Stochastic response surface method for reliability problems involving correlated multivariates with non-Gaussian dependence structure: Analysis under incomplete probability information

Computers and Geotechnics 89 (2017) 22–32

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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Stochastic response surface method for reliability problems involving correlated multivariates with non-Gaussian dependence structure: Analysis under incomplete probability information Fan Wang a,b,c,⇑, Heng Li a a b c

Department of Building and Real Estate, The Hong Kong Polytechnic University, Kowloon, Hong Kong Department of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Resource and Civil Engineering, Wuhan Institute of Technology, Wuhan 430073, PR China

a r t i c l e

i n f o

Article history: Received 23 September 2016 Received in revised form 9 January 2017 Accepted 9 February 2017

Keywords: Reliability Stochastic response surface method Pair-copula Vines Rosenblatt transformation Tunnels

a b s t r a c t This paper aims to provide a stochastic response surface method (SRSM) that can consider non-Gaussian dependent random variables under incomplete probability information. The Rosenblatt transformation is adopted to map the random variables from the original space into the mutually independent standard normal space for the stochastic surrogate model development. The multivariate joint distribution is reconstructed by the pair-copula decomposition approach, in which the pair-copula parameters are retrieved from the incomplete probability information. The proposed method is illustrated in a tunnel excavation example. Three different dependence structures characterized by normal copulas, Frank copulas, and hybrid copulas are respectively investigated to demonstrate the effect of dependence structure on the reliability results. The results show that the widely used Nataf transformation is actually a special case of the proposed method if all pair-copulas are normal copulas. The effect of conditioning order is also examined. This study provides a new insight into the SRSM-based reliability analysis from the copula viewpoint and extends the application of SRSM under incomplete probability information. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction There is an intrinsic uncertainty associated with the properties of geo-materials, and deterministic approaches to the stability analysis for geotechnical engineering cannot take these uncertainties into consideration. To address this problem, probabilistic methods, such as the first/second-order-order reliability method (FORM/SORM) and Monte Carlo simulation (MCS), have received a great deal of attentions because they are not only capable of handling uncertainty but also have the potential to support rational decision-making from a risk perspective [1]. One major challenge when applying FORM/SORM or MCS is to define the limit state function. Due to the complex underground conditions, the structure response has to be investigated using numerical approaches in many cases, which results in an implicit limit state function. To represent the limit state function in an explicit form, the response surface method (RSM) is widely adopted [2]. Various techniques, such as quadratic polynomials, ⇑ Corresponding author at: Department of Building and Real Estate, The Hong Kong Polytechnic University, Kowloon, Hong Kong. E-mail address: [email protected] (F. Wang). http://dx.doi.org/10.1016/j.compgeo.2017.02.008 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.

artificial neural networks and support vector machines, are used to construct the response surface (e.g., [3–6]). Unlike conventional deterministic RSM, the stochastic RSM (SRSM) uses polynomial chaos expansion (PCE) to model the input-output relationships in the standard random space [7]. The approximation by SRSM is accomplished by determining the coefficients associated with the PCE, which can be achieved by a probabilistic collocation method [8–10]. Generally, the approximation by PCE is valid across the entire random space [11], which is the major difference between the SRSM and the deterministic RSM. Li et al. [12] further extended the SRSM to consider correlated non-normal input variables for geotechnical reliability analysis by using the Nataf transformation [13]. It has been acknowledged that the Nataf transformation inherently assumes a Gaussian dependence structure for correlated multivariates [14]. From the copula viewpoint, it adopts a normal copula to characterize the underlying dependence structure [15,16]. However, recent investigations have demonstrated that this assumption does not always hold [17]. For non-Gaussian dependence structure cases, other copula functions should be used [18–20]. Unfortunately, in many reliability problems, the probabilistic description of the random vector is given in terms

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F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

of marginal distributions and correlations (referred to as the incomplete probability information [13]). Under such condition, the joint distribution cannot be uniquely determined because the dependence structure is not known [14,17,21]. In other words, the reliability evaluation based on Nataf transformation (normal copula) is only one of the various possible solutions. To consider non-Gaussian dependent random variables under incomplete probability information, the joint distribution is constructed based on non-normal copulas with its parameter related to the correlation coefficient [14,17,19,20,22,23]. Since the dependence structure is unknown, there is no guideline on the copula selection. Several works [17,22–24] have examined the impact of different copulas on the reliability evaluations, and it is concluded that the results could be differed in a non-trivial way under different selections of copula. However, most of the extant studies are restricted to the bivariate cases [14,17,22–24]. If more than two random variables are mutually correlated, it is difficult to establish a one-to-one relationship between the pair-wise correlation coefficients and the multivariate copula parameters because their numbers are generally not equivalent [14]. This limitation can be attributed to the inflexibility of the conventional multivariate copulas in representing multivariate joint distributions with complex dependence structure. Recently, construction of multivariate joint distribution by paircopulas has drawn great attentions because it is highly flexible in modeling complex patterns of dependence by taking bivariate copulas as building blocks [25–28]. In this study, the pair-copula decomposition approach is adopted to represent the multivariate joint distribution. As a result, the number of pair-copula parameters is equivalent to the pair-wise correlation coefficients so that it is possible to relate them one by one. After the construction of the multivariate joint distribution, the Rosenblatt transformation [29] is used to establish a mapping relationship between the original space and the mutually independent standard normal space (i.e., U-space) for the SRSM model development. The proposed method is illustrated in a tunnel excavation example and it is compared to the SRSM with the Nataf transformation. The aim of this paper is to extend the SRSM to correlated multivariates with any dependence structure under incomplete probability information.

Generally, the accuracy of approximation by Eq. (2) increases as the order p increases; however, higher order Hermite polynomials will incur a rapid increase of expansion terms. Hence, Eq. (2) is truncated at a specific order to achieve an accurate approximation, while the workload of deriving algebraic expressions is acceptable. For reference, Table 1 summarizes the closed-form of Eq. (2) from order 2 to 4. Li et al. [12] summarizes the four major steps for establishing a stochastic surrogate model: (1) represent the input random variables by the independent standard normal random variables; (2) represent the output using the Hermite polynomials; (3) determine the coefficients associated to the Hermite polynomials using the collocation method; and (4) estimate the failure probability by available reliability techniques, e.g., MCS or FORM/SORM. The first step is critical particularly for correlated non-normal input random variables because the valid representation generally requires a nonlinear transformation from the original space to the U-space [12]. In Section 3, this transformation will be discussed in detail for correlated multivariates with non-Gaussian dependence structure. 2.2. Selection of collocation points The unknown coefficients ai1 i2 ...in can be determined by the stochastic collocation method [30]. Similar to the deterministic collocation method, the roots of the next higher order Hermite polynomial are used as the stochastic collocation points. Then, the system responses are evaluated (e.g., by numerical approaches), and the coefficients are computed by: 1

a ¼ ðHT HÞ HT F

ð4Þ

2. Collocation-based stochastic response surface method

where H and F are the matrix of Hermite polynomials and the vector of system responses at the collocation points, respectively, and a is the coefficient vector to be solved. For n-dimensional problems, the candidate collocation points for p-order polynomials are the combinations of the (p + 1)-order Hermite polynomial roots. Note that the origin should be incorporated if it is not a collocation point because it captures the region of high probability in the standard normal space [7–9]. Thus, the number of candidate collocation points is:

2.1. Stochastic response surface method

Ncp ¼

In the SRSM, Hermite polynomials are widely adopted for functional approximation. Suppose that Y is the output random variable (i.e., system responses) and X ¼ ½x1 ; x2 ; . . . ; xn  is the n-dimensional input random variable vector represented by a vector of independent standard normal variables U ¼ ½U 1 ; U 2 ; . . . ; U n  as X ¼ TðUÞ. Then, the limit state function can be written as:

If all the candidate collocation points are adopted to determine the polynomial coefficients, a relatively large number of real model runs have to be implemented, particularly when n and p are high. The minimum number of collocation points needed to determine the coefficients is:

Y ¼ GðXÞ ¼ GðTðUÞÞ ¼ HðUÞ

Na ¼

(

ð1Þ

Using the Hermite polynomials, Y ¼ HðUÞ can be written as:

Y ¼ a0 þ

i1 n n X X X ai1 C1 ðU i1 Þ þ ai1 i2 C2 ðU i1 ; U i2 Þ i1 ¼1

i1 ¼1 i2 ¼1

i1 X i2 n X X ai1 i2 i3 C3 ðU i1 ; U i2 ; U i3 Þ þ . . . þ

ð2Þ

i1 ¼1 i2 ¼1 i3 ¼1

where ai1 i2 ...in are unknown coefficients, and Cp ðÞ is the multidimensional p-order Hermite polynomials given by: 1 TU

Cp ðU i1 ; . . . ; U ip Þ ¼ ð1Þp e2U

p

@ 1 T e2U U @U i1 ; . . . ; @U ip

ð3Þ

ðp þ 1Þn ;

if p is even

ðp þ 1Þn þ 1; if p is odd

ðn þ pÞ! n!p!

ð5Þ

ð6Þ

Generally, N a  N cp . Thus, the number of realizations can be reduced if the collocation points are selected appropriately. Li et al. [12] noted that the collocation points should be selected to ensure that the Hermite polynomial matrix H has a full rank. 3. Transformation for correlated multivariates with nonGaussian dependence structure under incomplete probability information In engineering practice, correlations may exist among various random variables [31]. For example, the Young’s modulus can be simultaneously correlated with the uniaxial compressive strength and the geological strength index of rock mass [32]. To consider

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F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

where Fðx1 ; x2 Þ is the bivariate joint probability distributions; u1 ¼ Fðx1 Þ and u2 ¼ Fðx2 Þ are marginal cumulative distribution functions (CDFs) of x1 and x2 , respectively; C is the copula function with parameter h. Table 2 lists a few commonly used bivariate copula functions and the range of the corresponding copula parameter. Each copula function represents a specific dependence structure. For example, the normal and Frank copulas are used to represent joint distributions without tails, while the joint distributions represented by the Clayton copula has an asymmetric lower tail [34]. If the multivariate joint PDF of X exists, it can be decomposed as:

Table 1 Closed-form expressions of the limit state function approximation by the Hermite polynomials from order 2 to 4. Order Closed-form expression P P P Pn 2 H2 ðUÞ ¼ a0 þ ni¼1 ai U i þ ni¼1 aii ðU 2i  1Þ þ n1 i¼1 j>i aij U i U j Pn 3 3 H3 ðUÞ ¼ H2 ðUÞ þ i¼1 aiii ðU i  3U i Þ P Pn1 Pn P P þ ni¼1 nj¼1 aiij ðU 2i U j  U j Þ þ n2 i¼1 j>i k>j aijk U i U j U k j–i Pn 4 2 4 H4 ðUÞ ¼ H3 ðUÞ þ i¼1 aiiii ðU i  6U i þ 3Þ P Pn P Pn 2 2 2 2 aiiij ðU 3i U j  3U i U j Þ þ n1 þ ni¼1 j¼1 i¼1 j>i aiijj ðU i U j  U i  U j þ 1Þ j–i Pn3 Pn2 Pn1 Pn Pn Pn1 Pn 2 þ i¼1 j¼1 k>j aiijk ðU i U j U k  U j U k Þ þ i¼1 j>i k>j l>k aijkl U i U j U k U l j–i

k–i

f ðXÞ ¼ f ðx1 Þf ðx2 jx1 Þ . . . f ðxn jx1 ; . . . ; xn1 Þ

wherein the conditional PDF f ðxn jx1 ; . . . ; xn1 Þ can be represented as [25]:

the effects of correlations, the Nataf transformation is commonly adopted to map the collocation points from the U-space into the original space, where the physical model computations are implemented to determine the coefficients for the stochastic surrogate model [12]. However, as mentioned before, the Nataf transformation is valid only if the dependence structure is Gaussian. For non-Gaussian cases, a novel transformation method based on Rosenblatt transformation [29] and pair-copula construction of multivariate joint distribution is proposed in this study.

f ðxjv Þ ¼ cx;v j jvj ðFðxjv j Þ; Fðv j jv j ÞÞf ðxjv j Þ where

R

0

U1 ðz1 Þ C B 1 B U ðz2 Þ C C B

Fðxjv Þ ¼

U ¼ T ðXÞ ¼ B B @

.. .

B B B C¼B C B A @

U1 ðzn Þ

1

U1 ðFðx1 ÞÞ U1 ðFðx2 jx1 ÞÞ

C C C C C A

.. .

is an arbitrarily selected element of a vector

v

excluding

v j , cx;v jv j

j

¼

v

while

@C x;v jv ðFðxjv j Þ;Fðv j jv j ÞÞ j j @Fðxjv j Þ@Fðv j jv j Þ

@C x;v j jvj ðFðxjv j Þ; Fðv j jv j ÞÞ @Fðv j jv j Þ

Particularly, when

ð7Þ

Fðxjv Þ ¼

U1 ðFðxn jx1 ; . . . ; xn1 ÞÞ

v j is a

ð11Þ

v is a univariate, Fðxjv Þ is simplified to:

@C x;v ðFðxÞ; Fðv ÞÞ @Fðv Þ

ð12Þ

The right hand side of Eq. (12) is the first derivative of a bivariate copula with respect to the second parameter, which is referred to as the conditional copula [26]. Recall that FðxÞ and Fðv Þ are uniform on [0, 1], the conditional copula can be denoted as:

where U1 is the inverse function of the cumulative distribution function (CDF) of a univariate standard Gaussian distribution; Fðxn jx1 ; . . . ; xn1 Þ is the marginal conditional distributions. Compared to the Nataf transformation, the Rosenblatt transformation provides a more general way for representing the input random variables by the independent standard normal random variables; however, it requires the joint probability density function (PDF) to be known so that Fðxn jx1 ; . . . ; xn1 Þ can be derived. Unfortunately, the incomplete probability information is more likely to be available in practice. To remove this limitation, the next subsection shows how to reconstruct the joint PDF from the incomplete probability information by means of pair-copulas.

@CðFðx1 Þ; Fðx2 ÞÞ ¼ hðu1 ; u2 ; hÞ @Fðx2 Þ

ð13Þ

with u2 always the conditioning variable. Table 2 also presents the corresponding h-functions for the listed copulas. There are many different representations of f ðXÞ using paircopulas. To help sort them, Bedford and Cooke [36,37] proposed a hierarchical graphical model known as regular vines. There are two major types of regular vines: (1) the canonical vine (C-vine) in which there is a key node that dominates the connections with all the other ones; and (2) the drawable vine (D-vine) which treats all the nodes equivalently. The joint PDF f ðXÞ corresponding to a C-vine can be written as:

3.2. Pair-copula construction of multivariate joint distributions Copulas have gained great popularity due to its efficiency in modeling joint probability distributions [33]. In particular, it enables the representation of dependent variables by their marginal distributions and the underlying dependence structure characterized by a copula function:

Fðx1 ; x2 Þ ¼ CðFðx1 Þ; Fðx2 Þ; hÞ ¼ Cðu1 ; u2 ; hÞ

ð10Þ

bivariate copula density function of C x;v j jvj (see Table 2). Using Eq. (10) recursively, a multivariate joint PDF can be represented by a product of bivariate copula density functions and marginal conditional distributions in the form of Fðxjv Þ.According to Joe [35], Fðxjv Þ can be expressed as:

The Rosenblatt transformation T R : Rn ! Rn is defined as [29]:

1

vj

denotes the vector

3.1. Rosenblatt transformation

0

ð9Þ

f ðXÞ ¼

n Y

f ðxk Þ

nj n1Y Y

cj;jþij1;...;j1 fFðxj jx1 ; . . . ; xj1 Þ; Fðxjþi jx1 ; . . . ; xj1 Þg

j¼1 i¼1

k¼1

ð14Þ

ð8Þ

while f ðXÞ corresponding to a D-vine is given as:

Table 2 Examples of bivariate copulas. Copula

Copula function Cðu1 ; u2 ; hÞ

Normal

Uh ðn1 ; n2 Þa

Frank

  1 ðehu1  1Þðehu2  1Þ  ln 1 þ h eh  1

Clayton a

Copula density function cðu1 ; u2 ; hÞ 2

1=h

h ðuh 1 þ u2  1Þ

1 ðhn1 Þ  2hn1 n2 þ ðhn2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2ð1  h2 Þ 1  h2 hehðu1 þu2 Þ ðeh  1Þ

2

h-function hðu1 ; u2 ; hÞ ! n  hn U p2 ffiffiffiffiffiffiffiffiffiffiffiffiffi1ffi 1  h2 ehu1 ðehu2  1Þ ðeh  1Þ þ ðehu1  1Þðehu2  1Þ

!

2

½ðeh  1Þ þ ðehu1  1Þðehu2  1Þ h ðh þ 1Þu1h1 uh1 ðuh 2 1 þ u2  1Þ

21=h

11=h

h uh1 ðuh 1 1 þ u2  1Þ 1

1

Uh is the bivariate standard Gaussian distribution function with correlation coefficient of h, and n1 ¼ U ðu1 Þ, n2 ¼ U ðu2 Þ.

Range of h ½1; 1 ð1; 1Þ n f0g ð0; 1Þ

25

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32 nj n n1Y Y Y f ðxk Þ ci;iþjjiþ1;...;iþj1

f ðXÞ ¼

Therefore, for example, the Rosenblatt transformation based on the C-vine models can be written as:

j¼1 i¼1

k¼1

fFðxi jxiþ1 ; . . . ; xiþj1 Þ; Fðxiþj jxiþ1 ; . . . ; xiþj1 Þg

ð15Þ

3.3. Relating the pair-copula parameters to the correlation coefficients By using the pair-copula approach to constructing joint PDF, the number of pair-copula parameters is equivalent to the correlation coefficients. To achieve a one-to-one retrieval of the pair-copula parameters from the pair-wise correlation coefficients, the bivariate joint PDF has to be derived from the multivariate joint PDF:

f ðxi ; xj Þ ¼

f ðXÞ f ðXi;j jxi ; xj Þ

ð16Þ

where Xi;j denotes the random vector excluding the i-th and j-the variables. By analogy with Eq. (9), the conditional PDF f ðXi;j jxi ; xj Þ can be written as:

f ðXi;j jxi ; xj Þ ¼ f ðx1 jxi ; xj Þf ðx2 jx1 ; xi ; xj Þ . . . f ðxn jXi;j;n ; xi ; xj Þ

ð17Þ

wherein the marginal conditional distributions in the right hand side of Eq. (17) can also be represented by using Eq. (10) recursively. Note both Eqs. (9) and (17) are only one of the various possible factorizations. The Pearson’s correlation coefficient has the following expression [38]:

qij ¼

Z

þ1

1

Z

þ1 1

lNi lNj f ðxi ; xj Þdxi dxj

ð18Þ

where lNi ¼ ðxi  li Þ=ri and lNj ¼ ðxj  lj Þ=rj ; li , lj and ri , rj are the means and the standard deviations of xi and xj , respectively. By representing Eq. (16) using pair-copula decomposition and inserting Eq. (16) into Eq. (18), the pair-copula parameters are related to the correlation coefficients. In particular, if f ðXÞ is a bivariate distribution, then the copula parameter can be solely determined by the correlation coefficient:

q12 ¼

Z

þ1

1

Z

þ1

1

l l N 1

N 2 f ðx1 Þf ðx2 ÞcðFðx1 Þ; Fðx2 Þ; h12 Þdx1 dx2

ð19Þ

Unlike bivariate distribution in which only the bivariate copula parameters are used to construct the joint PDF, for multivariate cases, both the bivariate copula parameters and the bivariate conditional copula parameters are needed to accomplish the construction. Unfortunately, as shown in the example in Appendix A, these bivariate conditional copula parameters cannot be solely determined by the corresponding correlation coefficients. Moreover, it is argued that the analytical approach to these copula parameters could be challenging because additional copula parameters that are irrelevant to the desired construction have to be estimated (see Appendix A). To address the problem, a simulation-based method is proposed in Appendix B to approximate the solutions. Once the pair-copula model is established, the marginal conditional distributions in Eq. (7) can be obtained using Eq. (11) recursively. However, different choice of v j in Eq. (11) can result in different conditioning order in the Rosenblatt transformation. For simplicity, for the C-vine models, v j is always chosen so that

Fðxj jx1 ; . . . ; xj1 Þ ¼

U 1 ¼ U1 ðFðx1 ÞÞ U 2 ¼ U1 ðhðFðx2 Þ; Fðx1 Þ; h12 ÞÞ U 3 ¼ U1 ðhðhðFðx3 Þ; Fðx1 Þ; h13 Þ; hðFðx2 Þ; Fðx1 Þ; h12 Þ; h23j1 ÞÞ

ð22Þ

... Then, the collocation points in the U-space are mapped into the original space using the inverse of the Rosenblatt transformation for evaluating the corresponding system responses. That is,

x1 ¼ F 1 ðUðU 1 ÞÞ 1

x2 ¼ F 1 ðh ðUðU 2 Þ; Fðx1 Þ; h12 ÞÞ 1

1

1

x3 ¼ F ðh ðh ðUðU 3 Þ; hðFðx2 Þ; Fðx1 Þ; h12 Þ; h23j1 Þ; Fðx1 Þ; h13 ÞÞ ... ð23Þ 1

where h ðÞ is the inverse function of Eq. (13). 4. An example application in a circular tunnel 4.1. Case description The proposed method is applied to a circular tunnel excavation stability reliability problem under hydrostatic stress field, which has been investigated by several researchers (e.g., [40–42]). Assume that the rock mass is homogeneous, continuous, and isotropic, due to the excavation of circular tunnel of a radius of R, the inward displacement of the tunnel lining uip under a hydrostatic stress field with a pressure of p0 and a radial uniform support pressure of pi , as shown in Fig. 1, are given by:

#  "  2 Rpl uip 1þt  ð1  2tÞðp0  pi Þ ¼ 2ð1  tÞðp0  pcr Þ R E R  1=ðk1Þ Rpl 2ðp0 þ sÞ ¼ ðk þ 1Þðpi þ sÞ R

ð24Þ

ð25Þ

p0

Plastic zone

Elastic zone

p0

Rpl R pi

@C j;j1j1;...;j2 ðFðxj jx1 ; . . . ; xj2 Þ; Fðxj1 jx1 ; . . . ; xj2 ÞÞ @Fðxj1 jx1 ; . . . ; xj2 Þ ð20Þ

while for the D-vine models,

Fðxj jx1 ;. .. ;xj1 Þ ¼

v j is always chosen so that

@C j;1j2;...;j2 ðFðxj jx2 ;. .. ;xj1 Þ;Fðx1 jx2 ;. . .; xj1 ÞÞ @Fðx1 jx2 ;. .. ;xj1 Þ

ð21Þ Fig. 1. Schematic diagram of a circular tunnel excavation in a rock mass.

26

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

to construct the joint PDF. Consequently, the joint PDF is expressed as

Table 3 Statistics of random variables. Parameter

Distribution

Mean

Standard deviation

E (MPa) c (MPa) u (°)

Lognormal Lognormal Lognormal

373 0.23 22.85

48 0.068 1.31

where Rpl is the radius of the plastic zone, E is the Young’s modulus,

t is the Poisson’s ratio, and pcr indicates the occurrence of plastic

yielding if pi < pcr , which is given by:

2p0  rc kþ1

ð26Þ

1 þ sin u 1  sin u

ð27Þ

cðk  1Þ tan u

ð28Þ

pcr ¼

k¼

rc ¼

where u is the friction angle, c is the cohesion, and in Eq. (25), rc s ¼ k1 . Here, p0 , t, and pi are considered to be deterministic with values p0 ¼ 2:5 MPa, t ¼ 0:3, and pi ¼ 0:868 MPa. The other factors, namely E, u, and c, are considered as random variables and their statistics are listed in Table 3. Moreover, The following correlation coefficients are assumed: qE;c ¼ 0:2, qc;u ¼ 0:5, and qE;u ¼ 0. Suppose the permissible inward displacement is 1% of the tunnel radius. Thus the performance function is defined as:

g ¼ 0:01 

uip R

ð29Þ

4.2. Normal copula case The bivariate normal copula is first adopted for all pair-copulas to illustrate the reliability evaluations under the assumption of Gaussian dependence structure. The conditioning order in the Rosenblatt transformation is taken as E, c, u, and the C-vine is used 10

P (uip /R>x)

10

10

10

10

f ðE; c; uÞ ¼ f ðEÞf ðcÞf ðuÞcE;c ðFðEÞ;FðcÞÞcE;u ðFðEÞ;FðuÞÞcc;ujE ðFðcjEÞ;FðujEÞÞ:

The same expression and conditioning order are adopted in Section 4.3 and 4.4, and the effect of conditioning order will be discussed in Section 4.5. According to Lebrun and Dutfoy [16], the Rosenblatt transformation and Nataf transformation are equivalent in the normal copula case. Consequently, for Gaussian dependence structure cases, the reliability results based on the two transformations should be consistent. Fig. 2 presents the CDFs of the relative inward displacement for both transformations (the Nataf transformation with a trivariate normal copula and the Rosenblatt transformation with three nested bivariate normal copulas) based on 106 realizations of independent normal random variables. The curves based on the two transformations are general consistent for each stochastic surrogate model, which shows the feasibility of the transformation method. Table 4 summarizes the failure probabilities obtained by different methods. The 4-order SRSM is selected to be integrated with the available FORM [43], SORM [44], and MCS. The results based on both transformations are consistent and close to the MCS result, which validates the accuracy of the proposed method. 4.3. Frank copula case The proposed method can deal with correlated multivariates with non-Gaussian dependence structure under incomplete probability information. To illustrate the impact of non-Gaussian dependence structure on reliability evaluation, the bivariate Frank copula is taken as the pair-copulas. Fig. 3 shows the CDFs of the relative inward displacement based on the SRSM from order 2 to 4. The curve plotted by a crude MCS with 106 runs of the physical model is taken as a benchmark. It is evident that the result based on a 3-order SRSM is sufficiently accurate. Table 5 tabulates the failure probabilities using the performance function. Compared to the normal copula case, the expected failure probability is slightly smaller if the Frank copula actually captures the underlying dependence structure. In other words, the reliability results can be biased

0

-1

-2

Rosenblatt+2-order SRSM Nataf+2-order SRSM Rosenblatt+3-order SRSM Nataf+3-order SRSM Rosenblatt+4-order SRSM Nataf+4-order SRSM

-3

-4

4

5

6

7

8

x

9

10

11

12 x 10

-3

Fig. 2. CDFs of the relative inward displacement under different transformations and SRSM (normal copula case).

27

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32 Table 4 Comparison of tunnel excavation failure probabilities obtained by different methods. Crude MCS (N = 106)

4-order SRSM + Nataf transformation

0.39% ±

4-order SRSM + Rosenblatt transformation

FORM

SORM

MCS (N = 10 )

FORM

SORM

MCS (N = 106)

0.43%

0.39%

0.39% ±

0.43%

0.39%

0.39% ±

10

P (uip /R>x)

10

10

10

10

6

0

2-order SRSM 3-order SRSM 4-order SRSM crude MCS

-1

-2

-3

-4

4

5

6

7

8

9

10

11

x

12 x 10

-3

Fig. 3. CDFs of the relative inward displacement for the Frank copula case. Table 5 Comparison of failure probabilities for the Frank copula case.

0.33% ±

Crude MCS (N = 106)

4-order SRSM + Rosenblatt transformation SORM

MCS (N = 10 )

0.34%

0.33%

0.33% ±

10

10

10

10

10

4-order SRSM + Rosenblatt transformation

6

FORM

P (uip /R>x)

Crude MCS (N = 106)

Table 6 Comparison of failure probabilities for the hybrid copula case.

0.70% ±

FORM

SORM

MCS (N = 106)

0.75%

0.71%

0.70% ±

0

2-order SRSM 3-order SRSM 4-order SRSM crude MCS

-1

-2

-3

-4

4

5

6

7

8

9

10

11

12

x Fig. 4. CDFs of the relative inward displacement for the hybrid copula case.

13

14 x 10

-3

28

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

based on the Nataf transformation if the underlying dependence structure is not Gaussian. 4.4. Hybrid copula case The potential of the proposed method lies in its ability of modeling complex dependence structure under incomplete probability information. Here, a hybrid copula case with unconditional copula function C E;c ¼ clayton, C E;u ¼ normal, and conditional copula function C c;ujE ¼ Frank is assumed. Similar to the normal copula and the Frank copula cases, the curves generated by the 3-order

4.5. Comparisons and discussion Comparisons of the investigated cases illustrate that the reliability results can be influenced considerably by the adopted dependence structure. Fig. 5 shows the generated triplets under different

E, c

0.9 0.8

and 4-order SRSM agree well with the crude MCS result as shown in Fig. 4. The corresponding failure probabilities for the hybrid copula case are summarized in Table 6. Note the expected failure probability increases significantly up to 0.70% only because of the assumption of dependence structure.

c, φ 30

r=0.2000

r= -0.4998

28

0.7 26

Normal

0.6 0.5

24

0.4

22

0.3

20

0.2

18

0.1 0 200

250

300

350

400

450

500

550

600

650

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

30

0.9 r=0.1964

0.8

r= -0.5033

28

0.7

26

0.6

Frank

16

0.5

24

0.4

22

0.3

20

0.2 18

0.1 0 200

250

300

350

400

450

500

550

600

650

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

30

0.9 r=0.1939

0.8

r= -0.5036

28

0.7

26

0.6

Hybrid

16

0.5

24

0.4

22

0.3

20

0.2 18

0.1 0 200

250

300

350

400

450

500

550

600

650

16

0

0.1

0.2

0.3

0.4

Fig. 5. Generated triplets of E, c, and u under different copula cases (N = 105).

0.5

0.6

0.7

0.8

0.9

29

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

copulas in the 2-dimensional space (pairs of E and u are not presented because the corresponding joint PDF is identical due to the assumption of zero correlation coefficient). Although the marginal distributions and correlations are identical, the shape of the joint PDF can differ significantly due to the assumption of different copulas. For example, there is no tail dependence between E and c for the nested bivariate normal and Frank copula case while an asymmetric lower left tail dependence is observed for the hybrid copula case. As a result, the failure domain bounded by the same limit state surface can be different, which gives rise to the different probabilistic results (i.e., 0.39% and 0.33% for the normal and Frank copula case, and 0.70% for the hybrid copula case). Note the validity of the SRSM in the entire random space is not affected by the isoprobabilistic transformation method. After all, it is the independent standard normal random variables that are used to represent the system output. The transformation method is merely used to map the probabilistic collocation points in the U-space into the original space for the physical model evaluation [12]. Fig. 6 presents the SORM results as a function of the correlation coefficient. Generally, the failure probability increases as the corre-

10

lation coefficient becomes larger. However, the magnitude of variation is different. In particular, when qE;c approaches to 1 (qc;u is fixed to 0.5), the failure probability based on the hybrid copula decreases whereas the results for the other two cases still increases. Therefore, the dependence structure is a critical factor that can impact the reliability results in a complicated and nontrivial way. The effect of conditioning order is summarized in Table 7. For each illustrated case, there are 3 different pair-copula decompositions of the multivariate joint distribution (i.e., fC E;c ; C E;u ; C c;ujE g; fC c;E ; C c;u ; C E;ujc g; and fC u;E ; C u;c ; C E;cju g) and each representation has 2 different conditioning orders if the Rosenblatt transformation is implemented by taking the pair-copulas as C-vine models (e.g., E ! c ! u or E ! u ! c for the first representation). The same is true for D-vine models because the decomposition for this 3-dimensional problem is both a C-vine and D-vine structure if we re-organize the random variables. For the cases that all pair-copulas are normal or Frank, the conditioning order has a minor effect on the FORM/SORM results. This effect can be attributed to the highly nonlinearity of the limit state surface after

(a)

-1

10

(b)

-1

normal Frank hybrid

10

10

failure probability

failure probability

normal Frank hybrid

-2

-3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10

-2

10

-3

-0.9

-0.8

-0.7

-0.6

ρE,c

-0.5

-0.4

-0.3

-0.2

-0.1

ρc,φ

Fig. 6. Failure probability as a function of (a) qE;c (qc;u is fixed to 0.5), and (b) qc;u (qE;c is fixed to 0.2).

Table 7 Effects of the conditioning order for the investigated cases. Pair-copulas

C E;c ¼ C E;u ¼ C c;ujE ¼ normal C E;u ¼ C E;c ¼ C c;ujE ¼ normal C c;E ¼ C c;u ¼ C E;ujc ¼ normal C c;u ¼ C c;E ¼ C E;ujc ¼ normal C u;E ¼ C u;c ¼ C E;cju ¼ normal C u;c ¼ C u;E ¼ C E;cju ¼ normal C E;c ¼ C E;u ¼ C c;ujE ¼ Frank C E;u ¼ C E;c ¼ C c;ujE ¼ Frank C c;E ¼ C c;u ¼ C E;ujc ¼ Frank C c;u ¼ C c;E ¼ C E;ujc ¼ Frank C u;E ¼ C u;c ¼ C E;cju ¼ Frank C u;c ¼ C u;E ¼ C E;cju ¼ Frank C E;c ¼ ClaytonC E;u ¼ normalC c;ujE ¼ Frank

Conditioning order

E, c, u E, u, c c, E, u c, u, E u, E, c u, c, E E, c, u E, u, c c, E, u c, u, E u, E, c u, c, E E, c, u

FORM (%)

0.4282 0.4281 0.4278 0.4278 0.4281 0.4284 0.3404 0.3401 0.3382 0.3288 0.3410 0.3349 0.7528

SORM (%)

0.3937 0.3911 0.3915 0.3926 0.3948 0.3908 0.3281 0.3314 0.3420 0.3337 0.3414 0.3169 0.7101

Design point E

c

u

270.1051 270.1157 270.1135 270.1131 270.1157 270.1352 267.6289 267.2946 267.4416 266.7067 267.9516 266.7420 274.2702

0.1650 0.1650 0.1649 0.1649 0.1650 0.1649 0.1943 0.1736 0.1694 0.1701 0.1713 0.1707 0.1478

22.2883 22.2895 22.2906 22.2906 22.2895 22.2892 21.7096 22.3993 22.4559 22.4384 22.4071 22.4470 22.4778

C E;u ¼ normalC E;c ¼ ClaytonC c;ujE ¼ Frank

E, u, c

0.7387

0.6917

273.7574

0.1459

22.5423

C c;E ¼ ClaytonC c;u ¼ FrankC E;ujc ¼ normal

c, E, u

0.4551

0.5756

273.6962

0.1335

22.8689

C c;u ¼ FrankC c;E ¼ ClaytonC E;ujc ¼ normal

c, u, E

0.4472

0.5780

273.5487

0.1329

22.8736

C u;E ¼ normalC u;c ¼ FrankC E;cju ¼ Clayton C u;c ¼ FrankC u;E ¼ normalC E;cju ¼ Clayton

u, E, c u, c, E

0.7701 0.6809

0.7394 0.7100

275.0628 277.0456

0.1450 0.1403

22.4194 22.3768

30

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

transformation and the possibly local optimization during the search of the design point along this limit state surface. As a result, the design point and the curvature of the limit state surface in the vicinity of the design point can be different, which leads to the different reliability results. For the hybrid copula case, the results correspond to the same pair-copula decomposition but different conditioning orders have relatively small differences. In contrast, the results correspond to different pair-copula decompositions has a larger discrepancy. However, this larger discrepancy is not caused by the so-called ‘‘false design point” in the FORM/SORM approximations [45,46]. In fact, the dependence structure represented by different pair-copula decomposition is different. For example, for the conditioning orders E ! c ! u and c ! E ! u, the unconditional pair-copulas are Clayton and normal copula, and Frank and normal copula, respectively, while the conditional pair-copula is Frank copula and Clayton copula, respectively. Although the same bivariate copula function is adopted for each pair of random variables (i.e., Clayton, Frank and normal copula for E and c, u and c, and E and u, respectively), the unconditional and conditional copulas represent different assumptions of dependence structure. Moreover, the results based on the crude MCS are 0.0070 ± 0.00008 and 0.0060 ± 0.00007 after 30 repeated runs for the canonical order and inverse order, respectively, which also verifies that the underlying dependence structures for the two conditioning orders are actually different.

the results obtained based on the normal copula can overestimate or underestimate the failure probability if the Frank copula or the hypothetical hybrid copula should be respectively used to characterize the underlying dependence structure for the investigated problem. Practitioners must be aware of the possibly biased results by relying heavily on the Nataf transformation. However, if the dependence structure is not known or there is no guideline on copula selection, identifying the most appropriate reliability result remains a challenging task. (3) The conditioning order in the proposed method can influence the reliability results. In the studied example, the conditioning order has a minor effect on the failure probability due to the FORM/SORM approximations when the normal or Frank copula is adopted. For the hybrid copula case, the results based on different pair-copula decompositions differ significantly. This larger discrepancy, however, is mainly caused by the different dependence structures represented by the different unconditional and conditional pair-copulas. In other words, if several different bivariate copulas have to be used to characterize the complex dependence structure, great attentions must be paid to the hierarchical level of pair-copula model construction because different hierarchical structures actually represent different dependence structures.

5. Conclusions

Acknowledgments

The stochastic response surface method is widely adopted to address the reliability problems involving implicit limit state surface. When the probabilistic description comes in terms of marginal distributions and correlations, the Nataf transformation is commonly adopted to map the random variables into the Uspace for the development of the stochastic surrogate model. However, it has been acknowledged that the Nataf transformation inherently assumes a Gaussian dependence structure for the random variables, which can be inappropriate in some cases. To consider non-Gaussian dependence structure cases, this paper proposes a novel isoprobabilistic transformation method based on the pair-copula construction of the multivariate joint distribution and the Rosenblatt transformation. The pair-copula parameters are retrieved from the incomplete probability information when specific copula functions are adopted to characterize the underlying dependence structure. The improved SRSM is illustrated in a tunnel excavation example involving three random variables with prescribed marginal distributions and correlation coefficients. Although the limit state surface in the example is analytical, the method can be applied to the problems involving numerical codes because the validity of the SRSM is independent of the adopted isoprobabilistic transformation method. The following conclusions can be summarized:

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51608399.

(1) The comparison of the SRSM with the Nataf transformation and the proposed transformation method shows no major difference if the normal copula is assumed for all bivariate dependence modeling. The corresponding reliability results are close to each other and both agree well with the crude MCS result. However, the proposed method has the strength to consider the non-Gaussian dependence structure cases under incomplete probability information. In other words, the Nataf transformation is actually a special case of the proposed method when all pair-copulas are normal. (2) The adopted dependence structure can impact the reliability results in a complicated and non-trivial way. In spite of the identical marginal distributions and correlation coefficients,

Appendix A. Relating the pair-copula parameters with correlation coefficients for a trivariate case Consider three random variables fx1 ; x2 ; x3 g with marginal PDFs f ðx1 Þ, f ðx2 Þ, f ðx3 Þ, marginal CDFs Fðx1 Þ, Fðx2 Þ, Fðx3 Þ, and pair-wise Pearson’s linear correlation coefficients q12 , q13 , q23 . The joint PDF can be decomposed as:

f ðx1 ; x2 ; x3 Þ ¼ f ðx1 Þf ðx2 Þf ðx3 Þc1;2 ðFðx1 Þ; Fðx2 ÞÞ  c1;3 ðFðx1 Þ; Fðx3 ÞÞc2;3j1 ðFðx2 jx1 Þ; Fðx3 jx1 ÞÞ

ðA:1Þ

or alternatively,

f ðx1 ; x2 ; x3 Þ ¼ f ðx1 Þf ðx2 Þf ðx3 Þc1;2 ðFðx1 Þ; Fðx2 ÞÞ  c2;3 ðFðx2 Þ; Fðx3 ÞÞc1;3j2 ðFðx1 jx2 Þ; Fðx3 jx2 ÞÞ

ðA:2Þ

The conditional PDF f ðx3 jx1 ; x2 Þ can be written as:

f ðx3 jx1 ; x2 Þ ¼ c3;1j2 ðFðx1 jx2 Þ; Fðx3 jx2 ÞÞc2;3 ðFðx2 Þ; Fðx3 ÞÞf ðx3 Þ

ðA:3Þ

or alternatively,

f ðx3 jx1 ; x2 Þ ¼ c3;2j1 ðFðx3 jx1 Þ; Fðx2 jx1 ÞÞc1;3 ðFðx1 Þ; Fðx3 ÞÞf ðx3 Þ

ðA:4Þ

Similarly, f ðx2 jx1 ; x3 Þ and f ðx1 jx2 ; x3 Þ can be expressed as:

f ðx2 jx1 ; x3 Þ ¼ c2;3j1 ðFðx2 jx1 Þ; Fðx3 jx1 ÞÞc1;2 ðFðx1 Þ; Fðx2 ÞÞf ðx2 Þ

ðA:5Þ

f ðx1 jx2 ; x3 Þ ¼ c1;3j2 ðFðx1 jx2 Þ; Fðx3 jx2 ÞÞc1;2 ðFðx1 Þ; Fðx2 ÞÞf ðx1 Þ

ðA:6Þ

Without loss of generality, suppose Eq. (A.1) is adopted to construct the joint PDF. The aim is to retrieve the parameters h12 , h13 , h23j1 corresponding to the bivariate copulas C 1;2 , C 1;3 , C 2;3j1 . Inserting Eqs. (A.1) and (A.5) into Eq. (16) and recall Eq. (18), we have:

q13 ¼

Z

þ1

1

Z

þ1

1

lN1 lN3 f ðx1 Þf ðx3 Þc1;3 ðFðx1 Þ; Fðx3 ÞÞdx1 dx3

ðA:7Þ

F. Wang, H. Li / Computers and Geotechnics 89 (2017) 22–32

which indicates that the copula parameter h13 can be determined solely by q13 . Similarly, by combining Eqs. (A.1) and (A.4) and using Eq. (18), the copula parameter h12 can be obtained from q12 :

q12 ¼

Z

þ1

1

Z

þ1

1

lN1 lN2 f ðx1 Þf ðx2 Þc1;2 ðFðx1 Þ; Fðx2 ÞÞdx1 dx2

ðA:8Þ

If we combine Eqs. (A.1) and (A.3), then we have: Z þ1 Z þ1 q12 ¼ lN1 lN2 f ðx1 Þf ðx2 Þ 1

ðA:9Þ Eq. (A.9) indicates that the copula parameter h23j1 cannot be obtained from a single correlation coefficient like h13 or h12 . Using Eqs. (A.2) and (A.4), another expression of q12 is given: Z þ1 Z þ1 q12 ¼ lN1 lN2 f ðx1 Þf ðx2 Þ 1

c1;2 ðFðx1 Þ;Fðx2 ÞÞc2;3 ðFðx1 Þ; Fðx3 ÞÞc1;3j2 ðFðx1 jx2 Þ;Fðx3 jx2 ÞÞ  dx1 dx2 c2;3j1 ðFðx2 jx1 Þ;Fðx3 jx1 ÞÞc1;3 ðFðx2 Þ; Fðx3 ÞÞ ðA:10Þ

Moreover, by analogy with Eq. (A.7), if Eqs. (A.2) and (A.6) are used, the copula parameter h23 corresponding to C 2;3 can be obtained:

q23 ¼

Z

þ1

1

Z

þ1

1

lN2 lN3 f ðx2 Þf ðx3 Þc2;3 ðFðx2 Þ; Fðx3 ÞÞdx2 dx3

ðA:11Þ

Thus, by inserting the computed h13 , h12 , and h23 into Eqs. (A.9) and (A.10), and using Eq. (11) recursively to represent the marginal conditional distributions, the parameters h23j1 can be determined. Note h23j1 can also be determined by combining other expressions. For example, the following expressions can be generated based on Eqs. (A.1), (A.6) and Eqs. (A.2), (A.5), respectively:

q23 ¼

q13 ¼

Z

þ1

1

Z

þ1

1

lN2 lN3 f ðx2 Þf ðx3 Þ



c1;3 ðFðx1 Þ; Fðx3 ÞÞc2;3j1 ðFðx2 jx1 Þ; Fðx3 jx1 ÞÞ dx2 dx3 c1;3j2 ðFðx1 jx2 Þ; Fðx3 jx2 ÞÞ

Z

þ1

1



pair-copula model. This simulation can be accomplished by the conditional distribution approach [26]. 3. Obtain the random variables X using the inverse of marginal CDFs xi ¼ F 1 ðui Þ. Calculate the linear correlation coefficients ^ ij from the samples. q ^ ij ¼ qij , where 4. Search the desired pair-copula parameters until q qij is the targeted correlation coefficients. The search process can be achieved using programs for solving system of nonlinear equations (e.g., the Matlab ‘fsolve’ command).

1

c1;2 ðFðx1 Þ;Fðx2 ÞÞc1;3 ðFðx1 Þ;Fðx3 ÞÞc2;3j1 ðFðx2 jx1 Þ;Fðx3 jx1 ÞÞ  dx1 dx2 c3;1j2 ðFðx1 jx2 Þ; Fðx3 jx2 ÞÞc2;3 ðFðx2 Þ;Fðx3 ÞÞ

1

31

Z

þ1

1

ðA:12Þ

lN1 lN3 f ðx1 Þf ðx3 Þ

c2;3 ðFðx2 Þ; Fðx3 ÞÞc1;3j2 ðFðx1 jx2 Þ; Fðx3 jx2 ÞÞ dx1 dx3 c2;3j1 ðFðx2 jx1 Þ; Fðx3 jx1 ÞÞ

ðA:13Þ

The parameters h23j1 can thus be determined by collecting Eqs. (A.7) and (A.11)–(A.13). However, no matter what expressions are used, additional parameters that are irrelevant to the desired pair-copula construction of the multivariate joint distribution have to be estimated (e.g., h23 and h13j2 ). Appendix B. Simulation-based method for retrieving the paircopula parameters The simulation-based method for approximating the paircopula parameters is as follows: 1. Choose a pair-copula model to represent the multivariate joint distribution, e.g., use Eq. (A.1) to represent a trivariate distribution; 2. Initialize the model by arbitrarily setting its pair-copula parameters (e.g., h12 , h13 , h23j1 ) within their ranges. Generate a large size of copula random numbers u from the tentative

Compared to the numerical or analytical methods (e.g., [14,17,22,23,39]), the simulation-based method directly searches the desired copula parameters without estimating irrelevant parameters, which can be advantageous in high dimensions.

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