Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties

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ScienceDirect Comput. Methods Appl. Mech. Engrg. 355 (2019) 1–26 www.elsevier.com/locate/cma

Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties X.Y. Longa , D.L. Maoa , C. Jianga ,∗, F.Y. Weib , G.J. Lib a

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, PR China b Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, Sichuan, 621000, PR China Received 22 March 2019; received in revised form 15 May 2019; accepted 24 May 2019 Available online xxxx

Highlights • • • •

Unified uncertainty analysis (UUA) under probabilistic, evidence, fuzzy and interval uncertainties is performed. Taylor expansion-based UUA method for small uncertainty problems is proposed. Dimensional reduction/efficient global optimization-based UUA method for large uncertainty problems is established. Effectiveness of the proposed methods is verified by three numerical examples.

Abstract The uncertainty analysis of structures generally involves uncertain parameters of different types. In order to derive predictions regarding uncertain structural responses, it is crucial to represent the uncertainty appropriately according to the underlying information available. This paper presents a unified framework for uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties, by which the quantities with sufficient data, sparse data, and subjective information can be simultaneously considered. A Taylor expansion-based unified uncertainty analysis (T-UUA) method is first proposed for small uncertainty problems. By temporarily neglecting the evidence, fuzzy and interval variables, the probability-evidence-interval-fuzzy model is degraded into a random problem, in which the expectations and variances of responses can be obtained as functions in terms of evidence, interval and fuzzy uncertainties. Then, through dealing with the evidence variables, the previous expectations and variances are further expressed as a summation of functions in terms of fuzzy and interval variables with basic probability assignments (BPAs). The fuzziness is then discretized by using α-cut technique and thus the expectations and variances are further expressed as functions of only intervals. Afterwards, by reconsidering the interval uncertainties, the bounds of the expectations and variances are computed via combining Taylor expansion with interval arithmetic. In addition, a dimensional reduction (DR)/efficient global optimization (EGO)-based unified uncertainty analysis (DR/EGO-UUA) method is also presented to solve the large uncertainty problems. The framework of DR/EGO-UUA is similar as T-UUA. However, in DR/EGO-UUA, the second moments of responses are computed by DR integrations, and their upper and lower bounds are calculated by the EGO. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed methods. c 2019 Published by Elsevier B.V. ⃝ Keywords: Unified uncertainty analysis; Evidence theory; Fuzzy; Interval; Dimensional reduction; Efficient global optimization

∗

Corresponding author. E-mail address: [email protected] (C. Jiang).

https://doi.org/10.1016/j.cma.2019.05.041 c 2019 Published by Elsevier B.V. 0045-7825/⃝

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1. Introduction Due to the influences of manufacturing errors and unpredictability of environment, uncertainties generally exist in practical engineering structures, including the model inaccuracies, physical imperfections, multiphase characteristics of materials, etc. It is necessary to take these uncertainties into consideration to guarantee the reliability, robustness and safety of structures. To this end, the uncertainty analysis approaches have been increasingly researched and attended [1–5]. In the uncertainty analysis, probabilistic approaches have been vastly investigated. Representative but not exhaustive methods include robust design [6–9], reliability-based design [10–13], and multidisciplinary optimization under uncertainty [14,15]. The required probabilistic modeling can be realized via classical mathematical statistics if sufficient data with high quality is available. In engineering practice, however, the data available can sometimes be quite limited. Uncertainty quantification based on probabilistic models may encounter difficulty since there are not enough samples to construct the precise probability distribution functions [16]. The information of uncertainties can appear as imprecise, incomplete, ambiguous, or linguistic. Besides, the information may variously be obtained from objective random sample data and subjective expert opinion or team consensus. To quantify these uncertainties suitably, besides the probabilistic model, many other uncertainty models have been developed, including the evidence theory [17], interval model [18], and fuzzy set [19], etc. Evidence theory often provides a suitable basis for an appropriate quantification when the information available is recorded with coarse specifications. For example, when the status of a system is monitored at discrete time instants, we can only collect information to estimate the probability of the failure occurrence over each time interval [20]. There is typically no probabilistic information available to specify distribution functions for these coarse specifications, so that modeling such quantities as sets is most appropriate. Interval model [18] is applicable to the uncertain-but-bounded quantities which only require the upper and lower bounds. For example, due to the manufacturing error, generally only the upper and lower bounds of a practical structural size after processing can be known through its nominal value and manufacture tolerance. The fuzzy model, however, is suitable for problems when only fuzzy information is available [19,21]. For example, when the characteristic of yield stress is expressed in the form of linguistic variables such as “bad”, “good”, and “excellent”, it is typically described by fuzzy sets so as to be formulated mathematically and processed by computers [22]. Though there exist different uncertain models, whether we are using probability, evidence theory, interval, or fuzzy set for uncertainty quantification, we are drinking the same water of truth from different sides of the same well. When different kinds of uncertainties are involved in a problem, it seems a challenge to formulate an effective analysis method, on one hand, without ignoring significant information and, on the other hand, without introducing unwarranted assumptions. If this balance is violated or not achieved, computational results may deviate significantly from reality, and the associated decisions may lead to serious consequences [16]. For these reasons, many unified uncertainty analysis (UUA) methods which are capable of integrating different kinds of uncertainties into a unified framework, have been proposed in recent years. For examples, Du [23] presented an approach to quantify the effects of mixed random and interval inputs on the reliability of a structure. Elishakoff and Colombi [24] combined probabilistic and convex models to deal with the uncertain acoustic excitation problem when knowledge is scarce. Jiang et al. [25] developed a hybrid probability and interval reliability model where the probability distribution parameters were treated as intervals, and further applied it to the fracture reliability analysis existing epistemic uncertainty [26]. Du [27] proposed a unified uncertainty analysis method considering probability and epistemic uncertainties by using evidence theory, based on which the corresponding sensitivity analysis on reliability [20] was further conducted. Xiao et al. [28] then provided a unified uncertainty analysis method based on the mean value first order saddle point approximation. Kang and Luo [11] developed a hybrid reliability model based on probability and convex model. Balu and Rao [29] presented an efficient uncertainty analysis method for estimating structural reliability in the presence of mixed random and fuzzy uncertainties. Wang et al. [30] proposed a unified uncertainty analysis method based on fuzzy and random variables. L¨u et al. [31] presented a unified stability analysis approach with two types of random-fuzzy uncertainties. Wang et al. [32] proposed a novel methodology of reliability-based multidisciplinary design optimization under hybrid interval and fuzzy uncertainties. Though important progress has been made in UUA, it also should be pointed out that most of the existing works focus on the problems with mixture of only two kinds of uncertainties, such as probability and interval, probability and evidence theory, random and fuzzy, interval and fuzzy. And so far there exist very few studies focusing on the UUA analysis which involves more than two kinds of uncertainties. However, with the development of modern industry, the encountered problems become more and more complex. In many cases, there will be more than two or even three kinds of uncertainties

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in the same problem or structure. In order to make the uncertainty analysis results of such engineering problems more accurate and credible, it is very important to develop a more powerful UUA method that can deal with more types of uncertainties at the same time. This paper thus aims to develop two more powerful UUA methods for complex engineering problems, in which the four kinds of uncertainties, namely probability, evidence theory, interval and fuzzy set could be treated in a framework. Firstly, a first order Taylor expansion-based unified uncertainty analysis (T-UUA) approach is presented for problems with small uncertainty. Secondly, by introducing dimensional reduction (DR) integration and efficient global optimization (EGO), another method named DR/EGO-UUA is also proposed for problems with relatively large uncertainty. The remainder of this paper is organized as follows. Section 2 summarizes the basic concepts of different kinds of uncertainties; Sections 3 and 4 present the formulation of the proposed T-UUA and DR/EGO-UUA methods; Section 5 conducts the analysis of three numerical examples; Section 6 gives the conclusion of the paper. 2. Basics of three typical uncertain models 2.1. Evidence theory Evidence theory, also known as Dempster–Shafer theory or D–S theory, was first proposed by Dempster [33] and Shafer [17]. Evidence theory is the generalization of probability theory and possibility theory [34,35]. To introduce the basic concept of evidence theory, for convenience, we will use Y to denote an uncertain parameter modeled by evidence theory as well as its sample space that contains all possible values of Y, and the evidence theory-based variable is abbreviated The { as } “evidence { }variable”. { } {sample space is called } a frame of discernment. Possible subsets of Y include {∅}, CYi , CY1 , CY2 , CY2 , CY3 , CY1 , CY2 , . . . , CY N , where {∅} is an empty set, and N is the { } total number of independent subset CYi . Suppose that {A} denotes a possible subset of Y. A probability or belief, m Y (A), can be assigned to the subset {A} based on statistical data or engineering judgment. m Y (A) is called a basic probability assignment (BPA). The BPA is a mapping function, and it satisfies the following three conditions: m (A) ≥ 0 ∀A ∈ Y m ∑(∅) = 0 m (A) = 1

(1)

A∈Y

The BPA structure can be formed from one source or multiple sources. When information is from multiple sources, the multiple BPA structures must be aggregated by so called rules of combination [17]. In addition, if multiple uncertain variables are involved and are independent, then a joint BPA is defined by: { m Y1 (A) m Y2 (B) , when C = A × B m Y (C) = (2) 0, otherwise where A is a set over the frame Y1 , B is a set over the frame Y2 , Y =Y1 × Y2 , and C is a set over the frame Y . 2.2. Fuzzy Set The fuzzy set theory first proposed by Zadeh [19] is a class of objects with a continuum of grades of membership. Such set is characterized by a membership function which assigns each object a grade of membership ranging between zero and one. Here, the fuzzy set theory and its some basic concepts are briefly presented. A fuzzy set is called a fuzzy number when it is normal and convex. For a detailed discussion on fuzzy number, one can refer to Ref. [36]. A fuzzy number is an ideal way to describe an uncertain parameter. A fuzzy number Z can be uniquely defined by the following set of pairs: ⏐ { } Z = (χ , µ Z (χ)) ⏐χ ∈ Ωχ (3) where χ is an element of the fuzzy set, Ωχ is the domain of the fuzzy set, and µ Z (χ ) ∈ [0, 1] is the membership function of the fuzzy set. A fuzzy number can be described as a family of α -level sets, via α-cut technique [37]. The α-cut technique is always used to conduct the fuzzy arithmetic operations. The α-cut of a fuzzy number Z can be defined as: { ⏐ } Z α = χ ⏐(µ Z (χ) ≥ α) , χ ∈ Ωχ , 0 ≤ α ≤ 1 (4)

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Fig. 1. A triangular fuzzy number and its α -cut.

For each α-cut level, an interval can be defined as: [ ] [ ] Z α = Z α , Z α = χαL , χαU

(5) χαL

χαU

where Z α and Z α are the lower and upper bounds of Z α , respectively. and are the χ − values of Z α and Z α , respectively. In addition, if the shape of membership function µ Z (χ) is a triangle, as shown in Fig. 1, the fuzzy number Z is called a triangular fuzzy number and it can be expressed as Z = (a, b, c), where a and c are the lower and upper bounds of the triangle at α = 0 and b is the crisp value at α = 1. 2.3. Interval The interval analysis was first systematically proposed by Moore [18]. In interval theory, the possible variation range of any uncertain parameter is represented by an interval, namely, we only need to know the upper and lower bounds of parameter, and do not need to know the exact probability distribution or fuzzy membership function [38,39]. Suppose that P denotes an interval variable. The interval variable P can be described by the following equations: [ ] P ∈ P, P (6) P+P r P−P Pr Pc = ,P = ,γ = c 2 2 P where P, P, P c , P r and γ represent the lower bound, the upper bound, the middle point, the radius and the degree of uncertainty of the interval variable P, respectively. Actually, the interval variable is a special case of the evidence variable or fuzzy variable. The evidence variable can be degraded into a series of intervals or sets by assigning BPA over the intervals, and the fuzzy variable can be simplified to intervals by α-cut technique. Therefore, the interval methods are always used to calculate the uncertain responses with evidence variables or fuzzy variables [40,41]. This also motivates us to translate the uncertainty problem with evidence and fuzzy variables to uncertainty problem with only intervals. 3. Taylor expansion based unified uncertainty analysis In solving engineering problems, it is extremely important to properly take uncertainty into consideration [42–44]. Theoretically, all types of uncertain models including evidence theory, fuzzy set, interval and probabilistic models can be applied to quantify the uncertainties. However, it is essential to quantify an uncertainty appropriately according to its own characteristic. Sometimes, we need to consider all types of uncertain models in a unified system

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in order to appropriately quantify the multi-source uncertainties, and this is rewarding since the advantages of these models can be utilized in a unity. For this reason, in this paper, we will propose a unified approach to solve the uncertainty analysis problems considering probabilistic, evidence, interval as well as fuzzy variables, simultaneously. It should be pointed out that our proposed method can be degenerated into unified uncertainty analysis approaches considering any two or three of these uncertain models. Let a response G be expressed abstractly by a performance function: G = g (X, Y, Z, P) (7) ( )T ( )T where X = X 1 , X 2 , . . . , X k1 denotes a k1 dimensional probabilistic vector, Y = Y1 , Y2 , . . . , Yk2 denotes a ( )T k2 dimensional uncertain vector modeled by evidence theory, Z = Z 1 , Z 2 , . . . , Z k3 denotes a k3 dimensional ( )T fuzzy vector, P = P1 , P2 , . . . , Pk4 denotes a k4 dimensional interval vector. All the uncertainties are assumed to be independent. The uncertainties associated with the model inputs X, Y, Z, and P will be propagated through the model g (•) to the performance (response) G. We are interested in knowing the effect of uncertainties in X, Y, Z, and P on the performance G. In the following content, a Taylor expansion based unified uncertainty analysis (T-UUA) method will be first formulated to solve this difficult problem. 3.1. Probability analysis By temporarily neglecting the evidence, fuzzy, and interval variables, g (X, Y, Z, P) can be simplified into a random function in terms of only probabilistic vector X. The expectation and variance of a random function can be then calculated by the random variable functional moment method based on Taylor expansion [45]. The Taylor expansion of g (X, Y, Z, P) can be expressed as: ⏐ k1 ∑ ∂g (X, Y, Z, P) ⏐⏐ g (X, Y, Z, P) = g (X, Y, Z, P)|X=E x p(X) + • (X i − E x p (X i )) ⏐ ⏐ ∂ Xi i=1 X=E x p(X) ⏐ (8) ⏐ k1 k1 ( 2 ∑ ∂ g (X, Y, Z, P) )⏐ ( ( )) 1 ∑ ⏐ • (X i − E x p (X i )) • X j − E x p X j . . . + ⏐ 2! i=1 j=1 ∂ Xi ∂ X j ⏐ X=E x p(X)

where E x p (X i ) is the expectation of random variable X i . When the degree of uncertainty is small, the high order terms can be neglected, and the expectation and variance of g (X, Y, Z, P) can be expressed as follows: E x p (g (X, Y, Z, P)) ≈ g (E x p (X) , Y, Z, P) ) ⏐ k1 ( ∑ ∂g (X, Y, Z, P) 2 ⏐⏐ V ar (g (X, Y, Z, P)) ≈ ⏐ ⏐ ∂ Xi i=1

(9) • V ar (X i )

(10)

X=E x p(X)

3.2. Evidence theory analysis The above E x p (g (X, Y, Z, P)) and V ar (g (X, Y, Z, P)) are still the functions of evidence vector Y, fuzzy vector Z, and interval vector P. To calculate them, further analysis is performed by reconsidering the evidence variables. According to Eq. (2), the joint BPA structure of Y can be expressed as: ⎧ ⎨ k2 Π m (yi ) , s = Y1 × Y2 · · · × Yk2 m Y (s) = i=1 (11) ⎩0, otherwise where s is the joint discernible frame, m Y (s) is the joint BPA, k2 is the number of epistemic variables modeled by evidence theory, yi is the focal element of the ith evidence variable Yi , m (yi ) is the BPA of yi . By temporarily treating X, Z, and P as constant vectors, the mean and variance of g (X, Y, Z, P) can be approximated as an expression in terms of Y [40]: E x p (g (X, Y, Z, P)) =

l ∑ ( ) g X, Ysi , Z, P • m Y (si ) i=1

(12)

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X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

V ar (g (X, Y, Z, P)) =

l ∑ ( ( ) ( ( )))2 g X, Ysi , Z, P − E x p g X, Ysi , Z, P • m Y (si )

(13)

i=1

where Ysi ∈ si , si is the ith subset (focal element) over frame s, m Y (si ) is the joint BPA of si , and l is the total number of focal elements. Note that the probabilistic variables can be considered as evidence variables with infinite BPAs. Thus, the mean of g (X, Y, Z, P) under mixed X and Y can be rewritten as: ∞ ∑ l ∑ ( ) ( ) g Xsi , Ys j , Z, P • m X (si ) • m Y s j

E x p (g (X, Y, Z, P)) =

i=1 j=1 (∞ ) l ∑ ∑ ( ) ( ) = g Xsi , Ys j , Z, P • m X (si ) • m Y s j j=1 l ∑

=

(14)

i=1

( ( )) ( ) E x p g X, Ys j , Z, P • m Y s j

j=1

where m X (si ) is the ith joint BPA of X. Similarly, the variance of g (X, Y, Z, P) under mixed X and Y can be estimated as: l ∞ ∑ ∑ ( ( ) ( ( )))2 ( ) g Xsi , Ys j , Z, P − E x p g Xsi , Ys j , Z, P V ar (g (X, Y, Z, P)) = • m X (si ) • m Y s j i=1 j=1 (∞ ) l ∑ ∑( ( ) ( ( )))2 ( ) g Xsi , Ys j , Z, P − E x p g Xsi , Ys j , Z, P = • m X (si ) • m Y s j (15) j=1

=

l ∑ (

i=1

( ( ))) ( ) V ar g X, Ys j , Z, P • m Y s j

j=1

Further, substituting Eqs. (9) and (10) into Eqs. (14) and (15) yields: E x p (g (X, Y, Z, P)) =

l ∑ ( ) ( ) g E x p (X) , Ys j , Z, P • m Y s j

(16)

j=1

( ( ) )2 ⏐⏐ k1 ∑ ⏐ ∂g X, Ys j , Z, P ⏐ ⎣ V ar (g (X, Y, Z, P)) = ⏐ ∂ X i ⏐ j=1 i=1 l ∑

⎤

⎡

( ) • V ar (X i )⎦ • m Y s j

(17)

X=E x p(X)

3.3. Fuzziness discretization The above mean and variance in Eqs. (16) and (17) are still the functions of evidence subset Ys j , fuzzy vector Z and interval vector P. To discretize the fuzziness, the α-cut technique is one of the most common methods, by which each fuzzy variable is discretized into two crisp values at the α-cut level as described in Eq. (4). By using the α-cut technique, the expectation and variance of Zα can be expressed as: )) ( ( ) ( ) ( (18) E x p (Zα ) = E x p Zα,1 , E x p Zα,2 , . . . , E x p Zα,k3 ( ( ) ( ) ( )) V ar (Zα ) = V ar Zα,1 , V ar Zα,2 , . . . , V ar Zα,k3

(19)

where Zα can be regarded as an interval vector at α-cut level: [ ] Zα + Zα Zα − Zα Zα = Zα , Zα , Zcα = , Zrα = (20) 2 2 where Zα , Zα denote the lower and upper bounds. Zcα and ∆Zrα denote the middle point and radius vectors of Zα . Now, by reconsidering fuzzy variable, for a given membership, Eqs. (16) and (17) are further approximated as: E x p (g (X, Y, Zα , P)) =

l ∑ ( ) ( ) g E x p (X) , Ys j , Zα , P • m Y s j j=1

(21)

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

( ( ) )2 ⏐⏐ k1 ∑ ⏐ ∂g X, Ys j , Zα , P ⏐ ⎣ V ar (g (X, Y, Zα , P)) = ⏐ ∂ X i ⏐ j=1 i=1 l ∑

7

⎤

⎡

( ) • V ar (X i )⎦ • m Y s j

(22)

X=E x p(X)

3.4. Interval analysis Due to the interval nature of Ys j , Zα and P in Eqs. (21) and (22), a single measure of E x p (g (X, Y, Zα , P)) and V ar (g (X, Y, Zα , P)) is not available. Instead, two measures, belief and plausibility measures, are involved. In this paper, we consider that the BPAs of epistemic variables are from nonconflicting items of evidence and that only one BPA exists for one interval of an epistemic variable. Under these conditions, belief and plausibility of E x p (g (X, Y, Zα , P)) and V ar (g (X, Y, Zα , P)) can be considered as their lower and upper bounds [46]. Therefore, based on the first order Taylor expansion and the natural interval extension in interval arithmetic [47], the belief and plausibility of E x p (g (X, Y, Zα , P)) can be estimated as: Bel {E x p (g (X, Y, Zα , P))} = E x p (g (X, Y, Zα , P)) l [ ( )] ∑ ( ) = g E x p (X) , Ycs j , Zcα , Pc • m Y s j j=1

⎡

⎤ ⏐ ) k2 l ⎢∑ ⎥ ∑ ∂g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ − • Yi,s ⎢ ⏐ ⎥ • mY s j c j ⏐X = E x p (X) , Ys j = Ys ⎣ ⎦ ∂Yi j i=1 j=1 c c Zα = Zα , P = P ⎡ ⎤ ⏐ ( ) k3 l ⎢∑ ⎥ ∑ ∂g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ − • Z ⎢ ⏐ ⎥ • mY s j i,α c ⏐ ⎣ ⎦ ∂ Z X = E x p (X) , Ys j = Ys j i,α i=1 j=1 Zα = Zcα , P = Pc ⎡ ⎤ ⏐ ( ) k4 l ⎢∑ ⎥ ∑ ∂g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r⎥ − • P ⎢ ⏐ i ⎥ • mY s j c ⏐X = E x p (X) , Ys j = Ys ⎣ ⎦ ∂ Pi j i=1 j=1 Zα = Zcα , P = Pc Pl {E x p (g (X, Y, Zα , P))} = E x p (g (X, Y, Zα , P)) l [ ( )] ∑ ( ) = g E x p (X) , Ycs j , Zcα , Pc • m Y s j (

(23)

j=1 ⎡

⎤ ⏐ ) k2 l ⎢∑ ⎥ ∑ ∂g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ • Y + ⎢ ⏐ i,s j ⎥ • m Y s j c ⏐X = E x p (X) , Ys j = Ys ⎦ ⎣ ∂Yi j j=1 i=1 Zα = Zcα , P = Pc ⎡ ⎤ ⏐ ( ) k3 l ⎢∑ ⎥ ∑ ∂g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ + • Z i,α ⎥ • mY s j ⎢ ⏐ c ⏐X = E x p (X) , Ys j = Ys ⎣ ⎦ ∂ Z i,α j j=1 i=1 c c Zα = Zα , P = P ⎡ ⎤ ⏐ ( ) k4 l ⎢∑ ⎥ ∑ ∂g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r⎥ + • P ⎢ ⏐ ⎥ • mY s j i c ⏐ ⎣ ⎦ ∂ P X = E x p , Y = Y (X) i sj sj j=1 i=1 Zα = Zcα , P = Pc (

(24)

where Bel {•} and Pl {•} denote the belief and plausibility of “•”, respectively; E x p (•) and E x p (•) denote the upper and lower bounds of mean of “•”, respectively; Ycs j , Zcα , and Pc are the middle point vector of Ys j , Zα , and

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r r is the radius of Z i at α-cut level; Pir is the radius is the radius of Yi over the subset s j ; Z i,α P, respectively; Yi,s j of Pi . Similarly, the belief and plausibility of the variance of V ar (g (X, Y, Zα , P)) can be approximated as:

Bel {V ar (g (X, Y, Zα , P))} = V ar (g (X, Y, Zα , P)) ⎡ ⎤ ⏐ ) ( ( ) 2 ⏐ k1 l ⎢∑ ⎥ ∑ ⏐ ∂g X, Ys j , Zα , P ( ) ⎢ ⎥ ⏐ = • V ar (X ) ⎢ k ⎥ • mY s j ⏐ ⎣ ⎦ ∂ Xk ⏐X = E x p (X) , Ys j = Ycs j j=1 k=1 c c Zα = Zα , P = P ⎡

⎤ ⏐ ) k1 k2 l ⎢∑ ⎥ ∑ ∑ ∂g X, Ys j , Zα , P ∂ 2 g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ • −2 • Y ⏐ ⎢ i,s j ⎥ • m Y s j c ⏐X = E x p (X) , Ys j = Ys ⎦ ⎣ ∂ Xk ∂ X k ∂Yi j j=1 k=1 i=1 Zα = Zcα , P = Pc ⎤ ⎡ ⏐ ( ) ( ) k1 k3 l ⎢∑ ⎥ ∑ ∑ ∂g X, Ys j , Zα , P ∂ 2 g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ • −2 ⏐ ⎢ c • Z i,α ⎥ • m Y s j ⏐X = E x p (X) , Ys j = Ys j ⎦ ⎣ ∂ Xk ∂ X k ∂ Z i,α j=1 k=1 i=1 (

)

(

Zα = Zcα , P = Pc

⎡

⎤ ⏐ ) k1 k4 l ⎢∑ ⎥ ∑ ∑ ∂g X, Ys j , Zα , P ∂ 2 g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r⎥ −2 • • P ⎢ ⏐ i ⎥ • mY s j c ⏐X = E x p (X) , Ys j = Ys ⎣ ⎦ ∂ Xk ∂ X k ∂ Pi j j=1 k=1 i=1 Zα = Zcα , P = Pc (

)

(

(25) Pl {V ar (g (X, Y, Zα , P))} = V ar (g (X, Y, Zα , P)) ⎡ ⎤ ⏐ ( ) ) ( 2⏐ k1 l ⎢∑ ⎥ ∑ ⏐ ∂g X, Ys j , Zα , P ( ) ⎢ ⎥ ⏐ = • V ar (X k )⎥ • m Y s j ⎢ ⏐ ⎣ ⎦ ∂ Xk ⏐X = E x p (X) , Ys j = Ycs j j=1 k=1 c c Zα = Zα , P = P ⎤ ⎡ ⏐ ( ) ( ) k1 k2 l ⎢∑ ⎥ ∑ ∑ ∂g X, Ys j , Zα , P ∂ 2 g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ • • Y +2 ⏐ ⎢ i,s j ⎥ • m Y s j c ⏐X = E x p (X) , Ys j = Ys ⎦ ⎣ ∂ Xk ∂ X k ∂Yi j j=1 k=1 i=1 Zα = Zcα , P = Pc ⎡ ⎤ ⏐ ( ) ( ) k1 k3 l ⎢∑ ⎥ ∑ ∑ ∂g X, Ys j , Zα , P ∂ 2 g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r ⎥ +2 • • Z i,α ⎢ ⏐ ⎥ • mY s j c ⏐X = E x p (X) , Ys j = Ys ⎣ ⎦ ∂ Xk ∂ X k ∂ Z i,α j j=1 k=1 i=1 c c Zα = Zα , P = P ⎤ ⎡ ⏐ ( ) ( ) k1 k4 l ⎢∑ ⎥ ∑ ∑ ∂g X, Ys j , Zα , P ∂ 2 g X, Ys j , Zα , P ⏐⏐ ( ) ⎢ r⎥ +2 • • P ⏐ ⎥ • mY s j ⎢ i c ⏐ ⎦ ⎣ ∂ X ∂ X ∂ P X = E x p (X) , Ys j = Ys j k k i j=1 k=1 i=1 Zα = Zcα , P = Pc (26) where V ar (•) and V ar (•) denote the upper and lower bounds of variance of “•”, respectively. The mean and variance under different membership can be obtained by repeating the above calculation through substituting different values of membership level α. Then, the upper and lower boundary lines of E x p (g (X, Y, Z, P)) and

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

9

V ar (g (X, Y, Z, P)) can be composed by combining each E x p (g (X, Y, Zα , P)) or V ar (g (X, Y, Zα , P)) and the corresponding α. The above proposed method is simple and the expectation and variance are explicitly expressed. However, it should be noted that the computational accuracy of Eqs. (23)–(26) is limited to uncertain variables with small degree of uncertainty since the T-UUA is based on the first order Taylor expansion where the expanded high order terms are neglected as mentioned in Eqs. (9) and (10). For relatively larger uncertainty problems, new method should be proposed.

4. Dimension reduction/efficient global optimization based unified uncertainty analysis In this section, a dimension reduction (DR)/efficient global optimization (EGO) based unified uncertainty analysis (DR/EGO-UUA) method is further proposed on the foundation of the above formulated UUA framework. First, the expectation and variance in the probabilistic analysis is derived by dimension reduction integration instead of the Taylor expansion. The evidence theory and fuzzy analysis is the same with T-UUA. Then, the EGO is further applied to interval analysis over subsets. Finally, using the interval arithmetic, the belief and plausibility of expectation and variance of performance are estimated.

4.1. Probabilistic moment analysis by dimension reduction integration By temporarily neglecting the evidence, fuzzy, and interval variables, according to the statistical theory, the nth moment of g (X, Y, Z, P) can be expressed as: ∫ ∞ ∫ ∞ { } n m gn =E [g (X, Y, Z, P)] = ··· n = 1, 2, . . . (27) [g (X, Y, Z, P)]n f x (X) dX −∞

−∞

where E is the expectation operator, f x (X) is the joint density function. The above equation is a multi-dimensional integration whose analytical solutions are generally difficult to obtain. Therefore, numerical approaches [48–51] are often employed to solve the above integration when only probabilistic variables exist in the uncertainty problem. In these methods, the dimensional reduction (DR) method [50] is one of the most popular and practical approaches for its simplicity. The main idea of the DR approach is to approximate the high dimensional integrations through a series of lower dimensional integrations. Through introducing DR approach, the expectation and variance of uncertain responses in a random model can be generally calculated with a good accuracy even when the degree of uncertainties is relatively large. In this paper, the univariate DR [50] is introduced into the UUA for probability analysis. According to the DR, Eq. (27) can be rewritten as: ⎧⎡ ⎤n ⎫ k1 ⎨ ∑ ( ) ( ) ⎬ m gn =E ⎣ g µ1 , . . . , µ j−1 , X j , µ j+1 , . . . , µk1 , Y, Z, P − (k1 − 1) g µ1 , . . . , µk1 , Y, Z, P ⎦ (28) ⎭ ⎩ j=1

where µj, j = 1, 2, . . . , k1 denotes ( ) the mean value of the jth random variable X j , and g µ1 , . . . , µ j−1 , X j , µ j+1 , . . . , µk1 , Y, Z, P denotes structural response depending on X j . By using the binomial expansion, Eq. (28) can be expressed as follows: ⎡ ⎤i k1 n ( ) ∑ ∑ ( ) n m gn = E⎣ g µ1 , . . . , µ j−1 , X j , µ j+1 , . . . , µk1 , Y, Z, P ⎦ i i=0 j=1 [ ( )](n−i) (29) × − (k1 − 1) g µ1 , . . . , µk1 , Y, Z, P ( ) n ∑ n [ ( )](n−i) = Sik1 − (k1 − 1) g µ1 , . . . , µk1 , Y, Z, P i i=0

10

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

where Sik can be obtained by the following recursive function: 1 [( ( ))i ] Si1 = E g X 1 , µ2 , . . . , µk1 , Y, Z, P , i = 1, . . . , n ( ) [( ( ))(i−k) ] ∑ i Sij = ik=0 Skj−1 E g µ1 , . . . , µ j−1 , X j , µ j+1 , . . . , µk1 , Y, Z, P , k j = 2, . . . ( , k1)− 1, i = 1, . . . , n [( ( ))(i−k) ] ∑ i Sik1 = ik=0 Skk1 −1 E g µ1 , µ2 , . . . , µk1 −1 , X k1 , Y, Z, P , i = 1, . . . , n k

(30)

It should be noted that the above equations involve only one dimensional integration, namely: [{ ( )}i ] Igi = E g µ1 , . . . , µ j−1 , X j , µ j+1 , . . . , µk1 , Y, Z, P ∫ ∞ { ( )}i ( ) = g µ1 , . . . , µ j−1 , X j , µ j+1 , . . . , µk1 , Y, Z, P f x j X j dX j , −∞

(31) i = 1, 2, . . . , n, j = 1, . . . , k1 r { ( )}i 1 ∑ wq g µ1 , . . . , µ j−1 , X jq , µ j+1 , . . . , µk1 = √ π q=1 )) ( (√ 2h q , wq and h q are the qth Gauss weight and Gauss point [52]. Substituting Eqs. (30) where X jq = FX−1j Φ and (31) into Eq. (29), the moments of g (X, Y, Z, P) can be expressed as an integration function in terms of Y, Z, and P. Finally, the expectation of g (X, Y, Z, P) based on DR integration can be approximated as: ( ) E x p (g (X, Y, Z, P)) = m gn |n=1 = S1k1 − (k1 − 1) g µ1 , . . . , µk1 , Y, Z, P (32) In addition, the variance of g (X, Y, Z, P) can be estimated by: V ar (g (X, Y, Z, P)) = m gn |n=2 − m 2gn |n=1 ( ) ( )]2−i [ 1 ( )]2 ∑2 2 i [ = i=0 Sk1 − (k1 − 1) g µ1 , . . . , µk1 , Y, Z, P − Sk1 − (k1 − 1) g µ1 , . . . , µk1 , Y, Z, P i = S2k1 − (S1k1 )2

(33)

4.2. Evidence analysis and fuzzy discretization In DR/EGO-UUA, the evidence analysis is the same with T-UUA. By substituting Eqs. (32) and (33) into Eqs. (14) and (15), the second moments with BPAs can be then expressed as: E x p (g (X, Y, Z, P)) =

l [ ∑

⏐ ( )] ( ) ⏐ S1k1 ⏐Y=Ys j − (k1 − 1) g µ1 , . . . , µk1 , Ys j , Z, P • m Y s j

(34)

j=1

V ar (g (X, Y, Z, P)) =

l [ ∑

⏐

⏐ S2k1 ⏐Y=Ys j

( ⏐ )2 ] ( ) 1 ⏐ − Sk1 ⏐Y=Ys j • mY s j

(35)

j=1

Further, via the α-cut technique, for a given membership level α, the mean and variance can be approximated as: E x p (g (X, Y, Zα , P)) =

l [ ∑

⏐ ( )] ( ) ⏐ S1k1 ⏐Y=Ys j ,Z=Zα − (k1 − 1) g µ1 , . . . , µk1 , Ys j , Zα , P • m Y s j

(36)

j=1 l [ ⏐ ( ⏐ )2 ] ∑ ( ) 1 ⏐ 2 ⏐ V ar (g (X, Y, Zα , P)) = Sk1 ⏐Y=Ys j ,Z=Zα − Sk1 ⏐Y=Ys j ,Z=Zα • mY s j j=1

(37)

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

11

4.3. Efficient global optimization In Eqs. (36) and (37), E x p (g (X, Y, Zα , P)) and V ar (g (X, Y, Zα , P)) are still functions of interval vectors Ys j , Zα , and P. For convenience, we rewrite Eqs. (36) and (37) as: E x p (g (X, Y, Zα , P)) = V ar (g (X, Y, Zα , P)) =

l ∑ j=1 l ∑

( ) ( ) H Ys j , Zα , P • m Y s j

(38)

( ) ( ) ˜ Ys , Zα , P • m Y s j H j

(39)

j=1

where: ⏐ ( ) ( ) ⏐ H Ys j , Zα , P = S1k1 ⏐Y=Ys j ,Z=Zα − (k1 − 1) g µ1 , . . . , µk1 , Ys j , Zα , P , j = 1, 2 . . . , l

(40)

⏐ ( ⏐ )2 ( ) ˜ Ys , Zα , P = S2k ⏐⏐Y=Ys ,Z=Zα − S1k ⏐⏐Y=Ys ,Z=Zα , j = 1, 2 . . . , l H (41) j 1 1 j j ) ( ) ( ˜ Ys , Zα , P should be first calculated, and In the above equations, the variation ranges of H Ys j , Zα , P and H j then combining with the interval arithmetic, the bounds of E x p (g (X, Y, Zα , P)) and V ar (g (X, Y, Zα , P)) can be thus obtained. In this section, the celebrated efficient global optimization (EGO) method originally proposed by Jones et al. [53] is introduced to calculate the bounds of them to overcome the local optimization problem. In this way, the upper and lower bounds can be well captured, regardless of the large uncertainty of intervals and nonlinearity of performance functions. ( ) According to EGO [53], H Ys j , Zα , P can be approximated by a Gaussian process (GP) model: ( ) Hˆ (U) = Hˆ Ys j , Zα , P = h (U) β + z (U) (42) where U indicates all involved interval vectors Ys j , Zα , and P, h (U) is the trend of the model, β is the vector of trend coefficients, and z (U) is a stationary Gaussian process with zero mean. The expectation and variance of the GP model prediction can be expressed as [54]: ( ) µ Hˆ (U) =hT (U) β + rT (U) R−1 H − h (U) β (43) [ ] [ ] [ ] [ ] 0 FT −1 h (U) σ H2ˆ (U) =σ Z2 1 − hT (U) rT (U) (44) F R r (U) where r (U) is a vector containing the covariance between U and each of the N training points, R is an N × N matrix containing the correlation between each pair of training points, H is the vector of response outputs at each of the training points, and F is an N × 1 matrix with rows hT (Ui ). The parameter σz is the process standard deviation which is determined through maximum likelihood estimation [54]. To select new training points, an expected improvement function (EIF) is defined as: )⎞ )⎞ ⎛( ⎛( ( ) ( ) Hˆ min (U) − µ Hˆ (U) Hˆ min (U) − µ Hˆ (U) ⎠ + σ ˆ (U) φ ⎝ ⎠ E I Hˆ (U) = Hˆ min (U) − µ Hˆ (U) Φ ⎝ H σ Hˆ (U) σ Hˆ (U) (45) In the above, φ and Φ are the standard normal density and distribution function. µ Hˆ and σ Hˆ were given in Eqs. (43) and (44). The EIF is used to select the location at which a new training point should be added to the Gaussian process model by maximizing the amount of improvement function. It provides a balance between exploiting areas of the design space that predict good solutions and) exploring areas where more information is needed [53]. ( In summary, the lower bound of H Ys j , Zα , P can be calculated by the following procedures: ( ) (1) Generate a small number of samples of H Ys j , Zα , P using Latin hypercube sampling. ( ) (2) Construct an initial Gaussian process model Hˆ Ys j , Zα , P from these samples.

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(3) Find the point the EIF in Eq. (45). If the EIF value at this point is sufficiently small, stop. ( that maximizes ) (4) Evaluate H Ys j , Zα , P at this point. Add this new sample to the previous set and build a new GP model. Go to step 3. ( ) Through the above procedures, the lower bound of H ( Ys j , Zα ,)P can be approximated. Its upper bound can be estimated by (simply adding ) a negative sign before H Ys j , Zα , P . By varying j from 1 to l, the( upper and) lower ˜ Ys , Zα , P , j = bounds of H Ys j , Zα , P , j = 1, 2, . . . , l can be obtained. Similarly, the variation range of H j 1, 2, . . . , l can also be acquired. 4.4. Belief and plausibility of the moments Through using the interval arithmetic, and combining Eq. (38), the belief and plausibility of E x p (g (X, Y, Zα , P)) can be calculated as: Bel {E x p (g (X, Y, Zα , P))} = E x p (g (X, Y, Zα , P)) =

l ∑

( ) ( ) H Ys j , Zα , P • m Y s j

(46)

j=1 l ∑

Pl {E x p (g (X, Y, Zα , P))} = E x p (g (X, Y, Zα , P)) =

( ) ( ) H Ys j , Zα , P • m Y s j

(47)

j=1

( ( ) ) ( ) where H Ys j , Zα , P and H Ys j , Zα , P denote the upper and lower bounds of H Ys j , Zα , P , respectively. Simultaneously, the belief and plausibility of V ar (g (X, Y, Zα , P)) can be also computed as: Bel {V ar (g (X, Y, Zα , P))} = V ar (g (X, Y, Zα , P)) =

l ∑

( ) ( ) ˜ Ys , Zα , P • m Y s j H j

(48)

j=1

Pl {V ar (g (X, Y, Zα , P))} = V ar (g (X, Y, Zα , P)) =

l ∑

( ) ( ) ˜ Ys , Zα , P • m Y s j H j

(49)

j=1

( ( ) ) ( ) ˜ Ys , Zα , P and H ˜ Ys , Zα , P denote the upper and lower bounds of H ˜ Ys , Zα , P , respectively. Based where H j j j on the above processes, the expectation and variance of uncertain response considering probabilistic, evidence, fuzzy and interval variables simultaneously under relatively large degree of uncertainty can be obtained. 4.5. MCS-UUA For verification Monte Carlo Simulation (MCS) method is a traditional method to solve uncertainty problems. MCS is often used as a verification method to judge the correctness of other methods by comparing the reference solutions calculated by MCS with the results of other methods. In this paper, MCS-UUA method is given as a verification method. In summary, the computational procedure of the MCS-UUA can be described in the following points (1) According to the information on the uncertain parameters, determine the random parameters X, evidence parameters Y, fuzzy parameters Z and interval parameters P of structure; (2) Take the jth focal element of Y, and the ith fuzzy subset of Z under the membership level of αi . Set j=1 and i=1, and I=1; (3) Generate N1∗ samples of X using random sampling approach; (4) Generate N2∗ samples of [Ys j , Zαi , P] using Latin hypercube sampling approach; Take the )Ith sample of [Ys j , Zαi , P]. Calculate the performance function g for all samples in ( (5) [ ] X, Ys j , Zαi , P I ; ( [ ] ) (6) Estimate the expectation and variance of g X, Ys j , Zαi , P I with the N1∗ random samples. When I ≤ N2∗ , Go to step 5. Otherwise, go to step 7; (7) When j ≤ l, go to step 4. Otherwise, go to step 8;

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

13

Fig. 2. Flowchart of MCS-UUA.

( ( )) ( ( )) (8) Estimate the upper and lower bounds of E x p g X, Y, Zαi , P and V ar g X, Y, Zαi , P . When i ≤ l Z , l Z is the number of membership levels, go to step 4. Otherwise, end. Fig. 2 shows the detailed algorithm flow. In MCS-UUA, for each focal element of Y and membership level of Z, there exists a mixed probability-interval uncertainty analysis problem and thereby a double-nested loop is involved, where the inner layer is random analysis and the outer layer is interval analysis, as described in Steps 5–6. For the jth focal element of Y, and the ith fuzzy subset of Z under the membership level of αi , the sample [ ] [ ] space is [X]] × Ys j , Zαi , P . Firstly, the inner calculation is carried out, Ys j , Zαi , P is set to be a fixed value [ Ys j , Zαi , P I and the number of evaluations of performance function for computing the expectation and variance is N1∗ . Then, the inner calculation will be repeatedly carried out with I changing from 1 to N2∗ , which requires to evaluate performance function N1∗ ∗ N2∗ times. Considering there are a total of l focal elements and l z membership levels, the total number of MCS-UUA function evaluations is equal to l z ∗l∗N1∗ ∗ N2∗ .

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Fig. 3. The cantilever tube model[27].

5. Numerical examples 5.1. The cantilever tube Considering a cantilever tube, as shown in Fig. 3, which is modified from the numerical example in Ref. [27]. The tube is subjected to external forces F1 , F2 , and P, and torsion T . σmax represents the maximum Von Mises stress on the top surface of the tube at the origin, which is given by: √ 2 σmax = σx2 + 3τzx (50) where σx and τzx denote the normal stress and tensional stress, respectively. And they could be calculated by: Mc P + F1 sin θ2 + F2 sin θ1 + (51) σx = A I Td τzx = (52) 4I where the first term of Eq. (51) is the normal stress due to the axial forces, and the second term is the normal stress due to the bending moment M; A, c and I represent the area , radius and moment of inertia, respectively. They are given by: M = F1 L 2 cos (θ2 ) + F2 L 1 cos (θ1 )

(53)

] π[ 2 (54) d − (d − 2h)2 4 d c= (55) 2 ] π [ 4 I = d − (d − 2h)4 (56) 64 In this example, the load F1 , load F2 , thickness h, torsion T, load P, and diameter d are assumed to be independent uncertain variables. First, the uncertain structural problem under small degree of uncertainty is considered. The information of uncertain variables is given in Table 1. The maximum stress of the cantilever tube is calculated by the proposed UUA methods. In T-UUA, the forward difference method is adopted to estimate the partial derivatives of functions. For example, the partial derivative of σmax with respect to load F1 can be approximated as ∂σmax σmax (F1 + ∆F1 , F2 , h, T, P, d) − σmax (F1 , F2 , h, T, P, d) = (57) ∂ F1 ∆F1 The MCS-UUA method is taken as a reference method. In MCS-UUA, the number of samplings N1∗ in random analysis and N2∗ in interval analysis are chosen as 10 000 and 100 000, respectively. There are 3 focal elements in A=

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15

Table 1 The information of uncertain variables under small degree of uncertainty for the tube. Uncertain variable

Distribution type

Parameter 1

Parameter 2

Parameter 3

Uncertainty level

Load, F1 Load, F2 Thickness, h

Normal Normal Normal

Torsion, T

Evidence

Load, P Diameter, d

Interval Fuzzy

90 kN 90 kN 3 mm 38 kN m 39 kN m 41 kN m 20 kN 38 mm

0.9 kN 0.9 kN 0.06 mm 39 kN m 41 kN m 42 kN m 0.2 kN 40 mm

– – – 20% 30% 50% – 42 mm

1% 1% 2% 1.3% 2.5% 1.2% 1% 5%

Probabilistic variable: parameter 1 is the expectation; parameter 2 is the standard deviation; Evidence variable: parameter 1 is the lower bound of focal element, parameter 2 is the upper bound of focal element, parameter 3 is BPA; Interval variable: parameter 1 is the midpoint value; parameter 2 is the radius; Fuzzy variable: parameter 1 and parameter 3 are the lower and upper bounds of fuzzy number at α = 0; parameter 2 is the value of fuzzy number at α = 1. Table 2 The maximum relative errors and function calls of the proposed UUA methods under small degree of uncertainty for the tube. Method

Maximum relative error Expectation

T-UUA DR/EGO-UUA MCS

Function calls Variance

Lower bound

Upper bound

Lower bound

Upper bound

0.088% 0.054% –

0.624% 0.019% –

0.522% 0.914% –

0.523% 0.849% –

1056 17 490 3.3 E10

evidence variable and 11 membership levels in fuzzy variable. The results of the proposed methods are compared with that of MCS-UUA and shown in Fig. 4. Additionally, Table 2 shows the maximum relative errors of the expectation and variance bounds between the proposed methods and MCS-UUA, and the number of function evaluations which is used to measure computational efficiency is also presented. It could be found in Fig. 4 and Table 2 that the solutions obtained by the proposed approaches match the reference solutions very well. The solutions of the expectation of the response are almost the same as the reference ones, and the maximum relative errors of expectation and variance are equal to 0.624% and 0.914%, respectively, which occur at the level of α = 0 and α = 0.1. However, the computational efficiency of the proposed methods is much better than that of MCS-UUA. The function calls for calculating the above results using T-UUA and DR/EGO-UUA are 1056 and 17 490 respectively while the number of evaluations of performance functions for MCS-UUA is 11*3*10000*100000 (3.3*1010 ). Under the case of small level of uncertainty, it seems that the proposed T-UUA has the most significant computational efficiency, at the same time, it has a good accuracy. Then, the uncertain structural problem under relatively large degree of uncertainty is considered. The detailed values of uncertain variables are given in Table 3. Again, the proposed T-UUA, DR/EGO-UUA, and MCS-UUA are used to calculate the bounds of the maximum stress. In DR/EGO-UUA, the number of initial training sample points is 10, if the expected improvement is less than 0.001% of the best current function value, the EGO algorithms go to stop, and the number of iteration steps is about 10 for interval analysis at one subset (focal element). The results are shown in Fig. 5 and Table 4. As demonstrated in Fig. 5 and Table 4, the results obtained from DR/EGOUUA and MCS-UUA have very good agreement. The curves of expectation and variance by DR/EGO-UUA are quite close to those obtained by MCS-UUA and the largest relative errors of the expectation and variance bounds are less than 0.3% and 0.6%, respectively. While the results obtained from T-UUA have a large deviation from those of MCS-UUA, whose maximum relative error reaches up to 23.146%. The computational error of T-UUA mainly suffers from the effect of neglecting the higher order terms of Taylor series. Thus, with the increasing of the uncertainty levels of uncertain variables, the accuracy of T-UUA is likely to decrease gradually. Therefore, for the case under relatively large degree of uncertainty, DR/EGO-UUA should be selected rather than T-UUA. Though DR/EGO-UUA is less efficient than T-UUA, it is much more efficient than MCS-UUA.

16

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26 Table 3 The information of uncertain variables under large degree of uncertainty for the tube. Uncertain variable

Distribution type

Parameter 1

Parameter 2

Parameter 3

Uncertainty level

Load, F1 Load, F2 Thickness, h

Normal Normal Normal

Torsion, T

Evidence

Load, P Diameter, d

Interval Fuzzy

90 kN 90 kN 3 mm 34 kN m 37 kN m 43 kN m 20 kN 32 mm

18 kN 22.5 kN 0.3 mm 37 kN m 43 kN m 46 kN m 2 kN 40 mm

– – – 20% 30% 50% – 48 mm

20% 25% 10% 4.23% 7.5% 3.37% 10% 20%

Probabilistic variable: parameter 1 is the expectation; parameter 2 is the standard deviation; Evidence variable: parameter 1 is the lower bound of focal element, parameter 2 is the upper bound of focal element, parameter 3 is BPA; Interval variable: parameter 1 is the midpoint value; parameter 2 is the radius; Fuzzy variable: parameter 1 and parameter 3 are the lower and upper bounds of fuzzy number at α = 0; parameter 2 is the value of fuzzy number at α = 1. Table 4 The maximum relative errors and function calls of the proposed UUA methods under large degree of uncertainty for the tube. Method

Maximum relative error Expectation

T-UUA DR/EGO-UUA MCS

Function calls Variance

Lower bound

Upper bound

Lower bound

Upper bound

2.255% 0.268% –

9.503% 0.290% –

23.146% 0.346% –

16.782% 0.588% –

1056 16 360 3.3E9

Table 5 The information of uncertain variables of the trestle bridge. Uncertain variable

Distribution type

Parameter 1

Parameter 2

Parameter 3

Uncertainty level

Thickness, tw Thickness, t f Load, W

Normal Normal Normal

Temperature, T

Evidence

Load, M1 Load, M2

Interval Fuzzy

6.5 mm 4.5 mm 0.5N/ mm 7.5 ◦ C 11.25 ◦ C 18.75 ◦ C 0.6 N/mm 64 kN

0.65 mm 0.45 mm 0.1 N/mm 11.25 ◦ C 18.75 ◦ C 22.5 ◦ C 0.12 N/mm 80 kN

– – – 20% 30% 50% – 96 kN

10% 10% 20% 20% 25% 9.1% 20% 20%

Probabilistic variable: parameter 1 is the expectation; parameter 2 is the standard deviation; Evidence variable: parameter 1 is the lower bound of focal element, parameter 2 is the upper bound of focal element, parameter 3 is BPA; Interval variable: parameter 1 is the midpoint value; parameter 2 is the radius; Fuzzy variable: parameter 1 and parameter 3 are the lower and upper bounds of fuzzy number at α = 0; parameter 2 is the value of fuzzy number at α = 1.

5.2. The trestle bridge A trestle bridge built of piers, abutments and steel beams is shown in Fig. 6. Steel beams include bailey beam, support frame and distribution beam. The bailey beam consists of five bailey plates which contain chord members and web members. The maximum stress of the web members of bailey beams is considered in the situation where the trestle bridge is subjected to wind load W , pedestrian load M1 and vehicle load M2 . In addition, the system temperature T is applied on the trestle bridge. The finite element model of the trestle bridge is shown in Fig. 6(b). The section of the web member is I-shaped and the element type is beam element. The I-shaped section is shown in Fig. 6(c). Wind load W is equivalent to linear loading applied to the outermost bailey beam. In order to simplify

X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26

17

Fig. 4. Belief and plausibility of the maximum stress under small degree of uncertainty for the tube.

the model, no deck element is established and the lane load is allocated to the bailey beams through the distribution beams for meeting the actual load transfer situation. The web thickness tw and flange thickness t f of I-section of the web member, the wind load W are considered as random variables. The temperature T is treated by evidence theory. The pedestrian load M1 is assumed to be interval variable. The vehicle load M2 is modeled as fuzzy variable. The investigated parameters are regarded as independent variables and their information is shown in Table 5. The proposed UUA approaches are employed to estimate the maximum stress of the trestle bridge and the analyzed results are shown in Fig. 7. In DR/EGO-UUA, the number of initial training sample points is 10, if the expected improvement is less than 0.001% of the best current function value, the EGO algorithms go to stop, and the number of iteration steps is about 10 for interval analysis at one focal element. For an easy comparison, the results of MCS-UUA are drawn in Fig. 7 as well. In MCS-UUA, the number of samplings N1∗ in random analysis

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Fig. 5. Belief and plausibility of the maximum stress under large degree of uncertainty for the tube.

and N2∗ in interval analysis are chosen as 10 000 and 10 000, respectively. The fuzzy variable is discretized into 11 membership level. According to Fig. 7, it can be found that the proposed DR/EGO-UUA has a good precision in contrast to MCS-UUA, while T-UUA does not. Table 6 provides the maximum relative errors of expectations and variances between the proposed methods and MCS-UUA as well as their computational costs. As can be seen from Table 6, the maximum relative errors of expectation and variance boundaries computed by DR/EGO-UUA are 0.15% and 2.642% respectively, and the computational costs are far less than that of MCS-UUA because the number of function calls of DR/EGO-UUA is 17 790 while the one of MCS-UUA is 3.3*109 . On the contrary, though T-UUA has the fewest function calls of response function, the calculated results from T-UUA and MCSUUA are not consistent, whose maximum relative error of expectation is as large as 10.314% and the maximum relative error of variance is 5.485%. It indicates that the DR/EGO-UUA is more applicable to analyze the complex engineering problems under relatively large degree of uncertainty for its accuracy and relatively well efficiency.

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Fig. 6. The trestle bridge model. Table 6 The maximum relative errors and function calls of the proposed UUA methods for the trestle bridge. Method

Maximum relative error Expectation

T-UUA DR/EGO-UUA MCS

Function calls Variance

Lower bound

Upper bound

Lower bound

Upper bound

10.314% 0.150% –

7.854% 0.086% –

5.485% 2.642% –

4.338% 2.470% –

1056 17 790 3.3E9

5.3. The self-balancing vehicle A self-balancing vehicle [55,56] is investigated as shown in Fig. 8. It is a type of smart electric vehicle that controls the bodywork attitude by adopting self-balancing technology. As an important vehicle for short-distance transport, it is significant to guarantee the life safety of the drivers. To keep the stability and reliability of the performance of vehicle, the designer should take a full consideration of uncertainties’ effects on the performance.

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Fig. 7. Belief and plausibility of the maximum stress of the trestle bridge.

The chassis of the self-balancing vehicle provides the mounting platform for two driving wheels, one lighting module and one control unit. The finite element model of the self-balancing automobile chassis is established and the shell element is adopted, as shown in Fig. 8(c). In order to analyze the structural stability of the balancing vehicle under the condition of turning, two equal pressure P are applied to the chassis transversely and the driver’s pressure G on the chassis is analyzed. Due to the manufacture and measurement errors, the size of the chassis is marked by the basic size and deviation, and the value usually fluctuates irregularly within an interval. Therefore, the length a and width b are considered as interval variables. The Young’s modulus E and Poisson’s ratio v are assumed to be probabilistic variables since the relevant data is sufficient to obtain their probability distributions. Suppose that the human’s pressure G on the floor is evidence variable and the transverse load P is fuzzy variable, since only coarse information of them can be acquired. First, the degrees of uncertainties are assumed to be small. The specific information of uncertainties

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Fig. 8. The self-balancing vehicle model[56].

is given in Table 7. The proposed T-UUA, DR/EGO-UUA are used to estimate the deformation of the chassis after these uncertainties propagate through the FEM model. The numerical results of MCS-UUA are taken as the reference solutions in this case. In this example, N1∗ and N2∗ are chosen as 10 000 and 100 000, respectively, and the fuzzy variable is also discretized into 11 membership levels. The obtained results are given in Fig. 9 and Table 8. The same to numerical example 1, again, it can be found that both the results of T-UUA and DR/EGO-UUA agree well with the ones of MCS-UUA with the largest relative error less than 1.5% when the degrees of uncertainties are small. The numbers of function calls of the three methods using for computing the above solutions are 1056, 9856 and 3.3*1010 , respectively. Thus, the computational costs of both T-UUA and DR/EGO-UUA are much lower than that of MCS-UUA and the most efficient method is the proposed T-UUA. Then, the degrees of uncertainties are assumed to be relatively large. The specific information of uncertainties is given in Table 9. Again, the proposed methods T-UUA and DR/EGO-UUA are used to calculate the expectation and variance of the deformation of the chassis. The numerical results by MCS-UUA are also taken as the reference solutions in this case. Fig. 10 and Table 10 depict the obtained results which are bounded by the belief and plausibility measures. It can be found that compared to the case under small uncertainty degree, DR/EGO-UUA is more accurate than T-UUA when the degrees of uncertainties are relatively large. The results of DR/EGO-UUA are almost the same with the ones of MCS-UUA while the maximum relative error of T-UUA is 33.12%. Therefore, when resorting to relatively accurate results of the UUA, DR/EGO-UUA is more competent than T-UUA.

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X.Y. Long, D.L. Mao, C. Jiang et al. / Computer Methods in Applied Mechanics and Engineering 355 (2019) 1–26 Table 7 The information of uncertain variables under small degree of uncertainty for the chassis. Uncertain variable

Distribution type

Parameter 1

Parameter 2

Parameter 3

Uncertainty level

Length a Width b Young modulus E Poisson’ ratio ν

Interval Interval Normal Normal

Load, G

Evidence

Load, P

Fuzzy

450 mm 350 mm 70 000 MPa 0.33 0.0192 MPa 0.0196 MPa 0.0204 MPa 0.57 MPa

4.5 mm 10.5 mm 1400 MPa 0.0066 0.0196 MPa 0.0204 MPa 0.0208 MPa 0.6 MPa

– – – – 20% 40% 40% 0.63 MPa

1% 3% 2% 2% 1% 2% 1% 5%

Probabilistic variable: parameter 1 is the expectation; parameter 2 is the standard deviation; Evidence variable: parameter 1 is the lower bound of focal element, parameter 2 is the upper bound of focal element, parameter 3 is BPA; Interval variable: parameter 1 is the midpoint value; parameter 2 is the radius; Fuzzy variable: parameter 1 and parameter 3 are the lower and upper bounds of fuzzy number at α = 0; parameter 2 is the value of fuzzy number at α = 1. Table 8 The maximum relative errors and function calls of the proposed UUA methods under small degree of uncertainty for the chassis. Method

Maximum relative error Expectation

T-UUA DR/EGO-UUA MCS

Function calls Variance

Lower bound

Upper bound

Lower bound

Upper bound

0.613% 0.353% –

0.543% 0.446% –

1.368% 1.119% –

1.143% 0.457% –

1056 9856 3.3E9

Table 9 The information of uncertain variables under large degree of uncertainty for the chassis. Uncertain variable

Distribution type

Parameter 1

Parameter 2

Parameter 3

Uncertainty level

Length a Width b Young modulus E Poisson’ ratio ν

Interval Interval Normal Normal

Load, G

Evidence

Load, P

Fuzzy

450 mm 350 mm 70 000 MPa 0.33 0.018 MPa 0.019 MPa 0.021 MPa 0.54 MPa

27 mm 21 mm 7000 MPa 0.0099 0.019 MPa 0.021 MPa 0.022 MPa 0.6 MPa

– – – – 20% 40% 40% 0.66 MPa

6% 6% 10% 3% 2.7% 5% 2.3% 10%

Probabilistic variable: parameter 1 is the expectation; parameter 2 is the standard deviation; Evidence variable: parameter 1 is the lower bound of focal element, parameter 2 is the upper bound of focal element, parameter 3 is BPA; Interval variable: parameter 1 is the midpoint value; parameter 2 is the radius; Fuzzy variable: parameter 1 and parameter 3 are the lower and upper bounds of fuzzy number at α = 0; parameter 2 is the value of fuzzy number at α = 1.

6. Conclusion This paper proposes two unified uncertainty analysis methods which complement probability, evidence theory, interval and fuzzy set into a unified framework, by which the multi-source uncertainties can be quantified appropriately and considered simultaneously. Though temporarily neglecting and then reconsidering a part of uncertain variables, a T-UUA approach is first proposed for small uncertainty problems. Then, a modified DR/EGOUUA is further presented for the large uncertainty problems. The proposed T-UUA approach is simple, efficient and explicit, but it is limited to uncertain problems with small level of uncertainty and weak nonlinearity. Correspondingly, the proposed DR/EGO-UUA method can be applied to uncertain problems with relatively large

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Fig. 9. Belief and plausibility of the deformation under small degree of uncertainty for the chassis.

Table 10 The maximum relative errors and function calls of the proposed UUA methods under large degree of uncertainty for the chassis. Method

Maximum relative error Expectation

T-UUA DR/EGO-UUA MCS

Function calls Variance

Lower bound

Upper bound

Lower bound

Upper bound

5.095% 0.450% –

1.980% 0.637% –

33.115% 1.702% –

26.389% 1.230% –

1056 14 448 3.3E10

23

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Fig. 10. Belief and plausibility of the deformation under large degree of uncertainty for the chassis.

level of uncertainty and strong nonlinearity, but it is relatively more complex than the T-UUA. The proposed methods can also be simplified to address the hybrid uncertainty problems under any two or three kinds of these uncertainties. Three numerical examples are given to illustrate the validity of the proposed methods. The results indicate that the means and variances of uncertain responses after the propagation of probabilistic, evidence, interval and fuzzy variables, have two measures including their beliefs and plausibilities. When the uncertainty degree is relatively small, the calculated bounds of T-UUA and DR/EGO-UUA agree well with the ones of the MCS-UUA, while T-UUA fails to estimate accurate results when the uncertainty degree is relatively large. DR/EGO-UUA can well predict the results regardless of the uncertainty level of uncertainties. Though DR/EGO-UUA is less efficient than T-UUA, it is much more efficient than MCS-UUA. In the future, the proposed method is possible to be extended to predict the whole probability distributions of uncertain responses. In addition, it can also be used for robust optimization design by complementing probability, evidence theory, interval and fuzzy set into a unified model.

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