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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

A hybrid approach for global sensitivity analysis ⁎

Souvik Chakraborty , Rajib Chowdhury

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Department of Civil Engineering, IIT Roorkee, Roorkee, India

A R T I C L E I N F O

A BS T RAC T

Keywords: Global sensitivity analysis PCFE Uncertainty analysis Structural mechanics

Distribution based sensitivity analysis (DSA) computes sensitivity of the input random variables with respect to the change in distribution of output response. Although DSA is widely appreciated as the best tool for sensitivity analysis, the computational issue associated with this method prohibits its use for complex structures involving costly ﬁnite element analysis. For addressing this issue, this paper presents a method that couples polynomial correlated function expansion (PCFE) with DSA. PCFE is a fully equivalent operational model which integrates the concepts of analysis of variance decomposition, extended bases and homotopy algorithm. By integrating PCFE into DSA, it is possible to considerably alleviate the computational burden. Three examples are presented to demonstrate the performance of the proposed approach for sensitivity analysis. For all the problems, proposed approach yields excellent results with signiﬁcantly reduced computational eﬀort. The results obtained, to some extent, indicate that proposed approach can be utilized for sensitivity analysis of large scale structures.

1. Introduction All physical systems have inherent associated randomness. This randomness may exists either in the model formulation [1–4] or in model parameters (aleatoric uncertainty) [1,2]. Naturally these uncertainties propagate and is reﬂected in the model results and predictions [5]. The knowledge regarding the inﬂuence of the input uncertainties on the model response is extremely important to a design engineer. In this regard, the sensitivity analysis is an important tool which quantiﬁes the relative importance of the input variables. Given limited computational resource, sensitivity analysis may be used to identify and eliminate the lesser important variables. Due to this reason, sensitivity analysis has found wide application in stochastic computations [6–10], environmental science [11], geographic information systems [12], physiochemical systems [13,14] etc. The last two decades have witnessed signiﬁcant progress in sensitivity analysis. The most widely used sensitivity analysis tools are based on the diﬀerential methods [15,16], where the sensitivity analysis is performed by diﬀerentiating the output variable with respect to the inputs. The diﬀerentiation is generally performed by employing the ﬁnite diﬀerence method. Although easy to implement, this method is computationally expensive because determining stochastic response corresponding to each system parameter is demanding, speciﬁcally for large scale systems that involve costly ﬁnite element (FE) analysis. To overcome this issue, researchers have tried to formulate sensitivity index by utilizing direct diﬀerentiation [17]. However, it was observed that direct diﬀerentiation yields accurate result only when (a) the underlying function is either linear and (b) the design point [18] can be accurately traced. Other possible alternatives include ⁎

score function based approach [19,20] and perturbation method [21–23]. All these methods come under the broad category of local sensitivity analysis. A possible alternative to the local sensitivity analysis are the methods based on global sensitivity analysis (GSA) [24–28]. GSA is better suited for stochastic systems because, unlike local sensitivity analysis, GSA does not compute the sensitivity indexes based on only a few selected points. One of the popular approaches for GSA is the Morris method [29]. The basic idea is to vary only one input while keeping the other inputs constant. GSA is computed based on the local variations at diﬀerent points. However for systems involving uncertainties, only partial information is obtained by employing the Morris method. Another approach for GSA is the variance based sensitivity analysis proposed by Sobol and his associates [30,31]. As the name suggests, this approach utilizes the second moment properties to compute the sensitivity index of a variable. This method is quite popular because of its simplicity. Another group of methods for studying the sensitivity of a model inputs are the distribution based sensitivity analysis (DSA) tools [11,32–34]. In this method, the sensitivity index depends on the distribution (either probability density function or cumulative distribution function) of the response. It is a well-known fact that sensitivity index computed using DSA is more appropriate as it reﬂects the overall inﬂuence of the input variables on the output response. However, computational ineﬃciency of this method has prohibited its use in mechanics oriented problems. Motivated by the limitations highlighted above, present study focusses on developing an eﬃcient algorithm for distribution based GSA. To be speciﬁc, proposed approach couples polynomial correlated function expan-

Corresponding author. E-mail address: [email protected] (S. Chakraborty).

http://dx.doi.org/10.1016/j.ress.2016.10.013 Received 30 October 2015; Received in revised form 3 September 2016; Accepted 22 October 2016 Available online 25 October 2016 0951-8320/ © 2016 Elsevier Ltd. All rights reserved.

Reliability Engineering and System Safety 158 (2017) 50–57

S. Chakraborty, R. Chowdhury

sion (PCFE) [35–41] with DSA. This coupling results in a signiﬁcant reduction of the computational cost. Furthermore unlike most methods, proposed method is capable of performing sensitivity analysis of system entailing both dependent and independent random variables without the need of any ad hoc transformations. The rest of the paper is organised as follows. In Section 2, the GSA has been reviewed. Section 3 describes the basic formulation of PCFE. In Section 4, an algorithm for integrating PCFE into the framework of DSA has been proposed. Computational eﬃciency of the proposed approach has also been explained. Application of the proposed approach for sensitivity analysis has been demonstrated in Section 5. Finally, Section 6 provides the concluding remarks. 2. Global sensitivity analysis: a review In this section, a brief description of global sensitivity analysis has been provided. Suppose, the uncertain input variables of a system are represented by an N dimensional vector x = {x1, x 2 , …, xN } ∈ N . Given the uncertainty in x , it is obvious that the output response Y would also be uncertain with the variability of Y consisting of diﬀerent components of x . The objective of present study is to evaluate these contributions, such that one have a clear idea about the relative importance of the input variables. Saltelli and Tarantola [42] conferred that in sensitivity analysis based on variance “We are asked to bet on the factor that, if determined (i.e., ﬁxed to its true value), would lead to the greatest reduction in the variance of Y ”. Hence,

:i =

Var (Y ) − (Var (Y |xi )) x

Fig. 1. Visual representation of DSA. si is the diﬀerence between the unconditional PDF

f (Y ) and conditional PDF f (Y |xi ) , obtained by ﬁxing the variable xi .

3. Polynomial correlated function expansion Polynomial correlated function expansion (PCFE) [35–41] is a fully equivalent operational model recently developed for capturing the high dimensional relationship between sets of input and output model variables. It can be viewed as an extension of classical functional ANOVA decomposition [48] where the component functions are represented by utilizing the extended bases [49]. In literature, PCFE is also referred as generalised high dimensional model representation [50]. The unknown coeﬃcients associated with the bases are determined by employing a homotopy algorithm (HA) [51–53]. HA determines the unknown coeﬃcients by minimizing the least squared error and an objective function. The objective function deﬁnes an additional criteria that is enforced on the solution. In PCFE, the hierarchical orthogonality of the component functions is considered to be the additional criteria. Let, x = {x1, x 2 , …. xN } be a N dimensional vector, representing the input variables of a structural system. It is quite logical to express the output Y as a ﬁnite series as [31]

i

(1)

Var (Y )

where (•) and Var(•), respectively, denotes expectation and variance operators. :i in Eq. (1) denotes the sensitivity index of the ith variable. On contrary in DSA, “We are asked to bet on the factor that, if determined would lead to the greatest expected modiﬁcation in the distribution of Y ” [43]. Distribution based sensitivity measure was ﬁrst proposed by Park and Ahn [44]. The sensitivity measure considered in this study takes the following form [11]:

δi =

1 ⎛ ⎜ 2 ⎝

∫Ω

Y

⎞ ϖ (Y ) − ϖ (Y |xi ) dy⎟ ⎠x

i

(2)

where ΩY is the domain of the output response Y and ϖ(•) denotes the probability density function. A visual representation of the distribution based sensitivity measure is shown in Fig. 1.

N

Remark 1. As pointed out by Borgonovo [11,33], the sensitivity index described in Eq. (2) has two desirable properties. Firstly, δi is individually and jointly normalised, i.e., 0 ≤ δi ≤ 1 and δ12… N = 1. Secondly, δi is invariant to monotonic transformation.

Y (=g(x)) = g0 +

∑ ∑

gi i

(xi1, xi2 ,

∑

gij (xi , xj ) + ⋯ + g12… N (x1, x 2 , …, xN )

1 2 … iN

k =1 i1< i2 ⋯< ik N

… xiN ) = g0 +

∑ gi(xi ) + i1=1

1≤ i1< i2 ≤ N

Remark 2. If input variable xi and response Y are independent, δi = 0 [11,33].

(3) where g0 is a constant and termed as zeroth order component function.

Remark 3. Iman and Hora [45] pointed out that in presence of long input/output, the statistical quantiﬁes (such as variance) obtained might not be robust. As a consequence, variance based sensitivity measures loses its robustness. Distribution based sensitivityindices are free from this problem.

Deﬁnition 1. : Assume, two subspace R and B in Hilbert space are spanned by basis {r1, r2, …, r1} and {b1, b2 , …, bm} respectively. Now if (i) B ⊃ R and (ii) B = R + R⊥ where, R⊥ is the orthogonal complement subspace of R in B , we term B as extended basis and R as non-extended basis [50]. Now if ψ be some suitable basis for x ⊆ X , where X: = {1, 2, …, N}, Eq. (3) can be expressed as [35,49]:

Remark 4. Computation of sensitivity index by Eq. (2) involves extremely large number of actual function evaluations. Due to this reason, use of this method is limited to models where thousands of model evaluations are possible within a feasible computer time. One possible alternative for addressing the above mentioned issue is the application of surrogate model [11,46,47] to replace the original model. In this work, an eﬃcient fully equivalent operational model, referred here as PCFE, has been used to replace the original model. The detailed description of this method is provided in next section.

N

g(x) = g0 +

⎧ N − k +1 ⎪

∑ ⎨⎪ ∑ k =1

⎩

i1=1

N

⋯

k

⎡

∞

∞

∑ ∑ ⎢⎢ ∑ ∑

ik = ik −1 r =1

⎣ m1=1 m2=1

⎤⎫ ⎪ αm(i11mi22……ikm)irrψmi1 … ψmir ⎥⎬ 1 r ⎥⎪ ⎦⎭ mr =1 ∞

⋯

∑

(4) where α indicates the unknown expansion coeﬃcients. However, Eq. (4) represents an inﬁnite series and needs to be truncated. Considering upto 51

Reliability Engineering and System Safety 158 (2017) 50–57

S. Chakraborty, R. Chowdhury

⎧ ⎫T unknown coeﬃcients vector. Additionally if g = ⎨g1, g2 , …, gq ⎬ be ⎩ ⎭ the observed responses at q sets of training points, then (7)

d=g−g T

T

where, g = {g0 , g0 , …, g0} . Premultiplying Eq. (7) by ψ , we obtain (8)

Bα = C T

T

where, B = ψ ψ and C = ψ d . From Eq. (5), it is obvious that for N > 1, ψ have identical columns. Hence, matrix B has identical rows. These identical rows are the redundants and must be removed. Removing the redundants, we have (9)

B′α = C′

where B′ and C′ are, respectively, B and C after removing the redundants. Eq. (9) represents a set of underdetermined equations and naturally, there exists inﬁnite solutions:

α = (B′)−1C′ + (I − (B′)−1B′)v(s ) = (B′)−1C′ + Pv(s )

(10)

−1

where, (B′) denotes the generalised inverse of B′ satisfying part or all four Penrose condition [54]. I represents an identity matrix of dimension q × q . All the solution of α obtained from Eq. (10) compose a completely connected submanifold 4 ⊂ R q . We are interested in the solution that satisﬁes the orthogonality condition deﬁned in [35,50]. Employing homotopy algorithm [51–53], the unknown coeﬃcient vector α is obtained as [35,50]: −1

αHA = [Vq − r (UTq − r Vq − r ) UTq − r ]α0

(11)

where α0 denotes the solution of Eq. (9) obtained by least squared minimization. U and V in Eq. (11) represents the matrices obtained by singular value decomposition of PW where W denotes the weight matrix. In this work, a weight matrix that satisﬁes the hierarchical orthogonality criteria has been used. For details regarding the formulation of weight matrix interested readers may refer [35,50]. A pseudo code for determining unknown coeﬃcients of PCFE is shown in Algorithm 1. 4. Proposed algorithm In this section, an algorithm that integrates the PCFE into the framework of DSA has been presented. It has been demonstrated that by integrating PCFE with DSA, it is possible to signiﬁcantly reduce the computational eﬀort. Furthermore, using the proposed approach, it is

Fig. 2. Flowchart for DSA using PCFE..

the Mth order component function and sth order basis, Eq. (4) reduces to: M

gˆ(x) = g0 +

⎧ N − k +1 ⎪

∑ ⎨⎪ ∑ k =1

⎩

i1=1

N

⋯

k

⎡

s

s

∑ ∑ ⎢⎢ ∑ ∑

ik = ik −1 r =1

⎣ m1=1 m2=1

⎫ ⎤⎪ αm(i11mi22……ikm)irrψmi1 … ψmir ⎥⎬ 1 r ⎥⎪ ⎦⎭ mr =1 s

⋯

∑

(5) Once α is determined, Eq. (5) represents the basic functional form of PCFE. Remark 5. It is to be noted that the term ‘order’ in PCFE does not indicate the degree of nonlinearity. On contrast, it indicates the degree of cooperativity. An essential condition, associated with Eq. (5), is the hierarchical orthogonality of the component functions [35,39,50]. In order to satisfy this criteria, homotopy algorithm (HA) [51–53] has been used in this paper. HA determines the unknown coeﬃcients by minimizing the least squared error and satisfying the hierarchical orthogonality criteria. Eq. (5) can be rewritten in matrix form as

ψα = d

(6) Fig. 3. Variation of sensitivity index δi with increase in number of training points. MCS

where ψ represents the basis function matrix and α denotes the

is performed with 6 × 107 training points.

52

Reliability Engineering and System Safety 158 (2017) 50–57

S. Chakraborty, R. Chowdhury

determining the sensitivity index δi , ﬁrst model responses are generated at some preselected training points. Next, PCFE based FEOM is generated by mapping the input variables (at the training points) with the obtained model responses. Once the unknown coeﬃcients associated with the bases are determined, the step of the conventional approach described above are performed on the generated surrogate model. A ﬂowchart describing the procedure for performing DSA using PCFE is shown in Fig. 2. Suppose nSAMP , τ1 and tPA , respectively, denote the number of training points in PCFE, time needed for a single virtual simulation (simulation on generated model) and total time using the proposed approach. Hence,

Table 1 Sensitivity indexes obtained using proposed approach (PA) and conventional approach using MCS. Sensitivity index

MCS

PA

δ1 δ2

0.23 0.39

0.23 0.39

possible to perform sensitivity analysis of systems involving both correlated and uncorrelated random variables. Algorithm 1. Pseudo-code for PCFE.

tPA = nSAMPτ + nMCS(1 + Nnquad )τ1

1: Initialize: Specify PCFE order, distribution type of input variables and corresponding parameters. Generate training points and obtain response by actual analysis.

(var)i j ←

2: Scale the input variables as:

(13)

Comparing Eq. (13) and Eq. (12),

j

(var)i − min (var )i max (var)i − min (var)i

∀ i, j

3: Determine mean response as: 1

g0 ← n ∑s g(xs ) 4: for i = 1: n di ← g(xi ) − g0 end for T

5: ψ ← [ ψ (x1) ψ (x 2) ⋯ ψ (x N )] where, r T

ψ (x ) ←

⎡ 1 r 2 r 1 r 1 r ⎢ ψ1 (x1 ) ψ2 (x1 ) ⋯ ψk (x1 ) ψ1 (x2 ) ⋯ ⎣ ⎤ ψ11(x1r ) ⋯ ψmN −2(xNr −2)ψmN −1(xNr −1) ψmN −1(xNr −1)ψmN (xNr )⎥ ⎦

6: d ← [ d1 d 2 ⋯ dn ]T 7: B ← ψT ψ and C ← ψT d 8: [B′, d′] ← removeredundants(B, d) 9: P ← I − (B′)−1B′ 10: W ← formulateweightmatrix 11: [U, V] ← svd (PW) −1

12: αHA ← [Vq − r (UTq − r Vq − r ) UTq − r ]α0 Determination of the sensitivity index δi involves solving two numerical integrals. Using conventional approach/MCS, δi is determined as: i) Perform MCS to determine the response. Use kernel density function (ksdensity in MATLAB) to obtain the unconditional PDF. ii) Fixing the ﬁrst variable at a given value, generate the training points for other variables. Determine response and hence obtain conditional PDF. Using the conditional and unconditional PDF, obtain the relative change. Repeat this step for various values of the ﬁrst variable. Obtain δ1. iii) Repeat step (b) by keeping the other variables ﬁxed to obtain sensitivity index of the other variables. Now suppose τ , nMCS and tMCS, respectively, denotes the time needed for an actual simulation, number of sample points and total time. Also assume nquad represents the number of quadrature points. Hence,

tMCS = nMCSτ + NnquadnMCSτ = nMCS(1 + Nnquad )τ

(12)

Now for a standard problem, nMCS is around 10 5 − 106 . As a consequence, total time required for a complex system becomes tremendously large, making the process computationally cumbersome. In this context, we here propose to utilize PCFE to replace the actual model. As already observed in previous study [35], PCFE is capable of accurately predicting the PDF of response of systems having large number of random variables and high order of nonlinearity. Thus, it is expected that the proposed approach will yield accurate result. For

Fig. 4. (a) Four-storey building with isolator located at second-storey. (b) The system is subjected a single impulse sinusoidal loading.

53

Reliability Engineering and System Safety 158 (2017) 50–57

S. Chakraborty, R. Chowdhury

Table 2 Details of random variables considered in Example 2. Random variable

Units

Description

type

Mean

COV

mf

kg

Floor mass

LN

6000

0.1

kf

N/m

Floor stiffness

LN

3 × 107

0.1

cf

N/m/s

Floor damping coefficient

LN

6 × 104

0.2

fy

N

Isolation yield force

LN

4

0.2

dy

m

Isolation yield displacement

LN

kC

N/m

Isolation contact stiffness

LN

0.3

T A

s m/m/s

Force period Force amplitude

LN LN

3 × 107 1 1

2 × 10 0.05

0.2 0.3 0.5

nSAMPτ + nMCS(1 + Nnquad )τ1 nSAMP tPA τ = = + 1 tMCS nMCS(1 + Nnquad )τ nMCS(1 + Nnquad ) τ

(14)

Clearly nSAMP < < nMCS(1 + Nnquad ) and τ1 < < τ . Thus, the proposed approach is computationally eﬃcient as compared to MCS.

5. Numerical examples

Fig. 5. Variation of sensitivity index with increase in number of training points. The sensitivity of the random variables has been studied with respect to displacement of isolator.

In this section, the proposed approach has been utilized for performing sensitivity analysis of three problems. Although PCFE is invariably applicable with both uniformly and non-uniformly distributed training points, quasi random training points has been used in this work. Out of various available scheme for generating quasi random sequences, such as Halton sequence [55], Sobol sequence [56,57] and Faure sequence [58], Sobol sequence has been used due to its high convergence rate [59]. While ﬁrst problem investigates the performance of proposed approach in predicting sensitivity indexes of an explicit function. In the second example a four storey buildings with an isolator system has been considered. The third example investigates the performance of the proposed framework for sensitivity analysis of a twenty ﬁve element truss. The typicality of this problem resides in the fact that the system involves 33 correlated random variables. For all the problems, results obtained using proposed approach has been benchmarked against MCS solutions.

5.1. Example 1: Cubic function with two variables [19] In this example, an explicit function of the form:

g(x) = 2.2257 −

0.025 2 33 (x1 + x 2 − 20)3 + (x1 − x 2 ) 27 140

(15)

has been considered where x1 and x 2 are uncorrelated Gaussian variables with mean 10 and standard deviation 3. Proposed approach is employed to determine sensitivity index of x1 and x 2 . Fig. 3 shows the convergence of proposed approach (PA) with increase in training points. It is observed that the proposed approach converges with 64 training points only. The sensitivity indexes, as obtained from the proposed approach and conventional approach using MCS, are shown in Table 1. Excellent agreement among the results has been observed.

Table 3 Sensitivity indexes, with respect to inter-storey drift, obtained using proposed approach and conventional approach.

5.2. Example 2: Four storey building with bases isolator [60,61] In this example, a four-storey building with a base isolation system

Sensitivity index

MCS

PA

δ1 δ2 δ3 δ4 δ5 δ6 δ7 δ8

0.00225 0.00250 0.00141 0.00234 0.108 0.00211 0.00218 0.0324

0.00219 0.00225 0.00175 0.00209 0.102 0.00241 0.00334 0.00349

Table 4 Sensitivity indexes, with respect to displacement of isolator, obtained using proposed approach and conventional approach. Sensitivity index

MCS

δ1 δ2

0.0216

0.0249

5.82 × 10−4

5.44 × 10−4

2.45 × 10−4 0.0686 0.0473

2.34 × 10−4 0.0649 0.0469

4.26 × 10−4 0.290 0.298

4.21 × 10−4 0.291 0.290

δ3 δ4 δ5 δ6 δ7 δ8

PA

Fig. 6. Variation of sensitivity index with increase in number of training points. The sensitivity of the random variables has been studied with respect to inter-storey drift.

54

Reliability Engineering and System Safety 158 (2017) 50–57

S. Chakraborty, R. Chowdhury

Table 6 Sensitivity indexes, with respect to maximum lateral displacement, obtained using proposed approach (PA) and conventional approach. Sensitivity index

MCS

PA

δ1 δ2 δ3

0.0133 0.101

0.0171 0.106

9.88 × 10−3 0.0178

11.0 × 10−4 0.0210

δ4 δ5

5.71 × 10−5

6.12 × 10−5

δ6

11.2 × 10−5

9.88 × 10−5

δ7

−4

δ8 δ9 δ10 δ11 δ12 δ13 δ14 δ15 δ16 δ17 δ18

2.66 × 10 0.308 0.0545 0.0389 0.0316 0.0257 0.0213 0.0258 0.0327 0.0208 0.0279

2.9 × 10−4 0.311 0.0469 0.0336 0.0296 0.0266 0.0273 0.0277 0.0395 0.0230 0.0275

8.16 × 10−3

8.43 × 10−3

δ19

8 × 10−3

9 × 10−3

δ20 δ21 δ22

δ23 δ24

Table 5 Details of random variables for space truss defined in Example 3. Standard deviation

Distribution

1 2–5 6 7 8

P1 P2 P6 P7

100 500 60 50

E (N/m2)

1000 10000 600 500 107

A1 (m2)

0.4

5 × 105 0.04

Lognormal Normal Lognormal Lognormal Lognormal

9 10–13

A2–A5 (m2)

0.1

0.01

Lognormal

14–17

A6–A9 (m2)

3.4

0.34

Lognormal

18–19

A10–A11 (m2)

0.4

0.04

Lognormal

20–21

A12–A13 (m2)

1.3

0.13

Lognormal

22–25

A14–A17 (m2)

0.9

0.09

Lognormal

26–29

A18–A21 (m2)

1.0

0.1

Lognormal

30–33

A22–A25 (m2)

3.4

0.34

Lognormal

(N) - P5 (N) (N) (N)

8.65 × 10−3 0.0101

8.55 × 10−3 0.0130

7.05 × 10−3

6.59 × 10−3 9.75 × 10−3

δ26

5.86 × 10−3

5.49 × 10−3

δ27

−3

5.85 × 10

5.32 × 10−3

δ28

7.65 × 10−3

7.09 × 10−3

δ29

8.76 × 10−3 0.0231 0.0119

8.65 × 10−3 0.0195 0.0134

9.07 × 10−3 0.0217

9.21 × 10−3 0.0226

has been considered. The structure is subjected to a sinusoidal impulse of ground acceleration. Fig. 4 shows the building conﬁguration along with the impulse force. Base isolation bearings are provided that resists the motion of the ﬁrst ﬂoor. An additional stiﬀness force is activated only when the displacement dc exceeds 0.5 m. The mass, damping and stiﬀness of each ﬂoor is represented by m f , cf and k f respectively. The isolator on the second storey consists of two isolated mass which represents isolated, shock sensitive equipment. The larger mass m1(=500kg) is connected to the ﬂoor with a relatively ﬂexible spring having spring constant k1(=2500N/m). A stiﬀ spring having spring constant k 2(=10 5N/m) and a damper c2(=200N/m/s) is connected with the smaller mass m 2(=100kg). The system have eight random variables. Details of random variables are provided in Table 2. Proposed approach is utilized to determine sensitivity of the random parameters deﬁned in Table 2 with respect to inter-storey drift and displacement of isolator. Fig. 4 shows the convergence of proposed approach with increase in number of training points. It is observed that proposed approach yields computationally eﬃcient estimate of the sensitivity indexes as compared to conventional approach using MCS. Tables 3, 4 shows the sensitivity index obtained using proposed approach and conventional approach using MCS. Excellent agreement among the results has been observed (Figs. 5 and 6).

Fig. 7. Twenty ﬁve element truss considered in Example 3, (a) Nodes and members, (b) Loads.

Mean

9.66 × 10−3 0.0115

9.20 × 10−3

δ33

Variable

8.56 × 10 0.0105

δ25

δ30 δ31 δ32

Variable no.

−3

Lognormal

5.3. Example 3: Twenty ﬁve element truss [39] In this example, a twenty ﬁve element truss, as shown in Fig. 7, has been considered. The structure is having 10 nodes and 25 elements. It 55

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[7] Feng Z, Zhenzhou L, Lijie C, Shufang S. Reliability sensitivity algorithm based on stratiﬁed importance sampling method for multiple failure modes systems. Chin J Aeronaut 2010;23:660–9. [8] Iman RL, Matrix-Based A. Approach to Uncertainty and Sensitivity Analysis for Fault Trees. Risk Anal 1987;7:21–33. [9] Iman RL, Johnson ME, Schroeder TA. Assessing hurricane eﬀects. Part 1. Sensitivity analysis. Reliab Eng Syst Saf 2002;78:131–45. [10] Helton JC, Hansen CW, Sallaberry CJ. Uncertainty and sensitivity analysis in performance assessment for the proposed high-level radioactive waste repository at Yucca Mountain, Nevada. Reliab Eng Syst Saf 2012;107:44–63. [11] Borgonovo E, Castaings W, Tarantola S. Model emulation and moment-independent sensitivity analysis: an application to environmental modelling. Environ Model Softw 2012;34:105–15. [12] Crosetto M, Tarantola S. Uncertainty and sensitivity analysis: tools for GIS-based model implementation. Int J Geogr Inf Sci 2010;15:415–37. [13] Rabitz H. Systems analysis at the molecular scale. Science 1989;246:221–6. [14] Turanyi T. Sensitivity analysis of complex kinetic systems. Tools and applications. J Math Chem 1990;5:203–48. [15] Choi KK, Santos JLT. Design sensitivity analysis of non-linear structural systems part I: theory. Int J Numer Methods Eng 1987;24:2039–55. [16] Santos JLT, Choi KK. Sizing design sensitivity analysis of non-linear structural systems. Part II: numerical method. Int J Numer Methods Eng 1988;26:2097–114. [17] Haug EJ, Choi KK, Komkov V. Design sensitivity analysis of structural systems. New York: Academic Press; 1986. [18] Hohenbichler M, Rackwitz R. First-order concepts in system reliability. Struct Saf 1983;1:177–88. [19] Chowdhury R, Rao BN, Prasad AM. Stochastic sensitivity analysis using HDMR and score function. Sadhana 2010;34:967–86. [20] Zhu P, Shi L, Yang R-J, Lin S-P. A new sampling-based RBDO method via score function with reweighting scheme and application to vehicle designs. Appl Math Model 2015;39:4243–56. http://dx.doi.org/10.1016/j.apm.2014.11.045. [21] Gerstl S. Second-order perturbation-theory and its application to sensitivity studies in shield design calculations. Trans Am Nucl Soc 1973;16:342–3. [22] Kamiński M. Stochastic perturbation approach to the wavelet-based analysis, Numer. Linear Algebr Appl 2004;11:355–70. http://dx.doi.org/10.1002/nla.365. [23] Xiong M, Arnett FC, Guo X, Xiong H, Zhou X. Diﬀerential dynamic properties of scleroderma ﬁbroblasts in response to perturbation of environmental stimuli. PLoS One 2008;3:12. [24] Ma C, Li X, Wang S. A Global Sensitivity Analysis of Soil Parameters Associated With Backscattering Using the Advanced Integral Equation Model. IEEE Trans Geosci Remote Sens 2015;53:5613–23. http://dx.doi.org/10.1109/ TGRS.2015.2426194. [25] Burt JM, Josyula E. Global Sensitivity Analysis and Uncertainty Quantiﬁcation for a Hypersonic Shock Interaction Flow. J Thermophys Heat Transf 2015;29:439–49. http://dx.doi.org/10.2514/1.T4368. [26] Branger F, Giraudet L-G, Guivarch C, Quirion P. Global sensitivity analysis of an energy–economy model of the residential building sector, Environ. Model. Softw 2015;70:45–54. http://dx.doi.org/10.1016/j.envsoft.2015.03.021. [27] Boscato G, Russo S, Ceravolo R, Fragonara LZ. Global Sensitivity-Based Model Updating for Heritage Structures. Comput Civ Infrastruct Eng 2015;30:620–35. http://dx.doi.org/10.1111/mice.12138. [28] Bodjona K, Lessard L. Load sharing in single-lap bonded/bolted composite joints. Part II: global sensitivity analysis. Compos Struct 2015;129:276–83. http:// dx.doi.org/10.1016/j.compstruct.2015.03.069. [29] Campolongo F, Saltelli A. Sensitivity analysis of an environmental model: an application of diﬀerent analysis methods. Reliab Eng Syst Saf 1997;57:49–69. [30] Sobol IM, Levitan YL. On the use of variance reducing multipliers in Monte Carlo computations of a global sensitivity index. Comput Phys Commun 1999;117:52–61. [31] Sobol IM. Sensitivity estimates for nonlinear mathematical models, Math. Model. Comput Exp 1993;1:407–14. [32] Liu Q, Homma T. A new computational method of a moment-independent uncertainty importance measure. Reliab Eng Syst Saf 2009;94:1205–11. [33] Borgonovo E. A new uncertainty importance measure. Reliab Eng Syst Saf 2007;92:771–84. [34] Chun M-H, Han S-J, Tak N-I. An uncertainty importance measure using a distance metric for the change in a cumulative distribution function. Reliab Eng Syst Saf 2000;70:313–21. [35] Chakraborty S, Chowdhury R. Polynomial Correlated Function Expansion for Nonlinear Stochastic Dynamic. Anal, J Eng Mech 2014;141:11. [36] Chakraborty S, Chowdhury R. Assessment of polynomial correlated function expansion for high-ﬁdelity structural reliability analysis. Struct Saf 2016;59:9–19. [37] Chakraborty S, Chowdhury R. Polynomial correlated function expansion. Model Simul Tech Struct Eng, IGI Glob 2017:348–73. [38] Chakraborty S, Mandal B, Chowdhury R, Chakrabarti A. Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion. Compos Struct 2016;135:236–49. [39] Chakraborty S, Chowdhury R. A semi-analytical framework for structural reliability analysis. Comput Methods Appl Mech Eng 2015;289:475–97. [40] Chakraborty S, Chowdhury R. Sequential experimental design based generalised ANOVA. J Comput Phys 2016;317:15–32. http://dx.doi.org/10.1016/ j.jcp.2016.04.042. [41] Chakraborty S, Chowdhury R. Modelling uncertainty in incompressible ﬂow simulation using Galerkin based generalised ANOVA. Comput Phys Commun 2016. [42] Saltelli A, Tarantola S. On the Relative Importance of Input Factors in Mathematical Models. J Am Stat Assoc 2002;97:702–9.

is subjected to seven loads, namely P1, P2, P3, P4, P5, P6 and P7. Fig. 7(b) shows the loading details. All the loads are considered to be random. Moreover, cross-sectional area of all the twenty ﬁve members are considered to be random. Therefore, the system is having 33 random variables. Details of the random variables are provided in Table 5. Unlike previous examples, the random variables are considered to be correlated. For this example, the elements of the correlation matrix are

ρ12 = ρ13 = ρ14 = ρ15 = ρ26 = ρ27 = ρ36 = ρ46 = ρ56 = ρ37 = ρ47 = ρ57 = ρ67 = 0.15 ρ23 = ρ24 = ρ25 = ρ33 = ρ35 = 0.95 ρ16 = ρ17 = 0.75 ρij = 0.45 i, j = 9, 10, …, 33,

i≠j

ρii = 1, ∀ i ρij = 0,

elsewhere (16)

Sensitivity analysis have been carried out with respect to the max lateral displacement of the system. Table 6 shows the results obtained using the proposed approach and conventional approach using MCS. For all the variables, excellent agreement among the results has been observed. Furthermore, while only 2179 actual simulations is required for proposed approach, conventional approach requires 3,400,000 simulations (10,000 simulations per MCS run). 6. Conclusion This study proposes a novel approach for DSA of structural systems. It is a well-known fact that distribution based sensitivity index reﬂects the change in distribution of the output response and is thus considered to be a better indicator of variable sensitivity. However, conventional DSA requires large number of training points making it prohibitive for complex structural systems. In order to address this problem, we propose an approach that couples polynomial correlated function expansion with DSA. We have also presented a ﬂowchart that shows the step by step procedure for performing DSA following the proposed approach. Proposed approach has been utilized for sensitivity analysis of three problems. For all the problems, results obtained using proposed approach are in excellent agreement with the results of conventional approach. Furthermore, number of training points using the proposed approach is signiﬁcantly less. The results obtained to some extend indicate that the proposed approach can be utilized for sensitivity analysis of large scale structures. Acknowledgement SC acknowledges the support of MHRD, Government of India. RC acknowledges the support of CSIR via grant no. 22(0712)/16/EMR-II. References [1] Der Kiureghian A, Ditlevsen O. Aleatory or epistemic? Does it matter?. Struct Saf 2009;31:105–12. [2] Ross JL, Ozbek MM, Pinder GF. Aleatoric and epistemic uncertainty in groundwater ﬂow and transport simulation. Water Resour Res 2009;45, [n/a–n/a]. [3] Chen X, Park EJ, Xiu D. A ﬂexible numerical approach for quantiﬁcation of epistemic uncertainty. J Comput Phys 2013;240:211–24. [4] Jakeman J, Eldred M, Xiu D. Numerical approach for quantiﬁcation of epistemic uncertainty. J Comput Phys 2010;229:4648–63. http://dx.doi.org/10.1016/ j.jcp.2010.03.003. [5] Yu D. Parallelization of a two-dimensional ﬂood inundation model based on domain decomposition, Environ. Model. Softw 2010;25:935–45. [6] Arwade SR, Moradi M, Louhghalam A. Variance decomposition and global sensitivity for structural systems. Eng Struct 2010;32:1–10. http://dx.doi.org/ 10.1016/j.engstruct.2009.08.011.

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Reliability Engineering and System Safety 158 (2017) 50–57

S. Chakraborty, R. Chowdhury

parameters more than observation data. J Math Chem 2010;48:1010–35. [52] Li G, Rey-de-Castro R, Rabitz H. D-MORPH regression for modeling with fewer unknown parameters than observation data. J Math Chem 2012;50:1747–64. [53] Chakraborty S, Chowdhury R. Multivariate function approximations using DMORPH algorithm. Appl Math Model 2015;39:7155–80. [54] Rao CR, Mitra SK. Generalized inverse of matrix and its applications. New York, U.S.A: Wiley; 1971. [55] Halton JH. On the eﬃciency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 1960;2:84–90. [56] Sobol IM. Uniformly distributed sequences with an additional uniform property. USSR Comput Math Math Phys 1976;16:236–42. [57] Bratley P, Fox BL. Implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Softw 1988;14:88–100. [58] Faure H. Good permutations for extreme discrepancy. J Number Theory 1992;42:47–56. [59] Galanti S, Jung A. Low-discrepancy sequences: Monte Carlo simulation of option prices. J Deriv 1997;5:63–83. [60] Chowdhury R, Rao BN, Prasad AM. High-dimensional model representation for structural reliability analysis. Commun Numer Methods Eng 2009;25:301–37. [61] Gavin HP, Yau SC. High-order limit state functions in the response surface method for structural reliability analysis. Struct Saf 2008;30:162–79.

[43] Borgonovo E, Tarantola S. Moment independent and variance-based sensitivity analysis with correlations: an application to the stability of a chemical reactor. Int J Chem Kinet 2008;40:687–98. [44] Park CK, Ahn K-I. A new approach for measuring uncertainty importance and distributional sensitivity in probabilistic safety assessment. Reliab Eng Syst Saf 1994;46:253–61. [45] Iman RL, Hora SC. A Robust Measure of Uncertainty Importance for Use in Fault Tree System Analysis. Risk Anal 1990;10:401–6. [46] Caniou Y, Sudret B. Distribution-based global sensitivity analysis using polynomial chaos expansions, Procedia - Soc. Behav Sci 2010;2:7625–6. http://dx.doi.org/ 10.1016/j.sbspro.2010.05.149. [47] Y. Caniou, B. Sudret, Distribution-based global sensitivity analysis in case of correlated input parameters using polynomial chaos expansions, in: Proceedings of the 11th International Conference Appl. Stat. Probab. Civ. Eng., 2011. [48] Rabitz H, Alis OF. General foundations of high-dimensional model representations. J Math Chem 1999;25:197–233. [49] Li G, Hu J, Wang S-W, Georgopoulos PG, Schoendorf J, Rabitz H. Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its diﬀerent order component functions. J Phys Chem A 2006;110:2474–85. [50] Li G, Rabitz H. General formulation of HDMR component functions with independent and correlated variables. J Math Chem 2012;50:99–130. [51] Li G, Rabitz H. D-MORPH regression: application to modeling with unknown

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