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Collocation methods for fuzzy uncertainty propagation in heat conduction problem
International Journal of Heat and Mass Transfer 107 (2017) 631–639
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Collocation methods for fuzzy uncertainty propagation in heat conduction problem Chong Wang a, Zhiping Qiu a,⇑, Menghui Xu b a b
Institute of Solid Mechanics, Beihang University, Beijing 100191, PR China Faculty of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, Zhejiang 315211, PR China
a r t i c l e
i n f o
Article history: Received 10 April 2016 Received in revised form 8 October 2016 Accepted 25 October 2016
Keywords: Fuzzy uncertainty propagation Heat conduction Legendre polynomial series Collocation technology Smolyak algorithm
a b s t r a c t Based on the combination of collocation technology and fuzzy theory, this paper proposes a full grid fuzzy collocation method (FGFCM) and a sparse grid fuzzy collocation method (SGFCM) for fuzzy uncertainty propagation in heat conduction problem. Converting fuzzy parameters into interval variables by levelcut strategy, the Legendre polynomial series provides a surrogate function for temperature response. To calculate the expansion coefficients, FGFCM evaluates the deterministic solutions directly on the full tensor product grids, whereas Smolyak algorithm is introduced in SGFCM to reduce the number of collocation points. According to the smoothness property of surrogate function and fuzzy decomposition theorem, the interval bounds and membership functions of uncertain temperature response are derived, respectively. Comparing result with traditional Monte Carlo simulation and parameter perturbation method, two numerical examples evidence the remarkable accuracy and effectiveness of proposed methods for fuzzy temperature field prediction in engineering. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In the traditional numerical calculation, the parameters of computational model are always treated as deterministic values. However, for most practical engineering cases, various uncertainties representing the system variability are unavoidable due to the aggressive environment factors, incomplete knowledge and inevitable measurement errors. The uncertainty propagation analysis, which can effectively assess the impact of input uncertainties on the system output responses, has received an increasing amount of attention in recent years [1,2]. In probabilistic framework, the numerical approaches, such as Monte Carlo simulation, spectral analysis method and so on, can be considered as the most valuable strategies, where uncertainties are quantified as random variables or stochastic processes by using a great amount of sample statistical information [3,4]. But as pointed in Ref. [5], some important uncertain characteristics cannot be handled appropriately by probability theory, mainly because the concept of randomness is not necessarily the uncertainty source. Consequently, nonprobabilistic interval model, introduced by Moore [6], provides another method to deal with the imprecise uncertainties whose bounds are well-defined but sufficient information about the probability density function is missing. Interval analysis is capable ⇑ Corresponding author. E-mail address: [email protected] (Z. Qiu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.083 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
to predict the guaranteed changing ranges of output responses with respect to the interval input data [7,8]. In recent years, various interval approaches have been presented for the interval uncertainty propagation in engineering [9–11]. The fuzzy set theory, emerged from the work of Zadeh [12], is another efficient category to model the system uncertainty based on the subjective opinions. In practice, fuzzy sets can be viewed as the generalization of interval variables by introducing a membership function [13]. In other words, the level-cut strategy is usually adopted to subdivide the membership function range into some cut levels and convert the original fuzzy sets into some interval numbers. Up to now, the existing fuzzy analysis methods can be grouped into two categories. The first one is known as intervalbased approach, where the fuzzy parameter is converted into an interval variable using the cut-level operation, and then the uncertainty analysis is evaluated based on the classical interval arithmetic [14,15]. The computational cost of interval-based method is small, but its inherent disadvantage is the imprecise outputs caused by neglecting the correlation between many operands, especially for the case with large uncertainty level [16]. The second one is known as optimization-based approach, in which two optimization problems aiming at the maximum and minimum values of the output response will be solved [17,18]. Although the results are more accurate and the optimization tools can be employed in a straightforward manner, the huge computational cost for a large number of optimization problems embarrasses its application in
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complex engineering areas [19]. Considering both computational accuracy and computational cost, it is necessary to develop new numerical methods for fuzzy uncertainty propagation. Traditional numerical thermal analysis with given deterministic parameters has undergone a rapid development in engineering. But considering the unavoidable uncertainties in material properties, external loads and boundary conditions, the nondeterministic numerical models are more feasible [20]. Based on the generalized polynomial chaos, Xiu and Karniadakis presented a random spectral decomposition method for the solution of transient heat conduction subjected to random inputs [21]. For the uncertain heat convection-diffusion problem without sufficient information, Xue and Yang estimated the temperature intervals by using the finite element method and expansion techniques [22]. Combining the fuzzy theory with perturbation analysis, the fuzzy perturbation methods are proposed to predict the membership functions of temperature responses [23,24]. But it should be pointed that the low computational accuracy caused by the first-order Taylor series is the inherent disadvantage of fuzzy perturbation methods, especially for the nonlinear problem with large uncertainty level. Comparatively speaking, the collocation methods represent great superiority in computational accuracy and efficiency by using the high-order polynomial series and a set of deterministic computing procedure [25]. According to the easy implement, several collocation schemes have been developed to the engineering problems [26,27]. Nevertheless, current research on collocation methods is mainly concentrated in the stochastic field, while the combination of collocation method and fuzzy theory is promising but mostly unexplored [28]. The purpose of our present study is to develop new collocation methods combined with fuzzy theory to deal with the uncertain heat conduction problems with different uncertainty dimensions. The paper is structured as follows. The governing equation of heat conduction problem with fuzzy input parameters is firstly established in Section 2. Then, the basic theory of level-cut operation and polynomial approximation is introduced in Section 3. Subsequently, two fuzzy collocation methods are presented in the next two sections. The first one is FGFCM where the collocation points are directly constructed by the full tensor product grids. To improve the computational cost for high-dimensional problems, the Smolyak algorithm is adopted in SGFCM to reconstruct the sparse grids. In Section 6, two numerical examples about a 2D thermal plate and a 3D sandwich structure are provided to verify the feasibility of proposed methods, and we conclude the paper with a brief discussion at last. 2. Governing equation of fuzzy heat conduction problem
ð1Þ
where x 2 Rn ðn ¼ 1; 2; 3Þ denotes the spatial coordinate; X is a bounded domain; TðxÞ stands for the temperature response; kðxÞ represents the heat conductivity, and f ðxÞ is the intensity of heat source. For the interior domain X bounded by C, the following three kinds of boundary conditions are shown in Fig. 1
TjC1 ¼ T s @T k ¼ qs @n C2 @T k ¼ hðT T e Þ @n C3
where T s is the boundary temperature; n denotes the normal vector; qs stands for the heat flux; h and T e represent the heat transfer coefficient and ambient temperature, respectively. For the engineering heat conduction problem, due to the vaguely defined system characteristics, insufficient information and judgment subjectivity, uncertainties in material properties, external loads and boundary conditions are unavoidable. In this study, all the uncertain input parameters whose membership functions can be defined subjectively based on the expert opinions are modeled as n fuzzy parameters
a ¼ ðai Þn ¼ ða1 ; a2 ; . . . ; an Þ
ð3Þ
Then the governing Eq. (1) with multiple fuzzy inputs can be rewritten as
@ @Tðx; aÞ þ f ðx; aÞ ¼ 0 ðx; aÞ 2 X V kðx; aÞ @x @x
ð2Þ
ð4Þ
where V denotes the fuzzy uncertain space, and the temperature response TðxÞ become uncertain with respect to the fuzzy parameter vector a. 3. Level-cut operation and polynomial approximation Based on the level-cut strategy [29], each fuzzy parameter ai can be intersected to yield an interval variable aIi;k with respect to the cut level k 2 ½0; 1
aIi;k ¼ ½ai;k ; a i;k ¼ aci;k þ Dai;k dIi
ð5Þ
i;k are the lower and upper bounds of the interval where ai;k and a i;k þ ai;k Þ=2 and Dai;k ¼ ða i;k ai;k Þ=2 are called variable aIi;k ; aci;k ¼ ða the midpoint and the radius, respectively, dIi represents the standard interval dIi ¼ ½1; 1. Using vector denotation, all the k-cut interval variables can be written as aIk ¼ aIi;k . n
According to the definition of level-cut set, we can derive the following property
aIi;kj aIi;kk if kj 6 kk
In heat transfer analysis, the governing equation of steady heat conduction problem with a heat source can be expressed as
@ @TðxÞ kðxÞ þ f ðxÞ ¼ 0 x 2 X @x @x
Fig. 1. Three kinds of boundary conditions in heat conduction.
ð6Þ
Thus, by setting the cut level be zero, i.e. k = 0, the 0-cut bounded interval aIi;0 can be considered as the compact support of the fuzzy parameter ai , which means all the samples of fuzzy parameter ai belong to the closed interval aIi;0 . Subsequently, the product of different compact supports can be defined as the support box VI of the fuzzy uncertain space V
VI ¼
n Y
aIi;0 ¼ aI1;0 aI2;0 aIn;0
ð7Þ
i¼1
Under the cut level k = 0, Eq. (4) is converted into an interval equation with respect to the 0-cut interval vector aI0
@T x; aI0 @ k x; aI0 þ f x; aI0 ¼ 0 x; aI0 2 X VI @x @x
ð8Þ
where the temperature response T x; aI0 will change in a certain interval range.
C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
As we know, the traditional interval analysis methods based on Taylor series only use the midpoint information to approximately calculate the maximum and minimum values, whose accuracy for the nonlinear function becomes unacceptable if the uncertainty level of expansion parameters is not small enough. In order to overcome this shortcoming, the high-order polynomial series will be introduced as the surrogate function in this paper. Considering the uniform distribution of interval variables, the weight function is a constant, and the Legendre-type polynomials can be selected as the basis functions [30]. Let i ¼ ði1 ; i2 ; . . . ; in Þ be a multi-index, and the n-variate basis functions Ui aI0 of Legendre polynomial chaos are the products of the univariate basis functions
Ui aI0 ¼ Ui1 aI1;0 Ui2 aI2;0 Uin aIn;0
ð9Þ
where Ui1 aI1;0 ; . . . ; Uin aIn;0 denote the univariate basis functions of polynomial chaos. Based on Eq. (9), the interval temperature response T x; aI0 in Eq. (8) can be expanded as
X T x; aI0 ¼ wi ðxÞUi aI0
ð10Þ
i
where wi ðxÞ is the corresponding expansion coefficient. Considering the global approximation of Legendre surrogate model, optimal convergence can be achieved by using the polynomial chaos with proper basis [31]. In practice, finite terms are often selected to truncate above polynomial series. With the following N-order basis functions of total degree less than or equal to N
Ui aI0 ¼ Ui1 aI1;0 Ui2 aI2;0 Uin aIn;0
0 6 jij 6 N
ð11Þ
Eq. (10) can be concisely rewritten as
X T x; aI0 T N x; aI0 ¼ wi ðxÞUi aI0
ð12Þ
jij6N
where jij ¼ i1 þ i2 þ . . . þ in , and the total number of polynomial expansion terms can be calculated by C nnþN ¼ ðn þ NÞ!=ðn!N!Þ with respect to the number of variables n and the order of polynomial series N. Different from the spectral analysis in stochastic systems where the probabilistic moments such as expectation and variance are calculated by the orthogonal relationship of polynomial bases, the eventual aim of the present study is to predict the membership functions of fuzzy output responses. If the surrogate function of temperature response as shown in Eq. (12) is derived, it is easy to obtain its extreme points by making the first-order derivatives to be zero
I
@T N x; a0 ¼ 0 i ¼ 1; 2; . . . ; n @ aIi;0
ð13Þ
Based on the continuously-differentiable property of surrogate function T N x; aI0 , the extreme points and boundary points in the
I support box V are assumed to be a1 ; a2 ; . . . ; ar , and then the interval temperature response T x; aI0 can be approximately yielded by iterating through the finite points
I
I
T x; a0 ¼ ½T x; a0 ; T x; aI0 T x; aI0 T N x; aI0 ¼ min T N x; aj j¼1;...;r I I T x; a0 T N x; a0 ¼ max T N x; aj
function T N x; aI0 derived in the support box VI can still be used to approximate the temperature response in the subspace VIk
0 < k 6 1. Similarly, by iterating through the finite extreme
points and boundary points in VIk , we can obtain the interval temper ature response T x; aIk ¼ T x; aIk ; T x; aIk under the cut level k. Eventually, based on the fuzzy decomposition theorem, the membership function of uncertain temperature response with respect to original fuzzy parameter vector a can be reconstructed by connecting the different interval solutions for all cut levels kq
n o Tðx; aÞ ¼ [ kq T x; aIkq q
j¼1;...;r
When the cut level k is set to be other value between 0 and 1, we can derive the k-cut interval uncertain space VIk with the property VIk ¼ aI1;k aI2;k aIn;k # VI . It indicates that above surrogate
ð15Þ
From above derivation we can see that the pre-process of selecting polynomial bases and the post-process of calculating interval bounds and membership function of temperature response can be easily implemented by existing computing procedures. The crucial issue for fuzzy temperature field prediction is to calculate the expansion coefficients of surrogate function T N x; aI0 in Eq. (12). 4. Full grid fuzzy collocation method (FGFCM) The aim of collocation method is to calculate expansion coefficients by using the deterministic responses at preselected points in uncertain space. The main advantage is that the existing deterministic model and computing procedure can be fully utilized without any further modification. In this section, we will extend the collocation technology based on full grids to the fuzzy uncertainty analysis. The first important aspect of collocation method is the selection of nodal set. The Clenshaw-Curtis nodes, which are the extreme points of Chebyshev polynomials, are one of the many kinds of collocation points in computational mathematics [32]. They can minimize the collocation error with the same number of nodes under the maximum norm. For the univariate case, the Clenshaw-Curtis i;0 of the ith fuzzy nodes distributed in the 0-cut interval ½ai;0 ; a parameter are defined as
(
ðiÞ bj
¼
aci;0 if mi ¼ 1 aci;0 cos pmðj1Þ Dai;0 j ¼ 1; 2; . . . ; mi if mi > 1 i 1
ð16Þ
where mi is the predefined number of collocation points. It is obvious that the collocation points locate in the entire interval range i;0 , and the higher the value of mi is, the more intensive the ½ai;0 ; a collocation points will be. For the multivariate case, the most straightforward approach to extend the collocation points defined in one dimension to the entire support box VI is to use the tensor product method. For each n o ðiÞ ðiÞ m m and an operator Oi i dimension, a nodal set Hi i ¼ b1 ; b2 ; . . . ; bðiÞ mi are introduced to denote the collocation points and the approximate function, respectively. Using the tensor product method directly, the operator O in the n-dimensional support box VI can be constructed as m2 mn 1 O ¼ Om 1 O2 On
ð17Þ
On this basis, one can obtain the entire nodal set H m2 mn 1 H ¼ Hm 1 H2 Hn
ð14Þ
633
ð18Þ
where the total number of collocation points M in the full grids is
M¼
n Y
mi ¼ m1 m2 mn
ð19Þ
i¼1
For large dimensions n 1, the number M will be particularly large even though the number of collocation points mi in each one-dimensional space is small, which poses a numerical challenge
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called the curse of dimensionality because each collocation point needs a simulation of the full-scale underlying deterministic system. Therefore, the collocation point construction of full grids is mostly used for the low-dimensional uncertain engineering problems. After selecting the nodal set, another important aspect of collocation method is to calculate the expansion coefficients. A set n o is given to denote the M collocation H ¼ bnode ; bnode ; . . . ; bnode 1 2 M is used to represent the deterpoints specifically, and T x; bnode j ministic solution to the governing Eq. (8) at the preselected point
For the two-dimensional case n ¼ 2, the operator O in Eq. (22) can be written as
X
O¼
i1 þi2 6L
X
¼
ð1Þ
i1 þi2 ¼L
1 L jij
For the practical engineering heat conduction problem, the deterministic solutions T x; bnode at all collocation points can be j calculated by the existing efficient finite element approach [33]. Based on the surrogate function in Eq. (12), a group of linear equation with respect to the unknown expansion coefficients wi ðxÞ can be derived as
U1 bnode 1 B B B U1 bnode 2 B B B. B .. @ U1 bnode M
1 U2 bnode U3 bnode UC nnþN bnode 1 1 1 C C C U2 bnode U3 bnode UC nnþN bnode 2 2 2 C C C .. .. . . .. C . . . . A node node node U2 bM U3 bM UC nnþN bM 1 0 1 0 T x; bnode 1 w1 ðxÞ B C C C B B C B w2 ðxÞ C B T x; bnode 2 C C B B C C¼B B .. C C B. B. C A B .. @ @ A wC nnþN ðxÞ node T x; bM
0
Oi11 Oi22
X
i1 þi2 ¼L1
ð23Þ
ð1ÞLjij
n1 L jij
Oi11 Oi22 Oinn
[
H¼
Hi11 Hi22 Hinn
ð25Þ
Lnþ16jij6L
By iterating through all the nodes on full grids, only the ones whose sum order jij ¼ i1 þ i2 þ . . . þ in across all dimensions is between L n þ 1 and L are retained. Compared to the full grids, the Smolyak algorithm provides an interpolation strategy with potentially orders of magnitude reduction in the number of support nodes required, and thus we may as well call the new tensor product grids ‘sparse grids’. For the jth one-dimensional space with the collocation level ij , i
of the Clenshaw-Curtis nodes the number mjj and positions bðjÞ s j;0 can be defined as distributed in the 0-cut interval ½aj;0 ; a
(
Once the polynomial bases Ui and the collocation nodal set H are provided, the expansion coefficients wi ðxÞ in Eq. (12) can always be determined by solving Eq. (21). To prevent the problem from becoming undetermined, the number of collocation points is required not to be smaller than the number of polynomial expansion terms, i.e. M P C nnþN . In this case, the generalized matrix inversion approach based on the least square principle is usually adopted to solve above linear equations. 5. Sparse grid fuzzy collocation method (SGFCM)
mjj ¼ bsðjÞ
¼
1
if ij ¼ 1
2ij 1 þ 1 if ij > 1
8 c < aj;0
: acj;0 cos
if ij ¼ 1 pðs1Þ ij
mj 1
Daj;0 s ¼
i 1; 2; . . . ; mjj
With different level indices, Fig. 2 depicts the one-dimensional i
i þ1
nodal sets, where it can be seen Hjj Hjj
ij ¼ 1; 2; . . ., which
means the Clenshaw-Curtis nodal sets are nested. Since the construction in Eq. (24) employs one-dimensional interpolations with various numbers of nodes, it is preferable the one-dimensional
i 1
Assumed Dik ¼ Okk Okk and O0k ¼ 0, the multivariate interpolation formula can be defined as follows based on the Smolyak algorithm
X i D 1 Di 2 Di n
ð22Þ
jij6L
where L P n is an integer representing the overall construction level, and ij denotes the separate level along the jth direction.
ð26Þ
if ij > 1
As mentioned above, the huge computational cost is considered as the inherent disadvantage of FGFCM for high-dimensional problems. In order to avoid the curse of dimensionality, the sparse grids, which are still based on the tensor product construction but only a small subset of the full grids, will be reconstructed in this section by using the Smolyak algorithm [34].
O¼
ð24Þ
In the process of extending univariate interpolation formulae to multivariate case, the Smolyak algorithm provides a linear combination of product formulae, which is selected in such a way that the interpolation property in each dimension is preserved for higher dimensions. The specific derivation can refer to Refs. [35] and [36]. Subsequently, the new tensor product grids corresponding to the operator in Eq. (24) can be constructed as
i
i
Equivalently, the operator O in FGFCM for the n-dimensional sup-
Lnþ16jij6L
ð20Þ
Oi11 Oi22
X
port box VI becomes
O¼
1 node @ @ node @T x; bj A þ f x; bnode ¼ 0 j ¼ 1; 2; . . . ; M k x; bj j @x @x
Ljij
L16jij6L
bnode j
0
X i Di1 Di2 ¼ O11 Oi22
Fig. 2. One-dimensional nodal sets with different level indices.
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nodal sets be nested, where the total number of nodes in Eq. (25) can reach a minimum. That is the other reason why we chose Clenshaw-Curtis nodes as the collocation points in this study. In Eq. (25), there is no explicit formula to calculate the number M in terms of n and L. To simplify symbol expression, an additional index k is introduced to denote the level of collocation points uniformly. Let L ¼ n þ k, then the total number of collocation points M in sparse grids can be approximately expressed in terms of n and k for n 1
M ¼ dimðHÞ ¼ dim
[
Hi11 Hi22 Hinn
!
Lnþ16jij6L k
2 k n k!
if n 1
ð27Þ
Given the level index k = 3, the comparison between full girds and sparse girds in the two-dimensional 0-cut interval uncertain i;0 ½aj;0 ; a j;0 is shown in Fig. 3, in which we can space ½ai;0 ; a observe a significant reduction in the number of collocation points for sparse grids. Besides, with different dimensions and nodal level indices, the numbers of collocation points in the two proposed fuzzy collocation methods are listed in Table 1. Compared with the full grids in FGFCM, the curse of dimensionality for sparse grids in SGFCM, albeit still present, is significantly lessened.
In both FGFCM and SGFCM, the polynomial series and matrix inversion approach are adopted to construct the surrogate function of uncertain temperature response in the pre-process, while the smoothness property of polynomial function and the fuzzy decomposition theorem are used to calculate the temperature ranges and eventual membership function in the post-process. The main difference between the two fuzzy collocation methods is that the deterministic codes at collocation points for FGFCM are carried out on the full grids; whereas, only the sparse grids are retained in SGFCM by using Smolyak algorithm. Thus, if given the same nodal level, the computational accuracy of FGFCM will be higher than that of SGFCM, but the computational cost of SGFCM will be much smaller, especially for the engineering problems with large dimensions. 6. Numerical examples 6.1. Low-dimensional uncertain heat conduction problem In order to verify the performance of proposed methods for lowdimensional uncertain heat conduction problem with less than five uncertain parameters, consider a 2D thermal plate as shown in Fig. 4, where the rectangular-sectioned part and the circular part are discretized by 100 quadrilateral elements and 188 triangular elements, respectively. There exists a continuous volumetric heat in the shaded portion, and a heat flux is imposed on the bottom surface. Seven feature points are selected to characterize the structural temperature filed. Due to aggressive environment factors and inevitable measurement errors, material properties, external loads and boundary conditions contain some subjective uncertainties, where four fuzzy parameters with the triangular-type membership functions are considered as uncertain input variables, i.e. heat conductivity k = (174,204,234) W/(m°C), heat flux qs = (750,900,1050) W/m2, volumetric heat Q = (1700,2000,2300) W/m3, boundary temperature Ts = (42,50,58) °C. Based on the level-cut strategy, eleven cut levels kq ¼ ðq 1Þ=10 q ¼ 1; . . . ; 11 are used to transform all the original fuzzy parameters into interval variables. The third-order Legendretype polynomial series is adopted to construct the surrogate function of temperature response with respect to support box under the 0-cut level. Given the Clenshaw-Curtis collocation points at nodal level k = 3, simulations of the proposed fuzzy collocation methods are carried out by Matlab R2014. The lower bound (LB) and upper bound (UB) of interval temperature responses at seven feature points under cut levels k1 = 0 and k6 = 0.5 are listed in Tables 2 and 3, where the response intervals calculated by the traditional Monte Carlo method (MCM) with 105 samples are adopted as referenced results. Besides, the commonly used parameter perturbation method (PPM) is introduced for further comparison.
Table 1 Comparison between the numbers of collocation points. Dimension (n)
Nodal level index (k)
M1 (FGFCM)
M2 (SGFCM)
M1/M2
2
2 3 4 5 2 3 4 5 2 3 4 5
25 81 289 1089 625 6561 83521 1185921 390625 43046721 6.9758 109 1.4064 1012
13 29 65 145 41 137 401 1105 145 849 3937 15713
1.92 2.79 4.45 7.51 15.24 47.89 208.28 1.07 103 2.69 103 5.07 104 1.77 106 8.95 107
4
8
Fig. 3. Comparison between two-dimensional girds with nodal level index k = 3.
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C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
Fig. 4. Model of a 2D thermal plate.
From the apparent errors obtain by PPM, we can conclude that the traditional perturbation method based on the first-order Taylor series is not suitable any more to the problem with large uncertainty level. Conversely, the maximum error obtained by the proposed fuzzy collocation methods does not exceed 1.7%, which means that relatively precise results can be derived by using the high-order polynomial approximation and collocation points in the entire uncertain space. Furthermore, the accuracy of FGFCM based on the full grids is higher than that of SGFCM where only the Smolyak sparse grids are retained.
Besides the accuracy, computational cost is another significant index to evaluate the performances of numerical methods. As listed in the tables, the sample number and execution time of FGFCM are 6561 and 49 seconds respectively, which are much smaller than those of MCM. The computational burden of FGFCM using full grids is completely acceptable for this low-dimensional fuzzy heat conduction problem. Besides, the sample number and execution time of SGFCM can be further compressed by using Smolyak algorithm despite introducing a certain accuracy decline. It should be noted that the ratio of execution time does not match the ratio of sample numbers. It is because constructing tensor product grids and filtering collocation points take up certain execution time besides the multiple computations of finite element model. In order to further investigate the convergence of proposed collocation methods with respect to the nodal level, the relative errors of 0-cut interval temperature response at feature point 4 are plotted in Fig. 5, from which we can observe that the relative errors decrease fast to stable values with respect to the increase of nodal level. Because the Smolyak sparse grids are a subset of the full grids, the relative errors obtained by FGFCM are always smaller than those yielded by SGFCM when the same nodal level is adopted. In order to further compare the accuracy of FGFCM and SGFCM when number of collocation points is approximately same, we take FGFCM at nodal level k = 2 and SGFCM at nodal level k = 4 for instance, where corresponding sample numbers are 625 and 401, respectively. From Fig. 5 we can see that the computational accuracy of SGFCM at nodal level k = 4 is higher than that of FGFCM at nodal level k = 2. It indicates that given approximately same
Table 2 Interval bounds of temperature responses under cut level k1 = 0. Point
Bound
1
LB UB LB UB LB UB LB UB LB UB LB UB LB UB
2 3 4 5 6 7
MCM (568 s) (105 samples)
PPM (0.5 s)
(°C)
(°C)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
45.40 63.88 49.19 70.11 51.66 74.09 53.32 76.58 54.45 78.74 55.01 79.87 56.63 82.95
43.10 66.83 46.10 74.14 48.07 78.83 49.35 81.79 50.26 84.28 50.72 85.56 52.02 89.13
5.07 4.62 6.28 5.75 6.95 6.40 7.45 6.80 7.70 7.04 7.80 7.12 8.14 7.45
45.76 63.41 49.63 69.50 52.15 73.41 53.85 75.83 55.02 77.94 55.62 79.04 57.30 81.98
0.79 0.74 0.89 0.87 0.95 0.92 0.99 0.98 1.05 1.02 1.11 1.04 1.18 1.17
45.85 63.26 49.77 69.31 52.35 73.15 54.06 75.55 55.24 77.63 55.85 78.68 57.55 81.63
0.99 0.97 1.18 1.14 1.34 1.27 1.39 1.34 1.45 1.41 1.53 1.49 1.62 1.59
FGFCM (49 s) (6561 samples)
SGFCM (21 s) (137 samples)
Table 3 Interval bounds of temperature responses under cut level k6 = 0.5. Point
Bound
1
LB UB LB UB LB UB LB UB LB UB LB UB LB UB
2 3 4 5 6 7
MCM (568 s) (105 samples)
PPM (0.5 s)
(°C)
(°C)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
49.94 59.18 54.26 64.72 57.06 68.28 58.91 70.54 60.26 72.40 60.94 73.37 62.87 76.03
48.79 60.65 52.72 66.74 55.27 70.65 56.93 73.15 58.16 75.17 58.80 76.22 60.56 79.12
2.30 2.48 2.84 3.12 3.14 3.47 3.36 3.70 3.48 3.83 3.51 3.88 3.67 4.06
50.12 58.94 54.48 64.42 57.31 67.94 59.18 70.17 60.54 72.00 61.25 72.96 63.20 75.54
0.36 0.41 0.41 0.46 0.44 0.50 0.46 0.52 0.46 0.55 0.51 0.56 0.52 0.64
50.16 58.87 54.55 64.32 57.41 67.81 59.28 70.03 60.65 71.85 61.36 72.78 63.33 75.37
0.44 0.52 0.53 0.62 0.61 0.69 0.63 0.72 0.65 0.76 0.69 0.80 0.73 0.87
FGFCM (49 s) (6561 samples)
SGFCM (21 s) (137 samples)
C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
637
Fig. 7. Model of a 3D sandwich structure. Fig. 5. Relative errors with respect to nodal level.
sample number, SGFCM with higher nodal level can obtain more accurate results before the convergence. Higher nodal level can obtain more accurate results, but huge additional computing cost will be unavoidably introduced because of the large number of collocation points. Thus, it is necessary to select a proper nodal level to balance the accuracy requirement and computational burden. Based on the fuzzy decomposition theorem, the membership functions of fuzzy temperature responses can be eventually reconstructed by connecting the different interval solutions for all cut levels. Take the fuzzy temperature responses at feature points 2 and 6 for instance, whose membership functions are plotted in
Fig. 6. The fuzzy input uncertainties cause a profound impact on output temperature responses. Specifically speaking, the temperature ranges derived by PPM contain the referenced results obtained by MCM, and the difference is particularly evident. The considerably overestimated results mean that the perturbation method neglecting high-order terms in Taylor series and Neumann series cannot be efficiently used in the heat conduction problem with large uncertainty level. However, non-conservative approximations are produced by the proposed fuzzy collocation methods, and match the referenced results perfectly. It demonstrates again that relatively precise results can be obtained based on the combination of Legendre-type surrogate function and collocation technology. 6.2. High-dimensional uncertain heat conduction problem
Fig. 6. Membership functions of fuzzy temperature responses.
In this numerical example, a 3D sandwich structure as shown in Fig. 7 is presented to evidence the feasibility of proposed methods for high-dimensional uncertain heat conduction problem with no less than five uncertain parameters. A total of 1280 quadrilateral elements are used to model the outer and inner faceplates, while the inclined webs are discretized into 720 quadrilateral elements. Two kinds of material are adopted to construct faceplates and inclined webs, respectively. Considering the manufacturing errors, the two heat conductivities are treated as Quasi-Gaussian fuzzy numbers k1 = h120,5,3i W/(m°C) and k2 = h204,8,3i W/(m°C). Besides, two fuzzy heat fluxes q1 = h15,0.5,3i W/m2 and q2 = h10,0.4,3i W/m2 are applied on the inclined webs. A part of heat transfers from outer faceplate to the ambient air with fuzzy temperature Te,1 = h25,1,3i °C and transfer coefficient h1 = h5,0.2,3i W/(m2°C), while another part of heat transfers from inner faceplate to the ambient air with fuzzy temperature Te,2 = h10,0.4,3i °C and transfer coefficient h2 = h20,1,3i W/(m2°C). All the uncertain parameters here are supposed to be independent of each other. Four nodes along the inclined web are selected as the feature points. Similar to the 2D thermal plate model, eleven values kq ¼ ðq 1Þ=10 q ¼ 1; . . . ; 11 between 0 and 1 are selected as the cut levels. Based on the third-order Legendre-type polynomial series and Clenshaw-Curtis nodes at nodal level k = 3, simulations for the high-dimensional heat conduction problem with eight uncertain parameters are carried out by Matlab R2014. The interval bounds of temperature responses at feature points under cut levels k1 = 0 and k8 = 0.7 are listed in Tables 4 and 5, in which the responses calculated by MCM with 108 samples and PPM are used for comparison. From the relative errors we can see that the temperature response ranges calculated by FGFCM and SGFCM match the referenced results perfectly. The accuracy of FGFCM in which the full grids are considered is higher than that of SGFCM where only a small subset of full grids is retained. However, the sample
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Table 4 Interval bounds of temperature responses under cut level k1 = 0. Point
Bound
1
LB UB LB UB LB UB LB UB
2 3 4
MCM (9685 s) (108 samples)
PPM (1.2 s)
(°C)
(°C)
68.98 103.03 63.57 95.74 51.10 77.17 46.06 70.43
61.41 112.02 57.47 103.18 45.52 83.88 41.68 75.80
FGFCM (4190 s) (43046721 samples)
SGFCM (208 s) (849 samples)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
10.97 8.73 9.60 7.77 10.92 8.70 9.51 7.62
69.71 102.04 64.21 94.92 51.63 76.46 46.51 69.84
1.06 0.96 1.01 0.86 1.04 0.92 0.98 0.84
70.14 101.48 64.59 94.36 51.95 76.04 46.81 69.43
1.68 1.50 1.60 1.44 1.66 1.46 1.63 1.42
Table 5 Interval bounds of temperature responses under cut level k8 = 0.7. Point
Bound
1
LB UB LB UB LB UB LB UB
2 3 4
MCM (9685 s) (108 samples)
PPM (1.2 s)
(°C)
(°C)
79.90 89.49 73.85 82.91 59.51 66.85 53.89 60.75
77.77 92.02 72.13 85.00 57.94 68.74 52.66 62.26
FGFCM (4190 s) (43046721 samples)
SGFCM (208 s) (849 samples)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
2.67 2.83 2.33 2.52 2.64 2.83 2.28 2.49
80.11 89.21 74.03 82.68 59.66 66.65 54.02 60.59
0.26 0.31 0.24 0.28 0.25 0.30 0.24 0.26
80.23 89.05 74.14 82.52 59.75 66.53 54.10 60.47
0.41 0.49 0.39 0.47 0.40 0.48 0.39 0.46
number adopted in FGFCM is 43046721, which is almost 5.07 104 times larger than that of SGFCM. Compared with the prominent improvement in computational cost, the little decrease of accuracy in SGFCM is completely acceptable. Therefore, in terms of the computational efficiency, SGFCM has a broader application prospect than FGFCM for the high-dimensional uncertain engineering problems. By connecting the different interval solutions, Fig. 8 depicts the membership functions of the fuzzy temperature responses at feature points 2 and 3. Influenced by the Quasi-Gaussian fuzzy input variables, the membership functions of output system responses obtained by the four numerical methods also satisfy the QuasiGaussian distribution. However, the results calculated by PPM go beyond the referenced bounds seriously, which indicates that the unacceptable errors caused by neglecting high-order is the inherent disadvantage of PPM for the uncertain problem with a mass of fuzzy parametric dimensions and a large uncertainty level. Relatively speaking, the proposed fuzzy collocation methods provide much better approximations for eventual temperature responses. Compared with FGFCM, the accuracy calculated by SGFCM is a little lower, but the sparse grids can avoid the huge computational cost in FGFCM caused by full grids, especially for the highdimensional case with a large number of uncertain parameters.
7. Conclusions
Fig. 8. Membership functions of fuzzy temperature responses.
This paper aims at developing new efficient methods for the fuzzy uncertainty propagation in heat conduction problems, where the multiple input uncertainties in material properties, external loads and boundary conditions are quantified as fuzzy variables. For this purpose, two fuzzy collocation methods, named as FGFCM and SGFCM, are presented based on the combination of collocation analysis technology and fuzzy modeling theory. In the 0-cut support box, the high-order Legendre polynomial series is adopted as the surrogate function of interval temperature response, which can be directly extended to other interval uncertain spaces under different cut levels. Different from the traditional stochastic spec-
C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
tral analysis where the probabilistic moments are calculated by the orthogonal relationship of polynomial bases, the temperature response ranges are evaluated using the smoothness property of surrogate function in this study. By preselecting Clenshaw-Curtis nodes in the support box and running deterministic codes, FGFCM and SGFCM are easy to implement for the expansion coefficients. However, the collocation points in FGFCM are directly constructed by the full grids. To overcome the drawback of FGFCM arising from the heavy computational burden for high-dimensional uncertain problems, the Smolyak algorithm is adopted to reconstruct the sparse grids in SGFCM. Comparing the computational accuracy and cost with traditional Monte Carlo method and parameter perturbation method, numerical results on a 2D thermal plate and a 3D sandwich structure demonstrate the remarkable feasibility and efficiency of proposed methods to solve fuzzy heat conduction problems. According to the polynomial approximation and collocation points distributed in the entire uncertain space, the applicability of fuzzy collocation methods is not affected by the uncertainty level of input parameters. Generally speaking, FGFCM adopting the full grids is usually applied in the low-dimensional uncertain problems due to its high computational accuracy; whereas SGFCM based on the sparse grids is more suitable to the highdimensional uncertain problems owing to its excellent computational efficiency. Besides the heat transfer problems, the proposed methods can be efficiently extended to other engineering problems such as structural analysis, acoustic field prediction and so on. Acknowledgements The project is supported by 111 Project (No.B07009), National Natural Science Foundation of the P.R. China (No.11432002, No.11372025) and Defense Industrial Technology Development Program (No.JCKY2013601B). Appendix: Fuzzy set theory Definition 1 (Triangular fuzzy number). A fuzzy number p is called triangular fuzzy number if the membership function lp ðxÞ can be defined as following form by three parameters
lp ðxÞ ¼
8 0 > > > xa <
ba
cx > > > : cb 0
x6a a6x6b b6x6c xPc
A parameter triplet p ¼ ða; b; cÞ is used to denote the triangular fuzzy number. Definition 2 (Quasi-Gaussian fuzzy number). The Quasi-Gaussian fuzzy number’s membership function is assumed to satisfy the Gaussian shape in a bounded domain
(
lp ðxÞ ¼
2 0Þ if jx x0 j < r r exp ðxx 2r2
0
if jx x0 j P rr
A parameter triplet p ¼ hx0 ; r; ri Quasi-Gaussian fuzzy number.
is
used
to
denote
the
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Collocation methods for fuzzy uncertainty propagation in heat conduction problem Chong Wang a, Zhiping Qiu a,⇑, Menghui Xu b a b
Institute of Solid Mechanics, Beihang University, Beijing 100191, PR China Faculty of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, Zhejiang 315211, PR China
a r t i c l e
i n f o
Article history: Received 10 April 2016 Received in revised form 8 October 2016 Accepted 25 October 2016
Keywords: Fuzzy uncertainty propagation Heat conduction Legendre polynomial series Collocation technology Smolyak algorithm
a b s t r a c t Based on the combination of collocation technology and fuzzy theory, this paper proposes a full grid fuzzy collocation method (FGFCM) and a sparse grid fuzzy collocation method (SGFCM) for fuzzy uncertainty propagation in heat conduction problem. Converting fuzzy parameters into interval variables by levelcut strategy, the Legendre polynomial series provides a surrogate function for temperature response. To calculate the expansion coefficients, FGFCM evaluates the deterministic solutions directly on the full tensor product grids, whereas Smolyak algorithm is introduced in SGFCM to reduce the number of collocation points. According to the smoothness property of surrogate function and fuzzy decomposition theorem, the interval bounds and membership functions of uncertain temperature response are derived, respectively. Comparing result with traditional Monte Carlo simulation and parameter perturbation method, two numerical examples evidence the remarkable accuracy and effectiveness of proposed methods for fuzzy temperature field prediction in engineering. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In the traditional numerical calculation, the parameters of computational model are always treated as deterministic values. However, for most practical engineering cases, various uncertainties representing the system variability are unavoidable due to the aggressive environment factors, incomplete knowledge and inevitable measurement errors. The uncertainty propagation analysis, which can effectively assess the impact of input uncertainties on the system output responses, has received an increasing amount of attention in recent years [1,2]. In probabilistic framework, the numerical approaches, such as Monte Carlo simulation, spectral analysis method and so on, can be considered as the most valuable strategies, where uncertainties are quantified as random variables or stochastic processes by using a great amount of sample statistical information [3,4]. But as pointed in Ref. [5], some important uncertain characteristics cannot be handled appropriately by probability theory, mainly because the concept of randomness is not necessarily the uncertainty source. Consequently, nonprobabilistic interval model, introduced by Moore [6], provides another method to deal with the imprecise uncertainties whose bounds are well-defined but sufficient information about the probability density function is missing. Interval analysis is capable ⇑ Corresponding author. E-mail address: [email protected] (Z. Qiu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.083 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
to predict the guaranteed changing ranges of output responses with respect to the interval input data [7,8]. In recent years, various interval approaches have been presented for the interval uncertainty propagation in engineering [9–11]. The fuzzy set theory, emerged from the work of Zadeh [12], is another efficient category to model the system uncertainty based on the subjective opinions. In practice, fuzzy sets can be viewed as the generalization of interval variables by introducing a membership function [13]. In other words, the level-cut strategy is usually adopted to subdivide the membership function range into some cut levels and convert the original fuzzy sets into some interval numbers. Up to now, the existing fuzzy analysis methods can be grouped into two categories. The first one is known as intervalbased approach, where the fuzzy parameter is converted into an interval variable using the cut-level operation, and then the uncertainty analysis is evaluated based on the classical interval arithmetic [14,15]. The computational cost of interval-based method is small, but its inherent disadvantage is the imprecise outputs caused by neglecting the correlation between many operands, especially for the case with large uncertainty level [16]. The second one is known as optimization-based approach, in which two optimization problems aiming at the maximum and minimum values of the output response will be solved [17,18]. Although the results are more accurate and the optimization tools can be employed in a straightforward manner, the huge computational cost for a large number of optimization problems embarrasses its application in
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complex engineering areas [19]. Considering both computational accuracy and computational cost, it is necessary to develop new numerical methods for fuzzy uncertainty propagation. Traditional numerical thermal analysis with given deterministic parameters has undergone a rapid development in engineering. But considering the unavoidable uncertainties in material properties, external loads and boundary conditions, the nondeterministic numerical models are more feasible [20]. Based on the generalized polynomial chaos, Xiu and Karniadakis presented a random spectral decomposition method for the solution of transient heat conduction subjected to random inputs [21]. For the uncertain heat convection-diffusion problem without sufficient information, Xue and Yang estimated the temperature intervals by using the finite element method and expansion techniques [22]. Combining the fuzzy theory with perturbation analysis, the fuzzy perturbation methods are proposed to predict the membership functions of temperature responses [23,24]. But it should be pointed that the low computational accuracy caused by the first-order Taylor series is the inherent disadvantage of fuzzy perturbation methods, especially for the nonlinear problem with large uncertainty level. Comparatively speaking, the collocation methods represent great superiority in computational accuracy and efficiency by using the high-order polynomial series and a set of deterministic computing procedure [25]. According to the easy implement, several collocation schemes have been developed to the engineering problems [26,27]. Nevertheless, current research on collocation methods is mainly concentrated in the stochastic field, while the combination of collocation method and fuzzy theory is promising but mostly unexplored [28]. The purpose of our present study is to develop new collocation methods combined with fuzzy theory to deal with the uncertain heat conduction problems with different uncertainty dimensions. The paper is structured as follows. The governing equation of heat conduction problem with fuzzy input parameters is firstly established in Section 2. Then, the basic theory of level-cut operation and polynomial approximation is introduced in Section 3. Subsequently, two fuzzy collocation methods are presented in the next two sections. The first one is FGFCM where the collocation points are directly constructed by the full tensor product grids. To improve the computational cost for high-dimensional problems, the Smolyak algorithm is adopted in SGFCM to reconstruct the sparse grids. In Section 6, two numerical examples about a 2D thermal plate and a 3D sandwich structure are provided to verify the feasibility of proposed methods, and we conclude the paper with a brief discussion at last. 2. Governing equation of fuzzy heat conduction problem
ð1Þ
where x 2 Rn ðn ¼ 1; 2; 3Þ denotes the spatial coordinate; X is a bounded domain; TðxÞ stands for the temperature response; kðxÞ represents the heat conductivity, and f ðxÞ is the intensity of heat source. For the interior domain X bounded by C, the following three kinds of boundary conditions are shown in Fig. 1
TjC1 ¼ T s @T k ¼ qs @n C2 @T k ¼ hðT T e Þ @n C3
where T s is the boundary temperature; n denotes the normal vector; qs stands for the heat flux; h and T e represent the heat transfer coefficient and ambient temperature, respectively. For the engineering heat conduction problem, due to the vaguely defined system characteristics, insufficient information and judgment subjectivity, uncertainties in material properties, external loads and boundary conditions are unavoidable. In this study, all the uncertain input parameters whose membership functions can be defined subjectively based on the expert opinions are modeled as n fuzzy parameters
a ¼ ðai Þn ¼ ða1 ; a2 ; . . . ; an Þ
ð3Þ
Then the governing Eq. (1) with multiple fuzzy inputs can be rewritten as
@ @Tðx; aÞ þ f ðx; aÞ ¼ 0 ðx; aÞ 2 X V kðx; aÞ @x @x
ð2Þ
ð4Þ
where V denotes the fuzzy uncertain space, and the temperature response TðxÞ become uncertain with respect to the fuzzy parameter vector a. 3. Level-cut operation and polynomial approximation Based on the level-cut strategy [29], each fuzzy parameter ai can be intersected to yield an interval variable aIi;k with respect to the cut level k 2 ½0; 1
aIi;k ¼ ½ai;k ; a i;k ¼ aci;k þ Dai;k dIi
ð5Þ
i;k are the lower and upper bounds of the interval where ai;k and a i;k þ ai;k Þ=2 and Dai;k ¼ ða i;k ai;k Þ=2 are called variable aIi;k ; aci;k ¼ ða the midpoint and the radius, respectively, dIi represents the standard interval dIi ¼ ½1; 1. Using vector denotation, all the k-cut interval variables can be written as aIk ¼ aIi;k . n
According to the definition of level-cut set, we can derive the following property
aIi;kj aIi;kk if kj 6 kk
In heat transfer analysis, the governing equation of steady heat conduction problem with a heat source can be expressed as
@ @TðxÞ kðxÞ þ f ðxÞ ¼ 0 x 2 X @x @x
Fig. 1. Three kinds of boundary conditions in heat conduction.
ð6Þ
Thus, by setting the cut level be zero, i.e. k = 0, the 0-cut bounded interval aIi;0 can be considered as the compact support of the fuzzy parameter ai , which means all the samples of fuzzy parameter ai belong to the closed interval aIi;0 . Subsequently, the product of different compact supports can be defined as the support box VI of the fuzzy uncertain space V
VI ¼
n Y
aIi;0 ¼ aI1;0 aI2;0 aIn;0
ð7Þ
i¼1
Under the cut level k = 0, Eq. (4) is converted into an interval equation with respect to the 0-cut interval vector aI0
@T x; aI0 @ k x; aI0 þ f x; aI0 ¼ 0 x; aI0 2 X VI @x @x
ð8Þ
where the temperature response T x; aI0 will change in a certain interval range.
C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
As we know, the traditional interval analysis methods based on Taylor series only use the midpoint information to approximately calculate the maximum and minimum values, whose accuracy for the nonlinear function becomes unacceptable if the uncertainty level of expansion parameters is not small enough. In order to overcome this shortcoming, the high-order polynomial series will be introduced as the surrogate function in this paper. Considering the uniform distribution of interval variables, the weight function is a constant, and the Legendre-type polynomials can be selected as the basis functions [30]. Let i ¼ ði1 ; i2 ; . . . ; in Þ be a multi-index, and the n-variate basis functions Ui aI0 of Legendre polynomial chaos are the products of the univariate basis functions
Ui aI0 ¼ Ui1 aI1;0 Ui2 aI2;0 Uin aIn;0
ð9Þ
where Ui1 aI1;0 ; . . . ; Uin aIn;0 denote the univariate basis functions of polynomial chaos. Based on Eq. (9), the interval temperature response T x; aI0 in Eq. (8) can be expanded as
X T x; aI0 ¼ wi ðxÞUi aI0
ð10Þ
i
where wi ðxÞ is the corresponding expansion coefficient. Considering the global approximation of Legendre surrogate model, optimal convergence can be achieved by using the polynomial chaos with proper basis [31]. In practice, finite terms are often selected to truncate above polynomial series. With the following N-order basis functions of total degree less than or equal to N
Ui aI0 ¼ Ui1 aI1;0 Ui2 aI2;0 Uin aIn;0
0 6 jij 6 N
ð11Þ
Eq. (10) can be concisely rewritten as
X T x; aI0 T N x; aI0 ¼ wi ðxÞUi aI0
ð12Þ
jij6N
where jij ¼ i1 þ i2 þ . . . þ in , and the total number of polynomial expansion terms can be calculated by C nnþN ¼ ðn þ NÞ!=ðn!N!Þ with respect to the number of variables n and the order of polynomial series N. Different from the spectral analysis in stochastic systems where the probabilistic moments such as expectation and variance are calculated by the orthogonal relationship of polynomial bases, the eventual aim of the present study is to predict the membership functions of fuzzy output responses. If the surrogate function of temperature response as shown in Eq. (12) is derived, it is easy to obtain its extreme points by making the first-order derivatives to be zero
I
@T N x; a0 ¼ 0 i ¼ 1; 2; . . . ; n @ aIi;0
ð13Þ
Based on the continuously-differentiable property of surrogate function T N x; aI0 , the extreme points and boundary points in the
I support box V are assumed to be a1 ; a2 ; . . . ; ar , and then the interval temperature response T x; aI0 can be approximately yielded by iterating through the finite points
I
I
T x; a0 ¼ ½T x; a0 ; T x; aI0 T x; aI0 T N x; aI0 ¼ min T N x; aj j¼1;...;r I I T x; a0 T N x; a0 ¼ max T N x; aj
function T N x; aI0 derived in the support box VI can still be used to approximate the temperature response in the subspace VIk
0 < k 6 1. Similarly, by iterating through the finite extreme
points and boundary points in VIk , we can obtain the interval temper ature response T x; aIk ¼ T x; aIk ; T x; aIk under the cut level k. Eventually, based on the fuzzy decomposition theorem, the membership function of uncertain temperature response with respect to original fuzzy parameter vector a can be reconstructed by connecting the different interval solutions for all cut levels kq
n o Tðx; aÞ ¼ [ kq T x; aIkq q
j¼1;...;r
When the cut level k is set to be other value between 0 and 1, we can derive the k-cut interval uncertain space VIk with the property VIk ¼ aI1;k aI2;k aIn;k # VI . It indicates that above surrogate
ð15Þ
From above derivation we can see that the pre-process of selecting polynomial bases and the post-process of calculating interval bounds and membership function of temperature response can be easily implemented by existing computing procedures. The crucial issue for fuzzy temperature field prediction is to calculate the expansion coefficients of surrogate function T N x; aI0 in Eq. (12). 4. Full grid fuzzy collocation method (FGFCM) The aim of collocation method is to calculate expansion coefficients by using the deterministic responses at preselected points in uncertain space. The main advantage is that the existing deterministic model and computing procedure can be fully utilized without any further modification. In this section, we will extend the collocation technology based on full grids to the fuzzy uncertainty analysis. The first important aspect of collocation method is the selection of nodal set. The Clenshaw-Curtis nodes, which are the extreme points of Chebyshev polynomials, are one of the many kinds of collocation points in computational mathematics [32]. They can minimize the collocation error with the same number of nodes under the maximum norm. For the univariate case, the Clenshaw-Curtis i;0 of the ith fuzzy nodes distributed in the 0-cut interval ½ai;0 ; a parameter are defined as
(
ðiÞ bj
¼
aci;0 if mi ¼ 1 aci;0 cos pmðj1Þ Dai;0 j ¼ 1; 2; . . . ; mi if mi > 1 i 1
ð16Þ
where mi is the predefined number of collocation points. It is obvious that the collocation points locate in the entire interval range i;0 , and the higher the value of mi is, the more intensive the ½ai;0 ; a collocation points will be. For the multivariate case, the most straightforward approach to extend the collocation points defined in one dimension to the entire support box VI is to use the tensor product method. For each n o ðiÞ ðiÞ m m and an operator Oi i dimension, a nodal set Hi i ¼ b1 ; b2 ; . . . ; bðiÞ mi are introduced to denote the collocation points and the approximate function, respectively. Using the tensor product method directly, the operator O in the n-dimensional support box VI can be constructed as m2 mn 1 O ¼ Om 1 O2 On
ð17Þ
On this basis, one can obtain the entire nodal set H m2 mn 1 H ¼ Hm 1 H2 Hn
ð14Þ
633
ð18Þ
where the total number of collocation points M in the full grids is
M¼
n Y
mi ¼ m1 m2 mn
ð19Þ
i¼1
For large dimensions n 1, the number M will be particularly large even though the number of collocation points mi in each one-dimensional space is small, which poses a numerical challenge
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called the curse of dimensionality because each collocation point needs a simulation of the full-scale underlying deterministic system. Therefore, the collocation point construction of full grids is mostly used for the low-dimensional uncertain engineering problems. After selecting the nodal set, another important aspect of collocation method is to calculate the expansion coefficients. A set n o is given to denote the M collocation H ¼ bnode ; bnode ; . . . ; bnode 1 2 M is used to represent the deterpoints specifically, and T x; bnode j ministic solution to the governing Eq. (8) at the preselected point
For the two-dimensional case n ¼ 2, the operator O in Eq. (22) can be written as
X
O¼
i1 þi2 6L
X
¼
ð1Þ
i1 þi2 ¼L
1 L jij
For the practical engineering heat conduction problem, the deterministic solutions T x; bnode at all collocation points can be j calculated by the existing efficient finite element approach [33]. Based on the surrogate function in Eq. (12), a group of linear equation with respect to the unknown expansion coefficients wi ðxÞ can be derived as
U1 bnode 1 B B B U1 bnode 2 B B B. B .. @ U1 bnode M
1 U2 bnode U3 bnode UC nnþN bnode 1 1 1 C C C U2 bnode U3 bnode UC nnþN bnode 2 2 2 C C C .. .. . . .. C . . . . A node node node U2 bM U3 bM UC nnþN bM 1 0 1 0 T x; bnode 1 w1 ðxÞ B C C C B B C B w2 ðxÞ C B T x; bnode 2 C C B B C C¼B B .. C C B. B. C A B .. @ @ A wC nnþN ðxÞ node T x; bM
0
Oi11 Oi22
X
i1 þi2 ¼L1
ð23Þ
ð1ÞLjij
n1 L jij
Oi11 Oi22 Oinn
[
H¼
Hi11 Hi22 Hinn
ð25Þ
Lnþ16jij6L
By iterating through all the nodes on full grids, only the ones whose sum order jij ¼ i1 þ i2 þ . . . þ in across all dimensions is between L n þ 1 and L are retained. Compared to the full grids, the Smolyak algorithm provides an interpolation strategy with potentially orders of magnitude reduction in the number of support nodes required, and thus we may as well call the new tensor product grids ‘sparse grids’. For the jth one-dimensional space with the collocation level ij , i
of the Clenshaw-Curtis nodes the number mjj and positions bðjÞ s j;0 can be defined as distributed in the 0-cut interval ½aj;0 ; a
(
Once the polynomial bases Ui and the collocation nodal set H are provided, the expansion coefficients wi ðxÞ in Eq. (12) can always be determined by solving Eq. (21). To prevent the problem from becoming undetermined, the number of collocation points is required not to be smaller than the number of polynomial expansion terms, i.e. M P C nnþN . In this case, the generalized matrix inversion approach based on the least square principle is usually adopted to solve above linear equations. 5. Sparse grid fuzzy collocation method (SGFCM)
mjj ¼ bsðjÞ
¼
1
if ij ¼ 1
2ij 1 þ 1 if ij > 1
8 c < aj;0
: acj;0 cos
if ij ¼ 1 pðs1Þ ij
mj 1
Daj;0 s ¼
i 1; 2; . . . ; mjj
With different level indices, Fig. 2 depicts the one-dimensional i
i þ1
nodal sets, where it can be seen Hjj Hjj
ij ¼ 1; 2; . . ., which
means the Clenshaw-Curtis nodal sets are nested. Since the construction in Eq. (24) employs one-dimensional interpolations with various numbers of nodes, it is preferable the one-dimensional
i 1
Assumed Dik ¼ Okk Okk and O0k ¼ 0, the multivariate interpolation formula can be defined as follows based on the Smolyak algorithm
X i D 1 Di 2 Di n
ð22Þ
jij6L
where L P n is an integer representing the overall construction level, and ij denotes the separate level along the jth direction.
ð26Þ
if ij > 1
As mentioned above, the huge computational cost is considered as the inherent disadvantage of FGFCM for high-dimensional problems. In order to avoid the curse of dimensionality, the sparse grids, which are still based on the tensor product construction but only a small subset of the full grids, will be reconstructed in this section by using the Smolyak algorithm [34].
O¼
ð24Þ
In the process of extending univariate interpolation formulae to multivariate case, the Smolyak algorithm provides a linear combination of product formulae, which is selected in such a way that the interpolation property in each dimension is preserved for higher dimensions. The specific derivation can refer to Refs. [35] and [36]. Subsequently, the new tensor product grids corresponding to the operator in Eq. (24) can be constructed as
i
i
Equivalently, the operator O in FGFCM for the n-dimensional sup-
Lnþ16jij6L
ð20Þ
Oi11 Oi22
X
port box VI becomes
O¼
1 node @ @ node @T x; bj A þ f x; bnode ¼ 0 j ¼ 1; 2; . . . ; M k x; bj j @x @x
Ljij
L16jij6L
bnode j
0
X i Di1 Di2 ¼ O11 Oi22
Fig. 2. One-dimensional nodal sets with different level indices.
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C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
nodal sets be nested, where the total number of nodes in Eq. (25) can reach a minimum. That is the other reason why we chose Clenshaw-Curtis nodes as the collocation points in this study. In Eq. (25), there is no explicit formula to calculate the number M in terms of n and L. To simplify symbol expression, an additional index k is introduced to denote the level of collocation points uniformly. Let L ¼ n þ k, then the total number of collocation points M in sparse grids can be approximately expressed in terms of n and k for n 1
M ¼ dimðHÞ ¼ dim
[
Hi11 Hi22 Hinn
!
Lnþ16jij6L k
2 k n k!
if n 1
ð27Þ
Given the level index k = 3, the comparison between full girds and sparse girds in the two-dimensional 0-cut interval uncertain i;0 ½aj;0 ; a j;0 is shown in Fig. 3, in which we can space ½ai;0 ; a observe a significant reduction in the number of collocation points for sparse grids. Besides, with different dimensions and nodal level indices, the numbers of collocation points in the two proposed fuzzy collocation methods are listed in Table 1. Compared with the full grids in FGFCM, the curse of dimensionality for sparse grids in SGFCM, albeit still present, is significantly lessened.
In both FGFCM and SGFCM, the polynomial series and matrix inversion approach are adopted to construct the surrogate function of uncertain temperature response in the pre-process, while the smoothness property of polynomial function and the fuzzy decomposition theorem are used to calculate the temperature ranges and eventual membership function in the post-process. The main difference between the two fuzzy collocation methods is that the deterministic codes at collocation points for FGFCM are carried out on the full grids; whereas, only the sparse grids are retained in SGFCM by using Smolyak algorithm. Thus, if given the same nodal level, the computational accuracy of FGFCM will be higher than that of SGFCM, but the computational cost of SGFCM will be much smaller, especially for the engineering problems with large dimensions. 6. Numerical examples 6.1. Low-dimensional uncertain heat conduction problem In order to verify the performance of proposed methods for lowdimensional uncertain heat conduction problem with less than five uncertain parameters, consider a 2D thermal plate as shown in Fig. 4, where the rectangular-sectioned part and the circular part are discretized by 100 quadrilateral elements and 188 triangular elements, respectively. There exists a continuous volumetric heat in the shaded portion, and a heat flux is imposed on the bottom surface. Seven feature points are selected to characterize the structural temperature filed. Due to aggressive environment factors and inevitable measurement errors, material properties, external loads and boundary conditions contain some subjective uncertainties, where four fuzzy parameters with the triangular-type membership functions are considered as uncertain input variables, i.e. heat conductivity k = (174,204,234) W/(m°C), heat flux qs = (750,900,1050) W/m2, volumetric heat Q = (1700,2000,2300) W/m3, boundary temperature Ts = (42,50,58) °C. Based on the level-cut strategy, eleven cut levels kq ¼ ðq 1Þ=10 q ¼ 1; . . . ; 11 are used to transform all the original fuzzy parameters into interval variables. The third-order Legendretype polynomial series is adopted to construct the surrogate function of temperature response with respect to support box under the 0-cut level. Given the Clenshaw-Curtis collocation points at nodal level k = 3, simulations of the proposed fuzzy collocation methods are carried out by Matlab R2014. The lower bound (LB) and upper bound (UB) of interval temperature responses at seven feature points under cut levels k1 = 0 and k6 = 0.5 are listed in Tables 2 and 3, where the response intervals calculated by the traditional Monte Carlo method (MCM) with 105 samples are adopted as referenced results. Besides, the commonly used parameter perturbation method (PPM) is introduced for further comparison.
Table 1 Comparison between the numbers of collocation points. Dimension (n)
Nodal level index (k)
M1 (FGFCM)
M2 (SGFCM)
M1/M2
2
2 3 4 5 2 3 4 5 2 3 4 5
25 81 289 1089 625 6561 83521 1185921 390625 43046721 6.9758 109 1.4064 1012
13 29 65 145 41 137 401 1105 145 849 3937 15713
1.92 2.79 4.45 7.51 15.24 47.89 208.28 1.07 103 2.69 103 5.07 104 1.77 106 8.95 107
4
8
Fig. 3. Comparison between two-dimensional girds with nodal level index k = 3.
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C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
Fig. 4. Model of a 2D thermal plate.
From the apparent errors obtain by PPM, we can conclude that the traditional perturbation method based on the first-order Taylor series is not suitable any more to the problem with large uncertainty level. Conversely, the maximum error obtained by the proposed fuzzy collocation methods does not exceed 1.7%, which means that relatively precise results can be derived by using the high-order polynomial approximation and collocation points in the entire uncertain space. Furthermore, the accuracy of FGFCM based on the full grids is higher than that of SGFCM where only the Smolyak sparse grids are retained.
Besides the accuracy, computational cost is another significant index to evaluate the performances of numerical methods. As listed in the tables, the sample number and execution time of FGFCM are 6561 and 49 seconds respectively, which are much smaller than those of MCM. The computational burden of FGFCM using full grids is completely acceptable for this low-dimensional fuzzy heat conduction problem. Besides, the sample number and execution time of SGFCM can be further compressed by using Smolyak algorithm despite introducing a certain accuracy decline. It should be noted that the ratio of execution time does not match the ratio of sample numbers. It is because constructing tensor product grids and filtering collocation points take up certain execution time besides the multiple computations of finite element model. In order to further investigate the convergence of proposed collocation methods with respect to the nodal level, the relative errors of 0-cut interval temperature response at feature point 4 are plotted in Fig. 5, from which we can observe that the relative errors decrease fast to stable values with respect to the increase of nodal level. Because the Smolyak sparse grids are a subset of the full grids, the relative errors obtained by FGFCM are always smaller than those yielded by SGFCM when the same nodal level is adopted. In order to further compare the accuracy of FGFCM and SGFCM when number of collocation points is approximately same, we take FGFCM at nodal level k = 2 and SGFCM at nodal level k = 4 for instance, where corresponding sample numbers are 625 and 401, respectively. From Fig. 5 we can see that the computational accuracy of SGFCM at nodal level k = 4 is higher than that of FGFCM at nodal level k = 2. It indicates that given approximately same
Table 2 Interval bounds of temperature responses under cut level k1 = 0. Point
Bound
1
LB UB LB UB LB UB LB UB LB UB LB UB LB UB
2 3 4 5 6 7
MCM (568 s) (105 samples)
PPM (0.5 s)
(°C)
(°C)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
45.40 63.88 49.19 70.11 51.66 74.09 53.32 76.58 54.45 78.74 55.01 79.87 56.63 82.95
43.10 66.83 46.10 74.14 48.07 78.83 49.35 81.79 50.26 84.28 50.72 85.56 52.02 89.13
5.07 4.62 6.28 5.75 6.95 6.40 7.45 6.80 7.70 7.04 7.80 7.12 8.14 7.45
45.76 63.41 49.63 69.50 52.15 73.41 53.85 75.83 55.02 77.94 55.62 79.04 57.30 81.98
0.79 0.74 0.89 0.87 0.95 0.92 0.99 0.98 1.05 1.02 1.11 1.04 1.18 1.17
45.85 63.26 49.77 69.31 52.35 73.15 54.06 75.55 55.24 77.63 55.85 78.68 57.55 81.63
0.99 0.97 1.18 1.14 1.34 1.27 1.39 1.34 1.45 1.41 1.53 1.49 1.62 1.59
FGFCM (49 s) (6561 samples)
SGFCM (21 s) (137 samples)
Table 3 Interval bounds of temperature responses under cut level k6 = 0.5. Point
Bound
1
LB UB LB UB LB UB LB UB LB UB LB UB LB UB
2 3 4 5 6 7
MCM (568 s) (105 samples)
PPM (0.5 s)
(°C)
(°C)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
49.94 59.18 54.26 64.72 57.06 68.28 58.91 70.54 60.26 72.40 60.94 73.37 62.87 76.03
48.79 60.65 52.72 66.74 55.27 70.65 56.93 73.15 58.16 75.17 58.80 76.22 60.56 79.12
2.30 2.48 2.84 3.12 3.14 3.47 3.36 3.70 3.48 3.83 3.51 3.88 3.67 4.06
50.12 58.94 54.48 64.42 57.31 67.94 59.18 70.17 60.54 72.00 61.25 72.96 63.20 75.54
0.36 0.41 0.41 0.46 0.44 0.50 0.46 0.52 0.46 0.55 0.51 0.56 0.52 0.64
50.16 58.87 54.55 64.32 57.41 67.81 59.28 70.03 60.65 71.85 61.36 72.78 63.33 75.37
0.44 0.52 0.53 0.62 0.61 0.69 0.63 0.72 0.65 0.76 0.69 0.80 0.73 0.87
FGFCM (49 s) (6561 samples)
SGFCM (21 s) (137 samples)
C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
637
Fig. 7. Model of a 3D sandwich structure. Fig. 5. Relative errors with respect to nodal level.
sample number, SGFCM with higher nodal level can obtain more accurate results before the convergence. Higher nodal level can obtain more accurate results, but huge additional computing cost will be unavoidably introduced because of the large number of collocation points. Thus, it is necessary to select a proper nodal level to balance the accuracy requirement and computational burden. Based on the fuzzy decomposition theorem, the membership functions of fuzzy temperature responses can be eventually reconstructed by connecting the different interval solutions for all cut levels. Take the fuzzy temperature responses at feature points 2 and 6 for instance, whose membership functions are plotted in
Fig. 6. The fuzzy input uncertainties cause a profound impact on output temperature responses. Specifically speaking, the temperature ranges derived by PPM contain the referenced results obtained by MCM, and the difference is particularly evident. The considerably overestimated results mean that the perturbation method neglecting high-order terms in Taylor series and Neumann series cannot be efficiently used in the heat conduction problem with large uncertainty level. However, non-conservative approximations are produced by the proposed fuzzy collocation methods, and match the referenced results perfectly. It demonstrates again that relatively precise results can be obtained based on the combination of Legendre-type surrogate function and collocation technology. 6.2. High-dimensional uncertain heat conduction problem
Fig. 6. Membership functions of fuzzy temperature responses.
In this numerical example, a 3D sandwich structure as shown in Fig. 7 is presented to evidence the feasibility of proposed methods for high-dimensional uncertain heat conduction problem with no less than five uncertain parameters. A total of 1280 quadrilateral elements are used to model the outer and inner faceplates, while the inclined webs are discretized into 720 quadrilateral elements. Two kinds of material are adopted to construct faceplates and inclined webs, respectively. Considering the manufacturing errors, the two heat conductivities are treated as Quasi-Gaussian fuzzy numbers k1 = h120,5,3i W/(m°C) and k2 = h204,8,3i W/(m°C). Besides, two fuzzy heat fluxes q1 = h15,0.5,3i W/m2 and q2 = h10,0.4,3i W/m2 are applied on the inclined webs. A part of heat transfers from outer faceplate to the ambient air with fuzzy temperature Te,1 = h25,1,3i °C and transfer coefficient h1 = h5,0.2,3i W/(m2°C), while another part of heat transfers from inner faceplate to the ambient air with fuzzy temperature Te,2 = h10,0.4,3i °C and transfer coefficient h2 = h20,1,3i W/(m2°C). All the uncertain parameters here are supposed to be independent of each other. Four nodes along the inclined web are selected as the feature points. Similar to the 2D thermal plate model, eleven values kq ¼ ðq 1Þ=10 q ¼ 1; . . . ; 11 between 0 and 1 are selected as the cut levels. Based on the third-order Legendre-type polynomial series and Clenshaw-Curtis nodes at nodal level k = 3, simulations for the high-dimensional heat conduction problem with eight uncertain parameters are carried out by Matlab R2014. The interval bounds of temperature responses at feature points under cut levels k1 = 0 and k8 = 0.7 are listed in Tables 4 and 5, in which the responses calculated by MCM with 108 samples and PPM are used for comparison. From the relative errors we can see that the temperature response ranges calculated by FGFCM and SGFCM match the referenced results perfectly. The accuracy of FGFCM in which the full grids are considered is higher than that of SGFCM where only a small subset of full grids is retained. However, the sample
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Table 4 Interval bounds of temperature responses under cut level k1 = 0. Point
Bound
1
LB UB LB UB LB UB LB UB
2 3 4
MCM (9685 s) (108 samples)
PPM (1.2 s)
(°C)
(°C)
68.98 103.03 63.57 95.74 51.10 77.17 46.06 70.43
61.41 112.02 57.47 103.18 45.52 83.88 41.68 75.80
FGFCM (4190 s) (43046721 samples)
SGFCM (208 s) (849 samples)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
10.97 8.73 9.60 7.77 10.92 8.70 9.51 7.62
69.71 102.04 64.21 94.92 51.63 76.46 46.51 69.84
1.06 0.96 1.01 0.86 1.04 0.92 0.98 0.84
70.14 101.48 64.59 94.36 51.95 76.04 46.81 69.43
1.68 1.50 1.60 1.44 1.66 1.46 1.63 1.42
Table 5 Interval bounds of temperature responses under cut level k8 = 0.7. Point
Bound
1
LB UB LB UB LB UB LB UB
2 3 4
MCM (9685 s) (108 samples)
PPM (1.2 s)
(°C)
(°C)
79.90 89.49 73.85 82.91 59.51 66.85 53.89 60.75
77.77 92.02 72.13 85.00 57.94 68.74 52.66 62.26
FGFCM (4190 s) (43046721 samples)
SGFCM (208 s) (849 samples)
Error (%)
(°C)
Error (%)
(°C)
Error (%)
2.67 2.83 2.33 2.52 2.64 2.83 2.28 2.49
80.11 89.21 74.03 82.68 59.66 66.65 54.02 60.59
0.26 0.31 0.24 0.28 0.25 0.30 0.24 0.26
80.23 89.05 74.14 82.52 59.75 66.53 54.10 60.47
0.41 0.49 0.39 0.47 0.40 0.48 0.39 0.46
number adopted in FGFCM is 43046721, which is almost 5.07 104 times larger than that of SGFCM. Compared with the prominent improvement in computational cost, the little decrease of accuracy in SGFCM is completely acceptable. Therefore, in terms of the computational efficiency, SGFCM has a broader application prospect than FGFCM for the high-dimensional uncertain engineering problems. By connecting the different interval solutions, Fig. 8 depicts the membership functions of the fuzzy temperature responses at feature points 2 and 3. Influenced by the Quasi-Gaussian fuzzy input variables, the membership functions of output system responses obtained by the four numerical methods also satisfy the QuasiGaussian distribution. However, the results calculated by PPM go beyond the referenced bounds seriously, which indicates that the unacceptable errors caused by neglecting high-order is the inherent disadvantage of PPM for the uncertain problem with a mass of fuzzy parametric dimensions and a large uncertainty level. Relatively speaking, the proposed fuzzy collocation methods provide much better approximations for eventual temperature responses. Compared with FGFCM, the accuracy calculated by SGFCM is a little lower, but the sparse grids can avoid the huge computational cost in FGFCM caused by full grids, especially for the highdimensional case with a large number of uncertain parameters.
7. Conclusions
Fig. 8. Membership functions of fuzzy temperature responses.
This paper aims at developing new efficient methods for the fuzzy uncertainty propagation in heat conduction problems, where the multiple input uncertainties in material properties, external loads and boundary conditions are quantified as fuzzy variables. For this purpose, two fuzzy collocation methods, named as FGFCM and SGFCM, are presented based on the combination of collocation analysis technology and fuzzy modeling theory. In the 0-cut support box, the high-order Legendre polynomial series is adopted as the surrogate function of interval temperature response, which can be directly extended to other interval uncertain spaces under different cut levels. Different from the traditional stochastic spec-
C. Wang et al. / International Journal of Heat and Mass Transfer 107 (2017) 631–639
tral analysis where the probabilistic moments are calculated by the orthogonal relationship of polynomial bases, the temperature response ranges are evaluated using the smoothness property of surrogate function in this study. By preselecting Clenshaw-Curtis nodes in the support box and running deterministic codes, FGFCM and SGFCM are easy to implement for the expansion coefficients. However, the collocation points in FGFCM are directly constructed by the full grids. To overcome the drawback of FGFCM arising from the heavy computational burden for high-dimensional uncertain problems, the Smolyak algorithm is adopted to reconstruct the sparse grids in SGFCM. Comparing the computational accuracy and cost with traditional Monte Carlo method and parameter perturbation method, numerical results on a 2D thermal plate and a 3D sandwich structure demonstrate the remarkable feasibility and efficiency of proposed methods to solve fuzzy heat conduction problems. According to the polynomial approximation and collocation points distributed in the entire uncertain space, the applicability of fuzzy collocation methods is not affected by the uncertainty level of input parameters. Generally speaking, FGFCM adopting the full grids is usually applied in the low-dimensional uncertain problems due to its high computational accuracy; whereas SGFCM based on the sparse grids is more suitable to the highdimensional uncertain problems owing to its excellent computational efficiency. Besides the heat transfer problems, the proposed methods can be efficiently extended to other engineering problems such as structural analysis, acoustic field prediction and so on. Acknowledgements The project is supported by 111 Project (No.B07009), National Natural Science Foundation of the P.R. China (No.11432002, No.11372025) and Defense Industrial Technology Development Program (No.JCKY2013601B). Appendix: Fuzzy set theory Definition 1 (Triangular fuzzy number). A fuzzy number p is called triangular fuzzy number if the membership function lp ðxÞ can be defined as following form by three parameters
lp ðxÞ ¼
8 0 > > > xa <
ba
cx > > > : cb 0
x6a a6x6b b6x6c xPc
A parameter triplet p ¼ ða; b; cÞ is used to denote the triangular fuzzy number. Definition 2 (Quasi-Gaussian fuzzy number). The Quasi-Gaussian fuzzy number’s membership function is assumed to satisfy the Gaussian shape in a bounded domain
(
lp ðxÞ ¼
2 0Þ if jx x0 j < r r exp ðxx 2r2
0
if jx x0 j P rr
A parameter triplet p ¼ hx0 ; r; ri Quasi-Gaussian fuzzy number.
is
used
to
denote
the
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