Unified polynomial expansion for interval and random response analysis of uncertain structure-acoustic system with arbitrary probability distribution

For structure-acoustic system with uncertainties, the interval model, the random model and the hybrid uncertain model have been introduced. In the interval model and the random model, the uncertain parameters are described as either the random variable with well defined probability density function(PDF) or the interval variable without any probability information, whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously. For response analysis of these three uncertain models of structure-acoustic problem involving arbitrary PDFs, a unified polynomial expansion method named as the Interval and Random Arbitrary Polynomial Chaos method(IRAPCM) is proposed. In IRAPCM, the response of the structure-acoustic system is approximated by APC expansion in a unified form. Particularly, only the weight function of polynomial basis is required to be changed to construct the APC expansion for the response of different uncertain models. Through the unified APC expansion, the uncertain properties of the response of three uncertain * Corresponding author. Tel.: +86 731 88821915; fax: +86 731 88823946. E-mail address: djyu@hnu.edu.cn *Manuscript Click here to download Manuscript: Manuscript.doc Click here to view linked References

 A unified polynomial expansion is established for interval model, random model and hybrid uncertain model;  The arbitrary polynomial chaos is extended for interval analysis and hybrid uncertain analysis;  The method is applied to structure-acoustic problem with interval/random variables involving complex probability distribution;  The proposed method has been compared with the hybrid perturbation method;  The proposed method for three uncertain models has been compared with several widely used polynomial chaos methods.

Introduction
The response analysis of structural-acoustic system is a key procedure for the control and optimization of the vibration and noise behaviors of engineering products, such as automobiles, steamships, aircrafts, submarines and spacecrafts. Traditional methods for response analysis of structural-acoustic system are deterministic numerical methods by assuming that all input parameters are fixed [1]. However, uncertainties related to material properties, boundary conditions and surrounding environment are unavoidable in the real engineering practices. Without considering these uncertainties, the results obtained by using deterministic numerical methods may be unreliable. Therefore, there is a growing interest for developing numerical methods for the response analysis of structural-acoustic system with uncertainties [2][3][4][5][6].
The most widely used technique for uncertainty quantification is the probabilistic method, in which the uncertain parameter is represented by the random variable with well defined probability density function(PDF). During past decades, lots of methods have been proposed for random uncertainty quantification, such as the Monte Carlo method [7][8][9], the perturbation probabilistic method [10][11][12][13] and the polynomial chaos method [14,15]. Among these methods, the Monte Carlo method is the simplest and the most versatile method for uncertain problems. However, the Monte Carlo method hybrid uncertain model, in which some uncertain parameters with well defined PDFs are treated as random variables, whereas the others are described as interval variables [39]. The uncertainty quantification of the hybrid uncertain model is more challenging than the interval uncertainty quantification and the random uncertainty quantification, as the approximation for the response related to different types of uncertainty in the hybrid uncertain model should be properly integrated [40]. Up to now, the studies for uncertainty quantification of the hybrid uncertain model are relatively small. The perturbation technique is a general choice for the hybrid uncertain analysis in the last decades, but it is limited to hybrid uncertain problems with small uncertainty level [41][42][43][44]. Recently, the polynomial chaos method has been developed for hybrid uncertain analysis. By integrating the Chebyshev polynomial with the generalized Polynomial Chaos(gPC), Wu et al. proposed a hybrid method for uncertainty quantification and robust topology optimization [45,46]. Subsequently, Yin et al. employed the Gegenbauer polynomial of gPC to construct a unified polynomial chaos expansion for structure-acoustic problems with interval and/or random uncertainties [47]. Wang et.al developed a response surface method for structural-acoustic systems with random and interval parameters based on the gPC [48]. To improve the computational efficiency for interval analysis of gPC expansion, Xu et. al developed a hybrid uncertainty analysis method by introducing the dimension wise analysis [49]. Compared with the perturbation technique based method, these gPC based methods have shown better accuracy for hybrid uncertain problem with large uncertainty level.
The random model, the interval model and the hybrid uncertain model listed above can be used to describe the uncertain system with interval and/or random variables in different cases according to the available information. For the uncertainty quantification of these three uncertain models, the polynomial chaos method can be effectively used for the uncertain problem with large uncertainty level and the efficiency is much higher than the Monte Carlo method. Thus, this paper will focus on the application of polynomial chaos method for uncertainty quantification of these three uncertain models. From the overall perspective, though the polynomial chaos method has gained a great achievement for uncertainty analysis, some important issues still remain unresolved. Firstly, as we mentioned before, the polynomial chaos methods for hybrid uncertain model are generally developed based on the polynomial basis of gPC. However, the accuracy and efficiency of these gPC based methods may be deteriorated for hybrid uncertain problem with the probability distribution out of Askey scheme, as the optimal polynomial basis of polynomial chaos expansion for uncertainty analysis with the probability distribution out of Askey scheme cannot be obtained by using gPC [50]. Secondly, there is little research on developing the unified polynomial expansion method for interval model, random model and hybrid uncertain model, especially when the random parameter of these uncertain models is following an arbitrary probability distribution. Recently, the Gegenbauer polynomial has been developed to construct the unified polynomial expansion for interval model, random model and hybrid uncertain model [47]. By using the unified Gegenbauer expansion, the response for these three uncertain models can be obtained by using a common numerical algorithm. However, unified Gegenbauer expansion method is only suitable for the uncertain problem with the bounded random variable following mono-valley or mono-peak probability distributions [47]. As regarding the engineering application, the PDF of random variable can be an arbitrary function, sometimes may be very complex. Therefore, it is desirable to develop new unified polynomial expansion method that can be used for three uncertain models with interval variable and/or random variable following arbitrary probability distributions.
The aim of the present study is to develop a new unified polynomial expansion method for response analysis of structure-acoustic systems with interval and/or random variables. For structure-acoustic systems with interval and/or random variables, three uncertain models will be considered, namely the interval model, the random model and the hybrid uncertain model. In order to construct the unified polynomial expansion for these three uncertain models, the Arbitrary Polynomial Chaos(APC) which has been successfully applied to uncertainty analysis with random variable following arbitrary probability distributions [51][52][53], will be developed for the uncertainty quantification of interval model and hybrid uncertain model. With this development, the unified Interval and Random Arbitrary Polynomial Chaos method(IRAPCM) is proposed to predict the response of three uncertain models of structure-acoustic system. In IRAPCM, the response of three uncertain models is approximated by the APC expansion in a unified form. For different uncertain models, only the weight function of polynomial basis is changed to construct the APC expansion. The coefficients of APC expansion are calculated though the Gauss integration. Once the APC expansion for uncertain models is obtained, the uncertain properties of response can be easily computed. The proposed IRAPCM is applied to a simple mathematical problem and a structure-acoustic problem. The effectiveness of IRAPCM for response analysis of interval model, random model and hybrid uncertain model has been investigated by comparing it with the hybrid first-order perturbation method and several existing polynomial chaos methods.

Fundamentals of the arbitrary polynomial chaos expansion
This section will briefly summarize the fundamentals of APC theory. Besides, the Gauss integration will be introduced to compute the coefficient of APC expansion due to its robustness and good efficiency. Furthermore, in order to efficiently calculate the weights and nodes of Gauss integration, the polynomial basis of APC expansion is constructed based on the recursive relations of the monic orthogonal polynomial.

Arbitrary polynomial chaos expansion for a function
A function () Y  approximated by the APC expansion can be expressed as where N is the retained order of APC expansion, i y represents the expansion w  is the weight function. () w  in the framework of APC theory can be an arbitrary continuous or discrete function, such as the piecewise function.
The free choice of the weight function of polynomial basis is the main advantage of APC expansion.
For multi-dimension uncertain problems, () Y ξ can be approximated by using the tensor order APC expansion as follows

Construction of polynomial basis for arbitrary given weight functions
In APC expansion, the polynomial basis for a given weight function can be 8 numerically obtained based on several numerical theories, such as the Gram-Schmidt orthogonalization [51] and the recursive relations of monic orthogonal polynomials [54]. Gram-Schmidt orthogonalization is the most widely used technique to construct the polynomial basis of APC expansion. However, the polynomial basis obtained by using Gram-Schmidt orthogonalization is not unique for a given weight function. As a comparison, the unique polynomial basis that is orthogonalized to a given weight function can be obtained based on the recursive relations of monic orthogonal polynomials. In addition, the Gauss integration formula for calculating the coefficients of APC expansion can be easily computed according to the coefficients of recursive relations of monic orthogonal polynomials. Therefore, the polynomial basis of APC expansion will be constructed based on the recursive relations of monic orthogonal polynomials in this paper.  9 with the coefficient 0 b being arbitrary and set by convention such that

Calculation of the expansion coefficient by using the Gauss integration
Owing the orthogonality of the polynomial basis, the expansion coefficient i y in Eq. (1) can be obtained via the following expression [54] ( ), ( ) Lots of integration techniques can be employed to calculate the integral in the above equation, such as the Gauss integration technique [54], the Clenshaw-Curtis integration technique [58] and the Newton-Cotes integration technique [59]. The Gauss integration technique is a widely used integration method for calculating the coefficient of the tensor-order polynomial chaos expansion [16]. This is because the Gauss integration technique can generally achieve high accuracy for determining the integral of the polynomial function, when the number of Gauss nodes is up to a certain value [54]. In this paper, the Gauss integration technique is introduced to calculated the integral in Eq.(9) due to its robustness. By using Gauss integration rule, i y in Eq.(9) can be expressed as a weighted sum of a finite set of function evaluations, that is[54]   (11) In particular, if where 1 1 ,..., ,..., 1 1 , By using the Gauss integration,  , , , In the above equations, ˆk j  denotes the k j th integration nodes for k  , and where ω is the angular frequency of external excitation; f , , In the above equations, Z , U and F denote the dynamic stiffness matrix, the response vector and the force vector of the structure-acoustic system, respectively.
Due to the unpredictable environment and the manufacturing tolerance, the structure-acoustic system always involved uncertainties. By using the vector to represent the uncertain parameters, the dynamic equilibrium equation of the structure-acoustic system can be rewritten as Where () Zx and () Fx denote the uncertain structure-acoustic dynamic stiffness matrix and the uncertain force vector, respectively.

Definition of three uncertain models for uncertain structure-acoustic system
In this paper, the uncertain parameter of structure-acoustic system is treated as either random or interval variable. When there is sufficient data to construct the PDF . According to the available PDF of uncertain parameters, the interval model, the random model and the hybrid uncertain model will be introduced to treat with the uncertain parameters. 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   13 Case1: the interval model In the interval model, each of the uncertain parameters is described as the interval variable. Accordingly, the uncertain vector x can be described as an interval vector and expressed as 12 [ , ,..., ] Case2: the random model In the random model, all of the uncertain parameters are described as the random variables and the uncertain vector x can be the expressed as 12 [ , ,..., ] Case3: the hybrid uncertain model In the hybrid uncertain model, the interval variable and the random variable exist simultaneously. In this case, the uncertain vector x can be expressed as a hybrid vector, which can be expressed as

Arbitrary polynomial chaos expansion for response analysis of structure-acoustic system with interval and random variables
The APC has been previously applied for random analysis [18]. In this section, 3 4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 14 the APC will be developed for response analysis of hybrid uncertain model of structure-acoustic system. As we mentioned in Section3.2, both the interval model and the random model can be viewed as the special case of the hybrid uncertain model. Therefore, through the extension of APC expansion for hybrid uncertainty analysis, a unified polynomial expansion approach is consequently established for response analysis of the interval model, the random model and the hybrid uncertain model of structure-acoustic system. In the following subsections, the procedure of APC expansion for hybrid uncertainty analysis of structure-acoustic problem will be deduced in detail.

Determine the polynomial basis with respect to the random variable
Polynomial chaos method is an uncertainty propagation approach which has been used in many engineering problems. The key idea of polynomial chaos method for random analysis is to approximate the random response by a sum of orthogonal polynomials. In the infinite amount of orthogonal polynomials, there always exists an optimal orthogonal polynomial for a given random variable. In particular, the orthogonal polynomial whose weight function is identical to the PDF of random variable can be viewed as the optimal polynomial basis of the polynomial chaos expansion for the random variable [50]. When the optimal polynomial basis is obtained, the polynomial chaos method can achieve exponentially convergence rate for random problem. Thus, in this paper, the PDF is used as the weight function of the polynomial basis with related to the random variable. Once the weight function related to a random variable is determined, the polynomial basis can be calculated through Eqs.(6)~ (8).

Determine the polynomial basis with respect to the interval variable
Theoretically, an arbitrary orthogonal polynomial that is defined on a closed interval can be used as the polynomial basis of APC expansion for the approximation of response of uncertain system with interval variable. However, the accuracy of APC expansion for the interval problem may change with different polynomial bases.
Therefore, it is necessary to determine a suitable polynomial basis of APC expansion for interval analysis. According to Section 2.2, the polynomial basis of APC expansion is determined by its corresponding weight function. In order to determine the polynomial basis of APC expansion for the interval problem, the effect of the weight function of polynomial basis on the accuracy of the APC expansion will be firstly investigated and discussed by a simple example as follows.  function is defined as follows [55]   In particular, the weight function of Legendre polynomial and Chebyshev polynomial are , respectively. In this paper,  Fig.1.
Define the Relative error(Re) as Where, () Px denotes the APC expansion. The relative error of the fifth-order APC expansion with different weight functions is plotted in Fig.2.
It can be found from Fig.2  , will be adopted as the weight function of the polynomial basis of APC expansion for interval analysis.

Construct the arbitrary polynomial chaos expansion for response with interval and random variables
Based on the APC expansion, the response of the hybrid uncertain structure-acoustic system can be approximated as In the above equation, ˆI x and ˆR x denote the integration nodes related to the interval variables and random variables, respectively.
Based on the APC expansion, the expectation of k U can be determined by  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 19 (30) In the above equation, is the weight function of the polynomial basis related to R j x . As the polynomial basis is orthogonal with respect to the PDF of the random variable, the analytical solution of the expectation of the APC expansion can be readily obtained [14]. According to Ref. [14], the expectation of the response approximated by APC expansion can be expressed as Before calculating the variance of the response, the expectation of mean square response should be obtained, which can be written as   1 1 Based on the orthogonal relationship of polynomial basis, the expectation of mean square response can be finally obtained and written as [14]     Consequently, the variance of the response can be obtained and expressed as          (34) Owing to the orthogonality of the polynomial basis, the expectation and variance of the response can be determined and expressed as  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 To obtain the maximum and minimum of the APC expansion shown in Eq.(39), various methods can be employed, such as the conventional optimization method [54], the Monte Carlo method [47], the interval arithmetic [37], and the dimension wise analysis [49]. The Monte Carlo method is the most accurate approach for interval analysis. However, a large number of sampling points is required to achieve a prescribed accuracy by using the Monte Carlo method. The interval arithmetic is the most efficient method for interval analysis, but its accuracy can hardly be evaluated due to the wrapping effect. The dimension wise analysis can also achieve high efficiency for interval analysis. However, the main potential limitation for dimension wise analysis is that the cooperative effects of multiple interval parameters acting together upon the system response are ignored. Thus, the accuracy of dimension wise analysis may be decreased in some cases [49]. The Genetic Algorithm(GA) algorithm 21 is a widely used method for solving complex optimization problems. Generally, the GA algorithm can achieve a prescribed accuracy for the interval analysis through an iterative process, and the computational efficiency of GA algorithm is much higher than the Monte Carlo method. Due to the good accuracy of GA algorithm, the GA algorithm will be employed to calculate the maximum and minimum of the APC expansion in this paper. Note that the APC expansion is a simple function, thus the computational cost suffering the GA algorithm is acceptable. Step2. Construct the polynomial basis that is orthogonalized with respect to the weight function related to each variable through Eqs.(6)~(8);

Procedure of arbitrary polynomial chaos expansion for uncertainty analysis with interval and random variables
Step3. Compute the integration nodes and weights through Eq.(12); Step4. Calculate the response of structure-acoustic system at the interpolation points through Eq.(20); Step5. Calculate the coefficients of APC expansion through Eq. (27); Step6. Calculate the response of structure-acoustic systems with interval and random variables through Eq.(39).
The main difference between the proposed IRAPCM and the conventional gPC based method is that different types of orthogonal polynomials are used for the polynomial chaos expansion. In the gPC based method, the orthogonal polynomial is selected from the Askey scheme, while the orthogonal polynomial in the proposed IRAPCM is numerically generated. As the choice of polynomials in Askey scheme is limited to some well known orthogonal polynomials, the optimal polynomial basis of polynomial chaos expansion for a wide range of complex probability distributions is not available by using the gPC based methods [50]. As a comparison, the optimal polynomial basis for an arbitrary PDF can be constructed by using IRAPCM. In other words, the proposed IRAPCM has the ability to provide the optimal polynomial basis for the uncertain problem involving arbitrary probability distribution.

Numerical examples
In

Mathematical problem
Consider a function as follows 1 x and 2 x are assumed as uncertain parameters. Table1 listed three uncertain models to describe 1 x and 2 x .
In the interval model, both 1 x and 2 x are described as interval variables. As the PDF of the interval variable is not available, only the range of variation is given for the interval variable. In the random model, both 1 x and 2 x are described as random variables. In the hybrid uncertain model, 1 x is described as random variable, while 2 x is described as interval variable. In Table1, the PDFs of 1  and 2  are given as follows Specially,   with the  distribution is related to the Gegenbauer polynomial in the Askey scheme, and the PDF of   is given by In the proposed method, the Gauss integration method is adopted to calculate the   14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   25 integration method [54], and the reference result of the lower and upper bounds of y related to the interval variable is calculated by using the GA algorithm [57].
From Fig.4, we can find that the accuracy of IRAPCM is the same as that of the widely used ICM for interval analysis. The main reason may be that the weight function of the APC expansion in IRAPCM for interval analysis is approximately the same as the weight function of the Chebyshev polynomial in ICM. However, it should be noted that the polynomial basis of APC expansion in IRAPCM is different from that of the Chebyshev polynomial in ICM. For instance, the APC expansion is constructed based on the monic polynomial, while the high-order Chebyshev polynomial is not the monic polynomial [32]. Therefore, the Chebyshev polynomial cannot be viewed as a special case of the polynomial basis of the APC expansion.
When compared with the ILM, it can be observed from Fig.4 that the error of the lower bound obtained by using IRAPCM is smaller than that of the ILM, while the error of the upper bound obtained by using IRAPCM is slightly larger than that that of the ILM at several retained orders. This is mainly because that the upper bound of y is obtained at the bounds of x, where the APC expansion can achieve higher accuracy than the Legendre expansion(Refer to Fig.2 in Section 4.2); while the lower bound of y is obtained around the mind-point of x, where the accuracy of APC expansion may be lower than that of the Legendre expansion. Therefore, the accuracy of IRAPCM is higher than that of ILM for calculating the upper bound of y, but is slightly lower than that of ILM for calculating the lower bound of y.
From Fig.5, we can find that IRAPCM can converge exponentially. As a comparison, the convergence rate of gPCM is much slower than that of IRAPCM, and the accuracy of gPCM remains no longer changed when the retained order is up to 3. This is mainly because the weight function of polynomial basis of gPCM can not accurately represent the random variable whose PDF is a piecewise function. In other words, some errors have been introduced for the PDF of random variable by using gPCM. Consequently, the results obtained by gPCM cannot converge to the exact 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   26 result. In addition, the nonlinear transformation of the random variable may also degrade the convergence rate of the gPCM. Therefore, the accuracy of gPCM is much lower than that of the IRAPCM for random problems when the PDF of random parameters is out of the Askey scheme.
From Figs.6 and 7, we can find that IRAPCM can also converge exponentially for hybrid uncertain analysis, while the gPC-ICM and gPC-WDM converges very slowly, especially when the retained order is up to 3. In other words, the IRAPCM can achieve much higher accuracy than the gPC-ICM and the gPC-WDM for hybrid uncertain analysis. As is addressed before, the accuracy of IRAPCM is the same as that of ICM for interval analysis, but is much higher than that of gPCM for random analysis. It indicates that the deterioration of the accuracy of gPC-ICM may be mainly caused by the use of the gPCM for random analysis. Therefore, it is more desirable to use the APC in the proposed method rather than the gPC for uncertainty quantification involving random variables.
As a conclusion from Figs.4~7, the proposed IRAPCM can achieve the same accuracy as the widely used ICM for interval analysis, whereas the accuracy of IRAPCM is much higher than that of the gPC based methods for random analysis and hybrid uncertain analysis. In other words, the proposed IRAPCM can not only keep the good accuracy of ICM for interval analysis, but can also improve the accuracy of the gPC based method for random analysis and hybrid uncertain analysis.

Compared with the hybrid perturbation method
In the last decade, the perturbation method and the polynomial chaos method have been widely used for uncertainty analysis of structure-acoustic system with interval and random variables [3,5,33,40]. Both the perturbation method and the polynomial chaos method have their own merits and application scope. The perturbation method can achieve high computational efficiency, but it is limited to uncertain problem with small uncertainty level. The polynomial chaos method can be employed to solve uncertain problem with large uncertainty level. However, the computational efficiency of the polynomial chaos method is lower than that of the perturbation method. The comparison between the perturbation method and the polynomial chaos method for structure-acoustic system with pure interval uncertainty has been fully discussed in the previous study [5,33]. In this paper, the proposed IRAPCM will be compared to the perturbation method for uncertainty analysis with both interval and random variables. Particularly, the Hybrid First-order Perturbation Method(HFPM) in Ref. [3] will be introduced for comparison. For uncertainty analysis of structure-acoustic system with interval and random variables, two cases with different uncertainty level will be considered. In case 4, the uncertainty level of the interval and random variables is very small, while the uncertainty level of the interval and random variables of case3 is much larger than that of case4. The first-order IRAPCM and the HFPM is employed to calculate the response of case3 and case4 at f=300Hz. In the first-order IRAPCM, the retained order of APC expansion for each uncertain variable is one. The lower and upper bounds of the expectation and variance of sound pressure distributing on the middle section obtained by the HFPM and the first-order IRAPCM are plotted in Figs.9 and 10. The reference results are obtained by using the Monte Carlo simulation. In Monte Carlo simulation, the sampling points for the random variables are 100000, and 10 uniformly distributed sampling points are used for each interval variable. 29 From Fig.9, we can find that the results obtained by the HFPM and the first-order IRAPCM are very close to the reference results. It indicates that both the HFPM and the first-order IRAPCM can achieve high accuracy for hybrid uncertainty analysis of structure-acoustic problem with small uncertainty level.
From Fig.10, we can find that both the HFPM and the first-order IRAPCM will lead to large errors. Namely, the HFPM and the first-order IRAPCM are not suitable to solve the structure-acoustic problem with large uncertainty level. It can be seen from Figs.6 and 7 that the accuracy of the IRAPCM for hybrid uncertainty analysis can be improved by increasing the retained order. To reduce the computational error of IRAPCM, the high-order IRAPCM will be employed to calculate the response of case3. In the high-order IRAPCM, the retained orders of the APC expansion of IRAPCM are 3, 2, 1 and 5 for E, t, f  and c , respectively. The results obtained by the high-order IRAPCM are plotted in Fig.11.
It can be seen from Fig.11 that the result obtained by high-order IRAPCM is very close to the reference result. It indicates that the proposed IRAPCM can achieve high accuracy for uncertainty analysis with large uncertainty level if the retained order is sufficiently large.
Theoretically, the accuracy of the hybrid perturbation method can also be improved by using high-order expansion. However, the computation of the derivatives of the high-order expansion of perturbation method for engineering problem is rather difficult and extremely cumbersome. Thus, the perturbation method for most of engineering problems is developed by using the low-order expansion, such as the first-order expansion and the second-order expansion. For uncertainty analysis of structure-acoustic problem with large uncertainty level, the accuracy of perturbation method cannot be significantly improved by using the second-order expansion instead of the first-order expansion. Therefore, up to now, the hybrid perturbation method is limited for structure-acoustic problem with small uncertainty level.
The computational time of the HFPM and the first-order IRAPCM is 12.8s and 75.2s, respectively. Namely, the efficiency of the HFPM is much higher than that of the first-order IRAPCM for structure-acoustic problem with interval and random uncertainties. Note that the HFPM can achieve high accuracy for uncertain structure-acoustic problem with small uncertainty level. Therefore, to save the computational cost, it is more reasonable to use the HFPM rather than the IRAPCM for response analysis of structure-acoustic problem with small uncertainty level.

Compared with several widely used polynomial chaos methods
In this subsection, the proposed IRAPCM will be compared to several widely From Table3, we can find that the computational time of the RRSS of each polynomial chaos method is close to its total computational time. It indicates that the computational costs of the polynomial chaos methods for three cases mainly suffer from the RRSS. Besides, we can find from Table3 that the computational time of the RRSS by using different polynomial chaos methods are almost the same. This is mainly because the same retained order is used in these polynomial chaos methods.
According to Eqs. (24) and (25), the total number of RRSS is determined by the retained order of polynomial chaos expansion. Therefore, computational time of RRSS by using different polynomial chaos methods will be very close when the same retained order is used in polynomial chaos expansion.

Conclusion
Through an extension of the APC expansion for interval analysis and hybrid uncertain analysis, a unified polynomial chaos method named as IRAPCM, is proposed for response analysis of the interval model, random model and hybrid uncertain model of structure-acoustic system. In IRAPCM, the response of three uncertain models is approximated by the APC expansion in a unified form. Based on the unified APC expansion, the uncertainty property of the response of structure-acoustic system can be efficiently obtained. In the procedure to construct the APC expansion for different uncertain models, only the weight function of polynomial basis is required to be changed. In particular, the  function with a small value of  is used as the weight function of polynomial basis for the interval variable, while the weight function of polynomial basis for the random variable is the same as the PDF. For a given weight function, the polynomial basis of APC expansion 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   33 is determined based on the recursive relations of monic orthogonal polynomials. As the weight function of polynomial basis of the APC expansion can be an arbitrary continuous or discrete function, the unified APC expansion can be effectively used for response analysis of three uncertain models of structure-acoustic system involving the random variable with arbitrary PDFs.
The proposed IRAPCM has been employed to calculate the response of three uncertain models of a mathematical problems and a structure-acoustic problem. the proposed IRAPCM can achieve higher accuracy than the gPC based methods for random analysis and hybrid uncertain analysis; (4) the computational efficiency of IRAPCM is lower than that of the hybrid first-order perturbation method, but the hybrid first-order perturbation method is limited for structure-acoustic problem with small uncertainty level.