a flexible polynomial expansion method for response analysis with random...

Research ArticleA Flexible Polynomial Expansion Method for Response Analysiswith Random Parameters

Rugao Gao ,1,2 Keping Zhou,1,2 and Yun Lin 1,2

1Central south university, School of resources and safety engineering, Hunan Changsha, Hunan, 410083, China2Hunan Key Laboratory of Mineral Resources Exploitation and Hazard Control for Deep Metal Mines, Changsha 410083, China

Correspondence should be addressed to Yun Lin; [email protected]

Received 20 July 2018; Revised 11 October 2018; Accepted 11 November 2018; Published 3 December 2018

Academic Editor: Michele Scarpiniti

Copyright © 2018 Rugao Gao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The generalized Polynomial Chaos Expansion Method (gPCEM), which is a random uncertainty analysis method by employingthe orthogonal polynomial bases from the Askey scheme to represent the random space, has been widely used in engineeringapplications due to its good performance in both computational efficiency and accuracy.But in gPCEM, a nonlinear transformationof random variables should always be used to adapt the generalized Polynomial Chaos theory for the analysis of random problemswith complicated probability distributions, whichmay introduce nonlinearity in the procedure of random uncertainty propagationas well as leading to approximation errors on the probability distribution function (PDF) of random variables. This paper aimsto develop a flexible polynomial expansion method for response analysis of the finite element system with bounded randomvariables following arbitrary probability distributions. Based on the large family of Jacobi polynomials, an Improved Jacobi ChaosExpansion Method (IJCEM) is proposed. In IJCEM, the response of random system is approximated by the Jacobi expansion withthe Jacobi polynomial basis whose weight function is the closest to the probability density distribution (PDF) of the randomvariable. Subsequently, the moments of the response can be efficiently calculated though the Jacobi expansion. As the IJCEMavoids the necessity that the PDF should be represented in terms of the weight function of polynomial basis by using the varianttransformation, neither the nonlinearity nor the errors on randommodelswill be introduced in IJCEM.Numerical examples on tworandom problems show that compared with gPCEM, the IJCEM can achieve better efficiency and accuracy for random problemswith complex probability distributions.

1. Introduction

Nowadays, design and optimization of engineering systemsheavily rely on numerical methods in industry. This isespecially true in the acoustic field, where both time andeconomic costs of experiment are particularly expensive. Inthe past decade, the uncertainty analysis methods in theprediction of acoustic field has been greatly developed [1–5]. Generally, uncertainty analysis methods can be classifiedas two distinct groups: probabilistic methods [6, 7] andnonprobabilistic methods [8–12]. The probabilistic methodis the optimal choice for uncertainty analysis once the PDFis derived [13].

The Monte Carlo method (MCM) is a simple and robustprobabilistic method, and thus it has been widely used forrandom analysis [14]. However, the convergence rate ofMCM

is rather slow and the corresponding computational cost ofMCM for the uncertainty analysis of massive engineeringprojects is tremendous. In order to improve the efficiencyof MCM, some advanced MCMs have been developed,such as the Latin hypercube sampling simulation method[15], the quasi Monte Carlo method [16], and the otherextended methods [17–19].The convergence rate of advancedMCMs will become faster than the ordinary MCM. However,additional restrictions are posed on the design of theseadvanced MCMs and their applicability is often limited[20]. Generally, in order to verify the accuracy of uncertainmethods, ordinaryMCM is still considered as themost robustsampling method.

The perturbation stochastic method is also a key categoryof probabilistic methods, where the response of randomsystems is expanded as the Taylor series around the mean

HindawiComplexityVolume 2018, Article ID 7471460, 14 pageshttps://doi.org/10.1155/2018/7471460

http://orcid.org/0000-0002-1840-5298http://orcid.org/0000-0002-8887-6343https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/7471460

2 Complexity

value of random variables and truncated at a certain order[21–23]. Typically, the first-order expansion is employed inTaylor series, as application of high-order Taylor expansiondoes not allow for significant improvements in accuracy.Due to the high computational efficiency, the perturbationstochasticmethod has been used in various engineering fields[24–26]. An inherent limitation of perturbation stochasticmethods is that it is limited to random problem with smalluncertainty level.

During the past decades, the generalized PolynomialChaos Expansion Method (gPCEM) has gained a largepopularity in the field of random uncertainty analysis [27].The development of gPCEM started with the work of R.Ghanem et al., who used Hermite polynomial as the basisfor the analysis of stochastic processes [28, 29]. Subsequently,the gPCEM which is an extension of Hermite expansionmethod, was derived based on the Askey Scheme [30–32]. Within gPCEM, different kinds of orthogonal poly-nomials can be chosen as basis to establish the surrogatemodel for the random response. Thus, the gPCEM canachieve higher accuracy for some random problems thanthe Hermite polynomial expansion, such as the randomproblems with Beta distribution and uniform distribution.The variance and standard deviation of gPCEM can beobtained efficiently when compared to the conventionalMCM. In addition, the law of approximation of generalizedpolynomial chaos expansion is rather mature [33–35]. Dueto the advantages mentioned above, the gPCEM is veryimportant probabilistic method for solving randomproblems[36–39].

Generally speaking, the gPCEM has been widely popu-larized in random problem analysis. However, it seems thegPCEMcanonly be directly employed to expand the responseof random systems with the so called basic random variable,whose PDF can be orthogonal with respect to the polynomialin Askey Scheme. For the random problems with non-basic random variable, a non-linear transformation shouldbe employed, through which the non-basic random variablecan be approximately represented as a nonlinear function ofbasic random variables. Nevertheless, two inherent issues willbe encountered in the transformation of nonbasic randomvariables. First, the non-linearity will be introduced byusing the nonlinear transformation, which may lead to slowconvergence of the expansion. Second, even if the high-ordernonlinear transformation is used to approximate the realPDF in terms of the weight function, some fitting errorsbetween the approximated PDF and the real PDF may beintroduced, especially when the real PDF of random variableis not a smooth function. If there is large fitting error, themoment of response obtained by gPCEM may be failed toconverge to the exact result. To decrease the fitting error byusing the nonlinear transformation, Multielement gPCEMis proposed by Xiu et al. The Multielement gPCEM canbe applied for random problems with arbitrary probabilitydistributions by discretizing the random space into piecewiserandom elements and construct the polynomial expansionlocally in each random element [39–43]. However, the dis-cretizing of probability distributions can lead to tremendouscomputational effort for the random problems with multiple

input variables [44]. As regarding the solution of engineeringproblems, the uncertain problems always involve complexprobability distribution as well as large number of variables.Thus, it is desirable to improve the application value ofgeneralized polynomial theory in stochastic problems, withfine ruggedness and strong adaptability.

Moreover, though the gPCEM has received much atten-tion lately, few engineering problems are analyzed withgeneralized polynomial chaos theory [45]. The analysis ofthe effects of vibration and noise is one of the key pointsfor the design of engineering systems with high level NVHperformance. During the last decades, lots of uncertaintyanalysis methods based on the perturbation theory have beenproposed for random response analysis of acoustic problem[46–50]. However, the perturbation based method can onlysolve random problems with small uncertainty level [50].Recently, in the interval and random analysis of acousticsystem, the Gegenbauer series was introduced to establishthe surrogate model [45]. The accuracy of Gegenbauer seriesexpansion is significantly improved.Nevertheless, theGegen-bauer series expansion method is only suitable for simpleuncertain problems, which may limit its applications in thereal acoustic problems.

The main purpose of this paper is to develop a powerfulpolynomial expansion method for the response analysis ofacoustic systems with random variables following arbitraryprobability distribution. As the uncertain parameter of acous-tic systems is always bounded in real engineering, the Jacobipolynomial that is defined in a bounded interval will beintroduced. Based on the Jacobi polynomial, an ImprovedJacobi Chaos Expansion Method (IJCEM) is proposed in thispaper. In IJCEM, the response of acoustic systems in the rangeof variation of randomvariables is approximated by the Jacobiexpansion. A polynomial selection criterion is established tofind the best polynomial basis from the large family of Jacobipolynomials for the approximation of response of acousticsystems with different probability distributions of randomvariables. The mean and variance of response are calculatedthrough the Jacobi expansion. For the random variable withBeta distribution, the mean and variance of Jacobi expansioncan be obtained analytically and expressed by the coefficientsof Jacobi expansion. For the random variable with non-Betadistribution, the mean and variance of Jacobi expansion areobtained by using the Monte Carlo simulation. Note that theJacobi expansion is a simple function, thus the computationalburden suffering from the Monte Carlo simulation of Jacobiexpansion is acceptable.The proposedmethod was employedto predict the response of acoustic systems with random vari-ables following complex probability distributions. Numericalresults have shown that the proposed IJCEM can achievehigher accuracy and efficiency than that of the gPCEM.

The main contents of this paper are as follows: Sec-tion 2 presents a stochastic model of acoustic response.Section 3 briefly summarizes the features and deficienciesof gPCEM analysis. In Section 4, the operation of ITJEM isillustrated. The characteristics and feasibility of this methodare described in Section 5. Section 6 summarizes the researchwork in this paper.

Complexity 3

2. Random Model of UncertainAcoustic Problems

The equilibrium equation affected by harmonics can beexpressed as

(K − 𝑘2M + j𝑘C) p = F (1)where j = √−1 is an imaginary unitary, 𝑘 = 𝜔/𝑐 is the wavenumber, 𝜔 is the angular frequency, 𝑐 is the acoustic speedof fluid, and K is the acoustic system stiffness matrix; M isthe acoustic system mass matrix; C is the acoustic systemdamping matrix; p stands for the sound pressure vector; Fstands for the acoustic system load vector. Equation (1) canbe queried in [47].

There are many inherent uncertainties in the engineeringsound field, such as uncertain physical property of acousticmedium caused by the variation of temperature and theuncertain velocity exciting produced by unpredictable envi-ronment. In most cases, these uncertainties can be modeledby random variables. Besides, the uncertain parameters ofacoustic systems are always bounded in practice and theuncertain parameters of acoustic field can be described asbounded random variables. Using the bounded randomvariable vector x = [𝑥1, 𝑥2, . . . , 𝑥𝐿], (1) can be rewritten as

Z (x)p (x) = F (x) (2)where F(x) is the random force vector and Z(x) is the

uncertain dynamic stiffness matrix, which can be expressedas follows:

Z (x) = K (x) − 𝑘 (x)2M (x) + j𝑘 (x)C (x) (3)where 𝑘(x) stands for the randomwave number andK(x),

M(x), and C(x) stand for the random stiffness, mass, anddamping matrix, respectively.

For convenience, we can transform the random variable𝑥𝑖 (𝑖 = 1, 2, . . . , 𝐿) into an unitary bounded random variable𝜉𝑖 (𝑖 = 1, 2, . . . , 𝐿) that is defined in [−1, 1]. The lineartransformation for a random vector x can be expressed as

x = x (𝜉) , 𝜉 = [𝜉1, 𝜉2, . . . , 𝜉𝐿]𝑥𝑖 = 𝑥𝑖 (𝜉𝑖) = 𝑥𝑖,0 + 𝑥𝑖,1𝜉𝑖, 𝑖 = 1, 2, . . . , 𝐿 (4)

where

𝑥𝑖,0 = 𝑥𝑖 + 𝑥𝑖2 ,𝑥𝑖,1 = 𝑥𝑖 − 𝑥𝑖2

(5)

𝑥𝑖 and 𝑥𝑖 stand for the upper and lower bounds of 𝑥𝑖,respectively. As a result, the response of the random modelof acoustic systems can be expressed as

p (𝜉) = Z−1 (x (𝜉)) F (x (𝜉)) (6)

3. gPCEM for Random Variables

In gPCEM, the Askey polynomial is used as the basis forinterpretation. Xiu et al. proposed that any second-orderprocess of L2 sense can be realized with such methods. [20].In order to analytically calculate the moment of response, thePDF in gPCEM is expressed by the polynomial basis weightfunction. For convenience, the probability distribution whosePDF is the same as one of the weight function of polynomialbasis in the Askey scheme is named as the basic probabilitydistribution in this paper.

3.1. gPCEM for RandomUncertainty Analysis with Basic Prob-ability Distribution. For the random problems with basicprobability distribution, the response of random system canbe expanded as

𝑌 (𝜉) = ∞∑𝑖=0

𝑐𝑖𝜙𝑖 (𝜉) (7)where 𝑐𝑖 represents the expansion coefficients to be

estimated and 𝜙𝑖(𝜉) denotes the generalized polynomial basisbelonging to the Askey scheme. The type of 𝜙𝑖(𝜉) varies fordifferent random problems. For random variables defined in[−∞,+∞], the bases are the generalized Hermite polynomi-als; for random variables defined in a semibounded domain[0, +∞], the bases are the generalized Laguerre polynomials;for random variables defined in a bounded interval [−1, 1],the bases are the Jacobi polynomials; andmore bases for othertypes of random variables can be found in [27].

Random variables of sound are usually constrained. Theone-dimensional Jacobi polynomials can be given as

𝐽𝛼,𝛽0 (𝜉) = 1,𝐽𝛼,𝛽1 (𝜉) = 12 (𝛼 + 𝛽 + 2) 𝜉 + 12 (𝛼 − 𝛽) ,𝐽𝛼,𝛽𝑛 (𝜉) = 𝐽

𝛼,𝛽𝑛−1 (𝜉) 𝐵0 + 𝐽𝛼,𝛽𝑛−2 (𝜉) 𝐶0𝐴0 , 𝑛 = 2, 3, . . .

(8)

where

𝐴0 = 2𝑛 (𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽 − 2)𝐵0 = (2𝑛 + 𝛼 + 𝛽 − 1)

⋅ {(2𝑛 + 𝛼 + 𝛽) (2𝑛 + 𝛼 + 𝛽 − 2) 𝜉 + 𝛼2 − 𝛽2}𝐶0 = −2 (𝑛 + 𝛼 − 1) (𝑛 + 𝛽 − 1) (2𝑛 + 𝛼 + 𝛽)

(9)

𝛼 and 𝛽 are the Jacobi parameters that satisfied 𝛼 >−1, 𝛽 > −1. For L-dimensional random variables, the Jacobipolynomials can be expressed as the tensor product of theone-dimension Jacobi polynomial, which can be given by

𝐽𝛼,𝛽𝑘1,...,𝑘𝐿

(𝜉) = 𝐿∏𝑖=1

𝐽𝛼,𝛽𝑘𝑖

(𝜉𝑖) , 𝑘𝑖 = 0, 1, 2, . . . (10)

4 Complexity

The Jacobi polynomials are orthogonal with respect to𝑤𝛼,𝛽(𝜉), which can be expressed as⟨𝐽𝛼,𝛽𝑖 (𝜉) , 𝐽𝛼,𝛽𝑘 (𝜉)⟩𝑤 = {{{

ℎ𝑘, 𝑘 = 𝑖0, 𝑘 ̸= 𝑖 (11)

where

ℎ𝑘= Γ (𝑘 + 𝛼 + 1) Γ (𝑘 + 𝛽 + 1) Γ (𝛼 + 𝛽 + 1)(2𝑘 + 𝛼 + 𝛽 + 1) Γ (𝑘 + 1) Γ (𝑘 + 𝛼 + 𝛽 + 1) Γ (𝛼 + 1) Γ (𝛽 + 1)

(12)

⟨⋅, ⋅⟩ denotes the ensemble average which is the innerproduct in the Hilbert space of the random variable 𝜉, whichcan be expressed as

⟨𝐽𝛼,𝛽𝑖 (𝜉) , 𝐽𝛼,𝛽𝑘 (𝜉)⟩𝑤= ∫1

−1𝑤(𝛼,𝛽) (𝜉) 𝐽(𝛼,𝛽)𝑖 (𝜉) 𝐽(𝛼,𝛽)𝑘 (𝜉) d𝜉

(13)

𝑤𝛼,𝛽(𝜉) is the weight function related to the Jacobipolynomial, which is given by

𝑤(𝛼,𝛽) (𝜉)= Γ (𝛼 + 𝛽 + 1)2𝛼+𝛽+1Γ (𝛼 + 1) Γ (𝛽 + 1) (1 + 𝜉)𝛼 (1 − 𝜉)𝛽

(14)

Based on the Jacobi polynomial, the response of interestfor a one-dimension problem can be rewritten as the Jacobiexpansion

𝑌 (𝜉) = ∞∑𝑖=0

𝑐𝑖𝐽𝛼,𝛽𝑖 (𝜉) (15)Based on the theory of the Jacobi expansion, the coeffi-

cient ci can be derived as

𝑐𝑖 = ⟨𝑌 (𝜉) , 𝐽𝛼,𝛽𝑖 (𝜉)⟩

⟨(𝐽𝛼,𝛽𝑖 )2 (𝜉)⟩= 1⟨(𝐽𝛼,𝛽𝑖 (𝜉))2⟩ ∫𝑌 (𝜉) 𝐽

𝛼,𝛽𝑖 (𝜉) 𝑤𝛼,𝛽 (𝜉) d𝜉

(16)

Since the weight function is equal to the PDF of randomvariables, the mean and variance of Y can be derived fromorthogonality analysis as follows [30]:

𝜇 (𝑌) = 𝑐0 (17)𝜎2 (𝑌) = ∞∑

𝑖=0

(𝑐𝑖)2 ℎ𝑖 − (𝑐0)2 (18)

3.2. gPCEM for Random Uncertainty Analysis with NonbasicProbability Distribution. In a wide range of engineeringapplications, the basic probability distribution is insufficientto describe the random process. To deal with other boundedprobability distributions, a nonlinear variable transformationis employed such that a complex probability distribution canbe represented by the weight function of polynomial basis.For instance, the nonlinear transformation for an arbitraryrandom variable x can be written as

𝑥 ≈ 𝑥 = 𝑏0 + 𝑏1𝜉 + 𝑏2𝜉2 + ⋅ ⋅ ⋅ (19)where 𝜉 is the random variables with basic probability

distribution and 𝑏0, 𝑏1, 𝑏2, . . . are the constant coefficients ofnonlinear transformation, which can be given by

To find 𝑏𝑘 (𝑘 = 0, 1, 2, . . .)min ∫1

−1{𝑃𝑋 (𝑥) − 𝑃�̃� (𝑥)} d𝑥 (20)

In the above equation, 𝑃𝑋(𝑥) is the original PDF of 𝑥 and𝑃�̃�(𝑥) is the PDF of function 𝑥.By using the variable transformation, the gPCEM can

also be extended to calculate the uncertain questions withnonbasic variables. Nevertheless, the computational effi-ciency of gPCEM will be seriously deteriorated when thevariable is transformed as a high-order polynomial functionof the basic random variable. The main reason is that thevariable transformation can introduce the nonlinearity inthe process of uncertainty propagation. Besides, the variabletransformations may lead to some fitting errors between theapproximated PDF and the original PDF, especially when thePDF of random variable is not a smooth function. If thereis large fitting error, the response average value and variancederived by gPCEMmay failed to converge to the exact result.

4. IJCEM for Random UncertaintyAnalysis of Sound with ArbitraryProbability Distributions

To avoid the variable transformation when using generalizedPolynomial Chaos theory to solve random problems withcomplicated probability distributions, an Improved JacobiChaos expansion Method (IJCEM) is developed in this sec-tion. In the developed method, the sound response is directlyexpanded in the application of the Jacobi polynomials. Thecore of the method is to realize the inversion of randomproblems in the acoustic system by polynomial expansion.

4.1. Determine Polynomial Parameters of Jacobi Expansion.Theoretically, all Jacobian polynomials defined on [−1, 1] canbe used for expansion and then invert the uncertain responseof the acoustic system, but the accuracy of the Jacobi methodwith different parameters of the corresponding polynomialbases may vary greatly as the weighted projection is a wayfor controlling the residual deviation of Jacobi method [30].Reference [24] has shown that if the type of polynomial isused to match the import variable distribution, the fastest

Complexity 5

PDF==9.5==5==0

00.20.40.60.8

11.21.41.61.8

PDF

and

Wei

ght f

unct

ions

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

Figure 1: PDF of a random variable and the weight function withdifferent polynomial parameters.

convergence rate will be implemented. In other words, theoptimal polynomial basis can be determined by the sameamount as the PDF for random problem analysis. However,in most engineering cases, the variable weighting value thatequals the PDF of uncertain variable cannot be found fromthe Askey sckeme. As a result, the weighting function ofpolynomials basis of the Jacobi method to solve a randomproblemmay be always deviated from the PDF of real uncer-tain variables if the variable transformation is not employed.In this paper, the deviation between the weight function andthe PDF is allowed and a polynomial selection criterion willbe established to choose the basis of Jacobi expansion. Beforeestablishing a polynomial selection criterion, the influenceof the residual between the variable weighting value andthe PDF on the expansion will be discussed. The degree ofresidual between the weighting dependent variable and thePDF of a random variable can be define by a proximity indexas follows:

𝛿 = 𝛿 (𝛼, 𝛽) = lg(∫Ω

𝑃𝑋 (𝑥) − 𝑤𝛼,𝛽 (𝑥)𝑤𝛼,𝛽 (𝑥)

d𝑥) (21)𝑃𝑋(𝑥) denotes the PDF of x; 𝑤(𝛼,𝛽)(𝑥) is the weight

function related to the Jacobi polynomial with the Jacobiparameters 𝛼 and 𝛽; 𝛿 denotes the proximity index of𝑤(𝛼,𝛽)(𝑥) for 𝑃𝑋(𝑥).

A simple mathematic example will be considered toinvestigate the effect of 𝛿 on the convergence behavior ofJacobi expansion for random uncertainty qualification.

Example. Consider a function 𝑦 = tan 𝑥, where x is auncertainty limited in [−1, 1]. The probability distributionfunction of x is shown Figure 1. The Jacobi expansion withdifferent polynomial parameters are used to approximate thefunction y, and then themean and variance of y are calculatedthrough MCM of y. Three cases when the Jacobi parametersare 𝛼 = 𝛽 = 0, 𝛼 = 𝛽 = 5 and 𝛼 = 𝛽 = 9.5, will beconsidered. For comparison, the weight function of Jacobi

Table 1: Proximity index of the weight functions with differentJacobi parameters for a PDF.

Polynomial parameters 𝛼 = 𝛽 = 9.5 𝛼 = 𝛽 = 5.5 𝛼 = 𝛽 = 0Proximity index 𝛿 0.9 4.6 9.7

polynomials related to each group of Jacobi parameters isalso plotted in Figure 1. According to Eq. (21), the proximityindex of theweight functionswith different Jacobi parametersfor the PDF shown in Figure 1 can be obtained, which arelisted in Table 1. The Jacobi expansions with different Jacobiparameters are used to derive the mean and square deviationof y. The convergence behavior of the expansion is shown inFigure 2.

From Figure 2, we can find that the convergence speedof Jacobi expansion is improved with the decrease of 𝛿. Theerror of the mean and variance of y calculated by usingJacobi expansion with any 𝛿 can eventually decrease to 1e-03 when the retained order is up to 7. Therefore, we canconclude from Figure 2 that (1) the Jacobi expansion with anyJacobi parameters can achieve high accuracy if the reservedsequences are large enough; (2) the converge speed of Jacobiexpansion method can be improved by reducing 𝛿.

Based on the above conclusions, we can define thepolynomial selection criterion by an optimization solution asfollows:

min 𝛿 (𝛼, 𝛽)s.t. 𝛼, 𝛽 > −1 (22)

Through the above optimization procedure, the Jacobiparameters of the Jacobi expansion for random uncertaintyanalysis of acoustic systems with arbitrary probability distri-butions can be obtained.

4.2. Invert Response Law by Establishing Jacobian Extension.Random sound models can be fitted through the evolution ofrelated parameters of variables, as follows:

𝑝𝑘 (𝜉) = ∞∑𝑖1=0

⋅ ⋅ ⋅ ∞∑𝑖𝐿=0

𝑓𝑘𝑖1,...,𝑖𝐿𝐽(𝛼,𝛽)𝑖1,...,𝑖𝐿 (𝜉) ,𝑘 = 1, 2, . . . , 𝑁𝑡𝑜𝑡

(23)

where 𝑝𝑘(𝜉) (𝑘 = 1, 2, . . . ,𝑁𝑡𝑜𝑡) denotes the soundpressure of acoustic field at the kth node,𝑁𝑡𝑜𝑡 is the numberof nodes of the sound field, and 𝑓𝑘𝑖1 ,...,𝑖𝐿 is the coefficientof Jacobi expansion to be estimated. In practice, the Jacobiexpansion should be truncated at a finite order. Generally,the Jacobi expansion can be truncated by using total orderexpansion theory or the tensor order expansion theory.The tensor product expansion theory can effectively solveuncertain problems when the relevant retention order of

6 Complexity

=0.9=4.6=9.7

2 3 4 5 6 71Retained order

10−4

10−3

10−2

10−1

100Re

lativ

e err

or

(a)

=0.9=4.6=9.7

2 3 4 5 6 71Retained order

10−8

10−6

10−4

10−2

100

Rela

tive e

rror

(b)

Figure 2: Convergence behavior of Jacobi expansion with different Jacobi parameters for the calculation of the mean and variance of y: (a)mean; (b) variance.

random variable changes [50]. The Jacobi expansion for theresponse of random acoustic model can be expressed as

𝑝𝑘 (𝜉) = 𝑁1∑𝑖1=0

⋅ ⋅ ⋅ 𝑁𝐿∑𝑖𝐿=0

𝑓𝑘𝑖1,...,𝑖𝐿𝐽(𝛼,𝛽)𝑖1,...,𝑖𝐿 (𝜉) ,𝑘 = 1, 2, . . . , 𝑁𝑡𝑜𝑡

(24)

where 𝑁𝑖 (𝑖 = 1, 2, . . . , 𝐿) is the retained order of Jacobiexpansion related to 𝜉𝑖.

For the type of tensor product expansion, the Gaussianintegral method is very suitable to estimate the expansioncharacteristics [51]. Generally, the Gauss-Jacobi integrationmethod is able to realize a reliable precision for the estimationof the expansion coefficient of Jacobi expansion When theGauss-Jacobi integration point is equal to the expansionamount [35]. The coefficient 𝑓𝑘𝑖1 ,...,𝑖𝐿 can be derived from theGauss-Jacobi integral as

𝑓𝑘𝑖1 ,...,𝑖𝐿 ≈ 1ℎ1 × ⋅ ⋅ ⋅ × ℎ𝐿𝑚1∑𝑗1=1

⋅ ⋅ ⋅ 𝑚𝐿∑𝑗𝐿=1

𝑝𝑘 (𝜉𝑗1 , . . . , 𝜉𝑗𝐿)⋅ 𝐽(𝛼,𝛽)𝑖1,...,𝑖𝐿 (𝜉𝑗1 , . . . , 𝜉𝑗𝐿)𝐴𝑗1 ,...,𝑗𝐿

(25)

where (𝜉𝑗1 , . . . , 𝜉𝑗𝐿) are the interpolating points andmi (i = 1, 2, . . . , L) is the amount of points of 𝜉𝑖. To reducethe deviation, the parameter mi is set as Ni+1. 𝐴𝑗1,𝑗2,...,𝑗𝐿 isthe weighting value of Gauss-Jacobi integration, which canbe expressed as

𝐴𝑗1 ,...,𝑗𝐿 = 𝐴𝑗1 × 𝐴𝑗2 × ⋅ ⋅ ⋅ × 𝐴𝑗𝐿 (26)

In the above equation, the weight 𝐴𝑗𝑖 (i = 1, 2, . . . ,L) isgiven by [35]

𝐴𝑗𝑖 = 2𝛼+𝛽+1 Γ (𝑚𝑖 + 𝛼 + 1) Γ (𝑚𝑖 + 𝛽 + 1)Γ (𝑚𝑖 + 1) Γ (𝑚𝑖 + 𝛼 + 𝛽 + 1) (1 − 𝑥2𝑗)−1

⋅ {𝑃(𝛼,𝛽)𝑚𝑖 (𝑥𝑗)}−2 , 𝑖 = 1, 2, . . . , 𝑚𝑖(27)

4.3. Estimation of Average and Variance. After the coefficientof Jacobi expansion is obtained, the average and variance ofthe response can be calculated approximately through theJacobi method. The weight function of a Jacobi polynomialbasis can be the same as the PDF of random variable. In thiscase, the IJCEM is identical to the traditional gPCEM. Thus,the mean and variance can be calculated as [30]

𝜇𝑝𝑘 = 𝑓𝑘0,0,...,0 (28)𝜎2𝑝𝑘 = E [(𝑝𝑘)2] − (𝜇𝑝𝑘)2

= 𝑁1∑𝑖1=0

⋅ ⋅ ⋅ 𝑁𝐿∑𝑖𝐿=0

(𝑓𝑘𝑖1 ,...,𝑖𝐿)2 ℎ𝑖1,...,𝑖𝐿 − (𝑓𝑘0,0,...,0)2(29)

For the random problems with nonbasic probabilitydistribution, the mean and variance of response cannot beobtained analytically. To obtain the mean and variance ofJacobi expansion with uncertain variables, some numericalsolvers can be utilized. In this paper, the mean and varianceof the Jacobi expansion related to the random variablewith nonbasic probability distribution are calculated throughMonte Carlo simulation due to its simplicity. The additionalcomputational cost suffering from MCM can be accepted inmost practical engineering applications due to the simplifica-tion of the function.

Sometimes, both basic and nonbasic probability dis-tributions may exist simultaneously. The response analysis

Complexity 7

of random problems with basic and nonbasic probabilitydistributions includes two main steps. First, the randomvariable with nonbasic probability distribution is viewed asa fixed value, and the conditional mean and variance ofJacobi expansion with basic probability distributions can beobtained. For clarity, the Jacobi expansion with basic andnonbasic probability distributions can be expressed as

𝑝𝑘 = 𝑀1∑𝑗1=1

⋅ ⋅ ⋅ 𝑀𝐿2∑𝑗𝐿2=1

( 𝑁1∑𝑖1=0

⋅ ⋅ ⋅ 𝑁𝐿1∑𝑖𝐿1=0

𝑓𝑘𝑖1 ,...,𝑖𝐿1 ,𝑗1,...,𝑗𝐿2𝐺𝑖1,...,𝑖𝐿1 (𝜂))

⋅ 𝐺𝑗1,...,𝑗𝐿2 (𝛽) =𝑀1∑𝑗1=1

⋅ ⋅ ⋅ 𝑀𝐿2∑𝑗𝐿2=1

𝑧𝑘𝑖1 ,...,𝑖𝐿1 ,𝑗1,...,𝑗𝐿2𝐺𝑗1,...,𝑗𝐿2 (𝛽)(30)

where

𝑧𝑘𝑖1 ,...,𝑖𝐿1 ,𝑗1,...,𝑗𝐿2 =𝑁1∑𝑖1=0

⋅ ⋅ ⋅ 𝑁𝐿1∑𝑖𝐿=0

𝑓𝑘𝑖1 ,...,𝑖𝐿1 ,𝑗1,...,𝑗𝐿2𝐺𝑖1,...,𝑖𝐿1 (𝜂) (31)In above equations, 𝜂 and 𝛽 denote the non-basic and

basic randomvariable, respectively; 𝐿1 is the number of 𝜂 and𝐿2 for 𝛽.Owing to the orthogonality of Jacobi polynomials, the

mean and variance of 𝑝𝑘 related to the basic random variable𝛽 are shown below

𝜇 (𝜂) = 𝑁1∑𝑖1=0

⋅ ⋅ ⋅ 𝑁𝐿1∑𝑖𝐿1=0

𝑓𝑘𝑖1 ,...,𝑖𝐿1,0...,0𝐺𝑖1,...,𝑖𝐿1 (𝜂) (32)𝜎2 (𝜂)

= 𝑀1∑𝑗1=1

⋅ ⋅ ⋅ 𝑀𝐿2∑𝑗𝐿2=1

( 𝑁1∑𝑖1=1

⋅ ⋅ ⋅ 𝑁𝐿1∑𝑖𝐿1=1

𝑓𝑘𝑖1 ,...,𝑖𝐿1 ,𝑗1,...,𝑗𝐿2𝐺𝑖1,...,𝑖𝐿1 (𝜂))2

⋅ ℎ𝑗1,...,𝑗𝐿2 − (𝑁1∑𝑖1=0

⋅ ⋅ ⋅ 𝑁𝐿1∑𝑖𝐿1=0

𝑓𝑘𝑖1 ,...,𝑖𝐿1,0...,0𝐺𝑖1,...,𝑖𝐿1 (𝜂))2

(33)

Then, the unconditional mean 𝜇 and the mean squarevalue (𝜇2 + 𝜎2) of the response over the complete randomensemble can be evaluated using the laws of conditionalprobability

𝜇 = ∫𝜇 (𝜂) 𝑝 (𝜂) 𝑑𝜂 = E [𝜇 (𝜂)] (34)𝜇2 + 𝜎2 = ∫ [𝜇2 (𝜂) + 𝜎2 (𝜂)] 𝑝 (𝜂) 𝑑𝜂

= E [𝜇2 (𝜂)] + E [𝜎2 (𝜂)] (35)where 𝜂 are now considered to be random and described

by a joint PDF 𝑝(𝜂). Equation (35) can be rewritten as𝜎2 = E [𝜇2 (𝜂)] − E2 [𝜇 (𝜂)] + E [𝜎2 (𝜂)] (36)

By substituting 𝜎2[𝜇(𝜂)] = E[𝜇2(𝜂)] − E2[𝜇(𝜂)] into (36),the unconditional variance is shown below

𝜎2 = 𝜎2 [𝜇 (𝜂)] + E [𝜎2 (𝜂)] (37)

Subsequently, the mean and variance expressed in (34)and (37) can be derived with the Monte Carlo method. Inparticular, each item of (34) and (37) can be calculated by (32)or (33).

4.4. Procedure of IJCEM for Random Analysis of Soundwith Arbitrary Probability Distributions. The procedures ofIJCEM for the response analysis of acoustic systems withrandom parameters are shown below:

(1) Calculating the Jacobi parameter 𝛼 and 𝛽 with respectto each random variable through the optimization solutionshown in (22);

(2) The expansion coefficient is solved by (25)(3) The integral weight is solved by (27)(4) The solution of the response is given by (6)(5) Calculate the mean and variance of response of sound

field with different types of random variables. For the randomvariables with basic distributions, the mean and varianceof responses are calculated through (28) and (29); for therandom variables with nonbasic distributions, the mean andvariance of responses are calculated through the MonteCarlo simulation; for the random variables with both basicand nonbasic distributions and the mean and variance ofresponses are calculated through (34) and (37).

5. Numerical Examples

5.1. Acoustic Tube

5.1.1. Random Model of an Acoustic Tube. As shown inFigure 3, a rigid two-dimensional sound tube finite elementmodel is built and intermittent loads are applied to the insideof the model.

Considering the unpredictability of the environmenttemperature, 𝜌, 𝑐, and V𝑛 are considered as uncertain factors.For simplicity, these factors are considered as a function offixed variables within a certain interval.

𝑐 = (340 + 30𝜉1)m/s𝜌 = [1.225 + 0.367 (0.35𝜉2 + 0.15𝜉32)] kg/m3V𝑛 = [0.01 + 0.003 (0.15𝜉3 + 0.35𝜉53)]m/s

(38)

In the above equations, 𝜉1, 𝜉2, and 𝜉3 are unitary randomvariables with Beta distribution. The equation for the Betadistribution is

𝑃(𝑎,𝑏) (𝜉) = Γ (𝑎 + 𝑏 + 1)2𝑎+𝑏+1Γ (𝑎 + 1) Γ (𝑏 + 1) (1 + 𝜉)𝑎 (1 − 𝜉)𝑏 ,−1 ≤ 𝜉 ≤ 1

(39)

where a and b are the Beta parameters, the Beta param-eters of 𝜉1, 𝜉2, and 𝜉3 are 𝑎1 = 𝑏1 = 4.5, 𝑎2 = 𝑏2 = 3, and𝑎3 = 𝑏3 = 5, respectively.5.1.2. Comparison of the Accuracy of IJCEM and gPCEMunder the SameRetainedOrder. According to the polynomialselection criterion of IJCEM, the Jacobi parameters of the

8 Complexity

1m

Central axis x

0.1m x

n

Figure 3: Mesh distribution of two-dimensional model.

Reference solutionIJCEMgPCEM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10x (m)

−15

−10

−5

0

5

10

15

Mea

n of

soun

d pr

essu

re (P

a)

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10x (m)


0

1

2

3

4

5

6

Varia

nce o

f sou

nd p

ress

ure (

P；2)

(b)

Figure 4: Mean and variance of the sound pressure’s imaginary parts of acoustic tube along the central axis at f=150Hz: (a) mean and (b)variance.

Jacobi expansion for 𝜌, 𝑐, and V𝑛 can be obtained, namely:𝛼𝜌 = 𝛽𝜌 = 4.5, 𝛼𝑐 = 𝛽𝑐 = 3.7, and 𝛼V𝑛 = 𝛽V𝑛 = 7.2. Arelative improvement criterion will be introduced for randomuncertainty analysis with IJCEM [51]. For multivariate ran-domproblems, the relative improvement of responses is givenby [51]

Ir (k, 𝑗) = 𝑈 (k + e𝑗) − 𝑈 (k)

𝑈 (k) (40)

where k = [𝑘1, 𝑘2, . . . , 𝑘𝐿] is the retained order vector, 𝑘𝑗jth retention order, and𝑈(k) denotes the response calculatedby using the Jacobi expansion with the k th retained orders.e𝑗 is a L-dimensional vector in which only the jth elementis equal to 1, while the others are equal to zero. IJCEM isconsidered to be converged when Ir(k, 𝑗) (j = 1, 2, . . . , L)is smaller than the customary prescribed tolerance. Moredetailed procedure of the promotion criteria for determiningthe retained order can be found in [51]. Assuming that thepermissible relative error is 10−2, by using the promotioncriteria, the order of IJCEM for random analysis of the modelat f=150Hz and f=300Hz is

N𝑟𝑒𝑞 = [𝑁𝜌, 𝑁𝑐, 𝑁V𝑛] = [1, 2, 1] (41)The results of the sound pressure calculation with the

retained order [1, 2, 1] are shown in Figures 4 and 5, respec-tively. The reference solution is obtained by using the MCM,

in which the number of sampling points is 105. Note that themean and variance of Jacobi expansion are also calculatedby using the Monte Carlo simulation. In the Monte Carlosimulation of Jacobi expansion, the number of samplingpoints is also 105. Besides, to show the accuracy of IJCEMand gPCEMmore clearly, the relative errors of the mean andvariance of responses integrated in different ways at f=150Hzare listed in Table 2.

From Figures 4 and 5 and Table 2, we can find that theIJCEM is a high precisionmethod. Besides, Table 2 has shownthat the maximum value of relative error yielded by IJCEM isno more than 2%, which retains high reliability. It illustratesthe feasibility of applying the relative improvement criteria.By a comparison of the results obtained through the IJCEMand the gPCEM, it is proved that IJCEM is superior to gPCEMin the same condition. From Table 2, the relative error of thevariance of responses yielded by gPCEM is more than 10%.It indicates that higher order polynomial should be retainedwhen the gPCEM is used to derive the variance of responseof the model with random variables.

5.1.3. Comparison of the Efficiency of IJCEM and gPCEMunder the Same Accuracy. For gPCEM, the required retainedorder can also be verified by using optimization guidelines.For random analysis of the acoustic tube at f=150Hz, therequired retained order of gPCEM estimated is [2, 4, 3].

Complexity 9


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10x (m)

−8

−6

−4

−2

0

2

4

6

8M

ean

of so

und

pres

sure

(Pa)

(a)


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10x (m)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Varia

nce o

f sou

nd p

ress

ure (

Pa)

(b)

Figure 5: Mean and variance of the sound pressure’s imaginary parts of acoustic tube along the central axis at f=300Hz: (a) mean and (b)variance.

Table 2: Relative error of mean and variance of the sound pressure’s imaginary parts.

x Relative error of mean Relative error of varianceIJCEM gPCEM IJCEM gPCEM

0m 0.01% 0.01% 1.1% 13.4%0.2m 0.01% 0.01% 0.6% 13.0%0.5m 0.01% 0.01% 1.3% 13.5%1m 0.01% 0.01% 1.0% 13.2%

Table 3: Relative error of variance of the sound pressure’s imaginary parts yield by gPCEM.

MCM(Pa) gPCEM(Pa) Relative error0m 1.568 1.554 0.9%0.2m 0.195 0.193 1.0%0.5m 1.211 1.196 1.2%1m 1.102 1.087 1.4%

The variances of responses of the acoustic tube obtained bygPCEM are listed in Table 3.

From Table 3, we can find that the maximum valueof relative errors yielded by gPCEM is 1.4%. Thus, we canconclude that the criterion can be also used in gPCEM.Besides, the results shown in Table 3 indicate that the gPCEMcan also achieve high accuracy for the response analysis ofacoustic tube with random variables if the retained order issufficiently large. However, the required retained order ofgPCEM is higher than that of the IJCEM when the sameaccuracy is achieved.

The detailed execution time of gPCEM and IJCEM tocalculate the response of acoustic tube at f=150Hz is listedin Table 4. It can be found from Table 4 that the compu-tational efficiency of IJCEM is greatly improved comparedwith the gPCEM. The main reason is that the requiredretained order of IJCEM is lower than that of the gPCEM,thus the corresponding computational time to constructthe polynomial expansion can be decreased compared with

the gPCEM. Besides, it can be seen from Table 4 that thecomputational time of Monte Carlo simulation of Jacobiexpansion is much shorter than the total execution time ofIJCEM. It indicates that the main computational burden ofthe proposed IJCEM for random problems suffers from theconstruction of the polynomial expansion. In other words,the additional time encountered from MCM of the Jacobiexpansion is acceptable.

5.2. Acoustic Cavity of a Car. A finite element model wasbuilt, as shown in Figure 6.The acoustic cavity is surroundedby air with the density 𝜌 and the acoustic speed 𝑐. At thefront windshield, the admittance coefficient along the Robinboundary is 𝐴𝑛. According to the characteristics of the frontengine, the discontinuous normal velocity V𝑛 is imposed atthe outside of the vocal cavity. The remaining edges areperfectly rigid. The finite element method is used to analyzethe sound pressure of the acoustic cavity of the car.

10 Complexity

Table 4: Execution time comparison.

Methods Retained orderExecution time of theconstruction of the

polynomial expansion

Execution time of the MonteCarlo simulation of thepolynomial expansion

Total execution time

gPCEM [2, 4, 3] 60.2s 0s 60.5sIJCEM [1, 2, 1] 12.8s 2.5s 15.3s

2.672m

1.072mAn

A

n

Figure 6: The 2D finite element mesh model of the acoustic cavity of a car.

The random parameters 𝜌, 𝑐,𝐴𝑛, and V𝑛 are assumed as afunction of the unitary random variable defined in [−1, 1] asfollows:

Density of air: 𝜌 = 1.225 + 0.367𝜉4Speed of sound: 𝑐 = 340 + 20.5𝜉4Admittance coefficient: 𝐴𝑛 = 0.0014 + 0.00042𝜉4Velocity of exciting: V𝑛 = 0.01 + 0.003𝜉5

In order to compare the accuracy of the proposedmethodand gPCEM for the response analysis of random acousticproblems with complex PDFs, the PDF of 𝜉4 and 𝜉5 areassumed as a piecewise function. The original PDFs of 𝜉4 and𝜉5 are shown in Figure 7. For 𝜉4 and 𝜉5, the weight functionfrom Askey scheme is difficult to approximate the originalPDF. As a result, some errors may be introduced on thePDF of random variables by using gPCEM. For comparison,the PDF approximated by the weight function of polynomialbasis in gPCEM is also plotted in Figure 7.

The proposed IJCEM and the gPCEM are employed tocalculate the response of acoustic cavity of the car with ran-domvariables.Theprescribed accuracy is set as 10−2 . By usingthe relative improvement criterion, the required retainedorders of IJCEMand gPCEM for randomuncertainty analysisof the acoustic cavity of the car can be obtained as follows:

N𝐼𝐽𝐶𝐸𝑀𝑟𝑒𝑞 = [𝑁𝜌,𝑁𝑐,𝑁𝐴𝑛 , 𝑁V𝑛] = [1, 2, 1, 1] ,N𝑔𝑃𝐶𝐸𝑀𝑟𝑒𝑞 = [𝑁𝜌,𝑁𝑐,𝑁𝐴𝑛 , 𝑁V𝑛] = [1, 3, 1, 1] .

(42)

In the proposed IJCEM, the Jacobi parameters obtainedaccording to the polynomial selection criterion are 𝛼1 =𝛽1 = 6.2 for 𝜌, 𝑐, and 𝐴𝑛 and 𝛼2 = 𝛽2 = 5.5 for V𝑛. Thesimulation results obtained by IJCEMand gPCEMare plottedin Figure 8 at 100Hz and Figure 9 at 200Hz. The reference

results obtained by using the MCMwith 105 sampling pointsare also plotted in Figures 8 and 9.

From Figures 8 and 9, we can find that the IJCEMcan complete high precision calculation of acoustic cavityof car with random variables following piecewise PDFs. Incontrast, the variance of response obtained by the gPCEMdeviates significantly from the reference solution. The mainreason is that the approximated PDF is used in gPCEM tocalculate the mean and variance of the response of acousticsystems. Therefore, we can conclude that that the proposedIJCEM ismore suitable for the randomuncertainty analysis ofacoustic problems than the gPCEM, especially when the PDFof random variables cannot be approximated by the weightfunction perfectly.

6. Conclusion

On the basis of the large family of Jacobi polynomials, theIJCEM is proposed to solve the random acoustic problemwith arbitrary probability distributions. Jacobi expansion isused to fit the random problem. The coefficients of Jacobiexpansion are calculated by the Gauss-Jacobi integrationdue to its robustness as well as its good efficiency for theestimation of the coefficients of tensor order polynomialexpansion. By determining the effective polynomial selectioncriteria, the correlation weight function and the polynomialbasis are obtained, and the response result can be effectivelycalculated. As the PDF of the random variable is not requiredto be expressed as the weight function of polynomial basis,the proposed IJCEM can avoid nonlinear transformation ofvariables for random problems with any complicated PDFs.Another advantage of the proposed IJCEM is that only theJacobi parameters should be changed to construct the Jacobiexpansion for the response analysis of acoustic systems with

Complexity 11

Real PDFApproximated PDF in gPCEM

0

0.2

0.4

0.6

0.8

1

1.2

1.4PD

F

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

(a)

Real PDF Approximated PDF in gPCEM

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

PDF

(b)

Figure 7: The original PDF of random variables and the approximated PDF in gPCEM: (a) 𝜉4 and (b) 𝜉5.


0.5 1 1.50x (m)

−1

−0.5

0

0.5

1

1.5

Mea

n of

imag

inar

y pa

rt o

f sou

nd p

ress

ure (

Pa)

(a)

Reference solutiongPCEMIJCEM

0.5 1 1.50x (m)

0

0.05

0.1

0.15

0.2

0.25

Varia

nce o

f im

agin

ary

part

of s

ound

pre

ssur

e (P；

2)

(b)

Figure 8: Mean and variance of sound pressure of car along the bottom boundary at f=100Hz: (a) mean and (b) variance.

different random variables. Thus, the proposed method canbe easily processed in the numerical implementation.

Two typical acoustic problems are used to investigate theeffectiveness of the proposed methodology. The numericalresults show that both IJCEM and gPCEM can achieve adesirable accuracy for the random uncertainty analysis ofacoustic systems when the PDF of random variables can berepresented in terms of the weight functions of polynomialbasis. However, the computational efficiency of gPCEM islower than that of the proposed IJCEM as the non-lineartransformation is used for random variables in gPCEM. Inaddition, it can be concluded from the numerical studyof random acoustic problems with complicated piecewiseprobability distributions that the accuracy of the gPCEMmay

deteriorate when some errors are introduced for PDFs byusing the transformation of random variables. In contrast,the proposed IJCEM can achieve high accuracy for therandom problem with arbitrary probability distributions ifthe retained order of Jacobi expansion is sufficiently large.Note that the mean and variance of Jacobi expansion arecalculated by the Monte Carlo simulation when the weightfunction of polynomial basis is not identical to the PDF ofrandomvariables.However, comparedwith the improvementof the accuracy and efficiency in constructing the Jacobiexpansion, the additional computational cost suffering fromtheMonte Carlo simulation of Jacobi expansion is acceptable,especially for large scale engineering problems. It should bealso pointed out that the proposed IJCEM is not limited to the

12 Complexity

Reference solution IJCEMgPCEM

0.5 1 1.50x (m)

−4

−3

−2

−1

0

1

2

3

4M

ean

of im

agin

ary

part

of s

ound

pre

ssur

e (Pa

)

(a)

Reference solution IJCEMgPCEM

0.5 1 1.50x (m)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Varia

nce o

f im

agin

ary

part

of s

ound

pre

ssur

e (P；

2)

(b)

Figure 9: Mean and variance of sound pressure of car along the bottom boundary at f=200Hz: (a) mean and (b) variance.

response analysis of acoustic systems. With a suitable exten-sion, it can be used for response analysis in multidisciplinarycomputational mechanics.

It is noted that the computational cost of IJCEM willincrease rapidly with the number of random variables. Thus,the IJCEM is not suitable for random problem with largenumber of random variables.

Data Availability

The data used by the researchers are for scientific researchpurposes rather than commercial purposes. Most of the datagenerated or analysed during this study are included inthis manuscript and all of the data are available from thecorresponding author on reasonable request.

Conflicts of Interest

The authors of this study state that there are no conflicts ofinterest to disclose.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (51774323).

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