A new evidence-theory-based method for response analysis of acoustic system with epistemic uncertainty by using Jacobi expansion

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ScienceDirect Comput. Methods Appl. Mech. Engrg. 322 (2017) 419–440 www.elsevier.com/locate/cma

A new evidence-theory-based method for response analysis of acoustic system with epistemic uncertainty by using Jacobi expansion Shengwen Yin, Dejie Yu ∗, Hui Yin, Baizhan Xia State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, People’s Republic of China Received 21 September 2016; received in revised form 21 March 2017; accepted 20 April 2017 Available online 5 May 2017

Highlights • The Jacobi polynomial is applied for evidence-theory-based uncertainty analysis. • The merit of the proposed method has been numerically verified. • The proposed method is employed to solve acoustic problem with epistemic uncertainty.

Abstract Evidence theory has strong ability to handle epistemic uncertainties whose precise probability distributions cannot be obtained due to limited information. However, the excessive computational cost produced by repetitively extreme value analysis severely influences the practical application of evidence theory. This paper aims to develop an efficient algorithm for epistemic uncertainty analysis of acoustic problem under evidence theory. Based on the orthogonal polynomial approximation theory, a numerical approach named as the evidence-theory-based Jacobi expansion method (ETJEM) is proposed. In ETJEM, the response of acoustic system with evidence variables is approximated by Jacobi expansion, through which the repetitively extreme value analysis needed in evidence theory can be efficiently performed. The parametric Jacobi polynomial of Jacobi expansion holds a large number of polynomials as special cases, such as the Legendre polynomial and Chebyshev polynomial. Thus, the ETJEM permits a much wider choice of polynomial bases to control the error of approximation than the traditional evidence-theory-based orthogonal polynomial approximation method, in which only the Legendre polynomial is used for approximation. Three numerical examples are employed to demonstrate the effectiveness of the proposed methodology, including a mathematic problem with explicit expression and two

∗ Corresponding author. Fax: +86 731 88823946.

E-mail address: [email protected] (D. Yu). http://dx.doi.org/10.1016/j.cma.2017.04.020 c 2017 Elsevier B.V. All rights reserved. 0045-7825/⃝

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engineering applications in acoustic field. In these three numerical examples, efficiency and accuracy are fully studied by comparing with Legendre expansion method as well as Monte Carlo simulations. c 2017 Elsevier B.V. All rights reserved. ⃝

Keywords: Epistemic uncertainty; Evidence theory; Jacobi expansion; Gauss–Jacobi integration formula; Uniformity approach; Acoustic system

1. Introduction Uncertainties are widely involved in engineering due to the manufacturing error, unpredictable environment and other factors. Without considering these uncertainties, the results obtained via deterministic analysis may not well satisfy the desired goal or even become unfeasible. Normally, uncertainties can be categorized as two distinct groups [1]: aleatory uncertainty and epistemic uncertainty. Aleatory uncertainty, also referred as objective or stochastic uncertainty, is defined as the inherent variation of nature, which is generally modeled as random variable or random process using probability theory. Conversely, the epistemic uncertainty is due to the incomplete knowledge in quantifying the uncertain system. This type of uncertainty is subjective and reducible, meaning that the uncertainty level can be reduced by modeling process itself. Up to now, researches have mainly concentrated on aleatory uncertainties and thereby a large number of stochastic analysis methods have been developed [2–4]. In particular, the stochastic methods based on the orthogonal polynomial can achieve good efficiency and accuracy in evaluating the statistic property of response, which is widely used in the field of stochastic uncertainty analysis [5,6]. In comparison to the stochastic uncertainty analysis, the quantification of epistemic uncertainty may be more challenging. Probability representation is inappropriate for the epistemic uncertainty, since some assumptions for the probability density function (PDF) of variables should be made by using the probability approach. Ben-Haim and Elishakoff have addressed that even small deviation from the real PDF may lead to relatively large errors of the statistic response [7]. To quantify the epistemic uncertainty in a more effective way, lots of non-probabilistic uncertain methods have emerged, such as the convex models [8], the fuzzy sets [9], the possibility theory [10] and the evidence theory [11]. Among these methods, it seems that the evidence theory has a much more flexible framework than other modeling techniques [12]. Evidence theory starts from basic probability assignments on the input variables and produces estimates of the lowest and highest probabilities of the model observables. Under different cases, the evidence theory can provide equivalent formulations to classical probability theory, possibility theory and convex models. However, as the evidence variable is represented by many subintervals that is called as focal element, onerous computational cost is inevitable in uncertainty quantification within evidence theory [13]. Thus, there is ever a growing demand on the development of more accurate and faster methods for the evidence-theory-based uncertainty analysis. In the uncertainty propagation under evidence theory, we need to find the extreme value of the system response over each focal element, which is very computationally expensive. In previous years, extensive research has been dedicated to reduce the excessive computational cost associated with repetitively extreme value analysis. One typical technique is to use advanced interval analysis method for the extreme value analysis in each focal element [14–16]. However, the computational burdens of this kind of technique will increase rapidly with the increasing number of focal elements, as the interval analysis of the original system should be repeated for all focal elements. Thus, the evidence-theory-based uncertainty propagation by using the advanced interval approach still suffers from large computational burdens, especially when the total number of focal elements is very large. Recently, the global surrogate models have been introduced for evidence-theory-based uncertainty analysis in order to reduce the computational cost [17–19]. The surrogate model allows one to emulate the output of a complex computational model with a simple function. Based on this simple function, the extreme value of the system response over each focal element can be efficiently obtained. Therefore, the evidence-theory-based method by using the global surrogate model can achieve high computational efficiency even there are a large number of focal elements. This paper focuses on the development of a global surrogate model for evidence based uncertainty analysis. Of great interest here is the so-called orthogonal polynomial expansion [20,21]. The orthogonal polynomial expansion is primarily used for stochastic analysis, which constitutes a surrogate model of the original problem using the orthogonal polynomial basis from Askey family. Due to the good accuracy, the orthogonal polynomial expansion has also been adapted for epistemic uncertainty analysis and hybrid

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stochastic/epistemic uncertainty analysis in recent years. Based on the Legendre polynomial, S. Adhikari, et al. proposed a spectral approach for fuzzy uncertainty propagation in finite element analysis [22]. For the uncertain problem with interval uncertainties, the Chebyshev interval method [23] and the Legendre interval method [24] have been proposed. More recently, E. Jacquelin, et al. gave a general framework to derive the response of a linear dynamical system with hybrid random/fuzzy uncertainty by using polynomial chaos theory [25]. Yin. et al. proposed a unified interval and random analysis method based on the Gegenbauer polynomial [26]. Harsheel et al. employed the stochastic expansions to quantify the margins and mixed aleatory/epistemic uncertainties under evidence theory [27]. Zhang et al. applied the stochastic expansions for robust design optimization under mixed aleatory/epistemic uncertainties, in which the Legendre polynomial is used to establish the global response surface related to epistemic uncertainties [28]. To the best knowledge of authors, the common choice of orthogonal polynomial for epistemic uncertainty analysis under evidence theory is the Legendre polynomial [27]. At the same time, according to the well-established stochastic polynomial approximation theory, if the type of polynomial basis used to approximate a stochastic solution is chosen according to the PDF of random variables, better approximation accuracy can be achieved than using other types of polynomial bases [29]. The evidence theory starts from basic probability assignments (BPA) which is very closely related to the probability distribution in probability theory. Therefore, we have reason to believe that the optimal choice of polynomial basis can improve the convergence properties of orthogonal polynomial expansion for evidencetheory-based uncertainty analysis. Moreover, with the increasing of customer expectations about vibrational and acoustical behavior of products, researches on response analysis of acoustic systems under uncertainties have been undergone a rapid development in engineering recently [30–33]. The aim of this paper is to develop an orthogonal polynomial expansion method for evidence-theory-based uncertainty analysis of acoustic system with epistemic uncertainties. As regarding the engineering cases, the uncertain parameters of acoustic system are always bounded in nature. Thus, the Jacobi polynomial which holds a large number of bounded polynomial bases from Askey family will be introduced. Note here that the widely used Legendre polynomial, Chebyshev polynomial and Gegenbauer polynomial are all the special cases of Jacobi polynomial. Based on the Jacobi polynomial, an Evidence-theory-based Jacobi Expansion method (ETJEM) is proposed for epistemic uncertainty analysis under evidence theory. In ETJEM, the response of acoustic system with evidence variables is approximated by Jacobi expansion, through which the extreme value analysis in each focal element can be efficiently performed. As there is a wide choice of polynomial bases by using Jacobi expansion, the uniformity approach [18] is introduced to seek the optimal polynomial basis for the approximation of system response with evidence variables. In particular, the uniformity approach is utilized to create a PDF for the evidence variable. Then, the optimal polynomial basis related to each evidence variable can be determined according to the transformed PDF. In the traditional orthogonal polynomial approximation method, the Legendre polynomial is the common choice for the approximation of response in terms of evidence variables [27]. To show the effectiveness of the proposed ETJEM, the analysis results of the proposed ETJEM are compared with that of the Legendre expansion method. The reference solution is obtained by using the Monte Carlo method. Results on three numerical examples show that the proposed ETJEM by using the optimal polynomial basis of Jacobi expansion can achieve higher accuracy than that of the Legendre expansion method. This study is organized as follows. Some basic conceptions of evidence theory are summarized in Section 2. Section 3 briefly described the main procedure and computational difficulty in uncertainty analysis of acoustic field using evidence theory. In Section 4, the detailed procedure of the proposed ETJEM is presented. The efficiency and accuracy of ETJEM is demonstrated in Section 5. Section 6 concludes the finding of the current paper. The fundamental idea of approximation theory of Jacobi expansion is introduced in Appendix A. 2. Evidence theory Evidence theory is also known as the Dempster–Shafer theory. Major concepts of this theory are summarized in this section. 2.1. Fundamental theory of evidence theory Evidence theory starts on the specification of a frame of discernment (FD), which is a nonempty finite set of mutually exclusive and exhaustive hypotheses Θ = {θ1 , θ2 , . . .}. Under evidence theory, a probability can be assigned

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for any possible subset of the FD based on the experimentation or expert opinion. This probability is called as Basic Probability Assignment (BPA). The BPA can not only be assigned on a single event but also a set of events, thus it is able to represent the imprecise probability information. The BPA of a event can be denoted by a mapping function m: 2Θ → [0, 1], where 2Θ represents all possible subsets of Θ. For a given evidential event A, the BPA should satisfy the following three axioms ⎧ m (A) ≥ 0, ∀A ∈ 2Θ ⎪ ⎪ ⎨ m∑ (∅) = 0, (1) ⎪ m (A) = 1 ⎪ ⎩ A∈2Θ

where each subset m(A) ∈ 2Θ satisfying m(A) > 0 is called as the focal element. Due to the lack of knowledge or information, the evidence theory cannot provide a precise probability for any possible proposition B. Therefore, an interval that consists of the belief and plausibility measures is used to treat the uncertainty of probability of the system response. These two measures can be defined as ∑ Bel(B) = m (A) (2) A⊆B

Pl(B) =

∑

m (A) .

(3)

A∩B̸=∅

In the above equations, the belief measure Bel(B) is obtained by summing the BPA of proposition which are totally included in B and it indicates the minimum amount of likelihood that could be associated with the event B. Whereas, the plausibility measure Pl(B) is the summation of BPA of propositions which are totally or partially included in B and it indicates the maximum amount of likelihood associated with B. Therefore, Bel and Pl can be viewed as the lower and upper bounds of the probability measure, which bracket the true probability of a proposition. Since no assumptions were made to obtain these measures, Bel and Pl are reasonably consistent with the given partial evidences. 2.2. Uncertainty qualification of a function using evidence theory Consider a general function with q-dimensional independent evidence variables Y = g (U) ,

U = [U1 , U2 , . . . , Uq ].

(4)

Similar to the joint PDF in probability theory, the joint FD for evidence variables vector U can be defined using the Cartesian product and denoted by S as follows S =U { 1 × U[ 2 × · · · × Uq ] } = sk = u 1 , u 2 , . . . , u q , u j ∈ U j , j = 1, 2, . . . , q, k = 1, 2, . . . , n s

(5)

where u j denotes the focal element of U j , sk denotes the focal element of the joint FD, n s is the total number of sk . Suppose the number of focal elements for the jth evidence variable is l j ( j = 1, 2, . . . , q), then the total number of sk is n s = l1 × l2 × · · · × lq . In practice, the evidence variables of engineering problems are generally continuous. Thus, U j is considered as a continuous interval in this paper and there is u 1 ∪ u 2 · · · ∪ u q = U j . The joint BPA for sk can be obtained by ⎧ q ∏ ( ) ⎪ ⎪ ⎨ m u j , u j ∈ sk m s (sk ) = j=1 (6) ⎪ ⎪ ⎩0, else. As a function of evidence variables, the output g(U) for each joint focal element sk can be represented by an interval with its corresponding BPA. The interval of g(U) over sk can be expressed as ] ] [ [ (7) YsIk = Y sk , Y¯sk = min g (U) , max g (U) . U∈sk

U∈sk

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And the BPA for YsIk is determined by ( ) {m(s ) k m Y YsIk = 0, else.

(8)

Based on the BPA of g(U), the Bel and Pl for g(U) can be obtained according to Eqs. (2) and (3). Similar to the probability theory, we can also use the moments to describe the distribution nature of response with evidence variables. The moments of g(U) can be defined as [34] µ(Y ) =

ns ∑

YsIk m(YsIk )

k=1 ns ( ∑

var(Y ) =

)2 YsIk − µ(Y ) m(YsIk ).

(9) (10)

k=1

3. Epistemic uncertainty analysis of acoustic system using evidence theory Assuming that the external excitation is harmonic, the acoustic dynamic equilibrium equation can be written as (K − k 2 M + jkC)p = F, (11) √ where, j = −1 is an imaginary unitary, k = ω/c is the wave number, ω is the angular frequency, and c is the acoustic speed of fluid, K is the acoustic system stiffness matrix; M is the acoustic system mass matrix; C is the acoustic system damping matrix; p stands for the sound pressure vector; F stands for the acoustic system load vector. The detailed derivation of Eq. (11) can be found in Ref. [31]. In real engineering, the acoustic field contains many inherent uncertainties, such as the uncertain physical property of acoustic medium caused by the variation of temperature and the uncertain velocity exciting produced by unpredictable environment. On the other hand, the precise probability density function of uncertain variables is always difficult to be obtained due to limited information and knowledge. This paper introduces the evidence theory to treat with the uncertainty of acoustic field. Using the evidence variable vector U to represent the uncertain parameters of acoustic field, the acoustic dynamic equilibrium equation can be rewritten as Z(U)p(U) = F(U)

(12)

where Z(U) and F(U) denote the total uncertain-but-bounded dynamic stiffness matrix and the uncertain-but-bounded force vector respectively, Z(U) can be expressed as Z(U) = K(U) − k(U)2 M(U) + jk(U)C(U),

(13)

where, k(U) stands for the uncertain-but-bounded wave number, K(U), M(U) and C(U) stand for the uncertain-butbounded stiffness, mass and damping matrix, respectively. Once the uncertain model of an acoustic system is established under evidence theory, the evidence measures such as the Bel, Pl and moments of the sound pressure, can be calculated according to the theory in Section 2. It can be found from Section 2 that we should calculate the extreme value of sound pressure over each joint focal element. To reduce the computational cost, the first-order perturbation method [15,16] can be employed for extreme value analysis, in which only the sound pressure at the mid-point should be calculated for each joint focal element. However, the use of the first-order perturbation method still suffers from large number of response reanalyses in practical applications. For instance, supposing that there exist 4 evidence variables in an acoustic system and each variable contains 8 focal elements, then a total number of 84 = 4096 joint focal elements will be involved. It means that we should recalculate the response in Eq. (12) for 4096 times. Thus, the evidence-theory-based uncertainty analysis by using the first-order perturbation technique is also time consuming especially when there is a large number of focal elements in epistemic uncertainty problems. To overcome the computational difficulty caused by large amount of focal elements, the evidence-theory-based uncertainty analysis method by using surrogate model seems more effective. Of great interest here is to use the orthogonal polynomial theory to construct the surrogate model. As the uncertain parameter is always bounded in

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practical acoustic field, the orthogonal polynomials that are defined in bounded intervals can be used to approximate the response of acoustic field. In these bounded polynomials, it seems that the Jacobi expansion is the most appropriate one to control the error of approximation of system response with evidence variables. This is because the parametric Jacobi polynomial contains a large number of bounded polynomials as special cases. Therefore, based on the above analysis, this paper will employ the Jacobi expansion to approximate the response of acoustic system with evidence variables. The fundamental theory of Jacobi expansion can be found in the Appendix. 4. ETJEM for epistemic uncertainty analysis under evidence theory The Jacobi expansion is based on the large family of Jacobi polynomials. Theoretically, any Jacobi polynomial can be used for uncertain problems with bounded evidence variables. However, the choice of polynomial basis may have great effect on the accuracy of Jacobi expansion. The key idea of the proposed method is to find the optimal Jacobi polynomial for epistemic uncertainty analysis under evidence theory. 4.1. Determine the Jacobi polynomial parameter In the orthogonal polynomial expansion method, the weight function related to the polynomial basis is always used to control the error of approximation. In practice, the choice of weight function associated with the polynomial basis is an important issue, which varies from problem to problem. It is shown in the stochastic polynomial approximation theory that the best approximation for the stochastic quantities of response can be achieved when the weight function is chosen to match the PDF of the random variables. That is to say, for stochastic uncertainty analysis, it is desirable to assign more weight in the region where there is larger probability density. For the case of interval uncertainty qualification, as there is no justification to weight errors unequally within the interval, the uniform weight associated with the Legendre polynomial is the nature choice [24]. In fact, we can assume the interval variable as a uniform random variables at the step of polynomial selection. Based on this uniform randomization assumption, the optimal polynomial for interval analysis (namely the Legendre polynomial) can also be obtained according the stochastic polynomial approximation theory. In other words, the stochastic polynomial approximation theory can provide a guidance for the choice of optimal polynomial for both interval and stochastic uncertainty problems if the uniform probability distribution is assumed for interval variables. The evidence theory starts from the basic probability assignments which are very closely related to the probability distribution in probability theory. Therefore, we have reason to believe that the accuracy of orthogonal polynomial expansion for evidence-theory-based uncertainty analysis can be also improved if we assign more weight to the region with large basic probability assignments by using the optimal polynomial basis. This paper will utilize the stochastic polynomial approximation theory to seek the optimal polynomial basis for evidence-theory-based uncertainty analysis. As we mentioned before, the weight function is chosen according to the PDF of the input variable under the stochastic polynomial approximation theory. However, the probability in evidence theory is generally assigned in an interval rather than a point. It means the probability distribution over the focal element is unavailable. Thus, in order to adapt the stochastic polynomial approximation theory for the polynomial selection of evidence-theory-based uncertainty problem, a PDF should be assumed over each focal element. Recently, a uniformity approach has been proposed by Jiang et al. to find the most probability focal element for evidence-theory-based reliability analysis [18]. In the uniformity approach, the uniform distribution probability is assumed over each focal element according to its BPA and correspondingly a transformed PDF can be obtained for the evidence variable. In fact, the transformed PDF obtained by using the uniformity approach would be very suitable to find the optimal Jacobi polynomial for evidencetheory-based uncertainty analysis. On the one hand, the focal element is very similar to the interval variable whose probability distribution over the interval is completely missing. On the other hand, the previous analysis has addressed that the uniform randomization assumption is appropriate for the interval variable when the stochastic polynomial approximation theory is used as guidance for polynomial selection. Therefore, it is desirable to assume a uniform PDF for each focal element at the step of polynomial selection. Based on the above analysis, the uniformity approach will be introduced to create a PDF for the evidence variable. To simplify the description, the evidence variable x can be regularized to a unitary variable defined in [−1, 1] through the following linear transformation x = X (ξ ) = b0 + b1 ξ.

(14)

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˜ which satisfies the following conditions Meanwhile, an arbitrary focal element A˜ of x will transform to B, { A˜ = b0 + b1 B˜ ˜ = m( B) ˜ . m( A)

425

(15)

For an evidence variable ξ defined on [−1, 1], a probability density function f ξ can be created through the uniformity approach, which can be expressed as [18] ∑ f ξ (ξ ) = δ j (ξ )m( B˜ j )/(U j − L j ) (16) j=1

where, B j is the jth focal elements of ξ , L j and U j are the lower and upper bounds of B j , δ j (y) = 1 if y ∈ B˜ j and 0 otherwise. Based on the transformed PDF in Eq. (16), the optimal polynomial of ETJEM for evidence-theory-based uncertainty analysis can then be determined according to the stochastic polynomial approximation theory. In particular, the polynomial whose weight function is the closest to the transformed PDF of evidence variable is chosen as the optimal polynomial of ETJEM for evidence-theory-based uncertainty analysis. As a result, the Jacobi parameter can be determined through the optimization as below To find Min

α, β [⏐ ⏐] l ⏐ f ξ (ηi ) − w(α,β) (ηi )⏐ 2 ∑

(17)

w(α,β) (ηi )

i=1

where ηi (i = 1, 2, . . . , l) are discrete points in the domain of definition of evidence variable. f ξ (ηi ) is the value of original PDF at the point of ηi . A lot of optimization methods can be adopted to obtain the solution of Eq. (17). 4.2. Jacobi expansion approximation for the response of acoustic problem with evidence variables Based on the linear transformation in Eq. (14), the response of acoustic system with evidence variables in Eq. (12) can be rewritten as u = u(ξ ) = Z−1 (X(ξ ))f(X(ξ )).

(18)

Then the Jacobi expansion can be used to approximate the response function u(ξ ) in the range of variation of ξ , which can be expressed as ˜ )= u(ξ ) ≈ u(ξ

N1 ∑ i 1 =0

···

NL ∑

fi1 ,...,i L Pi1 ,...,i L (ξ )

(19)

i L =0

where, Pi1 ,...,i L (ξ ) is the L-dimensional Jacobi polynomials, fi1 ,...,i L is the expansion coefficient to be estimated. According to Eq. (A.14), fi1 ,...,i L can be calculated by fi1 ,...,i L ≈

m1 mL ( ) ∑ ∑ 1 u X(ξˆ j1 , . . . , ξˆ jL ) Pi1 ,...,i L (ξˆ j1 , . . . , ξˆ jL )A j1 ,..., jL . ... h 1 × · · · × h L j =1 j =1 1

(20)

L

In the above equation, ξˆ ji ( ji = 1, 2, . . . , m i ; i = 1, 2, . . . , L) denotes the integration points, which are the roots (α,β) of Pm i (ξi ); m i is the total number of integration points related to ξi ; A j1 ,..., jL denotes the weight of Gauss-Jacobi integration, which is given by Eqs. ( (A.13) and (A.15); ) more detailed definition and derivation of the Jacobi expansion can be found in the Appendix. u X(ξˆ j1 , . . . , ξˆ jL ) denotes the responses of acoustic field at the interpolation points of Gauss–Jacobi integration, which can be calculated by ) ( ) ( ) ( u X(ξˆ j1 , . . . , ξˆ jL ) = Z−1 X(ξˆ j1 , . . . , ξˆ jL ) F X(ξˆ j1 , . . . , ξˆ jL ) . (21)

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4.3. Evidential measures of response of the acoustic problem with evidence variables Based on the Jacobi expansion shown in Eq. (19), the uncertainty property of response of the acoustic problem with evidence variables can be easily obtained. Mainly, there are four evidential measures to describe the uncertainty property of response of the acoustic problem with evidence variables, including the Bel, the Pl, the mean value and variance of response. For a given interval β I , the Bel(β I ) and the Pl(β I ) for u˜ i (ξ ) (i = 1, 2, . . . , Ncell ) can be calculated according to Eqs. (2)–(3) and expressed as ∑ ∑ m(dk ), k = 1, 2, . . . , n d . (22) m(dk ), Pl(β I ) = Bel(β I ) = {

⏐ } ⏐ dk ⏐β I ⊃u˜ iI (dk )

{

⏐ } ⏐ dk ⏐β I ∩u˜ iI (dk )̸=∅

In the above equation, dk (k = 1, 2, . . . , n d ) denotes the kth joint focal elements of ξ ; n d is the total number of the joint focal elements; m(dk ) is the BPA of dk ; u˜ iI (dk ) denotes the interval of u˜ i (ξ ) over the joint focal element dk . According to Eqs. (9)–(12), we can also calculate the mean value and variance of response, which can be written as µ(u i (ξ )) =

nd ∑

u˜ iI (dk )m(dk )

(23)

k=1 nd

var(u i (ξ )) =

∑( )2 u˜ iI (dk ) − µ(u˜ i (ξ )) m(dk ).

(24)

k=1

It can be seen from Eqs. (22)–(24) that we should calculate u˜ iI (dk ) to obtain the evidential measures. Namely, the extreme value of u˜ i (ξ ) over the joint focal element dk should be obtained. A lot of optimization methods can provide accurate results for the extreme value analysis. This paper employs the Genetic Algorithm [35] to calculate the extreme value of u˜ iI (dk ). 4.4. Procedure of ETJEM In a brief description, the proposed method can approximate the response of acoustic systems with epistemic uncertainties by Jacobi expansion, which is a simple function. Consequently, the evidence-theory-based uncertainty analysis can be easily processed through the Jacobi expansion. The detailed procedure of the proposed ETJEM was represented by a flow chart shown in Fig. 1. 5. Numerical example and discussions This paper developed the Jacobi expansion for epistemic uncertainty analysis under evidence theory. Traditionally, the Legendre expansion is the common choice to construct the surrogate model of response for epistemic uncertainty analysis and it is a special case of Jacobi expansion when the Jacobi parameters are α = β = 0. This section provides several numerical tests to compare the accuracy of the proposed ETJEM and the Legendre expansion method. The reference solution is calculated by the Monte Carlo method, in which the extreme value of response over each joint focal element is obtained by Monte Carlo simulation of the original response. In the following subsection, a mathematic problem is firstly considered to investigate the convergence property of the proposed method. Then, the proposed method is employed for epistemic uncertainty analysis of two acoustic problems. 5.1. A mathematic problem 5.1.1. Problem definition The following function is considered y = arctan x.

(25)

The range of variation of x is [−1, 1]. In order to illustrate the advantages of the proposed method, two different cases that both contain 10 focal elements are investigated. The detailed BPAs of two cases are listed in Table 1.

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Fig. 1. Flow chart of the proposed ETJEM. Table 1 BPA of evidence variable for two different cases. Case 1

Case 2

Interval

BPA (%)

Interval

BPA (%)

[−1.0, −0.8] [−0.8, −0.6] [−0.6, −0.4] [−0.4, −0.2] [−0.2, 0.0] [0.0, 0.2] [0.2, 0.4] [0.4, 0.6] [0.6, 0.8] [0.8, 1.0]

23.97 9.27 6.49 4.41 6.13 6.13 4.41 6.44 9.12 23.63

[−1.00, −0.78] [−0.78, −0.62] [−0.62, −0.4] [−0.4, −0.15] [−0.15, 0.08] [0.08, 0.25] [0.25, 0.48] [0.48, 0.66] [0.66, 0.80] [0.80, 1.00]

0.02 0.43 7.31 22.22 29.78 19.63 15.95 3.74 0.89 0.03

5.1.2. Choice of the Jacobi polynomial basis As a result of the theory stated in Section 2.2, one can obtain a transformed PDF for the evidence variable. Then, the Jacobi polynomial basis whose weight function is the closest to the transformed PDF will be chosen as the optimal polynomial basis for each evidence variable. For clarity, the transformed PDF and the weight function of optimal Jacobi polynomial basis for case 1 and case 2 are plotted in Figs. 2 and 3 respectively. In Figs. 2 and 3, there is deviation between the transformed PDF and the weight function, especially at the bounds of each focal element. The main reason is that the transformed PDF is a piecewise continuous function. Thus it is

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Fig. 2. The transformed PDF and the weight function of optimal polynomial basis for case 1.

Fig. 3. The transformed PDF and the weight function of optimal polynomial basis for case 2.

difficult to find a weight function that match the transformed PDF perfectly. However, unlike the stochastic polynomial expansion method in which the weight function related to the selected polynomial will be used as the PDF to yield the statistical results of response, the only purpose of polynomial selection in the proposed method is to assign more weight at the focal element where there is larger BPA. It is shown in Figs. 2 and 3 that the change of weight value is consistent with the variation of the transformed PDF. Thus, some errors between the weight function and transformed PDF are acceptable. It should be noted that in practical engineering, the interval of focal element of evidence variable will get narrower with the improvement of human cognitive level and the development of science and technology. When the focal element becomes a very narrow interval, the weight function may have the ability to match the transformed PDF perfectly well. 5.1.3. Comparison with the Legendre expansion method It was shown in Ref. [29] that, in the context of aleatory uncertainty quantification, if the polynomial basis used to approximate a stochastic solution is chosen according to the distribution of the underlying random variables, the best approximation accuracy can be achieved. Here, we investigate the effect of the choice of Jacobi polynomial basis

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Fig. 4. The convergence property of ETJEM and Legendre expansion method for case 1: (a) mean value (b) variance.

of ETJEM on the convergence of the mean and variance of solutions subject to epistemic uncertainty under evidence theory. The mean value and variance of response obtained by using evidence theory is an interval rather than a single value. For simplicity, the relative error of an interval of response is defined as the maximum relative errors at the lower and upper bounds of the interval. Thus, the relative error of the mean value and variance of response in this paper can be defined as ⏐} ⏐} {⏐ {⏐ ⏐ ⏐ ⏐ ⏐ ⏐ σ¯ − σ¯ r e f ⏐ ⏐⏐ σ − σ r e f ⏐⏐ ⏐ µ¯ − µ¯ r e f ⏐ ⏐⏐ µ − µr e f ⏐⏐ ⏐,⏐ ⏐,⏐ eµ = max ⏐⏐ (26) ⏐ , eσ = max ⏐⏐ ⏐ ⏐ µ¯ r e f ⏐ ⏐ µ σ¯ r e f ⏐ ⏐ σ r e f ⏐ ref

where, u and u¯ are the lower and upper bounds of mean value, σ and σ¯ are the lower and upper bounds of variance, (•)r e f denotes the results obtained by reference solution. Figs. 4 and 5 show the rate of convergence of ETJEM and the Legendre expansion method in estimating the mean value and variance for the uncertain problem defined in Section 5.1.1. From Figs. 4 and 5, we can find that the relative error of ETJEM and Legendre expansion method can eventually decrease to 10−4 when the retained order is up to 10. However, the convergence rate of the Legendre expansion method is slower than that of the ETJEM. Therefore, we can conclude from Figs. 4 and 5 that: (1) both the ETJEM and the Legendre expansion method can achieve high accuracy for the evidence-theory-based uncertainty analysis if the retained order is sufficiently large; (2) the ETJEM can achieve higher accuracy for the evidence-theory-based uncertainty analysis than the Legendre expansion method. Besides, it can be seen from Fig. 5(a) that the relative error of ETJEM may also fluctuate with the increase of the retained order. The main reason is that the weight function of Jacobi expansion that is the same as the transformed PDF of evidence variables cannot be available. As a result, the exponential convergence rate may not be achieved by using ETJEM. However, the key idea of the proposed method is to find the optimal polynomial basis from the large family of Jacobi polynomial basis for evidence-theory-based uncertainty analysis. As is shown in Figs. 4 and 5, the accuracy of ETJEM is much higher than that of Legendre expansion method. Note that the Legendre expansion method can be considered as the special case of ETJEM when the Jacobi parameters are α = β = 0. Therefore, it is reasonable to conclude that when the optimal Jacobi polynomial basis is chosen according to the distribution of the input variables through Eq. (17), higher accuracy can be achieved than the Legendre expansion method. Namely, the optimal choice of polynomial basis can improve the accuracy of ETJEM for the evidence-theory-based uncertainty analysis. 5.2. Acoustic tube Fig. 6 depicts a 2D acoustic tube model of dimensions 1 × 0.1 m. At the inner side of the acoustic tube, a discontinuous normal velocity is imposed, whereas the remaining edges are perfectly rigid. The sound pressure of

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Fig. 5. The convergence property of ETJEM and Legendre expansion method for case 2: (a) mean value (b) variance.

Fig. 6. The finite element mesh model of the tube cavity. Table 2 BPAs of two evidence variables of tube cavity. Density ρ

Speed of velocity c

Interval (kg/m3 )

BPA (%)

Interval (m/s)

BPA (%)

[1.103, 1.121] [1.121, 1.145] [1.145, 1.176] [1.176, 1.207] [1.207, 1.225] [1.225, 1.249] [1.249, 1.267] [1.267, 1.304] [1.304, 1.328] [1.328, 1.347]

1.60 6.52 13.48 17.23 11.17 14.80 10.38 16.71 6.51 1.60

[306.2, 310.7] [310.7, 316.7] [316.7, 324.1] [324.1, 331.5] [331.5, 335.9] [335.9, 341.9] [341.9, 346.4] [346.4, 354.4] [354.4, 360.4] [360.4, 365.9]

0.01 0.03 2.11 21.08 26.76 33.64 12.37 3.96 0.02 0.02

acoustic cavity is analyzed by using the Finite Element (FE) method. The finite element model of this tube cavity consists of 640 isoparametric elements and 729 nodes. Considering the unpredictability of the environment temperature, two uncertain parameters (ρ and c) are treated as the independent evidence variable. The environment temperature is assumed between −20 and 40 ◦ C. According to the change in temperature, the variation ranges of ρ and c can approximately be obtained: ρ = 1.103–1.347 kg/m3 , c = 306.2–365.9 m/s. The assumed BPAs of these two evidence variables are listed in Table 2. We employ the ETJEM, the Legendre expansion method and the Monte Carlo method for uncertainty analysis of acoustic tube with evidence variables. In ETJEM, the optimal polynomial basis of Jacobi expansion should be determined for each evidence variable through Eqs. (16) and (17). The Jacobi parameters of optimal polynomial basis for ρ and c are: αρ = βρ = 0.6 and αc = βc = 9.7. The same orders are retained in the proposed ETJEM and the Legendre expansion method, namely the second order for ρ and the third order for c. In the Monte Carlo method, symmetrical 5 uniformly distributed sampling points are applied to each uncertain parameter. The lower and upper bounds of the mean and variance of the imaginary part of sound pressure along the central axis obtained by using the

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431

Fig. 7. The lower and upper bounds of the mean and variance of the imaginary part of sound pressure along the central axis at f = 200 Hz: (a) mean value (b) variance.

ETJEM, the Legendre expansion method and the Monte Carlo method are plotted in Fig. 7 at f = 200 Hz and Fig. 8 at f = 300 Hz. It is shown in Figs. 7 and 8 that the computational results obtained by ETJEM are very close to the reference solutions obtained by Monte Carlo method. In contrast, there is large deviation between the results obtained by the Legendre expansion method and the Monte Carlo method. These results indicate that the proposed ETJEM is more accurate than the Legendre expansion method for response analysis of acoustic system with evidence variables when the same expansion orders are retained. In engineering practices, the cumulative probability of the response can be of guidance for design and optimization of the acoustic system with random variables. Inspired by the cumulative distribution functions (CDF) p(X ≤ x) in probability theory, the cumulative belief function (CBF) and the cumulative plausibility function (CPF) are introduced and defined as CBF(x) = Bel(X ≤ x)

(27)

CPF(x) = Pl(X ≤ x)

(28)

where x is an arbitrary value, and X denotes the response of interest such as the sound pressure. The CBF and CPF of sound pressure for the acoustic tube model at f = 300 Hz are calculated by using the ETJEM, the Legendre expansion method, as well as the Monte Carlo method. In the Monte Carlo method, 5 uniformly distributed sampling points are generated for each uncertain parameter of the joint focal element.

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Fig. 8. The lower and upper bounds of the mean and variance of the imaginary part of sound pressure along the central axis at f = 300 Hz: (a) mean value (b) variance.

In ETJEM and Legendre expansion method, the second order is retained for ρ, while the fourth order is retained for c. The corresponding results are shown in Fig. 9. It can be seen from Fig. 9 that the ETJEM can achieve better accuracy than the Legendre expansion method in the calculation of CBF and CPF. As a conclusion, the proposed ETJEM can achieve desirable accuracy not only for the prediction of the mean value and variance of response but also for the calculation of the CBF and CPF of response. To evaluate the computational efficiency of the proposed method, execution time of ETJEM, Legendre expansion method and Monte Carlo method to calculate the CBF and CDF of the sound pressure at f = 300 Hz are listed in Table 3. All of the computational results are obtained by using MATLAB R2014a on a 3.20 GHz Intel(R) Core (TM) CPU i5-3470. It can be seen from Table 3 that the computational time of the proposed ETJEM is much less than that of the Monte Carlo method. Meanwhile, we can find that the execution time of the proposed ETJEM and the Legendre expansion method are almost the same. This is mainly because the same expansion order is retained in ETJEM and Legendre expansion method. As we mentioned before, the computational burdens of ETJEM and Legendre expansion method mainly suffer from the calculation of the expansion coefficients related to the multi-dimension polynomial terms. When the same expansion order is retained, the number of multi-dimension polynomial terms of ETJEM is the same as that of Legendre expansion method. As a result, there is almost no difference between ETJEM and Legendre expansion method on efficiency. It should be noted that the total number of multi-dimension polynomial terms will increase rapidly with the increasing number of uncertain parameters (see in Eq. (A.10)). Thus, the ETJEM is more suitable to the acoustic problems with a modest number of uncertain parameters. But compared with the Monte Carlo

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433

Fig. 9. The CBF and CPF of sound pressure at different points of cavity tube when f = 300 Hz: (a) x = 0 m (b) x = 0.5 m.

method, in which the number of sampling points will increase exponentially with the increasing number of uncertain parameters, the ETJEM can achieve much higher computational efficiency for acoustic problems with large number of uncertain parameters. 5.3. Acoustic cavity of a car Consider a simplified 2D acoustic cavity of a car with dimensions 2.672 × 1.072 m, as shown in Fig. 10. The acoustic cavity is surrounded by air with the density ρ and the acoustic speed c. At the front windshield, the admittance coefficient along the Robin boundary is An . According to the characteristics of the front engine, the discontinuous normal velocity vn is imposed at the outside of the acoustic cavity. The remaining edges are perfectly rigid. The FE

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Fig. 10. The 2D finite element mesh model of the acoustic cavity of a car. Table 3 Execution time of the ETJEM, Legendre expansion method and Monte Carlo method. Method

ETJEM

Legendre expansion method

Monte Carlo method

Execution time

15.3 s

14.7 s

2476 s

Table 4 BPAs of the evidence variables of the car. ρ

c

vn

An

Interval (kg/m3 )

BPA (%)

Interval (m/s)

BPA (%)

Interval (m/Pa s)

BPA (%)

Interval (m/s)

BPA (%)

[1.128, 1.147] [1.147, 1.172] [1.172, 1.203] [1.203, 1.234] [1.234, 1.253] [1.253, 1.278] [1.278, 1.296] [1.296, 1.334] [1.334, 1.359] [1.512, 1.54]

25.05 7.19 6.80 5.77 4.11 5.48 4.28 9.82 6.45 25.05

[311.7, 317.2] [317.2, −321.2] [321.2, 326.7] [326.7, 332.9] [332.9, 338.7] [338.7, 343.0] [343.0, 348.8] [348.8, 353.3] [353.3, 358.8] [358.8, 363.7]

0.03 0.66 6.30 22.70 31.48 20.40 14.93 3.14 0.35 0.02

[1.26, 1.274] [1.274, 1.302] [1.302,1.337] [1.337, 1372] [1.372, 1.400] [1.400.1.428] [1.428, 1.449] [1.449, 1.491] [1.491, 1.512] [1.512, 1.54]

0.06 3.33 18.15 30.20 22.87 15.46 6.18 3.60 0.14 0.01

[8.0, 8.2] [8.2, 8.6] [8.6, 9.1] [9.1, 9.6] [9.6, 10.0] [10.0, 10.4] [10.4, 10.7] [10.7, 11.3] [11.3, 11.6] [11.6, 12.0]

6.70 22.19 27.45 20.76 11.02 6.70 2.83 2.15 0.19 0.02

method is used to analyze the sound pressure of the acoustic cavity of the car. In the FE model of this acoustic cavity, the numbers of elements and nodes are 710 and 806, respectively. In this numerical example, the uncertain parameters are modeled by an evidence-theory-based uncertain model, in which ρ, c, An and vn are treated as evidence variables due to the limited probability distribution information. The BPAs for ρ, c, An and vn are listed in Table 4. The proposed ETJEM and the Legendre expansion method are employed to calculate the CBF and CPF of response for the acoustic cavity of the car. The retained orders of ETJEM and Legendre expansion method for ρ, c, An and vn are 2, 4, 2 and 2 respectively. In ETJEM, the Jacobi parameters of the optimal polynomial basis for ρ, c, An and vn determined through Eqs. (16) and (17) are: (αρ , βρ ) = (−0.6, −0.6), (αc , βc ) = (5.7, 5.7), (α An , β An ) = (3.2, 4.6) and (αvn , βvn ) = (0.3, 2.9). The Bel and Pl of sound pressure obtained by ETJEM and Legendre expansion method at point A of the car is plotted in Fig. 11 when f = 120 Hz and Fig. 12 when f = 200 Hz. For this numerical example, the Monte Carlo method is not used as the reference method due to the tremendous computational burden. In the extreme value analysis over each joint focal element, suppose the number of sampling points of Monte Carlo simulation for each uncertain parameter is 5, the total number of sampling points will be 54 = 625. As the number of joint focal elements is 104 = 10 000, the total number of sampling points of Monte Carlo method for the evidence-theory-based uncertain model will come to 625 × 10 000 = 6.25 × 106 . Thus the Monte Carlo method for evidence-theory-based response analysis of the acoustic cavity of the car could lead to large computational cost. It has been concluded in Section 5.1 that the Legendre expansion method can achieve high accuracy for the evidence-theory-based uncertainty analysis if the retained order is sufficiently large. Besides, the high order Legendre polynomial expansion is widely used as the reference solution to validate the accuracy of other polynomial based methods for epistemic uncertainty analysis [36]. Therefore, due to the tremendous computational cost of Monte Carlo method for the evidence-theory based uncertainty analysis of the acoustic cavity of a car, the high

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435

Fig. 11. CBF and CPF of the real and imaginary part of sound pressure at point A of car when f = 120 Hz: (a) real part (b) imaginary part.

order Legendre expansion method will be used as the reference method in this numerical example. In the high order Legendre expansion method, the retained orders for ρ, c, An and vn are 4, 8, 4 and 4 respectively, which are much larger than the retained orders of the ETJEM and the Legendre expansion method in this numerical example. For comparison, the reference solution obtained by the high order Legendre expansion method is also plotted in Figs. 11 and 12. From Figs. 11 and 12, we can see that the accuracy of the ETJEM is higher than that of the Legendre expansion method when the same expansion orders are retained. It further verifies the good accuracy of ETJEM. Thus, it is desirable to use the proposed ETJEM with the optimal Jacobi polynomial rather than the Legendre polynomial for the epistemic uncertainty analysis of acoustic problem under evidence theory. Besides, these results also indicate that

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Fig. 12. CBF and CPF of the real and imaginary part of sound pressure at point A of car when f = 200 Hz: (a) real part (b) imaginary part.

relatively high orders should be retained to reduce the errors by using the Legendre expansion method. As a result, the computational efficiency by using the proposed ETJEM can be improved compared with the Legendre expansion method if we want to achieve a prescribed accuracy. 6. Conclusion In engineering practice, acoustic problems always contain epistemic uncertainties due to the lack of knowledge. These uncertainties may lead to significant changes of acoustical behavior of the engineering system. Evidence theory has originated from probability theory and has recently shown as a powerful tool to deal with the epistemic uncertainties. To efficiently calculate the response of acoustic system under evidence theory, a new method named

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as ETJEM is presented based on the Jacobi expansion. The parametric polynomial basis of Jacobi expansion we introduced includes the Legendre polynomial as a subset, thus the ETJEM can be viewed as the extension of traditional evidence-theory-based orthogonal polynomial approximation method, in which only the Legendre polynomial is used for approximation. The important feature of the new extension framework is that it incorporates a large number of polynomial bases to handle the evidence variables with different BPA information. In ETJEM, a transformed PDF that can approximately reflect the probability distribution property of evidence variables is obtained through the uniformity approach. Inspired by the stochastic polynomial expansion theory, the Jacobi polynomial basis whose weight function is the closest to the transformed PDF, is chosen as the optimal polynomial basis of Jacobi expansion for evidence-theory-based uncertainty analysis. The coefficients of Jacobi expansion are calculated by the Gauss–Jacobi integration method, which is more efficient and convenient to be implemented in numerical algorithm compared with the Galerkin technique. By using the ETJEM, the response of acoustic system with evidence variables is represented by the Jacobi expansion, thus the extreme value of response over each joint focal element can be calculated efficiently. A mathematic problem and two acoustic problems are used to demonstrate the effectiveness of the proposed method. Numerical results show that the ETJEM is much more efficient than the Monte Carlo method. Compared with the Legendre expansion method, the proposed ETJEM can achieve higher accuracy when the same expansion order is retained. Namely, ETJEM can control the errors of the solution more effectively and approximates the response of acoustic systems with evidence variables more accurately than the Legendre expansion method. Furthermore, two cumulative distribution curves, namely CPF and CBF of response, can be obtained by using ETJEM. In engineering practice, the CPF and CBF of response can be of guidance for risky design and conservative design of an acoustic field with epistemic uncertainties. It should be noted that the proposed ETJEM is not limited to the response analysis of the uncertain acoustic system. With the suitable extension, it may be applied to the response analysis of the dynamical structures and the multiphysical coupling fields. Acknowledgments The paper is supported by National Natural Science Foundation of China (Nos. 11572121, and 11402083). The author would also like to thank reviewers for their valuable suggestions. Appendix A. Jacobi expansion approximation theory In this section, we briefly summarize the definitions of Jacobi expansion. Besides, the Gauss–Jacobi integration method is introduced to calculate the coefficients of Jacobi expansion due to its robustness. A.1. Jacobi polynomials (α,β)

The standard Jacobi polynomial Pn (ξ ) with the Jacobi parameters α > −1, β > −1 can be defined by the recurrence relations as follows [21] ⎧ 1 1 ⎪ ⎪ P0(α,β) (ξ ) = 1, P1(α,β) (ξ ) = (α + β + 2)ξ + (α − β), ⎨ 2 2 (A.1) (α,β) (α,β) P B + P C ⎪ 0 0 n−2 ⎪ P (α,β) (ξ ) = n−1 ⎩ , n = 2, 3, . . . n A0 where ⎧ ⎨ A0 = 2n(n + α + β)(2n{+ α + β − 2) } (A.2) B0 = (2n + α + β − 1) (2n + α + β)(2n + α + β − 2)ξ + α 2 − β 2 . ⎩ C0 = −2(n + α − 1)(n + β − 1)(2n + α + β) The Jacobi polynomials are orthogonal on ξ ∈ [−1, 1] with respect to the weight function w (α,β) (ξ ) = Γ (α+β+1) (1 + ξ )α (1 − ξ )β , that is 2α+β+1 Γ (α+1)Γ (β+1) { ∫ 1 h , k= j (α,β) (α,β) w(α,β) (ξ )Pk (ξ )P j (ξ )dξ = k (A.3) 0, k ̸= j −1

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where Γ (k + α + 1)Γ (k + β + 1)Γ (α + β + 1) . (A.4) (2k + α + β + 1)Γ (k + 1)Γ (k + α + β + 1)Γ (β + 1)Γ (α + 1) In the above equations, Γ (•) is the Gamma function. In fact, Jacobi polynomials hold several famous orthogonal polynomials as special cases in terms of Jacobi parameters α and β, including the Gegenbauer polynomials (α = β, α > −1, β > −1), the Chebyshev polynomial of the first kind (α = β = −1/2), the Chebyshev polynomial of the second kind (α = β = 1/2) and the Legendre polynomial (α = β = 0). hk =

A.2. The Jacobi expansion of a function Based on the Jacobi polynomials, a continuous function f (ξ ) defined on ξ ∈ [−1, 1] can be approximated by the Jacobi expansion f (ξ ) =

∞ ∑

(α,β)

f i Pi

(ξ )

(A.5)

i=0

where f i is the ith (i = 0, 1, . . .) constant coefficient to be determined. For L-dimensional problems, the Jacobi expansion of the function f (ξ ) in terms of an L-dimensional vector ξ can be given as f (ξ ) =

∞ ∑ i 1 =0

...

∞ ∑

f i1 ,...,i L Pi1 ,...,i L (ξ )

(A.6)

i L =0

where, f i1 ,...,i L is the constant coefficient, and Pi1 ,...,i L (ξ ) denotes the L-dimensional Jacobi polynomials, which is the Lth-order tensor product of each one-dimensional polynomial and can be expressed as (α,β)

Pi1 ,...,i L (ξ ) = Pi1

(α,β)

(ξ1 )...Pi L

(ξ L ).

(A.7)

In practice, the infinite sum in Eq. (A.6) obviously should be truncated at a finite expansion order. Traditionally, the polynomial chaos expansion truncated the high order polynomial basis when the total order of multi-dimension polynomials is up to a certain value [6]. This traditional expansion is the so-called total order expansion. For the r th total order expansion with L variables, the multi-index i k (k = 1, 2, . . . , L) is constrained by i i + i 2 + · · · + i L ≤ r.

(A.8)

The total number of multi-dimensional polynomial terms Ntol for the total order expansion is given by (n + r )! . (A.9) n!r ! In the total order expansion, the truncation criterion is specified on the total order of multi-dimensional polynomial rather than on the one-dimensional polynomial. Thus, it is difficult to employ different retained order for the input variable. This difficulty can be overcome by another form of expansion named as the tensor product expansion, in which the polynomial order is applied on a per-dimension basis and all combinations of the one-dimensional polynomials are included. In other words, all of the multi-dimension polynomial satisfied i k ≤ rk (k = 1, 2, . . . , L) can be retained, where rk is the expansion order for the kth variable. Within the tensor product expansion, the total number of multi-dimension polynomial terms Nten is ∏ Nten = (ri + 1). (A.10) Ntol =

i

In the tensor product expansion, the polynomial order for each variable is specified independently. The tensor product expansion is particularly suitable for the uncertain problem when the required polynomial order varies significantly for different uncertain parameters. This is because the minimum required retained order can be used for each variable and thereby the total number of multi-dimension polynomial terms can be reduced [19]. In acoustic field, the response is much more sensitive to the velocity of sound than the other parameters [31]. It indicates the required retained order for uncertain input parameters may have great difference when the Jacobi expansion is applied

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for acoustic problem. Thus, the tensor product expansion is adopted in this paper. On the basis of tensor product expansion, the Jacobi expansion of a function f (ξ ) can be expressed as f (ξ ) =

N1 ∑ i 1 =0

...

NL ∑

f i1 ,...,i L Pi1 ,...,i L (ξ )

(A.11)

i L =0

where Nl (l = 1, 2, . . . , L) is the expansion order of polynomial expansion related to ξl . A.3. Computation of the coefficients of Jacobi expansion A lot of well established methods can be employed to calculate the coefficients of Jacobi expansion, including the stochastic response surfaces approach, the Galerkin technique and the Gauss–Jacobi integration method. By using the Galerkin technique [21], the dimension of the transformed equations is much larger than the degrees of freedom of the original system, which restrict its scope of application to large-scale systems. For the tensor product expansion, the number of Gauss–Jacobi integration points is the same as the number of expansion coefficients. In contrast, the sample points of stochastic response surfaces approach should be larger than the expansion coefficients in order to achieve a desirable accuracy. On the other hand, the Gauss–Jacobi integration method is relatively stable in accuracy compared with the stochastic response surfaces approach. Therefore, the Gauss–Jacobi integration method will be used to estimate the coefficient f i of Jacobi expansion. Based on the Gauss–Jacobi integration formula [21], the coefficient f i in Eq. (A.5) can be calculated and written as ⎛ ⎞ ∫ m 1 ⎝∑ 1 1 (α,β) (α,β) (α,β) w (ξ ) f (ξ )Pi (ξ )dξ ≈ f (ξˆ j )Pi (ξˆ j )A j ⎠ . (A.12) fi = h i −1 h i j=1 (α,β)

In the above equation, ξˆ j ( j = 1, 2, . . . , m) denote the interpolation points, which are the roots of Pm the total number of interpolation points, and the weight A j ( j = 1, 2, . . . , m) are given by [21] { }−2 ′ Γ (m + α + 1)Γ (m + β + 1) (1 − x 2j )−1 Pm(α,β) (x j ) , j = 1, 2, . . . , m. A j = 2α+β+1 Γ (m + 1)Γ (m + α + β + 1) To minimize the integral error, the parameter m is usually no less than N +1. Similarly, by using the Gauss–Jacobi integration formula, f i1 ,...,i L can be obtained and expressed as ∫ 1 ∫ 1 1 wi1 ,...,i L (ξ ) f (ξ ) Pi1 ,...,i L (ξ )dξ1 ...dξ L f i1 ,...,i L = ··· h 1 × · · · × h L −1 −1 m1 mL ( ) ∑ ∑ 1 ≈ ··· f ξˆ j1 , . . . , ξˆ jL Pi1 ,...,i L (ξˆ j1 , . . . , ξˆ jL )A j1 ,..., jL h 1 × · · · × h L j =1 j =1 1

(ξ ), m is

(A.13)

(A.14)

L

where ξˆ ji , . . . , ξˆ jL are the interpolation points, the weight A j1 , j2 ,..., jL can be expressed as A j1 ,..., jL = A j1 × A j2 × · · · × A jL .

(A.15)

By using the Gauss–Jacobi integration method, the Jacobi expansion can be derived by several iterative computations of the original functions at the interpolation points (see Eq. (A.14)). As a result, the Jacobi expansion method is non-intrusive, which can be easily processed in the numerical implementation even if the high order polynomial of Jacobi expansion is retained for approximation. References [1] F.O. Homan, J.S. Hammonds, Propagation of uncertainty in risk assessment: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability, Risk. Anal. 14 (5) (1994) 707–712. [2] G. Stefanou, The stochastic finite element method: past, present and future, Comput. Methods Appl. Mech. Engrg. 198 (9–11) (2009) 1031–1051. [3] S.S. Isukapalli, A. Roy, P.G. Georgopoulos, Stochastic response surface methods for uncertainty propagation application to environmental and biological systems, Risk Anal. 18 (1998) 351–363.

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