and funded by the Region Pays-de-la-Loire, via the fellowship of the

Iw o u l dl i k ea l s ot oe x p r e s sm ys i n c e r eg r a t i t u d et ot h ec o -d i r e c t o ro fm yt h e s i s Prof. Vitalyi GUSEV for all his advises and encouragements during the three years of PHD which are of great significance to me. I especially appreciated for his patience, enthusiasm, and immense knowledge that he shared with me all along my PhD. I am very lucky to have Vincent and Vitalyi as supervisors, they enlighten me the first glance of research. Without their help, I would not have achieved this work. Ih o p ea l s ot h e r eh a so t h e ro p p o r t u n i t i e si nt h ef u t u r et h a tw ec a nw o r kt o g e t h e r on other subjects. I would like to acknowledge Dr. Badreddine ASSOUAR and Dr. Bruno LOM-BARD for having accepted to be the reviewers of this thesis. I am thankful to Dr. Agnès MAUREL, Dr. Georgios THEOCHARIS and Dr. Hélène PICHARD for their agreements to be the examiners of my thesis.

My sincere thanks also goes to the former post-doc of LAUM, Dr. Alejendro CE-BRECOS RUIZ, who was responsible for the experimental part of LEGOs project presented in Chapter 4. Considering that this PhD thesis is largely a theoretical and numerical work, the exchange with Alejendro enabled to open my mind of experiments. I am also grateful to Dr. Georgios THEOCHARIS and Dr. Vassos ACHILLEOS, with whom I have had interesting discussions about the numerical

Conclusion and prospects

AS o l v i n gn o n l i n e a re q u a t i o ns y s t e mb yh a r m o n i cb a l a n c em e t h o d (HBM) A.1 HBM to solve the equations in lumped element modeling . . . . . . . A.2 HBM to solve the equations in rotating-element metasurface and meta-interface designs . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Figures

6

(a) Bifurcation-based acoustic switching and rectification achieved with a granular crystal chain. 1), Reverse configuration: driving far from the defect, the bandgap filters out vibrations at frequencies in the gap. 2), Forward configuration: driving near the defect, nonlinear modes are generated that transmit through the system. (b)Tunable magneto-granular phononic crystals composed of a chain of spherical steel beads inside a properly designed magnetic field. . . . . . . . . .

Meta-structures with amplitude gaps for elastic solitons, providing new opportunities to manipulate highly nonlinear elastic pulses, as demonstrated by the designed soliton diodes (left) and splitters (right). where nonlinear effects develop efficiently.. . . . . . . . . . . . . . . . 18 1.4 Spectrogram and waveform of (a) normalized incident wave σ inc {σ 0 , of (b) reflected wave normalized by incident amplitude σ ref {σ 0 ,o f (c) and (d) displacements of the two masses, respectively u 1 and u 2 , normalized by the maximum displacement of the first mass maxpU 1 q. These results have been obtained numerically by means of the fourthorder Runge-Kutta method (RK4) with a wave packet source of dimensionless width !T " 2000. The illustrated waveforms have been extracted around the time center t 0 of the source (⌧ " !t 0 " 6000). System parameters are fixed at γ " 0.0162, ⌘ " 0.0088 (corresponding to Q " 80), B 1 " 0.002 and B 2 " 0.. . . . . . . . . . . . . . . . . .21 1.5 Magnitude of the reflection coefficient at the fundamental frequency (three left-hand figures) and the second harmonic frequency (three right-hand figures) obtained from the Fourier spectrum for the entire reflected wave taken at Ω " 2, with various wave packet source widths (N T denotes the number of carrier wave periods within the width at half height of the wave packet source). In all graphs, the default parameters are impedance parameter γ 0 " 0.0162,qualit yfactorQ " 80 (ô ⌘ 0 " 0.0088), and nonlinear parameters B 1 " B 0 1 " 0.002, B 2 " 0. Otherwise, all parameter values are indicated in the graph legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [START_REF] Agranovich | Linear and nonlinear wave propagation in negative refraction metamaterials[END_REF] 1. [START_REF] Liu | Locally resonant sonic materials[END_REF] The Fast Fourier Transform (FFT) of normalized reflected wave σ ref {σ 0 present around the fundamental harmonic ! for various source widths, with N T denoting the number of periods at half height of theincident stress wave and using parameters of the system are fixed as above: γ " 0.0162, ⌘ " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.. . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.7 The FFT of normalized reflected wave σ ref {σ 0 present around the second harmonic 2! for various source widths, with N T denoting the number of periods at half height of the incident stress wave.and using parameters of the system are fixed as above: γ " 0.0162, ⌘ " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.. 24

8S o f t a r c h i t e c t e d m a t e r i a l s t h a t s u p p o r t t h e

1.8 Temporal signals of the wave packet source σ inc with the number of periods at half height equal to N T " 4,o ft h ec o r r e s po n d i n gn o r m a lized reflected wave σ ref {σ 0 "a n do ft h en o r m a l i z e dd i s p l a c e m e n t so f two masses U 1 and U 2 (with normalization U i " u i {u 0 , pi " 1, 2q). The second harmonic component (in red lines) for the reflected wave and for the displacements σ 2ω ref and U 2ω i are obtained by applying around 2! (from 1.5! to 2.5!)ab a n d p a s sfi l t e rt oe a c ho r i g i n a lt e m p o r a l signals respectively. Using the parameters of the system are fixed as above: γ " 0.0162, ⌘ " 0.0088 (corresponding to Q factor equal to 80), B 1 " 0.002 and B 2 " 0.. . . . . . . . . . . . . . . . . . . . . . [START_REF] Li | Double-negative acoustic metamaterial[END_REF] 1.9 Temporal signals of the wave packet source σ inc along with the number of periods at the half height equal to N T " 20,o ft h ec o rresponding normalized reflected wave σ ref {σ 0 ,a n do ft h en o r m a lized displacements of two masses U 1 and U 2 (with normalization U i " u i {u 0 , pi " 1, 2q). The second harmonic component (shown in red lines) for both the reflected wave and for the displacements σ 2ω ref and U 2ω i is obtained by applying around 2! (from 1.5! to 2.5!) ab a n d p a s sfi l t e rt oe a c ho r i g i n a lt e m po r a ls i g n a l ,r e s pe c t i v e l y ,i nu sing the system parameters fixed as above: γ " 0.0162, ⌘ " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.. 26 2.1 Nonlinear metasurface design: (a) Single layer of periodically arranged rigid squares sandwiched between a moving rigid plate and a fixed wall, with elastic springs running between all the elements. The propagation medium (1) in front of the designed metasurface is assumed to be semi-infinite. (b) The metasurface unit cell is composed of two identical squares with elementary mass m. The front rigid plate has an elementary mass 2m 0 .( c )D u et os y m m e t r y ,t a king into account the motion of just one square of mass m and the face plate with a mass per unit length of m 0 is demonstrated to be sufficient for the considered reflection problem at normal incidence. . [START_REF] Cai | Optical cloaking with metamaterials[END_REF] 2.2 Eigenfrequencies and eigenvectors of the considered metasurface. In the linear dissipative regime, three resonance frequencies ! i (i " 1, 2, 3)a r ep r e s e n t e d( a ) ,r e l a t e dt oe i t h e rat r a n s l a t The ratio of ✓ to U 1 is examined as a function of ✓ 0 and K s " K θ as well, at resonance frequencies ! θ and ! u in (c) and (d), respectively. The optimal value range of stiffness and initial angle of rotation is indicated by the white dotted line in both (c) and (d). . . . . . . . . [START_REF] Craster | Acoustic Metamaterials-Negative Refraction, Imaging, Lensing and Cloaking[END_REF] 2.3 Absolute reflection coefficients of the fundamental and second harmonic components, denoted R 1 and R 2 respectively, as a function of both the dimensionless impedance parameter γ and the normalized excitation frequency detuning ∆Ω. The latter is defined as the difference between the excitation frequency ! and the linear resonance frequency ! θ ,subsequen tlynormalizedb y! 0 ,i.e.,∆Ω "p! ´!θ q{! 0 .

When the input intensity is relatively weak, with a magnitude A inc " 5 ¨10 ´6,t h er e q u i r e de x c i t a t i o nd e t u n i n gi sl e s s( ∆Ω "1 ¨10 ´4), as the maximum value of R 2 exceeds 0.4 (a) and (b). Whereas with as t r o n g e rs o u r c eo fm a g n i t u d eA inc " 5 ¨10 ´5,af r e q u e n c yd e t u n i n g of around ∆Ω "1.7 ¨10 ´3 is needed to totally absorb R 1 , which does not alter the amplitude of the second harmonic R 2 to reach a maximum value of nearly 0.8 (c) and (d). . . . . . . . . . . . . . . . . [START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF] 2.4 Kinetic energy of the metasurface at various excitation levels, from a linear configuration with A inc " 10 ´7 to a weakly nonlinear configuration with A inc " 5 ¨10 ´6 and a highly nonlinear configuration A inc " 5¨10 ´5,r espectively ,forthecasesof: (a)excitationfrequencies close to the linear rotation-dominated resonance frequency ! θ ,a n d (b) excitation frequencies close to the linear translation-dominated resonance frequency ! u verifying ! u " 2! θ .D u r i n gt h ek i n e t i ce nergy test, the metasurface is excited, at each excitation frequency, by 1,000 periods of a sine signal. The dotted black line in (a) indicates the resonance shift under excitation A inc " 5¨10 ´5, which corresponds exactly to the optimal excitation detuning introduced in Fig. 2.3. . . [START_REF] Li | A sound absorbing metasurface with coupled resonators[END_REF] 2.5 Absolute reflection coefficient of fundamental (R 1 )a n ds e c o n dh a rmonics (R 2 ), as investigated by varying the excitation amplitude from al i n e a rl e v e l( A inc " 10 ´7)t oan o n l i n e a rl e v e l( A inc " 10 ´4). Frequency detuning is introduced in order to eliminate reflection of the fundamental wave at the desired excitation amplitude, such that (a) A inc " 5 ¨10 ´6 and (b) A inc " 5 ¨10 ´5, with the impedance parameter defined as γ " 0.008 and γ " 0.0195,r e s pe c t i v e l y . . . . . . . . . . . . [START_REF] Pichard | Surface waves in granular phononic crystals[END_REF] 2.6 Theoretical and numerical results for the nonlinear metasurface reflection, as obtained with the Harmonic Balance Method (HBM) and the fourth-order Runge-Kutta (RK4) method, respectively, for the case of a relatively weak excitation with amplitude A inc " 5 ¨10

´6

(a) and (b), and for the case of a stronger excitation with amplitude A inc " 5 ¨0´5 (c) and (d). The frequency axes are normalized by the detuned excitation frequency. By considering a wave packet source with characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results. Magnitudes of Short-Term Fourier Transforms taken at fundamental and second harmonic frequencies, by showing the temporal variation in the reflected wave spectrum (a) and (c), and by exploring the Fourier Transform of the central 1,000 periods of the reflected temporal wave (b) and (d), respectively. . . .

2.7

Optimal frequency conversion effect achieved for various physical properties of the proposed metasurface, under an excitation level of A inc " 5 ¨10 ´5. The maximum absolute value of the reflection coefficient of the second harmonic R 2 is identified by varying the impedance parameter and excitation frequency detuning simultaneously, for stiffnesses in the range K s " K θ Pp 0, 0.1q and an initial angle of rotation set at 10 o and 20 o in (a), and for an initial angle of rotation in the range ✓ 0 Pp0 o , 30 o q and stiffnesses set at 0.02 and 0.06 in (b). Shaded zones indicate the optimal ranges of initial angle of rotation and stiffnesses, i.e. K s " K θ Pp 0, 0.04q and ✓ 0 Pp 3 o , 15 o q,t h u sy i e l d i n ga ne ffi c i e n t second harmonic reflection with R 2 greater than 0.4. The results presented have been output by HBM. . . . . . . . . . . . . . . . . . . . .

2.8

Optimal frequency conversion effect achieved for various metasurface unit cell shapes, i.e. for different inertial moments ↵ of rotating elements. The maximum conversion is determined as a function of ↵ over the range of r0.02, 0.66s by varying the impedance parameter γ, excitation frequency detuning ∆Ω and stiffness K s " K θ simultaneously. In order to lessen the calculation burden, the initial angle of rotation has been set at 10 o and 8 o respectively, as these values are found to be favorable for producing the desired reflection over the entire inertial moment range. . . . . . . . . . . . . . . . . . . . . . .

x LIST OF FIGURES 3.1 Different types of structures and different excitation configurations under consideration for the proposed meta-interface design. The symmetrical structure (with identical mass of the two moving plates) is investigated in Section 3.3, the configurations of a single excitation at one side (a) and of a dual-excitation at two sides (b) are taken into account and explored respectively in Subsections 3.3.1 and 3.3.2. For the asymmetrical structure of meta-interface characterized by moving plates with different masses, the investigations performed in Section 3.4 are similar to the symmetrical structure, with consideration of single excitation (Subsection 3.4.1) (c) and of dual-excitation (Subsection 3.4.2) (d), respectively. Throughout this chapter, we are interested in the maximum energy conversion from fundamental incoming wave to the transmitted/reflected second harmonic. . . . . . . . . . . [START_REF] Mahmoud | All-passive nonreciprocal metastructure[END_REF] 3.2 Nonlinear meta-interface design, by (a) a monolayer of periodically arranged rigid squares sandwiched between two moving rigid plates, with elastic contact between all the elements. The designed metainterface is inserted between two identical semi-infinite propagation media p1q and p2q. The unit cell of the meta-interface is made of two identical squares with elementary mass m and two moving plates connected in front and back having elementary mass 2m 1 and 2m 2 respectively (b). Due to the symmetry, taking into account the movement of only one square m and the face blocs m 1 and m 2 is demonstrated sufficient for the considered 1D scattering problem at normal incidence (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [START_REF] Cabaret | Nonlinear hysteretic torsional waves[END_REF] 3.3 Eigenfrequencies and eigenvectors of the considered meta-interface in the linear lossless regime. When the eigenfrequencies Ω 2 and Ω 3 satisfy the condition Ω 3 " 2Ω 2 ,t h ea b s o l u t ev a l u eo fd i s p l a c e m e n t ratio between ✓ and U n (n=1,2) is investigated by varying the initial angle ✓ 0 and the stiffness K s " K θ ,a tf r e q u e n c y( a )Ω 2 and (b) Ω 3 ,r e s p e c t i v e l y . I nt h ec o n s i d e r e dv a l u er a n g eo f✓ 0 Pp 0 o , 30 o q and K s " K θ Pp 0, 0.2q,t h ee i g e n f r e q u e n c yΩ 2 corresponds always to a rotation-dominated movement since the minimum absolute value of ✓{U n is around 2.7 at this frequency, whereas the eigenfrequency Ω 3 relates either to a translation-dominated movement or to a rotationdominated movement depending on the values of system parameters. The white dotted lines in (a) and (b) show the threshold of displacement ratio ✓{U n equal to 15 at Ω 2 , while the red dotted lines represent the case of ✓{U n " 1 at Ω 3 .. . . . . . . . . . . . . . . . . . . . . . . . 52 xi 3.4 Maximum kinetic energy of the meta-interface with symmetrical structure, under monochromatic excitation of weak magnitude A inc " 10 ´7 (linear configuration), and of significant magnitude A inc " 5 ¨10

´5

(nonlinear configuration), respectively, in the case of (a) excitation frequencies close to the linear rotation-dominated resonance frequency Ω θ ,a n do f( b )e x c i t a t i o nf r e q u e n c i e sc l o s et ot h el i n e a rt r a n s l a t i o ndominated resonance frequency Ω u verifying Ω u " 2Ω θ .D u r i n gt h e kinetic energy test, the meta-interface is excited, for each excitation frequency, by a single monochromatic source of 1000 periods length at the left side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Magnitude of reflection and transmission coefficients for the fundamental wave (R 1 and T 1 )a n df o rt h es e c o n dh a r m o n i c( R 2 and T 2 ), investigated by varying the excitation amplitude from linear level (A inc " 10 ´7)t os t r o n gn o n l i n e a rl e v e l( A inc " 10 ´4). The frequency detuning is introduced to minimize the transmission of fundamental wave at excitation magnitude A inc " 5 ¨10 ´5,p r o v i d e dt h a tt h e transmission of second harmonic can not be reduced. Impedance parameter is chosen at γ " 0.013 enabling the maximization of T 2 in the condition that the stiffness and the initial rotation angle are defined as K s " K θ " 0. to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the fundamental reflected wave at excitation magnitude A inc " 5 ¨10 ´5,p r o v i d e dt h a tt h er e fl e c t i o no fs e c o n dh a rmonic is not reduced. Impedance parameter is chosen at γ " 0.02 enabling the maximization of T 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and ✓ 0 " 10 o respectively. . . . 64 3.12 Theoretical and numerical 1D scattering results of the proposed nonlinear meta-interface with symmetrical structure, obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, under two identical excitations with same magnitude of A inc " 5 ¨10 ´5 at two sides, in the case of K s " K θ " 0.05, ✓ 0 " 10 o and γ " 0.02. The optimal conversion is determined as the function of ↵ in the value range of r0.03, 0.66s, by varying simultaneously all other physical parameters, i.e., the impedance parameter γ,t h ee x c i t a t i o nf r e q u e n c yd e t u n i n g∆Ω,t h e stiffness K s " K θ and the initial angle ✓ 0 .. . . . . . . . . . . . . . . . 67 3.15 Absolute reflection and transmission coefficients of fundamental (R 1 and T 1 )andsecondharmonic(R 2 and T 2 ), investigated by varying the excitation amplitude from linear level (A inc " 10 ´7)t ostrongnonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the transmission of fundamental wave at excitation magnitude A inc " 5 ¨10 ´5,p r o v i d e dt h a tt h et r a n s m i s s i o no fs e c o n dh a r m o n i ci s not reduced. Impedance parameter is chosen at γ " 0.013 enabling the maximization of T 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and ✓ 0 " 10 o respectively. . . . . . . . . . 69 3.16 Theoretical and numerical 1D scattering results of the asymmetrical nonlinear meta-interface (r m " 5), obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, for the configuration of a single side excitation with magnitude of A inc " 5 ¨10 ´5, in the case where K s " K θ " 0.05, ✓ 0 " 10 o and γ " 0.013, and where the introduced detuning is ∆Ω " 6 ¨10 ´4. and with mass ratio between two moving plates defined as r m " 5. The maximum conversion is determined as the function of ↵ in the range of r0.03, 0.66s,b yv a r y i n ga l lt h eo t h e r physical parameters, i.e., the impedance parameter γ,t h ee x c i t a t i o n frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle ✓ 0 . 73 3.20 Optimal frequency conversion achieved for the asymmetrical rotatingsquare architected meta-interface with moving plates of different mass in front and back defined by the mass ratio r m " ↵ m2 {↵ m1 ,u n d e r as i n g l es i d ee x c i t a t i o no fm a g n i t u d eA inc " 5 ¨10 ´5,t h ei n e r t i a l moment ↵ is fixed for homogeneous square ↵ " 1{3. The maximum conversion is determined as the function of r m in the range of r1.5, 10s, by varying all the other parameters, i.e., the impedance parameter γ, the excitation frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle ✓ 0 in their considered value ranges respectively. . . . . . 74 The frequency detuning is introduced with an amount of ∆Ω " 1.1 ¨10 ´3 to minimize the reflected fundamental wave at excitation magnitude A inc " 5 ¨10 ´5,p r o v i d e d that the reflection of second harmonic is not lowered (for both sides). Impedance parameter is chosen at γ " 0.0228 yielding the maximization of R 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and ✓ 0 " 10 o respectively. . . . . . . . . . . . . . . . 77

Theoretical and numerical results of the nonlinear meta-interface 1D

scattering, obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, for the configuration of two excitations with both a magnitude of A p1q inc " 5 ¨10 ´5, in the case where K s " K θ " 0.05, ✓ 0 " 10 o and γ " 0.0228, and where the introduced detuning is ∆Ω " 1.1 ¨10 ´3. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). By considering a wave packet source with a characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 p erio ds of reflected temp oral wave (b) and (d). All incident and reflected waves are normalized by the square root of input intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.24 Optimal frequency conversion achieved for different physical properties of the proposed asymmetrical meta-interface excited from both sides. The maximum reflection coefficient of second harmonic R p2q 2 at right side of meta-interface is identified by varying the impedance parameter, the excitation magnitude ratio and the excitation frequency detuning simultaneously, with stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively. . . . . . . . . . . . 79 3.28 Optimal frequency conversion achieved for the asymmetrical rotatingsquare architected meta-interface with moving plates of different masses in front and back defined by the mass ratio r m " ↵ m2 {↵ m1 ,undert w o excitations at both sides, the inertial moment ↵ is fixed for homogeneous square ↵ " 1{3. The maximum conversion is determined as the function of r m in the range of r1.5, 10s,b yv a r y i n ga l lt h eo t h e r physical parameters, i.e., the impedance parameter γ,t h ee x c i t a t i o n magnitude ratio R amp ,t h ee x c i t a t i o nf r e q u e n c yd e t u n i n g∆Ω,t h e stiffness K s " K θ and the initial angle ✓ 0 .. . . . . . . . . . . . . . . .

Introduction

Phononic crystals and acoustic metamaterials are artificially structured composite materials which have been proposed and developed at an increasing pace over the past two decades. They can manipulate and control the wave propagation in ways that were not possible in conventional materials. Phononic crystals are identified by a periodically arranged structure made of scatterers with dimensions and periods comparable to the wavelength. This type of designed materials possesses a number of important properties such as band gaps [1] (see Fig. 1), i.e., waves do not propagate over specific frequency ranges. Compared to the phononic crystals, one can define acoustic metamaterials as structured on a scale that is significantly smaller than the wavelength of the affected waves. Acoustic metamaterials exhibit new features, most of the time associated to local resonances, that can lead for instance to zero or negative values for the effective mass density as well as bulk modulus. Consequently, the acoustic metamaterials offer new opportunities for unusual wave control at subwavelength scales due to their novel and unique properties as well as the modeling and concepts developed in this framework [2,3,4,5]. The first generation of sonic metamaterials, fabricated by Z. Liu et al. [START_REF] Liu | Locally resonant sonic materials[END_REF] with al a t t i c ec o n s t a n tt w oo r d e r so fm a g n i t u d es m a l l e rt h a nt h ei n c o m i n gw a v e l e n g t h (see Fig. 2), exhibits full spectral gaps, resulting from the local resonator-based construction that enables negative elastic constants in certain frequency ranges. Following on this seminal research, a plethora of novel acoustic phenomena have been demonstrated in appropriately designed metamaterials made of different local resonators. For instance, the phenomenon of acoustic transparency and slow sound INTRODUCTION propagation can be realized by a wave guide periodically loaded with Helmholtz resonators [START_REF] Santillán | Acoustic transparency and slow sound using detuned acoustic resonators[END_REF][START_REF] Theocharis | Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures[END_REF][START_REF] Santillán | Demonstration of slow sound propagation and acoustic transparency with a series of detuned resonators[END_REF], the perfect sound absorption in a broadband frequency range is achievable via different sub-wavelength devices or metamaterials based on resonators such as Helmholtz or membrane-type resonators [START_REF] Romero-García | Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators[END_REF][START_REF] Jiang | Ultra-broadband absorption by acoustic metamaterials[END_REF][START_REF] Yang | Subwavelength perfect acoustic absorption in membrane-type metamaterials: a geometric perspective[END_REF][START_REF] Mei | Dark acoustic metamaterials as super absorbers for low-frequency sound[END_REF][START_REF] Li | Acoustic metasurface-based perfect absorber with deep subwavelength thickness[END_REF][START_REF] Yang | Subwavelength total acoustic absorption with degenerate resonators[END_REF][START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-films[END_REF][START_REF] Sheng | Optimal sound-absorbing structures[END_REF][START_REF] Yang | Sound absorption structures: From porous media to acoustic metamaterials[END_REF], ap r o p e r l yd e s i g n e dm e t a -s t r u c t u r em a d eo fp e r i o d i c a l l ya r r a n g e dr e s o n a n tu n i t si s capable of guiding the acoustic wave propagation [START_REF] Devaux | Asymmetric acoustic propagation of wave packets via the self-demodulation effect[END_REF][START_REF] Lemoult | Wave propagation control at the deep subwavelength scale in ametamaterials[END_REF][START_REF] Lemoult | Soda cans metamaterial: A subwavelength-scaled phononic crystal[END_REF], anomalous refractions such as negative refraction (wave refracted on the same side of the surface normal as the incoming wave) can be induced using a metamaterial designed to realize a 'double negative' material (negative values for both the effective mass density and the effective bulk modulus) [START_REF] Agranovich | Linear and nonlinear wave propagation in negative refraction metamaterials[END_REF][START_REF] Eleftheriades | Negative-Refraction Metamaterials: Fundamental Principles and Applications[END_REF][START_REF] Yao | Optical negative refraction in bulk metamaterials of nanowires[END_REF][START_REF] Li | Double-negative acoustic metamaterial[END_REF][START_REF] Smith | Metamaterials and negative refractive index[END_REF][START_REF] Christensen | Metadevices for the confinement of sound and broadband double-negativity behavior[END_REF][START_REF] Kaina | Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials[END_REF][START_REF] Christensen | Anisotropic metamaterials for full control of acoustic waves[END_REF]. Furthermore, the artificial metamaterials allow as well the design of cloaking devices that render a sub-wavelength or macroscopic object invisible [START_REF] Cai | Optical cloaking with metamaterials[END_REF][START_REF] Valentine | An optical cloak made of dielectrics[END_REF][START_REF] Zhu | Designing a square invisibility cloak using metamaterials made of stacked positive-negative index slabs[END_REF][START_REF] Zigoneanu | Three-dimensional broadband omnidirectional acoustic ground cloak[END_REF][START_REF] Craster | Acoustic Metamaterials-Negative Refraction, Imaging, Lensing and Cloaking[END_REF]. Acoustic metasurfaces are metamaterials extending mostly in two-dimensions, with advantages of deep sub-wavelength spatial features and sometimes easier fabrication compared to the 3D structures. They have shown their ability to manipulate waves in a much smaller volume than bulk metamaterials, through the reflection, transmission and refraction processes. This family of planar metamaterials possessing average thickness drastically reduced, favors development of metamaterial-based devices, applications and basic scientific research. For instance, a perfect absorption at a desired frequency range can be realized via different ultra-thin meta-films without yielding major obstacles in the real applications [START_REF] Lagarrigue | Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions[END_REF][START_REF] Lagarrigue | Sustainable sonic crystal made of resonating bamboo rods[END_REF][START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF][START_REF] Li | Acoustic metasurface-based perfect absorber with deep subwavelength thickness[END_REF][START_REF] Li | A sound absorbing metasurface with coupled resonators[END_REF], as the example illustrated in Fig. 3. Metamaterials owning a 2D structure are also particularly advantageous for the investigations of surface wave propagation [START_REF] Pichard | Surface waves in granular phononic crystals[END_REF]. Another prominent role of metasurfaces is to reveal abnormal wave scattering phenomena by taking advantage of their simple constructions, such as the aforementioned negative refraction [START_REF] Kaina | Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials[END_REF][START_REF] Li | Theory of metascreen-based acoustic passive phased array[END_REF][START_REF] Eleftheriades | Negative-Refraction Metamaterials: Fundamental Principles and Applications[END_REF][START_REF] Smith | Metamaterials and negative refractive index[END_REF][START_REF] Wang | Subwavelength diffractive acoustics and wavefront manipulation with a reflective acoustic metasurface[END_REF], acoustic superlens (focusing beyond the diffraction limit) or wave conversion (propagative to evanescent for example) [START_REF] Kaina | Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials[END_REF][START_REF] Li | Theory of metascreen-based acoustic passive phased array[END_REF][START_REF] Christensen | Metadevices for the confinement of sound and broadband double-negativity behavior[END_REF][START_REF] Gusev | Resonant mechanical meta-interface suppressing transmission of acoustic waves without mode conversion[END_REF]. Different from all above anomalous wave manipulations performed by adjusting the shape and the arrangement of the constituting linear resonant units, O. R. Bilal et al. have started recently to target the nonlinear phononic metasurface [START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF], which opens up avenues for novel acoustic wave control. It is important to notice that the nonlinear metamaterials can offer a rich and diverse set of non-trivial acoustic phenomena. For instance, coupling a periodic or superlattice structure (a wave filter) with a nonlinear medium layer (or element) INTRODUCTION enables an asymmetric transmission of acoustic wave (transmission only allowed in one direction, yielding total reflection in the other direction) [START_REF] Liang | An acoustic rectifier[END_REF][START_REF] Devaux | Asymmetric acoustic propagation of wave packets via the self-demodulation effect[END_REF][START_REF] Boechler | Bifurcation-based acoustic switching and rectification[END_REF][START_REF] Mahmoud | All-passive nonreciprocal metastructure[END_REF], as shown in Fig. 4. Phononic crystals containing specific nonlinearity offer the opportunity to support nonlinear pulse and soliton propagation [START_REF] Cabaret | Nonlinear hysteretic torsional waves[END_REF][START_REF] Daraio | Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals[END_REF]. Moreover, other nonlinear phenomena such as higher harmonic generation / enhancement [START_REF] Grubsky | Glass micro-fibers for efficient third harmonic generation[END_REF][START_REF] Sun | Enhanced second-harmonic generation from nonlinear optical metamagnetics[END_REF][START_REF] Shadrivov | Second-harmonic generation in nonlinear left-handed metamaterials[END_REF] and breathers [START_REF] Wang | Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear schrödinger equation[END_REF][START_REF] Van Gorder | Breathers and nonlinear waves on open vortex filaments in the non relativistic abelian higgs model[END_REF] are revealed as well via metamaterials containing different types of nonlinearities, although rarely reported for acoustic waves in air [START_REF] Quan | A nonlinear acoustic metamaterial: Realization of a backwards-traveling second-harmonic sound wave[END_REF][START_REF] Quan | Quasi-phase-matched backward secondharmonic generation by complementary media in nonlinear metamaterials[END_REF][START_REF] Rudenko | A nonlinear screen as an element for sound absorption and frequency conversion systems[END_REF]. Acoustic metamaterials with tailored nonlinear responses, yet providing unexplored wave control opportunities, have received particular attention in this PhD thesis work. Nevertheless, regarding planar or two-dimensional acoustic metamaterials, the nonlinear studies are quite scarce in comparison to the linear analysis which is carried out extensively. Indeed, the nonlinear metasurfaces have been firstly investigated and developed in optics since they can exhibit relatively strong nonlinear optical responses [START_REF] Chen | Enhanced Second-Harmonic Generation by Metasurface Nanomixer and Nanocavity[END_REF][START_REF] Chandrasekar | Second harmonic generation with plasmonic metasurfaces: direct comparison of electric and magnetic resonances[END_REF][START_REF] Agranovich | Linear and nonlinear wave propagation in negative refraction metamaterials[END_REF][START_REF] Biris | Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires[END_REF][START_REF] Li | Nonlinear photonic metasurfaces[END_REF]. Researchers in optics have devoted special attention to the quest of an appropriate-structured metasurface providing nonlinearity enhancement at moderate input intensities [START_REF] Lee | Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions[END_REF][START_REF] Hsiao | Fundamentals and applications of metasurfaces[END_REF][START_REF] Krasnok | Nonlinear metasurfaces: a paradigm shift in nonlinear optics[END_REF]. The main approach to achieve this issue focuses on judiciously engineering the geometry, the interaction and the arrangement of meta-atom building blocks that determine the nonlinear properties of synthesized metamaterials, i.e., what can be called the nonlinearity management. Figure 5 illustrates two examples of second harmonic enhancement via different metasurface constructions. Therefore, following the current progress concerning nonlinear metasurfaces in optics, this PHD research is dedicated to initiate the design of acoustic nonlinear metasurfaces that enable unusual wave control. This requires to develop or use strategies for elastic or acoustic nonlinearity management, for the design of planar metamaterial structures exhibiting specific nonlinear scattering processes.

1D 2D 3D a Γ Γ M X Wavevector, k PBG Δ ω ω a/2π c
The key limitations in the development of nonlinear acoustic metamaterials and of their 2D planar counterparts, nonlinear acoustic metasurfaces, are related to the generally weak efficiency of their nonlinear response, together with the lack of control on this nonlinearity as mentioned above. Previous examples of tailoring the acoustic or elastic wave nonlinearity of a system are found in granular crystals [START_REF] Boechler | Bifurcation-based acoustic switching and rectification[END_REF][START_REF] Allein | Tunable magnetogranular phononic crystals[END_REF][START_REF] Cabaret | Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime[END_REF]. The nonlinear properties of granular chains rely on the interaction between the grains or beads, exhibited as the Hertz-Mindlin contact behavior. Figure 6 presents two examples of such granular crystal chains composed of spherical beads, that either enable the achievement of acoustic switching and rectification of mechanical waves, or ensure their suitability as nonlinear and tunable mechanical metamaterials for use in controlling elastic wave propagation. However, the nonlinearity tunability of this type of granular structure is intrinsically limited due to the specific nonlinear contact behavior among the building units. Hence, being able to manage the wave nonlinearity of a system and enhance the associated wave phenomena, over a wider parameter space, appears as the main challenge for the development of nonlinear acoustic metamaterials. It constitutes as well the subject of the proposed research on nonlinear acoustic metasurface designs.

(b) (a)

f c f d f dr f dr f d f c
2), 1), Figure 6: (a) Bifurcation-based acoustic switching and rectification achieved with a granular crystal chain [START_REF] Boechler | Bifurcation-based acoustic switching and rectification[END_REF]. (b)Tunable magneto-granular phononic crystals composed of a chain of spherical steel beads inside a properly designed magnetic field. [START_REF] Allein | Tunable magnetogranular phononic crystals[END_REF].

INTRODUCTION

and acoustic waves [START_REF] Hussein | Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook[END_REF][START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Shan | Multistable architected materials for trapping elastic strain energy[END_REF][START_REF] Florijn | Programmable mechanical metamaterials[END_REF][START_REF] Bertoldi | Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[END_REF][START_REF] Wang | Harnessing buckling to design tunable locally resonant acoustic metamaterials[END_REF][START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF]. The intrinsic structure and property of this class of architected materials are not only changeable by harnessing elastic buckling resulting from different statically produced pre-deformations [START_REF] Shan | Multistable architected materials for trapping elastic strain energy[END_REF][START_REF] Wang | Harnessing buckling to design tunable locally resonant acoustic metamaterials[END_REF][START_REF] Bertoldi | Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[END_REF], but are also configurable over a range of behaviors by taking advantage of geometric nonlinearities presented by the basic building blocks [START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF][START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF]. For instance, the latest research by Deng et al. [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] shows that the soft metamaterial composed of periodically arranged rotating crosses possesses a robust feature of amplitude gaps (ranges of pulse amplitudes where solitons are not stable) with gap width tunable via both the structural property variation of the units and the symmetry breaking in the underlying geometry. This enables the manipulation of highly nonlinear elastic pulses, e.g. soliton splitters and diodes, as illustrated in Fig. 7. Hence, the architected soft materials provide the opportunity to expanding the ability of existing metamaterial, and make them capable of supporting a wide variety of dispersive and nonlinear wave propagation. They are definitely a source of inspiration for the present research work.

Amplitude A Figure 7: Meta-structures with amplitude gaps for elastic solitons, providing new opportunities to manipulate highly nonlinear elastic pulses, as demonstrated by the designed soliton diodes (a) and splitters (b) [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF].

1 2 2N 2N + 1 2N + 2N a + 1 4N + 2N a + 1 20°- 20°0 °Unit number i 2 × N 2 1 + × N a 2 × N T =
Throughout this manuscript, as mentioned previously, we focus on the design of 2D nonlinear metamaterials, i.e., the nonlinear metasurface or meta-interface, targeting the enhancement of a given nonlinear response. More specifically we analyze the efficient energy conversion from a fundamental incoming wave towards the second harmonic wave during the scattering process of the acoustic wave by the metasurface. This considered phenomenon could have potential applications in various fields, for instance, for the construction of the harmonic images in medical imaging [START_REF] Meairs | 44 -ultrasonography[END_REF][START_REF] Hedrick | Tissue harmonic imaging: A review[END_REF] and for improving the noise control with strategies that are different from absorption alone [START_REF] Yang | Sound absorption structures: From porous media to acoustic metamaterials[END_REF]. However, these nonlinear metasurfaces are not limited to this specific effect and several other nonlinear effects could be studied as well. Nevertheless, the conversion to second harmonic wave is the starting point of this PhD and requires nonlinearity management of the metasurface, design of a realistic structure, development of analysis tools, as would require the study of other nonlinear effects. As the usually first manifesting feature of general classical nonlinearity, the quadratic nonlinearity has been given widespread attention and has been analyzed recently only in optics and electromagnetics in the context of metasurfaces [START_REF] Shadrivov | Second-harmonic generation in nonlinear left-handed metamaterials[END_REF][START_REF] Konishi | Polarization-Controlled Circular Second-Harmonic Generation from Metal Hole Arrays with Threefold Rotational Symmetry[END_REF][START_REF] Zeng | A classical theory for second-harmonic generation from metallic nanoparticles[END_REF][START_REF] Xiong | Strongly enhanced and directionally tunable secondharmonic radiation from a plasmonic particle-in-cavity nanoantenna[END_REF][START_REF] Sajedian | High efficiency second and third harmonic generation from magnetic metamaterials by using a grating[END_REF]. The desired acoustic nonlinear metasurface focusing on the second harmonic enhancement, has hardly been explored in acoustics, thus requiring first and foremost a theoretical comprehension and investigation.

Unusual effective properties leading to interesting wave effects or to wave control capabilities are one of the first motivations for the study of metamaterials and in particular acoustic metamaterials [4,2,[START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF][START_REF] Popa | Tunable active acoustic metamaterials[END_REF][START_REF] Craster | Acoustic Metamaterials-Negative Refraction, Imaging, Lensing and Cloaking[END_REF]. The correspondence between the architecture at the micro-scale and the desired effective properties at macro-scale should be constantly achieved. This can be achieved for instance by taking the longwavelength limit of a periodic system with a known behavior of the unit cell [START_REF] Colestock | Modeling of active and passive nonlinear metamaterials[END_REF][START_REF] Filonov | Resonant meta-atoms with nonlinearities on demand[END_REF][START_REF] Bobrovnitskii | A discrete model of damped acoustic metamaterials[END_REF][START_REF] Kosevich | Resonance absorption, reflection, transmission of phonons and heat transfer through interface between two solids[END_REF][START_REF] Bobrovnitskii | Models and general wave properties of two-dimensional acoustic metamaterials and media[END_REF][START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-films[END_REF]. Another way is to apply homogenization methods, as proposed for acoustic metasurfaces [START_REF] Marigo | An Interface Model for Homogenization of Acoustic Metafilms,c h .1 4[END_REF][START_REF] Mercier | Influence of the neck shape for helmholtz resonators[END_REF][START_REF] Lombard | Numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain[END_REF]. For the analysis of nonlinear resonant acoustic metasurfaces, little has been done so far, and the questions we wanted to address at the beginning of this PhD work can be formulated as follows: what sort of theoretical and numerical tools can be used to analyze "model" nonlinear metasurfaces? Are nonlinear acoustic metasurfaces plausible and what kind of nonlinear effects can be targeted? What are the conditions and limits to observe nonlinear effects at an acoustic metasurface? Can we manage the elastic nonlinearity and propose a realistic design for a nonlinear metasurface structure?

To this end, based on resonant nonlinear elastic elements, we start in Chapter 1b yp r o p o s i n gal u m p e d -e l e m e n tt h e o r e t i c a lm o d e lo fam e t a s u r f a c e . Ac l a s s ical quadratic nonlinearity is assumed to be carried directly by the elastic elements (springs) composing the metasurface resonating units. Within this chapter, under the hypothesis that the considered metasurface is connected to a rigid wall at one end, we try to investigate and characterize the nonlinear reflection process in order to determine the operating conditions enabling the second harmonic enhancement. However, it is necessary to notice that the modeling is based on lump elements, requiring to be tuned finely to achieve the desired effects. Indeed, the realization of the described metasurface model is relatively difficult, since the way of designing such elastic springs with effective quadratic nonlinearity remains to be explored.

Therefore, in the following Chapter 2, inspired by the recent research on architected soft materials [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] illustrated in Fig. 7 and Fig. 8, we present a realistic design of a nonlinear metasurface comprising rotating elements. In this case, the nonlinearity is of geometrical nature, and can be controlled by the rational conception of the structure. The specific nonlinear phenomenon reported in Chapter 1, i.e., complete conversion from fundamental incoming wave to the reflected second harmonic achieved through the reflection process, is further analyzed by considering the here-proposed metasurface design. The characteristic parameters of excitation that affect the nonlinearity implementation, such as the excitation magnitude and the excitation frequency detuning, are herein taken into account and evaluated in realistic value ranges. At the end of this chapter, the parameter space of considered problem is explored in the aim of determining the favorable range that enables the desired nonlinear wave manipulation. We then show that elastic nonlinearity INTRODUCTION management can be finely achieved in a metasurface configuration.

(a) (b)

Figure 8: Soft architected materials that support the propagation of elastic vector solitons [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF].

Following the work of metasurface design of Chapter 2 in which only the reflection process is considered, we continue in Chapter 3, by introducing a similar rotating-element based meta-interface design, to investigate several nonlinear frequency conversion effects via both wave reflection and transmission processes. The analysis is divided into two parts depending on the meta-interface structures, i.e., symmetrical or asymmetrical. For both the discussed types of meta-interface designs, as i n g l ee x c i t a t i o nc o n fi g u r a t i o na to n es i d eo fm e t a -i n t e r f a c ea n dad u a l -e x c i t a t i o n performed at both sides are studied respectively. The parameter space evaluation adopted in Chapter 2 is re-implemented and completed by additional consideration of other parameters, such as the mass ratio between plates contained in the designed structure and the magnitude difference between sources in dual-excitation configuration, contributing to expand the optimal parameter conditions that provide a considerable nonlinear effect.

The actually very range of possible nonlinear phenomena is hitherto restricted to the frequency conversion of fundamental wave to the second harmonic through scattering process, chosen since the second harmonic generation is a classical effect representative of nonlinear wave behaviors. Nevertheless, a variety of non-trivial acoustic phenomena can be observed with the reported rotating-element architected metamaterial. Thus in chapter 4, a meta-structure composed of rotating crosses is constructed. The theoretical and numerical analysis of the proposed artificial material is carried out in parallel with experiments conducted by other members of LAUM I collaborated with. One of the targeted effects among the complex nonlinear dynamics of the considered material, is the DC shrinking of the chain when excited harmonically. I have contributed to this preliminary work on the theoretical and numerical aspects, to derive the dispersion properties of the infinitely-long structure, and describe the shrinking phenomenon. These preliminary results are presented and discussed. The experimental and analysis tasks are still ongoing and should be completed in the next months in a collaborative work.

Conclusions and prospects of this PhD work are finally presented at the end of the document.

INTRODUCTION

Chapter 1

Manipulating acoustic wave reflection by a nonlinear elastic metasurface In this first chapter, the modeling of a nonlinear elastic metasurface based on elastic resonators is developed. In the conventional theoretical analysis of metamaterials, the nonlinear behaviors are exhibited via either resonant units owning particular shapes [START_REF] Zeng | A classical theory for second-harmonic generation from metallic nanoparticles[END_REF][START_REF] Litchinitser | Optical meta-atoms: Going nonlinear[END_REF][START_REF] Lan | Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered Helmholtz resonators[END_REF][START_REF] Mercier | A two-way model for nonlinear acoustic waves in a non-uniform lattice of helmholtz resonators[END_REF], or a nonlinear type of elastic contact between building blocks [START_REF] Cveticanin | Negative effective mass in acoustic metamaterial with nonlinear mass-in-mass subsystems[END_REF][START_REF] Lan | Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered Helmholtz resonators[END_REF][START_REF] Bonello | Ultra-low and ultra-broadband nonlinear acoustic metamaterials[END_REF][START_REF] Rudenko | A nonlinear screen as an element for sound absorption and frequency conversion systems[END_REF][START_REF] Cabaret | Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime[END_REF][START_REF] Boechler | Bifurcation-based acoustic switching and rectification[END_REF]. Accordingly, for the metasurface modelling performed in this chapter, we adopt a direct introduction of elastic elements that act nonlinearly. More specifically, we propose an architected metasurface model composed of a two degree-of-freedom mass-spring system with quadratic elastic nonlinearity carried by the springs. The possibility of converting, during the reflection process, most of the fundamental incoming wave energy into the second harmonic wave is shown, both theoretically and numerically, by means of the proposed proper design of metasurface. The theoretical results from the harmonic balance method for a monochromatic source are compared with time domain simulations for a wave packet source. The following presentation corresponds to the published article [START_REF] Guo | Manipulating acoustic wave reflection by a nonlinear elastic metasurface[END_REF]. Furthermore, the executed protocol allows analyzing the dynamics of the nonlinear reflection process in the metasurface as well as exploring the limits of the operating frequency bandwidth. The reported methodology can be applied to a wide variety of nonlinear metasurfaces, thus possibly extending in the future the family of exotic nonlinear reflection processes.

Introduction

The ability of locally-resonant architected materials to achieve wave control at wavelengths much longer than the dimensions of the resonant elements has been demonstrated and utilized extensively over the past several years [2,[START_REF] Craster | Acoustic Metamaterials-Negative Refraction, Imaging, Lensing and Cloaking[END_REF]3,[START_REF] Oudich | A sonic band gap based on the locally resonant phononic plates with stubs[END_REF][START_REF] Lagarrigue | Sustainable sonic crystal made of resonating bamboo rods[END_REF][START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF]. Slow sound [START_REF] Santillán | Acoustic transparency and slow sound using detuned acoustic resonators[END_REF][START_REF] Theocharis | Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures[END_REF][START_REF] Santillán | Demonstration of slow sound propagation and acoustic transparency with a series of detuned resonators[END_REF], negative refraction [START_REF] Agranovich | Linear and nonlinear wave propagation in negative refraction metamaterials[END_REF][START_REF] Eleftheriades | Negative-Refraction Metamaterials: Fundamental Principles and Applications[END_REF][START_REF] Yao | Optical negative refraction in bulk metamaterials of nanowires[END_REF][START_REF] Li | Double-negative acoustic metamaterial[END_REF][START_REF] Smith | Metamaterials and negative refractive index[END_REF][START_REF] Christensen | Metadevices for the confinement of sound and broadband double-negativity behavior[END_REF], sub-wavelength wave guiding and multiplexing [START_REF] Devaux | Asymmetric acoustic propagation of wave packets via the self-demodulation effect[END_REF][START_REF] Lemoult | Soda cans metamaterial: A subwavelength-scaled phononic crystal[END_REF], are all among the recently reported effects of significant interest. This sub-wavelength range of operations is especially pertinent for layers made of locally-resonant elements [START_REF] Li | Theory of metascreen-based acoustic passive phased array[END_REF][START_REF] Lagarrigue | Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions[END_REF], denoted as metasurfaces. As such, the average thickness can be drastically reduced, which is advantageous e.g. for sound absorption [START_REF] Romero-García | Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators[END_REF][START_REF] Jiang | Ultra-broadband absorption by acoustic metamaterials[END_REF][START_REF] Yang | Subwavelength perfect acoustic absorption in membrane-type metamaterials: a geometric perspective[END_REF][START_REF] Mei | Dark acoustic metamaterials as super absorbers for low-frequency sound[END_REF][START_REF] Li | Acoustic metasurface-based perfect absorber with deep subwavelength thickness[END_REF][START_REF] Yang | Subwavelength total acoustic absorption with degenerate resonators[END_REF][START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-films[END_REF][START_REF] Sheng | Optimal sound-absorbing structures[END_REF], carpet cloaking [START_REF] Cai | Optical cloaking with metamaterials[END_REF][START_REF] Valentine | An optical cloak made of dielectrics[END_REF][START_REF] Zhu | Designing a square invisibility cloak using metamaterials made of stacked positive-negative index slabs[END_REF] or other purposes. The key challenges ahead in improving and applying the proposed wave control designs, based on metamaterials, are mainly: i) the operating bandwidth, which is often limited to the resonance frequency range; ii) the tunability of the metamaterial response; and iii) the nonlinear (amplitude-dependent) response, as found to be particularly relevant for intense sound waves. Recent research has primarily sought to overcome the first two of these listed challenges [START_REF] Popa | Tunable active acoustic metamaterials[END_REF][START_REF] Bilal | Bistable metamaterial for switching and cascading elastic vibrations[END_REF][START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF], whereas this paper focuses on the third challenge, i.e. the nonlinear amplitude-dependent response of metamaterials.

Compared to the linear dispersive properties of acoustic metamaterials, the nonlinear wave interaction processes in metamaterials have been studied less extensively. Nevertheless, granular crystals and granular metamaterials are structures whose contact interaction nonlinearity may be efficiently mobilized to produce nonlinear wave processes, such as asymmetric transmission [START_REF] Boechler | Bifurcation-based acoustic switching and rectification[END_REF][START_REF] Devaux | Asymmetric acoustic propagation of wave packets via the self-demodulation effect[END_REF][START_REF] Li | Broadband asymmetric acoustic transmission in a gradient-index structure[END_REF][START_REF] Chen | Asymmetric transmission of acoustic waves in a layer thickness distribution gradient structure using metamaterials[END_REF][START_REF] Christensen | Anisotropic metamaterials for full control of acoustic waves[END_REF], nonlinear pulse and soliton propagation [START_REF] Cabaret | Nonlinear hysteretic torsional waves[END_REF][START_REF] Daraio | Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals[END_REF][START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF], harmonic generation [START_REF] Grubsky | Glass micro-fibers for efficient third harmonic generation[END_REF][START_REF] Sun | Enhanced second-harmonic generation from nonlinear optical metamagnetics[END_REF] and breathers [START_REF] Wang | Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear schrödinger equation[END_REF][START_REF] Van Gorder | Breathers and nonlinear waves on open vortex filaments in the non relativistic abelian higgs model[END_REF] ... While these granular structures are among the most widely studied nonlinear elastic engineered materials for waves and despite their rich behavior, the nonlinear parameter space of granular systems is highly constrained by the intrinsic Hertz-Mindlin contact nonlinearity. Moreover, the metasurfaces configuration, in the form of a sub-wavelength layer, does not favor a priori the accumulation of nonlinear effects along distances, as classically observed in homogeneous media [START_REF] Biris | Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires[END_REF][START_REF] Hamilton | Nonlinear Acoustic[END_REF]. Recent results on architected soft solids [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Shan | Multistable architected materials for trapping elastic strain energy[END_REF][START_REF] Florijn | Programmable mechanical metamaterials[END_REF][START_REF] Bertoldi | Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[END_REF][START_REF] Wang | Harnessing buckling to design tunable locally resonant acoustic metamaterials[END_REF][START_REF] Cabaret | Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime[END_REF] however have demonstrated some ways of managing the dynamic elastic nonlinearity and offered other ways of designing nonlinear resonating elements for elastic and acoustic wave control in propagation or in metasurface configurations.

This part of work sets out to show that unusual reflection effects by a nonlinear metasurface can indeed be modeled and predicted. More specifically, it demonstrates the ability to avoid reflection at the fundamental incident frequency and to convert most of the energy in the reflection process into the second harmonic wave. The metasurface configurations explored are found to be realistic for subsequent implementation in experimental testing. The theoretical analysis methodology developed can be applied to other nonlinear metasurface designs and other nonlinear effects. The first part of this paper studies the case of a reflected monochromatic incident stress wave, while the second part numerically analyzes the nonlinear reflection of a wave packet, in addition to studying the frequency bandwidth character (or timedomain effects) of the nonlinear reflection process.

The problem under consideration and the corresponding metasurface design

We consider herein the problem of wave reflection by a sub-wavelength thickness metasurface, in a one-dimensional configuration, i.e. with normal incidence on the flat surface. The incoming wave is a longitudinal scalar wave, such as an acoustic wave in a fluid or a pure longitudinal stress wave in a homogeneous solid. The propagation medium is assumed to be semi-infinite. The unit cell of the metasurface is composed of two elementary masses (m 1 and m 2 )connectedtotwononlinearsprings (K NL 1 and K NL 2 )a n dv i s c o u sd a m p e r s( Γ), as shown in Fig. 1.1. The metasurface thickness is assumed to be much less than the wavelength in the propagation medium (1). Springs and dampers are regularly positioned over the metasurface, with each occupying a lateral surface S.

A quadratic nonlinearity is considered for both springs: this nonlinearity follows af o r c e -d i s p l a c e m e n tl a we x p r e s s e da sF i " K i p∆`iq`β i K i p∆`iq 2 (i=1,2) with ∆`i being the elongation of spring i and β i the quadratic nonlinear parameter.

The metasurface is inserted between a semi-infinite propagation medium p1q and ar i g i dw a l lp2q.L e t ' sc o n s i d e rap l a n es t r e s s -w a v eσ inc of amplitude σ 0 incoming from ´8 and propagating along the positive x direction. The problem is therefore one-dimensional, and the incident and reflected waves can be written as a function of x ´ct and of x `ct,r e s pe c t i v e l y( u s i n gt h et i m ec o n v e n t i o ni!t), with c the wave velocity in the propagation medium. The total stress σ can be decomposed into an incoming stress-wave and a reflected stress-wave σ " σ inc `σref ; the following can now be written: Bσ Bx "1 c

Bσ inc

Bt `1 c

Bσ ref

Bt . The one-dimensional wave equation, ⇢ B 2 ux Bt 2 " Bσ Bx , with ⇢ the mass density of the propagation medium, must be satisfied on the metasurface at x " 0, which leads to the following useful relation:

σ ref " σ inc `⇢c Bu 1
Bt . The system of metasurface motion equations can thus be written in the following form:

$ & % m 1 B 2 u 1 Bt 2 "´`2σ inc `⇢c Bu 1 Bt ˘S ´K1 pu 1 ´u2 q´Γ Bpu 1 ´u2 q Bt ´β1 K 1 pu 1 ´u2 q 2 , m 2
B 2 u 2 Bt 2 " K 1 pu 1 ´u2 q`Γ Bpu 1 ´u2 q Bt `β1 K 1 pu 1 ´u2 q 2 ´K2 u 2 ´Γ Bu 2 Bt ´β2 K 2 u 2 2 , (1.1)
with S being the characteristic lateral surface area of each metasurface element, and

u i (i=1,2) the displacement of mass m i .
The analysis is carried out here for metasurface parameters defined using ratios between the two masses, i.e. m 2 {m 1 " 2,a n dt h et w ol i n e a rs p r i n gc o n s t a n t s , i.e. K 2 {K 1 " 2, while the dashpots are characterized by a damping coefficient Γ identical for both. Consequently, the proposed interface design, in the linear case, leads to a dual-resonance system characterized by the following relation between the two angular resonance frequencies

! 2 " 2! 1 " 2 a K 1 {2m 1 .
A n e x a m p l e o f such a metasurface response function in the case without coupling in present of a propagation medium (vacuum) is shown in Fig. 1.1(c). In using the first resonance frequency ! 1 ,let'sdefinethedimensionlessimpedanceparameterγ representing the ratio of the impedance of the propagation medium to the mechanical impedance of the metasurface as follows:

γ " ⇢cS 2m 1 ! 1 . (1.2)
The metasurface absorption parameter is defined as,

⌘ " Γ 2m 1 ! 1 . (1.3)
Using this expression, let's now define the quality factor Q " 1{ ? 2⌘ which quantifies the effect of viscous damping in the metasurface based on the expression for a single damped mass-spring system. Moreover, let's define the dimensionless nonlinear parameters (or amplitude parameters of our problem): B i " β i u 0 , with u 0 " σ 0 S{K 1 . The motion equation system Eq. (1.1) can then be rewritten with dimensionless parameters as follows:

$ & % 1 2 Ω 2 B 2 U 1 Bτ 2 "´2f p⌧ q´γΩ BU 1 Bτ ´pU 1 ´U2 q´⌘Ω BpU 1 ´U2 q Bτ ´B1 pU 1 ´U2 q 2 , Ω 2 B 2 U 2 Bτ 2 "pU 1 ´U2 q`⌘Ω BpU 1 ´U2 q Bτ `B1 pU 1 ´U2 q 2 ´2U 2 ´⌘Ω BU 2 Bτ ´2B 2 U 2 2 , (1.4
) where ⌧ " !t is the dimensionless time, Ω " !{! 1 is the normalized excitation frequency, f p⌧ q"σ inc p⌧ q{σ 0 is the normalized incident stress wave at the interface x " 0,a n dU i " u i {u 0 (i=1,2) is the normalized displacement of each mass m i .

In the weakly nonlinear regime of the metasurface operation, let's assume that the reflected wave spectrum from a monochromatic incident wave will contain, at the first order, combination frequencies of !,i . e . h a r m o n i c so ft h ei n c i d e n tw a v e . Consequently, at the boundary x " 0,t h ec o m p l e xa m p l i t u d eo ft h er e fl e c t e ds t r e s s wave is written as r

σ R " r σ 0 ∞ N n"1 r
R n pnΩqe inΩτ , with r σ 0 being the complex amplitude of the incident wave. Here, r R n pnΩqp1 § n § N q actually corresponds to the complex amplitudes of each reflected harmonic relative to the incident wave amplitude. In the following, for the sake of simplicity, r R n pnΩq will denote the complex reflection coefficient of the n-th harmonic.

Theoretical results and parametric analysis: case of a monochromatic source

In the case of a monochromatic source, i.e. f p⌧ q"cosp⌧ q,t h ec o n s i d e r e dm o t i o n equation system Eq. (1.4) can be solved by using the Harmonic Balance Method (HBM) [START_REF] Marinca | The Method of Harmonic Balance[END_REF] (see Appendix A.1). According to this method, the solution U i is developed in the form of a sum of all harmonics generated:

U i p⌧ q"U i0 `N ÿ n"1 rC in cospn⌧ q`S in sinpn⌧ qs, (1.5)
with U i0 indicates the constant terms, C in and S in the magnitudes of the sinusoidal terms cos and sin,r e s pe c t i v e l y ,a n dN the finite number of harmonics being considered. In the present study, which deals with weak quadratic nonlinearity, we verified that N " 10 is always sufficient since it yields results with relative error of less than 10 ´15 as compared to N " 9.B ym e a n so ft h i se x p l i c i te x p r e s s i o no ft h es o l u t i o n , the system Eq. (1.4) is simplified and capable of being solved numerically by applying the classical Newton-Raphson method. The complex reflection coefficients of each harmonic component n are then deduced as follows:

r R n " δ n1 `iγnΩpC 1n ´iS 1n q, (1.6)
where δ n1 is the Delta function, which is always zero except when n " 1. The results obtained are considered to be the theoretical. The Section 1.4 will compare these results to the case of a wave packet source in order to study the effects of finite bandwidth.

In the present study, the excitation frequency ! is always set equal to the first resonance frequency ! 1 of the linearized metasurface, i.e. the normalized excitation frequency is Ω " 1.A c c o r d i n g t o t h e t h e o r e t i c a l r e s u l t s p r o d u c e d b y t h e HBM method (Fig. 1.2), in order to obtain an optimal generation of the second or third harmonic component, the nonlinear parameters B i need to be carefully chosen. When the two springs of the model have the same nonlinearity (B 1 " B 2 ), higher harmonics are not necessarily generated during the reflection process, see for exam-

ple the value of | r R 2 | along the diagonal B 1 " B 2 in Fig. 1.2(b).
To enhance the nonlinear process of second harmonic generation, the difference between nonlinear parameters B 1 and B 2 must be as large as possible. In the following study therefore, we have set B 2 " 0,m o r e o v e r ,t h em a x i m u mv a l u eo fB 1 is defined such that the ratio of the nonlinear part of the elastic force to its linear part is approx. 0.1, which means that the nonlinearity remains weak. For the illustrated case in Fig. 1.2 with an impedance parameter γ " 0.0162 and an absorption parameter ⌘ " 0.0088,t h e defined maximum value of B 1 equal roughly 0.002.

Furthermore, for the prop osed linear prop erties of the metasurface, the second resonance lies at a frequency corresponding to twice that of the first resonance, i.e. ! 2 " 2! 1 . Consequently, when the system is excited at the first resonance frequency ! " ! 1 , the second harmonic, which is generated at 2! (and thus effectively "reflected") due to the quadratic nonlinearity, coincides with the second resonance frequency of the metasurface. Thanks to this selected resonance frequency matching and with appropriate nonlinear parameters (e.g. B 2 ! B 1 † 0.02 see Fig. 1.2(c), the reflected second harmonic can thus be well amplified. Simultaneously, we have found that the other higher harmonics are nearly all missing, even more interestingly, the fundamental wave has almost been entirely eliminated during the reflection, i.e.

| r R 1 |!1.
Let's now examine the role of the impedance parameter γ and the absorption parameter ⌘ on this nonlinear process of reflection, which converts a large amount of the energy from the incoming fundamental wave into the second harmonic reflected wave. By studying the linear case (B i " 0 with i " 1, 2)f o rt h ed e s i g n e di n t e r f a c e , it is possible to determine the characteristic times of each resonance: at the first resonance frequency ! 1 ,t h ed i m e n s i o n l e s sc h a r a c t e r i s t i ct i m e so fa b s o r p t i o n( l o s s e s due to the dashpots) and impedance (losses due to radiation in the propagation medium) are ⌧ abs

1 ⌧ i " 1 ⌧ abs i `1 ⌧ imp i .
(1.7)

In the linear case, the reflection coefficient of the fundamental wave can be obtained analytically in the following form:

Ä R 1 " p1{2qpΩ 2 ´1qpΩ 2 ´4q`i⌘Ωp´2Ω 2 `3q´iγΩp´Ω 2 `3q´⌘p⌘ ´2γqΩ 2 p1{2qpΩ 2 ´1qpΩ 2 ´4q`i⌘Ωp´2Ω 2 `3q`iγΩp´Ω 2 `3q´⌘p⌘ `2γqΩ 2 .
(1.8) Hence, without nonlinearity, when the excitation occurs at the first resonance frequency (Ω " 1), the reflection can be eliminated if the characteristic impedance time is equal to the characteristic absorption time ⌧ imp 1 " ⌧ abs 1 ,i . e . e q u i v a l e n tt o ⌘ " 2γ. This condition is highlighted in Fig. 1.3(a) with a dashed line, and the corresponding computed values of | r R 1 | are observed to be very low. For the studied quadratic nonlinear case (with B 1 ‰ 0 and B 2 " 0,t h ed i m e n s i o n l e s sc h a r a c t e ristic time of nonlinearity has also been defined as: ⌧ NL " 1{ ? B 1 , which conveys an analogous physical meaning to the shock formation characteristic distance for a nonlinear propagating wave [START_REF] Hamilton | Nonlinear Acoustic[END_REF]: the nonlinear effects can efficiently develop for characteristic times of metasurface vibration longer than ⌧ NL .O n ec o n s e q u e n c eo f this approach is that ⌧ NL † ⌧ i pi " 1, 2q is required for a nonlinear effect to efficiently develop, i.e. before the resonance vanishes. This condition for the significant nonlinear effect development can be verified with the results from Fig. 1.3(a),(b). In the cross-hatched region of Fig. 1.3(a), where

⌧ NL † ⌧ 1 , | r R 1 | is no longer zero along the dashed line ⌧ imp 1 " ⌧ abs
1 ,t h u sd e v i a t i n gf r o mt h el i n e a rc a s e . I nF i g .1 . 3 ( b ) , the greatest magnitudes for | R2 | occur in the lower left part of the graph, in the cross-hatched region where the inequality ⌧ NL † ⌧ 2 is satisfied.

More precisely, when

B 1 is set at B 1 " 0.002, ⌧ NL " ⌧ abs 1 (respectively ⌧ NL " ⌧ imp 1
) when ⌘ « 0.134 (respectively γ « 0.067), and

⌧ NL " ⌧ abs 2 (respectively ⌧ NL " ⌧ imp 2 )f o r⌘ « 0.027 (respectively γ « 0.134).
Hence, in order to satisfy the condition ⌧ NL † ⌧ i pi " 1, 2q, γ " ρcS 2m 1 ω 1 and ⌘ " Γ 2m 1 ω 1 should be much less than 1. Physically, this condition means that the propagation medium should actually be much softer than the metasurface. It also means that the metasurface should be weakly dissipative, i.e. the quality factor Q should not be too low, and typically much greater than unity.

In the results presented in Fig. 1.2, the value γ " 0.0162 has been chosen. Considering air as the propagation medium, this value of γ leads to a resonance frequency f r " 2 kHz for a metasurface with a mass per unit area equal to one, which can be achieved with a solid like balsa wood (density of 130kg{m 3 ), and a thickness of 7.7mm.S i m i l a r l y , t h e c h o i c e ⌘ " 0.0088 used for Fig. 1.2 corresponds to a quality factor Q=80. This configuration example, based on such realistic parameter values, shows the potential for applying the presented concept to the nonlinear manipulation of airborne sound. Note that in the linear case (i.e. B 1 " B 2 " 0), the assigned values of parameters γ and ⌘ lead to a fundamental reflection coefficient | r R 1 |«0.57. However, when the nonlinear parameter B 1 is nonzero and limited such that the ratio of the nonlinear part of the elastic force to its linear part is at most 0.1, e.g. B 1 " 0.002, the fundamental reflection can nearly vanish (with | r R 1 |«0.07), while the second harmonic can be efficiently generated and reflected with a reflection coefficient greater than 0.45.A ss u c h ,t h er a t i ob e t w e e n| r R 2 | and | r R 1 | exceeds 6. Therefore, even with very limited nonlinearity (e.g. a nonlinear elastic force ten times smaller than the linear elastic force), a nearly full conversion from the fundamental incoming energy to the second harmonic reflection can be achieved by the proposed metasurface design. The conversion result presented herein can be further improved if the impedance parameter is changed to

γ " 0.0176,t h u sp r o v i d i n gaf u n d a m e n t a l reflection coefficient of | r R 1 |«0.005 and a second harmonic reflection coefficient of | r R 2 |«0.46.
This theoretical study based on the HBM demonstrates a valuable energy transfer, from a fundamental wave to its second harmonic in the reflection process by means of a nonlinear metasurface. The preconditions for efficient conversion are now in place and provide the design rules for metasurface element characterization. These results remain valid for a monochromatic incident wave. The Section 1.4 will focus on analyzing the case of a finite-length wave packet in order to extend the operating conditions of such a nonlinear metasurface and verifying the robustness of the highlighted effects.

Numerical results and parametric analysis with aw a v ep a c k e ts o u r c e

The following discussion will consider a Gaussian modulated wave packet of the form, σ inc p⌧ q{σ 0 " f p⌧ q"sinp⌧ qe

´pτ´τ 0 q 2 pωT q 2 ,
as the incident wave, with ⌧ " !t, T the characteristic temporal width of the wave packet, and ⌧ 0 the dimensionless time center of the packet. A classical fourth-order Runge-Kutta integration method (RK4) [START_REF] Hairer | Runge-Kutta and Extrapolation Methods[END_REF] is used to solve the system of temporal equations, in Eq. (1.1), for all cases presented in this Section. Other numerical integration methods have been implemented to verify these RK4 results, i.e. 6th order Runge-Kutta, Matlab functions ODE45 and ODE133, and Adams methods. By introducing the relation

σ ref " σ inc `⇢c Bu 1
Bt ,thereflectedw a v esignalisobtained once the temporal displacements u i have been determined. The time-frequency analysis of the reflected signals can then be performed using the spectrogram method, yielding in particular a reflected time-and frequency-dependent magnitude | r R|. When the metasurface is excited by a wave packet with a carrier frequency equal to the first resonance frequency of the metasurface (! " ! 1 ), the two masses of the metasurface start vibrating with the same phase and at an amplitude ratio of 2 (corresponding to the eigenmode tu 1 ,u 2 u T 1 "t2, 1u T of the first resonance). During the increase in metasurface vibration amplitude, i.e. as the displacement magnitudes of both masses are rising, higher harmonics are gradually being generated, to an increasing extent, and the mass displacement waveforms are being distorted (see Fig. 1.4). More specifically, as observed in the Section 1.3, among all the higher harmonics generated, energy is mainly converted to the second harmonic component due to frequency matching with the second metasurface resonance, i.e. 2! " ! 2 . At 2! the displacement relationship between the two masses follows the eigenmode tu 1 ,u 2 u T 2 "t 1, 1u T of the second metasurface resonance, i.e. the same displacement magnitude for both masses yet with out-of-phase motion. In Fig. 1.4, the spectrograms and zooms of the waveforms of both the incident and reflected stress waves are plotted, along with the displacements of the two masses.

If the incident wave packet lasts long enough, the theoretical results derived via the HBM in the Section 1.3 should be replicated. This outcome can be verified by monitoring the maximum of | r RpΩq| and of | r Rp2Ωq| from the spectrogram contained in Fig. 1.4 for various temporal widths T of the incident wave packet. A good level of agreement has been obtained between the theoretical HBM results and the temporal simulation for a wave packet when the dimensionless characteristic width !T of the wave packet is much larger than the characteristic lifetime of the metasurface resonances, i.e. !T " ⌧ i with i " 1, 2.W i t h t h e c h o s e n v a l u e s o f i m p e d a n c e parameter γ " 0.0162 and absorption parameter ⌘ " 0.0088,t h el i f e t i m e so ft h e first and second resonances are ⌧ 1 « 72.82 and ⌧ 2 « 49.83.W i t h B 1 " 0.002, the characteristic time of nonlinearity ⌧ NL " 1{ ? B 1 « 22.36, which satisfies the condition ⌧ NL † ⌧ i for high nonlinear effect efficiency. In turn the inequality !T " ⌧ NL with i " 1, 2 needs to be satisfied in order to retrieve the HBM results for continuous excitation with a wave packet of temporal width T . In Fig. 1.4(b), it is observed that the steady state regime is reached at ⌧ " 6000, where the amplitude of the fundamental reflected wave is at a minimum and the amplitude of the reflected second harmonic wave is at a maximum of the two masses, respectively u 1 and u 2 , normalized by the maximum displacement of the first mass maxpU 1 q. These results have been obtained numerically by means of the fourth-order Runge-Kutta method (RK4) with a wave packet source of dimensionless width ωT " 2000. The illustrated waveforms have been extracted around the time center t 0 of the source (τ " ωt 0 " 6000). System parameters are fixed at γ " 0.0162, η " 0.0088 (corresponding to Q " 80), B 1 " 0.002 and B 2 " 0.

widths !T ,w ep e r f o r m e dan u m b e ro fn u m e r i c a ls i m u l a t i o n sf o r15 § !T § 566, i.e. equivalent to 4 § N T § 150, where N T is the number of fundamental carrier wave periods within the packet width at half its maximum amplitude. For an N T typically less than 10 however, the frequency width of each contribution (whether fundamental or second harmonic) cannot be easily separated in the time frequency analysis. Consequently, we opted to monitor the values at Ω " 1 and at Ω " 2 of the The FFT of normalized reflected wave σ ref {σ 0 present around the second harmonic 2ω for various source widths, with N T denoting the number of periods at half height of the incident stress wave.and using parameters of the system are fixed as above: γ " 0.0162, η " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.

The reflection coefficient magnitude at the second harmonic frequency is analyzed for the same metasurface configurations as that of the fundamental frequency. In all cases, the reflection coefficient magnitude | R2 | starts at a value close to zero for small N T values and increases to reach a plateau after N T " 25. The values attained for large N T depend on the configuration and among the presented set of metasurface parameters, the largest | R2 | is obtained for the default parameters γ 0 " 0.0162, Q " 80 and B 0 1 " 0.002.

In the aim of illustrating the spectral and temporal characteristics of the wave packet reflection process, the total signal spectra have been plotted in Fig. 1.6 for frequencies " Ω and in Fig. 1.7 for frequencies " 2Ω.F o u rc h a r a c t e r i s t i cw a v ep a c k e t widths are considered, namely: N T " 4, 20, 100,a n d150.I nF i g .1 . 6 ,t h ee n e r g y absorption and nonlinear energy transfer by the nonlinear metasurface toward the harmonics in the reflection process is displayed by a dip at Ω " 1 in the initial Gaussian spectrum. The nonlinear energy transfer toward the second harmonic is observed in Fig. 1.7 with the spectra displayed for Ω " 2,i . e . o v e raf r e q u e n c y range in which no energy is present in the incident wave packet. The temporal signals associated with these spectra are shown in Figs. 1.8 and 1.9, respectively. It can be observed that for the default set of metasurface parameters, delays occur when establishing the resonances in displacements U 1 and U 2 relative to the incident wave packet, as would be expected for the transient excitation of a resonant system. Consequently, the local minimum in the reflected wave amplitude is also delayed with the respect to the central time of the incident stress wave packet. Lastly, the maximum of the temporal wave packet filtered at the second harmonic frequency is even more heavily delayed, thus demonstrating the additional time required for the nonlinear energy transfer (or nonlinear accumulation time ⌧ NL )i nt h em e t a s u r f a c e . ref and U 2ω i are obtained by applying around 2ω (from 1.5ω to 2.5ω) a bandpass filter to each original temporal signals respectively. Using the parameters of the system are fixed as above: γ " 0.0162, η " 0.0088 (corresponding to Q factor equal to 80), B 1 " 0.002 and B 2 " 0.

Conclusion

In conclusion, through modeling a nonlinear metasurface with a dual-resonance mass-spring system, we have proven both theoretically and numerically the possibility of achieving a near perfect absorption of the incoming fundamental wave together with its efficient conversion into the second harmonic frequency. If the metasurface lies between a relatively soft propagation medium (air for instance) and a rigid wall and moreover if the metasurface exhibits weak intrinsic dissipation (Q " 80), our results indicate that even with a small quadratic nonlinearity (B 1 " 0.002), a reflection amplitude at the fundamental incoming wave frequency of | r RpΩq| « 0.05 is obtained and a reflected second harmonic of amplitude | r Rp2Ωq| « 0.46 can be reached. In order to study the characteristic frequency bandwidth character of this effect, the nonlinear reflection of a wave packet has also been examined via the numerical integration of the metasurface system of nonlinear motion equations. When the characteristic temporal width of the wave packet signal is large in comparison to the lifetimes of the metasurface two resonances (!T " ⌧ i with i " 1, 2), a good level of agreement between the theoretical results obtained by HBM and the implemented numerical results is found, in accordance with expectations. For smaller width however, deviations from the HBM results are observed, indicating that they tend toward the linear reflection results. This funding is explained by the fact that i is obtained by applying around 2ω (from 1.5ω to 2.5ω) a bandpass filter to each original temporal signal, respectively, in using the system parameters fixed as above: γ " 0.0162, η " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.

the excitation time is shorter than the time necessary to accumulate nonlinear effects, i.e. the characteristic time ⌧ NL .

The potentially very wide metasurface design space is limited here to the configuration of a relative abstract lumped-element model, chosen such that its first resonance frequency equals to the excitation frequency and half the second resonance frequency. Consequently, a number of interesting configurations still need to be studied with detuning, for example between the interface resonances or between the excitation frequency and the first resonance frequency. Also, as recently demonstrated in [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF], it is possible to design architected materials in order to achieve the desired type (quadratic, cubic) and amount of elastic wave nonlinearity, in addition to designing the linear dispersive properties. This approach opens up avenues for enhancing the possible wave phenomena induced during the reflection process by a nonlinear metasurface, including but not limited to the wave manipulation of intense sounds, energy mitigation, and the linearization of intense sound resonators.

Chapter 2

Frequency doubling effect in the acoustic reflection by a nonlinear rotating-square architected metasurface

CHAPTER 2. SOFT METASURFACE DESIGN

In the previous Chapter 1, via a lumped-element modeling based on nonlinear local resonators, the ability of an appropriately structured metasurface to achieve unusual wave manipulation, in particular to enable the energy conversion from the fundamental wave to the second harmonic through the reflection process, has been demonstrated. Yet the realization of the given metasurface model is relatively difficult, since the way of introducing the definite type of nonlinearity into elastic elements remains to be explored. Furthermore, as mentioned at the end of Chapter 1, the parameter space of the nonlinear metasurface reflection problem has been limited to a few aspects. The excitation properties that affect nonlinearity manifestation, such as excitation frequency detuning (compared to the metasurface resonance frequency) and the excitation magnitude, need to be taken into account for instance.

Inspired by recent research on soft architected rotating-square structures [START_REF] Guo | Manipulating acoustic wave reflection by a nonlinear elastic metasurface[END_REF], we propose in the present Chapter 2 a realistic design of nonlinear elastic metasurface composed of a single layer of rotating squares connected via thin and highly deformable ligaments placed between them and also to a rigid plate and a wall. During the process of reflection at normal incidence, the designed metasurface is evaluated to achieve the same nonlinear acoustic wave reflection control as previously, i.e., convert most of the incoming fundamental wave energy into the second harmonic wave. We expect herein to extend the reported design of nonlinear acoustic metasurface to a large family of architected structures, by considering an improved parameter space that includes the just-emphasized excitation parameters, with intention to open new ways for realistic metasurface designs that provide nonlinear or amplitude-dependent wave tailoring. The following contents are given in form of the article submitted in Physical Review E.

Introduction

Acoustic metamaterials composed of local resonators have proven to be of great interest, due to their ability to perform a variety of wave control functionalities at wavelengths much longer than the dimensions of the resonant elements. A wide array of novel acoustic phenomena such as slow sound [START_REF] Santillán | Acoustic transparency and slow sound using detuned acoustic resonators[END_REF][START_REF] Theocharis | Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures[END_REF][START_REF] Santillán | Demonstration of slow sound propagation and acoustic transparency with a series of detuned resonators[END_REF], negative refraction [START_REF] Agranovich | Linear and nonlinear wave propagation in negative refraction metamaterials[END_REF][START_REF] Eleftheriades | Negative-Refraction Metamaterials: Fundamental Principles and Applications[END_REF][START_REF] Yao | Optical negative refraction in bulk metamaterials of nanowires[END_REF][START_REF] Li | Double-negative acoustic metamaterial[END_REF][START_REF] Smith | Metamaterials and negative refractive index[END_REF][START_REF] Christensen | Metadevices for the confinement of sound and broadband double-negativity behavior[END_REF][START_REF] Kaina | Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials[END_REF], subwavelength wave guiding [START_REF] Devaux | Asymmetric acoustic propagation of wave packets via the self-demodulation effect[END_REF][START_REF] Lemoult | Wave propagation control at the deep subwavelength scale in ametamaterials[END_REF], sound absorption [START_REF] Romero-García | Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators[END_REF][START_REF] Jiang | Ultra-broadband absorption by acoustic metamaterials[END_REF][START_REF] Yang | Subwavelength perfect acoustic absorption in membrane-type metamaterials: a geometric perspective[END_REF][START_REF] Mei | Dark acoustic metamaterials as super absorbers for low-frequency sound[END_REF][START_REF] Li | Acoustic metasurface-based perfect absorber with deep subwavelength thickness[END_REF][START_REF] Yang | Subwavelength total acoustic absorption with degenerate resonators[END_REF][START_REF] Duan | Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-films[END_REF][START_REF] Sheng | Optimal sound-absorbing structures[END_REF] and cloaking [START_REF] Cai | Optical cloaking with metamaterials[END_REF][START_REF] Valentine | An optical cloak made of dielectrics[END_REF][START_REF] Zhu | Designing a square invisibility cloak using metamaterials made of stacked positive-negative index slabs[END_REF][START_REF] Zigoneanu | Three-dimensional broadband omnidirectional acoustic ground cloak[END_REF] have been demonstrated in appropriately designed metamaterials. Compared to the metamaterials composed of linear resonators, nonlinear metamaterials offer a rich and diverse set of nontrivial acoustic phenomena, including asymmetric transmission [START_REF] Boechler | Bifurcation-based acoustic switching and rectification[END_REF][START_REF] Devaux | Asymmetric acoustic propagation of wave packets via the self-demodulation effect[END_REF][START_REF] Li | Broadband asymmetric acoustic transmission in a gradient-index structure[END_REF][START_REF] Chen | Asymmetric transmission of acoustic waves in a layer thickness distribution gradient structure using metamaterials[END_REF][START_REF] Christensen | Anisotropic metamaterials for full control of acoustic waves[END_REF], nonlinear pulse and soliton propagation [START_REF] Cabaret | Nonlinear hysteretic torsional waves[END_REF][START_REF] Daraio | Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals[END_REF][START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF], harmonic generation [START_REF] Grubsky | Glass micro-fibers for efficient third harmonic generation[END_REF][START_REF] Sun | Enhanced second-harmonic generation from nonlinear optical metamagnetics[END_REF] and breathers [START_REF] Wang | Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear schrödinger equation[END_REF][START_REF] Van Gorder | Breathers and nonlinear waves on open vortex filaments in the non relativistic abelian higgs model[END_REF]. Nevertheless, the design of nonlinear metamaterials, which was initially investigated in optics for the purpose of enhancing the higher harmonic generation [START_REF] Lee | Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions[END_REF][START_REF] Chen | Enhanced Second-Harmonic Generation by Metasurface Nanomixer and Nanocavity[END_REF][START_REF] Chandrasekar | Second harmonic generation with plasmonic metasurfaces: direct comparison of electric and magnetic resonances[END_REF], has been studied much less extensively in the acoustic field.

The key limitations in developing nonlinear acoustic metamaterials pertain to the typically weak efficiency of their nonlinear response, combined with a lack of control over this nonlinearity. Examples of tailoring the acoustic or elastic wave nonlinearity of a system are found in granular crystals, yet the tunability is intrinsically limited due to the Hertz-Mindlin contact behavior [START_REF] Allein | Tunable magnetogranular phononic crystals[END_REF][START_REF] Cabaret | Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime[END_REF]. Being able to manage the wave nonlinearity of a system, over a wider parameter space, thus appears as the main challenge to developing nonlinear acoustic metamaterials.

In studying a lumped-element model of a nonlinear metasurface [START_REF] Guo | Manipulating acoustic wave reflection by a nonlinear elastic metasurface[END_REF], we recently demonstrated that nonlinear acoustic effects can be enhanced in a subwavelength metasurface comprising nonlinear oscillators, thanks to the resonance process. This process intrinsically increases the characteristic interaction times as well as local wave amplitudes. We have reported a nonlinear frequency conversion effect from the incoming fundamental wave to the reflected second harmonic. However, the key link between the lumped-element model of this nonlinear metasurface and a realistic structure is missing. More specifically, the method of designing elastic springs with an effective quadratic nonlinearity still needs to be determined.

Recent research has demonstrated that soft architected materials enable manipulating and controlling elastic and acoustic waves [START_REF] Hussein | Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook[END_REF][START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Shan | Multistable architected materials for trapping elastic strain energy[END_REF][START_REF] Florijn | Programmable mechanical metamaterials[END_REF][START_REF] Bertoldi | Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[END_REF][START_REF] Wang | Harnessing buckling to design tunable locally resonant acoustic metamaterials[END_REF][START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF]. The intrinsic structure and property of this class of architected materials are not only modifiable by harnessing the elastic buckling resulting from different staticallyproduced pre-deformations [START_REF] Shan | Multistable architected materials for trapping elastic strain energy[END_REF][START_REF] Wang | Harnessing buckling to design tunable locally resonant acoustic metamaterials[END_REF][START_REF] Bertoldi | Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures[END_REF], but also dynamically tunable over a broad range of frequencies by taking advantage of geometric nonlinearities in the basic building blocks [START_REF] Bilal | Reprogrammable Phononic Metasurfaces[END_REF][START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF]. As such, these nonlinearities provide the opportunity to expand the ability of existing metamaterials and enable them to support a wide variety of dispersive and nonlinear wave propagation.

Inspired by the latest research on the dynamics of soft architected materials comprising rotating units [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], our attention has been drawn to the fact that the local rotational degree of freedom necessarily leads to the presence of sinusoidal functions of the angle of rotation in the motion equations. These nonlinear functions of wave variables constitute geometric type sources of wave nonlinearity and are found to depend on the building blocks (elasticity, geometry, inertia) of the architected structure. Consequently, in the aim of proposing a realistic design of a nonlinear elastic metasurface that accomplishes the same nonlinear conversion as in [START_REF] Guo | Manipulating acoustic wave reflection by a nonlinear elastic metasurface[END_REF] but with a higher efficiency and over a much larger parameter space, the present paper analyzes a metasurface composed of a single layer of rotating squares connected with thin and highly deformable ligaments and placed between a rigid plate and a wall. Special focus is placed on the nonlinear reflection process, thus leading to an optimal conversion from an incident sinusoidal wave towards its reflected second harmonic. By adjusting the physical properties of the metasurface, the desired nonlinear conversion is demonstrated to be feasible over a wide parameter space, hence enabling the extension of the proposed single design to a family of dynamic rotating-element metastructures. The predictive theoretical framework developed is also expected to help manage the wave nonlinearity by metamaterials and moreover guide future experiments in this field.

Nonlinear elastic metasurface design and the reflection problem at normal incidence

From the previous studies on wave propagation in soft architected materials made of rotating square units [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], we propose herein a realistic design of a nonlinear metasurface. As shown in Fig. 2.1, this design comprises periodically arranged rigid squares, connected via ligaments at their corners to a moving rigid plate (at the front end of the single square layer) and a fixed wall (at the back end of this square layer). The ligaments are considered to be thin, massless and highly deformable, thus playing the role of elastic springs. The metasurface unit cell is composed of two identical squares with elementary masses m sandwiched between the solid plate with a surface mass density of 2m 0 and the rigid wall. The two unit cell squares, featuring the same initial angle of rotation ✓ 0 as defined in Fig. 2.1, are placed in symmetrical positions at rest. Since a horizontal force applied to the plate produces simultaneous square translation and rotation, three different springs are taken into account at each square vertex, i.e. a longitudinal (compression or tension) spring with stiffness k l , a shear spring with stiffness k s , and a bending spring with stiffness k θ .

The proposed metasurface structure is assumed to be infinitely long along the vertical direction y, while the plate thickness along x is assumed to be significantly less than that of the single square layer. The considered design then is a 2D metastructure in the px, yq plane; moreover, the elementary lateral surface area of the metasurface unit cell is denoted S.

Throughout this paper, focus is placed on the reflection at normal incidence by the designed metasurface, with the propagation medium (1) in front of the metasurface assumed to be semi-infinite. The metasurface width h along the x direction is assumed to be much smaller than the acoustic wavelength in medium p1q,i . e . , h ! λ.L e t ' sn o wc o n s i d e rap l a n es t r e s sw a v eσ inc of amplitude A inc incident from ´8 and propagating along the positive x direction. The problem therefore is onedimensional, and the incident and reflected waves can be written as a function of x ´ct and x `ct,respectiv ely(usingthetimecon v en tioni!t), with c denoting wave velocity in the propagation medium. The total stress σ can be decomposed into an incoming stress wave and a reflected stress wave σ " σ inc `σref . The one-dimensional wave equation,

⇢ B 2 u x Bt 2 " Bσ Bx (2.1)
with ⇢ as the mass density of the propagation medium and u x the displacement along the x direction, must be satisfied everywhere and especially on the metasurface at x " 0, which leads to the following relationship between the incident and reflected waves for the considered problem:

σ ref " σ inc `⇢c Bu 1 Bt (2.2)
where u 1 denotes the displacement of the plate with a surface mass density of m 0 . Since the single square layer is periodically arranged and assumed to be infinitely long, with homogenous excitation along y,thet w osquaresofeac hunitcelltranslate with the same displacement and moreover rotate with the same dynamic angle yet in opposite directions. Consequently, the motions of just one square and of its face plate are sufficient to describe the full dynamics.

For a systematic analysis, we introduce the following dimensionless parameters: normalized displacements U i " u i {2l,( i = 1 , 2 )o ft h ep l a t ea n ds q u a r e s ,r e s pe c t i v e l y , with 2l denoting the diagonal length of the squares, pulsation Ω " !{! 0 with ! 0 " a k l {m,t i m e⌧ " ! 0 t,i n e r t i a lm o m e n to fs q u a r e s↵ " J{ml 2 ,n o r m a l i z e ds h e a r , bending stiffnesses K s " k s {k l and K θ " k θ {k l l 2 respectively, and lastly mass ratio ↵ m " m 0 {m.

Based on previous results and validations [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], the springs are assumed to behave linearly and dissipation is accounted for via linear viscous damping associated with the respective translation and rotation motions of each square. The characteristic dissipation parameters Γ u and Γ θ are normalized as ⌘ u " Γ u {m! 0 and ⌘ θ " Γ θ {m! 0 for the translation and rotation, respectively. In the present work, it is considered that the dissipation remains relatively weak with a dimensionless value of ⌘ u " ⌘ θ " 0.001. Thus, for each square and the front plate occupying a lateral surface area S in the py, zq plane, the governing motion equations are written as: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

↵ m B 2 U 1 Bτ 2 "2f inc pΩ,⌧q´γ BU 1 Bτ ´U1 `U2 ´⌘u BU 1 Bτ `⌘u BU 2 Bτ `1 2 `cosp✓ 0 q´cosp✓ 0 `✓q`⌘ θ sin ✓ 0 Bθ Bτ ˘, B 2 U 2 Bτ 2 " U 1 ´2U 2 `⌘u BU 1 Bτ ´2⌘ u BU 2 Bτ , ↵ B 2 θ Bτ 2 "6K θ `✓ `⌘θ Bθ Bτ ˘`2pU 1 `⌘u BU 1 Bτ q sinp✓ 0 `✓q `6sinp✓ 0 `✓q `cosp✓ 0 `✓q´cosp✓ 0 q´⌘ θ sin ✓ 0 Bθ Bτ 2K s cosp✓ 0 `✓q `sinp✓ 0 `✓q´sinp✓ 0 q`⌘ θ cos ✓ 0 Bθ Bτ ˘(2.3)
where f inc pΩ,⌧q"σ inc pΩ,⌧qS{2k l l denotes the normalized force applied to the plate due to the incident stress wave, γ " ⇢cS{m! 0 the dimensionless impedance parameter representing the ratio of the propagation medium p1q impedance to the mechanical impedance of the metasurface. When a normal incident stress wave interacts with the metasurface, and under the condition that the squares are initially rotated at nonzero angles ✓ 0 ,thetranslation of the front plate induces both translation and rotation of the squares, along the x direction and around the z direction, respectively. Under a linear assumption for all springs, the elastic forces applied to each metasurface element are proportional to the spring elongations. However, since the square units rotate, the geometric nonlinearity of the structure is activated due to the sinusoidal dependence of spring deformations on the angle of rotation of the squares, as shown in Eq. (2.3). Interestingly, as a consequence of this geometric nonlinearity dependent on structural design, it can be tuned along with the linear elastic properties in order to produce specific nonlinear wave effects.

In the presence of nonlinearity, the reflected wave spectrum from a monochromatic incident wave at frequency ! may contain harmonics of the incident wave. Consequently, it is assumed that at the boundary x " 0,t h er e fl e c t e dw a v ei s composed of harmonics with the complex amplitude r R n (1 § n § N )r e l a t i v et o the incident wave amplitude, denoted by r A inc . In the following discussion and for the sake of simplicity, r R n and R n will be used to represent the complex reflection coefficient of the n-th harmonic and its magnitude, respectively.

Analysis of the linearized metasurface: Parameter definitions and frequency response

In the linear and weakly dissipative configuration, i.e. with fixed dissipation parameters ⌘ u " ⌘ θ " 0.001 and a linear approximation of trigonometric functions as cosp✓ 0 `✓q«cos ✓ 0 ´sinp✓ 0 q✓ and sinp✓ 0 `✓q«sin ✓ 0 `cosp✓ 0 q✓,t h er e s o n a n c e frequencies ! i (i " 1, 2, 3)o ft h ec o n s i d e r e dm e t a s u r f a c ed e p e n do na l lt h ei n t r i nsic parameters, i.e. the initial angle of rotation ✓ 0 ,t h em a s sr a t i o↵ m ,t h ei n e r t i a l moment ↵, the normalized shear stiffness K s and the bending stiffness K θ .U s i n g realistic materials studied earlier in [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], it is assumed here that the normalized shear and bending stiffnesses are both less than 0.1 and lie at the same value, i.e. K s " K θ § 0.1.A d d i t i o n a l l y ,t h ei n i t i a la n g l eo fr o t a t i o n✓ 0 is set smaller than 30 o . In Sections 2.3 and 2.4" the focus is placed on the case of homogeneous squares, i.e.

↵ " 1{3, though other types of rotating elements with different inertial moments are considered in Section 2.5.

In a previous theoretical study of a lumped-element, dual-resonance elastic metasurface model [START_REF] Guo | Manipulating acoustic wave reflection by a nonlinear elastic metasurface[END_REF], it was demonstrated that to conduct the optimal frequency conversion from fundamental wave to second harmonic through the reflection process, a ratio of 2 between the two linear resonance frequencies of the metasurface is needed. The targeted conversion takes place with an excitation at the first resonance frequency. Regarding the current metasurface design with three degrees of freedom (rotation and translation of the squares, plus translation of the front plate), three resonance frequencies are involved ! i (i " 1, 2, 3) with ! 1 † ! 2 † ! 3 .S i n c et h e (geometric) nonlinearity is primarily excited by the rotation of squares, the excitation frequency ! should coincide with the resonance frequency, denoted as ! θ , which corresponds to a rotation-dominated mode. Moreover, one of the other resonance frequencies, denoted here as ! u ,s h o u l dm a t c h2! θ ,i no r d e rt oa p p r o x i m a t et h e optimal conversion efficiency.

To satisfy the condition ! u " 2! θ ,t h em a s sr a t i o↵ m can be determined in the linear and weakly dissipative case (see Fig. 2.2(b)). By simultaneously varying stiffnesses pK s ,K θ q and initial angle of rotation ✓ 0 in their considered intervals, the eigenmodes can be characterized by the magnitude of the ratio ✓{U 1 taken at the different resonance frequencies. A ratio ✓{U 1 with a magnitude greater than unity indicates a rotation-dominated mode, whereas a translation-dominated mode occurs with a ratio less than unity. Fig. 2.2(c) and Fig. 2.2(d) illustrate this ratio at resonance frequencies ! θ and ! u , respectively. It has been verified that within the considered range of metasurface parameters, the absolute ratio ✓{U 1 is maintained above 3 at ! θ and below 0.5 at ! u ,i.e.,! θ (resp. ! u )correspondstoarotation(resp. translation)-dominated mode.

However, once the displacement ratio ✓{U 1 deviates from unity (with an absolute value becoming much smaller or much larger than unity), the rotation motion and translation motion turn out to be weakly coupled; consequently, the energy transfer from fundamental harmonic to higher harmonics becomes inefficient during the reflection process. In order to excite the rotation mode of the metasurface as much as possible while enhancing the intended nonlinear conversion, the ranges of stiffness and initial angle of rotation are limited, thus allowing for an absolute ratio ✓{U 1 less than 10 at frequency ! θ and greater than 0.1 at frequency ! u .A m o n gt h e chosen displacement ratio threshold values, the optimal range of stiffness and initial angle of rotation values can be obtained, i.e. defined as K s " K θ Pp 0, 0.04q and ✓ 0 Pp 3 o , 15 o q, as enclosed by the white dotted line in Fig. 2.2(c) and Fig. 2

.2(d).

The discussion in Section 2.5 verifies that the above choice of parameter space is indeed realistic and yields a high efficiency for the desired nonlinear conversion.

For the study of the nonlinear case in the following Section 2.4, both the stiffness and initial angle of rotation of the metasurface are set as K s " K θ " 0.02 and ✓ 0 " 10 o , which corresponds approximately to the center of the optimal parameter space region. The corresponding resonance frequencies in the linear regime, as normalized by ! 0 ,a r er e s pe c t i v e l yΩ 1 " 0.7145, Ω 2 " 1.0858,a n dΩ 3 " 2.1716. Furthermore, we found that as the incident amplitude increases, resulting in the activation of nonlinear effects, the metasurface resonance frequencies start shifting relative to the linear frequencies. Hence, one possible way to improve the second harmonic conversion effect, combined with absorption of the fundamental reflected wave, would be to detune the excitation frequency with respect to the linear resonance frequency, thus yielding better coincidence with the shifted nonlinear resonance frequencies. This applied detuning will also be discussed in Section2.4. To complete the analysis, the stiffness and initial angle of rotation, along with the inertial moment of the rotating masses, which remain set until Section 2.4, will then be varied and analyzed in 2.5.

Nonlinear reflection by the designed metasurface: Optimal frequency conversion

Let's now consider a monochromatic source f inc pΩ,⌧q"A inc cospΩ⌧ q in order to solve semi-analytically the considered problem Eq. ( 2.3) comprising nonlinear terms in the form of sin and cos functions. An expansion up to the fourth order of all sinusoidal terms is then applied:

$ ' ' ' ' & ' ' ' ' % cosp✓ 0 `✓q«cos ✓ 0 ´sin ✓ 0 ✓ ´1 2 cos ✓ 0 ✓ 2 `1 6 sin ✓ 0 ✓ 3 `1 24 cos ✓ 0 ✓ 4 , sinp✓ 0 `✓q«sin ✓ 0 `cos ✓ 0 ✓ ´1 2 sin ✓ 0 ✓ 2 ´1 6 cos ✓ 0 ✓ 3 `1 24 sin ✓ 0 ✓ 4 .
In the present study, which deals with the case of dynamic angles comparable to the initial angle of rotation, the considered expansion is determined to be sufficient since it yields reflection coefficient results with an accuracy to within 0.01 when compared to the numerical integration of the full problem described further below. The system of equations approximated by a polynomial form can now be solved using the Harmonic Balance Method (HBM) [START_REF] Marinca | The Method of Harmonic Balance[END_REF] (more details in Appendix A.2).

According to HBM, the solution tqu"t U 1 , U 2 ,✓u T is developed as the sum of all generated harmonics:

tqu"tq 0 u `N ÿ n"1 rtC n u cospnΩ⌧ q`tS n u sinpnΩ⌧ qs (2.4)
with tq 0 u"tC 1 0 , C 2 0 , C 3 0 u T indicating the constant terms of variables U 1 , U 2 and ✓, and tC n u (resp. tS n u)groupingthemagnitudesofcos (resp. sin)termsofthethree variables. N denotes the finite number of harmonics under consideration, which is set at N " 10,t h u sc o r r e s p o n d i n gt oar e l a t i v ee r r o ro fl e s st h a n10 ´15 ,c o m p a r e d to the solution for N " 9.A c c o r d i n g t o t h e v e c t o r i a l f o r m o f t h e s o l u t i o n , t h e approximated polynomial form governing the system of equations can be rewritten in a matrix form that is numerically solvable by applying the classical Newton-Raphson method. Once displacement U 1 is determined, the complex reflection coefficient of the n-th harmonic frequency component can be deduced as:

r R n " δ n1 `iγnΩpC 1 n ´iS 1 n q, (2.5)
where δ n1 is the Delta function, which always equals zero except when n " 1. C 1 n and S 1 n denote the magnitudes of sinusoidal terms cos nΩ⌧ and sin nΩ⌧ of displacement U 1 . The reflection coefficients obtained by HBM are considered as theoretical results and will be compared with the time domain simulation results at the end of the current section.

When the incident amplitude is relatively weak, the reflection coefficients obtained by HBM are close to the linear analytical solution. For instance, an excitation of dimensionless magnitude A inc " 10 ´7 leads to an absolute difference in the reflection coefficient of less than 0.01%,c o m p a r e dt ot h el i n e a ra n a l y t i c a ls o l u t i o n . Therefore, for the following discussion of nonlinear phenomena, the excitation magnitude range considered extends from A inc " 10 ´7 to A inc " 10 ´4,i . e . f r o mt h e linear case to amplitudes 3 orders of magnitude greater.

As mentioned at the end of Section 2.3, once the excitation level is significant, the nonlinear resonance frequencies of the metasurface shift relative to the linear frequencies. Consequently, taking into account excitation frequency detuning becomes necessary for the considered input amplitude range. In addition to the intrinsic parameters of the metasurface that have already been defined in Section 2.3, the nonlinear reflection also depends on the propagation medium. By choosing herein two different excitation amplitudes, i.e. a relatively weak one with magnitude A inc " 5 ¨10 ´6 and a stronger one with A inc " 5 ¨10 ´5,t h en o n l i n e a rr e fl e c t i o ni s thus being investigated simultaneously as a function of both the excitation frequency detuning ∆Ω (normalized by ! 0 )andthemediumimpedanceparameterγ, as shown in Fig. 2.3.

Through the reflection process and depending on input intensity, the frequency conversion can be achieved for a specific impedance value and for appropriate frequency detuning (see Fig. 2.3). In the case of A inc " 5 ¨10 ´6,b ys e t t i n gt h e impedance parameter at γ " 0.008 and considering a very small frequency detuning of ∆Ω "10 ´4,as e c o n dh a r m o n i cr e fl e c t i o nc o e ffi c i e n to fR 2 " 0.418 along with an e a r -z e r of u n d a m e n t a lc o e ffi c i e n tR 1 " 0.0024 are obtained. In comparison, as the source amplitude increases to A inc " 5 ¨10 ´5,t h ef r e q u e n c yd e t u n i n gn e c e s s a r y to reduce reflection at the fundamental frequency becomes ∆Ω "1.7 ¨10 ´3,t h u s yielding a second harmonic reflection coefficient of R 2 " 0.786,a c c o m p a n i e db ya fundamental coefficient R 1 " 0.006 at γ " 0.0195.F u r t h e r m o r e ,f o rt h ep a r a m e t e r ranges presented in Fig. 2.3, the reflection coefficients of harmonics higher than the second order are all found to be negligible compared to the second harmonic coefficient, with absolute values consistently less than 0.001; hence, these values will not be discussed any further in the present work.

The magnitude of excitation frequency detuning needed to minimize incident fundamental wave reflection corresponds to the resonance frequency shift of the metasurface with respect to the linear resonance frequency, under the considered level of excitation. More specifically, in order to analyze the resonance frequency shifts for the various source amplitudes indicated herein, i.e. weak level A inc " 10 ´7 corresponding to the linear configuration and nonlinear levels A inc " 5 ¨10 ´6 and A inc " 5 ¨10 ´5,t h em e t a s u r f a c ek i n e t i ce n e r g ya tf r e q u e n c i e sc l o s et ot h el i n e a r resonance frequencies ! θ and ! u has been introduced. For excitation around the rotation-dominated resonance frequency ! θ ,i th a sb e e nv e r i fi e dt h a tt h em a x i m u m kinetic energy, which indicates the frequency position of the nonlinear resonance, actually shifts with increasing excitation amplitude, as illustrated in Fig. 2.4. The frequency shift between the nonlinear and linear resonance frequencies coincides exactly with the optimal excitation detuning, as introduced previously in Fig. 2.3, in order to minimize reflection of the fundamental wave. Nevertheless, for excitation around ! u corresponding to a translation-dominated motion (Fig. 2.4(b)), as opposed to excitation around ! θ ,t h ee x c i t a t i o nl e v e ld o e s not influence the kinetic energy curve. Hence, when the excitation frequency is detuned to compensate for the frequency shift of resonance ! θ ,t h er e fl e c t i o no ft h e fundamental wave can become minimized, whereas the second harmonic (which is detuned twice as fast as the fundamental harmonic) will barely change its reflection coefficient R 2 . This result is due to the fact that the corresponding frequency detuning around resonance ! u does not introduce as much of a variation in kinetic energy as the detuning around ! θ .A c c o r d i n g l y ,t h ee x c i t a t i o nd e t u n i n gs i m u l t a n eously enables minimizing the reflection of the fundamental wave while maintaining the nonlinear conversion efficiency into the reflected second harmonic wave. In Fig. 2.5, the evolution of both the fundamental and second harmonic reflection coefficients are examined over the gradual increase in excitation amplitude, from the linear case to the case enabling activation of nonlinear effects. The comparison between cases, whether or not excitation detuning has been taken into consideration, is presented as well. These findings serve to confirm that the excitation detuning primarily affects the fundamental wave reflection and much less so the conversion towards second harmonic frequency. Frequency detuning is introduced in order to eliminate reflection of the fundamental wave at the desired excitation amplitude, such that (a) A inc " 5 ¨10 ´6 and (b) A inc " 5 ¨10 ´5, with the impedance parameter defined as γ " 0.008 and γ " 0.0195, respectively.

When excitation detuning is not introduced (the source f inc " A inc cos !t at frequency ! " ! θ coincides with the linear rotation-dominated resonance), the reflection coefficients R 1 and R 2 are plotted in Fig. 2.5 for an excitation amplitude range starting from the linear configuration A inc " 10 ´7.W i t ha ni n c r e a s i n ge x c i t ation amplitude, due to the introduced frequency matching, i.e., 2! " 2! θ " ! u ,t h e quadratic nonlinear effect is significantly amplified and appears first, thus yielding an efficient growth of R 2 .

When the source amplitude is further increased, cubic nonlinear effects start to appear, stemming from both the cubic nonlinear terms of the expansions in the full problem Eq. (2.3) and the nonlinear cascade process (next-order interaction) from the quadratic terms. Cubic nonlinear effects induce a variation of R 1 via self-action of ! while the quadratic via interactions between ! and 2!,r e s pe c t i v e l y . G i v e nt h e result of excitation matching ! " ! θ ,t h en o n l i n e a rp a r to fR 1 is magnified, yielding ac l e a ri n c r e a s ei nR 1 following an initial decrease, as shown in Fig. 2.5. However, the third harmonic component, which is also generated due to the nonlinear effect yet mismatched with the system resonances, remains negligible with a magnitude of less than 0.001.

By taking advantage of excitation detuning, the extreme value of R 1 ,o c c u r r i n g due to the activation of cubic nonlinear effects, can thus be minimized or even eliminated under a specific impedance parameter value γ.N e v e r t h e l e s s ,s i n c et h eh i g h e r resonance ! u is less sensitive to excitation detuning than the rotation-dominated resonance ! θ ,t h es e c o n dh a r m o n i cr e fl e c t i o nc o e ffi c i e n tR 2 is not influenced to the same extent as R 1 for the fundamental wave. For excitation level A inc " 5 ¨10 ´6, by introducing frequency detuning ∆Ω "1 ¨10 ´4 and setting γ " 0.008,t h ef u ndamental wave reflection coefficient is minimized to R 1 " 0.0024, while the second harmonic can reach a reflection coefficient of R 2 " 0.418, as shown in Fig. 2.5(a). In contrast, a frequency detuning of ∆Ω "´1.7 ¨10 ´3 enables R 1 " 0.006 at excitation level A inc " 5 ¨10 ´5 under γ " 0.195, along with a second harmonic reflection coefficient as high as R 2 " 0.786,s e eF i g .2 . 5 ( b ) . N o t et h a tt h e s ep a r a m e t e r sa r e the same as those used in Fig. 2.3.

Since an analytical solution does not exist for the full nonlinear problem (HBM is applied to the problem approximated by a polynomial expansion of all the nonlinear terms), numerical solutions to the full nonlinear problem with HBM results can now be compared. System [Eq. (2.3)] is solved numerically using the classical fourth-order Runge-Kutta (RK4) integration method [START_REF] Hairer | Runge-Kutta and Extrapolation Methods[END_REF], and the excitation is a Gaussianmodulated wave packet source of the form, σ inc pΩ,⌧q"A inc f inc pΩ,⌧q"A inc sinpΩ⌧ qe ´pτ´τ 0 q 2 pωT q 2 , with ⌧ " !t, T the characteristic temporal width of the wave packet, and ⌧ 0 the dimensionless time center of the packet. The reflected wave signal is obtained with the help of relation Eq. (2.2) once the temporal displacement u 1 of the front plate has been determined.

The time-frequency analysis of the reflected signals can therefore be performed using the spectrogram method, in yielding a reflected time-dependent spectrum at the fundamental and second harmonic frequencies, i.e. ! and 2!, as shown in Fig. 2.6(a) and Fig. 2.6(c), for the considered excitation amplitudes 5 ¨10 ´6 and 5 ¨10 ´5, respectively. The wave packet source has a characteristic half-height duration of 4000 periods. It has been demonstrated that the RK4 simulation reaches the theoretical results of HBM when the excitation amplitude lies close to the maximum magnitude A inc .F i g .2 . 6 ( b )a n dF i g .2 . 6 ( d )p r e s e n tt h ec o m p a r i s o nb e t w e e nH B M results and RK4 simulations when the signal spectrum is computed over the 1000 center periods of the reflected temporal signal. A good level of agreement is observed between the two methods for both plotted excitation amplitudes, each with an absolute difference of less than 0.01.

Discussion

According to the proposed metasurface design, which comprises rotating squares, the desired frequency conversion from the incoming fundamental wave to the reflected second harmonic can in fact be achieved. In addition to the specific design presented above in Section 2.4 (with fixed intrinsic parameters such as inertial moment ↵ " 1{3, shear and bending stiffnesses K s " K θ " 0.02 and initial angle of rotation ✓ 0 " 10 o ), it is possible to explore an even wider parameter space, offering greater tunability opportunitie.

With an initial angle of rotation set at ✓ 0 " 10 o and ✓ 0 " 20 o ,r e s p e c t i v e l y ,a n d for stiffness in the range K s " K θ Pp 0, 0.1q,t h eo p t i m a lc o n v e r s i o ni ss o u g h tb y varying the impedance parameter γ and introducing the excitation frequency shift ∆Ω. The optimal results of reflection coefficient R 2 as a function of stiffness K s " K θ are shown in Fig. 2.7(a). Note that the impedance parameter γ " ⇢cS{m! 0 can be modified by changing the size or the mass of the squares, which allows for impedance tuning. The final parameter of the metasurface is the mass ratio between front plate and rotating squares, denoted ↵ m . This parameter is defined to be positive and such that it satisfies the necessary frequency condition ! u " 2! θ in the linear regime.

A similar analysis has been repeated for stiffnesses set at K s " K θ " 0.02 and K s " K θ " 0.06, respectively. The maximum second harmonic reflection coefficient has been estimated for initial angles of rotation ✓ 0 Pp 0 o , 30 o q,a si l l u s t r a t e di n Fig. 2.7(b). The excitation level is set at A inc " 5 ¨10 ´5 for both parametric studies conducted in Fig. 2.7(a) and Fig. 2.7(b). It is shown that in the case of rotating square masses, i.e. ↵ " 1{3,thegenerationofasecondharmonicremainssignifican t over the entire optimal value ranges for stiffness K s " K θ Pp 0, 0.04q and initial angles of rotation ✓ 0 Pp 3 o , 15 o q, as enclosed by the white dotted lines in Fig. 2.2 of Section 2.3. Within this optimal parameter range, according to the results in Fig. 2.7, R 2 is observed to be greater than 0.4, while R 1 remains less than 0.05.

Furthermore, the desired nonlinear phenomena can b e derived for various rotating unit shapes, as characterized by different inertial moments ↵.L e t ' s r e c a l l herein that a point mass corresponds to a zero inertial moment ↵ " 0, whereas a hollow square (the entire mass distributed at the edges) has an inertial moment of ↵ " 2{3.A ni n e r t i a lm o m e n t↵ Pr0.02, 0.66s is thus considered (although in theory this moment could be removed from the value range by, for example, using gyroscopes in the design). Within the considered range of inertial moment and for the sake of computational efficiency, the initial angle of rotation has been set at 10 o and 8 o ,r e s pe c t i v e l y ,a st h e s ev a l u e sa r ef o u n dt obef a v o r a b l ef o rg e n e r a t i n gt h ed e s i r e d reflection over the entire range of inertial moment. The maximum generation of reflected second harmonic along with the minimum fundamental reflected wave can be determined by simultaneously varying all other intrinsic metasurface parameters, i.e. stiffness K s " K θ ,i m p e d a n c ep a r a m e t e rγ and required excitation detuning ∆Ω.L i k ei nt h ep r e v i o u ss t u d yp r e s e n t e di nF i g .2 . 7 ,t h el a s tp a r a m e t e r↵ m ,i . e . the mass ratio, is chosen so that the necessary condition ! u " 2! θ is satisfied.

According to the results shown in Fig. 2.8, R 2 is always greater than 0.4 over the entire range of inertial moment ↵ and possibly greater than 0.74 for inertial moment in the range ↵ Pp 0.2, 0.66q.H e n c e ,n o to n l yc a nt h ep r o p o s e dd e s i g nc o m p o s e do f homogeneous squares having an inertial moment ↵ " 1{3 produce the desired nonlinear frequency conversion, but other rotating periodic structures are also capable of efficiently generating the second harmonic through the reflection process, provided that the inertial moments of their unit cells are included in the above value range ↵ Pp0.2, 0.66q. The possible metasurface rotating element shapes may, for instance: be square, rectangular or diamond-shaped; contain holes or additional masses; constitute the centrally symmetric four-corner structure such as a cross (the inertial moment ↵ depends on the exact length-to-width ratio of each edge but is generally slightly less than 1{3); or even extend to other novel shapes based on regular polygons or centro-symmetric structures. Optimal frequency conversion effect achieved for various metasurface unit cell shapes, i.e. for different inertial moments α of rotating elements. The maximum conversion is determined as a function of α over the range of r0.02, 0.66s by varying the impedance parameter γ, excitation frequency detuning ∆Ω and stiffness K s " K θ simultaneously. In order to lessen the calculation burden, the initial angle of rotation has been set at 10 o and 8 o respectively, as these values are found to be favorable for producing the desired reflection over the entire inertial moment range.

Conclusion

In conclusion, through a realistic metasurface design inspired from recent results on the dynamics of soft architected rotating square structures [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], the possibility of achieving near-perfect absorption of the incoming fundamental wave has been demonstrated herein, along with an efficient conversion into the second harmonic frequency. By introducing appropriate excitation frequency detuning, which compensates for the nonlinear frequency shift of one rotation-dominated resonance ! θ of the metasurface, the reflection of the incoming fundamental wave can be as low as R 1 † 0.05.S i n c et h et r a n s l a t i o n -d o m i n a t e dr e s o n a n c eh a sab r o a d e rf r e q u e n c y response and a barely noticeable nonlinear frequency shift compared to the rotationdominated resonance, the efficiency of the conversion towards the second harmonic frequency is much less influenced by the introduced frequency detuning. The corresponding reflection coefficient R 2 can be consistently maintained above 0.4 and even reach values exceeding 0.8 depending on the excitation level.

In order to validate the theoretical results obtained with HBM in considering a monochromatic source, the nonlinear reflection of a wave packet has also been examined by numerically integrating the system of fully nonlinear motion equations. An excellent level of agreement has been obtained between the theoretical results output by HBM and the implemented numerical results, provided the characteristic temporal width of the wave packet signal is large enough, in accordance with expectations. Moreover, the value ranges of intrinsic metasurface parameters that efficiently lead to the desired frequency conversion have also been determined, i.e. for the metasurface unit cell consisting of homogeneous rotating squares (↵ " 1{3), the stiffness and initial angle of rotation in the range of K s " K θ Pp 0, 0.04q and ✓ 0 Pp 3 o , 15 o q, respectively, thus allowing for efficient second harmonic generation with a reflection coefficient R 2 always greater than 0.4.

The proposed metasurface design, which is capable of enhancing the nonlinear effect, has been found to be extendable to a series of designs with rotating unit cells, not only of a square shape but other available structures as well, possessing an inertial moment within the value range of ↵ Pr0.02, 0.66s. The second harmonic reflection coefficient may indeed exceed 0.74 if ↵ Pr 0.2, 0.66s.N e v e r t h e l e s s , t h e reported acoustic / elastic wave control by the scattering process is limited herein to reflection at normal incidence. Consequently, the considered nonlinear conversion would need to be investigated in other configurations, such as transmission by a thin and resonant meta-interface. The presented types of designs with rotating units, given the possibility of managing their dispersive and nonlinear elastic properties, open avenues for enhancing nonlinear wave control. By considering a larger number of layers with varying properties, the rather rudimentary scattering process studied herein could potentially be extended to broader operating frequency ranges as well as to other nonlinear processes. Moreover, such nonlinear wave scattering properties could become useful for applications in wave pulse mitigation, acoustic diode design and non-reciprocal transmission systems.

Introduction

In the previous Chapter 2, only the reflection process at normal incidence was investigated together with the metasurface design. However, in the current research trends (in optic for instance), and as could be expected in several applications, the transmission through the nonlinear metasurface appears attractive too [START_REF] Achouri | Homogenization and Scattering Analysis of Second-Harmonic Generation in Nonlinear Metasurfaces[END_REF], and would need to be studied with our rotating-element based structure. Thus, in the present chapter, we construct our meta-interface structure with the same monolayer of periodically arranged rigid squares, but connected with two parallel rigid moving plates in the front and the back. Ligaments between discrete elements are considered always to be thin and highly deformable, as introduced in the previous study. Since both the reflection and the transmission process are considered from now on, the conceived meta-structure thus corresponds to a meta-interface, as shown in the Fig. 3.2.

NL meta-interface

NL meta-interface NL meta-interface

(a) (b) (c) (d) S S

AS AS

NL meta-interface The present chapter is organized as follows: firstly in the Section 3.2, the pro-posed meta-interface structure is depicted and the considered 1D scattering problem is described by a nonlinear motion equation system. Then we divide the metainterface design into symmetrical and asymmetrical ones, distinguished by identical or different masses of the two moving plates in front and back. These two types of structures are analyzed in Section 3.3 and Section 3.4 respectively. We consider for each type of designs, both the configuration of a single excitation at only one side and the configuration of dual-excitation at respectively two sides of meta-interface. The Fig. 3.1 summarize all the considered cases (different structures and different excitation configurations) and the associated desired nonlinear conversion. The objective of each studied case is to determine the parameter space within which the transmitted/reflected second harmonic becomes optimal.

3.2 Nonlinear meta-interface design and the considered one-dimensional scattering problem

(1) (2)
meta-interface The previously studied 1D nonlinear metamaterial construction with rotating elements is still considered in the present chapter. The meta-interface is constituted by an infinite-area monolayer of periodically arranged rigid squares and by two moving plates connected in the front and the back, as illustrated in Fig. 3.2. The unit cell of the meta-interface is composed of two identical squares with elementary masses m, sandwiched between two rigid plates having elementary mass 2m 1 and 2m 2 respectively.

All the previous assumptions and initial conditions in Chapter 2 are retained herein, i.e., at each vertex of squares, three linear springs are taken into account, a compression or a tension spring with stiffness k l , a shear spring with stiffness k s and a bending spring with stiffness k θ . The two squares of the unit cell, having the same non-zero value ✓ 0 of initial rotation angle with respect to the vertical directions, are placed in symmetric positions at rest. In the present chapter, we focus on the reflection and the transmission by the meta-interface at normal incidence, assuming that the two semi-infinite propagation media in front and back are homogeneous and identical owning the same impedance parameters (same mass density ⇢ and same wave velocity c). The meta-interface width h along x direction is assumed to be much less than the acoustic wavelength in the propagation media p1q and p2q,i . e . , h ! λ.

Let's consider firstly a plane stress-wave σ inc of amplitude coming from ´8 and propagating along the positive x direction. The one-dimensional wave equation, i.e.,

⇢ B 2 u x Bt 2 " Bσ Bx , (3.1)
with ⇢ being the mass density of the propagation medium and u x denoting the displacement along x direction, must be satisfied everywhere and in particular on the meta-interface at both x " 0 and x " h, which leads to the following relations among the incident, reflected and transmitted wave, in the left and right medium respectively:

σ ref " σ inc `⇢c Bu 1 Bt ,σ tr "´⇢c Bu 2 Bt (3.2)
where u 1 and u 2 denote the displacement of the left plate m 1 and the right plate m 2 ,r e s pe c t i v e l y .

In the case where the meta-interface is excited from both sides, with two plane stress-waves σ p1q inc and σ p2q inc ,r e s p e c t i v e l y ,t h eo n e -d i m e n s i o n a lw a v ee q u a t i o nl e a d st o the following relations between the incident and the reflected wave, in the left and right medium respectively:

σ p1q ref " σ p1q inc `⇢c Bu 1 Bt ,σ p2q ref " σ p2q inc ´⇢c Bu 2 Bt , (3.3)
where σ As explained in Chapter 2, since the square monolayer is periodically arranged and supposed to be infinitely long, with homogenous excitations along y,t h et w o squares of each unit cell translate with the same displacement and rotate also with the same dynamic angle but in the opposite directions. Consequently, the motions of only one square and of its two connected plates are sufficient for describing the full dynamics.

For a systematic analysis, we introduce the following dimensionless parameters: the normalized displacements U i " u i {2l,( i = 1 , 2 )o ft h et w om o v i n gp l a t e sa n d U " u{2l the normalized displacement of the squares, with 2l denoting the diagonal length of the squares, the pulsation Ω " !{! 0 with ! 0 " a k l {m,t h et i m e⌧ " ! 0 t, the inertial moment of squares ↵ " J{ml 2 ,t h en o r m a l i z e ds h e a ra n dt h eb e n d i n g stiffnesses K s " k s {k l and K θ " k θ {k l l 2 respectively, and finally the two mass ratios ↵ mi " m i {m with i " 1, 2.

During the movement of the meta-interface elements, we assume that the springs behave linearly and the dissipation is accounted for via linear viscous damping associated with the respective translation and rotation motions of each square. The characteristic dissipation parameters Γ u and Γ θ are normalized as ⌘ u " Γ u {m! 0 and ⌘ θ " Γ θ {m! 0 for the translation and the rotation. As in the previous chapter, we consider herein that the dissipations are relatively weak with dimensionless value of ⌘ u " ⌘ θ " 0.001. Thus, for each square and the face plates occupying a lateral surface S in the py, zq plane, the governing motion equations are written as: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

↵ m1 B 2 U 1 Bτ 2 "2f p1q inc pΩ,⌧q´γ BU 1 Bτ ´U1 `U ´⌘u BU 1 Bτ `⌘u BU Bτ `1 2 `cos ✓ 0 ´cosp✓ 0 `✓q`⌘ θ sin ✓ 0 Bθ Bτ B2 U Bτ 2 " U 1 ´2U `U2 `⌘u BU 1 Bτ ´2⌘ u BU Bτ `⌘u BU 2 Bτ . ↵ B 2 θ Bτ 2 "6K θ `✓ `⌘θ Bθ Bτ ˘`2pU 1 ´U2 `⌘u BU 1 Bτ ´⌘u BU 2
Bτ q sinp✓ 0 `✓q

`6sinp✓ 0 `✓q `cosp✓ 0 `✓q´cos ✓ 0 ´⌘θ sin ✓ 0 Bθ Bτ 2K s cosp✓ 0 `✓q `sinp✓ 0 `✓q´sin ✓ 0 `⌘θ cos ✓ 0 Bθ Bτ ↵m2 B 2 U 2 Bτ 2 " 2f p2q inc pΩ,⌧q´γ BU 2 Bτ ´U2 `U ´⌘u BU 2 Bτ `⌘u BU Bτ ´1 2 `cosp✓ 0 q´cosp✓ 0 `✓q`⌘ θ sin ✓ 0 Bθ Bτ ˘(3.4)
where f piq inc pΩ,⌧q"σ piq inc pΩ,⌧qS{2k l l with i " 1, 2 denote the normalized incident stress-waves, coming from ´8 and `8 respectively. Since the two propagation media p1q and p2q are assumed to be identical, identified by the same mass density ⇢ and the same wave velocity c,t h ed i m e n s i o n l e s si m p e d a n c ep a r a m e t e ri sd e fi n e d as γ " ⇢cS{m! 0 representing the ratio of the impedance of the propagation media p1q and p2q to the mechanical impedance of the meta-interface.

When the excitation waves interact with the meta-interface, geometrical nonlinearity of the structure is activated once the square units start to rotate significantly. Accordingly, the scattered wave spectrum from a monochromatic incident wave contains new frequencies compared to the fundamental harmonic !.I nt h e following Sections 3.3 and 3.4, two different configurations are considered, i.e., one single excitation hits the meta-interface at only the left side or two excitations performed simultaneously at both sides. In the case of single side excitation, Ä R n and Ä

T n (1 § n § N )a

r ee m p l o y e dt od e s c r i b er e s p e c t i v e l yt h ec o m p l e xr e fl e c t i o na n dt h e

transmission of n-th harmonic, whereas in the case of dual-excitation, we use Ä R n p1q and Ä R n p2q to represent the complex reflection coefficients of the n-th harmonic propagating respectively in the left and right propagation medium, i.e., medium p1q and medium p2q. For the sake of simplicity, in the whole present chapter, the absolute

value of Ä R n , Ä T n , Ä R n p1q and Ä R n p2q are denoted as R n , T n , R p1q
n and R p2q n ,r e s pe c t i v e l y . We have shown in Chapter 2 that the linear frequency response of the metasurface design is insightful for the prediction of its future nonlinear property. Thus for the proposed meta-interface, before investigating the nonlinear scattering process in different configurations, the linear resonance frequencies of meta-interface should be determined and characterized in order to facilitate the analysis of meta-interface nonlinear dynamics. By applying the series expansions of the trigonometric functions as cosp✓ 0 `✓q«cos ✓ 0 ´sinp✓ 0 q✓ and sinp✓ 0 `✓q«sin ✓ 0 `cosp✓ 0 q✓ in the nonlinear equation system Eq. (3.4), and by removing the dissipation effects, i.e., ⌘ u " ⌘ θ " 0, the linearized motion equations of the considered problem in the free regime take the matrix form of » ---

´↵m1 Ω 2 `1 ´1 ´1 2 sin ✓ 0 0 ´1 ´Ω2 `20 ´1 ´2sin✓ 0 0 ´↵Ω 2 `A 2sin✓ 0 0 ´1 1 2 sin ✓ 0 ´↵m2 Ω 2 `1 fi ffi ffi fl $ ' ' & ' ' % U 1 U ✓ U 2 , / / . / / - " $ ' ' & ' ' % 0 0 0 0 , / / . / / - . (3.5)
with A " 6K θ `6sin 2 ✓ 0 `2K s cos 2 ✓ 0 . The eigenfrequencies of the meta-interface are obtained by setting the determinant of matrix in Eq. (3.5) to zero, which leads to

Ω 2 " ↵ m1 Ω 2 ´p2↵ m1 `1q ‰" p´↵Ω 2 `Aqp´↵ m2 Ω 2 `1q´2sin✓ 2 0 ‰ `Ω2 p↵ m1 ´↵m2 q " p´↵ `sin 2 ✓ 0 qΩ 2 `A ´2sin 2 ✓ 0 ‰ " 0. (3.6)
Thus, the linearized meta-interface has three non-zero eigenfrequencies ! i (i " 1, 2, 3). With dimensionless notation Ω i " ! i {! 0 , they obey the following relations:

$ ' ' ' ' ' & ' ' ' ' ' % Ω 2 1 `Ω2 2 `Ω2 3 " 2 `A α `1 α m1 `1 α m2 , Ω 2 1 Ω 2 2 `Ω2 2 Ω 2 3 `Ω2 3 Ω 2 1 " 2A α `1 α m1 α m2 `p 1 α m1 `1 α m1 qp A´sin 2 θ 0 α `1q, Ω 2 1 Ω 2 2 Ω 2 3 "p 1 α m1 `1 α m2 `1 α m1 1 α m2 qp A´2sin 2 θ 0 α q.
(3.7)

As the nonlinearity of the system is primarily stimulated by the rotation of the squares, the excitation frequency should be close to the resonance frequency denoted as Ω θ related to a rotation-dominated mode, i.e., resonance frequency with corresponding eigenvector enabling a displacement ratio | ✓{U n | (n " 1, 2)b i g g e rt h a n unity. According to the previous study of metasurface design, in order to maximize the second harmonic generation through the scattering process, it is necessary to make another resonance frequency of the meta-interface denoted as Ω u ,c o i n c i d e with the second harmonic frequency. The resonance frequencies Ω u and Ω θ satisfying Ω u " 2Ω θ are identified, amongst all the three resonances, by properly defining the intrinsic parameters of the meta-interface. The third resonance frequency normally related to a translation-dominated mode, is not considered here since we are interested only in the transform between the fundamental wave and the second harmonic. Depending on different meta-interface designs, i.e., symmetrical structure or asymmetrical structure, the above necessary frequency condition leads to different parameter relations and parameter evaluation methodologies. Thus we continue our meta-interface study by distinguishing the case of symmetrical design (Section 3.3) and the case of asymmetrical design (Section 3.4).

Furthermore, regarding the value range definitions of intrinsic parameters of the meta-interface, in all the following configurations, we consider the initial angle of rotating squares in the range of ✓ 0 Pp 0 o , 30 o q,a sa s s u m e di nt h ep r e v i o u ss t u d yo f Chapter 2, the bending and the shear stiffness are assumed always to be identical but the value range is herein extended to K s " K θ Pp 0, 0.2q. The inertial moment parameter ↵ is considered being included between 0 (point mass) and 2{3 (mass distributed at the edges). The mass ratio parameters ↵ m1 and ↵ m2 can be determined together with the resonance frequencies Ω i (i " 1, 2, 3)o n c et h er a t i obe t w e e nt h e m r m " ↵ m2 {↵ m1 is defined and the frequency matching condition Ω u " 2Ω θ is fulfilled.

Nonlinear scattering by a symmetrical metainterface design

When the proposed meta-interface has a symmetrical structure, i.e., the contained moving plates in the front and the back have the same mass ↵ m1 " ↵ m2 " ↵ m ,t h e equation of eigenfrequencies Eq. (3.5) can be simplified as the following:

pΩ 2 i ´2↵ m `1 ↵ m q " p´↵Ω 2 i `Aqp´↵ m Ω 2 i `1q´2sin✓ 2 0 ‰ " 0 (3.8)
Therefore, the symmetrical meta-interface has a resonance frequency Ω 1 "p 2↵ m 1q{↵ m .A tt h i se i g e n f r e q u e n c ya n di nt h el i n e a rl o s s l e s sr e g i m e ,t h es q u a r e sm move without rotation, as exhibited by the eigenvector tU 1 : U : ✓ : U 2 u T :

tU 1 : U : ✓ : U 2 u T Ω"Ω 1 " 1:´↵ m Ω 2 1 `1:0:1 ( T .
The nonlinear behaviors being mainly due to the rotation of squares, it cannot be efficiently excited at Ω 1 ; consequently, this frequency will not be considered in the following nonlinear study. At the other two eigenfrequencies denoted as Ω 2 and Ω 3 pΩ 3 °Ω2 q, which satisfy the relation p´↵Ω 2 i `Aqp´↵ m Ω 2 i `1q´2sin✓ 2 0 " 0,t h ee i g e n v e c t o rbe c o m e s :

tU 1 : U : ✓ : U 2 u T Ω"Ω i " " 1:0: 2p´↵ m Ω 2 i `1q sin ✓ 0 : ´1* T ,i " 2, 3.
It is shown that whatever the system parameters are defined, the two moving plates m 1 and m 2 move always with the same magnitude but in phase opposition at the eigenfrequencies Ω 2 and Ω 3 ,i . e . ,U 1 {U 2 "1. The rotation motion of squares is herein coupled with the translation of the two plates, the absolute ratio between rotation and translation displacement being | ✓{U n | Ω"Ω i " 2 |↵ m Ω 2 i `1{ sin ✓ 0 | with n " 1, 2 depends on the amount of mass ratio ↵ m ,o nt h ei n i t i a la n g l e✓ 0 and on the eigenfrequencies Ω i pi " 2, 3q.

Within the given value ranges of ✓ 0 and of K s " K θ ,theeigenfrequencyΩ 2 corresponds to a rotation-dominated mode (Ω 2 " Ω θ )s i n c et h em i n i m u ma b s o l u t ev a l u e of obtained ratio ✓{U n is around 2.7 at this frequency, whereas the eigenfrequency Ω 3 obeying Ω 3 " 2Ω 2 relates either to a translation-dominated movement or to a rotation-dominated movement depending on the values of meta-interface intrinsic parameters, as illustrated in Once the displacement ratio ✓{U n deviates from unity (with absolute value much smaller or much bigger than one), the rotation motion and translation motion turn to be weakly coupled, thus the energy transfer from fundamental harmonic to higher order ones becomes inefficient during the scattering process. Hence, according to the frequency response of the rotation-dominated resonance Ω θ ,w ed e fi n eat h r e s h o l d of displacement ratio value at ✓{U n " 15 related to the initial angle ✓ 0 around 5 o , as presented by the white dotted lines in Fig 3 .3. We demonstrate in the following nonlinear analysis that in order to achieve a considerable nonlinear conversion, the above defined threshold should not be exceeded, i.e., the initial angle should not be smaller than 5 o . The red dotted lines in Fig 3 .3 indicate the position of ✓{U n " 1 for resonance frequency Ω 3 , which examines whether Ω 3 corresponds to a translationdominated resonance. The presented results on linear frequency response of metainterface is considered as an indicator for the following investigation of scattering process with the aim of exploring the parameter space for an optimal nonlinear conversion.

Since the studied problem contains geometric nonlinearity, as in the Chapter 2, we perform firstly an expansion up to the fourth order of all the presented sinusoidal terms in Eq. (3.4):

$ & % cosp✓ 0 `✓q«cos ✓ 0 ´sin ✓ 0 ✓ ´1 2 cos ✓ 0 ✓ 2 `1 6 sin ✓ 0 ✓ 3 `1 24 cos ✓ 0 ✓ 4 , sinp✓ 0 `✓q«sin ✓ 0 `cos ✓ 0 ✓ ´1 2 sin ✓ 0 ✓ 2 ´1 6 cos ✓ 0 ✓ 3 `1 24 sin ✓ 0 ✓ 4 .
Thus, the nonlinear equation system Eq. (3.4) which is herein approximated into ap o l y n o m i a lf o r m ,c a nb es o l v e ds e m i -a n a l y t i c a l l yb yu s i n gt h eH a r m o n i cB a l a n c e Method (HBM) [START_REF] Marinca | The Method of Harmonic Balance[END_REF]. We ensure that, in the present work which deal with the case of dynamic angle comparable to the initial rotation angle, the considered expansion is sufficient since it yields scattering results with an accuracy better than 0.01 when compared to the numerical integration of the full problem described later on.

According to the HBM, the solution of the considered problem tqu"tU 1 , U,✓,U 2 u T is developed as the sum of all generated harmonics:

tqu"tq 0 u `N ÿ n"1
rtC n u cospnΩ⌧ q`tS n u sinpnΩ⌧ qs , (3.9)

with tq 0 u"t C 1 0 , C 2 0 , C 3 0 , C 4 0 u T indicating the constant terms of variables U 1 , U , ✓ and U 2 , tC n u and tS n u grouping the magnitudes of sinusoidal terms, cos and sin respectively, of the four variables at frequency nΩ. N denotes the finite number of harmonics being considered, which is fixed at N " 10 corresponding to a relative error of less than 10 ´15 as compared to N " 9.B ym e a n so ft h ev e c t o rf o r mo f the solution, the approximated polynomial-form governing equation system can be re-written in a matrix form and solved numerically by applying the classical Newton-Raphson method. More details are available in appendix A.

For the symmetrical meta-interface design with moving plates of the same mass, we investigate the nonlinear scattering process by considering respectively the configuration of a single monochromatic source and the configuration of two excitations simultaneously applied at both sides of the meta-interface. Since the meta-interface has a symmetrical structure with respect to y direction and the plates move with the same magnitude and in phase opposition, when it is excited by two sources at both sides, the simplest case is to take into account two identical excitations. Thus in the following parts, we consider firstly in 3.3.1 the case of f p1q inc pΩ,⌧q"A inc cospΩ⌧ q and f p2q inc pΩ,⌧q"0,f o l l o w e db yt h ec a s eo ff p1q inc pΩ,⌧q"f p2q inc pΩ,⌧q"A inc cospΩ⌧ q presented secondly in 3.3.2.

3.3.1

Configuration of a single side excitation: f p1q inc pΩ,⌧q" A inc cospΩ⌧ q and f p2q inc pΩ,⌧q"0

We consider in the current subsection the prop osed symmetrical meta-interface excited by a monochromatic source only at the left side, i.e., f p1q inc pΩ,⌧q"A inc cospΩ⌧ q and f p2q inc pΩ,⌧q"0,a n dw ef oc u so nt h ee n e r g yc o n v e r s i o nf r o mf u n d a m e n t a li n c o ming wave to transmitted second harmonic. When the HBM is performed to solve the concerned problem (Eq. (3.4)), the reflected and the transmitted waves satisfying relation Eq. (3.2) are developed into the sum of all the generated harmonics. The complex reflection and transmission coefficients of n-th harmonic (1 § n § N ), denoted as Ä R n and Ä T n respectively, can be reduced from the HBM results of U 1 and

U 2 : $ & % Ä R n " δ n1 `iγnΩrC n p1q´iS n p1qs{A inc Ä T n "íγnΩrC n p4q´iS n p4qs{A inc , (3.10)
where δ n1 is the Delta function, which is always zero except when n " 1. rC n p1q, S n p1qs and rC n p4q, S n p4qs denote the magnitudes of sinusoidal terms rcos nΩ⌧, sin nΩ⌧ s of displacement U 1 and U 2 ,r e s pe c t i v e l y .

In order to efficiently stimulate the geometric nonlinearity of meta-interface, the excitation frequency Ω should coincide with the rotation-dominated resonance frequency Ω 2 " Ω θ . Thus when the desired frequency condition Ω 3 " 2Ω 2 is satisfied, the displacement relation U 1 {U 2 "1 between two moving plates verified at resonance frequencies Ω 2 and Ω 3 in the linear regime, is fulfilled at frequencies of the fundamental wave and the second harmonic as well. Consequently, regarding to the second harmonic reflection and transmission, according to Eq. (3.10), the coefficients Ä R 2 and r T 2 remain to be the same, i.e., Ä R 2 " r T 2 .

In addition to the intrinsic parameters of meta-interface, i.e., stiffness K s " K θ , initial rotation angle ✓ 0 ,inertialmomen t↵ and the mass ratio ↵ m ,someotherph ysical parameters such as the impedance parameter γ and the excitation frequency detuning ∆Ω,p l a ya l s oa ni m p o r t a n tr o l eo nt h es e c o n dh a r m o n i cg e n e r a t i o n . I ndeed, the impedance parameter γ can be defined to be favorable for the second harmonic generation. Moreover, we have demonstrated in the previous metasurface study (see Chapter 2) that the introduction of frequency detuning enables the absorption of fundamental wave, resulting from the resonance frequencies shift due to the nonlinearity activation. Accordingly, in the current work, for the purposes of improving the desired nonlinear conversion (amplification of second harmonic and weakening of the fundamental wave, both at once), it is necessary to keep considering the contribution of these two physical parameters, especially of the excitation frequency detuning ∆Ω.

In order to estimate the detuning effect, by considering an example of K s " K θ " 0.05 and ✓ 0 " 10 o for rotating-square architected meta-interface (having ↵ " 1{3), we take advantage of the maximum kinetic energy of the meta-interface under monochromatic excitation at resonance frequencies Ω 2 and Ω 3 corresponding herein to the rotation-dominated mode and the translation-dominated mode respectively, as presented in Fig 3 .4. In the linear regime, the above choice of system parameters leads to a displacement ratio | ✓{U n | Ω"Ω 2 « 8.5 (n " 1, 2)attherotation-dominated resonance Ω 2 " Ω θ ,a n dar a t i oo f| ✓{U n | Ω"Ω 3 « 0.5 at the translation-dominated resonance Ω 3 " Ω u .B a s e do nt h ec h o s e ni n t r i n s i cp a r a m e t e rv a l u e s ,t h ei m p e d a n c e parameter γ is fixed at γ " 0.013,yieldinganoptimalgenerationofsecondharmonic on transmission. Figure 3.4: Maximum kinetic energy of the meta-interface with symmetrical structure, under monochromatic excitation of weak magnitude A inc " 10 ´7 (linear configuration), and of significant magnitude A inc " 5 ¨10 ´5 (nonlinear configuration), respectively, in the case of (a) excitation frequencies close to the linear rotation-dominated resonance frequency Ω θ , and of (b) excitation frequencies close to the linear translation-dominated resonance frequency Ω u verifying Ω u " 2Ω θ . During the kinetic energy test, the meta-interface is excited, for each excitation frequency, by a single monochromatic source of 1000 periods length at the left side.

By comparing the maximum kinetic energy curves under weak excitation level (A inc " 1¨10 ´7)a ndsignificantlevel(A inc " 5¨10 ´5), it is shown that for the considered symmetrical meta-interface, the resonance frequency shift, which is generated due to the nonlinearity, is weak for both Ω θ and Ω u when compared to the results of the metasurface obtained in Chapter 2. Furthermore, when the excitation level is important, the resonance frequency Ω θ and Ω u shift towards opposite directions, thus, it is unreasonable to introduce an excitation detuning only according to the shift amount of Ω θ .W er e m a r ka sw e l lt h a ta te x c i t a t i o nl e v e lA inc " 5 ¨10 ´5,t h e frequency response peak around Ω θ becomes broader, whereas that of Ω u becomes slightly narrower around Ω u . Consequently, it becomes difficult in this case to absorb efficiently the fundamental wave with a small amount of excitation detuning, and at the same time to keep an important level of second harmonic transmission.

In the present work, as we are interested in the maximum transmission of second harmonic through the proposed meta-interface, the excitation detuning should be introduced judiciously to weaken the transmitted fundamental wave, provided that the transmitted wave of second harmonic should not be reduced. With the value of system parameters given for producing Fig 3.4, we present in Fig 3 .5 the evolution of the reflection and the transmission coefficients of fundamental wave and of second harmonic, denoted as R i and T i (i " 1, 2), over the gradual increase of excitation amplitude, with and without excitation detuning. We confirm herein that the equality between the reflected and the transmitted wave of second harmonic, i.e., R 2 " T 2 , is verified by the performed nonlinear HBM testing. ) and for the second harmonic (R 2 and T 2 ), investigated by varying the excitation amplitude from linear level (A inc " 10 ´7) to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the transmission of fundamental wave at excitation magnitude A inc " 5¨10 ´5, provided that the transmission of second harmonic can not be reduced. Impedance parameter is chosen at γ " 0.013 enabling the maximization of T 2 in the condition that the stiffness and the initial rotation angle are defined as K s " K θ " 0.05 and θ 0 " 10 o .

Notice that the reflection and transmission coefficients obtained by HBM will be close to the linear analytical solution if the incident amplitude is relatively weak. In Fig 3 .5, we consider the same value range of excitation magnitude as in the Chapter 2, i.e., from A inc " 1 ¨10 ´7 which leads to an absolute error of fundamental reflection coefficient less than 0.01%,t oa m p l i t u d e st h a ta r e3o r d e r so fm a g n i t u d e larger enabling the efficient activation of nonlinear effects, i.e., A inc " 1 ¨10 ´4.

When the excitation detuning is not introduced, with the increase of the excitation amplitude, the quadratic nonlinear effect is significantly amplified at first due to the introduced frequency matching, i.e., Ω u " 2Ω θ , yielding the efficient growth of R 2 and T 2 .I ft h es o u r c ea m p l i t u d ei si n c r e a s e df u r t h e r ,t h ec u b i cn o n l i n e a re ffect starts to manifest. The combination of cascade quadratic and the direct cubic nonlinear effects both contribute to the amplified variation of reflected and transmitted fundamental waves, resulting in a considerable increase of R 1 and a significant decrease of T 1 , as shown in Fig 3 .5. However, the third harmonic, which is also generated due to the nonlinear effect, but which is mismatched with the resonances, remains negligible with a magnitude smaller than 0.001.

For the purp oses of weakening the fundamental transmitted wave under the excitation level A inc " 5 ¨10 ´5,a ne x c i t a t i o nd e t u n i n g∆Ω " 5 ¨10 ´4 is applied in Fig 3 .5, making sure that the transmitted second harmonic is not attenuated with this detuning. Nevertheless, we find that in the present configuration of symmetrical meta-interface excited by a single side excitation, the excitation detuning effect is relatively small. Hence, the introduction of excitation detuning does not promote the total nonlinear conversion from the fundamental wave to the second harmonic, as it has been achieved in the previous study with metasurface.

At excitation level A inc " 5 ¨10 ´5,f o rt h ep a r a m e t e r sd e fi n e da sγ " 0.013, K s " K θ " 0.05 and ✓ 0 " 10 o , the reflection and the transmission coefficients without detuning are R 1 " 0.5615, T 1 " 0.4427,a n dR 2 " T 2 " 0.3998.W h e r e a sb y introducing ∆Ω " 5¨10 ´4, T 1 has a possible minimum value of T 1 " 0.4416,together with maximized R 2 " T 2 " 0.4019 and with R 1 " 0.5584.D e s p i t et h ef a c tt h a t the detuning effect is relatively weak for the considered configuration, in order to complete the nonlinear scattering analysis, we keep taking into account this factor since it is still advantageous for optimizing the desired nonlinear transfer. Otherwise, considering the detuning effect in the current configuration enables as well the comparison with other configurations in which the detuning effect is relatively important.

In order to validate the above theoretical results obtained by HBM, the classical fourth-order Runge-Kutta (RK4) integration method [START_REF] Hairer | Runge-Kutta and Extrapolation Methods[END_REF] is applied with a Gaussian modulated wave packet source of the form σ inc pΩ,⌧q"A inc f inc pΩ,⌧q"A inc sinpΩ⌧ qe ´pτ´τ 0 q 2 pωT q 2 , with ⌧ " !t, T the characteristic temporal width of the wave packet, and ⌧ 0 the dimensionless time center of the packet. The reflected and transmitted wave signals are obtained with the help of the obtained relations σ ref " σ inc `⇢c Bu 1 Bt and σ tr " ´⇢c Bu 2 Bt ,o n c et h et e m p o r a ld i s p l a c e m e n t su i (i " 1, 2)o ft w om o v i n gp l a t e sa r e determined.

Thereafter, the time-frequency analysis of the scattering signals can be performed via the spectrogram method, yielding respectively the reflected and the transmitted time-dependent spectra at the fundamental and the second harmonic frequencies, i.e., Ω and 2Ω, as shown in Fig. 3.6(a) and Fig. 3.6(c). The wave packet source is defined of magnitude 5 ¨10 ´5 and with the periods number at the half-height of 4000 periods, and the time-dependent reflection results are also illustrated with the signal width of 4000 periods. It is demonstrated that the RK4 simulation reaches the theoretical results of HBM when the excitation amplitude reaches its maximum magnitude A inc . The Fig. 3.6(b) and Fig. 3.6(d) present the comparison between the HBM results and the RK4 simulations, by taking into account the center 1000 periods of temporal reflected and transmitted signals. A good agreement between the two methods is achieved for the considered configuration, with an absolute difference between them smaller than 0.01.

The above presented result is obtained in the specific case of K s " K θ " 0.05 and ✓ 0 " 10 o ,y e ti ti sn e c e s s a r yt oe s t i m a t et h en o n l i n e a rc o n v e r s i o ne ffi c i e n c yi n the defined whole parameter space, i.e., stiffness K s " K θ Pp0, 0.2q,i n i t i a lr o t a t i o n angle ✓ 0 Pp 0 o , 30 o q and inertial moment ↵ Pp 0, 2{3q. Firstly the stiffness K s " K θ and the initial rotation angle ✓ 0 are evaluated for the meta-interface composed of reflected ampl. homogeneous rotating squares (↵ " 1{3). The mass ratio ↵ m between the moving plates and the rotating squares can be determined by satisfying the necessary frequency matching condition Ω u " 2Ω θ . The optimal conversion into the transmitted second harmonic is identified herein via both the impedance parameter variation and the excitation frequency detuning.

Notice that the impedance parameter γ " ⇢cS{m! 0 can be modified by changing the size or the square mass, which enables the mentioned impedance adjustment. Indeed, when the stiffness and initial angle are definite, the impedance parameter γ is varied at first in order to obtain a maximum generation of second harmonic on transmission. The later-performed introduction of excitation detuning together with am o r ea c c u r a t ev a r i a t i o no fγ allow to reduce as much as possible the fundamental transmitted wave, provided that the second harmonic transmission coefficient T 2 is not decreased. With an initial angle set at ✓ 0 " 5 o , ✓ 0 " 10 o and ✓ 0 " 15 o respectively, the scattering investigation results throughout the stiffness variation within K s " K θ P p0, 0.2q and under the excitation level of A inc " 5 ¨10 ´5,a r ep r e s e n t e di nF i g3 . 7 . When the stiffness is set at K s " K θ " 0.05, K s " K θ " 0.1 and K s " K θ " 0.15 respectively, the evolution of scattering coefficients (R 1 , R 2 , T 1 and T 2 )c a nb ee xplored in a similar way over the variation of initial rotation angle in the range of ✓ Pp0 o , 30 o q and under the same excitation, as illustrated in the harmonic can reach an absolute value in the vicinity or even larger than 0.4, while the reflected and transmitted fundamental waves remain comparable within the above optimal value ranges of K s " K θ and ✓ 0 ,a l t h o u g ht h ee x c i t a t i o nf r e q u e n c y detuning is taken into account. Otherwise, we confirm here that the reflection and the transmission coefficients of second harmonic are always identical, as explained at the beginning of this section (two moving plates move with same displacement but in opposition of phase at Ω and 2Ω,u n d e rt h ec o n s i d e r a t i o no fs y m m e t r i c a l meta-interface structure). After investigating the influence of stiffness and initial angle on the second har-monic transmission for the proposed rotating-square meta-interface structure (↵ " 1{3), we evaluate the considered nonlinear scattering process for meta-interfaces composed of different rotating elements, i.e., different values of inertial moment parameter ↵. The maximum transmission of second harmonic is searched by varying all the other parameters of the system, i.e., initial angle ✓ 0 , stiffness K s " K θ , impedance parameter γ and excitation detuning ∆Ω.F o rt h es a k eo fe ffi c i e n tc a lculation, ✓ 0 and K s " K θ are varied by steps of 2 o and 0.01,r e s p e c t i v e l y ,i nt h e i r considered ranges ✓ 0 Pp 0 o , 30 o q and K s " K θ Pp 0, 0.2q. The excitation frequency detuning ∆Ω is counted with accuracy of 1 ¨10 ´4 which is already sufficient since the excitation detuning effect is relatively small. 0.2 0.4 0.6 0.4 0.5

Figure 3.9: Optimal frequency conversion achieved for different shapes of unit cell of the meta-interface with symmetrical structure, i.e., for different inertial moments α of rotating elements, in the configuration of single excitation performed at only left side. The maximum conversion to second harmonic wave is determined as a function of α in the range of r0.03, 0.66s, by varying the stiffness K s " K θ ,theinitialangleθ 0 , the impedance parameter γ and the excitation frequency detuning ∆Ω simultaneously.

As in the previous study, the impedance γ and the excitation detuning ∆Ω are identified in each configuration of p✓ 0 ,K s " K θ q,toac hiev eamaxim umsecondharmonic transmission providing as much as possible the reduction of the transmission coefficient of the fundamental wave T 1 .A f t e re x p l o r a t i o no ft h ed e s i r e dn o n l i n e a r process within the whole value ranges of K s " K θ and of ✓ 0 ,t h efi n a ls c a t t e r i n g result for a defined value of ↵ is identified by extracting the most advantageous configuration which enables the transmission coefficient T 2 to be the maximum. The optimization approach described here is summarized in the diagram pbq of Fig 3 .10.

The inertial moment evaluation is carried out with ↵ varying from 0.03 to 0.66 with a step of 0.03 (↵ " 0 corresponds to a point mass, ↵ " 2{3 corresponds to entire mass distributed at the edges), the Fig 3.9 shows the final result of transmission coefficient of second harmonic T 2 over the ↵ variation, achieved and optimized by ranging all the other physical parameters. We find that the transmission of second harmonic can always be considerable producing a transmission coefficient T 2 between 0.4 and 0.5, whereas the corresponding transmitted fundamental wave remains con-sistently comparable with the second harmonic one, having T 1 within the value range p0.4, 0.6q.D u r i n g↵ evaluation in the whole considered value range, the reported optimal results are achieved constantly with ✓ 0 " 6 o and with K s " K θ " 0.01, which constitute a parameter space contained exactly in the favorable interval predicted by the linear frequency analysis (Fig 3 .3) and also within that obtained previously in the nonlinear study presented in Fig 3 . In conclusion, for the proposed symmetrical rotating-element meta-interface design, we demonstrate that when the meta-interface is excited at only one side, the generation of second harmonic can be important, with reflection and transmission coefficients of second harmonic R 2 " T 2 within p0.4, 0.5q for the inertial moment ↵ Pp 0.03, 0.66q, while accompanied with significant fundamental reflection and transmission producing R 1 and T 1 included in p0.4, 0.6q. Thus it seems impossible for the present symmetrical meta-interface in the single side excitation configuration, to convert most of energy into the second harmonic. In the next Section 3.3.2, we will show that with two identical excitations performed from the two sides of the considered meta-interface, it becomes possible to achieve a total energy conversion from the fundamental incoming wave to the second harmonic through the reflections at both sides. The nonlinear analysis will be similar as in this section.

Configuration of two symmetrical excitations:

f p1q inc pΩ,⌧q" f p2q inc pΩ,⌧q"A inc cospΩ⌧ q
When the symmetrical meta-interface is excited from two sides near the rotationdominated resonance frequency Ω θ ,a st h et w om o v i n gp l a t e so fm e t a -i n t e r f a c eu n i t cell move always invariably with the same magnitude and in phase opposition at Ω θ and Ω u ,t h er e fl e c t i o nc o e ffi c i e n t so fs e c o n dh a r m o n i ca r ei d e n t i c a la tb o t hs i d e so f the meta-interface, i.e., Ä R n p1q " Ä R n p2q .I nt h i sc a s e ,w ec o n s i d e rt w os y m m e t r i c a l excitations, i.e., of the same magnitude but directed in opposite directions, yielding a perfect symmetrical configuration (symmetrical meta-structure with symmetrical excitations) which is investigated in the present section. Let's consider two identical monochromatic sources defined as

f p1q inc pΩ,⌧q"f p2q inc pΩ,⌧q"A inc cospΩ⌧ q,
the complex reflection coefficients of n-th harmonic (1 § n § N ) is defined with respect to the square root of input intensity ?

2A inc : $ ' & ' % Ä R n p1q " rδ n1 A inc `iγnΩ pC n p1q´iS n p1qqs {p ? 2A inc q, Ä R n p2q " rδ n1 A inc ´iγnΩ pC n p4q´iS n p4qqs {p ? 2A inc q, (3.11)
where δ n1 is the Delta function, which is always zero except when n " 1. rC n p1q, S n p1qs and rC n p4q, S n p4qs denote the magnitudes of sinusoidal terms rcos nΩ⌧, sin nΩ⌧ s of displacement U 1 and U 2 ,r e s pe c t i v e l y . As in the previous configuration of single side excitation, the same excitation magnitude range is considered in this configuration, i.e., from linear configuration with A inc " 1 ¨10 ´7 to strong nonlinear configuration having A inc " 1 ¨10 ´4.

The effect of excitation frequency detuning is exhibited in Fig 3 .11 by investigating the evolutions of reflection coefficients of both sides of the meta-interface over the gradual increase of excitation level, with or without the frequency detuning. The impedance parameter is chosen at γ " 0.02 so that the reflected second harmonic is maximum under the excitation level of A inc " 5 ¨10 ´5 and with the intrinsic parameters K s " K θ and ✓ 0 defined as K s " K θ " 0.05 and ✓ 0 " 10 o respectively (same parameter amounts as in the previous configuration 3.3.1).

We have shown in the previous Section 3.3.1 that for the considered symmetrical meta-interface, the resonance frequency shift due to the nonlinear effect is relatively small. Fortunately, in the present dual-excitation configuration, when the impedance parameter γ is chosen to highlight the second harmonic reflection, the corresponding fundamental reflected wave at both sides of the meta-interface is relatively weak with R 1 around 0.1.H e n c e ,t h ei n t r o d u c t i o no fe x c i t a t i o nd e t u n i n gc a nb ef a v o r a b l e to eliminate the fundamental reflected wave. By applying a frequency detuning of ∆Ω " 1 ¨10 ´3,t h ef u n d a m e n t a lr e fl e c t i o n sa tb o t hs i d e so ft h em e t a -i n t e r f a c ea r e ) and of second harmonic (R 2 ) of both sides of the meta-interface, investigated by varying the excitation amplitude from linear level (A inc " 10 ´7) to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the fundamental reflected wave at excitation magnitude A inc " 5 ¨10 ´5, provided that the reflection of second harmonic is not reduced. Impedance parameter is chosen at γ " 0.02 enabling the maximization of T 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively.

nearly vanishing with R p1q 1 " R p2q 1 " 0.0156, while the reflection of second harmonic can be significant with R p1q 2 " R p2q 2 " 0.6048. Therefore, it becomes possible to concentrate most of energy into the second harmonic via the performed dual-excitation configuration for the proposed symmetrical meta-interface design.

In order to validate the theoretical results obtained by HBM, we apply the spectrogram method by considering two identical Gaussian modulated wave packet sources in the form of σ p1q inc pΩ,⌧q"σ p2q inc pΩ,⌧q"A inc f inc pΩ,⌧q"A inc sinpΩ⌧ qe

´pτ´τ 0 q 2 pωT q 2 .
The comparison results are shown in Fig. 3.12 with system parameters defined as A inc " 5 ¨10 ´5, K s " K θ " 0.05, ✓ 0 " 10 o , γ " 0.02 and ∆Ω " 1 ¨10 ´3. The time-dependent reflection curves of Fig. 3.12(a) and Fig. 3.12(c) are extracted from the reflected time-dependent spectra at the fundamental and the second harmonic frequencies respectively. The wave packet source is defined with the periods number at the half-height of 4000 periods, and the time-dependent reflection results are also illustrated with the signal width of 4000 periods. It is demonstrated that the RK4 simulation reaches the theoretical results of HBM when the excitation amplitude reaches its maximum magnitude A inc .B y t a k i n g i n t o a c c o u n t t h e c e n t r a l 1000 periods of the temporal reflected signal, a good agreement between the two methods is found, with an absolute difference between them smaller than 0.01.

Through the proposed symmetrical meta-interface design and with two symmetrical excitations at both sides, we have demonstrated that the reflected second harmonic can be large, even a reflection coefficient R 2 around 0.6 with respect to Figure 3.12: Theoretical and numerical 1D scattering results of the proposed nonlinear meta-interface with symmetrical structure, obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, under two identical excitations with same magnitude of A inc " 5 ¨10 ´5 at two sides, in the case of K s " K θ " 0.05, θ 0 " 10 o and γ " 0.02. The introduced detuning is ∆Ω " 1 ¨10 ´3. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). By considering a wave packet source with characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d). All wave magnitudes are normalized by the square root of input intensity A inc .

the square average of input intensity is achievable, together with a fundamental reflection coefficient smaller than 0.02.H o w e v e r ,t h ed e s i r e dt o t a lc o n v e r s i o nc a nb e realized not only with the presented specific properties of meta-interface. In order to obtain the optimal value ranges of system parameters enabling the efficient nonlinear conversion, we analyze and investigate the meta-interface reflection process over the whole parameter space, i.e., K s " K θ Pp 0, 0.2q, ✓ 0 Pp 0 o , 30 o q,a sd o n ei n the previous study of Subsection 3.3.1.

With an initial angle set as ✓ 0 " 5 o , ✓ 0 " 10 o and ✓ 0 " 15 o ,r e s pe c t i v e l y ,a n df o r each value of stiffness included in the considered range, i.e., K s " K θ Pp0, 0.2q,w e seek the maximum conversion by performing the exploration process similar to that presented in Fig. 3 Optimal frequency conversion achieved for different physical properties of proposed symmetrical meta-interface, under dual-excitation, both with magnitude A inc " 5 ¨10 ´5 at the two sides of the meta-interface. The maximum absolute value of reflection coefficient of second harmonic R 2 of both sides is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of initial angle set at 5 o , 10 o and 15 o respectively (a), and the case of stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively (b).

case, considering the important role of excitation detuning ∆Ω on the elimination of fundamental reflected wave, the search process of Fig. 3.10(a) is improved by adding a supplementary stage of adapting individually the excitation detuning with am o d e r a t es t e p ,a p p l i e di m m e d i a t e l ya f t e rt h efi r s t -f u l fi l l e di m p e d a n c ep a r a m e t e r adjustment. The next (final) stage of monitoring simultaneously the detuning ∆Ω and the impedance γ is carried out with more accurate variation of these two parameters, with the goal to enhance the second harmonic generation and to provide am i n i m u mr e fl e c t i o no ff u n d a m e n t a lw a v e . The same analysis is repeated for stiffness fixed at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively, the maximum reflection of second harmonic is estimated for initial angle ✓ 0 Pp 0 o , 30 o q,a si l l u s t r a t e di nF i g .3 . 1 3 ( b ) . The excitation level is set at A inc " 5 ¨10 ´5 for both studies of Fig. 3.13(a) and Fig. 3.13(b). We show herein that, in the case of construction with homogeneous squares, i.e., ↵ " 1{3,t h eg e n e r a t i o no fs e c o n dh a r m o n i cr e m a i n sc o n s i d e r a b l ef o r K s " K θ † 0.05 and ✓ 0 Pp3 o , 30 o q. Within the defined whole value range of intrinsic parameter, according to the results of Fig. 3.13, the reflected second harmonic is significant, with reflection coefficients R piq 2 around 0.6 at both sides of the metainterface, together with a minor fundamental wave reflection (reflection coefficients R piq 1 less than 0.05).

Furthermore, for the different shapes of the rotating elements characterized by different inertial moments ↵,thetotalenergycon v ersionfromthefundamen talw a v e to the second harmonic can be demonstrated also achievable. Within the value range of ↵ Pp 0.03, 0.66q and under two symmetrical excitations with same magnitude A p1q inc " A p1q inc " 5 ¨10 ´5,t h ed e s i r e dn o n l i n e a rs c a t t e r i n gr e s u l t sc a nb ei d e n t i fi e d by varying simultaneously all other physical parameters of considered problem, i.e., stiffness K s " K θ ,i n i t i a la n g l e✓ 0 ,i m pe d a n c ep a r a m e t e rγ and excitation frequency detuning ∆Ω, and following the main steps of the approach carried in the last singleexcitation configuration (see Fig. The optimal conversion is determined as the function of α in the value range of r0.03, 0.66s,b y varying simultaneously all other physical parameters, i.e., the impedance parameter γ,t h e excitation frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle θ 0 . pK s " K θ ,✓ 0 q,w ed e t e r m i n et h em a x i m u mg e n e r a t i o no fs e c o n dh a r m o n i ct h a t enables a negligible fundamental reflected wave, via the same investigation process applied previously for obtaining Fig. 3.13. The optimal amount of R for each value of inertial moment ↵ is hereafter acknowledged by extracting the most favorable case amongst all scattering results obtained through the variation of pK s " K θ ,✓ 0 q.I na d d i t i o n ,t h ea c c u r a c yo fa l lc o n c e r n e dp a r a m e t e r sa r er e m a i n e d unchanged.

From the results presented in Fig. 3.14, it is shown that the reflected second harmonic can be important for the whole range of inertial parameter ↵ Pp0.03, 0.66q (↵ " 0 for point mass, ↵ " 2{3 for mass distributed at the edges), enabling the reflection coefficient of second harmonic between 0.6 and 0. are calculated with respect to the square root of input intensity, i.e., ? 2A inc , which means that they should be amplified by a factor of ? 2 when compared to the source magnitude A inc .

Otherwise, when the significant reflection of second harmonic is achieved, the accompanied fundamental reflected wave is nearly eliminated within the entire range of ↵ (the reflection coefficient of fundamental wave consistently weaker than 0.05). For all ↵ Pp 0.03, 0.66q,t h en o n l i n e a rc o n v e r s i o nr e s u l t sp r e s e n t e di nF i g .3 . 1 4a r e obtained with the unique initial angle value ✓ 0 " 8 o and the special stiffness amount K s " K θ " 0.02,c o n f o r m i n gt h ep r e d i c t i o no ft h el i n e a rf r e q u e n c yr e s p o n s ei l l u s -trated in Fig. 3.3 and corresponding to the nonlinear results of Fig. 3.13 as well.

After investigating the symmetrical design of meta-interface, in the next Section 3.4, our attention is devoted to the asymmetrical meta-interface design composed of the same monolayer of rotating elements but sandwiched between two moving plates of different mass. Thus, the additional intrinsic parameter, i.e., the mass ratio between two plates, should be taken into account. We are interested invariantly in the nonlinear conversion from fundamental wave to the second harmonic. The first configuration under consideration in the next Section 3.4 is the same as in this section, i.e., single excitation performed at one side of meta-interface. However, when two excitations at both sides of meta-interface are carried out, we will consider the same monochromatic sources but with different magnitudes in order to optimize the second harmonic generation. The role of the magnitude ratio between two excitations will be discussed in the corresponding part 3.4.2 as well.

Nonlinear scattering by an asymmetrical metainterface design

In the current section, we discuss the asymmetrical meta-interface structure, i.e., the proposed monolayer of rotating squares is kept, but two moving plates of different mass m 1 ‰ m 2 in front and back are now considered. The analysis is similar as the previous study of symmetrical meta-interface design, and the considered configurations are also similar, i.e., single excitation at only one side, or two excitations at respectively two sides. The value ranges of common intrinsic parameters are defined as previously as K s " K θ Pp0, 0.2q, ✓ 0 Pp0 o , 30 o q and ↵ Pp0, 2{3q. Comparing to the symmetrical meta-interface, additional parameters such as the mass ratio between two moving plates defined as r m " ↵ m1 {↵ m2 and the magnitude ratio between two excitations R amp " A p2q inc {A p1q inc , should be herein taken into account. They will be discussed according to the considered configuration.

3.4.1 Configuration of a single side excitation: f p1q inc pΩ,⌧q" A inc cospΩ⌧ q and f p2q inc pΩ,⌧q"0

In the case where the meta-interface is excited by a single source at the left side, the reflection and the transmission of n-th harmonic are defined identically as in Eq. (3.10). The intrinsic parameters in common with the previous symmetrical design, i.e., K s " K θ Pp0, 0.2q, ✓ 0 Pp0 o , 30 o q and ↵ Pp0, 2{3q,a r ec o n s i d e r e di nt h e same value ranges for the present study. For the sake of brevity, the linear analysis of frequency response will not be repeated herein, since the offered prediction will be verified by the following nonlinear scattering investigation.

The excitation magnitude range is counted invariantly from the linear level A inc " 1 ¨10 ´7 to strong nonlinear level A inc " 1 ¨10 ´4.W ei n t r o d u c eh e r e i na n o t h e r parameter, the mass ratio between two moving plates defined as r m " ↵ m2 {↵ m1 . For the asymmetrical meta-interface design, r m ‰ 1.W et a k efi r s t l ya ne x a m p l eo f r m " 5 for the nonlinear analysis, this parameter will be evaluated at the end of this ) and second harmonic (R 2 and T 2 ), investigated by varying the excitation amplitude from linear level (A inc " 10 ´7) to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the transmission of fundamental wave at excitation magnitude A inc " 5 ¨10 ´5, provided that the transmission of second harmonic is not reduced. Impedance parameter is chosen at γ " 0.013 enabling the maximization of T 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively.

subsection.

Throughout the current meta-interface analysis, we focus on the maximum conversion from the fundamental wave to the transmitted second harmonic. Apart from the intrinsic parameter, we keep monitoring the excitation frequency detuning ∆Ω and the impedance parameter γ in order to favor the second harmonic magnification and if possible the fundamental wave absorption.

In Fig. 3.15, the evolutions of the reflection and the transmission of fundamental wave (R 1 and T 1 )a n ds e c o n dh a r m o n i c( R 2 and T 2 ) are shown over the gradual increase of excitation amplitude from linear level (A inc " 10 ´7)t os t r o n gn o n l i n e a r level (A inc " 10 ´4), with parameter definitions as K s " K θ " 0.05 and ✓ 0 " 10 o . The physical parameters γ and ∆Ω are chosen at γ " 0.013 and ∆Ω " 6¨10 ´4 respectively so that the transmission of fundamental wave at excitation magnitude A inc " 5¨10 ´5 is minimum, provided that the transmission of second harmonic is not reduced. However, by comparing the case of introducing or not the excitation detuning ∆Ω, we found that the influence of ∆Ω is minor, as obtained with symmetrical structure of meta-interface under single side excitation.

With the above defined physical parameters, we illustrate in Fig. 3.16 the timedependent reflection results under the single side excitation of a wave packet source having a maximum magnitude of A inc " 5 ¨10 ´5. The wave packet source is defined with the periods number at the half-height of 4000 periods, the signal width of reflected waves presented in Fig. 3.16(a) and Fig. 3.16(c) are also chosen at 4000 periods. It is demonstrated that the RK4 simulation reaches the theoretical results of HBM when the excitation amplitude closes to its maximum magnitude A inc .B y taken into account the center 1000 periods of temporal reflected signal. A good agreement between the two methods is achieved for considered configuration, with an absolute difference between them smaller than 0.01.

After optimizing the second harmonic transmission with a specific parameter definition, we assess herein the whole parameter space for the current configuration. Firstly, we fix the value of stiffness K s " K θ at 0.05, 0.1 and 0.15 included in the considered value range p0, 0.2q,r e s pe c t i v e l y .

Through the variation of the initial rotation angle ✓ 0 between 0 o and 30 o ,themass ratios ↵ m1 and ↵ m2 is determined by satisfying the frequency condition Ω u " 2Ω θ in the linear regime under fixed ratio r m " 5. Then the impedance parameter γ and the excitation detuning ∆Ω are varied in the same way as in the single side excitation configuration of symmetrical metasurface summarized in Fig. 3.10(a), with the purposes of reducing as much as possible the fundamental reflected wave .17: Optimal frequency conversion achieved for different physical properties of proposed asymmetrical meta-interface, under single excitation with magnitude A inc " 5 10 ´5 at left side of meta-interface. The maximum absolute value of transmission coefficient of second harmonic T 2 is identified by varying the impedance parameter and the excitation frequency detuning, in the case of stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively. The Fig. 3.17 shows the final results of optimized transmission coefficient of fundamental and second harmonic (T 1 and T 2 )a n dt h e i rc o r r e s p o n d i n gr e fl e c t i o n coefficients (R 1 and R 2 ), over the increase of initial rotation angle from 0 o to 30 o . The same analysis is carried out in Fig. 3.18 for the initial angle ✓ 0 set at 5 o , 10 o and 15 o respectively, and the stiffness K s " K θ varying in the given value range p0, 0.2q. It is demonstrated that the transmission coefficient of second harmonic can reach an absolute value around 0.6 if the stiffness K s " K θ is included in p0, 0.05q and if the initial rotation angle ✓ 0 is around 10 o .

Notice that the resonance frequencies of meta-interface do not shift as much as those of metasurface, consequently the introduced excitation frequency detuning is not favorable to fundamental wave elimination. Hence, when the maximum transmission of second harmonic is achieved, the transmitted fundamental wave remains to be comparable. For instance when K s " K θ " 0.05 and ✓ 0 " 10 o , and with γ " 0.013,thetrans mis s ionc oe ffic ie n toff unda me n talw a v eandofs e c ondharmonic are T 1 " 0.4441 and T 2 " 0.6516. Reminding that during the investigation with symmetrical meta-interface design (↵ m1 " ↵ m2 ), we have defined identically the physical parameters K s " K θ , ✓ 0 and γ,y i e l d i n gT 1 " 0.4757 and T 2 " 0.455.B y c o m p a r i n g t h e s e t w o t r a n s m i s s i o n results both obtained with single side excitation of magnitude A inc " 5 ¨10 ´5,i ti s Nevertheless, when the transmitted second harmonic is significant, the related transmitted fundamental wave is still comparable with the second harmonic (although T 1 slightly weaken than T 2 ). After variation of all the physical parameters, the final transmission results of Fig. 3.19 are obtained with stiffness K s " K θ consistently being 0.02 for all amounts of ↵ but with initial angle ✓ 0 varied from 6 o to 18 o depending on the exact value of ↵, in accordance with nonlinear prediction of Fig. 3.17 and Fig. 3.18.

So far, the mass ratio between two moving plates is defined as r m " 5.W e presented in Fig. 3.20 the evolution of maximum T 2 for the rotating-square metainterface design (↵ " 1{3), via the variation of all other parameters, i.e., the impedance parameter γ,t h ee x c i t a t i o nf r e q u e n c yd e t u n i n g∆Ω, the stiffness K s " K θ and the initial angle ✓ 0 . The exploration process of optimal transmission is similar as in Fig. 3.19, except that we modify herein the mass ratio r m instead of varying the inertial moment ↵. Figure 3.20: Optimal frequency conversion achieved for the asymmetrical rotating-square architected meta-interface with moving plates of different mass in front and back defined by the mass ratio r m " α m2 {α m1 , under a single side excitation of magnitude A inc " 5 ¨10 ´5, the inertial moment α is fixed for homogeneous square α " 1{3. The maximum conversion is determined as the function of r m in the range of r1.5, 10s, by varying all the other parameters, i.e., the impedance parameter γ, the excitation frequency detuning ∆Ω,t h e stiffness K s " K θ and the initial angle θ 0 in their considered value ranges respectively.

For r m changing from 1.5 to 10 with step of 0.5, it is shown in Fig. 3.20 that the transmission of second harmonic can always be considerable yielding a transmission coefficient T 2 consistently between 0.5 and 0.7. The transmission results remain almost unchanged for r m bigger than 6,t h u st h ec h o i c eo fr m " 5 is sufficient for the reported nonlinear analysis. We supplement that for all considered value of r m , the presented results of Fig. 3.20 are obtained as previously with stiffness of K s " K θ " 0.02,s a m ea st h ep r e v i o u si n e r t i a lm o m e n ti n v e s t i g a t i o n( F i g .3 . 1 9 ) , whereas the needed amount of initial angle is changed from 10 o to 14 o depending on the mass ratio r m .

Hence, according to the current results with the proposed asymmetrical metainterface, for the purposes of second harmonic enhancement through the transmission process, it is more advantageous to construct the rotating-element metainterface with moving plates of different mass in the front and the back. In the following configuration where two excitations at both sides of the meta-interface are considered, by playing with an additional parameter, i.e., the amplitude ratio between the two excitations, the desired energy conversion on transmitted second harmonic is confirmed to be more efficient, when compared to the present single side excitation configuration and to the previous scattering results for symmetrical meta-interface structure. When the meta-interface structure is asymmetric, the two moving plates of different mass (↵ m1 ‰ ↵ m2 ) can not move consistently with U 1 {U 2 "´1 at frequencies Ω θ and Ω u . Thus, when two excitations are performed simultaneously, in order to efficiently achieve the desired nonlinear conversion from the fundamental wave to the reflected second harmonic, the difference between excitation magnitudes characterized by the magnitude ratio R amp " A p2q inc {A p1q inc should be investigated. In the current configuration, we define the complex reflection coefficients of n-th harmonic (1 § n § N )a tt h et w os i d e so fm e t a -i n t e r f a c ea st h ec o r r e s po n d i n gw a v e amplitude normalized by the square root of input intensity :

$ ' ' & ' ' % Ä R n p1q " " δ n1 A p1q inc `iγnΩ pC n p1q´iS n p1qq ı { b pA p1q inc q 2 `pA p2q inc q 2 Ä R n p2q " " δ n1 A p2q inc ´iγnΩ pC n p4q´iS n p4qq ı { b pA p1q inc q 2 `pA p2q inc q 2 , (3.12)
where δ n1 is the Delta function, which is always zero except when n " 1. rC n p1q, S n p1qs and rC n p4q, S n p4qs denote the magnitudes of sinusoidal terms rcos nΩ⌧, sin nΩ⌧ s of displacement U 1 and U 2 ,r e s pe c t i v e l y .

During the investigation of excitation magnitude ratio R amp ,f o rt h es a k eo f simplicity, we fix the amplitude of excitation at left side denoted as A p1q inc ,a n dw e look for the maximum transmitted second harmonic by varying the magnitude of A p2q inc . In order to compare with the previous symmetrical meta-interface design in which we have introduced two identical excitations of magnitude A inc " 5 ¨10 ´5,i n the presented asymmetrical design, we impose that the maximum value from A p1q inc and A p2q inc is equal to 5 ¨10 ´5. The obtained scattering results of the fundamental and the second harmonic at both sides of the meta-interface are shown in Fig. 3.21, over the gradual variation of excitation amplitude ratio R amp from 0.1 to 2,a n df o r impedance parameter γ of value 0.0228 enabling a maximum transmission of second harmonic in the case of intrinsic parameters defined as K s " K θ " 0.05 and ✓ 0 " 10 o respectively.

The optimal amplitude ratio R amp favorable for the desired nonlinear conversion is observed close to 1 under the above chosen specific parameter definitions. However, )a n do ft h e second harmonic (R piq 2 ) of both sides (i " 1, 2) of the proposed asymmetrical meta-interface structure with r m " 5, investigated over the gradual increase of amplitude ratio R amp within the value range of p0.1, 2q. The maximum excitation level between two sources is set at maxpA p1q inc ,A p2q inc q"5 ¨10 ´5. Impedance parameter is chosen at γ " 0.0228 enabling the maximization of R 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively. the appropriate amount of R amp highlighting the second harmonic transmission is proved to be different, depending on the system parameter values (stiffness, initial angle and impedance parameter). Thus, it should be judiciously defined during the later introduced investigation of parameter space targeting the realization of the desired nonlinear conversion.

Beside the excitation amplitude ratio, we take into account as in the previous works the excitation detuning effect. For the proposed meta-interface with symmetrical or asymmetrical structure, the excitation detuning effect is always weak. Nevertheless, in the configuration of two excitations performed at both sides of meta-interface, the impedance parameter γ enables to obtain a fundamental reflection relatively weak (with fundamental reflection coefficient around 0.1 for instance), thereafter, the desired total conversion from the fundamental incoming wave to the reflected second harmonic can be achieved by introducing properly the excitation detuning ∆Ω,p r o v i d e dt h a tt h er e fl e c t i o nc oe ffi c i e n to fs e c o n dh a r m o n i ci sn o tl e s sened.

By keeping the same parameter definitions as in the previous analysis of excitation amplitude ratio (Fig. 3.21), i.e., same excitation magnitude of sources (R amp " 1), intrinsic parameters with values of K s " K θ " 0.05 and ✓ 0 " 10 o respectively, and impedance parameter with amount of γ " 0.0228 leading to maximum transmission of second harmonic, we compare in Fig. 3.22 the reflection results at both sides of the meta-interface obtained with or without the excitation frequency detuning, over the excitation magnitude increasing from linear level (A p1q inc " A p2q inc " 10 ´7)t os t r o n gn o n l i n e a rl e v e l( A p1q inc " A p2q inc " 10 ´4). For the con-sidered excitation level A inc " 5 ¨10 ´5,t h en e c e s s a r ya m o u n to fe x c i t a t i o nd e t u n i n g for the fundamental wave absorption is ∆Ω " 1.1 ¨10 ´3,y i e l d i n gaf u n d a m e n t a lr eflection coefficient at right side being R p2q 1 " 0.0243, together with second harmonic reflection coefficient of R p2q 2 " 0.7908,a n da tt h el e f ts i d eo fm e t a -i n t e r f a c e ,w eh a v e R p1q 1 " 0.0799 and R p1q 2 " 0.1765.N o t et h a ta l lt h es c a t t e r i n gc o e ffi c i e n t sa r ed e fi n e d with normalization by the square root of input intensity b pA p1q inc q 2 `pA p2q inc q 2 . Thus, with the chosen amounts of all the physical parameters, it is possible to achieve the energy concentration on the reflected second harmonic at the right side of metainterface (in medium p2q). The frequency detuning is introduced with an amount of ∆Ω " 1.1 ¨10 ´3 to minimize the reflected fundamental wave at excitation magnitude A inc " 5 ¨10 ´5, provided that the reflection of second harmonic is not lowered (for both sides). Impedance parameter is chosen at γ " 0.0228 yielding the maximization of R 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively.

When the two excitations are both of magnitude A inc " 5 ¨10 ´5,t h ea b o v e obtained nonlinear conversion results can be verified by applying the time simulation via the RK4 method. With the chosen specific amount of physical parameters (K s " K θ " 0.05, ✓ 0 " 10 o , γ " 0.0228 and ∆Ω " 1.1 ¨10 ´3), the time-dependent reflection coefficients under two wave packet sources having maximum magnitude of A inc " 5 ¨10 ´5 are illustrated in Fig. 3.23. Under the definition of the wave packet sources both with a periods number at the half-height of 4000 periods, and the signal width of reflected waves presented in Fig. 3.23(a) and Fig. 3.23(c) of 4000 periods as well, we confirm that the RK4 simulation reaches the theoretical results of HBM when the excitation amplitude reaches its maximum magnitude A inc .B yc o n s i d e r i n gt h e center 1000 periods of temporal reflected signal, a good agreement between the two methods is achieved for this configuration, with an absolute difference between HBM and RK4 smaller than 0.01. After examining the theoretical HBM results by time simulation, we focus on the evaluation of parameter space that enables the desired nonlinear conversion into the second harmonic via dual-reflection process. As the previous studies, we consider firstly the proposed meta-interface composed of homogeneous rotating square (having inertial moment ↵ " 1{3)andcontainingtwomovingplatesofmassratior m " 5, by varying the physical parameters such as amplitude ratio R amp " A p2q inc {A p1q inc between two excitations, impedance parameter γ and the excitation detuning ∆Ω,t h e optimal value range of the stiffness K s " K θ (the initial angle ✓ 0 )thatenablesamaximum reflection coefficient of second harmonic R p2q 2 at right side of meta-interface, can be determined for defined values of initial angle ✓ 0 (stiffness K s " K θ ).

Throughout the above exploration, the mass ratios ↵ m1 and ↵ m2 are determined once the frequency matching condition Ω u " 2Ω θ between the two linear resonance frequencies is fulfilled, provided by fixed value of r m " ↵ m2 {↵ m1 " 5.W i t h t h e two excitation sources performed at the rotation-dominated frequency, as the first step, we vary the excitation magnitude ratio R amp then the impedance parameter γ, targeting a maximum reflected wave of second harmonic in medium p2q (maximum R 2 at right side of meta-interface is identified by varying the impedance parameter, the excitation magnitude ratio and the excitation frequency detuning simultaneously, with stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively. introducing the excitation frequency detuning. The detuning variation is carried out at first individually and then simultaneously with impedance parameter γ both at more accurate steps, enabling a minimization of fundamental wave subject however to the improvement of second harmonic reflection in medium p2q. The main stages of the applied optimization approach are illustrated later in a schematic form in Fig. 3 3.25 present the optimal conversion result after the described investigation, for initial angle set at 5 o , 10 o and 15 o , and for stiffness fixed as K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15,r e s p e c t i v e l y . N o t i c e that during the investigation, the excitation magnitude ratio R amp is varied by fixing the magnitude of A p1q inc and modifying the magnitude A p2q inc ,executedb yimposingthe maximum between two magnitudes as 5 ¨10 ´5.I nF i g .3 . 2 5 ,as i n g u l a rp o i n ta p p e a r s at the first sampled value of stiffness for initial angle of amount 5 o .I ty i e l d sz e r o s for all R i pi " 1, 2q,arisingfromthefactthatthefrequencyconditionΩ u " 2Ω θ can not be satisfied by the corresponding parameter definitions.

According to the results shown in Fig. 3.24 and Fig. 3.25, we find that in the considered configuration of dual-excitations, at the right side of meta-interface (in medium p2q), it is possible to achieve a reflection coefficient of second harmonic in the vicinity of 0.8 together with a nearly zero fundamental reflection coefficient (smaller than 0.03), resulting from stiffness K s " K θ smaller than 0.05 and the initial angle ✓ 0 included in p10 o , 20 o q.H e n c e ,t h ec u r r e n ta s y m m e t r i c a lm e t a -i n t e r f a c ei s demonstrated more efficient for the second harmonic generation on reflection, when compared to the symmetrical meta-interface design with dual-excitations of identical amplitude A p1q inc " A p2q inc " 5 ¨10 ´5 and under the same intrinsic parameter definitions yielding a reflection coefficient of second harmonic around 0.6 at the right side of meta-interface.

Furthermore, as the analysis rep orted in the previous sections, the achieved nonlinear conversion can be evaluated in the whole range of initial moment parameter ↵,f r o m0 (point mass) to 2{3 (masse concentrated at the edges), by varying all the system parameters, i.e., stiffness K s " K θ ,i n i t i a la n g l e✓ 0 ,i m pe d a n c ep a r a m e t e rγ, excitation magnitude ratio R amp and excitation frequency detuning ∆Ω. The maximum conversion is determined as the function of α in the range of r0.03, 0.66s,b y varying all the other physical parameters, i.e., the impedance parameter γ, the excitation magnitude ratio R amp , the excitation frequency detuning ∆Ω and the stiffness K s " K θ simultaneously.

Reminding that for obtaining Fig. 3.24 and Fig. 3.25, the exploration of maximum second harmonic reflection in the medium p2q is performed via the variations of impedance parameter γ,o fe x c i t a t i o nm a g n i t u d er a t i oR amp and of excitation frequency detuning. In the present assessment of inertial moment ↵,t h es a m ei nvestigation process is repeated for different amounts of stiffness K s " K θ and of initial angle ✓ 0 ,a sr e p r e s e n t e db yt h ed i a g r a mpbq of Fig. 3.27. Amongst all the results obtained by varying the stiffness K s " K θ and the initial angle ✓ 0 in their considered ranges, i.e., K s " K θ Pp 0, 0.2q and ✓ 0 Pp 0 o , 30 o q respectively, the final optimal scattering result for defined ↵ is identified by extracting the most favorable case enabling the maximum R 2 in medium p2q.

Figure 3.26 illustrates the final result of reflection coefficients of the fundamental wave and the second harmonic in both media p1q and p2q (R for medium p2q). During the scattering test, the parameters common to the previous configurations are varied with the same steps, i.e., stiffness K s " K θ and initial angle ✓ 0 with step of 0.01 and 2 o respectively, excitation detuning ∆Ω with accuracy of 1 ¨10 ´4, whereas the additional parameter, the excitation amplitude ratio R amp , is shifted in each iteration with rate of 0.05. The final results are obtained still for inertial moment ↵ changing from 0.03 to 0.66 with accuracy of 0.03.

According to Fig. 3.26, when the proposed asymmetrical meta-interface is excited from both sides, whatever the shape of the contained rotating element, it is always possible to convert most of the energy of fundamental incoming wave on the reflected second harmonic, especially on that at one side of meta-interface (at the side of medium p2q in presented case), provided that the inertial moment is included in ↵ P p0.03, 0.66q. The reflection coefficient of second harmonic at right side of the metainterface can reach an absolute value greater than 0.8 for whole range of ↵,itcanbe even in the vicinity of 0.9 for ↵ † 0.25.R e c a l l i n gt h a tt h e s er e fl e c t i o nc o e ffi c i e n t s are defined with respect to the square root of input intensity b pA p1q inc q 2 `pA p2q inc q 2 , thus they will be magnified when compared to the excitation magnitudes A p1q inc or A p2q inc . Otherwise, the accompanied fundamental reflection coefficient at the same side remains minor within the whole range of ↵ (consistently weaker than 0.05). At the other (left) side of meta-interface, both the reflected fundamental wave and the second harmonic are not considerable, although the reflected second harmonic owns ar e fl e c t i o nc oe ffi c i e n ta r o u n d0.2 for all considered amounts of ↵.

Therefore, for the proposed meta-interface with asymmetrical structure composed of rotating elements owning an inertial moment ↵ Pp0.03, 0.66q,i ti spo s s i b l e to achieve the energy conversion primarily into reflected second harmonic at only one side of meta-interface in the configuration of two excitations applied at both sides. Remember that for the configuration of symmetrical meta-interface excited from both sides (Fig. 3.14), the input energy can equally be concentrated on the reflected second harmonic, but splits to two sides of the meta-interface with same reflection coefficient R 2 between 0.6 and 0.7. Hence, with asymmetrical meta-interface, not only the desired nonlinear conversion can be achieved, the energy can be focused on one direction as well.

We set hitherto the mass ratio b etween moving plates contained in the metainterface at r m " 5,i no r d e rt oc o m p l e t et h ep a r a m e t e rs p a c ei n v e s t i g a t i o n ,w e evaluate herein r m for the rotating-square meta-interface (↵ " 1{3), the investigation process is similar to that of inertial moment as presented in Fig. 3.27(b), except that the mass ratio is varied instead of ranging ↵. The mass ratio r m is considered in the value range of p1.5, 10q and is varied with the step of 0.5. Figure 3.28 shows the obtained reflection coefficients of both sides of the meta-interface over the gradual change of r m .W ep r o v et h a tf o rt h em a s sr a t i o in the range of r m Pp 4, 10q,i ti sp o s s i b l et oc o n v e r tm o s t l yt h ee n e r g yo fi n c o m i n g waves to the second harmonic at only one side of meta-interface, i.e., at the interface with the heavier mass.

In addition, we remark that in order to achieve a considerable reflection of the second harmonic as presented in Fig. 3.26 and Fig. 3.28, the stiffness K s " K θ should be small. Since its value range is defined as p0, 0.2q and its variation accuracy is set at 0.01, the optimal results of Fig. 3.26 and Fig. 3.28 are realized with K s " K θ § 0.02 for the whole range of ↵ Pp0.03, 0.66q.W h e r e a st h en e c e s s a r ya m o u n to fi n i t i a la n g l e is changed from 8 o to 24 o depending on the definite value of ↵. Otherwise, the needed value of impedance parameter γ is always smaller than 0.03 for all ↵ Pp0.03, 0.66q, thus the medium in front and back the meta-interface should be much less resistant to wave propagation comparing to the meta-interface, it could be the air for instance. Optimal frequency conversion achieved for the asymmetrical rotating-square architected meta-interface with moving plates of different masses in front and back defined by the mass ratio r m " α m2 {α m1 , under two excitations at both sides, the inertial moment α is fixed for homogeneous square α " 1{3. The maximum conversion is determined as the function of r m in the range of r1.5, 10s, by varying all the other physical parameters, i.e., the impedance parameter γ, the excitation magnitude ratio R amp , the excitation frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle θ 0 .

Accordingly, the parameter space that enables the desired nonlinear conversion is relatively broad for the presented meta-interface with moving plates of different mass.

Conclusion

In conclusion, throughout this chapter, we have investigated the 1D scattering process through the rotating-element architected meta-interface with symmetrical structure or asymmetrical structure, i.e., with external moving plates of identical or different mass. We demonstrate that by satisfying the frequency condition Ω u " 2Ω θ between frequencies of rotation-dominated resonance Ω θ and of translation-dominated resonance Ω u ,i ti sc o n s i s t e n t l yp o s s i b l et or e a l i z eac o n s i d e r a b l es e c o n dh a r m o n i c through reflection / transmission processes. When a single side excitation is applied, a transmission coefficient of second harmonic in the vicinity of 0.5 or even up to 0.6 is achievable, whereas the accompanied fundamental transmitted wave remains always comparable with the second harmonic. Nevertheless, when the meta-interface is excited from both sides, by adjusting all the system parameters, i.e., stiffness, initial rotation angle, impedance parameter, excitation frequency detuning and excitation magnitude ratio, the nearly total conversion from the fundamental incoming wave to the reflected second harmonic is proved attainable, no matter if the meta-interface is symmetrical or asymmetrical. The corresponding reflection coefficient of second harmonic can be even greater than 0.9, together with a reflected fundamental wave nearly vanished (re-flection coefficient smaller than 0.05), in the case where the meta-interface structure contains two moving plates of different mass.

Furthermore, the meta-interface with asymmetrical structure is confirmed more advantageous for the second harmonic generation, especially for the case where only one excitation is applied. It enables a transmitted second harmonic with a transmission coefficient between 0.6 and 0.7, whereas the symmetrical meta-interface can allow a transmission coefficient of second harmonic just between 0.4 and 0.5.I n addition, for the case of dual-excitation, from both sides of the meta-interface, the asymmetrical structure provides not only a significant second harmonic reflection with a coefficient in the order of 0.9,b u ta l s ot h eg e n e r a t e dn o n l i n e a rw a v ei sp r edominantly emitted in one direction (emerging from the interface with the heavier mass). While, regarding to the meta-interface with symmetrical structure, the dualexcitation configuration can produce a reflection coefficient of second harmonic of about 0.6 for both sides, the converted energy is however split into two directions, even if the desired nonlinear conversion is produced as well.

The above obtained scattering results are verified to be achievable in a wide parameter space, i.e., the contained rotating elements having inertial moment in the value range ↵ Pp0, 2{3q (0 for point mass, 2{3 for mass concentrated at the edges), the mass ratio between two moving plates smaller than 10,p r o v i d e dt h a tt h es h e a r and the bending stiffness are much smaller than the longitudinal stiffness yielding K s " K θ in the vicinity of 0.01. Therefore, the proposed type of meta-interface design shows the ability to manipulate scattering of acoustic wave, and in particular to promote the nonlinear second harmonic wave generation. They should be tested with experiments and be further studied with multilayer structures as well, in order to achieve other specific nonlinear wave controls.

The construction of 2D architected rotating element metasurfaces is hitherto restricted to the periodically-arranged monolayer design, showing their ability for the nonlinear wave manipulation. Among the wide range of possibilities offered by nonlinear wave phenomena, only the conversion from the fundamental wave to the second harmonic through scattering process has been analyzed in details. Nevertheless, a variety of non-trivial acoustic phenomena can be observed, especially once the proposed meta-surface design is extended to a multilayer structure. Recent research on soft metamaterials [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] has demonstrated that a rotating-mass (a LEGO cross) meta-structure enables the propagation of elastic vector solitons, and showed several nonlinear wave phenomena such as soliton splitting by defects, a diode effect. Elastic nonlinearity can be designed by varying in a rational manner the structural properties of the units.

Accordingly, in the present Chapter 4, the same meta-structure as reported in [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] is exploited, yet a finite chain is considered rather than a long arrangement. This part of the work has been performed along the last year of my thesis, in collaboration with Alejandro Cebrecos, a post-doctoral researcher at LAUM at the time, who led the experiments. It has been also for me the occasion to interact with a Master student, Shilin QU, and guide him on the numerical aspects of his internship at LAUM. A first observation of shrinking phenomenon has been observed in experiments before the theoretical and numerical implementations of the effect. Thus, for a better understanding of the physical properties of the tested structure, we propose herein a theoretical study of the nonlinear dynamics of this system. To start with, we establish the nonlinear system of coupled equations of motion, and we focus on the dispersion curve of the infinitely long structure. Then, we analyze the shrinking phenomenon via numerical integration of the motion equations, while discussing the results and their manifestations. We consider herein an architected metasurface composed of two rows and N columns rotating crosses made of LEGO bricks [START_REF] Celli | Manipulating waves with lego bricks: A versatile experimental platform for metamaterial architectures[END_REF]. Figure 4.1 illustrates a sample with a structure of p2ˆ5q crosses as tested in experiments. Since all the elements are connected elastically with thin, massless and highly deformable ligaments, the same assumption as in the previous Chapters 2 and 3 is retained in the current study: three different springs are taken into account at each unit vertex, i.e. a longitudinal (compression or tension) spring with stiffness k l , a shear spring with stiffness k s ,and a torsional spring with stiffness k θ .

In addition, when the structure under consideration is activated, all the contained elements are assumed to have three degree of freedoms, i.e., translations along x and y directions, and a rotation around z direction. The displacements along x and y of element pp, jq localized at the row p and the column j are denoted as u p,j and v p,j respectively (with 1 † j † N and p " 1, 2 for the considered structure). ' p,j and ✓ p,j are employed to indicate respectively the initial rotation angle at rest and the dynamic rotation angle. The unit cell of the meta-structure consists of 4 adjacent elements forming a 2 by 2 matrix, as shown in Fig. 4.2. According to the chosen coordinate directions, the angles ' p,j and ✓ p,j are defined to be positive if the initial deviation and the dynamic rotation take place in the counterclockwise direction. $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % Bτ ´⌘u BU p,j`1 Bτ q `1 2 K θ rp✓ p,j´1 ´✓p,j`1 q sinp' p,j `✓p,j q`⌘ θ sin ' p,j p Bθ p,j´1 Bτ ´Bθ p,j`1 Bτ qs `1 2 p´1q p K θ rp✓ p,j ´✓p`1,j q cosp' p,j `✓p,j q`⌘ θ cos ' p,j p Bθ p,j Bτ ´Bθ p`1,j Bτ qs

B 2 V p,j Bτ 2 " K s pV p,j`1 ´2V p,j `Vp,j´1 q`⌘ v K s p BV p,j´1 Bτ ´2 BV p,j

Bτ

`BV p,j`1 Bτ q`1 2 K s rsin ' p,j`1 ´sinp' p,j`1 `✓p,j`1 q`sinp' p,j´1 `✓p,j´1 q´sin ' p,j´1 s`1 Bτ q`1 2 K θ rp✓ p,j`1 ´✓p,j´1 q cosp' p,j `✓p,j q `⌘θ cos ' p,j p Bθ p,j`1 Bτ ´Bθ p,j´1 Bτ qs `1 2 p´1q p K θ rp✓ p,j ´✓p`1,j q sinp' p,j `✓p,j q `⌘θ sin ' p,j p Bθ p,j Bτ ´Bθ p`1,j Bτ qs

↵ B 2 θ p,j Bτ 2
"śinp' p,j `✓p,j qr3cos' p,j `cos ' p,j´1 `cos ' p,j`1 `cos ' p`1,j ´2U p,j´1 `2U p,j`1 `p 1q p 2pV p`1,j ´Vp,j q´3cosp' p,j `✓p,j q´cosp' p,j´1 `✓p,j´1 q´cosp' p,j`1 `✓p,j`1 q ´cosp' p`1,j `✓p`1,j q`2⌘ u p Bτ qs ´Ks cosp' p,j `✓p,j qr2V p,j´1 ´2V p,j`1 `p´1q p 2pU p`1,j ´Up,j q`3sinp' p,j `✓p,j q `sinp' p,j´1 `✓p,j´1 q`sinp' p`1,j `✓p`1,j q`sinp' p,j`1 `✓p,j`1 q´3sin' The spatial positions of elements located at consecutive columns j ´1, j and j `1,v e r i f yt h er e l a t i o nx j ´xj´1 " x j`1 ´xj " d where d denotes the distance between two horizontally adjacent elements being 2l cos ' 0 . Thus, in the lossless case (⌘ u " ⌘ v " ⌘ θ " 0), the introductions of the above solution form tu j u L and of the here-mentioned displacement relations into the linearized motion equations of the LEGO brick pp, jq lead to the following equation system: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

´!2 U p,j "ṕ 4sin 2 pkd{2q`K s qU p,j `Ks U p`1,j `1 2 pK s ´Kθ q cos ' 0 ✓ p`1,j

´"i sin ' 0 p1 `Kθ q sinpkdq`1 2 pK θ `Ks q cos ' 0 ‰ ✓ p,j

´!2 V p,j ""4K s sin 2 pkd{2q`1 ‰ V p,j `Vp`1,j ´1 2 pK θ `1q sin ' 0 ✓ p`1,j

`"i cos ' 0 p´K s `Kθ q sinpkdq´1 2 pK θ `1q sin ' 0 ‰ ✓ p,j ´↵! 2 ✓ p,j "p 4i sin ' 0 sinpkdq´2K s cos ' 0 qU p,j ´ri4K s cos ' 0 sinpkdq`2sin' 0 s V p,j

´"sin 2 ' 0 p3 `2cospkdqq `Ks cos 2 ' 0 p3 ´2cospkdqq `Kθ p3 `2cospkdqq ‰ ✓ p,j

´"sin 2 ' 0 ´Ks cos

2 ' 0 `Kθ ‰ ✓ p`1,j `2K s cos ' 0 U p`1,j `2sin' 0 V p`1,j (4.5)
Similarly, for the other LEGOs in the same unit cell, i.e., LEGO pp, j `1q, pp `1,jq and pp `1,j`1q, their linearized motion equations can as well be written as a function of the displacements of the LEGOs in their corresponding column. The obtained systems of equations are shown in Eq. (4.6), Eq. (4.7) and Eq. (4.8) respectively. $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

´!2 U p,j`1 "ṕ 4sin 2 pkd{2q`K s qU p,j`1 `Ks U p`1,j`1 `1 2 pK θ ´Ks q cos ' 0 ✓ p`1,j`1 `"´i sin ' 0 p1 `Kθ q sinpkdq`1 2 pK θ `Ks q cos ' 0 ‰ ✓ p,j`1

´!2 V p,j`1 ""4K s sin 2 pkd{2q`1 ' ' ' ' ' ' ' ' % A 1 "´4sin 2 pkd{2q,A 2 "´i sin ' 0 p1 `Kθ q sinpkdq, A 3 "´"4K s sin 2 pkd{2q`2 ‰ ,A 4 "´p1 `Kθ q sin ' 0 , A 5 " 4i sin ' 0 sinpkdq,A 6 "´4sin✓ 0 , A 7 "´"sin 2 ' 0 p4 `2cospkdqq `Ks cos 2 ' 0 p2 ´2cospkdqq `Kθ p4 `2cospkdqq ‰ . (4.12) Thus, the relation between the pulsation ! (or the frequency f )a n dt h ew a v e number k can be obtained by solving the eigen problem presented in Eq. (4.10). For the LEGO structure used in the experiments, its physical properties are,

‰ V p,j`1 `Vp`1,j`1 ´1 2 pK θ `1q sin ' 0 ✓ p`1,j`1 `"i cos ' 0 pK s ´Kθ q sinpkdq´1 2 pK θ `1q sin ' 0 ‰ ✓ p,j`1 ↵! 2 ✓ p,j`1 "p 4i sin ' 0 sinpkdq´2K s cos ' 0 qU p,j`1 ´ri4K s cos ' 0 sinpkdq´2sin' 0 s V p,j`1 `r sin 2 ' 0 p3 `2cospkdqq `Ks cos 2 ' 0 p3 ´2cospkdqq `Kθ p3 `2cospkdqqs✓ p,j`1 `"sin 2 ' 0 ´Ks cos 2 ' 0 `Kθ ‰ ✓ p`1,j`1 `2K s cos ' 0 U p`1,j`1 ´2sin' 0 V p`1,
$ & % m " 4.52g, 2l " 42mm, k l " 71690N {m, K s " k s {k l " 0.01851, K θ " k θ {kl 2 " 1.534 ¨10´4,↵ " J{ml 2 " 1{p1.815 2 q (4.13)
By applying the parameter values in Eq. (4.13), Fig. 4.1 shows the obtained dispersion curve (real part of frequency as a fonction of the wavenumber) of the considered meta-structure for the case of initial angle equal to 0 o and 3 o respectively. The imaginary part of frequency is proved to be negligible comparing to its real part, with absolute values of the order of 10 ´15 . According to Fig. 4.4, the simplification from the 3 dof configuration to 2 dof one is quite reliable provided that the initial angle of meta-structure ' 0 is sufficiently small, i.e., in the vicinity or smaller than 1 o .W h e nt h ei n i t i a la n g l eb e c o m e sb i g g e r , with absolute value of 3 o as example shown in Fig. 4.4(b), the difference between the two cases is appreciable, especially for the low frequency range. Thus in the following nonlinear study, we keep taking into account the 3 dof leading to three modes.

Preliminary nonlinear study: observation of metastructure shrinking phenomenon

In this section, we present a preliminary nonlinear result obtained with the proposed metamaterial: the meta-structure exhibits a constant average shrinking at some specific frequencies of excitation. We assume that the two units in the first column of structure are fixed at the same angle (but in the opposite directions) and they are only allowed to move in the horizontal direction. The initial angles of all the elements are set with absolute value of 1 o and with alternating signs between two adjacent elements (corresponding to the natural deformation). The associated nonlinear motion equations described in Eq. unit and set them all with an identical value. In addition, during the theoretical and numerical study, we have set all the dissipation parameters at ⌘ u " ⌘ v " ⌘ θ " 0.001, and these values should be further tested by some additional numerical and experimental tests. Nevertheless, there is a good qualitative agreement between the numerical and experimental results, especially for the frequency dependence of the effect. These preliminary results set the basis for continuing the study, to get a deeper understanding of the nonlinear dynamics of such structures, where nonlinearity can be managed, and possibly turn to more complex structures and effects.

Conclusion

In this chapter, an architected soft meta-structure composed of periodically arranged rotating crosses has been proposed and investigated both in the linear and nonlinear regimes. In the linear regime, the dispersion properties have been determined assuming that the structure is infinitely long. The considered degrees of freedom, i.e., the vertical and the horizontal translation, and the rotation of the units, play a role, although it is found possible to reduced the problem to two degrees of freedom, neglecting the vertical movement, provided that the initial angles of the units are relatively small. When the meta-structure is finite, and excited by a finite amplitude, we applied time domain simulations of the coupled nonlinear motion equations, via the 4th order Runge-Kutta method. The structure shrinking behavior even under weak excitation amplitudes, is found to take place at specific frequencies, close to those of the rotation-dominated resonances. These are able to activate the geometrical nonlinearity of the structure, generating the shrinking effect. The chosen frequency range for observing the structure shrinking has been limited in the present Chapter 4 to relatively low frequencies, since the associated effect could be obtained in experiments in the same range, where rotations are found easy to excite.

However, the reported shrinking phenomena study is just preliminary, and there remains to understand the full picture leading to this nonlinear phenomenon. The relations between the amount of shrinking, the corresponding frequency position, the excitation signal properties, the initial rotation angles, and the structure length remain to be explored. Moreover, despite the presented theoretical approach is relatively complete, with considerations of dissipation for all introduced degrees of freedom and with adaptability to an extended structure, the comparison between the numerical results and the associated experiments still needs to be pushed. As mentioned at the end of the previous section, several difficulties still exist both in the theoretical aspects and in the experimental aspects of this work. For instance, how to manipulate or at least charaterize experimentally all the initial angles with the necessary accuracy and how to get the different dissipation parameters to approach the experimental configuration in the theory. A further research is certainly needed to understand deeper the physics of this problem and to overcome these challenges. We could envision that this effect, if sufficiently optimized, could lead to the design of actuators, capable of moving objects over a large distance, via the excitation by ah a r m o n i ca n dr e l a t i v e l ys m a l l e ra m p l i t u d ed i s p l a c e m e n t .

Conclusion

The presented PhD work has been dedicated to the design of nonlinear elastic metamaterial and metasurface architectures, enabling acoustic wave control in the nonlinear regime. Specifically, the conversion effect from a fundamental wave to its second harmonic has been studied through the one-dimensional scattering process (reflection and transmission) by appropriately designed metasurfaces. Based on nonlinear local elastic resonators, the aim of the theoretical work is primarily to determine the necessary conditions that yield a significant reflected / transmitted wave at the second harmonic frequency from a fundamental frequency incoming wave. After having found the necessary operating conditions and achieved the desired nonlinear conversion, our attention has been devoted to evaluate the roles of the structure parameters, i.e., the ones that dominate the scattering results in the linear regime such as impedance and dissipation, the ones that define the intrinsic properties of the metamaterial structure, the ones that affect the nonlinearity enhancement or those related to the incoming wave, such as excitation magnitudes and excitation frequency detuning. The objective is to explore most of the parameter space in order to efficiently promote the energy conversion into the second harmonic through the scattering process. The intention of the presented work is to expand the studied metasurface structure to a series of specific designs that favor the nonlinear wave control through other nonlinear effects and to pursue further researches and actual fabrications of nonlinear metasurfaces.

In Chapter 1, the metasurface design and study have been started by proposing a discrete lattice model made of a dual-resonance mass-spring system. Through the 1D reflection process, the possibility of achieving a near perfect absorption of the incoming fundamental wave together with its efficient conversion into the second harmonic frequency has been demonstrated both theoretically and numerically, provided that the first resonance frequency of system equals to the excitation frequency and half the second resonance frequency. The lifetimes of each resonance and the characteristic time of nonlinearity were determined as well, enabling to evaluate the operating conditions of nonlinear effects. With the proposed metasurface modeling, a reflection amplitude at the fundamental incoming wave frequency of | r RpΩq| « 0.05 was obtained and a reflected second harmonic of amplitude | r Rp2Ωq| « 0.46 was reached, under a relatively small quadratic nonlinearity (B 1 " 0.002). A good level of agreement between the theoretical results obtained by harmonic balance method (HBM) and the implemented numerical results was found, in accordance with expectations. However the key link between the proposed lumped-element model of nonlinear metasurface and a realistic structure is missing, more specifically, the method of designing elastic springs with the required effective quadratic nonlinearity still needs to be explored.

Therefore, in the following Chapter 2, a realistic metasurface design composed of rotating units was put forward, inspired by the recent research on the nonlinear dynamics of soft architected meta-structures [START_REF] Deng | Elastic vector solitons in soft architected materials[END_REF][START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF]. The same nonlinear frequency conversion as previously, i.e., from the fundamental incoming wave to the second harmonic, was demonstrated achievable through the reflection process. By introducing an appropriate excitation frequency detuning, which compensates for the nonlinear frequency shift of one rotation-dominated resonance ! θ of the metasurface, the reflection of the incoming fundamental wave can be as low as R 1 † 0.05,andthecorresponding reflection coefficient R 2 can reach values exceeding 0.8 depending on the excitation level. Moreover, the value ranges of the intrinsic metasurface parameters that efficiently lead to the desired frequency conversion have also been determined: the reflection coefficient of second harmonic R 2 is demonstrated to be greater than 0.4 for the stiffness and initial rotation angle in the ranges of K s " K θ Pp 0, 0.04q and ✓ 0 Pp 3 o , 15 o q respectively. Furthermore, the proposed metasurface design has been found to be extendable to a series of designs with rotating unit cells, not only with the considered square shape but other available structures as well, possessing an inertial moment within the value range of ↵ Pr0.02, 0.66s. The second harmonic reflection coefficient may even exceed 0.74 if ↵ Pr0.2, 0.66s accompanied with minor reflection of fundamental wave.

The reported acoustic wave control through the scattering process has been hitherto limited to the reflection at normal incidence. By connecting the employed monolayer of rotating units to two moving plates in front and back, a meta-interface is constructed enabling the study of nonlinear wave phenomena, through both the reflection and the transmission process. By considering a single source applied at one side of the meta-interface and two excitations at both sides respectively, the generation of second harmonic was shown to be consistently considerable. More specifically, a transmission coefficient of second harmonic in the vicinity of 0.5 or even up to 0.6 was proved achievable in the single excitation configuration, despite that the accompanied fundamental transmitted wave remains always comparable with the second harmonic amplitude. Nevertheless, when two excitations from both sides are applied, it was demonstrated that the previously obtained total conversion from the fundamental incoming wave to the reflected second harmonic was attainable at both sides of the meta-interface: the corresponding reflection coefficient of second harmonic can be even greater than 0.9, together with a reflected fundamental wave which nearly vanishes (reflection coefficient smaller than 0.05), in the case where the two moving plates of the meta-interface possess different masses.

In the meta-interface study presented in Chapter 3, two additional parameters, i.e., the excitation magnitude ratio between two monochromatic sources performed on fundamental frequency but at respectively both sides of the meta-interface, and the mass ratio between moving plates of meta-interface, are taken into account to expand the parameter space defined in Chapter 2. The obtained scattering results are verified to be achievable in a wide range of parameters, i.e., rotating elements of the meta-interface having inertial moment in the value range ↵ Pp 0, 2{3q (0 for point mass, 2{3 for mass concentrated at the edges), the mass ratio between two moving plates smaller than 10, provided that the shear and the bending stiffnesses are much smaller than the longitudinal stiffness, yielding K s " K θ in the vicinity of 0.01. In addition, the meta-interface with asymmetrical design is confirmed to be more advantageous when compared to the symmetrical structure, since it enables not only significant second harmonic scattering, but also the generated nonlinear wave can be predominantly emitted in one direction. Whereas with the symmetrical meta-interface, the converted energy is split into two directions although the desired nonlinear conversion is produced as well.

We have fo cused so far on the sp ecific nonlinear phenomena, i.e., the frequency conversion of fundamental wave into the second harmonic, chosen since the second harmonic generation is a typical nonlinear wave phenomenon, widely observed in acoustics. However, considering that the presented type of 2D metamaterial designs with rotating units gives the possibility of managing the nonlinear elastic properties, the rather rudimentary scattering process studied herein may be extended to other nonlinear processes. To this end, in Chapter 4, a meta-structure composed of rotating crosses forming a p2 ˆN q matrix is constructed. A preliminary study of the average shrinking phenomenon observed under harmonic driving has been carried out. After establishing the nonlinear coupled equations of motion describing the dynamics of this finite structure, the dispersion properties of the infinite structure was derived. it has been shown that in the specific structure under study, there exist conditions for reducing the description from 3 degrees of freedom to 2. By numerical implementation of the nonlinear motion equations, the nonlinear shrinking phenomenon was obtained numerically. The frequency and amplitude dependent effect is found in qualitative agreement with the carried experiments by colleagues of the research group. This work is currently ongoing, to make further progress in the understanding of the involved processes, and compare more accurately the experimental and theoretical/numerical results.

There are several perspectives to this PhD work. Because the metasurface designs presented in Chapters 2 and 3 have been studied only theoretically and numerically, such materials providing the desired nonlinear frequency conversion should be realized and tested with experiments. With 3D-printing technology of soft polymers, and referring to the determined optimal parameter space that efficiently provides the nonlinear conversion, the considered metamaterials should be relatively easy to fabricate. In the reported wave manipulation through the architected structure, the excitation is performed at only one frequency, while a broader operating frequency range is frequently required for a range of applications. Thus the proposed metasurface could be extended from a monolayer structure to a larger number of layers. After adjusting the resonance frequencies of the multi-layer architecture via variations of its intrinsic properties (graded variation of the sizes or the initial angles of unit cells for example), the distribution of frequencies could make the metasurface operating on a broader frequency range. This would require however a complex design of the structure for achieving any desired nonlinear wave effect, and this should be done with progressive increase of complexity. Only one-dimensional scattering processes have been studied here, and obviously, it would be interesting to derive solutions and tackle problems related to oblique plane wave incidence on the metasurfaces and meta-interfaces. The case of acoustic beam interactions with the nonlinear meta-interface is also open, and could lead to interesting findings related to exotic nonlinear refraction effects.

Since the presented type of rotating-element metasurfaces opens possibilities for enhancing nonlinear wave control, a variety of acoustic phenomena could be targeted. In the last Chapter 4, by considering a meta-structure made of rotating crosses, the shrinking phenomena was preliminarily observed yet it has not been discussed in depth, and it necessitates clearly to be further studied and analyzed. Besides the shrinking, other nonlinear wave manipulations can potentially be investigated as well with rotating-element metasurfaces. Amplitude gaps for elastic solitons have been demonstrated with such metamaterial chains [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF] enabling to construct soliton splitter and mechanical diode. Similar researches could thus be carried out by consider other nonlinear waves instead of solitons. For instance, it would be meaningful to explore meta-interfaces with nonlinear designs that can achieve the acoustic diode effect or the non-reciprocal transmission via the energy conversion from fundamental wave to higher harmonics. Furthermore, when the presented metasurfaces or meta-structures are extended to a 2D metamaterial arrayed in a pN ˆN q matrix, they will be favorable for studying not only the longitudinal wave propagation, but equally the shear and surface wave manipulation, probably enabling specific nonlinear wave control such as amplitude-dependent waveguiding, wave pulse mitigation and wave conversion (from longitudinal to shear for example). Rotating-element elastic structures have shown their potential for being buildings blocks of nonlinear structures, enabling the nonlinearity management. Although this idea and platform of rotating-elements may probably reveal rich processes, effects, and lead to various designs in the future, it would be desirable to find other ways to induce elastic nonlinearity with the same amount of tunability and control, for acoustic and elastic waves.

Thus, the considered system of motion equations Eq. (1.4) can be re-written in the following matrix form, Thereafter, the studied matrix-form motion equation system Eq. (A.4) can be projected onto the base rSs as, containing integrals that can be numerically evaluated. With all the explicit forms of the terms of Eq. (A.9), the considered problem can be solved numerically by the Newton-Raphson method.

The main steps of the numerical calculation are recalled here :

1, write the considering equation Eq. (A.9) in matrix form of Fpqq"0 (with N harmonics), calculate the linear solution tq L u (with B i " 0)ofthesystemandinput it as the initial value of tq 1 u"tq L u.

2,f o rt h en e x ti t e r a t i o nn `1, according to the Newton-Raphson method, we have q n`1 " q n ´rJpq n qs ´1 Fpq n q.

(A. [START_REF] Li | Acoustic metasurface-based perfect absorber with deep subwavelength thickness[END_REF] with rJpq n qs ´1 the inverse of Jacobian matrix rJpq n qs which can be obtained by rJpq n qs ij " BFpq n q i Bpq n q j . (A.15)

A.2 HBM to solve the equations in rotating-element metasurface and meta-interface designs

For the metasurface design by rotating elements presented in Chapter 2, we remind that the corresponding nonlinear equation system Eq. (2.3) is written as: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % Bτ q sinp✓ 0 `✓q `6sinp✓ 0 `✓q `cosp✓ 0 `✓q´cosp✓ 0 q´⌘ θ sin ✓ 0 Bθ Bτ

↵ m B 2 U

2K

s cosp✓ 0 `✓q `sinp✓ 0 `✓q´sinp✓ 0 q`⌘ θ cos ✓ 0 Bθ Bτ ˘(A.23)

This equation system possess trigonometric type of geometric nonlinearity, which can not be solved directly by HBM. Thus we propose to perform firstly the expansions up to fourth order of all the presented sinusoidal terms in Eq. (A.23) as:

$ & % cosp✓ 0 `✓q«cos ✓ 0 ´sin ✓ 0 ✓ ´1 2 cos ✓ 0 ✓ 2 `1 6 sin ✓ 0 ✓ 3 `1 24 cos ✓ 0 ✓ 4 sinp✓ 0 `✓q«sin ✓ 0 `cos ✓ 0 ✓ ´1 2 sin ✓ 0 ✓ 2 ´1 6 cos ✓ 0 ✓ 3 `1 24 sin ✓ 0 ✓ 4 . which leads to $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % Bτ qpsin ✓ 0 `cos ✓ 0 ✓ ´1 2 sin ✓ 0 ✓ 2 ´1 6 cos ✓ 0 ✓ 3 `1 24 sin ✓ 0 ✓ 4 q `6psin ✓ 0 `cos ✓ 0 ✓ ´1 2 sin ✓ 0 ✓ 2 ´1 6 cos ✓ 0 ✓ 3 `1 24 sin ✓ 0 ✓ 4 q ¨`´sin ✓ 0 ✓ ´1 2 cos ✓ 0 ✓ 2 `1 6 sin ✓ 0 ✓ 3 `1 24 cos ✓ 0 ✓ 4 ´⌘θ sin ✓ 0 Bθ Bτ

↵ m B 2 U 1 Bτ 2 "2f inc pΩ,

2K

s pcos ✓ 0 ´sin ✓ 0 ✓ ´1 2 cos ✓ 0 ✓ 2 `1 6 sin ✓ 0 ✓ 3 `1 24 cos ✓ 0 ✓ 4 q ¨`cos ✓ 0 ✓ ´1 2 sin ✓ 0 ✓ 2 ´1 6 cos ✓ 0 ✓ 3 `1 24 sin ✓ 0 ✓ 4 `⌘θ cos ✓ 0 Bθ Bτ ˘(A. [START_REF] Yao | Optical negative refraction in bulk metamaterials of nanowires[END_REF]

CONTENTS 3 . 3 . 2

 332 Introduction 1M a n i p u l a t i n g a c o u s t i c w a v e r e fl e c t i o n b y a n o n l i n e a r e l a s t i c m e t asurface 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The problem under consideration and the corresponding metasurface design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Theoretical results and parametric analysis: case of a monochromatic source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Numerical results and parametric analysis with a wave packet source 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2F r e q u e n c y d o u b l i n g e ff e c t i n t h e a c o u s t i c r e fl e c t i o n b y a n o n l i n e a r rotating-square architected metasurface 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonlinear elastic metasurface design and the reflection problem at normal incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Analysis of the linearized metasurface: Parameter definitions and frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Nonlinear reflection by the designed metasurface: Optimal frequency conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3A r e a l i s t i c n o n l i n e a r m e t a -i n t e r f a c e d e s i g n a c h i e v i n g a c o u s t i c w a v e scattering control 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlinear meta-interface design and the considered one-dimensional scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Nonlinear scattering by a symmetrical meta-interface design . . . . . 3.3.1 Configuration of a single side excitation: f p1q inc pΩ,⌧q"A inc cospΩ⌧ q and f p2q inc pΩ,⌧q"0 ......................... iii iv Configuration of two symmetrical excitations: f p1q inc pΩ,⌧q" f p2q inc pΩ,⌧q"A inc cospΩ⌧ q ..................... 3.4 Nonlinear scattering by an asymmetrical meta-interface design . . . . 3.4.1 Configuration of a single side excitation: f p1q inc pΩ,⌧q"A inc cospΩ⌧ q and f p2q inc pΩ,⌧q"0 ........................ 3.4.2 Configuration of dual-excitations: f p1q inc pΩ,⌧q"A p1q inc cospΩ⌧ q and f p2q inc pΩ,⌧q"A p2q inc cospΩ⌧ q .................. 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4P r e l i m i n a r y s t u d y o n t h e n o n l i n e a r d y n a m i c b e h a v i o r o f a s o f t rotating-square structure of connected LEGOs 4.1 Considered architected meta-structure and the corresponding governing motion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dispersion curve of the meta-structure . . . . . . . . . . . . . . . . . 4.3 Particular case of 2 dof . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Preliminary nonlinear study: observation of meta-structure shrinking phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1D, 2D and 3D phononic crystals made of two different elastic materials arranged periodically. Different colors represent materials with different elastic properties. (b) An example of a phononic band diagram !pkq for a two-dimensional phononic crystal. . . . . . . . . . 2S o n i c m e t a m a t e r i a l s c o n s t r u c t i o n b a s e d o n t h e i d e a o f l o c a l i z e d r e sonant structures with a lattice constant two orders of magnitude smaller than the relevant wavelength that exhibit spectral gaps. . . . 3 Acoustic metasurface-based perfect absorber with deep subwavelength thickness composed of a perforated plate with a hole and a coiled air chamber (yellow region). Figures at right show the absorption coefficient (↵)ofthepresen tedmetasurface(figureabo v e)andthenormalized specific acoustic reactance (red line) and resistance (blue dotted line) of the whole metasurface system (figure below) respectively. . . 4( a ) , A n a c o u s t i c d i o d e m a d e o f a 1 D p h o n o n i c c r y s t a l ( a l t e r n a t i n g layers of glass and water) coupled to a nonlinear acoustic medium [44]. (b), Nonreciprocal elastic wave transmission in a single-mode elastic waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5T w o o p t i c a l m e t a s u r f a c e s d e s i g n e d f o r g i a n t n o n l i n e a r r e s p o n s e f o r second harmonic generation (SHG) . . . . . . . . . . . . . . . . . . .

1

 1 p r o p a g a t i o n o f e l a s t i c v e ctor solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design of the nonlinear elastic metasurface by: (a) a vertically periodic structure at the sub-wavelength scale; in order to simplify the analysis, (b) a dual-resonance model with two mass-spring elements is implemented. A semi-infinite medium (1) and a rigid wall (2) are separated by the designed metasurface. It is assumed herein that all model elements of the model are capable of only moving along the x-direction, while the nonlinearities are only presented in the two springs; (c) presents the frequency response in the linear case of the first mass, with the proposed model featuring two resonance frequencies, i.e., ! 1 and ! 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . .14 1.2 Theoretical magnitudes of the reflection coefficients for, (a) the reflected fundamental wave, (b) the reflected second harmonic, and (d) the reflected third harmonic, derived by HBM, as a function of the nonlinear parameter values B 1 and B 2 . (c) shows an example of a special case with B 1 " 0.002 and B 2 " 0. The graphs have been produced with an impedance parameter γ " 0.0162 and an absorption parameter ⌘ " 0.0088.. . . . . . . . . . . . . . . . . . . . . . . . . .17 1.3 Magnitude of the theoretical reflection coefficient (a) at the incoming fundamental frequency, and (b) for the reflected second harmonic wave. | r R 1 | and | r R 2 | are obtained via the HBM with a monochromatic source and are evaluated as a function of both the impedance parameter γ and the absorption parameter ⌘. The nonlinear parameters are fixed at B 1 " 0.002 and B 2 " 0. The dashed lines show the characteristic parameter equalities. The cross-hatched regions in both (a) and (b) highlight the parameter space characterized by ⌧ NL † ⌧ 1,2

  i o n -d o m i n a t e d movement, denoted ! u ,o rar o t a t i o n -d o m i n a t e dm o v e m e n td e n o t e d ! θ ,o rac o m b i n a t i o no fb o t h . W h e nt h er e s o n a n c ef r e q u e n c yc o n d ition ! u " 2! θ is satisfied, the mass ratio ↵ m is determined for different values of initial angles of rotation ✓ 0 and stiffnesses K s and K θ (b).

  7 and Fig 3.8). The optimization outcome shown in Fig 3.9 during the inertial moment variation is obtained via the procedure illustrated in (b) where the parameter γ and ∆Ω are varied in the same way as process paq.. . . . . . . . . . . . . . . . . . . . . . . . 62 3.11 Absolute reflection coefficients of fundamental wave (R 1 )a n do fs e cond harmonic (R 2 )o fb o t hs i d e so ft h em e t a -i n t e r f a c e ,i n v e s t i g a t e d by varying the excitation amplitude from linear level (A inc " 10 ´7)

3. 21

 21 Absolute reflection coefficients of the fundamental wave (R piq 1 )a n do f the second harmonic (R piq 2 )o fb o t hs i d e s( i " 1, 2)o ft h ep r o p o s e d asymmetrical meta-interface structure with r m " 5,i n v e s t i g a t e do v e r the gradual increase of amplitude ratio R amp within the value range of p0.1, 2q. The maximum excitation level between two sources is set at maxpA p1q inc ,A p2q inc q"5 ¨10 ´5.I m p e d a n c e p a r a m e t e r i s c h o s e n at γ " 0.0228 enabling the maximization of R 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and ✓ 0 " 10 o respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.22 Magnitude of the reflection coefficients of fundamental (R 1 )a n ds e cond harmonic (R 2 )o fbo t hs i d e so ft h ep r o po s e da s y m m e t r i c a lm e t ainterface structure with r m " 5,i n v e s t i g a t e db ys e t t i n gt h ef a v o r a b l e amplitude ratio R amp " 1 between the two excitations. The excitation level is varied from linear level (A p1q inc " A p2q inc " 10 ´7)t os t r o n g nonlinear level (A p1q inc " A p2q inc " 10 ´4).

3. 25

 25 Optimal frequency conversion achieved for different physical properties of the proposed asymmetrical meta-interface excited from both sides. The maximum reflection coefficient of second harmonic R p2q 2 at right side of meta-interface is identified by varying the impedance parameter, the excitation magnitude ratio and the excitation frequency detuning simultaneously, in the case of initial angle set at 5 o , 10 o and 15 o ,r e s pe c t i v e l y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.26 Optimal frequency conversion achieved for different shapes of unit ceil of the asymmetrical meta-interface, i.e., for different inertial moments ↵ of rotational elements. The maximum conversion is determined as the function of ↵ in the range of r0.03, 0.66s,b yv a r y i n ga l lt h eo t h e r physical parameters, i.e., the impedance parameter γ,t h ee x c i t a t i o n magnitude ratio R amp ,theexcitationfrequencydetuning∆Ω and the stiffness K s " K θ simultaneously. . . . . . . . . . . . . . . . . . . . . 81 3.27 Schematic representation of performed optimization processes applied for obtaining maximum reflection of the second harmonic propagating in medium p2q. The diagram (a) shows the main steps of algorithm implemented for the case of definite inertial moment (calculation results presented in Fig 3.25 and Fig 3.24). The optimization outcome shown in Fig 3.26 during the inertial moment or mass ratio variation is obtained via the procedure illustrated in (b) where the parameter γ and ∆Ω are varied in the same way as process paq.. . . . . . . . .83

84 4. 1 A 88 4. 2 89 4. 3 95 4. 4 97 4. 5

 841882893954975 picture of the considered architected meta-structure, composed of two rows of rotating crosses with 5 elements for each row. This structure is placed horizontally and is connected to a low frequency shaker, seen on the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of the considered architected meta-structure, composed of two rows of rotating crosses with N elements for each row. The unit cell of the structure corresponds to a 2 ˆ2 ensemble of adjacent crosses pbq. The initial and the dynamic angles of element pp, jq are denoted as ' p,j and ✓ p,j respectively, which are defined as positive if they follow the counterclockwise direction pcq.. . . . . . .Dispersion curves of the considered meta-structure with infinite length, obtained by setting the initial rotation angle ' 0 at 0 o (a) and 3 o (b), respectively. The color gradient in each figure is employed to characterize the associated movement property of each possible mode p!, kq,thecolorsgreen,blueandredcorrespondtothev erticalmo v ement dominated, the horizontal movement dominated and the rotation dominated mode respectively. . . . . . . . . . . . . . . . . . . . Dispersion curve of the meta-structure with infinite length, determined by considering only the horizontal movement U and the translation ✓. The initial rotation angle ' 0 is set at 1 o (a) and 3 o (b), respectively. The obtained results (solid lines) are compared to that of 3 dof configuration (dotted lines). The color gradient in each figure is employed to characterize the associated movement property of each possible mode p!, kq,t h ec o l o r sb l u ea n dr e dc o r r e s p o n dt ot h e horizontal movement dominated and the rotation dominated mode respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of average (DC) horizontal displacement of the units in the last column of the meta-structure over the gradual increase of monochromatic excitation frequency, obtained with structure length of N " 6 (a) and N " 16 (c) respectively. Each average displacement is determined under a sinusoidal excitation of 40 period length and of normalized magnitude 1 ¨10 ´3 applied to the first column of metastructure. The excitation frequency range is considered between 10Hz to 100Hz for (a) and (c). In (b) and (d), for the last units of the metastructure, the magnitude of the Fourier spectrum of the displacement U along x and of the rotation ✓ around z are presented in the case of N " 6 and N " 16 respectively, under sweep frequency excitation of magnitude 1 ¨10 ´5 and within the whole available frequency band shown in dispersion curve (from 1Hz to 1300 Hz). The obtained spectrum results are illustrated in the range of p0, 300q Hz in (b) and (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 1 :

 1 Figure 1: (a) 1D, 2D and 3D phononic crystals made of two different elastic materials arranged periodically. Different colors represent materials with different elastic properties. (b) An example of a phononic band diagram ωpkq for a two-dimensional phononic crystal. [1].

Figure 2 :

 2 Figure2: Sonic metamaterials construction based on the idea of localized resonant structures with a lattice constant two orders of magnitude smaller than the relevant wavelength that exhibit spectral gaps[START_REF] Liu | Locally resonant sonic materials[END_REF].

Figure 3 :

 3 Figure 3: Acoustic metasurface-based perfect absorber with deep subwavelength thickness composed of a perforated plate with a hole and a coiled air chamber (yellow region). Figures at right show the absorption coefficient (α) of the presented metasurface (figure above) and the normalized specific acoustic reactance (red line) and resistance (blue dotted line) of the whole metasurface system (figure below) respectively. [14].

Figure 4 :

 4 Figure 4: (a), An acoustic diode made of a 1D phononic crystal (alternating layers of glass and water) coupled to a nonlinear acoustic medium [44]. (b), Nonreciprocal elastic wave transmission in a single-mode elastic waveguide [19].

Figure 5 :

 5 Figure 5: Two optical metasurfaces designed for giant nonlinear response for second harmonic generation (SHG). (a) [57] (b) [58].

Contents 1 . 1

 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The problem under consideration and the corresponding metasurface design . . . . . . . . . . . . . . . . . . . . 1.3 Theoretical results and parametric analysis: case of a monochromatic source . . . . . . . . . . . . . . . . . . . . 1.4 Numerical results and parametric analysis with a wave packet source . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Design of the nonlinear elastic metasurface by: (a) a vertically periodic structure at the sub-wavelength scale; in order to simplify the analysis, (b) a dual-resonance model with two mass-spring elements is implemented. A semi-infinite medium (1) and a rigid wall (2) are separated by the designed metasurface. It is assumed herein that all model elements of the model are capable of only moving along the x-direction, while the nonlinearities are only presented in the two springs; (c) presents the frequency response in the linear case of the first mass, with the proposed model featuring two resonance frequencies, i.e., ω 1 and ω 2 .

Figure 1 . 2 :

 12 Figure 1.2: Theoretical magnitudes of the reflection coefficients for, (a) the reflected fundamental wave, (b) the reflected second harmonic, and (d) the reflected third harmonic, derived by HBM, as a function of the nonlinear parameter values B 1 and B 2 . (c) shows an example of a special case with B 1 " 0.002 and B 2 " 0. The graphs have been produced with an impedance parameter γ " 0.0162 and an absorption parameter η " 0.0088.

1 " 3 η and ⌧ imp 1 " 2 " 3 5η and ⌧ imp 2 " 3 γ

 1312323 3 2γ ,respectively ,Atthesecondresonancefrequency ! 2 ,t h e yb e c o m e⌧ abs . These characteristic times lead to the

Figure 1 . 3 :

 13 Figure 1.3: Magnitude of the theoretical reflection coefficient (a) at the incoming fundamental frequency, and (b) for the reflected second harmonic wave. | r R 1 | and | r R 2 | are obtained via the HBM with a monochromatic source and are evaluated as a function of both the impedance parameter γ and the absorption parameter η. The nonlinear parameters are fixed at B 1 " 0.002 and B 2 " 0. The dashed lines show the characteristic parameter equalities. The cross-hatched regions in both (a) and (b) highlight the parameter space characterized by τ NL † τ 1,2 where nonlinear effects develop efficiently..

  . The local values of | R2 |"0.5 and | R1 |"0 closely correspond to the HBM results values (see Fig. 1.2(c)). To study the robustness of this effect for various signal characteristic

Figure 1 . 4 :

 14 Figure 1.4: Spectrogram and waveform of (a) normalized incident wave σ inc {σ 0 ,o f( b ) reflected wave normalized by incident amplitude σ ref {σ 0 , of (c) and (d) displacementsof the two masses, respectively u 1 and u 2 , normalized by the maximum displacement of the first mass maxpU 1 q. These results have been obtained numerically by means of the fourth-order Runge-Kutta method (RK4) with a wave packet source of dimensionless width ωT " 2000. The illustrated waveforms have been extracted around the time center t 0 of the source (τ " ωt 0 " 6000). System parameters are fixed at γ " 0.0162, η " 0.0088 (corresponding to Q " 80), B 1 " 0.002 and B 2 " 0.

Fourier

  spectrum for the entire reflected wave signals. These results are displayed in Fig. 1.5 for the reflection at the fundamental frequency | RpΩ " 1q| " | R1 | and in Fig. 1.6 for the reflection at the second harmonic frequency | RpΩ " 2q| " | R2 |,f o r various metasurface parameters.

Figure 1 . 5 :

 15 Figure1.5: Magnitude of the reflection coefficient at the fundamental frequency (three left-hand figures) and the second harmonic frequency (three right-hand figures) obtained from the Fourier spectrum for the entire reflected wave taken at Ω " 2, with various wave packet source widths (N T denotes the number of carrier wave periods within the width at half height of the wave packet source). In all graphs, the default parameters are impedance parameter γ 0 " 0.0162, quality factor Q " 80 (ô η 0 " 0.0088), and nonlinear parameters B 1 " B 0 1 " 0.002, B 2 " 0. Otherwise, all parameter values are indicated in the graph legend.

Figure 1 . 6 :

 16 Figure 1.6:The Fast Fourier Transform (FFT) of normalized reflected wave σ ref {σ 0 present around the fundamental harmonic ω for various source widths, with N T denoting the number of periods at half height of theincident stress wave and using parameters of the system are fixed as above: γ " 0.0162, η " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.

Figure 1 . 7 :

 17 Figure 1.7:The FFT of normalized reflected wave σ ref {σ 0 present around the second harmonic 2ω for various source widths, with N T denoting the number of periods at half height of the incident stress wave.and using parameters of the system are fixed as above: γ " 0.0162, η " 0.0088 (corresponding to a Q factor equal to 80), B 1 " 0.002 and B 2 " 0.

Figure 1 . 8 :

 18 Figure 1.8: Temporal signals of the wave packet source σ inc with the number of periods at half height equal to N T " 4, of the corresponding normalized reflected wave σ ref {σ 0 " and of the normalized displacements of two masses U 1 and U 2 (with normalization U i " u i {u 0 , pi " 1, 2q). The second harmonic component (in red lines) for the reflected wave and for the displacements σ 2ω ref and U 2ω i are obtained by applying around 2ω (from 1.5ω to 2.5ω) a bandpass filter to each original temporal signals respectively. Using the parameters of the system are fixed as above: γ " 0.0162, η " 0.0088 (corresponding to Q factor equal to 80), B 1 " 0.002 and B 2 " 0.

Figure 1 . 9 :

 19 Figure 1.9: Temporal signals of the wave packet source σ inc along with the number of periods at the half height equal to N T " 20, of the corresponding normalized reflected wave σ ref {σ 0 , and of the normalized displacements of two masses U 1 and U 2 (with normalization U i " u i {u 0 , pi " 1, 2q). The second harmonic component (shown in red lines) for both the reflected wave and for the displacements σ 2ω ref and U 2ω

Contents 2 . 1

 21 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonlinear elastic metasurface design and the reflection problem at normal incidence . . . . . . . . . . . . . . . . . 2.3 Analysis of the linearized metasurface: Parameter definitions and frequency response . . . . . . . . . . . . . . . 2.4 Nonlinear reflection by the designed metasurface: Optimal frequency conversion . . . . . . . . . . . . . . . . . . 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2 . 1 :

 21 Figure 2.1: Nonlinear metasurface design: (a) Single layer of periodically arranged rigid squares sandwiched between a moving rigid plate and a fixed wall, with elastic springs running between all the elements. The propagation medium (1) in front of the designed metasurface is assumed to be semi-infinite. (b) The metasurface unit cell is composed of two identical squares with elementary mass m. The front rigid plate has an elementary mass 2m 0 . (c) Due to symmetry, taking into account the motion of just one square of mass m and the face plate with a mass per unit length of m 0 is demonstrated to be sufficient for the considered reflection problem at normal incidence.

Figure 2 . 2 :

 22 Figure 2.2: Eigenfrequencies and eigenvectors of the considered metasurface.In the linear dissipative regime, three resonance frequencies ω i (i " 1, 2, 3) are presented (a), related to either a translation-dominated movement, denoted ω u , or a rotation-dominated movement denoted ω θ , or a combination of both. When the resonance frequency condition ω u " 2ω θ is satisfied, the mass ratio α m is determined for different values of initial angles of rotation θ 0 and stiffnesses K s and K θ (b). The ratio of θ to U 1 is examined as a function of θ 0 and K s " K θ as well, at resonance frequencies ω θ and ω u in (c) and (d), respectively. The optimal value range of stiffness and initial angle of rotation is indicated by the white dotted line in both (c) and (d).

Figure 2 . 3 :

 23 Figure2.3: Absolute reflection coefficients of the fundamental and second harmonic components, denoted R 1 and R 2 respectively, as a function of both the dimensionless impedance parameter γ and the normalized excitation frequency detuning ∆Ω. The latter is defined as the difference between the excitation frequency ω and the linear resonance frequency ω θ , subsequently normalized by ω 0 , i.e., ∆Ω "p ω ´ωθ q{ω 0 . When the input intensity is relatively weak, with a magnitude A inc " 5 ¨10 ´6, the required excitation detuning is less (∆Ω "1 ¨10 ´4), as the maximum value of R 2 exceeds 0.4 (a) and (b). Whereas with a stronger source of magnitude A inc " 5 ¨10 ´5, a frequency detuning of around ∆Ω "1.7 ¨10 ´3 is needed to totally absorb R 1 , which does not alter the amplitude of the second harmonic R 2 to reach a maximum value of nearly 0.8 (c) and (d).

Figure 2 . 4 :

 24 Figure 2.4: Kinetic energy of the metasurface at various excitation levels, from a linear configuration with A inc " 10 ´7 to a weakly nonlinear configuration with A inc " 5 ¨10 ´6 and a highly nonlinear configuration A inc " 5 ¨10 ´5, respectively, for the cases of: (a) excitation frequencies close to the linear rotation-dominated resonance frequency ω θ ,a n d (b) excitation frequencies close to the linear translation-dominated resonance frequency ω u verifying ω u " 2ω θ . During the kinetic energy test, the metasurface is excited, at each excitation frequency, by 1,000 periods of a sine signal. The dotted black line in (a) indicates the resonance shift under excitation A inc " 5 ¨10 ´5, which corresponds exactly to the optimal excitation detuning introduced in Fig.2.3.

Figure 2 . 5 :

 25 Figure2.5: Absolute reflection coefficient of fundamental (R 1 ) and second harmonics (R 2 ), as investigated by varying the excitation amplitude from a linear level (A inc " 10 ´7)t o a nonlinear level (A inc " 10 ´4). Frequency detuning is introduced in order to eliminate reflection of the fundamental wave at the desired excitation amplitude, such that (a) A inc " 5 ¨10 ´6 and (b) A inc " 5 ¨10 ´5, with the impedance parameter defined as γ " 0.008 and γ " 0.0195, respectively.

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure2.6: Theoretical and numerical results for the nonlinear metasurface reflection, as obtained with the Harmonic Balance Method (HBM) and the fourth-order Runge-Kutta (RK4) method, respectively, for the case of a relatively weak excitation with amplitude A inc " 5 ¨10 ´6 (a) and (b), and for the case of a stronger excitation with amplitude A inc " 5 ¨0´5 (c) and (d). The frequency axes are normalized by the detuned excitation frequency. By considering a wave packet source with characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results. Magnitudes of Short-Term Fourier Transforms taken at fundamental and second harmonic frequencies, by showing the temporal variation in the reflected wave spectrum (a) and (c), and by exploring the Fourier Transform of the central 1,000 periods of the reflected temporal wave (b) and (d), respectively.

Figure 2 . 8 :

 28 Figure2.8: Optimal frequency conversion effect achieved for various metasurface unit cell shapes, i.e. for different inertial moments α of rotating elements. The maximum conversion is determined as a function of α over the range of r0.02, 0.66s by varying the impedance parameter γ, excitation frequency detuning ∆Ω and stiffness K s " K θ simultaneously. In order to lessen the calculation burden, the initial angle of rotation has been set at 10 o and 8 o respectively, as these values are found to be favorable for producing the desired reflection over the entire inertial moment range.

Chapter 3 A 47 3. 3 51 3. 3 . 1 68 3. 4 . 1

 347351316841 realistic nonlinear meta-interface design achieving acoustic wave scattering control Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Nonlinear meta-interface design and the considered onedimensional scattering problem . . . . . . . . . . . . . . . Nonlinear scattering by a symmetrical meta-interface design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration of a single side excitation: f p1q inc pΩ,τq"A inc cospΩτ q and f p2q inc pΩ,τq"0 ...................... 54 3.3.2 Configuration of two symmetrical excitations: f p1q inc pΩ,τq" f p2q inc pΩ,τq"A inc cospΩτ q ................... 63 3.4 Nonlinear scattering by an asymmetrical meta-interface design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration of a single side excitation: f p1q inc pΩ,τq"A inc cospΩτ q and f p2q inc pΩ,τq"0 ...................... 68 3.4.2 Configuration of dual-excitations: f p1q inc pΩ,τq"A p1q inc cospΩτ q and f p2q inc pΩ,τq"A p2q inc cospΩτ q ................ 75 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 3 . 1 :

 31 Figure 3.1: Different types of structures and different excitation configurations under consideration for the proposed meta-interface design. The symmetrical structure (with identical mass of the two moving plates) is investigated in Section 3.3, the configurations of a single excitation at one side (a) and of a dual-excitation at two sides (b) are taken into account and explored respectively in Subsections 3.3.1 and 3.3.2. For the asymmetrical structure of meta-interface characterized by moving plates with different masses, the investigations performed in Section 3.4 are similar to the symmetrical structure, with consideration of single excitation (Subsection 3.4.1) (c) and of dual-excitation (Subsection 3.4.2) (d), respectively. Throughout this chapter, we are interested in the maximum energy conversion from fundamental incoming wave to the transmitted/reflected second harmonic.

Figure 3 . 2 :

 32 Figure 3.2: Nonlinear meta-interface design, by (a) a monolayer of periodically arranged rigid squares sandwiched between two moving rigid plates, with elastic contact between all the elements. The designed meta-interface is inserted between two identical semi-infinite propagation media p1q and p2q. The unit cell of the meta-interface is made of two identical squares with elementary mass m and two moving plates connected in front and back having elementary mass 2m 1 and 2m 2 respectively (b). Due to the symmetry, taking into account the movement of only one square m and the face blocs m 1 and m 2 is demonstrated sufficient for the considered 1D scattering problem at normal incidence (c).

p1q ref and σ p2q ref

 p2q denote the reflected wave in the left and right medium, respectively.

  Fig 3.3.

Figure 3 . 3 :

 33 Figure 3.3: Eigenfrequencies and eigenvectors of the considered meta-interface in the linear lossless regime. When the eigenfrequencies Ω 2 and Ω 3 satisfy the condition Ω 3 " 2Ω 2 ,t h e absolute value of displacement ratio between θ and U n (n=1,2) is investigated by varying the initial angle θ 0 and the stiffness K s " K θ , at frequency (a) Ω 2 and (b) Ω 3 , respectively.In the considered value range of θ 0 Pp 0 o , 30 o q and K s " K θ Pp 0, 0.2q, the eigenfrequency Ω 2 corresponds always to a rotation-dominated movement since the minimum absolute value of θ{U n is around 2.7 at this frequency, whereas the eigenfrequency Ω 3 relates either to a translation-dominated movement or to a rotation-dominated movement depending on the values of system parameters. The white dotted lines in (a) and (b) show the threshold of displacement ratio θ{U n equal to 15 at Ω 2 , while the red dotted lines represent the case of θ{U n " 1 at Ω 3 .

Figure 3 . 5 :

 35 Figure3.5: Magnitude of reflection and transmission coefficients for the fundamental wave (R 1 and T 1 ) and for the second harmonic (R 2 and T 2 ), investigated by varying the excitation amplitude from linear level (A inc " 10 ´7) to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the transmission of fundamental wave at excitation magnitude A inc " 5¨10 ´5, provided that the transmission of second harmonic can not be reduced. Impedance parameter is chosen at γ " 0.013 enabling the maximization of T 2 in the condition that the stiffness and the initial rotation angle are defined as K s " K θ " 0.05 and θ 0 " 10 o .

Figure 3 . 6 :

 36 Figure 3.6: Theoretical and numerical results of 1D scattering by the proposed nonlinear meta-interface with symmetrical structure, obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, under a single side excitation with magnitude A inc " 5 ¨10 ´5, and in the case of K s " K θ " 0.05, θ 0 " 10 o and γ " 0.013, the introduced detuning is ∆Ω " 5 ¨10 ´4. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). With a wave packet source of characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d).

Figure 3 . 7 :

 37 Figure 3.7: Optimal frequency conversion achieved for different properties of the proposed symmetrical meta-interface, under a single side excitation of magnitude A inc " 5¨10 ´5. The maximum absolute value of transmission coefficient of second harmonic T 2 providing the T 1 reduction is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of initial angle θ 0 set at 5 o , 10 o and 15 o , respectively. Presented results are obtained by HBM.

  Fig 3.8. The optimization process implemented for obtaining Fig 3.7 and Fig 3.8 is briefly summarized in the schematic paq of Fig 3.10 at the end of this section. According to Fig 3.7 and Fig 3.8, for the stiffness K s " K θ smaller than 0.05 and the initial angle ✓ 0 between 3 o and 15 o ,t h et r a n s m i s s i o nc o e ffi c i e n to fs e c o n d

Figure 3 . 8 :

 38 Figure 3.8: Optimal frequency conversion achieved for different properties of the proposed symmetrical meta-interface, under a single side excitation of magnitude A inc " 5¨10 ´5. The maximum absolute value of transmission coefficient of second harmonic T 2 providing the T 1 reduction is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of stiffness set at 0.05, 0.1 and 0.15 respectively. Presented results are obtained by HBM.

Figure 3 . 10 :

 310 Figure 3.10: Schematic representation of the performed optimization processes applied for obtaining maximum transmission of second harmonic. The diagram (a) shows the main steps of the implemented algorithm for the case of definite inertial moment (calculation results presented in Fig 3.7 and Fig 3.8). The optimization outcome shown in Fig 3.9 during the inertial moment variation is obtained via the procedure illustrated in (b) where the parameter γ and ∆Ω are varied in the same way as process paq.

Figure 3 . 11 :

 311 Figure 3.11: Absolute reflection coefficients of fundamental wave (R 1 ) and of second harmonic (R 2 ) of both sides of the meta-interface, investigated by varying the excitation amplitude from linear level (A inc " 10 ´7) to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the fundamental reflected wave at excitation magnitude A inc " 5 ¨10 ´5, provided that the reflection of second harmonic is not reduced. Impedance parameter is chosen at γ " 0.02 enabling the maximization of T 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively.

Figure 3 .

 3 Figure3.13: Optimal frequency conversion achieved for different physical properties of proposed symmetrical meta-interface, under dual-excitation, both with magnitude A inc " 5 ¨10 ´5 at the two sides of the meta-interface. The maximum absolute value of reflection coefficient of second harmonic R 2 of both sides is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of initial angle set at 5 o , 10 o and 15 o respectively (a), and the case of stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively (b).

Figure 3 . 14 :

 314 Figure 3.14: Optimal frequency conversion achieved with the proposed meta-interface with symmetrical structure composed of different rotating elements, i.e., for different inertial moments α, under two identical excitations with magnitude A p1q inc " A p1q inc " 5 ¨10 ´5.The optimal conversion is determined as the function of α in the value range of r0.03, 0.66s,b y varying simultaneously all other physical parameters, i.e., the impedance parameter γ,t h e excitation frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle θ 0 .

piq 1 and R piq 2

 2

  7.I n d e e d ,f o rt h ec o nsidered excitation level A p1q inc " A p1q inc " 5 ¨10 ´5,t h eo b t a i n e dc oe ffi c i e n t sR piq 1 and R piq 2

Figure 3 . 15 :

 315 Figure3.15: Absolute reflection and transmission coefficients of fundamental (R 1 and T 1 ) and second harmonic (R 2 and T 2 ), investigated by varying the excitation amplitude from linear level (A inc " 10 ´7) to strong nonlinear level (A inc " 10 ´4). The frequency detuning is introduced to minimize the transmission of fundamental wave at excitation magnitude A inc " 5 ¨10 ´5, provided that the transmission of second harmonic is not reduced. Impedance parameter is chosen at γ " 0.013 enabling the maximization of T 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively.

Figure 3 . 16 :

 316 Figure 3.16: Theoretical and numerical 1D scattering results of the asymmetrical nonlinear meta-interface (r m " 5), obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, for the configuration of a single side excitation with magnitude of A inc " 5 ¨10 ´5, in the case where K s " K θ " 0.05, θ 0 " 10 o and γ " 0.013, and where the introduced detuning is ∆Ω " 6 ¨10 ´4. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). For a wave packet source with characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d).

Figure 3

 3 Figure 3.17: Optimal frequency conversion achieved for different physical properties of proposed asymmetrical meta-interface, under single excitation with magnitude A inc " 5 10 ´5 at left side of meta-interface. The maximum absolute value of transmission coefficient of second harmonic T 2 is identified by varying the impedance parameter and the excitation frequency detuning, in the case of stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively.

Figure 3 . 18 :

 318 Figure 3.18: Optimal frequency conversion achieved for different physical properties of proposed asymmetrical meta-interface, under a single side excitation with magnitude A inc " 5 ¨10 ´5 at left side of meta-interface. The maximum absolute value of transmission coefficient of second harmonic T 2 is identified by varying the impedance parameter and the excitation frequency detuning, in the case of initial angle set at 5 o , 10 o and 15 o , respectively.

3. 4 . 2

 42 Configuration of dual-excitations: f p1q inc pΩ,⌧q"A p1q inc cospΩ⌧ q and f p2q inc pΩ,⌧q"A p2q inc cospΩ⌧ q

Figure 3 . 21 :

 321 Figure 3.21: Absolute reflection coefficients of the fundamental wave (R piq 1 )a n do ft h e second harmonic (R

Figure 3 . 22 :

 322 Figure 3.22: Magnitude of the reflection coefficients of fundamental (R 1 ) and second harmonic (R 2 ) of both sides of the proposed asymmetrical meta-interface structure with r m " 5, investigated by setting the favorable amplitude ratio R amp " 1 between the two excitations. The excitation level is varied from linear level (Ap1q inc " A p2q inc " 10 ´7)t o strong nonlinear level (A p1q inc " A p2q inc " 10 ´4).The frequency detuning is introduced with an amount of ∆Ω " 1.1 ¨10 ´3 to minimize the reflected fundamental wave at excitation magnitude A inc " 5 ¨10 ´5, provided that the reflection of second harmonic is not lowered (for both sides). Impedance parameter is chosen at γ " 0.0228 yielding the maximization of R 2 for the stiffness and the initial rotation angle defined as K s " K θ " 0.05 and θ 0 " 10 o respectively.

Figure 3 . 23 :

 323 Figure 3.23: Theoretical and numerical results of the nonlinear meta-interface 1D scattering, obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, for the configuration of two excitations with both a magnitude of A p1q inc " 5 ¨10 ´5, in the case where K s " K θ " 0.05, θ 0 " 10 o and γ " 0.0228, and where the introduced detuning is ∆Ω " 1.1 ¨10 ´3. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). By considering a wave packet source with a characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d). All incident and reflected waves are normalized by the square root of input intensity.

p2q 2 )

 2 . Thereafter the obtained maximum second harmonic result is optimized by

Figure 3 . 24 :

 324 Figure 3.24: Optimal frequency conversion achieved for different physical properties of the proposed asymmetrical meta-interface excited from both sides. The maximum reflection coefficient of second harmonic R p2q

Figure 3 . 25 :

 325 Figure 3.25: Optimal frequency conversion achieved for different physical properties of the proposed asymmetrical meta-interface excited from both sides. The maximum reflection coefficient of second harmonic R p2q 2 at right side of meta-interface is identified by varying the impedance parameter, the excitation magnitude ratio and the excitation frequency detuning simultaneously, in the case of initial angle set at 5 o , 10 o and 15 o , respectively.

Figure 3 .

 3 Figure 3.24 and Figure3.25 present the optimal conversion result after the described investigation, for initial angle set at 5 o , 10 o and 15 o , and for stiffness fixed as K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15,r e s p e c t i v e l y . N o t i c e that during the investigation, the excitation magnitude ratio R amp is varied by fixing the magnitude of A

Figure 3 . 26 :

 326 Figure 3.26: Optimal frequency conversion achieved for different shapes of unit ceil of the asymmetrical meta-interface, i.e., for different inertial moments α of rotational elements.The maximum conversion is determined as the function of α in the range of r0.03, 0.66s,b y varying all the other physical parameters, i.e., the impedance parameter γ, the excitation magnitude ratio R amp , the excitation frequency detuning ∆Ω and the stiffness K s " K θ simultaneously.

Figure 3 . 27 :

 327 Figure 3.27: Schematic representation of performed optimization processes applied for obtaining maximum reflection of the second harmonic propagating in medium p2q. The diagram (a) shows the main steps of algorithm implemented for the case of definite inertial moment (calculation results presented in Fig 3.25 and Fig 3.24). The optimization outcome shown in Fig 3.26 during the inertial moment or mass ratio variation is obtained via the procedure illustrated in (b) where the parameter γ and ∆Ω are varied in the same way as process paq.

Figure 3 .

 3 Figure3.28: Optimal frequency conversion achieved for the asymmetrical rotating-square architected meta-interface with moving plates of different masses in front and back defined by the mass ratio r m " α m2 {α m1 , under two excitations at both sides, the inertial moment α is fixed for homogeneous square α " 1{3. The maximum conversion is determined as the function of r m in the range of r1.5, 10s, by varying all the other physical parameters, i.e., the impedance parameter γ, the excitation magnitude ratio R amp , the excitation frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle θ 0 .

4. 1

 1 Considered architected meta-structure and the corresponding governing motion equations

Figure 4 . 1 :

 41 Figure 4.1: A picture of the considered architected meta-structure, composed of two rows of rotating crosses with 5 elements for each row. This structure is placed horizontally and is connected to a low frequency shaker, seen on the left.

Figure 4 . 2 :

 42 Figure 4.2: Schematic representation of the considered architected meta-structure, composed of two rows of rotating crosses with N elements for each row. The unit cell of the structure corresponds to a 2 ˆ2 ensemble of adjacent crosses pbq. The initial and the dynamic angles of element pp, jq are denoted as ϕ p,j and θ p,j respectively, which are defined as positive if they follow the counterclockwise direction pcq.

Figure 4 . 3 :

 43 Figure 4.3: Dispersion curves of the considered meta-structure with infinite length, obtained by setting the initial rotation angle ϕ 0 at 0 o (a) and 3 o (b), respectively. The color gradient in each figure is employed to characterize the associated movement property of each possible mode pω, kq, the colors green, blue and red correspond to the vertical movement dominated, the horizontal movement dominated and the rotation dominated mode respectively.

Figure 4 . 4 :

 44 Figure 4.4: Dispersion curve of the meta-structure with infinite length, determined by considering only the horizontal movement U and the translation θ. The initial rotation angle ϕ 0 is set at 1 o (a) and 3 o (b), respectively. The obtained results (solid lines) are compared to that of 3 dof configuration (dotted lines). The color gradient in each figure is employed to characterize the associated movement property of each possible mode pω, kq, the colors blue and red correspond to the horizontal movement dominated and the rotation dominated mode respectively.

  (4.1) are solved by implementing time

Figure 4 . 5 :

 45 Figure 4.5: Evolution of average (DC) horizontal displacement of the units in the last column of the meta-structure over the gradual increase of monochromatic excitation frequency, obtained with structure length of N " 6 (a) and N " 16 (c) respectively. Each average displacement is determined under a sinusoidal excitation of 40 period length and of normalized magnitude 1 ¨10 ´3 applied to the first column of meta-structure. The excitation frequency range is considered between 10Hz to 100Hz for (a) and (c). In (b) and (d), for the last units of the meta-structure, the magnitude of the Fourier spectrum of the displacement U along x and of the rotation θ around z are presented in the case of N " 6 and N " 16 respectively, under sweep frequency excitation of magnitude 1¨10 ´5 and within the whole available frequency band shown in dispersion curve (from 1Hz to 1300 Hz). The obtained spectrum results are illustrated in the range of p0, 300q Hz in (b) and (d).

6 )

 6 inc u"tF NL u, (A.4) with the source vector tU inc u written as tU inc u"t0, 2, 0, 0, ¨¨¨, 0u T p4N `2qˆ1 . The mass matrix rMs,s p r i n gm a t r i xrKs and the damping matrix rΓs are respectively rMs"The nonlinear force tF NL u takes the form tF NL u"

Ω 2 2 ⇢if m ‰ n ‰ 0 1 ,C

 221 xrSs T rMs " B 2 S B⌧ ytqu`ΩxrSs T rΓs " BS B⌧ ⇢ ytqux rSs T rKsrSsytqu`xrSs T rSstU inc uy " xrSs T F NL y, (A.9) where x¨¨¨y denotes the integral operator 1 π ≥ 2π 0 p¨¨¨qd⌧ .Thanks to this projection, the calculation is simplified because for all positive integers n et m,w eh a v e 1 π ≥ 2π 0 cospm⌧ q sinpn⌧ qd⌧ " if m " n ‰ 0 2, if m " n " 0 . (A.10)In the matrix form motion equation system Eq. (A.9), with these integral results, all the linear terms can be simplified and determined analytically. The nonlinear term xrSs T F NL y has the following explicit form, xrSs T F NL y"B 1 rAs"t Cut∆qu T tCutCu T , rBs"t Cutq 2 u T tCutCu T , and with t∆qu T " tq 1 u T ´tq 2 u T . i C m C j d⌧, (A.[START_REF] Mei | Dark acoustic metamaterials as super absorbers for low-frequency sound[END_REF]

  Optimal frequency conversion achieved for different properties of the proposed symmetrical meta-interface, under a single side excitation of magnitude A inc " 5 ¨10 ´5. The maximum absolute value of transmission coefficient of second harmonic T 2 providing the T 1 reduction is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of stiffness set at 0.05, 0.1 and 0.15 respectively. Presented results are obtained by HBM. . 60 3.9 Optimal frequency conversion achieved for different shapes of unit cell of the meta-interface with symmetrical structure, i.e., for different inertial moments ↵ of rotating elements, in the configuration of single excitation performed at only left side. The maximum conversion to second harmonic wave is determined as a function of ↵ in the range of r0.03, 0.66s, by varying the stiffness K s " K

	3.7 Optimal frequency conversion achieved for different properties of the
	proposed symmetrical meta-interface, under a single side excitation
	of magnitude A

05 and ✓ 0 " 10 o .. . . . . . . . . . . . . . . . . . . .56 3.6 Theoretical and numerical results of 1D scattering by the proposed nonlinear meta-interface with symmetrical structure, obtained with Harmonic Balance Method (HBM) and with the fourth order Runge-Kutta method (RK4), respectively, under a single side excitation with magnitude A inc " 5¨10 ´5,a ndinthecaseofK s " K θ " 0.05, ✓ 0 " 10 o and γ " 0.013,t h ei n t r o d u c e dd e t u n i n gi s∆Ω " 5 ¨10 ´4. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). With a wave packet source of characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d). . . . . inc " 5 ¨10 ´5. The maximum absolute value of transmission coefficient of second harmonic T 2 providing the T 1 reduction is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of initial angle ✓ 0 set at 5 o , 10 o and 15 o ,r e s p e c t i v e l y . P r e s e n t e dr e s u l t sa r eo b t a i n e db y HBM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 θ ,theinitialangle✓ 0 ,the impedance parameter γ and the excitation frequency detuning ∆Ω simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.10 Schematic representation of the performed optimization processes applied for obtaining maximum transmission of second harmonic. The diagram (a) shows the main steps of the implemented algorithm for the case of definite inertial moment (calculation results presented in Fig 3.

  Optimal frequency conversion achieved for different physical properties of proposed symmetrical meta-interface, under dual-excitation, both with magnitude A inc " 5 ¨10 ´5 at the two sides of the metainterface. The maximum absolute value of reflection coefficient of second harmonic R 2 of both sides is identified by varying the impedance parameter and the excitation frequency detuning simultaneously, in the case of initial angle set at 5 o , 10 o and 15 o respectively (a), and the case of stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively (b). . . . . . . . . . . . . . . . . . . . . 66 3.14 Optimal frequency conversion achieved with the proposed meta-interface with symmetrical structure composed of different rotating elements, i.e., for different inertial moments ↵,u n d e rt w oi d e n t i c a le x c i t a t i o n s with magnitude A

	The introduced detuning is ∆Ω " 1 ¨10 ´3. The frequency axes are normalized by the detuned excitation frequency in (b) and (d). By considering a wave packet source with characteristic width N p1q inc " A p1q inc " 5 ¨10 ´5.

T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d). All wave magnitudes are normalized by the square root of input intensity A inc .. . . . . . . . 65 xiii 3.13

  Optimal frequency conversion achieved for different physical properties of proposed asymmetrical meta-interface, under a single side excitation with magnitude A inc " 5 ¨10 ´5 at left side of meta-interface. The maximum absolute value of transmission coefficient of second harmonic T 2 is identified by varying the impedance parameter and the excitation frequency detuning, in the case of initial angle set at 5 o , 10 o and 15 o ,r e s pe c t i v e l y . . . . . . . . . . . . . . . . . . . . . . . . 72 3.19 Optimal frequency conversion achieved for the asymmetrical metainterface design composed of different shapes of rotating element, i.e., for different inertial moments ↵,u n d e ras i n g l es i d ee x c i t a t i o no f magnitude A inc " 5 ¨10 ´5

	. . 70
	3.17 Optimal frequency conversion achieved for different physical proper-
	ties of proposed asymmetrical meta-interface, under single excitation
	with magnitude A

The frequency axes are normalized by the detuned excitation frequency in (b) and (d). For a wave packet source with characteristic width N T " 4000T , the RK4 results are compared to the theoretical HBM results, by investigating the temporal variation of reflected wave spectrum (spectrogram) (a) and (c), and by exploring the Fourier Transform of the center 1000 periods of reflected temporal wave (b) and (d). . . . . . . . . . . . . . . . . . . . . . . inc " 5¨10 ´5 at left side of meta-interface. The maximum absolute value of transmission coefficient of second harmonic T 2 is identified by varying the impedance parameter and the excitation frequency detuning, in the case of stiffness set at K s " K θ " 0.05, K s " K θ " 0.10 and K s " K θ " 0.15 respectively. . . . . . . . . . . .

71 3.18

  Bτq`1 2 rcos ' p,j`1 ´cos ' p,j´1 `cosp' p,j´1 `✓p,j´1 q´cosp' p,j`1 `✓p,j`1 qs `1 2 ⌘ θ rsin ' p,j`1 p´1q p K s rsinp' p,j `✓p,j q`sinp' p`1,j `✓p`1,j q´sin ' p,j ´sin ' p`1,j s `1 2 p´1q p ⌘ θ K s rcos ' p,j Bτ s´K s pU p,j ´Up`1,j `⌘u

	B 2 U p,j Bτ 2	" U p,j´1 ´2U p,j `Up,j`1 `⌘u p	BU p,j´1 Bτ	´2 BU p,j Bτ	`BU p,j`1
						Bθ p,j`1 Bτ	´sin ' p,j´1	Bθ p,j´1 Bτ s
		`1 2 Bθ p,j Bτ `cos ' p`1,j	Bθ p`1,j	BU p,j

  2 ⌘ θ K s rcos ' p,j´1

				Bθ p,j´1
				Bτ
	´cos ' p,j`1	Bθ p,j`1
				Bθ p`1,j Bτ s
	´p V p,j ´Vp`1,j `⌘v	BV p,j Bτ ´⌘v	BV p`1,j

Bτ s`1 2 p´1q p rcos ' p,j `cos ' p`1,j ´cosp' p,j `✓p,j q ´cosp' p`1,j `✓p`1,j qs `1 2 p´1q p ⌘ θ rsin ' p,j Bθ p,j Bτ `sin ' p`1,j

  `cosp✓ 0 q´cosp✓ 0 `✓q`⌘ θ sin ✓ 0 Bτ 2 "6K θ `✓ `⌘θ

	1 Bτ 2	"2f inc pΩ,⌧q´γ BU 1 Bτ ´U1 `U2 ´⌘u `1 2 Bθ BU 1 Bτ `⌘u Bτ ˘,	BU 2 Bτ
	B 2 U 2 Bτ 2 ↵ B 2 θ	" U 1 ´2U 2 `⌘u Bθ BU 1 Bτ ´2⌘ u Bτ ˘`2pU 1 `⌘u BU 2 Bτ ,	BU 1

  ⌧q´γ BU 1 Bτ ´U1 `U2 ´⌘u Bτ `1 2 sin ✓ 0 ✓ `1 4 cos ✓ 0 ✓ 2 ´1 12 sin ✓ 0 ✓ 3 ´1 48 cos ✓ 0 ✓ 4 `1 2 ⌘ θ sin ✓ 0

			BU 1 Bτ `⌘u	BU 2
			Bθ
			Bτ
	B 2 U 2 Bτ 2 ↵ B 2 θ Bτ 2	" U 1 ´2U 2 `⌘u Bτ BU 1 Bτ ´2⌘ u "6 `Kθ ✓ `⌘θ Bθ 2pU BU 2 Bτ .
		1 `⌘u	BU 1

provided that the second harmonic generation is not weaken.

Acknowledgements ACKNOWLEDGEMENTS methods, their suggestions helped me to advance my thesis work. With a special mention to Dr. Katia BERTOLDI and her research group, It

This thesis was carried out in the Acoustic Laboratory of Le Mans University (LAUM)project "Acoustic HUB" of the Consortium de Le Mans Acoustic (LMAC), in Le Mans, for assistance on my study throughout last seven years in Le Mans, or for kindness brought to me during my stay in LAUM.

First of all, I would like to thank the director of my thesis Dr. Vincent TOUR-NAT for proposing this PHD, and for introducing me to Dr. Katia BERTOLDI, the professor of applied mechanics school of engineering and applied sciences of Harvard university. Thanks to Dr. Vincent TOURNAT for his cooperation with the research group of Dr. Katia BERTOLDI, the present PHD work was possible to be continued and further developed.

shown that the transmission of second harmonic is more efficient with the presented asymmetrical design. Optimal frequency conversion achieved for the asymmetrical meta-interface design composed of different shapes of rotating element, i.e., for different inertial moments α, under a single side excitation of magnitude A inc " 5 ¨10 ´5 and with mass ratio between two moving plates defined as r m " 5. The maximum conversion is determined as the function of α in the range of r0.03, 0.66s, by varying all the other physical parameters, i.e., the impedance parameter γ, the excitation frequency detuning ∆Ω, the stiffness K s " K θ and the initial angle θ 0 .

The above analysis on nonlinear scattering can be completed herein by the consideration of ↵ value expansion to its considered range p0, 2{3q.I nt h ec a s eo fr m " 5, the required second harmonic enhancement is acquired via the adjustment of all the other involved parameters (stiffness K s " K θ ,initialangle✓ 0 ,impedanceparameter γ and excitation frequency detuning ∆Ω). The approach for the single side excitation case of the symmetrical metasurface (Fig. 3.10(a)) is performed as previously. In short, from the desired scattering coefficients achieved by varying γ and ∆Ω under each amount of p✓ 0 ,K s " K θ q,t h eo p t i m a lr e s u l t( m a x i m u mT 2 )t h r o u g h o u tt h e variation of p✓ 0 ,K s " K θ q is accounted as the final optimized scattering coefficients for the definite value of ↵. The variation steps of all the parameters are retained as well (2 o and 0.01 for ✓ 0 and K s " K θ , 1 ¨10 ´4 for ∆Ω, ↵ with step of 0.03 from 0.03 to 0.66), The Fig. 3.19 presented the final result of transmission coefficients over the increase of ↵. Within the considered whole value range of ↵ Pp 0, 2{3q, the transmission of second harmonic is confirmed to be relevant, owning a transmission coefficient always more important than 0.5,e s p e c i a l l yf o r↵ Pp 0.2, 0.66q,t h e transmission coefficient of second harmonic can be larger than 0.65.

Moreover, we confirm with Fig. 3.19 that the proposed asymmetrical metainterface enables a second harmonic transmission larger than the symmetrical design. Indeed, in the present case, the transmission coefficient of second harmonic T 2 is demonstrated between 0.5 and 0.7 in the whole range of ↵, whereas for the symmetrical design with single side excitation, T 2 can just between 0.4 and 0.5 (see Fig. 3.9).

Chapter 4

Preliminary study on the nonlinear dynamic behavior of a soft rotating-square structure of connected LEGOs All the basic elements are identical, with mass m and inertial moment J.F o r a systematic analysis, we introduce the following dimensionless parameters: horizontal and vertical displacements normalized as U p,j " u p,j {2l and V p,j " v p,j {2l for the LEGO cross pp, jqp p " 1, 2, 1 § j § N q, respectively, with 2l denoting the length of the LEGO brick, pulsation Ω " !{! 0 with ! 0 " a k l {m,t i m e⌧ " ! 0 t,i n e r t i a l moment of each LEGO cross ↵ " J{ml 2 , normalized shear stiffness K s " k s {k l and normalized bending stiffness K θ " k θ {k l l 2 .

Based on results and validations shown in [START_REF] Deng | Metamaterials with amplitude gaps for elastic solitons[END_REF], the springs are assumed to behave linearly and dissipation is accounted for via linear viscous damping associated with the respective translation and rotation motions of each unit. The characteristic dissipation parameters Γ u , Γ v and Γ θ are normalized as ⌘ u " Γ u {m! 0 , ⌘ v " Γ v {m! 0 and ⌘ θ " Γ θ {m! 0 for the two translations and the rotation, respectively. The governing motion equations of element pp, jq with 1 † j † N and p " 1, 2 are written in Eq. (4.1) where the index p `1 should be changed to p ´1 for the second row p " 2. Despite that the meta-structure has only two rows p " 1, 2,w ek e e pu s i n g the notation pp, jq to carry out the following calculations instead of p1,jq and p2,jq, since the presented formulation will be helpful for a wider multi-row structure to be investigated perhaps in the future. The equations for the elements at the two ends (j " 1 and j " N )a r es i m i l a rt oE q .( 4 . 1 ) ,b u tf o ra ne a s i e rr e a d i n g ,t h e ya r e presented in Appendix B.

The meta-structure contains a geometric nonlinearity which is activated via the rotations of the LEGO units, while under a relatively weak input intensity, the structure behaves quasi-linearly. Thus, before investigating the complex nonlinear properties of the meta-structure, in order to understand step-by-step its behavior, we start with the dispersion properties. A linear study of dispersion is performed firstly in following Section 4.2 by assuming that the structure is infinitely long (j Ñ8).

Dispersion curve of the meta-structure

When the structure is infinitely long, the dispersion properties can be determined in the lossless configuration, by solving the linearized problem in one unit cell composed of 4 adjacent elements forming a 2 ˆ2 matrix, i.e., of LEGO bricks denoted as pp, jq, pp `1,jq, pp, j `1q and pp `1,j `1q.

Let's start with the LEGO brick of number pp, jq, the following linear approximation via the series expansions of the trigonometric functions is applied to the motion equations given in Eq. (4.1),

with pm, nq denoting the numbering of the considered LEGO pp, jq and of all the elements connected with it, i.e., elements pp, j ´1q, pp, j `1q and pp `1,jq. Then, the linearized motion equations for the LEGO pp, jq are obtained by setting ⌘ u " ⌘ v " ⌘ θ " 0: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

" U p,j´1 ´2U p,j `Up,j`1 `1 2 rsin ' p,j`1 ✓ p,j`1 ´sin ' p,j´1 ✓ p,j´1 s `1 2 p´1q p K s rcos ' p,j ✓ p,j `cos ' p`1,j ✓ p`1,j s´K s pU p,j ´Up`1,j q `1 2 K θ p✓ p,j´1 ´✓p,j`1 q sin ' p,j `1 2 p´1q p K θ p✓ p,j ´✓p`1,j q cos ' p,j

" K s pV p,j`1 ´2V p,j `Vp,j´1 q`1 2 K s r´cos ' p,j`1 ✓ p,j`1 `cos ' p,j´1 ✓ p,j´1 s `1 2 p´1q p rsin ' p,j ✓ p,j `sin ' p`1,j ✓ p`1,j s´pV p,j ´Vp`1,j q `1 2 K θ p✓ p,j`1 ´✓p,j´1 q cos ' p,j `1 2 p´1q p K θ p✓ p,j ´✓p`1,j q sin ' p,j

´2U p,j´1 `2U p,j`1 `p´1q p 2pV p`1,j ´Vp,j qs ´Ks cos ' p,j r2V p,j´1 ´2V p,j`1 `p´1q p 2pU p`1,j ´Up,j q`3cos' p,j ✓ p,j `cos ' p,j´1 ✓ p,j´1 `cos ' p`1,j ✓ p`1,j `cos ' p,j`1 ✓ p,j`1 s ´Kθ p3✓ p,j ´✓p,j`1 ´✓p`1,j ´✓p,j´1 q (4.

3) The motion equations of LEGOs pp`1,jq, pp, j`1q and pp`1,j`1q are linearized in the same way as LEGO cross pp, jq, which will not be repeatedly shown here.

The angles of each element (initial and dynamic) have been defined as positive or negative depending on the rotation direction, whereas in the present dispersion study, we specify that all the quantities ' p,j correspond to positive values, i.e. the alternating angle signs of each row is included in the definition of the angle. Similarly, the definition of the dynamic angle ✓ p,j is defined with alternating signs for adjacent elements, providing a continuous in space change of angle. We assume as well that at rest, the initial rotation angles of all the elements are identical having the value of ' 0 .I nt h i sc a s e ,c o n s i d e r i n gam o n o c h r o m a t i ce x c i t a t i o na tf r e q u e n c y! applied to the meta-structure, the complex solutions of the linearized motion equations for each unit cell can be included in two vectors tu j u L and tu j`1 u L for column j and j `1 respectively, taking the form of

/ / / / / / -e ipωt´kx j`1 q , (4.4)

with k denoting the wave number. The symbol i in the exponential parts denotes the complex number verifying i 2 "´1.

$ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

´!2 U p`1,j "ṕ 4sin 2 pkd{2q`K s qU p`1,j `Ks U p,j ´1 2 pK θ ´Ks q cos ' 0 ✓ p,j

´"i sin ' 0 p1 `Kθ q sinpkdq`1 2 pK θ `Ks q cos ' 0 ‰ ✓ p`1,j

´!2 V p`1,j ""4K s sin 2 pkd{2q`1

$ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

`"´i cos ' 0 pK s ´Kθ q sinpkdq`1 2 pK θ `1q sin ' 0 ‰ ✓ p`1,j`1 ´↵! 2 ✓ p`1,j`1 "p 4i sin ' 0 sinpkdq`2K s cos ' 0 qU p`1,j`1

´"sin 2 ' 0 ´Ks cos

the above systems of equations Eq. (4.5) to Eq. (4.8) can be rewritten into the matrix form:

´!2 rMs

-"

When the initial angle is set to zero, there is no coupling between the vertical movement V ,h o r i z o n t a lm o v e m e n tU and the rotation ✓.W h i l ea c c o r d i n gt o Fig. 4.1, for the initial angle set as ' 0 " 3 o ,t h eh o r i z o n t a lm o v e m e n tU and the rotation ✓ can be coupled depending on the excitation frequency and the wave number. Furthermore, if we consider only the excitation frequencies below 880Hz, it is impossible to excite the vertical movement with the given meta-structure. As a consequence, it seems sufficient to take into account only 2 degrees of freedom for the structure, i.e., the horizontal movement and the rotation. Thus, in the following Section 4.3, assuming that the meta-structure moves with the above mentioned two degrees of freedom, the motion equations are derived, the dispersion curve are determined and then compared to the 3 dof case.

Particular case of 2 dof

When only the horizontal movement and the rotation are taken into account, for the LEGO cross pp, jq, the system of motion equations Eq. (4.1) is simplified into the following one for 1 † j † N : $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

" U p,j´1 ´2U p,j `Up,j`1 `1 2 rcos ' p,j`1 ´cos ' p,j´1 `cosp' p,j´1 `✓p,j´1 q ´cosp' p,j`1 `✓p,j`1 qs `1 2 p´1q p K s rsinp' p,j `✓p,j q`sinp' p`1,j `✓p`1,j q ´sin ' p,j ´sin ' p`1,j s´K s pU p,j ´Up`1,j q `1 2 K θ p✓ p,j´1 ´✓p,j`1 q sinp' p,j `✓p,j q`1 2 p´1q p K θ p✓ p,j ´✓p`1,j q cosp' p,j `✓p,j q

"śinp' p,j `✓p,j qr3cos' p,j `cos ' p,j´1 `cos ' p,j`1 `cos ' p`1,j ´2U p,j´1 `2U p,j`1 ´3cosp' p,j `✓p,j q´cosp' p,j´1 `✓p,j´1 q ´cosp' p,j`1 `✓p,j`1 q´cosp' p`1,j `✓p`1,j qs ´Ks cosp' p,j `✓p,j qrp´1q p 2pU p`1,j ´Up,j q`3sinp' p,j `✓p,j q `sinp' p,j´1 `✓p,j´1 q`sinp' p`1,j `✓p`1,j q`sinp' p,j`1 `✓p,j`1 q ´3sin' p,j ´sin ' p,j´1 ´sin ' p,j`1 ´sin ' p`1,j s ´Kθ p3✓ p,j ´✓p,j`1 ´✓p`1,j ´✓p,j´1 q (4.14) where the index p `1 needs to be changed into p ´1 for the second row (p " 2). The motion equations of the elements at the two ends of the structure (j " 1 and j " N )a r ep r e s e n t e di nA p pe n d i xB .

The same approach to determine the dispersion properties is applied herein to the 2 dof configuration. Since all the modes are decoupled when the initial angles of the meta-structure units are equal to zero (' 0 " 0 o ), it is unnecessary in this case to domain simulation via the fourth order Runge-Kutta method (RK4 method), considering the meta-structure properties presented in Eq. (4.13). Due to the symmetry, each elements in the first and in the second row move symmetrically with respect to the direction x,i . e . ,h a v i n gt h es a m ed i s p l a c e m e n tu along the horizontal direction, vertically moved and rotated respectively at the same rate but in the opposite directions.

In the present work, it is considered that the dissipation remains relatively weak by taking the same dimensionless values of ⌘ u " ⌘ v " ⌘ θ " 0.001. In the case where the meta-structure is excited via a monochromatic source inducing a horizontal displacement to the units of the first column. The variation of the average displacement of the units in the last column j " N s is presented in Fig. 4.5(a) and Fig. 4.5(c), over the gradual increase of excitation frequencies from 10Hz to 100Hz, and with structure lengths set at N " 6 and N " 16 respectively. All the monochromatic excitations are set at a magnitude corresponding to a dimensionless displacement of U inc " 1 ¨10 ´3 (with respect to the distance between two units) and with a signal length of 40 periods.

It is shown that the horizontal displacement of the meta-structure goes to negative values for some specific frequencies, which corresponds to structure shrinking. In order to characterize the frequencies at which this effect occurs, we illustrate in Fig. 4.5(b) and Fig. 4.5(d) the Fourier transform of the horizontal displacement U and of the rotation ✓,o ft h es a m eu n i t sp r e s e n t e di nF i g .4 . 5 ( a )a n dF i g .4 . 5 ( c ) ,r espectively. Sweep frequency excitation varied from 1Hz to 1300 Hz and with a weak amplitude of 1 ¨10 ´5 is applied, enabling to determine the resonance frequencies of the structure detected at these end units, in a quasi-linear configuration.

When the structure is short, N " 6 for example, the shrinking phenomenon is observed at the frequency f " 53Hz, which corresponds to the first resonance frequency detected at the considered units. When the structure length is N " 16,t h e shrinking occurs at two specific frequencies (33Hz and 50Hz) and also at a wider frequency range between 65Hz and 75Hz. By comparing Fig. 4.5(c) and Fig. 4.5(d), the obtained first two frequencies relate to the first two resonance frequencies, whereas the following frequency range that allows for shrinking, covers exactly the third and the forth resonance frequencies which are found relatively close to each other. For the above mentioned resonance frequencies, the associated resonances are all rotation dominated favoring geometrical nonlinearity activation. Also, if we look at the static case, when a compressive force is applied on the structure, noticeable shrinking occurs when rotation (or equivalently, buckling of the plastic shims connecting the LEGO crosses) is activated.

Furthermore, we observe that the shrinking effect is accumulated along the structure. Under the applied excitation level U inc " 1 ¨10 ´3,f o rt h em e t a -s t r u c t u r eo f N " 6 columns, the average displacement of last units is in the vicinity of ´0.015mm at frequency f " 53Hz, while for N " 16,t h ea v e r a g ed i s p l a c e m e n ti si nt h er a n g e of p´0.5, ´0.2qmm. A difference is found when comparing the obtained numerical results with experiments in which an average shrinking displacement of ´4mm is achieved at a frequency around 40 Hz and under excitation level of 5V .I n d e e d ,i n the theoretical study, the initial angles of all the units are set at absolute value of 1 o , while in experiments, it is difficult to control accurately the initial angle of each Appendix A Solving nonlinear equation system by harmonic balance method (HBM)

A.1 HBM to solve the equations in lumped element modeling

According to the harmonic balance method (HBM), the solution U i pi " 1, 2q of the reflection problem by the metasurface presented in the first chapter Eq. (1.4) can be written in the form of a vector product U i p⌧ q"tCu T tqu i with tCu T the transpose of column vector tCu containing all the sinusoidal terms:

tCu" $ ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' % 1 cosp⌧ q sinp⌧ q cosp2⌧ q sinp2⌧ q . . . and tq i up i " 1, 2q another vector containing all the magnitude terms,

. The derivatives of displacement U i p⌧ q with respect to ⌧ are BU i Bτ "t BC Bτ u T tqu i and

We define the sinusoidal matrix as rSs"

and the magnitude vector as tqu"

, .

- The Jacobian matrix takes the form,

Notice that all the linear terms can be directly determined, only the nonlinear part

should be numerically calculated. By using the Einstein notation, we find finally that,

with

and

where ∆q T m " q T 1m ´qT 2m pm " 1, 2,...,2N `1q.

3,F o re a c hi t e r a t i o nn `1 °1,w ed e fi n et h er e l a t i v ee r r o ra s

An acceptable value of this relative error is put at ✏ c " 10 ´6 in the present study.

The loop goes on until the relative error is smaller than this given value.

4, when ✏ n § ✏ c ,t h el o o pi ss t o p p e da n dt h es o l u t i o nU i of the considered problem is obtained. Therefore the reflection coefficients r R n (1 § n § N )f o ra l lt h es i g n a l harmonics are determined : r R n " δ 1 n `inγΩpC in ´iS in q.

hence $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

Bτ q✓ 4 ´5 36 p3 ´Ks q sin ✓ 0 cos ✓ 0 ✓ 6 ´1 72 p3 ´Ks q cos 2✓ 0 ✓ 7 `1 288 p3 ´Ks q sin ✓ 0 cos ✓ 0 ✓ 8 (A.25) The obtained approximative equations Eq. (A.25) are in polynomial form, thus possible now to be solved with HBM. The calculations are similar to those for the lumped-element metasurface modeling (see A.1), except that the vector of nonlinear forces tF NL u become more complex:

The associated parts of nonlinear forces tF NL u presented in Jacobian matrix

BxrSs T F pnq NL y i Bpq n q j (A.27) should be identified term by term to enable the numerical resolution of considered problem via the Newton-Raphson.method.

For the nonlinear meta-interface design presented in Chapter 3, the HBM is performed in the same way as for the metasurface. Only the motion equation system is changed from a system of three equations to four equations, the solving process will not be repeated here.

Appendix B

Supplementary equations for structure of LEGOs

When the horizontal movement, the vertical movement and the rotation of metastructure (see Fig. 4.2) are all taken into account, the equations for the elements at the two ends (j " 1 and j " N ) are similar to those for the elements with 1 † j † N given by Eq. (4.1) in Chapter 4.

For the LEGOs bricks at the first column pp, 1q (p " 1, 2), the motion equations are written as: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

Bτ 2 " U p,2 ´2U p,1 `Up,0 `1 2 rcos ' p,2 ´cosp' p,2 `✓p,2 qs `1 2 p´1q p K s rsinp' p,1 `✓p,1 q´sin ' p,1 `sinp' p`1,1 `✓p`1,1 q´sin ' p`1,1 s ´Ks pU p,1 ´Up`1,1 q´1 2 K θ ✓ p,2 sinp' p,1 `✓p,1 q `1 2 p´1q p K θ p✓ p,1 ´✓p`1,1 q cosp' p,1 `✓p,1 q

"śinp' p,1 `✓p,1 qr3cos' p,1 `cos ' p`1,1 `cos ' p,2 `p´1q p 2pV p`1,1 ´Vp,1 q ´2U p,0 `2U p,2 ´3cosp' p,1 `✓p,1 q´cosp' p`1,1 `✓p`1,1 q´cosp' p,2 `✓p,2 qs ´Ks cosp' p,1 `✓p,1 qr3sinp' p,1 `✓p,1 q`sinp' p`1,1 `✓p`1,1 q`sinp' p,2 `✓p,2 q ´3sin' p,1 ´sin ' p`1,1 ´sin ' p,2 `p´1q p 2pU p`1,1 ´Up,1 q´2V p,2 s ´Kθ p3✓ p,1 ´✓p,2 ´✓p`1,1 q. (B.1) with the index p `1 needs to be changed to p ´1 for the second row p " 2.A n dU p,0 (p " 1, 2)denotethehorizon taldisplacemen tsthatapplytothet w oelemen tsofthe first column, respectively. In the particular case presented in Chapter 4 that only the horizontal movement is provided for the first column of symmetrical arranged meta-structure, we have

Whereas for the LEGOs of the last column pp, N q with p " 1, 2,t h em o t i o n equations are: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

`sinp' p,N ´1 `✓p,N´1 q´sin ' p,N ´1 `2V p,N ´1s `1 2 K θ p✓ p,N ´✓p,N´1 q cosp' p,1 `✓p,1 q`1 2 p´1q p K θ p✓ p,N ´✓p`1,N q sinp' p,N `✓p,N q

`p´1q p 2pU p`1,N ´Up,N q´2V p,N `2V p,N ´1s´K θ p2✓ p,N ´✓p,N´1 ´✓p`1,N q. (B.2) with the index p `1 needs to be changed to p ´1 for the second row p " 2 of meta-structure.

When only the horizontal movement and the rotation of elements are considered, the above equation systems containing three degrees of freedom can be simplified to the following ones, for the elements at the first column pp, 1q (p " 1, 2), we have $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

Bτ 2 " U p,2 ´2U p,1 `Up,0 `1 2 rcos ' p,2 ´cosp' p,2 `✓p,2 qs `1 2 p´1q p K s rsinp' p,1 `✓p,1 q´sin ' p,1 `sinp' p`1,1 `✓p`1,1 q´sin ' p`1,1 s ´Ks pU p,1 ´Up`1,1 q´1 2 K θ ✓ p,2 sinp' p,1 `✓p,1 q `1 2 p´1q p K θ p✓ p,1 ´✓p`1,1 q cosp' p,1 `✓p,1 q

Bτ 2 "śinp' p,1 `✓p,1 qr3cos' p,1 `cos ' p`1,1 `cos ' p,2

´2U p,0 `2U p,2 ´3cosp' p,1 `✓p,1 q´cosp' p`1,1 `✓p`1,1 q´cosp' p,2 `✓p,2 qs ´Ks cosp' p,1 `✓p,1 qr3sinp' p,1 `✓p,1 q`sinp' p`1,1 `✓p`1,1 q`sinp' p,2 `✓p,2 q ´3sin' p,1 ´sin ' p`1,1 ´sin ' p,2 `p´1q p 2pU p`1,1 ´Up,1 qs ´Kθ p3✓ p,1 ´✓p,2 ´✓p`1,1 q. (B.3) with the index p `1 changing to p ´1 in the case of p " 2.

For the elements at the last column pp, N q,w eh a v e $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' %

" U p,N ´1 ´Up,N ´1 2 rcos ' p,N ´1 `cos ' p,N ´cosp' p,N ´1 `✓p,N´1 q ´cosp' p,N `✓p,N qs `1 2 p´1q p K s rsinp' p,N `✓p,N q´sin ' p,N `sinp' p`1,N `✓p`1,N q´sin ' p`1,N s´K s pU p,N ´Up`1,N q ´1 2 K θ p✓ p,N ´✓p,N´1 q sinp' p,N `✓p,N q `1 2 p´1q p K θ p✓ p,N ´✓p`1,N q cosp' p,N `✓p,N q ↵ B 2 θ p,N Bτ 2 "śinp' p,N `✓p,N qr2cos' p,N `cos ' p`1,N `cos ' p,N

´1

´2U p,N ´1 `2U p,N ´2cosp' p,N `✓p,N q ´cosp' p`1,N `✓p`1,N q´cosp' p,N ´1 `✓p,N´1 qs ´Ks cosp' p,N `✓p,N qr2sinp' p,N `✓p,N q`sinp' p`1,N `✓p`1,N q `sinp' p,N ´1 `✓p,N´1 q´2sin' p,N ´sin ' p`1,N ´sin ' p,N ´1 `p´1q p 2pU p`1,N ´Up,N qs ´Kθ p2✓ p,N ´✓p,N´1 ´✓p`1,N q. (B.4) with the index p `1 changing to p ´1 in the case of p " 2.