An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters

Journal of Fluids and Structures 49 (2014) 441–449

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An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters Chong Wang n, Zhiping Qiu Institute of Solid Mechanics, Beihang University, Beijing 100191, PR China

a r t i c l e i n f o

abstract

Article history: Received 28 July 2013 Accepted 5 May 2014 Available online 12 June 2014

This paper proposes two interval analysis methods, called the first-order interval parameter perturbation method (FIPPM) and the modified interval parameter perturbation method (MIPPM), for use in exterior acoustic field prediction when there are uncertainties in both the material properties and the external load. Interval variables are used to quantitatively describe the uncertain parameters in the face of limited information. The conventional first-order Taylor expansion and perturbation terms are employed in the FIPPM, while the MIPPM introduces modified Taylor series to approximate the non-linear interval matrix and vector. The high-order terms of the Neumann expansion are retained to calculate the interval matrix inverse. A numerical example is given by comparing the results with a Monte Carlo simulation to demonstrate the feasibility and effectiveness of the proposed methods at evaluating the sound pressure ranges in an exterior acoustic field. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Exterior acoustic field prediction Parameter perturbation method Interval uncertainties High-order terms

1. Introduction In recent years, research on acoustic behavior has undergone rapid development in engineering with the increase in peoples' awareness of the environment. The finite element method (FEM), boundary element method (BEM) and infinite element method (IEM) are currently understood to be the most effective numerical strategies for solving low-frequency acoustic radiation problems (Antebas et al., 2013; Raveendra, 1999; Autrique and Magoules, 2007). Traditional acoustic analysis has been conducted under the assumption that the physical properties and boundary conditions are deterministic. However, uncertainties in material properties, geometric dimensions and boundary conditions are unavoidable due to model inaccuracies, physical imperfections and system complexities, and lead to uncertainty in the acoustic field. Probabilistic methods are popular to solve these problems of uncertainty, and the probability density functions are defined unambiguously. Allen and Vlahopoulos (2000) have combined boundary element methods and finite element methods with stochastic analysis to calculate the noise radiated from a structure subjected to random excitation. Chen et al. (2009a) have developed a computing technique for acoustic pressure spectral density and its sensitivity in coupled structural–acoustic systems subjected to stochastic excitation. James and Dowling (2005, 2008, 2011) have studied an approach to quantifying uncertainties and a method of determining the probability density functions of sound pressure's amplitude and phase in a predicted acoustic-field. A large amount of statistical information is required to construct the precise probability distribution

n Correspondence to: Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China. Fax: þ86 10 82313658. E-mail address: [email protected] (C. Wang).

http://dx.doi.org/10.1016/j.jfluidstructs.2014.05.005 0889-9746/& 2014 Elsevier Ltd. All rights reserved.

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C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449

functions of uncertain parameters in probabilistic methods. Unfortunately, it is often too difficult or costly to collect enough information about the uncertainty in many engineering problems. It is well known that interval technology is appropriate for the numerical analysis of non-probabilistic systems without sufficient information. Neumaier (1990) has investigated the use of the hypercube approximation for the united solution set of interval equations by the Gaussian elimination scheme, but this approximation is extremely conservative due to a large number of elimination operations (Berkel and Rao, 1997). The exact response intervals can be achieved by the vertex method, using all possible combinations of the interval parameters (Qiu et al., 2007). However, the computational effort of this method increases exponentially with an increase in the number of uncertain parameters. Compared with the above interval approaches, the interval perturbation method, proposed by Qiu et al. (1996) and Qiu and Elishakoff (1998), has been widely applied to structural response analysis due to its simplicity and efficiency (Sliva et al., 2010). From a high-level perspective, the current research on interval approaches has been concentrated mainly in the special field of uncertain structures. The interval analysis of uncertain structural-acoustic problems with interval parameters has attracted increased attention in recent years (Xia and Yu, 2012; Xia et al., 2013). Although the perturbation method procedure is easy to implement, it should be noted that the accuracy of this procedure's computational results becomes unacceptable if the nonlinear degree of the interval matrix and vector is high or a large number of uncertain parameters exist due to the unpredictable effect of neglecting the high-order terms in the Taylor series and perturbation calculation. The purpose of this paper is to propose a new numerical method with broader applicability to predict the response bounds of an exterior acoustic field with interval parameters. The interval equilibrium equation for the exterior acoustic field prediction is established in Section 2 based on a structural finite element model and an acoustic boundary element model. Two interval analysis methods are presented in the following two sections to obtain the acoustic response ranges. The first method is called FIPPM, in which the interval matrix and vector are expanded by the first-order Taylor series and the interval equation is solved based on the first-order perturbation theory. To overcome the shortcomings of FIPPM, another method called MIPPM is proposed by combining the modified Taylor expansion with the high-order Neumann series. The numerical results for a flexible plate with uncertain parameters are given for verification in Section 5, and we draw conclusions in the final section. 2. FEM/BEM equilibrium equation for a structural-acoustic system The FEM/BEM technique is widely used to predict the acoustic radiation from vibrating structures because it involves only surface discretization and solves exterior problems naturally. The key idea in this technique involves representing the acoustic field as a superposition of fields due to elementary sources located on the structural surface, as shown in Fig. 1, where the elementary solutions precisely satisfy the Helmholtz equation. In the frequency domain, the steady-state finite element equation of an elastic structure containing damping can be expressed as ðKs þjωCs ω2 Ms ÞUs ¼ Fs ; s

ð1Þ s

s

s

where K stands for the structural stiffness matrix; C is the damping matrix; M denotes thepmass ffiffiffiffiffiffiffiffi matrix; F gives the external load vector applied to the structure; Us is the structural displacement vector; and j ¼ 1 is an imaginary unit. In the steady-state form, the structural velocity vector Vs and displacement vector Us satisfy the following relationship: Vs ¼ jωUs :

ð2Þ

Multiplying both sides of Eq. (1) with the constant jω, the dynamic equilibrium equation for the structural vibration velocity can be written as ðKs þjωCs ω2 Ms ÞVs ¼ jωFs :

ð3Þ

Fig. 1. Exterior acoustic field of a vibrating structure.

C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449

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In addition to satisfying the Helmholtz equation, the acoustic field of the vibrating structure must also satisfy the applied boundary condition. In other words, the normal component of the acoustic particle velocity must be equal to the structure's velocity. Thus, after calculating the structural velocity vector, the boundary element equation of the sound pressure can be obtained simply as follows: APs ¼ GVsn ¼ GNVs ¼ ΒVs ;

ð4Þ

Vsn

where is the structural normal velocity vector; N denotes the conversion matrix from a velocity vector to a normal velocity vector; Ps stands for the sound pressure vector along the boundary of the structure; and A, G and Β ¼ GN are three complex coefficient matrices related to the structural surface shape, dimensions and selected interpolation functions. Combining Eq. (3) with Eq. (4), the dynamic equilibrium equation of the structural-acoustic system can be written as ZU ¼ F; where

"

Z¼

ð5Þ

Ks þ jωCs  ω2 Ms

0

Β

A

"

# ;

U¼

Vs Ps

"

# ;

F¼

jωFs 0

# :

If the structural velocity vector Vs and sound pressure vector Ps on the interface are obtained, the sound pressure P e at any point in the exterior field can be given by ! Vs T T T P e ¼ aT Ps þ b Vsn ¼ aT Ps þb NVs ¼ ðb N aT Þ ð6Þ ¼ cT U; Ps ! ΝT b are the interpolation coefficient vectors in the exterior acoustic field. In addition to being related where a, b and c ¼ a to the geometric shape of the closed structure, they are also related to the position of the selected field point. Therefore, by discretizing the structure and acoustic boundary with appropriate elements and by solving the FEM/BEM equilibrium equation, the sound pressure response can be efficiently obtained using Eq. (6). In an actual engineering problem, due to production limitations and measurement errors, uncertainties in material properties, external loads and boundary conditions are unavoidable. In this paper, the uncertainties, whose lower and upper bounds can be determined by the limited information, are quantitatively described as the interval parameters and assumed to belong to an uncertain-but-bounded parameter vector as follows: αI ¼ ½α; α ¼ ðαIi Þm ¼ ð½α i ; αi Þm ¼ ðαci þ ΔαIi Þm ¼ ðαci þ Δαi δÞm ¼ αc þΔαδ

i ¼ 1; 2; …; m;

ð7Þ

where α i and αi are the lower and upper bounds; αci ¼ ðαi þ α i Þ=2 and Δαi ¼ ðαi α i Þ=2 are called the midpoint and the radius, respectively; and the transition parameter δ denotes a fixed interval, i.e., δ ¼ ½  1; 1. Thus, the FEM/BEM equilibrium equation with respect to interval parameters can be transformed into ZðαI ÞUðαI Þ ¼ FðαI Þ:

ð8Þ

The theoretical solution set of Eq. (8) is defined as  Ω ¼ fUðαÞZðαÞUðαÞ ¼ FðαÞ; α A αI g;

ð9Þ

where Ω has a complicated region. A direct solution to Eq. (9) cannot be implemented in practice, and solving Eq. (9) in interval mathematics is synonymous to searching for a multi-dimensional rectangle UI ¼ ½U; U, which contains the above mentioned theoretical solution set Ω. With the approximate expression, Eq. (8) can be rewritten as ZðαI ÞUI ¼ FðαI Þ:

ð10Þ e

The acoustic pressure P in the exterior field is clearly also an interval number with respect to the interval parameter vector αI : ðP e ÞI ¼ cT UI :

ð11Þ

3. First-order interval parameter perturbation method (FIPPM) The first-order Taylor expansion for a linear function is known to be accurate, and its accuracy for non-linear functions is also acceptable if the interval parameter ranges are narrow enough. Ignoring high-order terms, the coefficient matrix and right-hand vector in Eq. (10) can be expanded into the following approximate expressions at the midpoints of the interval parameters: m ∂Zðαc Þ ∂Zðαc Þ I ðαi  αci Þ ¼ Zðαc Þ þ ∑ Δαi δi ¼ Zc þ ΔZI ; i ¼ 1 ∂αi i ¼ 1 ∂αi m

ZðαI Þ ¼ Zðαc Þ þ ∑

444

C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449 m ∂Fðαc Þ ∂Fðαc Þ I ðαi αci Þ ¼ Fðαc Þ þ ∑ Δαi δi ¼ Fc þΔFI : i ¼ 1 ∂αi i ¼ 1 ∂αi m

FðαI Þ ¼ Fðαc Þ þ ∑

ð12Þ

The response interval UI can be expressed as UI ¼ Uc þΔUI in interval mathematics. By substituting Eq. (12) into Eq. (10), one subsequently yields ðZc þΔZI ÞðUc þ ΔUI Þ ¼ Fc þ ΔFI :

ð13Þ

Ignoring the high-order cross terms, we can obtain the following equations based on the first-order perturbation theory: Zc Uc ¼ Fc ; Zc ΔUI þΔZI Uc ¼ ΔFI :

ð14Þ c

Considering that the matrix Z is always nonsingular, one obtains Uc ¼ ðZc Þ  1 Fc ; ΔUI ¼ ðZc Þ  1 ðΔFI  ΔZI ðZc Þ  1 Fc Þ:

ð15Þ

By substituting Eq. (15) into Eq. (11), we can derive the formula ðP e ÞI ¼ ðP e Þc þ ΔðP e ÞI ¼ cT ðUc þ ΔUI Þ ¼ cT ðZc Þ  1 Fc þ cT ðZc Þ  1 ðΔFI ΔZI ðZc Þ  1 Fc Þ;

ð16Þ

where ðP e Þc ¼ cT ðZc Þ  1 Fc ;

ð17Þ

ΔðP e ÞI ¼ cT ðZc Þ  1 ðΔFI  ΔZI ðZc Þ  1 Fc Þ:

ð18Þ

Using Eq. (12), Eq. (18) can be rewritten as " #   m ∂Fðαci Þ ∂Zðαci Þ c  1 c e I c 1 T ΔðP Þ ¼ c ðZ Þ  ðZ Þ F Δαi δi ¼ ΔP e U δ: ∑ ∂αi ∂αi i¼1

ð19Þ

It is clear that the partial derivative ∂ΔðP e ÞI =∂δi is a constant, which means that ΔðP e ÞI is a linear monotonic function with respect to δi . Therefore, the radius ΔP e can be calculated by     m   ∂Fðαci Þ ∂Zðαci Þ c  1 c  ðZ Þ F Δαi ; ð20Þ ΔP e ¼ ∑ cT ðZc Þ  1 ∂αi ∂αi i¼1 where jdj denotes the absolute value. Hence, the upper and lower bounds of the sound pressure at the selected point in the exterior acoustic field can be expressed as e

P ¼ ðP e Þc þ ΔP e ;

P e ¼ ðP e Þc  ΔP e :

ð21Þ

4. Modified interval parameter perturbation method (MIPPM) The interval extension problem can become so serious that the accuracy of the first-order Taylor expansion for a nonlinear function is unacceptable when many uncertain parameters exist with large interval ranges. Based on the work of Chen et al. (2009b) and Impollonia and Muscolino (2011), we present a modified Taylor expansion and a modified parameter perturbation method in this section. Using the rail generation method for an approximate surface, the non-linear interval coefficient matrix ZðαI Þ can be approximately expressed as m

~ I Þ  ðm  1Þ UZðαc Þ; ZðαI Þ ¼ ZðαI1 ; αI2 ; :::; αIm Þ ¼ ∑ Zðα i

ð22Þ

i¼1

where ~ I Þ ¼ Zðαc ; :::; αc ; αI ; αc ; :::; αc Þ; Zðα i 1 i1 i iþ1 m

i ¼ 1; 2; :::; m:

ð23Þ

~ I Þ at the midpoint of αI , the high-order terms are ignored, and an approximate expression of the Expanding the matrix Zðα i i interval coefficient matrix ZðαI Þ is obtained: " # " # ~ cÞ m m ∂Zðα I c c I i ~ Δαi ¼ Zc þ ΔZI ; ð24Þ Zðα Þ ¼ ∑ Zðαi Þ  ðm 1Þ U Zðα Þ þ ∑ i¼1 i ¼ 1 ∂αi

C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449

445

where ~ cÞ ~ cÞ m ∂Zðα ∂Zðα i i ΔαIi ¼ ∑ Δαi δi : ∂α i i¼1 i ¼ 1 ∂αi

m

m

~ c Þ  ðm 1Þ U Zðαc Þ; Zc ¼ ∑ Zðα i

ΔZI ¼ ∑

i¼1

ð25Þ

Similarly, the non-linear interval parameter vector FðαI Þ can be rewritten as "

# " # ~ cÞ m ∂Fðα c c I i ~ Δαi ¼ Fc þ ΔFI ; Fðα Þ ¼ ∑ Fðαi Þ  ðm  1Þ UFðα Þ þ ∑ i¼1 i ¼ 1 ∂αi I

m

ð26Þ

where ~ cÞ ~ cÞ m ∂Fðα ∂Fðα i i ΔαIi ¼ ∑ Δαi δi : i ¼ 1 ∂αi i ¼ 1 ∂αi

m

m

~ c Þ ðm  1Þ U Fðαc Þ; Fc ¼ ∑ Fðα i

ΔFI ¼ ∑

i¼1

ð27Þ

In terms of the modified first-order Taylor series, we can rapidly determine the interval characters of the non-linear coefficient matrix and vector, restraining the interval extension to a certain extent. By substituting Eqs. (24) and (26) into Eq. (10) and by pre-multiplying both sides with ðZc þ ΔZI Þ  1 , we can obtain Uc þ ΔUI ¼ ðZc þ ΔZI Þ  1 ðFc þ ΔFI Þ: c 1

If the spectral radius of ðZ Þ

ð28Þ I 1

c

ΔZ is less than 1, ðZ þ ΔZ Þ

can be expanded by the Neumann series:

1

ðZc þ ΔZI Þ  1 ¼ ðZc Þ  1 þðZc Þ  1 ∑ ð  ΔZI ðZc Þ  1 Þr :

ð29Þ

r¼1

Using Eq. (25), one obtains 1

1

r¼1

r¼1

~ cÞ ∂Zðα i ΔαIi ðZc Þ  1 i ¼ 1 ∂αi m

∑ ð  ΔZI ðZc Þ  1 Þr ¼ ∑

 ∑

!r

1

m

 ∑ ΔαIi Zi

¼ ∑

r¼1

i¼1

!r ;

ð30Þ

~ c ÞÞ=ð∂αi ÞÞðZc Þ  1 is a simplified expression. For different values of r, the terms in the summation sign can be where Zi ¼ ðð∂Zðα i rewritten in the following explicit forms: !r ¼ 1 m

 ∑ ΔαIi Zi i¼1 m

!r ¼ 2

m

¼  ∑ ΔαIi Zi ; i¼1

m

¼ ∑ ðΔαIi Zi Þ2 þ

 ∑ ΔαIi Zi i¼1

i¼1

m

∑ ΔαIi ΔαIj Zi Zj ; i; j ¼ 1 :i ¼ j

m

 ∑ ΔαIi Zi i¼1

!r ¼ 3

m

¼  ∑ ðΔαIi Zi Þ3 þ i¼1

m

ΔαIi ΔαIj ΔαIk Zi Zj Zk ; ∑ i; j; k ¼ 1 :i ¼ j ¼ k

::::::;

ð31Þ

where the expression :i ¼ j stands for any combination except i ¼ j. By substituting Eq. (30) and (31) into Eq. (29), one obtains m

1

ðZc þ ΔZI Þ  1 ¼ ðZc Þ  1 þðZc Þ  1 ∑ ∑ ð  ΔαIi Zi Þr þ ðZc Þ  1 i¼1r ¼1

m

∑ ΔαIi ΔαIj Zi Zj þ ⋯: i; j ¼ 1

ð32Þ

:i ¼ j r I If the condition ‖Δαi Zi ‖ o 1 is satisfied, the series ∑1 r ¼ 1 ð Δαi Zi Þ is convergent. The inversion of the interval matrix can then be approximated by retaining only the first two terms: m

1

ðZc þΔZI Þ  1  ðZc Þ  1 þ ðZc Þ  1 ∑ ∑ ð  ΔαIi Zi Þr i ¼ 1r ¼ 1

m  ΔαIi Zi ¼ ðZc Þ  1 þ ðZc Þ  1 ∑ EIi ; I i ¼ 1I þ Δαi Zi i¼1 m

¼ ðZc Þ  1 þ ðZc Þ  1 ∑ where EIi ¼ ð  ΔαIi Zi Þ=ðI þ ΔαIi Zi Þ.

ð33Þ

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C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449

Based on the interval algorithms, the following formulas for the median and radius about the interval matrix EIi can be obtained simply as     1 Δαi Zi Δαi Zi 1 Δαi Zi Δαi Zi  Eci ¼  þ ; ΔEi ¼  : ð34Þ 2 I Δαi Zi I þ Δαi Zi 2 I  Δαi Zi I þ Δαi Zi  By substituting Eqs. (28), (33) and (34) into Eq. (11), we can derive m

ðP e Þc þ ΔðP e ÞI ¼ cT ðZc Þ  1 Fc þ ∑ cT ðZc Þ  1 Eci Fc þ cT ðZc Þ  1 ΔFI i¼1

m

m

m

i¼1

i¼1

i¼1

þ ∑ cT ðZc Þ  1 ΔEIi Fc þ ∑ cT ðZc Þ  1 Eci ΔFI þ ∑ cT ðZc Þ  1 ΔEIi ΔFI :

ð35Þ

Neglecting the high-order cross small terms, we are left with m

ðP e Þc ¼ cT ðZc Þ  1 Fc þ ∑ cT ðZc Þ  1 Eci Fc ;

ð36Þ

i¼1

" e I

T

c 1

ΔðP Þ ¼ c ðZ Þ

I

m

ΔF þ ∑

i¼1

ΔEIi Fc þ

m

∑

i¼1

# Eci ΔFI

:

ð37Þ

Finally, using Eq. (27), Eq. (37) can be rewritten as " # ~ cÞ ~ cÞ m ∂Fðα m m m ∂Fðα j j Δαj δj þ ∑ ΔEi Fc δei þ ∑ Eci ∑ Δαj δj ΔðP e ÞI ¼ cT ðZc Þ  1 ∑ j ¼ 1 ∂αj i¼1 i¼1 j ¼ 1 ∂αj ( ¼ cT ðZc Þ  1

m

m

i¼1

j¼1

∑ ΔEi Fc δei þ ∑

" ~ cÞ ∂Fðα j ∂αj

m

þ ∑ Eci i¼1

# ~ cÞ ∂Fðα j ∂αj

) Δαj δj

¼ ΔP e δ:

ð38Þ

If δei and δj are considered independent from each other, it is clear that ΔðP e ÞI is a linear monotonic function with respect to δei and δj . Hence, the radius ΔP e can be expressed as   " # ~ c ~ cÞ  m m  m ∂Fðα  T c  1 ∂Fðαj Þ  j e c 1 c c T þ ∑ Ei ð39Þ ΔP ¼ ∑ jc ðZ Þ ΔEi F j þ ∑ c ðZ Þ Δαj :   ∂α ∂α j j i¼1 j¼1 i¼1 Therefore, the upper and lower bounds of the sound pressure at the selected point in the exterior acoustic field can be written as e

P ¼ ðP e Þc þ ΔP e ;

P e ¼ ðP e Þc  ΔP e :

ð40Þ

In the MIPPM, the non-linear matrix and vector are approximated by the modified Taylor expansion and the interval matrix inverse is calculated by retaining some high-order terms in the Neumann series. Therefore, the accuracy of the sound pressure bounds obtained with the MIPPM is higher than that calculated by the FIPPM, where only the first-order terms are considered. However, an additional numerical burden of computing the high-order terms is introduced in the MIPPM.

Fig. 2. The finite/boundary element mesh model.

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447

5. Numerical example Consider a 1.0 m  0.5 m rectangular-sectioned flexible plate with a thickness of 5 mm, as shown in Fig. 2. The structure is discretized into 200 quadrilateral elements, and the acoustic interface is discretized into the same number and type of elements. A normal harmonic excitation is imposed on the central point at the bottom, and the four vertices are set to be fixed. Due to production limitations and measurement error, the material properties and external load all contain a certain level of unpredictability. The exterior acoustic cavity is surrounded by air with a density ranging from 1.125 kg/m3 to 1.325 kg/m3, and the acoustic speed ranges from 330 m/s to 350 m/s. The density, elastic modulus and Poisson's ratio of the flexible plate are all considered to be uncertain-but-bounded parameters, i.e., ρA [2600,2800] kg/m3, EA [70,72] GPa, and μA [0.32,0.33]. The amplitude of the harmonic excitation fluctuates between 0.9 N and 1.1 N. Simulations of the 3D structural-acoustic system are carried out in Matlab R2012 on a 3.00 GHz Pentium(R) 4-CPU computer. For the nine selected frequencies, the lower bound (LB) and upper bound (UB) of the sound pressure amplitude at the observation point calculated by the FIPPM, MIPPM and Monte Carlo method (abbreviated as MCM) are listed in Table 1. The response ranges obtained by the MCM with 106 samples are used as reference results to validate the accuracy of the proposed methods, and the relative errors of the FIPPM and MIPPM are also listed in the table. It can be observed that the response bounds obtained by the FIPPM are different from those obtained by the MCM, and these differences are caused by the effects of neglecting certain terms in the Taylor series and perturbation calculations. However, the bounds yielded by the MIPPM are very close to the reference bounds. From a comparison of the computational errors, we know that the accuracy of the MIPPM, in which a modified Taylor expansion is introduced and the high-order terms are retained, is much higher than the accuracy of the FIPPM, where only the first-order Taylor series and first-order perturbation are considered. Computational cost is another index with which to evaluate the performance of these numerical methods. From the Table 1 Bounds and relative errors of the sound pressure amplitude results. Frequency

100 170 200 300 341 382 400 483 500

Bound

LB UB LB UB LB UB LB UB LB UB LB UB LB UB LB UB LB UB

MCM (Pa) (76.4 s)

5.37e  6 6.81e  6 6.59e  3 8.40e  3 6.76e  5 8.63e  5 5.55e  5 7.22e  5 1.25e  3 1.67e 3 2.72e 3 3.58e  3 1.84e  4 2.44e  4 6.00e  3 8.24e 3 7.26e  4 1.03e  3

FIPPM (0.37 s)

MIPPM (0.45 s)

Value (Pa)

Error (%)

Value (Pa)

Error (%)

5.05e  6 7.14e  6 6.20e  3 8.81e  3 6.36e  5 9.06e  5 5.21e  5 7.60e  5 1.17e  3 1.76e  3 2.55e  3 3.77e 3 1.73e  4 2.57e  4 5.60e  3 8.74e  3 6.75e  4 1.09e  3

5.96 4.85 5.92 4.88 5.92 4.98 6.13 5.26 6.40 5.39 6.25 5.31 5.98 5.33 6.67 6.07 7.02 5.83

5.31e  6 6.89e  6 6.51e  3 8.50e  3 6.68e  5 8.71e  5 5.48e  5 7.30e  5 1.23e  3 1.69e  3 2.69e  3 3.62e  3 1.82e  4 2.46e  4 5.92e  3 8.35e  3 7.17e  4 1.04e  3

1.12 1.17 1.21 1.19 1.18 0.93 1.26 1.11 1.60 1.20 1.10 1.12 1.09 0.82 1.33 1.33 1.24 0.97

Fig. 3. Bounds of the sound pressure amplitude in the frequency band ω ¼100–500 Hz.

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C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449

Fig. 4. Bounds of the sound pressure amplitude in the following partial frequency bands: (a) ω¼ 163–177 Hz; (b) ω¼ 334–348 Hz; (c) ω ¼374–388 Hz; and (d) ω ¼476–490 Hz.

computational execution time listed in Table 1, we can see that the MIPPM's execution is much shorter than that of the MCM but is a little longer than that of the FIPPM, and the exceeded time is implemented to approximate high-order terms. Compared with the improvement in accuracy, the increase in computational effort is deemed acceptable. The differences between the results obtained from the MIPPM and MCM are so small that the numerical results calculated by the MIPPM are completely reliable and greatly reduce the computational burden of very large samples. The intervals of the sound pressure amplitude at the observation point obtained by the MCM, FIPPM and MIPPM in the frequency band from 100 Hz to 500 Hz are plotted in Figs. 3 and 4. Compared with the frequency response intervals calculated by the FIPPM, the results given by the MIPPM match the reference results obtained by the MCM more precisely, which means the MIPPM is more appropriate for predicting the response ranges of an exterior acoustic field with uncertainbut-bounded parameters. To investigate the impact of uncertainties on the frequency response, the frequency evolutions of the sound pressure amplitude at the observation point are plotted in Fig. 5. The deterministic value is obtained under the assumption that the physical properties and external load are deterministic, while the mean value is calculated by the MIPPM in the case of uncertainties treated as interval variables. Compared with the deterministic values, it is apparent that the uncertainties in the system have an impact on the exterior acoustic field that cannot be ignored. In addition to the sound pressure amplitude, an obvious change in the characteristic frequencies is observed because the interval parameters considered here are closely related to the structural-acoustic dynamic stiffness matrix and the mass matrix described in Section 2. 6. Conclusions By combining interval analysis theory with FEM/BEM numerical techniques, this paper proposes two interval parameter perturbation methods, abbreviated as the FIPPM and MIPPM, for exterior acoustic field prediction with uncertain-butbounded parameters. The uncertainties in the material properties and external load are all fully considered to make the calculation more objective. In the FIPPM, the parameter matrix and vector are approximated by the first-order Taylor

C. Wang, Z. Qiu / Journal of Fluids and Structures 49 (2014) 441–449

449

Fig. 5. Frequency evolutions of the sound pressure amplitude.

expansion, and the response ranges are calculated by considering only the first-order terms. Unlike the FIPPM, the interval characters of the non-linear coefficient matrix and vector can be efficiently determined in the MIPPM by using the modified Taylor series. The interval matrix inverse in the MIPPM is conducted using a modified Neumann series in which some highorder terms are retained. The numerical results obtained for a flexible plate fully validate the feasibility and superiority of the proposed methods at evaluating exterior acoustic fields with interval uncertainties. Compared with the FIPPM, the MIPPM can obtain more accurate results and greatly reduces the large computational cost of Monte Carlo simulations. However, it should be noted that the additional computational burden of calculating the high-order terms in the matrix inverse is an inherent disadvantage of the MIPPM, especially in the presence of a large number of uncertain parameters. Acknowledgments The project is supported by the National Natural Science Foundation of China (No. 11002013), the Defense Industrial Technology Development Program (Nos. A2120110001 and B2120110011) and the 111 Project (No. B07009). References Allen, M.J., Vlahopoulos, N., 2000. Integration of finite element and boundary element methods for calculating the radiated sound from a randomly excited structure. Computers & Structures 77, 155–169. Antebas, A.G., Denia, F.D., Pedrosa, A.M., Fuenmayor, F.J., 2013. A finite element approach for the acoustic modeling of perforated dissipative mufflers with non-homogeneous properties. Mathematical and Computer Modelling 57, 1970–1978. Autrique, J.C., Magoules, F., 2007. Analysis of a conjugated infinite element method for acoustic scattering. Computers & Structures 85, 518–525. Berkel, L., Rao, S.S., 1997. Analysis of uncertain structure systems using interval analysis. AIAA Journal 35, 727–773. Chen, G., Zhao, G.Z., Chen, B.S., 2009a. Sensitivity analysis of coupled structural-acoustic systems subjected to stochastic excitation. Structural and Multidisciplinary Optimization 39, 105–113. Chen, S.H., Ma, L., Meng, G.W., Guo, R., 2009b. An efficient method for evaluating the natural frequency of structures with uncertain-but-bounded parameters. Computers & Structures 87, 582–590. Impollonia, N., Muscolino, G., 2011. Interval analysis of structures with uncertain-but-bounded axial stiffness. Computer Method in Applied Mechanics and Engineering 220, 1945–1962. James, K.R., Dowling, D.R., 2005. A probability the density function method for acoustic field uncertainty analysis. Journal of the Acoustical Society of America 118, 2802–2810. James, K.R., Dowling, D.R., 2008. A method for approximating acoustic-field-amplitude uncertainty caused by environmental uncertainties. Journal of the Acoustical Society of America 124, 1465–1476. James, K.R., Dowling, D.R., 2011. Pekeris waveguide comparisons of methods for predicting acoustic field amplitude uncertainty caused by a spatially uniform environmental uncertainty (L). Journal of the Acoustical Society of America 129, 589–592. Neumaier, A., 1990. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge. Qiu, Z.P., Chen, S.H., Elishakoff, I., 1996. Bounds of eigenvalues for structures with an interval description of uncertain-but-non-random parameters. Chaos Solitons & Fractals 7, 425–434. Qiu, Z.P., Elishakoff, I., 1998. Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis. Computer Method in Applied Mechanics and Engineering 152, 361–372. Qiu, Z.P., Xia, Y.Y., Yang, J., 2007. The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem. Computer Method in Applied Mechanics and Engineering 196, 4965–4984. Raveendra, S.T., 1999. An efficient indirect boundary element technique for multi-frequency acoustic analysis. International Journal for Numerical Methods in Engineering 44, 59–76. Sliva, G., Brezillon, A., Cadou, J.M., Duigou, L., 2010. A study of the eigenvalue sensitivity by homotopy and perturbation methods. Journal of Computational and Applied Mathematics 234, 2297–2302. Xia, B.Z., Yu, D.J., 2012. Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters. Journal of Sound and Vibration 331, 3774–3790. Xia, B.Z., Yu, D.J., Liu, J., 2013. Interval and subinterval perturbation methods for a structural-acoustic system with interval parameters. Journal of Fluids and Structures 38, 146–163.