Uncertainty analysis in vibroacoustic panels with band...

Uncertainty analysis in vibroacoustic panels with band gap

D. Beli1, A.T. Fabro2, M. Ruzzene3, J.R.F. Arruda1 1 University of Campinas, School of Mechanical Engineering, Department of Computational Mechanics,

Cidade Universitária Zeferino Vaz, Campinas-SP, 13083-080, Brazil

e-mail: [email protected] 2 University of Brasilia, Department of Mechanical Engineering,

Campus Universitário Darcy Ribeiro, Brasilia-DF, 70910-900, Brazil 3 Georgia Institute of Technology, School of Aerospace Engineering, School of Mechanical Engineering,

North Avenue NW, Atlanta, Georgia 30332, United States

Abstract Vibration and acoustic isolation by passive control methods have been extensively studied along the years.

In a recent past, phononic crystals and acoustic metamaterials have also been employed for this purpose.

These periodic structures present band gaps, which are used herein to reduce vibration and acoustic

radiation of panels. However, variability in the manufacturing process causes material and geometry

uncertainties that affect their band gap robustness and consequently their dynamic attenuation

performance. In this work, the robustness of the band gap phenomenon in vibroacoustic panels is

investigated by employing the WFE method together to WKB approximation and KL expansion. A two-

dimensional analysis where the panel is represented by either a phononic undulated beam or a

metamaterial straight beam with distributed resonators is performed. Spatial correlated variability of the

elastic modulus and resonator stiffness is considered. Band gap of both panels are shown to be robust with

respect to this variability. However, vibration and acoustic attenuation are reduced.

1 Introduction

Phononic crystals and acoustic metamaterials are periodic structures used to control and to manipulate

acoustic and elastic waves [1-3]. One of their properties is the occurrence of band gaps, which are the

range of frequencies where wave propagation is forbidden. Applications could involve vibration

attenuation [4, 5] and noise reduction [6, 7] as well as cloaking and tunneling of mechanical waves [8, 9].

While phononic crystals are produced by material or geometric periodic modulation, metamaterials have

inclusions that work like internal resonators [10]. Herein, these configurations are applied in vibroacoustic

panels to attenuate structural vibration and acoustic radiation by using undulated beams, which work as a

phononic crystal, and straight beams with attached resonators, which work as a metamaterial.

In undulated beams, the coupling between flexural and extensional waves opens the band gaps; moreover,

it was demonstrated that beam thickness and undulation are responsible, respectively, for the location and

width of the band gaps [11]. For a beam with attached resonators, the coupling between beam and

resonator produces a band gap in the tuned resonance [12]. This configuration leads an acoustic mode at

low-frequency, where beam and resonators move in phase, and an optical mode at higher frequencies,

where beam and resonators move out of phase [13]. Besides, a beam with internal resonance band gaps

presents a locally negative slope in the dispersion diagram that is related to vibration attenuation [14].

Herein, these structures interact with an external fluid at one side. This problem is solved using coupling

equations that provide better insight about the elastic and acoustic wave interaction. In addition, the

Sommerfeld radiation condition is satisfied. This general formulation can be used for light or heavy fluid

interaction and also for interior problems if the Sommerfeld radiation condition is not imposed.

1969

By using wave propagation methods, such as the Spectral Element (SE) method [15, 16] and the Wave

Finite Element (WFE) [17-21] method, dispersion diagrams and forced responses of periodic structures

can be obtained by modeling one unit cell. Dispersion diagrams are computed by considering free

traveling waves and by applying Block’s theorem in the homogeneous equations, while, for forced

response, the structure is finite with boundary conditions and applied external forces. SE and WFE save

computational time because they model a single unit cell of the whole structure, which leads to a suitable

framework for optimization and uncertainty analysis, where a large number of interactions is required. In

addition, by employing the Finite Element (FE) method to construct a unit cell [22], WFE provides

numerical solutions for problems where analytical solutions are not available [23].

Producing structures by 3D printing to control waves is one of the emerging topics in acoustic

metamaterials [24]. However, manufacturing processes produce material and geometrical uncertainties

[25] that modify the structural dynamic behavior affecting the stop band performance [26]. Herein,

uncertainty analyses are performed to predict the robustness of phononic and metamaterials panels to

variability by using the WKB method using the WFE approach [27].

The WKB approximation was proposed by Wentzel, Kramers, and Brillouin, who give their name to the

method, for solving the Schrodinger equation in quantum mechanics. It assumes slowly spatial variation of

the properties such that internal reflections due to local impedance changes are negligible [28]. Uniformly

valid solutions including the turning point can also be found (e.g. [29]).

In this paper a Gaussian spatial distribution for the elastic modulus is generated by the Karhunen-Loève

(KL) expansion [30]. Then, the forward and backward propagation matrices are computed via WKB

approximation, which allow obtaining the forced response with uncertainties via dynamic stiffness or

reflection matrix [31]. The KL expansion applied herein to decompose exponential decaying

autocorrelation functions provides analytical equations for the spectral uncertainty expansion. A numerical

approach is usually necessary [32].

Sections 2 and 3 present the theoretical background for the WFE and WKB approaches, respectively. In

the former, it is discussed how to obtain the wavenumbers and wave shapes by a generalized eigenvalue

problem, while in the latter the uncertainty expression obtained by the KL expansion is presented. A

spectral decomposition, as well as the dynamic stiffness matrix for a substructure with slowly spatial

variability are obtained. Then, in Section 4, results for homogeneous and non-homogenous cases are

discussed and a complete uncertainty analysis is performed. Finally, Section 5 draws some conclusion and

discusses possible next steps of this work.

2 Dynamic Behavior by the WFE Method

2.1 Unit Cell by the FE Method

In this work, a panel in contact with a fluid is represented, assuming in-plane behavior, by a beam with

two-dimensional fluid elements. The unit cell is composed by a beam element connected at one side to a

two-dimensional fluid domain (2D fluid elements). On the external fluid elements, at the top of the unit

cell, the Sommerfeld radiation condition is imposed through absorption elements, which truncate the

unbounded fluid domain, which necessary in exterior acoustic problems [33]. Infinite elements produce

the same effect [34]; however, cylindrical or spherical external geometry is needed, which makes their

application unfeasible in periodic structures in rectangular coordinates. The mass, damping and stiffness

finite element matrices for the kth unit cell, which are extracted from the FE software Ansys®, can be

written in symmetric form as [35]:

1970 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

0ff

uuu

K00

0K000K

uuu

C00

00C

0C0

uuu

M00

0M000M

f

s

a

f

s

af

ff

s

a

f

s

af

cf

cf

a

f

s

af

ff

S

, (1)

where the superscripts s, f and a correspond, respectively, to structure, fluid, and absorption degrees of

freedom. In the damping matrix are included the fluid-structure coupling matrices, cC , and the

Sommerfeld damping matrix, aC .

Thus, the dynamic stiffness matrix for one unit cell is kkkki KCMD 2*

, with ω the angular

frequency and )1( iss KK , where η is the structural damping. *

kD is reorganized in submatrices that

are associated to left (L) and right (R) cross-section or interior (I) degrees of freedom (DOF). Considering

that no external forces are applied to the interior nodes, these can be condensed [17]

****

IBIIBIBBBB DDDDD-1

with the B subscript being to L or R. This operation reduces the dimension of

the matrix used to compute the eigenvalue problem and gives the dynamic reduced matrix as:

k

kR

L

kR

L

kRRRL

LRLLkk

FF

F

U

UDDDD

UD ˆˆ

ˆ

ˆ

ˆˆ

. (2)

2.2 Dispersion Relation

Assuming the fluid-structure problem free of external forces, the compatibility of internal displacements

as well as internal forces at the interface of two consecutive unit cells is given by )1()(

ˆˆ

kLkR

qq and

)1()(

ˆˆ

kLkR

ff , respectively. In addition, considering the relations between internal and external

displacements and forces, kRLkRL ]ˆˆ[]ˆˆ[ UUqq and

kRLkRL ]ˆˆ[]ˆˆ[ FFff - , the state vector for two

consecutive cross-sections are related by a transfer matrix which is written in terms of the condensed

dynamic stiffness matrix as:

)(11

11

)1(

)1( ˆ

ˆ

ˆ

ˆkLk

kL

L

kLRRRLLLRRRRL

LRLLLR

kL

L

kLψS

f

q

DDDDDD

DDD

f

qψ

. (3)

If the dynamic stiffness matrix is symmetric, it can be proved the S is symplectic; moreover, S is Δ-

periodic (Δ is the length of unit cell), which makes this matrix invariant under a translation of length Δ

[19]. For a periodic structure, Bloch’s theorem relates the state vectors of two consecutive cross-sections

by the wavenumbers Λ, T

k

T

L

T

L

iT

k

T

L

T

Lke ]ˆˆ[]ˆˆ[

)1(fqfq

λ. Replacing this equation in Eq. (3) yields the

eigenvalue problem, kkkk

ψμψS , where T

k

T

L

T

Lk ]ˆˆ[ fqψ and

ki

ke

λμ . This eigenproblem provides

2n eigenvalues and respective eigenvectors, where n corresponds to the number of DOF associated with

each cross section. While the eigenvalues (µ) are associated to phase change or attenuation along the beam

length, the eigenvectors (Φ), or wave mode shapes, indicate the spatial distribution of the displacements

and forces on the cross section [17-18].

Because of the symplectic nature of matrix S, wave modes appear in pairs

kkΦμ , and

kkΦμ , that

correspond to n waves that propagate to the right and left, respectively. Also, the eigenvalues related to

forward and backward going waves are linked by kkμμ 1 , with 1

kμ and 1

kμ . Moreover, the

eigenvectors can be partitioned in displacement and force components as T

fq][ ΦΦΦ .

METAMATERIALS 1971

A modified eigenvalue problem can be written as proposed by Zhong and Williams [36] to avoid ill-

conditioning on the inversion of matrix DLR. The eigenproblem in Eq. (4) is well conditioned because the

eigenvectors are described just by the displacement components, which avoid the large discrepancy that

occur when they are considered together with the force components [20, 21].

μLwNw , where

LRLL

n

DD

0IL ,

RRRL

n

DD

I0N and

q

q

μΦ

Φw (4)

The relation between eigenvectors and wave mode shapes is given by LwΦ . The modified eigenvalue

problem of Eq. (4) can yield wave modes that do not satisfy exactly the relation kkμμ 1 , which results

in an ill-conditioned matrix system to compute the forced response. A regularization procedure was

proposed to solve this issue for symmetric unit cells with respect to their mid-plane [20], μμ 1 ,

qq

RΦΦ and ff

RΦΦ , where R is a symmetry transformation matrix.

3 Stochastic Behavior by WKB Approximation

The WKB approximation is generally used to solve differential equations with variable coefficients

[37,38]. Herein, this procedure is applied to calculate the forced response of a periodic structure with slow

material property variation, which leads to wavenumbers dependent of space, λ(x). The WKB

approximation provides two wave properties, which capture the spatial fluctuation - the phase change and

the amplitude change, used as inputs for wave propagation methods, such as the WFE. However, this

approach fails when the properties have abrupt changes or when the oscillating waves reach a cut-off

section, breaking the propagation [31]. In the next subsection, we introduce how to represent a random

field with correlated spatial properties by the KL Expansion. Then, the propagation matrix composed by

the phase change and amplitude change are obtained, and finally it is used to compute the forced response

in the presence of uncertainties.

3.1 KL Expansion

A spatial random distribution can be represented by an expansion composed by a complete set of

deterministic functions with corresponding random coefficients [32]. By using the KL expansion the total

mean square error is minimized. Thus, it is an optimal procedure to express the information in a random

field by a series. Otherwise, representing a random process by point discretization methods would need a

large number of samples for a good approximation. In the KL expansion, the eigenvalues and eigenvectors

of the covariance function provide the magnitudes and functions of the orthogonal deterministic basis,

respectively.

A random field, H(x,τ), defined in a finite domain D (x ϵ D) and in a probability space Ω (τ ϵ Ω) with mean

Ho(x) and finite variance E[H(x,τ)-Ho(x)]2, can be represented by a spectral decomposition (like a

generalized Fourier series) as in Eq. (12). Herein, Ho(x) describes the deterministic solution part of a

physical behavior [32].

1

0,

j

jjjxfxHxH (12)

where χ and f (x) are the eigenvalues and eigenvectors of the covariance function C(x1, x2) that satisfy the

Fredholm integral equation of the second kind with orthogonal eigenvectors, Eq. (13) [30]. The covariance

function is assumed bounded, symmetric and positive definite. In addition, ξ corresponds to uncorrelated

random coefficients.

)()(,ojj

Diijoi

xfdxxfxxC and ijD

jidxff (13)


Considering that the properties vary in a one-dimensional space, a Gaussian autocorrelation function can

be expressed by an exponential decay as bxx

oiexxC 21,

, with b the correlation length in the interval

L/2 < x < L/2, L is the domain length, and xi and xo are any two points within this interval, represented,

respectively, by the kth and kth+Nuc cross-sections, where Nuc is the number of unit cells between these

points. For a zero mean and NKL terms of the series, truncated because of numerical implementation

purposes, the analytical decomposition in descending order of eigenvalues is [30, 32]:

KLN

j

jjjjjjxwxxwxxH

1

2211cossin, (14)

where ξj are Gaussian random coefficients with mean and covariance function given by E[ξi(τ)] = 0 and

E[ξi(θ)ξj(τ)] = δij [31]. Moreover, αj, βj, w1j, w2j, are given by Eq. (15) and Eq. (16) obtained by the

analytical spectral decomposition of Eq. (13) [30].

j

j

jjw

LwL

1

1

12

sin

2/ , where

22

1

1

2

cw

c

j

j

and 02

tan11

jj

wL

wc (15)

and

j

j

jjw

LwL

2

2

22

sin

2/ , where

22

2

2

2

cw

c

j

j

and 02

tan22

c

Lww

jj (16)

with c = 1/b.

The random variation of physical properties such as elastic modulus, density and geometry in space, can

be represented by a KL expansion, ,1,0

xHx , where φ0 is the nominal value and σ the

standard deviation. The assumption of Gaussian probability distribution allows negative values for the

material and geometric properties that contradict the physical meaning; however, the random uncorrelated

coefficients obtained by Monte Carlo sampling can be chosen within an interval to avoid this issue. In

addition, the correlation length can be appropriately chosen to produce a smooth distribution, as the larger

the correlation length is, the smoother is the spatial distribution [30].

3.2 Propagation Matrix

The spatial part of the spectral solution can be written as a function of the local wavenumber. Then, the

eikonal function S(x) is employed to seek a wave solution as ˆ S x i xU x e U x e

, with

xixUxS ~

ln [27, 28]. The propagation matrices for a nonhomogeneous periodic structure with

slow variation of the properties are composed by the phase change and amplitude change. The phase

change for a substructure composed by Nuc unit cells is computed integrating the local wavenumbers on

their domain, Eq. (17). The Gauss-Legendre numerical integration is used; therefore, the wavenumbers

need to be evaluated only just at the integration points, which must to be kept a minimum to avoid excess

computation. This procedure considers a homogeneous cross-section with discretization in the mid-point.

GL

o

i

N

m

mm

x

xoi xWdxxxx )(,θ (17)

where xm and Wm are the sampling points and weights used in the Gauss-Legendre integration with NGL

integration points. Considering the energy conserving property as a consequence of WKB approximation,

the amplitude change is computed by Eq. (18).

io

oi

o

i

oixxi

xxi

Q

Qxx

q

H

f

q

H

f

ΦΦ

ΦΦγ

Re

Relog

2

1log, (18)

METAMATERIALS 1973

The local wavenumbers, λ (xm), as well as the wave shapes on Eq. (18) are obtained using the WFE

method, as described in subsection 2.2. Thus, considering the eikonal solution, forward and backward

propagation matrices with oscillating and evanescent waves are obtained by Eq. (19) [31].

oioioi

oioioi

xxxxidiagxx

xxxxidiagxx

,γ,θexp,Γ

,γ,θexp,Γ

(19)

3.3 Dynamic Stiffness Matrix by WFE Approach

In the WFE method, the dynamic stiffness matrix is obtained by relating the displacements and forces at

the ends of the element. Moreover, this matrix can be developed for an undefined number of unit cells [18-

21], and only at the boundary conditions or locations with external forces the geometry needs to be

meshed. By using Block’s theorem, the state vector can be expanded in terms of the complete wave basis,

j

k

jjjkQ )1(

Φψ . Considering a periodic system composed by Nuc unit cells between the points xo and

xi, the state vector can be evaluated at the ends of the system to write displacements and forces as a

function of the wavenumbers, wave modes and wave amplitudes [18-21]:

Q

Q

ΦΓΦ

ΓΦΦ

qq

qq

R

L

U

Uˆ

ˆ and

Q

Q

ΦΓΦ

ΓΦΦ

ff

ff

R

L

F

Fˆ

ˆ (20)

where L and R are the coordinates correspondent to kth and kth+Nuc cross-sections, respectively.

Eliminating the vectors of wave amplitudes, forces are related to displacements in the frequency domain

by a dynamic stiffness matrix, UDF ˆˆ .

4 Numerical Results

The unit cell geometries, with length Δ = 0.0252m and thickness h = 0.00252m, for the undulated beam

and for the straight beam with resonators are shown in Fig. 1. These structures are made of polyamide -

density ρ = 1000kg/m3, elastic modulus E = 1GPa and structural damping η = 0.01 - and interact with air -

density ρo = 1.2kg/m3 and sound speed c0 = 341m/s. The polyamide was chosen because it is a commonly

used material in 3D printing. For the undulated beam, the orthogonal undulation of the neutral beam axis

is given by l(x) = l0 cos(2πx/Δ) [11], where l0 is the undulation amplitude; l0/Δ = 0.015 is used in this work.

Moreover, a resonator with mass mr = 5mg (7.9% of the total weight), stiffness kr = 0.5MN/m and

damping ηr = 0.01 is placed in the middle of the unit cell. Timoshenko beam elements (BEAM188) and

acoustic fluid elements (FLUID29) are used. The fluid is truncated at y = 0.126m, which was a sufficient

distance to guarantee the Sommerfeld radiation condition. The mesh respects the criterion of 8 elements

per wavelength to reach numerical convergence, which gives 6 elements in the x-direction and 30

elements in the y-direction.

(a)

(b)

Figure 1: Unit cells for an undulated beam (a) and a straight beam with internal resonator (b) interacting

with acoustic elements and imposing the Sommerfeld radiation condition.


In the dispersion diagram for a straight beam with air interaction, Fig.2 (aP) and (aE), the two flexural

wavenumbers, green and red thick lines, the extensional wavenumber, orange thick line, as well as the

acoustic wavenumbers can be observed. These waves do not interact and this configuration does not

produce a band gap. The number of acoustic wavenumbers correspond to the number of fluid DOF

because a numerical WFE approach is employed. The speed of the acoustic waves approximate the sound

speed at higher frequencies, while their evanescent part decrease. They are always confined between the

sound speed, thick blue line, and the x-axis. In the undulated beam, Fig. 2(bP) and (bE), the flexural and

extensional wavenumbers present a coupling by the curvature that produces internal wave interference,

opening the band gap. This behavior appears as a straight band in both flexural and extensional elastic

waves; in addition, the acoustic waves are not directly modified by this interaction. The pure evanescent

flexural wave is not influenced, while the pure travelling elastic waves in the stop band zone have a small

attenuation part. Coupling between resonator and beam open the band gap at the tuned resonator

frequencyrrr

mk , Fig. 2(cP) and (cE). The coupling just modifies the two flexural wavenumbers,

the pure evanescent and the pure oscillating flexural waves become complex, with negative-positive

derivatives in the real and imaginary components. These effects on the dispersion diagram are associated

with the vibration attenuation zone on the forced response.

(aP)

(aE)

(bP)

(bE)

(cP)

(cE)

Figure 2: Propagative (P) and evanescent (E) parts of the dispersion diagram for a straight beam (a),

undulated beam (b) and straight beam with internally resonators (c) with weak interaction.

METAMATERIALS 1975

The whole structure is composed by 30 unit cells, L = 0.756m, and with baffle boundary conditions at the

ends; the structural excitation is imposed at x = 0.2268m, the structural and acoustic forced responses are

measure at x = 0.504 m, and for the acoustic response y = 0.0504m. For the non-homogenous analysis

using the WKB approach, this structure is portioned in three. Two integration points are employed for

each substructure, and also the waves need be computed at the ends, which gives nine WFE evaluation

points. In all the results, the correlation length b = L and a standard deviation σ = 0.05 were used.

Furthermore, variability in the elastic modulus is considered for straight and undulated beams, while just

variability in the resonator stiffness is assumed for straight beams with distributed resonators.

The forced response obtained by WKB approach was compared with the FE method for non-homogenous

cases using one sample. For the straight beam, Fig. 3, structural and acoustic responses are in agreement;

however, small errors at some frequency zones appear which can be explained by the approximation of the

WKB method or by ill-conditioning of the dynamic stiffness matrix obtained using the WFE approach.

The use of the WFE with reflection matrix formulation, which should provide well-conditioned dynamic

responses, is left for future work. Moreover, non-homogeneity in the elastic modulus produces more

vibration modes at higher frequencies, as well as a shift in the resonance frequencies. The acoustic

response also captures these modes, which propagate by the fluid generating more noise at higher

frequencies.

(a)

(b)

Figure 3: Structural (a) and acoustical (b) responses for homogeneous (green) and non-homogeneous, by

using WKB-WFE (red) and FE (black), straight beam using one sample with σ = 0.05 and b = L.

(a)

(b)


using WKB-WFE (red) and FE (black), undulated beam using one sample with σ = 0.05 and b = L.


The structural and acoustic forced responses for the undulated beam, Fig. 4, and straight beam with

periodically distributed resonators, Fig. 5, exhibit attenuations around 5000Hz and 1500Hz, respectively.

Even for non-homogenous spatial variation of these properties, the band gaps are opened. More evident

numerical issues could be observed mainly in the transition of stop to pass bands for both cases, which is

worse for the undulate beam. These transitions could produce internal reflections where the WKB

assumption is broken; however, for frequencies below or above as well as within the band gap, WKB and

FE results are in a good agreement. These results show that the WKB method gives a fairly accurate

response prediction in all frequency ranges for structures that present band gaps, which justifies its

application in uncertainty analysis, since this method provides an approximate response saving

computational time (around 30% of computational cost compared to the FE approach) for the examples

studied in this work.

(a)

(b)


using WKB-WFE (red) and FE (black), straight beam with internally attached resonators using one sample

with σ = 0.05 and b = L.

The complete uncertainty results performed by using 200 Monte Carlo samples are presented by

employing the mean and the 95% confidence bounds, Fig.6. Only the structural dynamic behavior is

presented because similar conclusions can be transferred to the fluid response. For the straight beam, Fig.

6a, WKB and FE results are in perfect agreement and the uncertainties influence mainly the higher

frequencies. After 3000Hz the location of the resonance peaks cannot be determined with accuracy.

However, the band gap created by the undulated beam, Fig. 6b, exhibits robustness to material variability

thought its width was reduced. The band gap robustness can be explained by the fact that it is opened by

geometry modulation that depends only on the beam thickness and undulation; thus, an elastic modulus

variation has little effect on the band gap characteristics, which present a low standard deviation in this

zone. For the variability only in the internal resonator spring, Fig. 6c, WKB and FE results are also in

good agreement, and the band gap is also robust to variability, but it becomes wider and shallower.

Moreover, the uncertainty causes variability in the dynamic response before and mainly after the band gap

zone.

The formulation presented here is straightforward to apply in cases with strong fluid-structure interaction.

To exemplify, the undulated structure was surrounded by water – density ρ0 = 1030kg/m3 and sound speed

c0 = 1460m/s. The WKB and FE results do not agree in the band gap zone; main divergences appear in the

upper and lower confidence bounds. This issue needs more investigation. The band gap location is shift to

lower frequencies because the fluid adds mass to the system; in addition, its width becomes narrower.

However, vibration and acoustic attenuation are also robust in the undulated case with uncertainties in the

elastic modulus for cases with strong fluid interaction.

METAMATERIALS 1977

(a)

(b)

(c)

Figure 6: Mean and 95% confidence bounds for structural response of the straight beam (a), undulated

beam (b) and straight beam with periodically attached resonators (c) by using 200 Monte Carlo samples, σ

= 0.05 and b = L. Legend: WFE-WKB (red), FE (black) and homogeneous properties (green).

(a)

(b)

Figure 7: Mean and 95% confidence bounds for structural (a) and acoustic (b) response of undulated with

water interaction by using 200 Monte Carlo samples, σ = 0.05 and b = L. Legend: WFE-WKB (red), FE

(black) and homogeneous properties (green).

5 Conclusions

The WFE approach was applied to compute the dispersion relations of an undulated beam (phononic

crystal) and a straight beam with internally distributed resonators (metamaterial) that interact with external

fluid. Band gaps are observed in the structural and acoustic responses, which make these configurations

applicable in acoustic panel to improve vibration and sound attenuation. This work also uses the WKB


approximation to analyze the dynamic response with correlated spatial variability, which reduces the

computational time when compared to the FE method. Results are in good agreement, in all the analyzed

range of frequencies, with the FE method. However, issues occur in the transition stop band -pass band

that are more evident for the undulated beam. An approach to better treat these transitions is a point to be

improved in future work. The WKB approach presents more issues for the strong fluid-structure case,

which is more sensitive in these transitions.

The uncertainties affect the band gaps that become narrower for the undulated beam and wider for the

straight beam with periodic resonators; moreover, vibration and acoustic attenuations are reduced in both

cases. However, the band gaps can be considered robust to variability in the elastic modulus and in the

resonator stiffness. The undulated beam showed superior robustness due to the nature of its band gap,

which is open by geometry modulation. The band gap robustness in both cases needs to be confirmed by

investigating variability in other physical properties, such as thickness and undulation, which may have a

more significant influence in the case of the undulated beam. Moreover, the manufacturing process could

present a higher standard deviation or a shorter correlation length compared to those used in this work. An

experimental study is necessary to confirm these parameters.

Acknowledgements

The authors are grateful to the São Paulo Research Foundation (FAPESP) Project numbers 2014/19054-6

and 2015/15718-0, to the Federal District Research Foundation (FAPDF) Project number

0193001040/2015, and to the Brazilian National Council of Research (CNPq) Project number

445773/2014-6 for the financial support.

References

[1] M.M. Sigalas, E.N. Economu, Elastic and Acoustic Wave Band Structure, Journal of Sound and

Vibration Vol. 158, No. 2, (1992), pp. 377-382.

[2] Z.Y. Liu, X.X. Zhang, Y.W. Mao, Y.Y. Zhu, Z.Y. Yang, C.T. Chang, P. Sheng, Locally Resonant

Sonic Materials, Science, Vol. 289, No. 5485, (2000), pp. 1734-1736.

[3] M.I. Hussein, M.J. Leamy, M. Ruzzene, Dynamic of Phononic Materials and Structures: Historical

Origins, Recent Progress, and Future Outlook, Applied Mechanics Reviews, Vol. 66, (2014), pp.

040802-1-38.

[4] J.O. Vasseur, P.A. Deymier, B. Chenni, B. Djafari-Rouhani, L. Dobrwnski, D. Prevost, Experimental

and Theoretical Evidence for the Existence of Absolute Acoustic Band Gaps in Two-Dimensional

Solid Phononic Crystals, Physical Review Letters, Vol. 86, No. 14, (2011), pp. 3012-3015.

[5] M. Maldovan, Sound and heart revolution in phononics, Nature, Vol. 503, (2013), pp. 209-217.

[6] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Dark acoustic metamaterials as super

absorbers for low-frequency sound, Nature Communications, Vol. 3, (2012), p.7.

[7] C. Clayes, E. Deckers, B. Pluymers, W. Desmet, A lightweight vibro-acoustic metamaterial

demonstrator: Numerical and experimental investigation, Mechanical Systems and Signal

Processing, Vol. 70-71, (2016), pp. 853-880.

[8] H. Chen, C.T. Chan, Acoustic Cloaking in Three Dimensions Using Acoustic Metamaterials, Applied

Physics Letters, Vol. 91, No. 18, (2007), pp. 183518-183518.

[9] A. Spadoni, M. Ruzzene, S. Gonella, F. Scarpa, Phononic properties of hexagonal chiral lattices,

Wave Motion, Vol. 46, No. 7, (2009), pp. 435-450.

[10] L. Liu, M.I. Hussein, Wave Motion in Periodic Flexural Characterization of the Transition Between

Bragg Scattering and Local Resonance, Journal of Applied Mechanics, Vol. 79, (2012), pp. 011003-

1-17.

METAMATERIALS 1979

[11] G. Trainiti, J. Rimoli, M. Ruzzene, Wave Propagation in Periodically Undulated Beams and Plates,

International Journal of Solids and Structures, Vol. 75-76, (2015), pp. 260-276.

[12] M. Ruzzene, Internally Resonating Metamaterials for Wave and Vibration Control, in Proceedings

of the Noise and Vibration – Emerging Technologies (NOVEM 2015), Dubrovnik, Croatia, (2015).

[13] P.F. Pai, Metamaterial-based Broadband Elastic Wave Absorber, Journal of Intelligent Material

Systems and Structures, Vol. 21, (2010), pp. 517-528.

[14] D. Bigoni, S. Guenneau, A.B. Movchan, M. Brun. Elastic metamaterials with inertial locally

resonant structures: Application to lensing and localization, Physical Review B, Vol. 87, (2013), pp.

174303-1-6.

[15] J.F. Doyle, Wave Propagation in Structures, 2nd Edition, Springer, N.Y., (1997).

[16] U. Lee, Spectral Element Method in Structural Dynamics, John Wiley & Sons, Singapore, 2009.

[17] B.R. Mace, D. Duhamel, M.J. Brennan, L. Hinke, Finite element prediction of wave motion in

structural waveguides, Journal of Acoustic Society of America, Vol. 117, (2005), pp. 2835-2842.

[18] S.-K Lee, B.R. Mace, M.J. Brennan, Wave propagation, reflection and transmission in non-uniform

one-dimensional waveguides, Journal of Sound and Vibration, Vol. 304, (2007), pp. 31-49.

[19] J.-M. Mencik, M.N. Ichchou, A substructuring technique for finite element wave propagation in

multi-layered systems, Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 6-8,

(2008), pp. 502-523.

[20] J-.M. Mencik, D. Duhamel, A wave-based model reduction technique for the description of the

dynamic behavior of periodic structures involving arbitrary-shaped substructures and large-sized

finite element models, Finite Element in Analysis and Design, Vol, 101, (2015), pp.1-14.

[21] P.B. Silva, J-.M. Mencik, J.R.F. Arruda, Wave finite element-based superelements for forced

response analysis of coupled systems via dynamic substructuring, International Journal for Numerical

Methods in Engineering, John Wiley & Sons Ltd. (2015), p. 24.

[22] M. Petyt, Introduction to Finite Element Vibration Analysis, Second Edition, Cambridge University

Press, Cambridge, 2010.

[23] D. Duhamel, B.R. Mace, M.J. Brennan, Finite element analysis of the vibrations of waveguides and

periodic structures, Journal of Sound and Vibration, Vol, 294, No.1-2, (2006), pp. 205-220.

[24] S.A. Cummer, J. Christensen, A. Alù, Controlling sound with acoustic metamaterials, Nature

Reviews, Vol. 1, (2016), p. 13.

[25] R. Hague, I. Campbell, P. Dickens, Implications on design of rapid manufacturing, Proceedings of

the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol.

217, No. 1, (2003), pp.25-30.

[26] D. Beli, J.R.F. Arruda, Influence of additive manufacturing variability in elastic band gaps of beams

with periodically distributed resonators, in Proceeding of the 3rd International Symposium on

Uncertainty Quantification and Stochastic Modeling, Maresias, Brazil, (2016).

[27] A. Fabro, N. Ferguson, B. Mace, Wave propagation in slowly varying one-dimensional random

waveguides using a finite element approach, in Proceeding of the 3rd International Symposium on

Uncertainty Quantification and Stochastic Modeling, Maresias, Brazil, (2016).

[28] A.D. Pierce, Physical Interpretation of the WKB or Eikonal Approximation for Waves and Vibrations

in Inhomogeneous Beams and Plates, The Journal of the Acoustical Society of America, Vol, 48, No.

1, (1970), pp. 275-284.

[29] N.C. Ovenden, A uniformly valid multiple scales solution for cut-on cut-off transition of sound in

flow ducts, Journal of Sound and Vibration, Vol. 286, (2005), pp. 403-416.

[30] R. G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Spring-Verlag, New

York, 1991.


[31] A.T. Fabro, N.S. Ferguson, T. Jain, R. Halkyard and B.R. Mace, Wave propagation in one-

dimensional waveguides with slowly varying random spatially correlated variability, Journal of

Sound and Vibration, Vol. 343, (2015), pp. 20-48.

[32] S.P. Huang, S.T. Quek and K.K. Phoon, Convergence study of the truncated Karhunen-Loeve

expansion for simulation of stochastic processes, International Journal for Numerical Methods in

Engineering, Vol. 52, (2001), pp. 1029-1043.

[33] E. Mesquita, R. Pavanello, Numerical methods for the dynamics of unbounded domains,

Computational & Applied Mathematics, Vol. 24, No. 1, (2005), pp. 1-26.

[34] R.J. Astley, G.J. Macaulay, Mapped wave Envelope Elements for Acoustical Radiation and

Scattering, Journal of Sound and Vibration, Vol, 170, No. 1, (1994), pp. 97-118.

[35] A. Bhuddi, M-.L. Gobert, J.-M. Mencik, On the acoustic radiation of axisymmetric fluid-filled pipes

using the wave finite element (WFE) Method, Journal of Computational Acoustics, Vol. 23, No. 3

(2015), p. 34.

[36] W.X. Zhong, F.W Williams, On the direct solution of the wave propagation for repetitive structures,

Journal of Sound and Vibration, Vol. 181, No. 3, (1995), pp. 485-501.

[37] R. Nielsen, S. Sorokin, The WKB approximation for analysis of wave propagation in curved rods of

slowly varying diameter, Proceedings of the Royal Society A, Vol. 240, No. 2167, (2014), p. 17.

[38] G. Stefanou, The stochastic finite element method: Past, present and future, Computer Methods in

Applied Mechanics and Engineering, Vol. 198, No. 9-12, (2009), pp. 1031-1051.

METAMATERIALS 1981

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