R P Singh

C Droz

S De Rosa

F Franco

O Bareille

M N Ichchou

A study of structured uncertainties in wave characteristic of one-dimensional periodic structures

Periodic structures have properties of controlling mechanical wave. These are used in aircraft, trains, submarines, space structures and demand high level of robustness, which can be ensured with consideration of the presence of uncertainty in the numerical models. The uncertainties, in terms of material properties and geometrical parameters, are mostly introduced in both the manufacturing and assembly process. In order to predict the wave characteristics of the periodic structures under the uncertainty, the stochastic wave finite element method has been extended to periodic structures; wave finite element method (WFEM) added with an additional dimension to model the uncertain parameter using the perturbation approach. Numerical experiments are performed to test the method with one-dimensional periodic rod and periodic beam with parametric uncertainty. The performance of the developed method is compared in terms of computational cost which offers computational advantages over the Monte Carlo Simulation.

Introduction

The vibroacoustic performance and dynamics of the structure are important subjects in aeronautics, transport, energy, and space where the periodic media can be used for vibration reduction, acoustic blocking, acoustic channeling and acoustic cloaking. Application for these unusual phenomena is mainly correlated to the inherent band gaps or frequency regions in which the propagation of acoustic/elastic waves in the periodic media can be manipulated. Inherent material inhomogeneity, manufacturing errors (ex. variation of geometry) and environmental influences are unavoidable.

To address this unavoidable actuality, the effects of uncertainties need to be considered when analyzing band structures (pass and stop band) and dynamics of the periodic structures. Also, to meet the regulatory compliance and user requirement, the designer accounts for variation in the input parameter at the design phase. For example, in the space industry designers consider the uncertainty in the system parameter to ensure that the during launch and orbital operation the vibration level are in a range that is acceptable. Three different frequency ranges of analysis can be defined for the typical dynamic response of weakly dissipative structures. At the low-frequency range, FE-like approaches are applicable because of the low order modes and low sensitivity to variability, contrary the high frequency with many high order modes involved, energy-based methods (e.g. Statistical Energy Analysis (SEA)) are well suited. The mid-frequency range affected by variability of the model parameter and the wave-based approaches have been developed by increasing computational efficiency. However, most of them assume that waveguide properties are homogeneous in the direction of the travelling wave, which is the limitation of such approaches for variety of application where uncertainty can be considered. Also the analytical solutions are available but limited to the particular cases (e.g. acoustic horns, ducts, rods and beams) [START_REF] Arenas | A note on a WKB application to a duct of varying cross-section[END_REF]. Manohar et al. [START_REF] Manohar | Axial vibrataion of a stochastic rod[END_REF] considered the randomness in the wave propagation in waveguides using spectral element analysis. Ichchou et al. [START_REF] Ichchou | Stochastic wave finite element for random periodic media through first-order perturbation[END_REF] proposed a numerical approach using the Wave Finite Element Method (WFEM) [START_REF] Mead | A general theory of harmonic wave propagation in linear periodic systems with multiple coupling[END_REF] considering spatially homogeneous variability in waveguides using a first order perturbation and extended with second order perturbation is proposed [START_REF] Bouchoucha | Stochastic wave finite element method in uncertain elastic media through the second order perturbation[END_REF].

When dealing with layered structures the material or geometrical uncertainty often exhibits spatial correlation and random fields theory can be used to model spatially distributed variability with probability measure [START_REF] Fabro | Wave propagation and response statistics of random waveguides with spatially correlated properties[END_REF]. In addition, a Karhunen-Loeve (KL) expansion provides characterization of the random field in terms of deterministic eigenfunctions weighted by uncorrelated random variables. Also the (after Wentzel, Kramers and Brillouin) WKB approximation are employed in the modelling the random media [START_REF] Steele | Application of the WKB method in solid mechanics[END_REF]. The limitation of the method is that it does not consider the internal reflections which occur due to any local changes in the material or geometrical properties. This assumption can be mitigated if one considers a waveguide with piecewise constant material variability, where internal reflections due to local changes in the impedance can be taken into account at the junctions of the sections and the piecewise constant separated into a finite number of discrete sections [START_REF] Langley | Wave Transmission Through One-Dimensional Near Periodic Structures: Optimum and To Random Disorder[END_REF]. Also, this would be an approximate representation for the spatially varying system [START_REF] Fabro | Uncertainty analysis of band gaps for beams with periodically distributed resonators produced by additive manufacturing[END_REF].

The literature reveals that the effects of uncertainty in the material properties, geometry, loading condition and model play significant role in altering wave states. The work reported so far in the literature with the best of authors knowledge, is limited to wave propagation (with structured and unstructured uncertainty) in elastic media and random media only. In the present work, a Stochastic Wave Finite Element Method (SWFEM) [START_REF] Ben Souf | Dynamics of random coupled structures through the wave finite element method[END_REF], [START_REF] Ben Souf | On the Variability of the Sound Transmission Loss of Composite Panels Through a Parametric Probabilistic Approach[END_REF] is employed to analyze the complex band gap in 1D periodic structures. The results of the stochastic formulation are compared those obtained with analytical and Monte Carlo Simulation (MCS) for the bar and beam.

Formulation of SWFEM

One dimensional periodic structures are obtained by formulating the unit cell and then repeating in the propagation direction. Then, the study of this structure can be converted into study of unit cell based on the Floquet-Bloch theorem [START_REF] Ichchou | Stochastic wave finite element for random periodic media through first-order perturbation[END_REF]. The schematic representation is shown in Fig. 1.

To consider the effect of the uncertainty on the response of the structure, the SWFEM is formulated which is extension of the classical WFEM to accommodate the uncertainty. The workflow of the SWFEM is depicted in Fig. 2.

A B A B Cell k Cell k+1 𝑙 1 𝑙 2

Figure 1: Schematic representation of periodic structure

To make the reader easily understand, the step in the formulation of SWFEM [START_REF] Ichchou | Stochastic wave finite element for random periodic media through first-order perturbation[END_REF] is explained here.

The WFEM is used to study the behavior of periodic structures. This is extended to apply the WFEM for the uncertainty study. This starts with consideration of the random field as a supplementary dimension through the spatial discretization using the finite element steps. Which start from discretization of one subelement of length (d). This discretization leads to stochastic dynamic equilibrium of any substructure in the following manner.

(𝐷 ̃) ( 𝑞 ̃𝐿 𝑘 𝑞 ̃𝑅 𝑘 ) = ( 𝐹 ̃𝐿𝑘 𝐹 ̃𝑅 𝑘 ) (1)
Here (𝐷 ̃) is the stochastic complex dynamic stiffness matrix of the substructure, condensed on left and right boundaries degree of freedom at the pulsation 𝜔:

(𝐷 ̃) = -𝜔 2 𝑀 ̃+ 𝐾 ̃(1 + 𝑖𝜂) (2)
Here 𝑀 ̃ 𝑎𝑛𝑑 𝐾 ̃ are stochastic mass and stiffness matrix respectively. In the probabilistic tools, the parametric approach allows to consider the uncertain parameters (material, geometrical properties etc.) as random quantities. This is modeled using the stochastic finite element discretization approaches as expressed in [START_REF] Ghanem | Numerical solution of spectral stochastic finite element systems[END_REF].

The random variables are modeled using the first order perturbation, as Gaussian variables, such that the dynamical equilibrium is expressed as:

[-𝜔 2 (𝑀 ̅ + 𝜎 𝑀 𝜀) + (𝐾 ̅ + 𝜎 𝐾 𝜀)(1 + 𝑖𝜂)] ( 𝑞 ̅ 𝐿 (𝑘) + 𝜎 𝑞 𝐿 (𝑘) 𝜀 𝑞 ̅ 𝑅 (𝑘) + 𝜎 𝑞 𝑅 (𝑘) 𝜀 ) = ( 𝐹 ̅ 𝐿 (𝑘) + 𝜎 𝐹 𝐿 (𝑘) 𝜀 𝐹 ̅ 𝑅 (𝑘) + 𝜎 𝐹 𝑅 (𝑘) 𝜀 ) (3)
The overbar . ̅̅̅ symbol denotes the mean of the random variable, σ; is standard deviation and 𝜀 is a Gaussian centered variable [START_REF] Sarkar | Mid-frequency structural dynamics with parameter uncertainty[END_REF]. In the expression 𝑀 ̅ , 𝐾 ̅ , 𝑞 ̅, 𝐹 ̅ are the mean quantities of mass matrix, stiffness matrix, displacement vector and load. 𝜎 𝑀 , 𝜎 𝐾 , 𝜎 𝑞 , 𝜎 𝐹 are their standard deviation.

As in the classical case, the stochastic problem in equation ( 1) can be partitioned in the following way:

( 𝐷 ̃𝐿𝐿 𝐷 ̃𝐿𝑅 𝐷 ̃𝑅𝐿 𝐷 ̃𝑅𝑅 ) ( 𝑞 ̃𝐿 𝑘 𝑞 ̃𝑅 𝑘 ) = ( 𝐹 ̃𝐿𝑘 𝐹 ̃𝑅 𝑘 ) (4)
It is to be noted that equation ( 4) is valid and can accommodate the stochastic behavior of the stiffness and mass matrix.

The stochastic kinematic variable, 𝑞 ̃ and 𝐹 ̃ are represented through stochastic state vector and related by the stochastic transfer matrix 𝑆 ̃.

𝑢 ̃𝑅 𝑘 = 𝑆 ̃. 𝑢 ̃𝐿 𝑘

Where

𝑆 ̃= 𝑆 ̅ + 𝜎 𝑠 𝜀 𝑆 ̅ = ( -𝐷 ̅ 𝐿𝑅 -1 𝐷 ̅ 𝐿𝐿 𝐷 ̅ 𝐿𝑅 -1 𝐷 ̅ 𝑅𝐿 -𝐷 ̅ 𝑅𝐿 𝐷 ̅ 𝐿𝑅 -1 𝐷 ̅ 𝐿𝐿 -𝐷 ̅ 𝑅𝐿 𝐷 ̅ 𝐿𝑅 -1 ) 𝜎 𝑠 = ( -𝐷 ̅ 𝐿𝑅 0 -𝐷 ̅ 𝑅𝑅 1 ) -1 ( 𝜎 𝐷 𝐿𝐿 𝜎 𝐷 𝐿𝑅 𝜎 𝐷 𝑅𝐿 𝜎 𝐷 𝑅𝑅 ) ( 1 0 -𝐷 ̅ 𝐿𝑅 -1 𝐷 ̅ 𝐿𝐿 𝐷 ̅ 𝐿𝑅 -1 )
Following the steps of the deterministic development, a stochastic eigenvalue problem formulated as follows:

𝑆 ̃𝜙 ̃𝑖 = 𝜇̅ 𝑖 . 𝜙 ̃𝑖 |𝑆 ̃-𝜇 ̃𝑖𝐼 2𝑛 | = 0 (6)
The polynomial chaos projection of the eigenvalue problem leads to find their mean and standard deviation.

The standard deviation of the propagation constant 𝜇 𝑖 (𝜎 𝜇 𝑖 )

𝜎 𝜇 𝑖 = [(𝜙 ̅ 𝑖 ) 𝑇 𝜎 𝑠 𝑇 (𝑆 ̅ 𝑇 -𝜇̅ 𝑖 𝐼 2𝑛 ) -1 𝐽 𝑛 (𝑆 ̅ -(𝜇̅ 𝑖 ) -1 𝐼 2𝑛 ) -(𝜙 ̅ 𝑖 ) 𝑇 𝐽 𝑛 𝜎 𝑠 )] x [(𝜙 ̅ 𝑖 ) 𝑇 (𝑆 ̅ 𝑇 -𝜇̅ 𝑖 𝐼 2𝑛 ) -1 𝐽 𝑛 (𝑆 ̅ -(𝜇̅ 𝑖 ) -1 𝐼 2𝑛 ) -(𝜇̅ 𝑖 ) -2 (𝜙 ̅ 𝑖 ) 𝑇 𝐽 𝑛 )] -1 (7)
similarly, the standard deviation of the eigenvectors (𝜎 𝜙 𝑖 ) can be written as

𝜎 𝜙 𝑖 = -[𝑆 ̅ -𝜇̅ 𝑖 𝐼 2𝑛 ] + [𝜎 𝑠 -𝜎 𝜇 𝑖 𝐼 2𝑛 ]𝜙 ̅ 𝑖 ( 8
)
where + is pseudo inverse.

Using equation ( 7) and ( 8), the statistics of the wave characteristics can be expressed in the standard deviation of the propagation constant.

Results

This section shows the validation of SWFEM formulation considering binary periodic rod and binary periodic beam respectively. Binary rod and beam consist of section A of length 𝑙 1 and section B of length 𝑙 2 as depicted in the Fig. 3. Here cells A and B are made of different materials. In the first case, the longitudinal waves are studies while in the second case the flexural waves in beam of circular section are studied.

Longitudinal waves in binary periodic rod with uncertainty

In this subsection, the SWFEM is used to study the effect of parametric uncertainty on the dispersion relation of the longitudinal wave in binary periodic rod with sections A and B made of epoxy and aluminum respectively. The length of 𝑙 1 and 𝑙 2 are 1m each with circular cross section of radius of 0.0644m.

For the validation purpose SWFEM results is compared with the analytical solution of the wave number K expressed [START_REF] Sun | Elastic wave propagation in periodic structures through numerical and analytical homogenization techniques[END_REF]:

cos(𝐾𝑙) =
Where 𝑐 𝑎 , 𝑐 𝑏 is the wave velocity in the section A and B respectively and expressed as 𝑐 𝑎 = √ 𝐸 𝑎 𝜌 𝑎 and

𝑐 𝑏 = √ 𝐸 𝑏 𝜌 𝑏
The uncertainty effect is studied considering variation of the Young's modulus with four percentage. The study frequency range are up to 1800 Hz. The analytical results are treated as reference result for validation purpose. The sampling method with 10000 samples are used to get the wave characteristics of the analytical wave number obtained using equation [START_REF] Langley | Wave Transmission Through One-Dimensional Near Periodic Structures: Optimum and To Random Disorder[END_REF].

In SWFEM formulation two-node rod elements are considered with the stiffness and mass matrices as following:

𝐾 𝑟𝑜𝑑 = 𝐸𝑆 𝑑 ( 1 -1 - 1 1
)

𝑀 𝑟𝑜𝑑 = 𝜌𝑆𝑑 6 ( 2 1 1 2 ) A B Unit cell 𝑙 1 𝑙 2 Figure 3: Symmetric unit cell
Where E is the Young's modulus, S is cross sectional area and d is element length.

The global stiffness and mass matrices are assembled in the MATLAB environment with 100 elements in the unit cell of binary periodic rod. The material and geometric properties are shown in Table 1.

Table 1: Geometric parameters and material properties of binary periodic rod Validation of SWFEM result in terms of dispersion equation is also computed using the MCS method with 10000 samples. As shown in Fig. 4, there exist one stop band in the considered frequency range according to the results obtained with SWFEM, analytical sampling and MCS in the considered frequency range. Also, the non-dispersive nature of propagation can be observed for the standard deviation. The comparison of the standard deviation is presented in the Fig. 5, obtained from analytical sampling, SWFEM and WFEM MCS.

The results are in good agreement. The SWFEM standard deviation is computed considering loss factor on the contrary the analytical sampling is based on without damping.

Flexural waves in binary periodic beam with uncertainty

In this subsection, the study of the uncertainty effect on the bending wave in binary periodic beam is performed. The Euler-Bernoulli beam theory is considered with material and geometrical properties listed in Table 2.

Table 2: Geometric parameters and material properties of binary periodic beam For the numerical formulation the two-node beam element with two degrees of freedom per node is considered. The mass and stiffness matrices of the beam are as follows:

𝐾 𝑏𝑒𝑎𝑚 = 𝐸𝐼 𝑑 3 [ 16 6𝑑 -12 6𝑑 4𝑑 2 -6𝑑 -12 -6𝑑 12 6𝑑 2𝑑 2 -6𝑑 6𝑑 2𝑑 2 -6𝑑 4𝑑 2 ] 𝑀 𝑏𝑒𝑎𝑚 = 𝜌𝑆𝑑 420 [ 156 22𝑑 54 22𝑑 4𝑑 2 13𝑑 54 -6𝑑 156 -13𝑑 -3𝑑 2 -22𝑑 -13𝑑 -3𝑑 2 -22𝑑 4𝑑 2 ]
The frequency range of computation is 1-1500 Hz. Considering the uncertainty in the Young's modulus of the binary periodic beam with four percent about the nominal value, the numerical experiment is performed with SWFEM formulation. To validate the obtained result, MCS study is carried out with 10000 samples. For the elapsed time computation, the MCS result obtained using homemade WFEM code which exploits the resources of the workstation with following characteristics, Intel® core™i7 7820 HQ CPU@2.90GHz with 32 GB RAM and data storage using solid state drive. The comparison is presented in the

Geometry/Property Value

Conclusion

This paper provides a computationally inexpensive stochastic numerical approach to study the uncertainty effect on free vibration of the 1D periodic structures. This formulation allows wave characteristic to be defined by stochastic finite element model using parametric probabilistic approach and which is easy to implement as it considers finite element model of the sample structural element. The SWFEM shows that the wave characteristic of the longitudinal and flexural wave can be obtained precisely. Main conclusion can be derived as follows:

(1) The formulation of the SWFEM is applied for the binary periodic rod and beam.

(2) Analytical sampling and numerical result showed the effectiveness of the formulation to predict the mean and standard deviation of the wave propagation. (3) The SWFEM has computational advantage over MCS results.

The saving the computational cost can be a good point for the optimization and reliability study under uncertainties of complex waveguides. The statistics of the response can also be useful for the damage detection and sensitivity analysis. Future investigation is under progress for the forced response computation, using the present formulation.

DynamicFigure 2 :

 2 Figure 2: Flow diagram for SWFEM

Figure 4 :Figure 5 :

 45 Figure 4: Comparison of mean of wave number

Fig. 6

 6 Fig.6shows the comparison of the mean value of the wave number obtained from SWFEM and MCS on the WFEM simulation. In Fig.7, the standard deviation comparison is shown from SWFEM and MCS results. The results are in good agreement which confirmed the validity of the SWFEM formulation applied to bending wave analysis.

Figure 6 :Figure 7 :

 67 Figure 6: Comparison of mean of wave number

Table 3 .

 3

		WFEM MCS	SWFEM
		(10000 Sample)	(single run)
	Elapse time (Seconds)	3840	9.56

Table 3 :

 3 Elapsed time comparison

Acknowledgements

This paper containing the work carried out in the framework of the VIPER project (VIbroacoustic of PERiodic media). This project has received funding from the European Union's Horizon 2020 research and innovation program under Marie Curie grant agreement No 675441.


R P Singh

S De Rosa

F Franco

M Ichchou

O Bareille
email: olivier.bareille@ec-lyon.fr

and Vibroacoustics Sevilla (Spain) A LITERATURE REVIEW FOR THE ANALYSIS OF STRUCTURED AND UNSTRUCTURED UNCERTAINTY EFFECTS ON VIBROACOUSTIC

A better knowledge of the structural response of periodic structures can be achieved through the classification of possible uncertainty, to be included in the predictive models. Nowadays, the wave finite element approach is used for the simulation of periodic structures to reduce the computational cost. On the contrary, it introduces ahigher complexity in the model formulation. Recently, the spectral approach has shown its relevance in the uncertainty analysis of structures. This paper presents the literature on uncertainty effect on vibroacoustic of periodic media. The present work is an initial literature study on available analytical and numerical approaches for capturing the parametric and nonparametric uncertainties.

.

INTRODUCTION

The research community has been developing mathematical representation in the form of a mathematical model for a wide class of physical system to understand the process. Moreover, these models are being used to make the decision for design and manufacturing. It true that no mathematical model can be an ideal representation of physical system it is intended to capture because of the modeling assumption and limits of digital computing machines where numerical simulation performed. Although the digital computing is extending its wing in last decade, there are constraints of time and cost of computation. To the best of authors understanding, all derived model involve uncertainty and which are bound to come. The reason is how to model is represented, and how the physical system behaves in reality, there is always the issue of the goodness of fit.

The vibroacoustic performance and dynamics of the structure areimportant subjects in the area of aeronautic, transport, energy and space. To meet the regulatory compliance and user requirement, the designer accounts for variation in the input parameter at the design phase. For example, in the space industry designers consider the uncertainty in the system parameter to ensure that the during launch and orbital operation the vibration level are in a range that is acceptable. Which open plethora of opportunity for considering the effect of uncertainty with theaim to improve and develop the model for reliable and safe design. Keeping the view this paper summaries uncertainty quantification, uncertainty classification, modeling of uncertainty system and the most commonly used technique to analyses uncertainty in the low to high-frequency model prediction.

UNCERTAINTY QUANTIFICATION:

In general term the uncertainty quantification involves the five steps:

MODELING OF UNCERTAINTY SYSTEM:

The uncertainty in structural response such as frequency response function, natural frequency, and mode shape are the result of propagation of uncertainty (may be parametrical or non-parametric). To capture the response with uncertainty in models and parameters, various

Identification

Characterization Propagation Analysis and reduction research group spread across the world using various approach to modeling the uncertainty. The basic uncertainty block of technique/tool available for uncertainty modeling in the structural dynamics [START_REF] Zhang | Interval Monte Carlo methods for structural reliability[END_REF][START_REF] Majba | A Review of Uncertainty Quantification of Estimation of Frequency Response Functions[END_REF][START_REF] Stefanou | The stochastic finite element method: Past, present and future[END_REF][START_REF] Soize | Random matrix theory and non-parametric model of random uncertainties in vibration analysis[END_REF][START_REF] Chen | Fuzzy finite-element approach for the vibration analysis of imprecisely-defined systems[END_REF][START_REF] Nannapaneni | Reliability analysis under epistemic uncertainty[END_REF][START_REF] Soize | A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics[END_REF] can be drawn (Figure 1).

Figure 1. Uncertainty modeling approaches

VIBROACOUSTIC ANALYSIS WITH UNCERTAINTY

In thevibroacousticproblems, the interaction between a solid and fluid field in the form of vibration and sound happen respectively. The use of the numerical method for the numerical modeling and simulation varies, and itdepends on the frequency range of interest.

Figure 2.Vibroacoustic analysis methods without uncertainty

A graphical representation is drawn (Figure 2) of the technique available for vibroacousticsimulation [START_REF] Mace | Uncertainty in structural dynamics[END_REF][START_REF] Cicirello | Efficient parametric uncertainty analysis within the hybrid Finite Element/Statistical Energy Analysis method[END_REF][START_REF] Ichchou | Stochastic wave finite element for random periodic media through first-order perturbation[END_REF][START_REF] Xu | Uncertainty propagation in SEA for structural-acoustic coupled systems with non-deterministic parameters[END_REF]

  1. Identification: Finding the source and location of uncertainties in the system. In reality many sources of uncertainty such asuncertainties due to variabilities in the design parameter values, environmental condition, initial conditions, boundary conditions, imprecise and simplified physics, missing physics, model implementation, numerical errors and most importantly due to lack of unavailability of sufficient data 2. Characterization: Finding the form they are available. Mostly, the parametric uncertainty is characterized and defined in the form of probability distribution and intervals bound. Whereas nonparametric uncertainty so-called model uncertainties can have their form of uncertainties. 3. Propagation: Understanding how uncertainties are transmitting and spreading in the model and finding a relation between parameter uncertainties and response of the model. 4. Analysis and reduction: Establishing the relation between the uncertainty and its influence on the system response and reasoning of the same. Once it is done, what corrective measure can be taken to have the reliability of the original system.

  .When uncertainty introduced in the modeling and simulation of periodic media the low-frequency domain is unaffected (assumption), the highfrequency domain is modeled using statistical energy approach which can accommodate uncertainty. However mid frequency domain is in question. A tree is drawn (Figure3) as based on available literature.

				Uncertainty Modelling
	Virbroacoustic	Parametric approach Probabilistic Wave Finite Element (WFE)	Non probabilistic Stochastic WFE Spectral Stochastic WFE	Non parametric approach Interval Approach
	Monte Carlo Simul ation with uncertanity Responce Surface Method	Stochastic Finite Element Method Hybrid approach Reliability of system using FORM/	Fuzzy Finite Element Method FE/SEA WF/SEA	Interval Finite Element Method	Bayesian Theory	Random Matrix Theory Random Matrix Theory
				SORM
			Figure 3.Vibroacousticanalysis methods with uncertainty
						Finite Element Method
				Low Frequency
						Boundary Element Method
							Spectral Methods
	Vibroacoustic	Mid Frequency
	Analysis Methods			Hybrid Approaches
							Statistical Energy Analysis
				High Frequency	Gaussian Orthogonal Ensemble and SEA variance theory
						Dynamic Energy Analysis

vibroacoustic

ACKNOWLEDGEMENTS

This paper containing the preliminary work carried out in the framework of the VIPER project (VIbroacoustic of PERiodic media). This project has received funding from the European Union's Horizon 2020 research and innovation program under Marie Curie grant agreement No 675441 and it has financed the Ph.D.program of the 1st author.