Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin

Applied Acoustics xxx (2017) xxx–xxx

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Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin K. Dammak a,b, S. Koubaa a,⇑, A. El Hami b, L. Walha a, M. Haddar a a b

Mechanical Modeling and Manufacturing Laboratory (LA2MP), National School of Engineers of Sfax, University of Sfax, BP. 1173, Sfax 3038, Tunisia LOFIMS, INSA de Rouen, BP: 8, Avenue de l’université, 76801 St Etienne de Rouvray, France

a r t i c l e

i n f o

Article history: Received 22 November 2016 Received in revised form 8 May 2017 Accepted 1 June 2017 Available online xxxx Keywords: Finite element simulation Uncertainty Generalized polynomial chaos Coupling fluid-structure Vibro-acoustic analysis

a b s t r a c t In this paper, the formulation of the finite element method for vibro-acoustic problem is applied to a vehicle to investigate the sound pressure level inside the cabin. This study is combined with a stochastic analysis to account for variability of different parameters, considered as random variables, which are related to the characteristics of materials and boundary conditions. The results show that the Generalized Polynomial Chaos (gPC) method is more efficient compared to the direct Monte Carlo simulation (MC) without causing significant loss of accuracy. It is also shown that uncertainty levels in the input data could result in large variability in the calculated interior sound pressure level. The result of this modeling strongly depends on the order of the Polynomial Chaos. An increase in this order is accompanied by a better projection of the solution. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction ‘Noise, vibration and harshness’ (NVH), is one of the most important and critical factors for automobile industries to improve the quality of their product and to reduce noise inside the vehicle cabin. This study ‘NVH’ represents the acoustic analysis of the cavity of the car. Although the nature of the excitation may be unique for each type of vehicle, the internal pressure field within the enclosed cavity is significantly affected by the acoustic modal characteristics of the cavity, by the dynamic behaviour of the surrounding structure, as well as the nature of the coupling. Most of the spectral noise energy is between 20 Hz and 200 Hz [1]. The vibration energy generated from various sources is transmitted into the compartment cavity through structural connections. Thus vibration characteristics of the cavity and its boundary are very important factors which dominate acoustic response in a vehicle passenger compartment. The knowledge of acoustic resonances and their own modes (i.e., sound pressure distributions) is an essential task in the study of the interaction between the vibrating surface and the enclosed cavity, and ultimately in determining the level of the resulting interior noise.

⇑ Corresponding author. E-mail addresses: [email protected] (K. Dammak), [email protected] (S. Koubaa), [email protected] (A. El Hami), [email protected] (L. Walha), [email protected] (M. Haddar).

Several experimental studies [2,3] were made in the past to determine the frequencies resonance of acoustic car cavity, and the associated sound pressure distributions. Many authors applied numerical techniques such as finite element [4,5] to determine the resonances of an irregularly shaped cavity, and have been reasonably successful. Craggs [6] presented a verification of finite element predictions of acoustic resonances in a vehicle cabin, while in [7], Nefsk and al presented an experimental verification of the finite element method to predict the noise reduction in the automobile. However, in all these works only the acoustic behaviour was studied and no analysis about the interaction with the structure is mentioned. It is plausible that the design of cars is a hard task to the complex structural acoustic systems. In [8] the author presented an active vibration control system for structural acoustic coupling of a 3D vehicle cabin model. The structural-acoustic coupling system is analyzed by combining the structural data from modal testing with the acoustic data from the finite element method. Several authors showed in their studies the interaction phenomenon fluid-structure applied to different applications [7,9]. All these works have a major drawback: none of them has shown the effect of the presence of uncertainties and their propagation in models. In general, it is well known that the numerical study of the frequency response of the acoustic field without considering the variability associated with different parameters has achieved great attention. However, there are many sources of uncertainty such as errors related to manufacturing, material characteristics, boundary

http://dx.doi.org/10.1016/j.apacoust.2017.06.001 0003-682X/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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conditions applied on systems, the unpredictability of the environment (etc.), that we owe to introduce in our analysis to ensure the robustness and reliability of results. Taking into account of uncertainties in systems analysis is a complex area that includes the following steps: identification and modeling of sources of uncertainty, uncertainty propagation and post-processing to measure the influence of these uncertainties on the behaviour of the system. The main approaches for the inclusion of these uncertainties are probabilistic methods with uncertain parameters which can be modeled as random variables through the description of their probability density functions. Among these probabilistic methods, the Monte Carlo (MC) technique [10,11] is widely used because of its ease of implementation, while its main drawback is the high time to achieve accurate results. Consequently, the MC process is generally used as a method for validating the accuracy and efficiency of other methods. In [12], the author used only MC technique to account for uncertainties in vibro-acoustic problem which took a huge computational cost in its development process. Several authors used the non-parametric probabilistic approach to model the effect of uncertainties in their vibroacoustic studies of car models [13–15]. The spectral stochastic method is also an alternative solution for stochastic problems. The basic concept of this method is to use a series expansion to model the relationship between the uncertainty of the input and output variability. The generalized polynomial chaos (gPC), one of the spectral stochastic methods, has already been applied in the modeling of uncertainty in various areas, such as structural dynamics [16], fluid dynamics [17] and acoustics [18–21]. In [18], the author applied the gPC to evaluate the vibro-acoustic performance of laminated composite plates with consideration uncertain elastic parameters. In [19,20] the authors applied the theory of gPC only on a purely acoustic event so the appearance of interaction with the structure is not presented. The method of the gPC is appropriate for the study of uncertainty propagation systems related to the model parameters, initial conditions, or inputs [22]. In several papers [23–27], the method of gPC was used and compared to other uncertainties. In this paper, we aim at increasing the robustness of performance prediction for vibro-acoustic propagation systems that will operate in the presence of the inevitable parameters uncertainty associated with the material properties and boundary conditions. The stochastic methods discussed above, MC and gPC, are implemented and integrated in finite element simulation for a vehicle cabin. This paper is organized as follows. First, the details of Vibro-acoustic finite element formulation are given is Section 2. The stochastic approach of MC and gPC with the algorithm implementation is presented in Section 3. Results of the deterministic coupled Vibro-acoustic simulations are described in Section 4. Finally, the robustness analysis and results including uncertainties are also detailed. Simulation results are discussed, compared and validated with literature. In Section 5, conclusions are drawn and further works are outlined.

and the pressure p, on the grids of the fluid excite the structural part.

@p ¼ qf x2 u; @n

on R

ð2Þ

where qf denotes the mass density of the fluid. For the case of a @p ¼ 0. rigid surface @n The structure is supposed elastic and it response can be described by the local equations [31,32]:

div rðuÞ þ qs x2 u ¼ 0;

ð3Þ

in V s

with qs is the mass density of the structure, tensor.

rðuÞ means the stress

rðuÞ  n ¼ p; on Cs u ¼ u0 ;

ð4Þ

on C0

ð5Þ

where Cs \ C0 ¼ f0g and Cs [ C0 ¼ R. The natural frequencies of the cavity can then be obtained from the eigenvalues, while the eigenfunctions are simply the acoustic modes. In most practical cases, it is necessary to solve Eq. (6) numerically, and the finite element method is well adapted for this purpose. This is a coupled problem between the vibration of the structure and the internal sound pressure. It has been shown that after finite elements discretization, the equations of motion for the fluid part and the structure part are accomplished with the following equations [33–38]:

x2 ½M f fPg þ ½K f fPg ¼ fF f g þ fF fs g

ð6Þ

x2 ½M s fUg þ ½K s fUg ¼ fF s g þ fF sf g

ð7Þ

where ½Mf  is the mass matrix of the fluid. ½K f  is the equivalent stiffness matrix of the fluid. fF f g is a vector of the external forces applied to the fluid. fPg is a vector of acoustic nodal pressures. ½Ms  is the mass matrix of the structure part. ½K s  is the equivalent rigidity matrix of the structural part. fF s g is a vector of external forces applied to the structure and fUg is a vector of nodal displacements. fF fs g ¼ x2 AT U is the vector of forces from the structure part that acting on the fluid portion. fF sf g ¼ AP is the vector of forces from the fluid portion that acting on the structure part when A represents the acousticstructure coupling matrix at the boundary surfaces. The differential equations of motion for the coupled fluidstructure system can be written in the following matrix form [38]:




Ks

A

0

Kf



 x2




Ms

0

AT

Mf



U P





¼

Fs Ff



ð8Þ

3. Stochastic approach 3.1. Monte Carlo method

2. Vibro-acoustic finite element formulation For an enclosed cavity, Vf, and for the case of a harmonic variation, the equation that governs the sound pressure p is the acoustic Helmholtz equation [28–30]: 2

Dp þ k p ¼ 0;

in V f

ð1Þ

where k is the wave number, k ¼ xc , c is the speed of sound in the environment and D is the Laplacian operator. At the interface between the structure and the fluid (R), the acceleration u, on the grid points of the structure excite the particles of the air in the fluid,

In this section, the Monte Carlo method is described. An excellent exposition of this approach is given by [39–41]. The MC method provides successive resolutions of a deterministic system incorporating uncertain parameters modeled by random variables. This technique is used when the problem to be dealt is too complex for an analytical resolution is considered. It generates, for all uncertain parameters and according to their probability distributions and their correlations, random draws. For each draw, a set of parameters is obtained and a deterministic calculation, following analytical or numerical models well defined, is made. The main advantage of this method is essentially related to its applicability. In fact, theoretically, such a method can be applied to any system,

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whatever are its size and complexity. Its results are statistically accurate, i.e. they have some uncertainty that decreases with increasing of the number of draws. A reasonable accuracy requires a large number of draws which makes the MC method prohibitive in terms of computational cost. The classical standard MC technique is employed in this paper as a referential method to validate the accuracy and efficiency of the generalized polynomial chaos approach.

h/i ; /j i ¼

X i ðx; nÞ ¼

Y ¼ MðXÞ;

ð9Þ

with

where M represents the model under consideration, X is a vector of uncertain input parameters modeled by a random vector X = [X1, X2, . . ., Xn]T and Y represents a vector of estimated outputs that will be a random vector. The algorithm of the Monte Carlo method can be summarized in 5 steps:

Np ¼

8 > > > > > <

lY ¼ N1

N X

MðX ðjÞ Þ

j¼1

N > X > 2 > 2 1 > ðMðX ðjÞ Þ  lY Þ > : rY ¼ N1

ð10Þ

j¼1

 Step 5: convergence analysis of the distribution of the model output. 3.2. Generalized polynomial chaos method In this section we will present the theory of generalized polynomial chaos (gPC). The basic concept of this method is that any stochastic process can be approximated by a linear combination of orthogonal polynomials of independent random variables [18]. The polynomial chaos formalises a separation between the stochastic components of a random function and its deterministic components. It is a powerful mathematical tool which was developed by Wiener [42] as part of his theory on homogeneous chaos. The approach was extended to gPC [25] to deal with more general random fields such as non-Gaussian random fields and multidimensional random fields. Here is a brief review of mathematical approach. For example, given any random variable (Xi) such as speed, density, or the pressure in a dynamic stochastic problem of fluids, we can write [43,44]:

X i ðx; nÞ ¼

1 X X i;j ðxÞ  /j ðnÞ

ð11Þ

j¼0

where n is the vector of random variables with a known function of joint density W(n), X i;j is the deterministic component and /j ðnÞ are the orthogonal polynomial functions satisfying the orthogonality relation:

/i /j WðnÞdn ¼

i–j

0 si

ð12Þ

h/i ; /i i si i ¼ j

hi is the internal product operator. As a series expansion to infinity cannot be used in practice, the sum is truncated to an order NP to limit the number of terms in the sum to a finite number. This one depends on the order of the polynomial chaos p and its dimension, r, denoting the number of uncertain parameters in the system:

3.1.1. Algorithmic implementation The standard Monte Carlo approach considers functions of the following form:

 Step 1: probabilistic identification of uncertain parameters in the model.  Step 2: sampling and random generation of achievements following identified probabilistic laws.  Step 3: spread of uncertainty i.e. of the data set resulting from step 2 into the model and determining the corresponding output set.  Step 4: estimate the output distribution law whose statistical characteristics are given by the mean value lc and standard deviation rc. These are calculated using a set of N simulation as follows:



Z

Np X X i;j ðxÞ  /j ðnÞ

ð13Þ

j¼0

ðp þ rÞ! 1 p!r!

ð14Þ

The calculation of a representation by the gPC requires the determination of Np + 1 stochastic modes. Two approaches are used: an intrusive and the other non-intrusive. The first one consists in substituting, in the stochastic model, random functions by their spectral representations and then in making a projection, called Galerkin [45,24], of the resulting system based on the gPC. The second project directly but only the stochastic solution based on the gPC. It appears more attractive in the sense that it requires no modification and no manipulation of the original model uncertain. For this, we will present in the following only the nonintrusive approach. Thus, the term ‘‘black box” is attributed to it in literature. This technique is especially advantageous because it only requires a few calculations for the deterministic solutions in a finite number of points in the field of probabilistic variation. The determination of these points depends on the method used to calculate the stochastic modes. The spectral projection (NISP) [43] and regression [46] are the two main methods of nonintrusive gPC used to calculate the stochastic modes. In the NISP process, once the solution expressed in the base of polynomial chaos following the general expression, it is projected in this base which determines stochastic coefficients by:

X i;j ðxÞ ¼

hX i ðx; nÞ; /j ðnÞi 1 ¼ 2 h/j ðnÞ; /j ðnÞi h/j i

Z X i ðx; nÞ  /j ðnÞ  WðnÞdn;

¼ 0; . . . ; Np

j ð15Þ

The numerator in Eq. (15) passes through the calculation of multidimensional integrals along the probabilistic support for the random vector n. Different methods are used to approximate these integrals such as numerical Monte Carlo techniques, Simpson methods or Gauss collocation techniques. For example, in the case of Gauss monodimensional integration, Eq. (15) can be put in the following form [43]:

X i;j ðxÞ ¼

Q 1 X X i ðx; nk Þ  /j ðnk Þ  Wðnk Þ; 2 h/j i k¼1

j ¼ 0; . . . ; Np

ð16Þ

The point n(k) are the Gaussian integration points also called Gauss Collocation points given by the roots of the polynomial /j of degree p while the W(k) represent the Gaussian integration weight. The regression method consists of calculating the stochastic modes so as to minimize the least squares sense, the difference e, between the solution of the stochastic model and its approximation in the base of generalized polynomial chaos [46].

" # Np Q X X ðkÞ ðkÞ e¼ X i ðx; n Þ  X i;j ðxÞ  /j ðn Þ k¼1

ð17Þ

j¼0

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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This technique requires a number, Q, of simulations higher than the number of stochastic coefficients: Np + 1 < Q. T

By designating X i;j ¼ ðX i;0 ; . . . ; X i;Np Þ the vector of modal coefficients,

Z

the

matrix

ð1Þ

of

elements

zq;l ¼ /j ðnðqÞ Þ

and

ðqÞ

X i ¼ ðX i ðx; n Þ; . . . ; X i ðx; n ÞÞ the vector corresponding to the game simulations fnðqÞ g and if the matrix (ZTZ) is non-singular then the optimal solution of the classical least squares problem is given by: 1

X i;j ¼ ðZ T ZÞ Z T X i

30 1 0 1 /0 ðn0 Þ /1 ðn0 Þ ::: /NP ðn0 Þ Uðx; n0 Þ U 0 ðxÞ 7B C B C 6 6 / ðn1 Þ / ðn1 Þ . . . / ðn1 Þ 7B U ðxÞ C B Uðx; n Þ C 1 NP 1 C 7B 1 C B 6 0 7B C B C 6 7B . C ¼ B C 6 . .. .. .. .. 7B . C B C 6 . 7B . C B C 6 . . . . . 5@ A @ A 4 Uðx; nQ Þ U NP ðxÞ /0 ðnQ Þ /1 ðnQ Þ . . . /NP ðnQ Þ 2

ð23Þ

ð18Þ

3.2.1. Algorithmic implementation A summary of the gPC procedure, as applied to vibro-acoustic analysis, is presented below:  Step 1: represent the uncertain input parameters that follow a uniform distribution. These parameters can be related to the characteristics of the structure and the fluid, the boundary conditions and the geometry of the model. Let a be the uncertain parameter of vibro-acoustic problem (density, impedance, elasticity, (etc.)) and it can be written as:

a ¼ a0 þ na1

ð19Þ

where a0 is the mean value, a1 is a convenient constant and n is a uniform variable U(0, 1).  Step 2: express the model output {U, P} of Eq. (9) under consideration in terms of the same set of random variables as:

P¼

and the same for the displacement of the structure:

NP X Pj  /j ðnÞ

 Step 4: once the stochastic coefficients are determined by one of developed techniques, statistics responses of outputs can be obtained by calculating the mean and variance:

8 > <

lðPÞ ¼ P0

2 > : r ðPÞ ¼

Np X

2

ðPj Þ h/j ; /j i

ð24Þ

j¼1

and for the displacement of the structure, we can write:

8 > <

lðUÞ ¼ U 0

2 > : r ðUÞ ¼

Np X

2

ðU j Þ h/j ; /j i

ð25Þ

j¼1

4. Numerical results The considered model consists on a simple vehicle cabin as shown in Fig. 1. The geometric characteristics of the studied cavity

ð20Þ

j¼0

U¼

NP X U j  /j ðnÞ

ð21Þ

j¼0

 Step 3: calculate the unknown stochastic coefficients of (Eq. (15)).

2

/0 ðn0 Þ

6 6 /0 ðn1 Þ 6 6 6 .. 6 . 4

/1 ðn0 Þ

:::

/1 ðn1 Þ

...

.. .

.. .

/NP ðn0 Þ

30

P0 ðxÞ

7B /NP ðn1 Þ 7B P1 ðxÞ 7B 7B 7B .. .. 7B . . 5@

/0 ðnQ Þ /1 ðnQ Þ . . . /NP ðnQ Þ

PNP ðxÞ

1

0

Pðx; n0 Þ

1

C B C C B Pðx; n1 Þ C C B C C¼B C C B C .. C B C . A @ A Pðx; nQ Þ

ð22Þ

Fig. 2. Ansys finite elements model of the acoustic cavity.

Fig. 1. 2D diagram of the cabin.

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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Fig. 3. Acoustic modal shapes for the 3D acoustic rigid cavity: (a) f1 = 78.84 Hz, (b) f2 = 122.76 Hz, (c) f3 = 145.89 Hz, (d) f4 = 145.93 Hz, (e) f5 = 160.17 Hz, (f) f6 = 181.71 Hz and (g) f7 = 190.70 Hz.

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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are the following: Lx = 2.51 m, Ly = 1.1 m and Lz = 1.4 m. A steel sheet with a thickness 8 mm is situated in front of the cavity. Numerical simulation of the vibro-acoustic response of the cavity are performed using the commercial software ANSYS. Regarding the discretization of the internal fluid and the sheet, 3808 and 70 elements were adopted respectively for the fluid and the structure. The internal fluid elements are 3D elements bounded by 6 faces, having 8 nodes with one degree of freedom per node: the acoustic pressure. The shell is a four-node elements with six degrees of freedom at each node: 3 translations in the nodal x, y and z directions and 3 rotations about the nodal x, y and z axes (see Fig. 2). The adopted boundary conditions were fluid-structure interface at all free nodes of the steel plate. All the nodes at the plate border were constrained [47]. The fluid inside the cabin is the air whose characteristics are: its density is 1.21 kg/m3 and the speed of sound is 343 m/s. The finite element analysis (FEA) is used to predict the sound pressure level (SPL) inside the cabin. This SPL can be determined using Eq. (26) [36,48]:

 SPLdB ¼ 20log10

P Pref

harmonic force in the frequency range 0–200 Hz is applied perpendicularly at the center of the front plate. Fig. 4 presents a comparison between the sound pressure levels in the two positions without considering the acoustic absorption materials. In Table 2, we present a comparison of the SPL values between the two positions. Fig. 5 shows the SPL response when the roof of the cab has the characteristics described in Table 3. The theoretical sound-absorbing coefficient is calculated as [50,51]:

  Z  Z 0   Z þ Z0 

ath ¼ 1  

ð27Þ

 ð26Þ

with Pref = 20 lPa is the acoustic pressure of human hearing. 4.1. Acoustic analysis (deterministic) In this section, a modal analysis is performed without considering uncertainty. Fig. 3 illustrates the acoustic mode shapes obtained by FE simulations. One can remark a neutral line of zero pressure around the middle of the cabin length at the first mode. We notice that, the effect of the seats on the modal analysis is ignored. This result is in accordance with [7,47]. 4.2. Vibro-acoustic analysis (deterministic)

Fig. 4. Sound pressure level inside the cabine without acoustic absorption materials.

Table 2 Characteristic values of SPL without acoustic absorption materials.

In this section the influence of the vibration of the structure on the acoustic response (SPL) is studied.

Position

Max SPL (dB)

Min SPL (dB)

Frequency (Hz)

DRE PRE

124.8 138.2

81.52 74.39

185/60 185/140

4.2.1. Modal analysis We consider that the vibration of the panels in the front of the structure is the dominated source of noise in the vehicle cabin [48]. Table 1 shows the results for the natural frequencies obtained from FEM simulations for both coupled and decoupled problems. It is plausible that the natural mode of vibration of the coupled system is a mixture of the isolated subsystem modes. Each mode of the isolated acoustic cavity interacts with each mode of the isolated structural plate resulting the coupled system vibration modes [49]. 4.2.2. Harmonic analysis In this section, we evaluate the sound pressure level (SPL) inside the cabin as a function of the frequency in two positions: the first at the driver right ear (DRE), located at (1.5, 0.8, 1.25) and the second at the passenger right ear (PRE) located at (0.6, 0.8, 1.25). A

Table 1 Vibro-acoustic natural frequency. Acoustic cavity

Structural plate

Coupled system

78.84 122.75 145.89 145.93 160.17 181.71 190.69 217.53

20.87 32.82 51.58 73.64 94.85 126.75 151.49 173.43

21.26 87.35 124.58 150.66 159.18 162.67 185.23 201.02

Fig. 5. Comparison of the SPL with and without the presence of absorption material on the roof.

Table 3 Acoustic material characteristics of the roof. Density (kg/m3) Velocity of sound (m/s) Sound absorption coefficient Normalized impedance

1100 2400 0.25 1

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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where Z 0 ¼ qf c is the characteristic impedance of the fluid and Z is the specific acoustic impedance of the absorbing surface. In terms of a real part and an imaginary part, the normalized specific acoustic impedance is given by:

Z ¼ R þ jX Z0

ð28Þ

Then the sound-absorbing coefficient can be written in the following form:

ath ¼

4R

ð29Þ

ðR2 þ X 2 Þ þ 2R þ 1

It is found that, with the presence of the acoustic absorption materials on the roof, the SPL decreased by 25 dB for both positions. Table 4 shows a comparison of the SPL values between the two positions. Fig. 6 shows the SPL response when the floor of the cab with the following properties (Table 5). In the presence of absorption material on the floor, it is noted that the SPL decreased by around 15 dB for both positions. Table 6 shows a comparison of the SPL values between the two positions. If we take into account both the absorption of the roof and also the floor, we obtain the results described in Fig. 7. With the full absorption, we find that:  For the DRE: an increased by 1 dB for the maximum value and a decreased by 2 dB for the minimum value of the SPL.  For the PRE: a reduction by 30 dB for the maximum value and by 15 dB for the minimum value of the SPL.

Table 4 Characteristics values of SPL with an absorption material on the roof. Position

Max SPL (dB)

Min SPL (dB)

Frequency (Hz)

DRE PRE

100 115.6

54.75 46.83

112.1/65 175/135

Fig. 6. Comparison of the SPL with and without the presence of absorption material on the floor.

Table 5 Acoustic material characteristics of the floor. Density (kg/m3) Velocity of sound (m/s) Sound absorption coefficient Normalized impedance

55 343 0.25 1

It can be noticed that the evolution of the SPL according to frequency presents less fluctuation and becomes more stable. One can also remark the convergence of the curve to two values of the SPL: 100 dB for the DRE position and 91 dB for the PRE position. Table 7 shows a comparison of the SPL values between the two positions.

4.3. Stochastic analysis (proposed) In this section, the variability of uncertain parameters in the vibro-acoustic study in case of full absorption is introduced. The stochastic method discussed above, gPC, is implemented and integrated in the FE simulation described in Section 4.2. The objective of this section is to improve the performance and robustness of vibro-acoustic systems that operate in the presence of inevitable uncertain parameters associated simultaneously to material properties and boundary conditions. These parameters are chosen to be random following a uniform distribution around their nominal values ±5%. Table 8 presents an idea about the different uncertain parameters chosen for the example of the cabin. Using the MC method to analyze the sound pressure level (SPL) inside the cabin consists on creating a grid of numerical values from the uncertain parameters and calculating the quantity of interest of the system linearized for each value of the grid. The quantity of interest is analyzed for 2000 simulations. Fig. 8 shows the distribution of the input variables (c, Z, coef-abs, E, qs) in the case of uniform distribution. Figs. 9 and 10 present the evolution of the SPL as a function of the frequency by two stochastic methods MC and gPC in the case of 5 uncertain parameters. It is of interest to note a remarkable dif-

Table 6 Characteristics values of SPL with an absorption material on the bottom. Position

Max SPL (dB)

Min SPL (dB)

Frequency (Hz)

DRE PRE

114.8 115.1

67.57 58.15

110/185 75/155

Fig. 7. Comparison of the SPL with and without the presence of absorption material on the roof and the floor (full absorption).

Table 7 Characteristics values of SPL with a full absorption. Position

Max SPL (dB)

Min SPL (dB)

Frequency (Hz)

DRE PRE

125.9 105.9

79.49 60

165/55 140/55

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K. Dammak et al. / Applied Acoustics xxx (2017) xxx–xxx

Table 8 Characteristics of uncertain parameters. Parameters

Values

Acoustic part

Structure part

c (m/s)

Z (Pas/m)

coef-Abs

E (GPa)

qs (kg/m3)

U [325.85, 360.15]

U [0.95, 1.05]

U [0.2375, 0.2625]

U [199.5, 220.5]

U [7457.5, 8242.5]

Fig. 8. Probability distribution of the inputs: (a) sound speed, (b) impedance, (c) density, (d) elasticity, and (e) absorption coefficient.

ference between the curves obtained by the gPC method and those obtained by MC simulations for low value of the order p. This seems plausible as there are not enough terms of chaos to correctly represent the random response of the system. This result is

assessed by [52]. By increasing the value of p = 6, the curves are similar to reference ones MC. For three uncertain parameters, we find that an order p = 2 of Chaos is sufficient given that the gPC follows exactly the response obtained by MC simulations with an

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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Fig. 9. Mean of the SPL for an order p = 6 of the chaos (full absorption).

Fig. 10. Standard deviation of the SPL for an order p = 6 of the chaos (full absorption).

error around of 10-5. Besides, we notice that the time of calculation has considerably reduced and become negligible. If the speed of sound (c) becomes uncertain, we must increase the order of the chaos to p = 4, so that the gPC can follow the MC response. If we take into account even the variability of the absorption coefficient, the system becomes larger in terms of size (5 parameters) and so gPC needs an order p = 6 to follow the evolution obtained by MC and to reduce the error with MC technique. One can also remark that when the ranges variation of uncertain parameters are assumed to be these in Table 9, the eigenfrequencies in frequency band [50–200] Hz are situated in frequency ranges f = 130–140 Hz, f = 160–180 Hz respectively. It also necessary to mention that with the presence of inevitable uncertainties, the evolution of the SPL as a function of the frequency becomes more stable and the resonance peaks becomes smoother. The probability distribution of the SPL of the two positions DRE and DRE are shown in Figs. 11 and 12. One can notice that the SPL distribution is very similar to the ones obtained when we use a deterministic model. It was also shown that the probability distributions of the SPL of the positions DRE and PRE behave as a Gaussian law. Table 9 presents a summary of all the simulations made by the two methods. It is worth mentioning that the MC technique is a well-known method to solve systems complexity with random parameters. For reasonable accuracy, it requires a great number of samples. In this work, 10,000 of samplings of 5 input variables are calculated and then the problem is solved for each sample of input variables. Nevertheless, it is well shown that this technique has poor convergence for mean and standard deviation of the solution, requiring a large number of samples to achieve good precision in results, resulting in costly computation. This result is in a good agreement with [43]. The gPC method is considered as the best framework in dealing with uncertainty quantification. This technique is more attractive and more efficient compared to MC approach.

Table 9 Summary of results for two positions DRE and PRE. Number of uncertain parameters NMC Order of Chaos Mean maxi of the SPL (MC/gPC) Std maxi of the SPL (MC/gPC) Time MC (h) Time gPC (s)

DRE PRE DRE PRE

r = 3 (z, E, qs) 500 2 125.36/124.57 105.19/106.48 10.54/10.68 13.28/14.36 4.16 1.24

r = 4 (z, c, E, qs) 1000 4 120.47/121.26 103.62/103.39 9.78/11.17 14.36/15.87 8.2 5.26

r = 5 (z, c, coef-abs, E, qs) 2000 6 113.28/113.75 100.14/102.26 9.45/12.24 16.32/12.48 25.43 45.18

Fig. 11. Probability distribution of the SPL-DRE, (a) MC and (b) gPC.

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

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Fig. 12. Probability distribution of the SPL-PRE (a) MC and (b) gPC.

5. Conclusion In this study, we have used the gPC method based finite element simulation for the solutions of vibro-acoustic problems when the material properties and the boundary conditions present uncertainty. The probabilistic nature of physical parameters is considered in analytical model, and the influences of these uncertain parameters on the sound pressure level field are particularly discussed. The simulation results show that uncertainty levels in the input data could result in large variability in the calculated pressure field in the vibro-acoustic domain. The case of a cavity of a vehicle was studied taking into account the fluid-structure interaction. Results using the polynomial chaos expansion method are compared with MC technique. Convergence is assessed when comparing exact solutions and solutions from Monte Carlo simulations. The effectiveness of the gPC method applied to vibro-acoustic problems in 3D vehicle cabin is demonstrated. The results of this modeling strongly depend on the order of the gPC. An increase in this order is accompanied by a better projection of the solution. As a future work, we will introduce the exterior boundary conditions and study their influence on the sound pressure level inside the cabin. References [1] Jee SH, Yi JC, Park JK. The comparison of the BEM and FEM techniques for the interior noise analysis of the passenger car. Inchon (Korea); 2000. [2] Priede T, Jha SK. Low frequency noise in cars-its origin and elimination. J Autom Eng 1970;15:17–21. [3] Stanef D, Hansen CH, Morgans R. Active control analysis of mining vehicle cabin noise using finite element modelling. J Sound Vib 2004;277(1– 2):277–97. [4] Shuku T, Ishihara K. The analysis of the acoustic field in irregularly shaped rooms by the finite element method. J Sound Vib 1973;29(1):67–76. [5] Petyt M, Lea J, Koopmann GHA. Finite element method for determining the acoustic modes of irregular shaped cavities. J Sound Vib 1976;45(4):495–502. [6] Craggs A. An acoustic finite element approach for studying boundary flexibility and sound transmission between irregular enclosures. J Sound Vib 1973;30 (3):343–57. [7] Nefske DJ, Wolf JA, Howell LJ. Structural-acoustic finite element analysis of the automobile passenger compartment: a review of current practice. J Sound Vib 1982;80(2):247–66. [8] Song CK, Hwang JK, Lee JM, Hedrick JK. Active vibration control for structural– acoustic coupling system of a 3-D vehicle cabin model. J Sound Vib 2002;267 (4):851–65. [9] Lim TC. Automotive panel noise contribution modeling based on Ònite element and measured structural-acoustic spectra. Appl Acoust 2000;60(4):505–19. [10] Hammersley JM, Handscomb DC. Monte Carlo methods. London: Methuen; 1964. [11] Shreider YA. The Monte Carlo method. New York: Pergamon; 1966.

[12] Barillon F, Boubaker M, Mordillat P, Lardeur P. Vibro-acoustic variability of a body-in-white using Monte Carlo simulation in a development process. In: PROC ISMA2012-USD2012, Leuven, Belgium; 2012. [13] Durand J, Gagliardini L. Random uncertainties modelling for vibroacoustic frequency response functions of cars. In: Proc ISMA 2004, Leuven, Belgium; 2004. [14] Fernandez C, Soize C, Gagliardini L. Sound-insulation layer modelling in car computational vibroacoustics in the medium-frequency range. Acta Acust United Acust 2010;96:437–44. [15] Gagliardini L, Durand J. Stochastic modeling of the vibro-acoustic behavior of production cars. In: Acoustics’08, Paris, Fance; 2008. [16] Sarkar A, Ghanem R. Mid-frequency structural dynamics with parameter uncertainty. Comput Methods Appl Mech Eng 2002;191(47–48):5499–513. [17] Knio OM, Le Maitre OP. Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid Dyn Res 2006;38(9):616–40. [18] Sepahvand K, Scheffler M, Marburg S. Uncertainty quantification in natural frequencies and radiated acoustic power of composite plates: analytical and experimental investigation. Appl Acoust 2015;87:23–9. [19] LePage KD. Estimation of acoustic propagation uncertainty through polynomial chaos expansions. In: Proc information fusion, Florence, Italy; 2006. [20] Creamer DB. On using polynomial chaos for modeling uncertainty in acoustic propagation. J Acoust Soc Am 2006;119:1979–94. [21] Finette S. Embedding uncertainty into ocean acoustic propagation models (L). J Acoust Soc Am 2005;117:997–1000. [22] Nagy ZK, Braatz RD. Distributional uncertainty analysis using power series and polynomial chaos expansions. J Process Control 2007;17(3):229–40. [23] Saha SK, Sepahvand K, Matsagar VA, Jain AK, Marburg S. Stochastic analysis of base-isolated liquid storage tanks with uncertain isolator parameters under random excitation. Eng Struct 2013;57:465–74. [24] Xiu D, Karniadakis GE. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 2002;24:619–44. [25] Xiu D, Karniadakis GE. Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 2003;187(1):137–67. [26] Sepahvand K, Marburg S, Hardtke H. Uncertainty quantification in stochastic systems using polynomial chaos expansion. Int J Appl Mech 2010;2:305–53. [27] Sepahvand K, Marburg S, Hardtke H. Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion. J Sound Vib 2012;331 (1):167–79. [28] Kinsler LB, Frey AR. Fundamental of acoustics. New York: John Wiley & Sons; 1962. [29] Brebbia C, Ciskowski R. Boundary element methods in acoustics. New York: Elsevier Science Publishing Company; 1991. [30] Morse PM. INGARDKU. Theoretical acoustics. New York: McGraw-Hill Book Company; 1968. [31] Mansouri M. Etude mécano-fiabiliste et réduction du modèle des problèmes vibro-acoustiques à paramètres aléatoires. Phd thesis. INSA ROUEN; 2013. [32] Larbi W, Deü JF, Ohayon R. Vibroacoustic analysis of double-wall sandwich panels with viscoelastic core. Comput Struct 2016;174:92–103. [33] Everstine GC, Schroeder EA, Marcus MS. The dynamic analysis of submerged structures. NASA TM; 1975. [34] Muthukrishnan A. Car cavity acoustics using ANSYS. In: International ANSYS conference; 2006. [35] Guan J. An innovative solution for true full vehicle NVH simulation. In: Altair technology conference global series; 2012. [36] Lupea I, Szatmari R. Vibroacoustic frequency response on a passenger compartment. J Vibroeng 2010;4:406–18.

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001

K. Dammak et al. / Applied Acoustics xxx (2017) xxx–xxx [37] EL Maani R, Radi B, EL Hami A. Reliability study of a coupled three-dimensional system with uncertain parameters. Adv Mater Res 2015;1099:87–93. [38] Corian A, Lupea I. Improving the sound pressure level for a simplified passenger cabin by using modal participation and size optimization. RJAV 2013;X(1). [39] Spanos PD, Kontsos A. A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites. Probab Eng Mech 2008;23(3):456–70. [40] Hurtado JE, Alvarez DA. The encounter of interval and probabilistic approaches to structural reliability at the design point. Comput Methods Appl Mech Eng 2012;225–228:74–94. [41] Fishman GS. Concepts and methods in discrete event simulation. New York: Wiley; 1973. [42] Wiener N. The homogeneous chaos. Am J Math 1937;60:897–936. [43] Nechak L, Berger S, Aubry E. Robust analysis of uncertain dynamic systems: combination of the centre. WSEAS Trans Syst 2013;98:386–95. [44] Guerine A, EL Hami A, Walha L, Haddar M. A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system. Eur J Mech A/Solids 2016;59:76–84.

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[45] Ghanem R, Spanos P. Stochastic finite elements: a spectral approach. Dover; 2003. [46] Blatman G, Sudret B. Sparse polynomial chaos expansions and adaptive stochastic finite. Compte Rendus Mécanique 2008;336(6):518–23. [47] Kruntcheva M. Acoustic – structural coupling of the automobile passenger compartment. In: Proc world congress on engineering, London; 2007. [48] Coroian A, Lupea I. Vibro – acoustic analysis of passenger compartment. Appl Math Mech 2012;55(4):831–6. [49] Deü J, Larbi W, Ohayon R, Sampaio R. Piezoelectric shunt vibration damping of structural-acoustic systems: finite element formulation and reduced-order model. J Vib Acoust 2014;136. [50] Mónica AL, Miguel Angel RP, Jose ADS. Acoustic absorption coefficient of opencell polyolefin-based foams. Mater Lett 2014;121:26–30. [51] Howard CQ, Cazzolato BS. Acoustic analyses using Matlab and Ansys. Boca Raton: CRC Press; 2015. [52] Beyaoui M, Tounsi M, Abboudi K, Feki N, Walha L, Haddar M. Dynamic behaviour of a wind turbine gear system with uncertainties. J Comptes Rendus Mecanique 2016;344(6):375–87.

Please cite this article in press as: Dammak K et al. Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin. Appl Acoust (2017), http://dx.doi.org/10.1016/j.apacoust.2017.06.001