciprocal responses were in either monochromatic or strongly modulated. We show here that similar responses can be realized in the strongly coupled nonlinear waveguide. As examples of the nonreciprocal responses, the equations of motion were simulated for the parameters shown in Table 1, a total normalized simulation time equal to 1500, and for 1000 oscillators in total. Also note that exchanging the excitation position from the oscillator Ep to Ap is equivalent to simply exchanging the detuning parameters σ1 and σ2; therefore, we will study the nonreciprocal acoustics by fixing the excitation position and comparing the responses when the detuning parameters (σ1,σ2) are interchanged.

Table 1

System parameters for the simulations of Eq. (2)

Parameterd{σ1,σ2}ζθαApp
System 10.5{1,1.5}0.03(1/6)π20.254
System 20.5{1,0}0.02(2/3)π20.254
Parameterd{σ1,σ2}ζθαApp
System 10.5{1,1.5}0.03(1/6)π20.254
System 20.5{1,0}0.02(2/3)π20.254

In Fig. 2, we consider the responses for the first set of parameters (System 1) listed in Table 1. The results for (σ1,σ2)=(1,1.5) are shown in Figs. 2(a), 2(c), and 2(e), while the results of the excitation from the other side, i.e., (σ1,σ2)=(1.5,1), are depicted in Figs. 2(b), 2(d), and 2(f). In Figs. 2(a) and 2(b), we compare the displacement time series of the two oscillators of the nonlinear gate, x0 and y0. For (σ1,σ2)=(1,1.5), the steady-state oscillation amplitudes of the gate oscillators are similar, so it is inferred that energy transmission occurs through the nonlinear gate. This is not the case when (σ1,σ2) are flipped though (i.e., interchanging the site of the excitation between the left and right subwaveguides), since drastically diminished energy propagates through the nonlinear gate as the amplitude of y0 is much smaller than the amplitude of x0 at steady-state. Therefore, we observe the realization of strong nonreciprocity in the steady-state acoustics, with energy being transmitted only in one direction.

Fig. 2
Numerical simulations of system 1 listed in Table 1: (a), (c), (e) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=−1 and σ2=−1.5, (b), (d), (f) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=−1.5 and σ2=−1 note the nonreciprocity in the steady-state acoustics
Fig. 2
Numerical simulations of system 1 listed in Table 1: (a), (c), (e) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=−1 and σ2=−1.5, (b), (d), (f) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=−1.5 and σ2=−1 note the nonreciprocity in the steady-state acoustics
Close modal
Next, we proceed to analyze the harmonic contents of the transmitted waves by showing the amplitude of the discrete Fourier transform of the response y0. In Figs. 2(c) and 2(d), only a single dominant harmonic is noted, which is the frequency of the exerted harmonic excitation; hence, the transmitted waves are monochromatic. Lastly, we evaluate the total work exerted by the external force as well as the energy transmitted to the downstream subwaveguide, which enables the study of nonreciprocity and transmissibility in an energy context. To this end, the exerted work (input energy by the force), Winput, and transmitted energy through the gate, Edown, is given by
Winput=0τ2dApcos(ω̂τ0)xpdτ0Edown=0τd(y0y1)y0dτ0
(4)

These energy measures are plotted as functions of time in Figs. 2(e) and 2(f) for (σ1,σ2)=(1,1.5) and (1.5,1), respectively. From Fig. 2(e), we note that the transmitted energy is about 25% of the input energy for (σ1,σ2)=(1,1.5), and almost zero in the opposite direction. Besides the time series and the frequency spectra, these energy measures provide another prospective to understand the physics of acoustic nonreciprocity in the gated waveguide.

Another case is shown in Fig. 3, for parameters indicated as System 1 listed in Table 1. The simulations for (σ1,σ2)=(1,0) are shown in Figs. 3(a), 3(c), 3(e), while for (σ1,σ2)=(0,1) in Figs. 3(b), 3(d), and 3(f). From Figs. 3(e) and 3(f), we note that again energy transmission is only allowed only in one direction, revealing strong nonreciprocity in terms of energy. Moreover, in this case, the responses of the waveguide are strongly modulated for (σ1,σ2)=(1,0); as in Ref. [28], this type of responses will be referred to as strongly modulated responses (SMRs). The amplitude of the discrete Fourier transformation of y0 is plotted in Fig. 3(c), showing equally-spaced peaks (side bands) centered at the excitation frequency which is the highest (dominant) harmonic; this indicates that the SMR is quasi-periodic. However, for flipped (σ1,σ2) the transmitted wave is monochromatic. In this case, the acoustics are nonreciprocal in terms of both transmitted energy and harmonic composition of the steady-state responses.

Fig. 3
Numerical simulations of system 2 listed in Table 1: (a), (c), (e) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=1 and σ2=0, (b), (d), (f) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=0 and σ2=1; note the nonreciprocity in the steady-state acoustics
Fig. 3
Numerical simulations of system 2 listed in Table 1: (a), (c), (e) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=1 and σ2=0, (b), (d), (f) Time series of x0 and y0, corresponding discrete Fourier spectra and input/downstream energy measures, respectively, for σ1=0 and σ2=1; note the nonreciprocity in the steady-state acoustics
Close modal
Motivated by these results it is natural to seek the conditions for designing the nonlinear gated waveguide for tailored nonreciprocity and energy transmissibility. To address this question, we need to start by quantifying the performance of the waveguide through appropriate transmissibility and nonreciprocity measures which, following Ref. [28], are defined as follows:
η(σ1,σ2,T)=Edown(σ1,σ2,T)Winput(σ1,σ2,T)δ(σ1,σ2,T)=log10Edown(σ1,σ2,T)Edown(σ2,σ1,T)
(5)

In Eq. (5), T denotes the time interval where the measurements are recorded, η is a measure of transmissibility and δ a measure of nonreciprocity. Note that in a reciprocal system, Edown(σ1,σ2,T)=Edown(σ2,σ1,T), which corresponds to δ=0 [1]. Therefore, δ measures the difference (in logarithmic scale) in the transmitted energy when the excitation is switched between the two subwaveguides. We also focus on optimizing the nonreciprocal transmissibility η, which is defined as the ratio of the transmitted energy to input energy. We aim at maximizing the nonreciprocal transmissibility from one direction with negligible nonreciprocal transmissibility from the other direction, thus requiring large transmissibility and nonreciprocity measures η and δ. As the acoustics reaches the steady-state, the energy measures η and δ converge for sufficiently large simulation times. Hence, the time interval T is selected such as the acoustics reach steady-state.

Intuitively, for sufficiently large damping to coupling ratio, ζ, the dissipation effect overweighs wave propagation and the energy is confined in the leading oscillators close to the excited oscillator Ep; therefore, this ratio is carefully selected and assigned moderate values. In Fig. 4, we show the nonreciprocal transmissibility (linear scale) and downstream energy (logarithmic scale) measures when the normalized damping ratio ζ varies from 0 to 0.07 for the parameters listed in Table 1. In the schematic shown in Fig. 1, the upstream subwaveguide is located at the left side, while the downstream subwaveguide at the right side. Hence for the frequency detuning values (σ1,σ2) listed in Table 1, we refer to the simulated responses as “left-to-right” propagating waves, whereas for the flipped values (σ2,σ1) as “right-to-left” propagating waves.

Fig. 4
Transmissibility and downstream energy measures in the gated waveguide for varying normalized damping ratio ζ and other parameters listed in Table 1: (a) and (b) simulation 1 and (c) and (d) simulation 2
Fig. 4
Transmissibility and downstream energy measures in the gated waveguide for varying normalized damping ratio ζ and other parameters listed in Table 1: (a) and (b) simulation 1 and (c) and (d) simulation 2
Close modal

Considering the results for system parameters corresponding to System 1 (cf. Figs. 4(a) and 4(b)), the transmissibility and downstream energy measures decrease when the normalized damping ratio ζ increases. Moreover, steep descents in downstream energy are observed for both “left-to-right” propagation and “right-to-left” propagation at certain critical thresholds of ζ, with the corresponding decreases being nearly two to three orders of magnitude. We note that the critical damping ratio ζ is around 0.05 for “left-to-right” propagation, and 0.015 for “right-to-left” propagation. This disparity in the critical damping ratios provides a region of strong nonreciprocity, where energy can propagate from left to right but is prohibited from propagating right to left.

Another interesting acoustic feature of the gated waveguide is observed for system parameters corresponding to Simulation 2. In Fig. 4(c), when ζ<0.02, the “left-to-right” transmissibility increases as the damping ratio increases, which is counterintuitive since damping typically restricts energy transfer. Moreover, In Fig. 4(d), we note that the “left-to-right” transmitted energy decreases linearly in the logarithmic plot for ζ>0.02, indicating that the transmitted energy exponentially decreases for increasing damping. This is not the case for ζ<0.02, where the transmitted energy is nearly constant. Therefore, we hypothesize that in this case the “left-to-right” acoustics exhibit drastically different features before and after the critical threshold ζ=0.02. To further explore this phenomenon, we investigate the responses of the gated waveguide by means of numerical simulations corresponding to values of ζ close to the critical threshold 0.02.

In Fig. 5, we plot the time series of x0, y0 and the corresponding Morlet wavelet transforms for system parameters corresponding to Simulation 2 of Table 1 with ζ=0.02,0.022, and 0.024. In these results the mother wavelet (normalized) frequency is equal to 16. We observe that a small variation in damping leads to different types of steady-state responses. As shown in Figs. 5(a) and 5(b) (ζ=0.02), a strongly modulated quasi-periodic response is generated, involving multiple harmonics at steady-state. However, in Figs. 5(e) and 5(f) (ζ=0.024) the steady-state response is monochromatic involving only one frequency component. A transition state is observed for ζ=0.022 in Figs. 5(c) and 5(d), where multiple harmonics are noted in the response; however, in this case it takes a relatively long duration for the response to converge to steady-state where a single harmonic survives. For this particular case, at the critical damping ratio there occurs a bifurcation (transition) from an SMR to a monochromatic steady-state response as damping increases. We then hypothesize that this critical damping ratio applies to other cases which support SMRs. This interesting feature paves the way for predictive design of the nonlinear nonreciprocal acoustics of the gated waveguide. To this end, we proceed to the analytical study of the strongly nonlinear and strongly coupled system (2) in order to recover the previous numerical results and study the governing mechanism of the different nonlinear acoustic regimes at steady-state; this will represent the first step toward predictive design and optimization of the nonreciprocal acoustics of the gated waveguide.

Fig. 5
Time series and corresponding wavelet transforms of x0 and y0 for System 2 with (a) and (b) ζ=0.02, (c) and (d) ζ=0.022, and (e) and (f) ζ=0.024
Fig. 5
Time series and corresponding wavelet transforms of x0 and y0 for System 2 with (a) and (b) ζ=0.02, (c) and (d) ζ=0.022, and (e) and (f) ζ=0.024
Close modal

3 Analytical Study of the Non-Reciprocal Acoustics

The analytically study of the normalized Eq. (2) is based on the complexification-averaging (CX-A) method which is suitable for the analysis of this strongly nonlinear problem [31,32]. Generally, the CX-A method requires that the responses only possess one or a finite number of “fast” dominant frequencies [33,34], so in this section, only monochromatic steady-state solutions are considered. To this end, we assume that the responses of all oscillators of the gated waveguide possess a single dominant harmonic component at the excitation frequency. Note, however, that this ansatz is not valid for SMRs with multiple harmonics (side bands).

Accordingly, the following new complex variables for the oscillator responses are introduced
ψx0=x0+iω̂x0=φx0eiω̂τψy0=y0+iω̂y0=φy0eiω̂τψxn=xn+iω̂xn=φxneiω̂τψyn=yn+iω̂yn=φyneiω̂τ
(6)
where φ denotes a “slow” complex amplitude modulating the “fast” oscillation eiω̂τ. The Eq. (2) is expressed in terms of the new complex variables (6), and then averaged with respect to the fast frequency ω̂. Then, the following slow flow is derived, governing the (slow) evolution of the complex amplitudes (or modulations)
φx0+(iω̂2+1+dσ12iω̂+dζ)φx03dα8ω̂3i(φx0φy0)2(φx0φy0)¯+d2iω̂(φx0φx1)=0φy0+(iω̂2+1+dσ22iω̂+dζ)φy03dα8ω̂3i(φy0φx0)2(φy0φx0)¯+d2iω̂(φy0φy1)=0
φxn+(iω̂2+12iω̂+dζ)φxn+d2iω̂[(φxnφx(n1))+(φxnφx(n+1))]=dApδ(np)φyn+(iω̂2+12iω̂+dζ)φyn+d2iω̂[(φynφy(n1))+(φynφy(n+1))]=0
(7)
where overbar denotes complex conjugate. At steady-state, the responses are monochromatic (based on the previous ansatz) so the complex amplitudes φx0, φy0, φxn, and φyn are constants; this yields the following set of nonlinear algebraic equations governing the monochromatic responses at the steady-state:
(iω̂2+1+dσ12iω̂+dζ)φx03dα8ω̂3i(φx0φy0)2(φx0φy0)¯+d2iω̂(φx0φx1)=0(iω̂2+1+dσ22iω̂+dζ)φy03dα8ω̂3i(φy0φx0)2(φy0φx0)¯+d2iω̂(φy0φy1)=0(iω̂2+12iω̂+dζ)φxn+d2iω̂[(φxnφx(n1))+(φxnφx(n+1))]=dApδ(np)(iω̂2+12iω̂+dζ)φyn+d2iω̂[(φynφy(n1))+(φynφy(n+1))]=0
(8)
The slow flow dynamics of the nonlinear gate are governed by first two equations in (8), while the slow flow acoustics of the semi-infinite linear waveguides are governed by the last two equations in (8). In order to analytically solve the slow flow we consider the linear part of Eq. (8) to express the variables φx1 and φy1 in terms of φx0 and φy0. To this end, we first focus on the last equation of (8), which is a linear homogeneous difference equation that admits a solution of the form
φyn=C1λ1n+C2λ2n
(9)
where C1 and C2 are undetermined coefficients and λ1 and λ2 are the roots of the following eigenvalue equation:
(iω̂2+12iω̂+dζ)λ+d2iω̂[(λ1)+(λλ2)]=0
(10)
Based on Vieta's formulas, it holds that λ1λ2=1 and λ1+λ2=(ω21)/d+2+2iω̂ζ. Since the frequency ω̂ and the damping ratio ζ are nonzero, the imaginary part of λ1+λ2 is nonzero and, therefore, λ1 and λ2 are not complex conjugate pairs. Without loss of generality, we assume that |λ1|<1 and |λ2|>1. Since the energy is finite, the amplitudes of the responses are bounded as n+ and therefore C2=0. The remaining coefficient C1 is determined by the boundary condition. Substituting φyn=φy0 into (9), we get C1=φy0 and, therefore, φyn is expressed as
φyn=φy0λ1n
(11)
Next, we proceed to the third equation of (8), which is a nonhomogeneous linear difference equation. Based on the linear superposition method, the solution of φxn is composed of two parts, namely, the solution of the homogeneous equation with a nonhomogeneous boundary condition and the solution of the nonhomogeneous equation with the homogeneous boundary condition. Therefore, the solution of φxn is expressed as
φxn=φx0λ1n+CnAp
(12)
where Cn is an undetermined coefficient dependent on n. Based on the linear superposition method, we can evaluate Cn by considering the solution of the nonhomogeneous equation with the homogeneous boundary condition, i.e., solving the third equation of (8) with the homogeneous boundary condition φx0=0. Since the boundary is homogeneous, we can mirror the linear semi-infinite subwaveguide to an infinite waveguide, and the forcing is mirrored on the other side with the same amplitude and different direction. Applying the linear superposition method again, we obtain solutions of φxn as linear summations of the following two nonhomogeneous linear difference equations
(iω̂2+12iω̂+dζ)φxn+d2iω̂[(φxnφx(n1))+(φxnφx(n+1))]=0,np,nZ(iω̂2+12iω̂+dζ)φxp+d2iω̂[(φxpφx(p1))+(φxpφx(p+1))]=dAp
(13)
(iω̂2+12iω̂+dζ)φxn+d2iω̂[(φxnφx(n1))+(φxnφx(n+1))]=0,np,nZ(iω̂2+12iω̂+dζ)φx(p)+d2iω̂[(φx(p)φx(p1))+(φx(p)φx(p+1))]=dAp
(14)
where Z denotes the integer set. To start with, we first solve (13) by prescribing φxp and solving for the other unknown variables based on the homogeneous equations. Based on (11), φxn is expressed in terms of φxp as
φx(p±n)=φxpλ1n,nN
(15)
where N denotes the natural number set. Substituting Eq. (15) into the second equation of Eq. (13), we derive the solution of φxp
φxp=dAp(iω̂2+12iω̂+dζ)+d(1λ1)iω̂
(16)
Substituting Eq. (16) into Eq. (15) we obtain the first nonhomogeneous component of φx1 as a function of Ap
φx1=λ1p1dAp(iω̂2+12iω̂+dζ)+d(1λ1)iω̂
(17)
Similarly, the second nonhomogeneous component of φx1 is given by the solution of (14) as follows:
φx1=λ1p+1dAp(iω̂2+12iω̂+dζ)+d(1λ1)iω̂
(18)
Adding Eqs. (17) and (18), we get that φx1=2iω̂λ1pAp for the nonhomogeneous problem with homogeneous boundary condition. Finally, we reconsider (12) and incorporate the solution of φx1 for the homogeneous problem with nonhomogeneous boundary condition. The final expression of φx1 is then expressed in terms of φx0 and Ap as follows:
φx1=φx0λ1+2iω̂λ1pAp
(19)
Substituting Eqs. (11) and (19) into the first two equations of (8), we obtain the final governing slow flow equations for the two oscillators of the nonlinear gate
[ω̂2+1d+σ1+2iω̂ζ+(1λ1)]φx0+3α4ω̂2(φx0φy0)2(φx0φy0)¯=2iω̂λ1pAp[ω̂2+1d+σ2+2iω̂ζ+(1λ1)]φy0+3α4ω̂2(φy0φx0)2(φy0φx0)¯=0
(20)

We note that due to the monochromatic feature of the applied excitation, the governing equation at the steady-state is ultimately reduced to a 2DOF reduced-order system; note that this is an exact reduction since no approximation was involved. Moreover, the reduced system has the form of two linear damped oscillators coupled with a cubic nonlinear spring, but the system parameters are dependent on λ1, which, in turn, is dependent on the excitation frequency. Therefore, the reduction of the acoustics of the infinite-dimensional gated waveguide to the 2DOF reduced system is only valid for monochromatic responses. This is not the case for nondispersive semi-infinite waveguides, which are equivalent to linear dampers with constant damping coefficients regardless of the excitation frequency [3].

The closed-form solutions of Eq. (20) are provided in Appendix  A. Based on these monochromatic solutions, we proceed to their stability analysis. To this end, we perform a variational analysis by reconsidering Eq. (7) and linearizing them in the neighborhood of the analytically computed steady-state solutions, which correspond to fixed points. Therefore, the following perturbations are introduced
φx0=φx0,0+ϕx0,φy0=φy0,0+ϕy0,φxn=φxn,0+ϕxn,φyn=φyn,0+ϕyn
(21)
where φx0,0, φy0,0, φxn,0, and φyn,0 denote the (fixed point) solutions of Eq. (8) for φx0, φy0, φxn and φyn, respectively, and ϕx0, ϕy0, ϕxn, and ϕyn are sufficiently small perturbations of φx0, φy0, φxn, and φyn in the neighborhood of the fixed point solutions. Neglecting the higher order terms of the perturbations, the resulting linearized slow-flow equations are given by
ϕx0+(iω̂2+1+dσ12iω̂+dζ)ϕx0+d2iω̂(ϕx0ϕx1)3dα8ω̂3i(φx0,0φy0,0)2(ϕx0ϕy0)¯3dα4ω̂3i(φx0,0φy0,0)(φx0,0φy0,0)¯(ϕx0ϕy0)=0ϕy0+(iω̂2+1+dσ22iω̂+dζ)ϕy0+d2iω̂(ϕy0ϕy1)3dα8ω̂3i(φy0,0φx0,0)2(φy0φx0)¯3dα4ω̂3i(φy0,0φx0,0)(φy0,0φx0,0)¯(ϕy0ϕx0)=0ϕxn+(iω̂2+12iω̂+dζ)ϕxn+d2iω̂[(ϕxnϕx(n1))+(ϕxnϕx(n+1))]=0ϕyn+(iω̂2+12iω̂+dζ)ϕyn+d2iω̂[(ϕynϕy(n1))+(ϕynϕy(n+1))]=0
(22)

Note that Eq. (22) is an infinite set of linear differential equations and represents the variational system for stability analysis. Therefore, we truncate the number of equations and consider the finite version of Eq. (22) as an approximation for the stability analysis. In this work, 1000 oscillators are considered in total for the stability analysis, and the eigenvalues of Eq. (22) are studied with the details provided in Appendix  B. Clearly, the monochromatic solution is deemed stable as long as the real parts of all eigenvalues are negative.

To compare the analytical stability predictions versus direct numerical solutions, we proceed to analytically approximate the transmitted energy at the steady-state. We start from the power transmitted downstream through the nonlinear gate, Pdown
Pdown=dy0(y0y1)
(23)
Substituting Eqs. (6) and (11) into Eq. (23), we note that the transmitted power Pdown is an oscillating function in time at the excitation frequency ω̂. Therefore, we express the averaged power at steady-state, P¯down, by averaging out the fast oscillatory terms
P¯down=ω̂2πτ0τ0+2πω̂Pdowndτ=Im(λ1)d2ω̂|φy0|2
(24)

From Figs. 2(e), 3(e), 2(f), and 3(f), we notice that the durations of the transient effects are much smaller compared with the total simulation time. Therefore, we approximate the analytical transmitted energy downstream by the product of the averaged power predicted in Eq. (24) and the total simulation time T, and compare it with the corresponding numerically computed transmitted energy, Edown, given by Eq. (4).

In Fig. 6, we reconsider the numerical results of the transmitted energy shown in Figs. 4(a) and 4(b) and superimpose onto the analytical predictions based on the previous variational analysis. We notice that the analytical results agree well with the numerical simulations in the regime of stable monochromatic steady-state solutions. S-shape curves are observed in Figs. 6(a) and 6(b) with three branches of solutions as predicted by the cubic algebraic equation (20). We notice that the middle branch is always unstable. As the damping ratio increases, the simulation results switch from the upper to the lower branch of stable analytical solutions, with the transmitted energies after the transitions being lower by nearly two orders of magnitude. The predicted bifurcations lead to the steep reduction of the transmitted energy at a critical damping ratio. The critical damping ratios vary when switching the harmonic excitation (i.e., flipping σ1,σ2), which gives rise to nonreciprocal acoustics for damping ratios between the critical values. It is interesting to point out that the disparities in the critical damping ratios observed in Figs. 6(a) and 6(b) are successfully captured by the analytical approximations.

Fig. 6
Comparison of the analytical and numerical downstream energy for the parameters given in Table 1: (a) and (b) simulation 1 for left-to-right and right-to-left propagation, respectively; (c) and (d) simulation 2 for left-to-right and right-to-left propagation
Fig. 6
Comparison of the analytical and numerical downstream energy for the parameters given in Table 1: (a) and (b) simulation 1 for left-to-right and right-to-left propagation, respectively; (c) and (d) simulation 2 for left-to-right and right-to-left propagation
Close modal

We also notice that the analytical solutions lose stability for sufficiently low damping ratios in all cases considered. This is because in these regimes there is a unique monochromatic solution and it is unstable. The most robust unstable region is observed in Fig. 6(c), with damping ratio ζ ranging from approximately 0 to 0.02, and the numerical simulations differ from the analytical results. Since the single monochromatic solution is unstable, the responses at the steady-state are not monochromatic but rather SMRs are generated. From Figs. 3(a) and 5(a) we notice that at ζ=0.02, SMRs are realized at the steady-state. In this case, the monochromatic ansatz is not valid and therefore the analytical prediction is not accurate (as expected). Moreover, when steady-state monochromatic solutions governed by Eq. (20) are realized, the transmissibility decreases as the damping ratio increases. This is not necessarily valid for the SMR responses, which are not governed by Eq. (20). Lastly, we conclude that the SMRs are generated due to the loss of stability of the monochromatic solutions. Hence, by increasing damping ratios, the monochromatic solutions tend to be stable and the SMRs vanish. Moreover, a critical damping ratio exists where a bifurcation occurs, and beyond which the monochromatic solutions become stable. These analytical results validate our hypothesis regarding the existence of a critical damping ratio that governs the transition between different steady-state acoustic regimes and enable the prediction of the corresponding downstream transmitted energies.

We note that the analytical method succeeds in predicting the stability of the monochromatic responses and the bifurcations. However, the SMRs cannot be modeled by this analysis, and are difficult to predict through analytical methods due to the dispersive nature of the linear subwaveguides. Therefore, in the next section we proceed to the machine learning approach first developed in Ref. [1] in order to computationally predict the performance in parameter space of the gated waveguide in terms of transmissibility and nonreciprocity.

4 Machine Learning Approach for Predicting the Non-Reciprocal Acoustics

In this section, we study the nonreciprocal acoustics of the gated waveguide in both regimes of SMRs and monochromatic responses. To the end, a learnable simulator is built to predict the transmissibility and nonreciprocity of the waveguide in the parameter space. We start by noting that system (2) can be further scaled as follows:
X0+2ξX0+(1+dσ1)X0+dαAp2(X0Y0)3+d(X0X1)=0Y0+2ξY0+(1+dσ2)y0+dαAp2(Y0X0)3+d(Y0Y1)=0Xn+2ξXn+Xn+d(XnXn1)+d(XnXn+1)=[deiω̂τ+c.c.]δ(np),n>0Yn+2ξYn+Yn+d(YnYn1)+d(YnYn+1)=0,n>0
(25)

where X0=x0/Ap, Y0=y0/Ap, Xn=xn/Ap, and Yn=yn/Ap. We note from Eq. (25) that the nonlinear acoustics of the gated waveguide are governed by six normalized independent parameters, that is, the normalized detuning parameters σ1, σ2, the excitation frequency ω̂, the normalized (strong) coupling d, the nonlinear coupling parameter αAp2, and finally the damping ratio ξ. Without loss of generality, in what follows we assume that α=0.5 and vary the amplitude Ap to tune the nonlinear coefficient. The aim for constructing the learnable simulator is to predictively design the parameters of the waveguide so that is suitable for experimental design or engineering applications. Since the precise control of damping in the experiments is difficult [29,30], in practice nonuniform damping coefficients are not avoidable. As a theoretical preliminary study, we assume that the damping ratios of the grounding stiffnesses are uniform along the waveguides with ξ= 0.01. Here, machine learning methods are employed to study the effects of the parameters σ1, σ2, d, Ap, and ω̂ on the previously defined transmissibility and nonreciprocity measures.

A five-dimensional dataset is created for varying parameters σ1, σ2, d, Ap, and ω̂. Since the excitation frequency ω̂ is located inside the passband, that is, in the range of Eq. (3) with the wavenumber θ ranging from 0 to π. If the excitation frequency is close to the boundary of the pass band, the group velocity approaches zero and it takes a long time for the system to reach the steady-state. Therefore, the values of ω̂ are selected for wavenumbers θ between 16π and 56π. The detuning parameters σ1, σ2 range from 2 to 2 with an increment of 0.2, while d and Ap range from 0.2 to 0.8 with an increment of 0.2. Since the grounding stiffnesses of the nonlinear gate are positive, the detuning parameters follow the constraints that (1+dσ1,2)>0. The dataset is generated by randomly sampling the coefficients with a uniform distribution in the five-dimensional domain. At the end, the random dataset incorporating the constraints involves 31,768 sets of parameters in total.

A machine learning model is then developed following a similar model developed for the weakly coupled case [1] that replaces the direct numerical simulations of the strongly coupled gated waveguide with multi-objective regression. It is shown in Ref. [1] that the trained machine learning model can drastically reduce the simulation time by 4 to 5 orders of magnitude with acceptable accuracy. The inputs to the machine learning model are the system and excitation parameters σ1, σ2, d, Ap, and ω̂, while the outputs are the nonreciprocal transmissibility η and the nonreciprocity δ measures. The machine learning model maps the input to the output variables with an artificial NN, which is constructed based on the available 31,768 sets of data. By performing direct numerical simulations of the governing equations of motion, we get inputs and outputs for the learnable simulator, which are normalized and assembled to vectors of dimensions (5×1) and (2×1), respectively. Each individual element of the input and the output vectors is linearly normalized so that the minimum is mapped to 1 and the maximum to +1. The normalized input is then mapped to the normalized output through four hidden layers of 50 neurons each, as per the schematic of Fig. 7 (the choice of architecture is explained in Appendix  C). The estimates of each layer are computed through the previous layer as
ai=f(wiai1+bi)
(26)

where ai denotes the vector of the values at the ith layer, wi and bi are the learnable weight matrix and the offset vector, respectively, and f is the activation function. In this work, the activation functions of the hidden layers are rectified linear unit (ReLU) functions, where f(x)=max(0,x), and the activation function of the output layer is linear (identity transformation).

Fig. 7
Schematic of the architecture of the NN model
Fig. 7
Schematic of the architecture of the NN model
Close modal

The overall dataset is randomly divided into three subsets with 50% used for training, 15% for validation and 35% for testing. The Levenberg–Marquardt backpropagation algorithm is adopted to train the weights wi and the offsets bi with the mean square error adopted as the loss function. The loss function for the validation subset is evaluated after each epoch (the number of passes of the entire training set), and the training stops if the loss function on the validation subset does not get improved in six consecutive epochs. The loss function at each epoch is shown in Appendix  D. The neural network is developed and trained based on the platform of matlab deep learning toolbox. In order to evaluate the efficacy of the trained simulator, the nonreciprocity and transmissibility measures δ and η are plotted in Figs. 8(a) and 8(b) with the comparison of the NN prediction and the direct numerical simulations on the testing subset. We notice that the neural network accurately predicts the nonreciprocity and the transmissibility, with the correlation R between the targets generated by the simulations and the outputs predicted by the NN simulator are close to 0.99.

Fig. 8
Comparison between the target and output (prediction) of the testing set: (a) Transmissibility measure, (b) nonreciprocity measure, (c) distribution based on direct numerical simulations, and (d)distribution based on NN predictions
Fig. 8
Comparison between the target and output (prediction) of the testing set: (a) Transmissibility measure, (b) nonreciprocity measure, (c) distribution based on direct numerical simulations, and (d)distribution based on NN predictions
Close modal
A good design of the nonreciprocal waveguide corresponds to a high nonreciprocity measure δ and a high transmissibility η in the direction where energy propagation occurs. Here, we consider η>0.2 and δ>1 as the desirable performance criteria for the waveguide, i.e., the transmitted energy should be over 20% of the input energy, and the transmitted energy should be smaller by at least one order of magnitude in the direction of wave propagation compared to the other direction. The nonreciprocity and transmissibility measures for the test set are plotted in Figs. 8(c) and 8(d) based on direct numerical simulations and NN results, respectively. The desired cases with η>0.2 and δ>1 are highlighted in red circles. Predicted by both the NN results and ODE simulations, the transmissibility and nonreciprocity measures in the desired region are approximately in the ranges 0.2<η<0.4 and 1<δ<3, respectively. The confusion matrix is shown in Table 2; it classifies the testing subset into four groups, namely, TP, TN, FP, and FN. TP denotes the amount of true positive values that are found to be desirable by both the NN simulator and the numerical simulations; TN denotes the amount of true negative values that are undesirable by both the NN simulator and the numerical simulation; FP is the amount of false positive values that are found to be desirable by the NN simulator but undesirable by the numerical simulations; and FN denotes the amount of false negative values that are undesirable by the NN simulator but desirable by the numerical simulations. The sensitivity, specificity and precision measures are then defined to evaluate the performance of this classification
Sensitivity=TPTP+FN,Specificity=TNTN+FP,Precision=TPTP+FP
(27)
Table 2

Confusion matrix for the predictions of the NN on the test subset

Data points in the test set: N=11,119Predicted conditionPerformance
Actual condition  Desirable Undesirable  
Desirable TP = 81 FN = 17 Sensitivity: 0.827 
Undesirable FP = 15 TN = 11,006 Specificity: 0.999 
   Precision: 0.844 
Data points in the test set: N=11,119Predicted conditionPerformance
Actual condition  Desirable Undesirable  
Desirable TP = 81 FN = 17 Sensitivity: 0.827 
Undesirable FP = 15 TN = 11,006 Specificity: 0.999 
   Precision: 0.844 

From Table 2, we note that the sensitivity, precision, and specificity measures are all close to 1, and therefore, the NN simulator well classifies the waveguides with large transmissibility and nonreciprocity.

The classification results are visualized in Fig. 9. Note that the performance is dependent on the five independent parameters σ1,σ2,Ap,θ,d. The desired region predicted by the NN simulator is shown in Fig. 9(a) by fixing Ap=0.4, θ=7/12π and d=0.6. Only the blue region in Fig. 9(a) is well defined due to the constraints 1+dσ1,2>0. A region with desired performance classified by the NN simulator where σ1 is close to zero and σ2 is close to −1. To visualize the desired region in higher dimensional space, we fit a square in the desired region. An auxiliary measure, the kernel size, is defined as the side length of the square, and is used to quantify the robustness of the desired performance given θ,Ap and d. In Fig. 9(b), we fix d=0.6 and plot the kernel size versus Ap and θ. We note that the most robust region occurs at around Ap=0.4 and θ=712π, where the kernel size is close to 1. In order to visualize classification results in the three-dimensional space involving varying Ap, θ, and d, we proceed to animate the contour plots of the kernel sizes with variable d where each frame depicts a contour of kernel sizes for fixed d similar to Fig. 9(b). The animation is available through the link given in Appendix  E.

Fig. 9
Desired region predicted by the neural net: (a) Desired region in the (σ1,σ2) plane for (θ,Ap,d)=(712π,0.4,0.6) and (b) kernel size of the desired region in (σ1,σ2) for d=0.6
Fig. 9
Desired region predicted by the neural net: (a) Desired region in the (σ1,σ2) plane for (θ,Ap,d)=(712π,0.4,0.6) and (b) kernel size of the desired region in (σ1,σ2) for d=0.6
Close modal
To evaluate the efficacy and accuracy of the NN simulator, we compare the classification results of the neural network with the analytical results as well as the direct numerical simulations. In Eq. (24), we provide an analytical approximation for the transmitted energy, based on which, the nonreciprocity measure is approximated by substituting Eq. (24) to Eq. (5). Besides the transmitted energy, the exerted work of harmonic excitation – the input energy, is also required to predict the nonreciprocal transmissibility. The averaged input power of the excitation, P¯input, is approximated as
P¯input=ω̂2πτ0τ0+2πω̂Pinputdτ=ω̂2πτ0τ0+2πω̂Ap(eiω̂τ+c.c.)xpdτApRe(ϕxp)
(28)
where Re() denotes the real part and ϕxp is computed based on (12):
φxp=φx0λ1p+2iω̂Apλ1(1λ12p)1λ12
(29)

The transmissibility and downstream energy are computed by substituting Eqs. (24) and (28) into Eq. (5). Note that the analytical approximations are only applicable for monochromatic steady-state responses. To this end, the analytical monochromatic desired responses are identified through the following two steps:

  • Step 1: Obtain the analytical solution from Eq. (20) and check its stability via Eq. (22). If there is only one stable solution, then we proceed to the next step.

  • Step 2: Compute the analytical approximation of the nonreciprocity measure δ and nonreciprocal transmissibility measure η, and check whether η>0.2 and δ>1.

Similar to the results of the NN prediction shown in Fig. 9, the kernel size of the (monochromatic) analytical predictions is defined as the side length of the square fit to the desired region in the (σ1,σ2) plane given the parameters Ap,θ and d. The animation of the analytical (monochromatic) kernel sizes for the desired regions is available in Appendix  E. In Fig. 10, we show the comparison of the analytical (monochromatic) results and the NN predicted results for the three representative values d=0.2,0.5, and 0.8. Since the analytical results are only applicable for the monochromatic response regime, we expect that the analytical desired region is a subset of the full desired region. Indeed, in Fig. 10, the monochromatic desired region is mainly located in the NN desired region with smaller kernel sizes, validating the previous inclusion predictions for the NN and the analytical predicted results.

Fig. 10
Kernel sizes for the desired region in the (σ1,σ2) plane: (a), (c), and (e) NN prediction for d=0.2, 0.5, and 0.8, respectively, (b), (d), (f) analytical prediction for d=0.2, 0.5, and 0.8, respectively, based on the monochromatic ansatz; points A, B, and C refer to the considered cases for the validation study
Fig. 10
Kernel sizes for the desired region in the (σ1,σ2) plane: (a), (c), and (e) NN prediction for d=0.2, 0.5, and 0.8, respectively, (b), (d), (f) analytical prediction for d=0.2, 0.5, and 0.8, respectively, based on the monochromatic ansatz; points A, B, and C refer to the considered cases for the validation study
Close modal

In a final validation step, we verify the predicted results by the NN output and the CX-A analysis by considering three cases denoted with crosses in Fig. 10 and labeled as A, B, and C, respectively. The corresponding system parameters for these cases are summarized in Table 3. Case A corresponds to (Ap,θ,d)=(0.35,13π,0.5) with NN and analytically predicted kernel sizes being close to 0.5. Case B corresponds to (Ap,θ,d)=(0.35,712π,0.5) and is predicted to be desirable by the analytical solution but undesirable by the NN output. Lastly, case C corresponds to (Ap,θ,d)=(0.35,712π,0.8) and the NN predicted kernel size is much larger than the analytically predicted kernel size.

Table 3

Simulation system parameters in the validation step

CasesCase ACase BCase C
Case parameters (Ap,θ,d)(0.35,13π,0.5)(0.35,712π,0.5)(0.35,712π,0.8)
SystemsSystem A1System B1System C1
Gate parameters (σ1,σ2)(1,1.5)(0,1.5)(0.3,0.5)
CasesCase ACase BCase C
Case parameters (Ap,θ,d)(0.35,13π,0.5)(0.35,712π,0.5)(0.35,712π,0.8)
SystemsSystem A1System B1System C1
Gate parameters (σ1,σ2)(1,1.5)(0,1.5)(0.3,0.5)

In Figs. 11(a) and 11(b), we compare the desired regions predicted by the CX-A method and the NN output for case A. The desired regions are in the third quadrant of the (σ1,σ2) domain in both cases with similar areas and positions. To verify the results of NN and analytical predictions, we present the numerical results of one representative case with (σ1,σ2)=(1,1.5) denoted as A1 in the desired regions. To analyze the nonreciprocal acoustics, the case of flipped (σ1,σ2)=(1.5,1) is simulated also, corresponding to the excitation moved to the opposite side. The time series of x0 and y0 are shown in Figs. 11(c) and 11(d) for the direct and flipped (σ1,σ2) cases, respectively. In both cases the system reaches the steady-state at approximately τ=400 and the amplitudes of the responses are not modulated. Hence, the monochromatic response ansatz is valid in this case. The amplitude of y0 is comparable to x0 for (σ1,σ2)=(1,1.5), while the amplitude of y0 is much smaller than that of x0 for (σ1,σ2)=(1.5,1). The corresponding input and transmitted energies are shown in Figs. 11(e) and 11(f). A strong disparity is observed between the two simulation cases in terms of the transmitted energy as well as the amplitudes of the oscillator after the gate, indicating the strong nonreciprocal acoustics, namely, with strong energy transfer from left to right, but weak from right to left. The resulting transmissibility and nonreciprocity measures η and δ are listed in Table 4 for the numerical simulations in comparison with the analytical and NN predictions. Based on the numerical simulations of system A1, η=0.395 and δ=1.923, indicating the strong nonreciprocity and transmissibility which agrees with the NN and analytical predictions. Since the steady-state responses are monochromatic, the analytical solutions accurately predict the transmissibility and nonreciprocity measures with small relative errors. As a comparison, the NN also captures the nonreciprocal results.

Fig. 11
Numerical comparison of case A of Fig. 10: (a) Desired region predicted by the NN, (b) desired region predicted by the analytical solution, (c) time series of x0, y0 at point A1 for τ∈[400,450], (d) time series of x0, y0 at the point A1 with flipped (σ1,σ2) for τ∈[400,450], (e) the input and downstream energies at point A1, and (f) the input and downstream energies at point A1 with flipped (σ1,σ2)
Fig. 11
Numerical comparison of case A of Fig. 10: (a) Desired region predicted by the NN, (b) desired region predicted by the analytical solution, (c) time series of x0, y0 at point A1 for τ∈[400,450], (d) time series of x0, y0 at the point A1 with flipped (σ1,σ2) for τ∈[400,450], (e) the input and downstream energies at point A1, and (f) the input and downstream energies at point A1 with flipped (σ1,σ2)
Close modal
Table 4

Nonreciprocity and transmissibility measures for different systems computed through direct numerical simulations, NN outputs, and analytical predictions

System A1System B1System C1
Performance of the waveguideηδηδηδ
Numerical simulations0.3951.9230.2482.5010.2201.159
NN outputs0.3811.8620.1642.2560.2121.122
Analytical predictions0.4101.9300.2532.460No stable solution
System A1System B1System C1
Performance of the waveguideηδηδηδ
Numerical simulations0.3951.9230.2482.5010.2201.159
NN outputs0.3811.8620.1642.2560.2121.122
Analytical predictions0.4101.9300.2532.460No stable solution

In Figs. 12(a) and 12(b), we plot the desired regions predicted by the analysis and the NN for case B. We observe a desired region predicted by the analytical results for σ1 close to zero and σ2 in the approximate range −1.3 to −1.8. This region, however, is not predicted by the NN. Therefore, we present numerical simulation results at one representative value in the analytical desired region denoted by B1 with (σ1,σ2)=(0,1.5). The time series of the displacements of the gate oscillators and the energy flow are shown in Figs. 12(c) and 12(e), respectively. To highlight the strong nonreciprocal acoustics in this case, the simulation results of B1 for the case of flipped (σ1,σ2) are shown in Figs. 12(d) and 12(f). We notice the nonreciprocal energy transfer from left to right but not for right-to-left. The numerical transmissibility and nonreciprocity measures are listed in Table 4 with η=0.248 and δ=2.501, indicating the desired nonreciprocal performance. We note that the desired performance is captured by the analytical solutions but not by the NN (with predicted η=0.164 and δ=2.256). Despite the false negative prediction at B1, we note that the predicted transmissibility measure is close to the threshold η=0.2. Since the sensitivity of the NN classification is 0.827, the NN captures most of the desired responses in the parameter domain, and the prediction of the major robust desired region should be acceptable. From Fig. 10(d), we note that the desired region at point B is sensitive to the excitation parameters, and therefore the NN prediction is not necessarily valid. This reveals the limitation of the NN in predicting the nonreciprocal responses that are sensitive to input coefficients, which does not conflict with its good overall performance.

Fig. 12
Numerical comparison of case B of Fig. 10: (a) Desired region predicted by the NN, (b) desired region predicted by the analytical solution, (c) time series of x0, y0 at point B1 for τ∈[400,450], (d) time series of x0, y0 at the point B1 with flipped (σ1,σ2) for τ∈[400,450], (e) the input and downstream energies at point B1, and (f) the input and downstream energies at point B1 with flipped (σ1,σ2)
Fig. 12
Numerical comparison of case B of Fig. 10: (a) Desired region predicted by the NN, (b) desired region predicted by the analytical solution, (c) time series of x0, y0 at point B1 for τ∈[400,450], (d) time series of x0, y0 at the point B1 with flipped (σ1,σ2) for τ∈[400,450], (e) the input and downstream energies at point B1, and (f) the input and downstream energies at point B1 with flipped (σ1,σ2)
Close modal

In Figs. 13(a) and 13(b), we consider the desired regions for case C. We note that the desired region predicted by the NN is mainly located in the fourth quadrant of the (σ1,σ2) domain. However, the major part of the desired region is not captured by the analytical prediction. Therefore, we present the numerical simulations for (σ1,σ2)=(0.3,0.5) denoted as C1, which is a representative case where the NN and analytical solutions provide different predictions. The direct numerical simulations are performed for C1 and the case of flipped (σ1,σ2). The time series of x0 and y0 are shown in Fig. 13(c) for left to right wave propagation. In this case, the steady-state amplitudes of both responses are strongly modulated. However, in Fig. 13(d) the amplitudes are not modulated for right to left wave propagation and flipped (σ1,σ2). Therefore, the nonreciprocity is observed in terms of the frequency contents of the responses. The energy flows for the excitation applied from both sides are shown in Figs. 13(e) and 13(f) with the strong transmissibility and nonreciprocity evidenced by the measures η=0.220 and δ=1.159 in Table 4. This desired performance is captured by the NN but not by the analytical solutions. Based on the eigenvalue analysis of Eq. (22), the analytical solution at point C1 is unstable. Therefore, the monochromatic ansatz is invalid, and the responses exhibit multiple frequency components since the response amplitudes are modulated as shown in Fig. 13(c).

Fig. 13
Numerical comparison of case C of Fig. 10: (a) Desired region predicted by the NN, (b) desired region predicted by the analytical solution, (c) time series of x0, y0 at point C1 for τ∈[400,450], (d) time series of x0, y0 at the point C1 with flipped (σ1,σ2) for τ∈[400,450], (e) the input and downstream energies at point C1, and (f) the input and downstream energies at point C1 with flipped (σ1,σ2)
Fig. 13
Numerical comparison of case C of Fig. 10: (a) Desired region predicted by the NN, (b) desired region predicted by the analytical solution, (c) time series of x0, y0 at point C1 for τ∈[400,450], (d) time series of x0, y0 at the point C1 with flipped (σ1,σ2) for τ∈[400,450], (e) the input and downstream energies at point C1, and (f) the input and downstream energies at point C1 with flipped (σ1,σ2)
Close modal

In synopsis, the NN simulator is trained and applied to predict the responses of the waveguide and classify the desired results based on the transmissibility and nonreciprocity measures. A high overall performance of the classification is achieved, which is verified through the correlations of the NN output and target data, as well as the confusion matrix. The analytical solutions are also adopted to classify the desired region for monochromatic responses. Through the comparison with direct numerical simulations, we observe that the results agree well. However, high accuracy is achieved in the analytical predictions only in the regime of monochromatic responses. As a comparison, the NN simulator predicts all types of desired responses, with lower accuracy, however, compared to the analytical results. The errors of NN prediction in sensitive regions represent a possible limitation of the machine learning approach.

5 Concluding Remarks

We studied the nonreciprocal acoustics of an infinite linear waveguide subject to a single harmonic excitation, and incorporating a local, asymmetric, nonlinear gate. The grounded oscillators comprising the linear waveguide were strongly coupled, and, as a result, it supported broadband wave transmission in its single passband. This posed distinct challenges in our study, compared to an earlier work [28] where the weakly coupled waveguide was considered, supporting narrowband wave transmission. A strongly nonlinear problem was formulated to explore the nonreciprocal acoustics. First, we numerically showed that this system enabled strong nonreciprocity resulting in energy propagation only in one direction. The transmitted waves exhibited either a single frequency (case of monochromatic response), or multiple frequency components (case of strongly modulated responses, SMRs). Transitions between these two response types were governed by bifurcations, which were analytically studied by CX-A analysis.

The CX-A method, valid for this strongly nonlinear problem, was employed based on a monochromatic response ansatz, yielding analytical approximations based on slow flow (modulation) equations. Stability analysis was also performed, and the stable solutions agreed well with the numerical results. The dramatic change in nonreciprocal transmitted energy originated from the transition between two branches of analytical solutions, while SMRs were generated at the loss of stability of the monochromatic results. The governing mechanism of the transition between different types of responses was clearly explained through the analysis, enabling predictive design of the nonreciprocal gated waveguide.

Since the analytical model was only applicable to monochromatic responses, we proceeded to predict and classify the general class of nonreciprocal waveguides through a machine learning approach which was applicable for both monochromatic responses and SMRs. The NN model first introduced for the case of weak coupling and weak nonlinearity [28] was adopted and extended to the case of strong coupling and strong nonlinearity. Reasonable damping ratios were taken into considerations as well. Following the preliminary training process, the machine learning model well captured the waveguide responses corresponding to strong nonreciprocity and transmissibility. This was verified through comparisons with analytical solutions and direct numerical simulations. Hence, the machine learning model, along with the analytical model, act as powerful tools for the predictive design of the nonreciprocal waveguide for desired nonreciprocity and transmissibility features. In the desired region, the transmissibility measure reaches 0.4 and the nonreciprocity measure reaches as high as 3.

This work demonstrates the design of an otherwise linear dissipative broadband waveguide featuring strong break of acoustic reciprocity, achieved merely by a local nonlinear gate. The machine learning approach can be extended to the design of nonreciprocal waveguides with multiple passbands, and in higher dimensions. The approaches developed in this work can be applied to a broad class of linear waveguides incorporating a local nonlinear element. Therefore, a promising prospect is to design the local dynamics of the nonlinear element to be capable of complicated functions, such as frequency conversion or acoustic logic operation. The main limitations in these studies would be the high dimensionality of the parameter spaces where the NN would be applied which requires a larger training set. Another promising prospect is to effectively presample the data through analytical predictions, thus optimizing the neural net efficiency and reducing the training costs.

Acknowledgment

We would like to acknowledge the useful discussions on the contents of the paper with Ali Kanj at the University of Illinois at Urbana-Champaign. Also, we appreciate the reviewers' comments that help us improve the performance of the neural networks.

Funding Data

  • Division of Emerging Frontiers (Award ID: 1741565; Funder ID: 10.13039/100000156).

A Closed Form Solution of the Slow Flow Equations at the Nonlinear Gate

To complete the analytical solution, we pursue the closed form solution of the gates' slow-flow equations. Without loss of generality, we consider the general form of Eq. (20)
a1x+C(xy)2(x¯y¯)=b1a2y+C(yx)2(y¯x¯)=b2
(A1)
where x and y are unknown complex variables, a1, a2, b1, and b2 are complex coefficients, and C is a real coefficient. Setting u=x+y and v=xy, eq. (A1) are transformed to
a1+a22u+a1a22v=b1+b2a1a22u+a1+a22v+2Cv2v¯=b1b2
(A2)
Eliminating u, we get the following nonlinear cubic equation for v
[a1+a22(a1a2)22(a1+a2)]v+2Cv2v¯=(b1b2)(a1a2)(b1+b2)(a1+a2)
(A3)
Denoting A=[a1+a24(a1a2)24(a1+a2)] and B=(b1b2)2(a1a2)(b1+b2)2(a1+a2), (A3) is rewritten as
Av+Cv2v¯=B
(A4)
Taking the squares of the moduli of both sides of (A4), we obtain
C2(vv¯)3+(A+A¯)C(vv¯)2+(AA¯)(vv¯)BB¯=0
(A5)

We note that Eq. (A5) is a real cubic equation, where the closed form solution for |v|2=vv¯ is given by Cardano's formula. Substituting the closed form solution for vv¯ into Eq. (A1) we get linear equations with respect to x and y, and the closed form solution of equation (A1) is completed.

Stability Analysis of the Truncated Version of Eq. (22)

Given the nonzero dissipation effects, the energy attenuation is not avoidable and the amplitudes at infinity tend to zero. Therefore, it is reasonable to assume the “truncated” boundary conditions ϕxN=ϕyN=0 for sufficiently large N. Hence, the infinite-dimensional dynamical system (22) is approximated with a dynamical system of finite dimensions.

Denoting by φ the vector composed of ϕyN, ϕy(N1), …, ϕy0, ϕx0, …, ϕxN, (22) can be rearranged into the matrix form
φ=Aϕ+Bφ¯
(B1)
where A and B denote matrix coefficients. Taking the complex conjugate of (B1) we obtain the following relation
φ¯=A¯φ¯+B¯φ
(B2)
and combine (B1) and (B2) we obtain
(φφ¯)=(ABB¯A¯)(φφ¯)
(B3)

where Eq. (B3) is in the general form of a linear autonomous dynamical system with constant coefficients. The response of the dynamical system is stable as long as all the eigenvalues of the matrix (ABB¯A¯) have negative real parts.

The Efficacy of Neural Nets With Different Hyperparameters

The performance of different NN configurations (in terms of layers and number of neurons in each layer) with different hyperparameters is considered here. Multiple NN simulators were trained, ranging from 2 to 4 layers, and 20 to 50 neurons per layer. The sensitivity, specificity, precision, and correlations for the different neural nets are listed in Tables 59.

Table 5

The sensitivity of the neural net prediction for the test set

Sensitivity20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.7140.7290.7460.773
3 Layers0.7430.8090.7430.752
4 Layers0.7950.8000.7500.827
Sensitivity20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.7140.7290.7460.773
3 Layers0.7430.8090.7430.752
4 Layers0.7950.8000.7500.827
Table 6

The specificity of the neural net prediction for the test set

Specificity20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.9980.9980.9980.999
3 Layers0.9980.9980.9990.998
4 Layers0.9990.9980.9980.999
Specificity20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.9980.9980.9980.999
3 Layers0.9980.9980.9990.998
4 Layers0.9990.9980.9980.999
Table 7

The precision of the neural net prediction for the test set

Precision20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.7890.7960.7730.860
3 Layers0.7810.7760.8400.759
4 Layers0.8240.7770.7960.844
Precision20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.7890.7960.7730.860
3 Layers0.7810.7760.8400.759
4 Layers0.8240.7770.7960.844
Table 8

The correlation of the transmissibility between the neural net prediction and the numerical simulation for the test set

Correlation of transmissibility20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.9900.9900.9900.990
3 Layers0.9890.9900.9900.988
4 Layers0.9920.9920.9920.992
Correlation of transmissibility20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.9900.9900.9900.990
3 Layers0.9890.9900.9900.988
4 Layers0.9920.9920.9920.992
Table 9

The correlation of the nonreciprocity between the neural net prediction and the numerical simulation for the test set

Correlation of nonreciprocity20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.9890.9890.9900.990
3 Layers0.9900.9900.9900.991
4 Layers0.9920.9920.9920.993
Correlation of nonreciprocity20 Neurons30 Neurons40 Neurons50 Neurons
2 Layers0.9890.9890.9900.990
3 Layers0.9900.9900.9900.991
4 Layers0.9920.9920.9920.993

From these tables, we note that the NN with 4 layers and 50 neurons per layer has the best overall performance in terms of the regression and classification measures. Therefore, the neural network with 4 layers and 50 neurons was adopted in this work to predict the acoustic transmissibility and nonreciprocity of the gated waveguide.

Note that by adding more layers and more neurons, the performance measures are not significantly enhanced. Therefore, we did not train deeper neural networks.

Animations of the Desired Regions

In Fig. 14, the loss function at each epoch is plotted. The loss function of the validation set does not decrease for consecutive 6 epochs after epoch 32. At epoch 32, the normalized mean square error decreases from 104 to 0.028, revealing that the neural network is well trained.

Fig. 14
Loss function of the neural network at each epoch
Fig. 14
Loss function of the neural network at each epoch
Close modal

Animations of the Desired Regions

Here, we provide the links for animations for the kernel sizes of the desired regions in the (σ1,σ2) domain for varying. (Ap,θ,d).

The link for the animations of the kernel sizes of the desired regions predicted by the NN simulator can be found below.1

The link for the animations of the kernel sizes of the desired regions predicted by the analytical solutions can be found below.2

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.10.1007/s11071-011-0028-0