Machine learning aided static structural reliability analysis for functionally graded frame structures

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Machine learning aided static structural reliability analysis for functionally graded frame structures Qihan Wang , Qingya Li , Di Wu , Yuguo Yu , Francis Tin-Loi , Juan Ma , Wei Gao PII: DOI: Reference:

S0307-904X(19)30595-5 https://doi.org/10.1016/j.apm.2019.10.007 APM 13068

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

23 January 2019 27 September 2019 1 October 2019

Please cite this article as: Qihan Wang , Qingya Li , Di Wu , Yuguo Yu , Francis Tin-Loi , Juan Ma , Wei Gao , Machine learning aided static structural reliability analysis for functionally graded frame structures, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.10.007

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Highlights 

A machine learning aided structural reliability analysis is proposed



A new kernel based support vector regression technique is developed



Static structural reliability of 3D functionally graded frame structures is studied



Effectiveness and robustness of the proposed method are well demonstrated

Machine learning aided static structural reliability analysis for functionally graded frame structures Qihan Wang1, Qingya Li1, Di Wu2, Yuguo Yu1, Francis Tin-Loi1, Juan Ma3, Wei Gao1, * 1

Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

2

Centre for Built Infrastructure Research (CBIR), School of Civil and Environmental Engineering, University of Technology Sydney, Sydney, NSW, Australia 3

Research Centre of Applied Mechanics, School of Electro-Mechanical Engineering, Xidian University, Xi'an, P.O.Box: 187, 710071, PR China

*

Corresponding author. E-mail: [email protected]

Abstract A novel machine learning aided structural reliability analysis for functionally graded frame structures against static loading is proposed. The uncertain system parameters, which include the material properties, dimensions of structural members, applied loads, as well as the degree of gradation of the functionally graded material (FGM), can be incorporated within a unified structural reliability analysis framework. A 3D finite element method (FEM) for static analysis of bar-type engineering structures involving FGM is presented. By extending the traditional support vector regression (SVR) method, a new kernel-based machine learning technique, namely the extended support vector regression (X-SVR), is proposed for modelling the underpinned relationship between the structural behaviours and the uncertain system inputs. The proposed structural reliability analysis inherits the advantages of the traditional sampling method (i.e., Monte-Carlo Simulation) on providing the information regarding the statistical characteristics (i.e., mean, standard deviations, probability density functions and cumulative distribution functions etc.) of any concerned structural outputs, but with significantly reduced computational efforts. Five numerical examples are investigated to illustrate the accuracy, applicability, and computational efficiency of the proposed computational scheme.

Keywords Functionally graded material; Structural reliability analysis; Machine learning; Uncertainty analysis. 1. Introduction Functionally graded material (FGM) is an engineered composite material whose properties can be manipulated through the gradation of microstructure to achieve some specific functionalities [1-5]. In 1984, the Japanese scientists invented the first FGM as the thermal barrier against high thermal environment in their space project. Moreover, with the dramatic development of the manufacturing technology (e.g., additive manufacturing), many different types of FGM have been developed with additional excellence in material properties. For example, by layering the boron carbide (B4C) with one type of the titanium alloy (Ti-6Al4V), a functionally graded armour tile with enhanced competence on ballistic protection is invented. This new engineered product combines the desirable properties of ceramic (i.e., high compressive strength, high hardness, and light weight) with the exceptional structural properties of titanium (i.e., high strength-to-weight ratio, high ballistic mass efficiency, and high corrosion resistance), and conceivably, a more effective ballistic armour can be manufactured with relatively lower cost. The applications of FGM in real-life engineering applications include aerospace, mechanical, biomedical, military, and automotive etc. [6-10] Even though the FGM has been widely implemented in modern engineering applications, there are still some issues associated with the FGM. From practical experiences, one frequently encountered phenomenon is that the actual gradation of the FGM is always different with the designation due to the presence of uncertainties within the intricate manufacturing process [11-13]. For example, the randomness of the local volume fraction of a ZrO2 chromium-nickel alloy FGM has been reported by Ilschner [14]. Consequently, the material properties of the actual FGM will always be different with the designated uniform gradation of the FGM. The effect of such unpleasant discrepancy is that the safety of the FGM structure may be compromised due to the invalid assumption of the uniform gradation of FGM. Consequently, it is requisite that all uncertainties involved in the structural system must be thoroughly considered in both structural analyses and designs so the integrity of the FGM structural system can be robustly maintained throughout its service life [15-20]. In order to achieve a more appropriate and robust analysis for FGM structures, there are numbers of research works that have been reported to investigate the effects of various

uncertain system parameters on the FGM structures. Yang et al. [21, 22] investigated the stochastic static and buckling behaviours of FGM plates against material uncertainties through the first-order matric perturbation method. Kitipornchai et al. [23] employed the combination of a semi-analytical method with the first-order matrix perturbation approach to investigate the dynamic behaviours of FGM subjected to random excitations. The stochastic free vibration analysis of FGM plate with random system parameters was investigated by Shaker et al [24]. Moreover, Jagtap et al. [25] implemented the first-order matrix perturbation approach to investigate the stochastic nonlinear natural frequencies of FGM plate resting on elastic foundation. The stochastic post-buckling behaviour of FGM plates was investigated by Lal et al. [26] through the stochastic finite element method (FEM). The stochastic nonlinear bending behaviour of piezoelectric functionally graded beams was investigated by Shegokar and Lal [27]. By employing the first-order perturbation theory, Talha and Singh [28] calculated the first two statistical moments of the linear buckling load of FGM plate. Moreover, Talha and Singh [29] implemented the same approach to determine the first two statistical moments of the natural frequencies of FGM plates under thermal working conditions. García-Macías et al. [30] implemented a surrogate model approach to investigate the free vibration of carbon nanotube reinforced FGM plates with various random variables. By combining the finite element method with the matrix perturbation theory, Wu et al. [31] studied the stochastic static responses of 2D FGM frame structures by calculating the first two statistical moments of the structural responses. In addition to the stochastic analysis, Wu et al [32] proposed a mathematical programming based approach to determine the extreme bounds of structural responses of FGM frame structures under static loading conditions. Later on, Wu et al [33] extended the mathematical programming based approach to analytically determine the upper and lower bounds of the natural frequencies of FGM frame structures involving interval Young’s moduli and densities. Recently, some of the authors of this work [34] developed an extended support vector regression (X-SVR) method for stochastic free vibration analysis for FGM frame structures. Furthermore, metamodel based approaches, neural network, Kriging model, etc., have been emerged as effective alternatives for solving computationally expensive stochastic problem involved in structural reliability analysis [3538]. Structural reliability analysis through machine learning approaches has revealed tremendous potential for practical implementation [39-41]. In this study, a static structural reliability analysis for 3D FGM frame structure is presented. The considered uncertain system parameters are including the material properties, dimensions of structural members, magnitudes of the applied loads, as well as the degree of

gradation of the FGM. In order to conquer the stochastic system of linear equations, the extended support vector regression (X-SVR) method, which was developed by some of the authors of this paper in [34], is adopted to fulfil the purpose of structural safety assessment. In [34], an X-SVR approach was particularly developed for the stochastic free vibration analysis of FGM frame structures, which involved uncertainty quantifications for the natural frequencies of the FGM structures when some of the system inputs were modelled as random variables. However, in this work, the previously proposed X-SVR approach is further implemented to tackle the structural reliability analysis of FGM structure subjected to static loadings. Robustly assessing the structural reliability of the FGM frame structures against static loadings forms the main target of the present work. From a theoretical perspective, a stochastic eigenvalue problem with random coefficient matrices was investigated in [34], whereas solving a stochastic linear system of equations with random coefficient matrix and right-hand side is the focus of this paper. In addition to [34], the X-SVR approach has been further improved for handling relatively high-dimensional random system inputs by optimizing the previously proposed algorithm. Such improvement of the X-SVR approach can be evidenced through one of the presented numerical examples. Therefore, the work presented in this paper is considered as a subsequent extension of the one presented in [34]. The adopted X-SVR method offers a sampling scheme which is capable of providing information regarding the statistical moments (i.e., means, standard deviation etc), as well as probability density and cumulative distribution functions (i.e., PDFs and CDFs) of any concerned structural responses. By adequately establishing the PDFs and CDFs, the static structural reliability of FGM frame structures can be effectively determined. Another advantage of the X-SVR method is that by utilizing identical size of training datasets, a higher-accuracy regression model between the uncertain system inputs and the concerned structural outputs can be established, in comparison to the traditional Support Vector Regression (SVR) method. This paper is structured as follows. Section 2.1 introduces the deterministic static analysis of 3D FGM frame structure through FEM. In Section 2.2, the static structural reliability of FGM frame structures is formally introduced through the stochastic static analysis of FGM frames. To adequately estimate the static structural reliabilities of FGM frame structures involving various uncertainties, the proposed X-SVR approach is comprehensively introduced in Section 3. Subsequently, in Section 4, five numerical examples are thoroughly investigated to evidently illustrate the applicability and efficiency of the proposed method. Finally, some conclusions are drawn in Section 5.

2. Preliminaries 2.1. Deterministic static analysis of 3D FGM frame structure through FEM In this paper, the adopted generic 3D FGM beam element e with length le, width be, and thickness he of the cross-section is defined in a Cartesian coordinate system (x, y, z), as illustrated in Figure 1. It is assumed that the material properties of the FGM are continuously varying in y-direction, which are governed by the power-law or exponential material constitutive relationship [42, 43]. More specifically, k  1 e e  y  PU  PL      PLe Power-law relationship   he 2  e P ( y)    1  PUe   2 y    e P  exp  log   e   1    Exponential relationship  U 2 he    PL    

(1)

where P e ( y ) denotes a material property (e.g., Young’s modulus, density, etc.) of the eth FGM beam element at location y, for 

he h  y  e ; PUe and PLe denote the generic material 2 2

properties at the upper and lower surfaces of the FGM beam element, respectively; and

k 0 : {k  | k  0} is the power-law index.

Figure 1. Geometric layout of a 3D FGM beam element Based on the linear analysis framework of FEM, the strain field of the beam element without the consideration of warping effects can be elaborated in the form of:

 e u e  xe  ye  y+ z  xx  x x x   e v e  xe  z   ze  xy  x x   e we  xe  y   ye  xz  x x 

(2)

e e e where  xx denotes the normal strain;  xy and  xz denote the shear strain components in xy-

and xz-planes, respectively; u e , v e , and we denote the structural deflections of the beam element in x-, y-, and z-directions, respectively;  xe ,  ye , and  ze denote the rotations about the x-, y-, and z-axes, respectively. Moreover, the combinations of Lagrange and Hermite interpolation polynomials have been implemented herein for modelling the displacement field [31]. By implementing the Hooke’s law, the constitutive relationship of the eth FGM beam element can be formulated as:

 e  u e  xe  ye  e e e  y+ z  xx  E ( y ) xx  E ( y )  x x    x   e  v e  xe  e e e e e  z   ze   xy  K sy G ( y ) xy  K sy G ( y )  x   x   e e  e  K e G e ( y ) e  K e G e ( y )  w   x y   e   sz xz sz y   xz x  x 

(3)

e where  xxe ,  xy , and  xze denote the normal and shear stresses of the beam element; E e ( y )

and G e ( y) denote the Young’s modulus and shear modulus of the beam element at location y, e respectively; K sy and K sze denote the shear correction factors, with well-known quantity as

5/6 for a rectangular cross-section. Also, G e ( y) can be calculated from:

E e ( y) G ( y)  2(1   e ) e

where

e

(4)

denotes the Poisson’s ratio of the eth FGM beam element. e

Consequently, the potential energy of the eth FGM beam element U can be formulated as:

1 le 1 le 1 le  xxe  xxe dAe dx     xze  xze dAe dx     xye  xye dAe dx   2 0 Ae 2 0 Ae 2 0 Ae 1 le 1 le 1 le    E e ( y )  ( xxe ) 2 dAe dx    K sye G e ( y )  ( xze ) 2 dAe dx+   K sze G e ( y )  ( xye ) 2 dAe dx 2 0 Ae 2 0 Ae 2 0 Ae (5)

Ue 

where Ae denotes the cross-sectional area of the beam element. Through the Castigliano’s theorem [44-46], the stiffness matrix of the 3D beam element can be expressed as:  k1,1e  e  0 k2,2 e  0 0 k3,3  e 0 0 k4,4  0 e e  0 0 k5,3 0 k5,5  e e e k6,2 0 0 0 k6,6 k k e   6,1 e e ke 0 0 0 0 k7,6 k7,7  7,1 e e e  0 k8,2 0 0 0 k8,6 0 k8,8  e e e 0 k9,3 0 k9,5 0 0 0 k9,9  0 e  0 0 0 k10,4 0 0 0 0 0  e e e 0 k11,3 0 k11,5 0 0 0 k11,9  0 e e e e k e 0 0 k12,6 k12,7 k12,8 0  12,1 k12,2 0 1212

where k  e

Symmetric

e k10,10 0 0

e k11,11 0

                 e  k12,12 

(6)

e denotes the stiffness matrix of the beam element; k(,) denotes the

component of k e at location (, ) . The detailed expression of each component within k e is presented in Appendix A. By considering the contribution of each structural element, the global stiffness matrix of d d

the 3D frame structure with d degrees-of-freedom, K 

K   k e   GTe k eG e e

can be explicitly formulated as: (7)

e

where G e 1212 denotes the transformation matrix of beam element from local to global coordinate systems. Consequently, the deterministic static analysis of the 3D FGM frame structure can be formulated into a generalized linear equation as:

F  KU

where F 

d

(8)

denotes the applied load vector and U 

d

denotes the structural

displacement vector. 2.2. Structural reliability analysis of 3D FGM frame structure Stimulated from practical engineering applications, uncertainties of structural system parameters are inevitable with various impacts on the system outputs. Some typical system uncertain parameters include the material properties, dimensions of structural members, and the applied loads etc. For a generic FGM frame structure involving d structural responses (e.g., structural displacement, stress, etc.), the non-deterministic structural responses are modelled as random





d variables, collected within a random vector χ  χˆ : χ   j ~ f  j ( x), for j  1, 2,..., d ,

where  j denotes the jth random system response, which has the PDF f  j ( x) . The limitstate function (LSF) is normally denoted as G(χ ) , which is defined as G(χ )  χ*  χ , where

χ * denotes the capacity of the concerned structural system, which is often considered as a deterministic parameter. Among all possible circumstances, the safe region (i.e.,

s : {χ d G(χ )  0} ) and the failure region (i.e.,  f : {χ d G(χ )  0} ) are separated by the limit state surface (i.e., G(χ )  0 ). Consequently, the structural reliability can be determined by its probability of survivor, which is defined as a multi-fold probability integral of: Pr  P(G(χ )  0)  P(χ  χ* )  



χ  χ*

f χ ( x)dx

(9)

Eq. (9) is a multi-dimensional integration which cannot be analytically solved, especially for very complex engineering system involving large-scale of uncertain parameters. To tackle such real-life inspired challenges, the sampling based computational techniques have been developed to estimate Eq. (9). The Monte Carlo simulation (MCS) is one of the simplest and most robust simulation methods for approximating structural reliability. The MCS provides estimations on the structural reliability for any types of performance function as:

Pr 

1 N  I [G(  j )] N j 1

(10)

where N denotes the number of the random sampling sets; the indicator function I [G(  j )] is equal to 0 for  j  f , and 1, otherwise. However, to obtain the structural reliability with high accuracy, the computational cost of MCS method is rather expansive due to the tremendous number of samples and performance function evaluations required. Sometimes, the implementation of this method may even become infeasible in practice especially when the single execution of the performance function is very time-consuming. In addition, the systematic outputs through this method are also limited. To overcome these hurdles, an alternative routine for structural reliability analysis through stochastic structural analysis is adopted herein. Let  R  Z() denote a random variable, in which Z() defines the set of all real random variables in a probability space ( Ω , A, P), and  denotes the set of real number. When all uncertain system parameters are considered simultaneously, the governing equation of the stochastic static analysis of FGM bar-type structures can be formulated as:

K (ξ R )U R  F(ξ R )   R R n R ξ  Ω : ξ   ~ f R ( x), for =1, 2,..., n





(11)

where K (ξ R ) d d denotes the stochastic system stiffness matrix which is the function of the random vector ξ R ; U  denotes the random displacement vector; F(ξ R ) d R

d

denotes the stochastic applied load vector, which is also a function of the random vector ξ ; R

and the random vector ξ R n is an n-dimension vector which collects all the uncertain input parameters that are presented in the system. That is, for the

th random variable  R , it

has the corresponding PDF f R ( x) . It should be noticed that Eq. (11) is a linear system which possesses random variables in both coefficient matrix and right hand-side. Theoretically, it is computationally intractable to solve for all solutions for Eq. (11) due to the infinite set of possible realizations of the random variables. Instead, the statistical characteristics, for example, the mean, standard deviation, PDF and CDF, of the structural responses are more meaningful in the context of real-life engineering application.

In order to establish a more generalized structural reliability analysis framework, a novel computational approach, namely the extended support vector regression (X-SVR) method, is proposed to estimate the structural reliability of FGM bar-type structures against various stochastic system inputs. 3. Structural reliability analysis through extended support vector regression (X-SVR) In this paper, a novel SVR method, namely the extended Support Vector Regression (XSVR), is adopted from [34] for the stochastic static analysis and subsequently structural reliability assessment of FGM frame structures. In this paper, the algorithm of the X-SVR approach has been further optimized to handle stochastic linear system of equations with relatively high-dimensional uncertain system inputs. To achieve a self-contained work, the XSVR approach is also presented in Sections 3.1-3.3. Subsequently, the algorithm of the structural reliability analysis of FGM frame structures through X-SVR approach is freshly presented in Section 3.4. 3.1. The linear X-SVR T In binary classification, given the training dataset with input xtrain  [x1 , x2 ,..., xi ,..., xm ]

mn ( xi n , i  1,2,..., m ) and output ytrain m , the hyperplane that is separating the two classes can be defined as [47, 48]:

fˆ (x)  wT x   where w  [w1 , w2 ,..., w ,..., wn ]T n ( w  ,

(12)

 1,2,...,n ) denotes the normal to the

hyperplane;   denotes the bias; m is the number of training samples; and n denotes the number of input variables. When the support vector theory is implemented for regression (i.e., SVR), Eq. (12) can be considered as the targeted regression function. By implementing the ε-insensitive loss function, as shown in Figure 2, where ε denotes the tolerable deviation between true value y train and model prediction fˆ (x) , the linear regression function Eq. (12) can be established by solving the optimization problem:

min * :

w , ,ξ ,ξ



m 1 2 w 2  C  i  i* 2 i 1



(13a)

wT xi    yi    i  s.t.  yi  wT xi      i*  * i , i  0

(13b)

where C  : {x  | x  0} is the penalty constant which determines the trade-off between the flatness of fˆ (x) and the amount up to which deviations larger than ε are tolerated, namely the empirical error; 

2

denotes the L2-norm of  ;

xi , xi* are the slack

variables, which denote the allowable negative and positive excessive deviations, respectively.

Figure 2. The ε-insensitive band for a one-dimensional linear SVR As an extension of the theory of support vector machine (SVM), the doubly regularized support vector machine (Dr-SVM) [49] is adopted to conduct the classification and feature selection simultaneously with a combination of elastic-net penalty that contains both L1- and L2-norm penalties with the hinge loss function. Furthermore, by extending the excellent binary classification feature of the pq-SVM [50], a decomposition process is adopted to eliminate the L1-norm w 1 computation. Subsequently, a quadratic ε-insensitive loss function (i.e., l2 ( ) ) is implemented herein to improve the numerical stability when solving the succeeding mathematical programs as:

l2  yi  fˆ (xi )   yi  fˆ (xi )  

2

(14)

Consequently, the governing formulation for the proposed X-SVR can be explicitly expressed as: min * :

p ,q , ,ξ ,ξ

p 2

1

2 2

 q

2 2

 e

T 2 n

p  q  



C T ξ ξ  ξ*T ξ* 2



(15a)

s.t.

xtrain (p  q)   em  y train   em  ξ  * y train  xtrain (p  q)   em   em  ξ  * p, q  0n ; ξ, ξ  0m

(15b)

where 1, 2  denote two tuning parameters which balance the classification performance and feature selection; ξ, ξ* m denote two non-negative vectors which collect slack variables; en  [1,1,...,1]T n and 0n  [0,0,...,0]T n denote ones and zeros vectors in dimensions of n, respectively; p, q n consist of non-negative variables such that:

 0, w  0  w , w  0 p : ( w )   and q : ( w )   , for  1, 2,..., n w , w  0  0, w  0

(16)

It is indicated by the definition in Eq. (16) that p q  0 is promised  . Thus, w 1 and w

2 2

can be alternatively expressed as: w 1  w1  w2  ...  wn  p1  q1  p2  q2  ...  pn  qn

(17)

 eTn  p  q  w

2 2

 pq

2 2

 p 2  q 2  2pT q 2

2

 p2 q

2

2

(18)

2

To achieve a more simplified formulation, Eq. (15) can be alternatively formulated as:

min : zˆ ,







1 Tˆ zˆ Czˆ   2  2aˆ T zˆ 2

 

(19a)



ˆ I ˆ ˆ ˆ ˆ s.t. A (2 m  2 n )(2 m  2 n ) z   I (2 m  2 n )(2 m  2 n )   G b  d  02 m  2 n

(19b)

(2 m 2 n )(2 m 2 n ) where I (2m2n)(2 m2 n)  denotes an identity matrix. Also, the square of the bias

parameter (i.e.,

2 )

is added to the objective function, which provides the benefits of

optimizing the orientation and location of the regression model simultaneously [50, 51]. The

ˆ (2n2m)(2n2m) are defined as: ˆ ,G ˆ , and A matrices C

ˆ  1I 2 n2 n C  

 , CI 2 m2 m 

02 n2 n ˆ  0 G  m2 n  0m2 n

02 nm I mm 0mm

02 nm   02 nn ˆ   x 0mm  , A  train  xtrain I mm 

02 nn 02 n2 m  xtrain 0m2 m  xtrain 0m2 m  (20)

and the vectors

aˆ , bˆ , dˆ , and zˆ 2n2m

are defined as:

p   en  02 n   02 n  q  aˆ   e n  , bˆ   e m  , dˆ   y train  , zˆ    ξ   y train  02 m   e m   *

(21)

ξ 

and the constraints that p and q are non-negative have been reinforced by Eq. (19b). Alternatively, Eq. (19) can be solved through its dual formulation, which can be expressed as: min : 

1 T  Q  mT  2

(22a)

s.t.  02m2n

(22b) (2 m 2 n )(2 m 2 n )

where  2 m 2 n denotes the Lagrange multiplier vector; Q 

and

m 2m2n are defined as:



 

ˆ I ˆ 1 A ˆ I Q A (2 m  2 n )(2 m  2 n ) C (2 m  2 n )(2 m  2 n )







T

ˆ ˆ ˆTG ˆ  Gbb

ˆ I ˆ 1 ˆ ˆ ˆ m  2 A (2 m  2 n )(2 m  2 n ) C a   b  d

(23) (24)

By using the dual formulation, the number of variables in the optimization problem is reduced from (2m  2n  1) in Eq. (15) to (2m  2n) in Eq. (22). Moreover, the constraint in the dual problem is much simpler than that in the primal problem. In order to demonstrate that the proposed X-SVR has the globally optimal solution, the following proposition can be put forward that the dual problem presented in Eq. (22) is a convex optimization problem.

Proposition. Given the training dataset with input xtrain mn and output y train m , with pre-defining the positive tuning parameters for X-SVR as 1 , 2 , C,   , the optimization problem defined in Eq. (22) is a convex quadratic programming problem. Proof. For quadratic programming expressed in Eq. (22), the proof of convexity is equivalent to proving that Q

ˆ definition, i.e., C

0 . Moreover, considering that Cˆ is a positive and diagonal matrix by

0 , and also Cˆ 1 0 . Let v 2m2n be a non-zero column vector, then:

T ˆ I ˆ 1 ˆ ˆ ˆ ˆT ˆ  vT Qv  vT (A (2 m  2 n )(2 m  2 n ) )C (A  I (2 m  2 n )(2 m  2 n ) )  Gbb G  v





T

T T ˆ I ˆT ˆ 2  ˆ 1  ˆ   (A (2 m  2 n )(2 m  2 n ) ) v  C  (A  I (2 m  2 n )(2 m  2 n ) ) v   (b Gv )

(25)

0

Therefore, Eq. (22) is a convex optimization. □

This concludes the proof.

Subsequently, the global optimum of the proposed X-SVR approach can be effectively determined by solving the associated dual problem by any available quadratic programming solvers. Let * 2 m 2 n be the solution of Eq.(22), the variables zˆ and  can be calculated as:



ˆ 1  A ˆ I zˆ  C (2 m  2 n )(2 m  2 n ) 



T

*  2aˆ  

(26)

ˆ *   bˆ T G

(27)

w  p  q  zˆ (1: n)  zˆ (n  1: 2n)

(28)

and the coefficient w can be obtained as:

Consequently, the established linear regression function by the proposed X-SVR approach can be formulated as: T ˆ  fˆ (x)   p  q  x  bˆ T G

(29)

3.2. Kernel based nonlinear X-SVR In addition, the proposed X-SVR can be extended to the nonlinear regression. To effectively transform the linear X-SVR to a kernelized learning approach, an alternative method, namely the empirical kernel map [52, 53], is employed herein. The implemented empirical kernelization can be expressed as:

xi  [ xi ,1 , xi ,2 ,..., xi ,n ]T

 Φ(x1 )T Φ(xi )    (x1 , xi )      Φ(x 2 )T Φ(xi )    (x 2 , xi )   κˆ ( xi )    , for i  1, 2,..., m        Φ(x m )T Φ(xi )   (x m , xi ) 

(30)

where Φ(xi ) denotes the appropriate mapping function which implicitly maps the ith input data xi n into a higher-dimensional Euclidian space or even infinite dimensional Hilbert feature space; κˆ (xi ) denotes the ith empirical feature vector with the empirical degree m which is equal to the number of training samples [53]. Such m-dimensional vector space is named as the empirical feature space [54]. Then, the empirical feature vector κˆ (xi ) is regarded as the ith training sample for constructing the learning model. The empirical feature space is finite-dimensional and jointly defined by the employed kernel function and training samples [53]. Consequently, for an arbitrary training dataset xtrain and a specific kernel function

 (, ) , the initial training samples can be transferred through the kernel matrix κ train mm as:

κ train

  (x1 , x1 )  (x1 , x 2 )   (x , x )  (x , x ) 2 1 2 2     (x m , x1 )  (x m , x 2 )

 (x1 , x m )   (x 2 , x m ) 

(31)

   (x m , x m ) 

and the kernel matrix κ train is utilized as the training dataset. Subsequently, the nonlinear XSVR problem can be formulated as: min

p ,q , ,ξ ,ξ

*

:

p 2

1

2

 2

 q

2 2

 e

T 2 m

 p  q  



C T ξ ξ + ξ*T ξ* 2



(32a)

s.t.

κ train (p  q )   em  y train   em  ξ  * y train  κ train (p  q )   em   em  ξ  * p , q , ξ, ξ  0m

(32b)

where p , q m serve the same functions as p and q for the linear X-SVR; the subscript

 indicates a kernelized learning model. Similar to Eq. (15), the kernelized X-SVR can also be reformulated into:

min : zˆ  ,







1 Tˆ zˆ  C zˆ    2  2aˆ T zˆ  2

(33a)



(33b)





ˆ I ˆ ˆ ˆ ˆ s.t. A  4 m4 m z   I 4 m4 m   G b  d  04 m ˆ , and A ˆ 4 m4 m are defined as: ˆ ,G where the kernelized matrices C   

ˆ  1I 2 m2 m C   

02 m2 m  ˆ  , G    0m2 m CI 2 m2 m   0m2 m

02 mm   02 mm  ˆ 0mm  , A   κ train  κ train I mm 

02 mm I mm 0mm

02 mm κ train κ train

02 m2 m  0m2 m  0m2 m  (34)

and the kernelized vectors aˆ  , bˆ  , dˆ  , and zˆ  4m are defined as:

02 m  e  ˆ aˆ    2 m  , b    ,  e2 m  02 m 

p 

q   02 m   , zˆ     ˆd   y   train   ξ   y train   *

(35)

ξ 

Once again, the mathematical program defined in Eq. (33) can be equivalently solved through its dual formulation by using the Lagrange method with the KKT conditions. By introducing the non-negative Lagrange multiplier

 4m , the proposed kernelized X-SVR

can be alternatively calculated through a quadratic program. That is, min : 

1 T  Q   mT  2

(36a)

s.t.  04m

(36b)

where Q 4 m4 m and m 4m are defined as:



 

ˆ I ˆ 1 ˆ Q  A  4 m4 m C A  I 4 m4 m





T

ˆ bˆ bˆ T G ˆ G    



ˆ I ˆ 1 ˆ ˆ ˆ m  2 A  4 m4 m C a   b  d

(37) (38)

Let φ* 4m be the solution of Eq.(36), the variables zˆ  and   can be determined as:



ˆ 1  A ˆ I zˆ   C    4 m4 m 



T

*  2aˆ   

  bˆ T Gˆ  *

(39) (40)

Subsequently, the coefficient w can be obtained as:

w  p  q  zˆ  (1: m)  zˆ  (m  1: 2m)

(41)

Consequently, the nonlinear regression function obtained by the proposed kernelized XSVR can be formulated as: T ˆ * fˆ (x)   p  q  κ(x)  bˆ T G  

(42)

The difference between the linear and nonlinear X-SVR is that the input dataset has been mapped into the empirical space by utilizing the specified kernel function within the nonlinear model. Subsequently, the kernelized X-SVR is equivalent to a linear X-SVR with a manipulated input samples and therefore, the convexity of the mathematical program is well preserved regardless of the type of kernel function. The series expansion of Gegenbauer polynomial [55, 56] is implemented herein as a new type of kernel for X-SVR. The univariate Gegenbauer polynomials denoted by Pdˆ ( x) with the order of the polynomial dˆ 

 0

:  x  | x  0 and positive parameter   , can be

formulated as:  P0 ( x)  1    P1 ( x)  2 x  1   Pdˆ ( x)   2 x dˆ    1 Pdˆ1 ( x)  dˆ  2  2 Pdˆ 2 ( x)  , for dˆ  2,3, 4,...   dˆ  









(43)

For a given  , the Gegenbauer polynomials are orthogonal on x [1,1] with respect to the weight function   ( x) , such that:



1

1

  ( x) Piˆ ( x) Pˆj ( x)dx  hiˆ  ijˆˆ , for iˆ, ˆj  0,1,..., dˆ

(44)

where   ( x) , hiˆ , and  ijˆˆ can be formulated as: 

 ( x)  (1  x ) 

hiˆ 

2

1 2

(45)

 212 (iˆ  2 ) iˆ !(iˆ   ) 2 ( ) 0, 1,

 ijˆˆ  

(46)

iˆ  ˆj iˆ  ˆj

(47)

In Eq. (46), () denotes the Gamma function. By adopting the strategy utilized for defining the generalized Chebyshev polynomial for vector inputs [57-59], the generalized Gegenbauer polynomials are defined recursively as following:

 P0 (x)  1    P1 (x)  2 x  1 ˆ    T  ˆ ˆ   Pdˆ (x)  dˆ  2 d    1 Pdˆ 1 (x) x  d  2  2 Pdˆ  2 (x)  , for d  2,3, 4,...







(48)



where x n denotes the column vector of input variables. Since the Gaussian kernel function has a better capability in capturing local information than the originally employed square root function [58, 60], the Gaussian kernel function is adopted herein as the weighting function for the proposed Generalized Gegenbauer Kernel (GGK). Thus, the proposed dˆ th ˆ

d (xi , x j ) of two arbitrary input vectors x i and x j is defined as: order GGK function  GGK

dˆ

 P ( x )

T

ˆ

d  GGK ( xi , x j ) 

kˆ  0

kˆ



i

Pkˆ (x j )

exp  xi  x j

2 2



(49)

where each element of x i and x j is defined in [-1,1]. Both  and  are regarded as the kernel scales of the proposed kernel function. It is worthy to address that the proposed GGK satisfies the Mercer Theorem [47, 52, 61, 62] which is a prerequisite for implementing the kernel function in SVM/SVR. Thus, not limited in the proposed X-SVR model, the GGK introduced in this study can also be employed in any other kernelized learning models which require the Mercer condition to be satisfied. 3.3. Selection of the X-SVR model parameters Within the proposed X-SVR with GGK, there are seven hyperparameters, including two regularization parameters  1 and  2 , the penalty parameter C, the insensitive tube width  , the polynomial order dˆ , and two positive kernel scale parameters  and  . The prediction accuracy of the proposed X-SVR with GGK strongly depends on the selection of these parameters. For machine learning approaches, the k-fold cross-validation (CV) over the training samples is an effective approach to ensure that the regression model has the generalized ability in accurately predicting the training dataset while checking if the selected parameters will result in overfitting [63]. Practically, k-fold is commonly set to 5-10 as a trade-off of computational cost and prediction accuracy. In the present work, the 5-fold CV error, denoted by Err5CV is employed as the measurement of the training error for X-SVR, which is defined as:

Err5CV 

1 5  err 5  1

(50)

where err denotes the mean squared error between y   y ,1 , y ,2 ,..., y ,m  m and T

T

fˆ (x )   fˆ (x ,1 ), fˆ (x ,2 ),..., fˆ (x ,m )  m ; y denotes the true function value and   fˆ (x ) denotes the model prediction by X-SVR in the  th fold, respectively. err is

expressed as:

err 

2 1 m  y ,  fˆ (x , )  , for   1, 2,...,5   m  1 

(51)

where m denotes the number of training samples in fold  ; y , denotes the  th component of y ; x m n collects the training samples of the  th fold; x , n denotes the  th component of x ; fˆ () denotes the model prediction in the  th fold. Typically, the Bayesian optimization constructs a probabilistic approximation of the objective function by using Gaussian process and then determines the next estimation point that results in the maximum of the acquisition function [64]. Relying on all the available information from previous evaluations of the objective function, the minimum of the objective function can be efficiently obtained with relative less iterations. Consequently, the Bayesian optimization method is integrated in the proposed metamodel for automatically selecting the learning parameters. 3.4. Machine learning aided static structural reliability analysis for FGM bar-type structures To more effectively illustrate the proposed machine learning aided structural reliability analysis for 3D FGM frame structures, the following algorithm presents the detailed steps involved in proposed method. Algorithm: A machine learning aided static structural reliability analysis for FGM frame structures Step 1 Generate m (where m

1 ) realizations for each n random variables through Sobol set

based Quasi-MCS sampling method, where m denotes the size of the training dataset. Let x i denote the ith realization, and xtrain mn denote the set containing all m realization points, such that:





xtrain : xi n , for i  1,..., m .

(52)

Step 2 Determine the considered structural displacement by solving the governing equation of the stochastic static analysis for FGM frame structures, which can be expressed as:

K (xi )UiR  F(xi )

(53)

where UiR d , for i  1,..., m denotes the structural displacement at the ith sampling point.

Step 3 Determine the training dataset with input x train and output y train m . Let yi denote the ith output and y train denote the set containing all m structural responses, such that:





y train : yi  usi  UiR , for s  1,..., d , i  1,..., m .

(54)

where usi denotes the structural displacement at sth degree of freedom within the ith sampling point. Step 4 Established the regression model by the X-SVR approach through the training dataset. The regression model, i.e., a kernelized function, can be formulated as: T ˆ * fˆ (x)   p  q  κ(x)  bˆ T G  

(55)

Step 5 Estimate the PDFs and CDFs for the concerned structural responses by using the established regression model fˆ (x) . Step 6 Evaluate the structural reliability for the concerned structural response against its the structural capacity, which can be calculated as: u*

Pr  P(u  u* )   fu ( x)dx -

(56)

*

where u denotes the dependent structural capacity for displacement; f u ( x ) denotes the estimated PDF of the concerned structural displacement u. 4. Numerical investigation In order to demonstrate the applicability, accuracy, and efficiency of the proposed machine learning aided structural reliability analysis, both analytical examples and practically motivated numerical examples involving random variables, are thoroughly investigated. Both the accuracy and computational efficiency of the proposed approach are compared with the MCS method with large simulation cycles. Besides, all random numbers are generated by the statistics toolbox of MATLAB R2017b [65]. Moreover, in order to quantitatively assess the performance of the X-SVR approach, the adopted estimation metrics are listed in Table 1. For the subsequent numerical investigations, all calculations were carried out on a computer equipped with Intel® Core(TM) i7-6700 CPU @ 3.40 GHz and 16 GB of RAM.

Table 1. The adopted estimation metrics Estimation metrics

Formulation*

 Y  Yˆ   1  Y  Y 

2

2

R-squared (R )

R

2

N

2

N

Relative Standard Deviation (RSD)

 Y  Yˆ 

2

N

N 1 Y

RSD 

RE 

Relative Error (RE)

Y  Yˆ Y

 100%

 100%

*where Y, Yˆ , and Y denote the true values, the modelled values, and the mean of the true values, respectively; and N denotes the number of samples.

4.1 First test function: The Borehole function The Borehole function is an 8-dimensional function which was initialized to model the water flow through a borehole. Due to its simplicity and ease of evaluation, it is commonly implemented as a benchmark function for emulation and prediction tests in the form of [6668]:

f ( x) 

2 x3 ( x4  x6 )  2 x7 x3 x  ln( x2 x1 ) 1   3 2  ln( x2 x1 ) x1 x8 x5 

(57)

where all input parameters are modelled by independent uniform variables within the ranges that are specified in Table 2 [68]. Table 2. Details of the input parameters for the Borehole function Input parameter

Range

x1

[0.05, 0.15]

x2

[100, 50000]

x3

[63070, 115600]

x4

[990, 1110]

x5

[63.1, 116]

x6

[700, 820]

x7

[1120, 1680]

x8

[9855, 12045]

The initial design of experiment consists of 5 training points that are augmented to 50 and further up to 100 points by the Sobol set based Quasi-MCS sampling method. The XSVR with GGK and the traditional SVR with Gaussian kernel are applied. For each given size of training dataset, mutually independent analysis is replicated for 50 times in order to verify the stability and the convergence of the regression models [68]. The corresponding boxplots of the computational results are illustrated in Figures 3 and 4. The validation metrics R2 and RSD for each calculation are computed for both regression techniques using 100,000 points. In addition, scatter plots with 100 training samples are also presented in Figures 3 and 4 to demonstrate the dispersion of the estimations against the true function values for both methods.

(a) (b) 2 Figure 3. Estimated R by (a) X-SVR and (b) SVR

(a) (b) Figure 4. Estimated RSD (%) by (a) X-SVR and (b) SVR As illustrated in Figures 3 and 4, an obvious convergence trend can be observed in both the proposed X-SVR method and the traditional SVR approach. However, the stability of the estimation through the X-SVR approach surpasses the traditional SVR approach by demonstrating a relatively less dispersion in all box plots presented in Figures 3 and 4.

Moreover, by comparing the R2, RSD, as well as the scatter subplots, it is evidently demonstrated that the proposed X-SVR is competence for estimating the Borehole function but with advantages on computational accuracy and stability over the traditional SVR approach. 4.2 Second test function: The 50-D function To further demonstrate the adequacy of the proposed X-SVR approach, the 50-D function with 50 random variables is thoroughly investigated in this subsection. The considered test function has an explicit formulation of [69, 70]: 50   f (x)  1  exp  0.01 xi2  i 1  

(58)

where all variables are assumed to be mutually independent random variables with uniform distribution within the range of [0, 1] [70]. The design of experiment consists of 100 training points which were generated by the Sobol set based Quasi-MCS sampling method. The comparison analysis between the proposed X-SVR and traditional SVR with Gaussian kernel function approaches is once again conducted for this numerical investigation. The validation metrics are computed for both X-SVR and SVR techniques using 100,000 points. The estimated PDFs and CDFs by the X-SVR and SVR methods in comparison to the MCS approach are reported in Figure 5(a, b). To more clearly illustrate the performance in computational accuracy, the RE of CDF has further been calculated by both machine learning methods, as plotted in Figure 5(c). Also the estimated function values at seven selected locations (i.e.,  ,    ,   2 , and   3 ) for both methods are listed in Table 3.

(a) (b) (c) Figure 5. Estimated (a) PDF and (b) CDF of f (x) ; (c) RE of CDF

Table 3. Estimations of f (x) Location MCS X-SVR RE (%)   3 0.0997 0.1000 0.2634   2 0.1176 0.1178 0.1669    0.1354 0.1356 0.0958  0.1533 0.1534 0.0412    0.1711 0.1711 0.0019   2 0.1890 0.1889 0.0370   3 0.2068 0.2067 0.0659 2

R RSD

N/A N/A

0.9999 0.1201

N/A N/A

SVR

RE (%)

0.1009

1.1810

0.1180 0.1352 0.1523 0.1694

0.3874 0.1970 0.6453 1.0001

0.1866

1.2878

0.2037

1.5258

0.9456 2.6383

N/A N/A

From both Figure 5 and Table 3, it is well-demonstrated that both methods can adequately establish the PDFs and CDFs when the size of the training sample reaches 100. However, the proposed X-SVR approach provides high-accuracy estimations for PDFs and CDFs by possessing an overlap with the estimations by the MCS method. By comparing with the MCS results, the proposed X-SVR approach is capable of providing a higher level of prediction accuracy for the 50-D function than the traditional SVR approach by having a much higher R2-value and smaller RSD-value when the size of training datasets is identical. Moreover, the superior computational accuracy of the proposed approach to the traditional SVR approach is intuitively demonstrated by possessing maximum RE of 0.2081% via XSVR approach. Unsurprisingly, the performance of the proposed X-SVR approach surpasses the traditional SVR method for the estimation of the 50-D test function. 4.3. 2D FGM frame screen example The third numerical example involves a 2D FGM frame structure whose general structural layout is illustrated in Figure 6. Each FGM structural member has an identical length of 1 m. The FGM structural member is made of steel and alumina (Al2O3) whose material property is governed by the exponential-law. As illustrated in Figure 6(a), the structural nodes on the bottom layer of the screen are clamped. The seven structural nodes on the top right layer are subjected to nodal compressive force, denoted as P1, and the remainder seven nodes on the top left layer are subjected to nodal tensile force, denoted as P2. According to the convergence analysis within the framework of FEM, the 2D FGM frame screen has been discretized into 130 nodes and 345 beam elements.

(a) (b) Figure 6. Structural layout of the frame screen (a) 3D view and (b) 2D view It is assumed that the Young’s moduli of the top and bottom layers of the FGM structural member (i.e., EU and EL ), the Poisson’s ratio of the FGM (i.e.,  ), the magnitudes of the applied loads (i.e., P1 and P2 ), as well as the dimensions of the cross-section of the structural member (i.e., b and h) are modelled as random variables. The associated statistical information of the concerned uncertain system parameters are presented in Table 4. Table 4. Statistical information of the considered random variables of the 2D FGM frame Random variables

Distribution type

Mean

Standard deviation

P1

Normal [71]

1 kN

25 N

P2

Normal [71]

2 kN

50 N

EU (Al2O3)

Lognormal [46, 72] 390 GPa

19.5 GPa

EL (Steel)

Lognormal [46, 72] 210 GPa

10.5 GPa

 b h

Beta Lognormal [46, 72] Lognormal [46, 72]

0.3 20 mm 40 mm

0.0065 0.4 mm 0.8 mm

To demonstrate the competence of the proposed X-SVR aided structural reliability analysis, four structural responses, namely the horizontal and vertical displacements of points A and B (i.e., uA , vA , uB , and vB ), have been selected for observation. Moreover, the MCS method with 1 million simulation cycles has been implemented for verifying the accuracy of the proposed approach. In addition, the traditional SVR approach with Gaussian kernel is also adopted so the effectiveness of the proposed X-SVR over the traditional SVR can be highlighted. Firstly, in order to demonstrate the impact of the training size on the probability estimation, the initial design of experiments consists of 5 training samples, which are

augmented to 50 and further up to 100. The estimated probability of uA and vB are summarized in Tables 5 and 6, respectively. Table 5. Estimated probability of uA (unit: m) Training size Location

P(uA  

 3

MCS uA

MCS MCS uA

5 X-SVR

50 X-SVR

100 X-SVR

100 SVR

)

0.000663 0.000075 0.000468 0.000514 0.000055

P(uA  uMCS  2 uMCS )

0.018943 0.008157 0.018378 0.018304 0.009169

P(uA  uMCS   uMCS )

0.158937 0.146771 0.159339 0.159406 0.130737

P(uA  uMCS )

0.509801 0.585198 0.507802 0.510329 0.504252

A

A

A

A

A

P(uA  uMCS   uMCS )

0.840833 0.919430 0.839801 0.840483 0.871294

P(uA  uMCS  2 uMCS )

0.973235 0.994124 0.974307 0.973448 0.990247

P(uA  uMCS  3 uMCS )

0.997617 0.999830 0.998066 0.997787 0.999931

A

A

A

A

A

A

uMCS = 3.1620×10-5 A  uMCS = 1.5771×10-6 A

Table 6. Estimated probability of vB (unit: m) Training size Location P(vB  

 3

MCS vB

MCS MCS vB

5 X-SVR

50 X-SVR

100 X-SVR

100 SVR

)

0.000690 0.000003 0.000867 0.000602 0.000075

P(vB  vMCS  2 vMCS )

0.018947 0.001269 0.020101 0.018717 0.010315

B

P(vB  

B

MCS vB



MCS vB

)

P(vB  vMCS )

0.158797 0.066353 0.157279 0.159162 0.133633 0.509567 0.541058 0.503753 0.509356 0.502156

B

P(vB  vMCS   vMCS )

0.840758 0.966716 0.841784 0.840655 0.867496

P(vB  vMCS  2 vMCS )

0.973323 0.999931 0.976535 0.973838 0.989595

P(vB  

0.997724 1.000000 0.998542 0.997924 0.999915

B

B

B

MCS vB

B

 3

MCS vB

)

vMCS = 3.4415×10-5 m B



MCS vB

= 1.7290×10-6 m

From Tables 5 and 6, it is demonstrated that by using 5 training samples, the accuracy of the regression model established by the X-SVR approach remains at a lower level with distinguished variation compared to the MCS method. This phenomenon is caused by the fact

that the machine leaning approach is based on data. The extremely small training sample size (i.e., 5 samples) cannot provide sufficient information to train the regression model for this example. Subsequently, by increasing the size of the training samples to 50, the accuracy of the proposed approach improves apparently. Further, when the training size reaches 100, the proposed approach has demonstrated excellent agreement with the MCS method. By closely examining Tables 5 and 6, the absolute relative errors between the proposed approach with 100 training samples and the MCS method are less than 0.2% at all selected locations, which are much lower than the ones of the traditional SVR approach with the same training sample size. Furthermore, by utilizing the regression models established by both the proposed X-SVR and traditional SVR approaches, the PDFs and CDFs of the selected structural responses can be constructed. To highlight the variation between the computational results through different methods, REs of CDF have also been presented. The corresponding results are summarized in Figures 7 and 8. In addition to the implementation of these two semi-simulative approaches, the exhaustive MCS method with 1 million simulations has also been employed as the benchmark.

(a) (b) (c) Figure 7. Estimated (a) PDF and (b) CDF of uA ; (c) RE of CDF

(a) (b) (c) Figure 8. Estimated (a) PDF and (b) CDF of vB ; (c) RE of CDF

As illustrated in Figures 7 and 8, both the proposed approach and the traditional SVR method can provide estimations for the PDFs and CDFs of the concerned structural responses by using 100 training samples. However, it is clearly observed that the accuracy of the estimations through the proposed approach surpasses the traditional SVR approach by possessing an overlap with the MCS results in PDF and CDF. The same phenomenon can also be noticed from the associated scatter subplots. Moreover, by comparing the R2-values and the REs of CDF, the superior prediction precision of the proposed X-SVR approach against the traditional SVR method with the identical training size has been once again confirmed. That is, by using relatively small size of the training set, the proposed approach is capable of establishing adequate relationships between the considered system uncertainties and the concerned structural responses. In addition to the effectiveness of the proposed approach, the high efficiency of the proposed approach against the MCS method should also be highlighted. Within the identical computational environment, the proposed X-SVR approach only took 56.96 seconds on average to train the regression model by using 100 samples, whereas the MCS method with 1 million simulations consumed approximately 10.59 hours. Therefore, the proposed X-SVR approach has superior computational efficiency over the exhaustive MCS approach in the calculations of the static structural reliability for the 2D FGM frame screen example. 4.4. 3D FGM frame tower example To further reveal the performance of the proposed approach, a numerical investigation of a 3D FGM frame tower is conducted in this section. The general structural layout of the frame tower is depicted in Figure 9. The total height of the frame tower is 5 m, and each layer has a height of 1 m. All structural members share the same rectangular cross-section which are made of FG Aluminium and Zirconia (i.e., Al/Zr2O). The governing relationship of FGM properties is presumed in power-law.

(a) (b) (c) Figure 9. General structural layout of the 3D frame tower (a) 3D view, (b) top view, and (c) 3D view of the top layer of the tower Regarding the boundary conditions, the structural nodes on the bottom layer of the frame tower are fixed. The nodal loads are applied on the top layer, where the 12 outer nodes are subjected to the compressive force, denoted as P1 , and the remainder 6 inner nodes are subjected to the tensile force, as P2 . In this numerical investigation, two distinctive cases (i.e., Case A and B) of analysis have been investigated. Although the power-law index (i.e., k) is considered as a random variable in both cases, the adopted statistical information of k is different. In Case A, the power-law index k is modelled as a lognormal random variable with mean of 2 and standard deviation of 0.04, whereas Case B assumes k in Beta distribution with mean of 0.2 and standard deviation of 0.004. For both Cases A and B, the remainder considered random variables are including the magnitudes of the applied loads (i.e., P1 and P2 ), Young’s moduli of the top and bottom layers of the FGM structural member (i.e., EU and EL ), the Poisson’s ratio of the FGM (i.e.,

 ), and the dimensions of the cross-section of the structural member (i.e., b and h). The detailed statistical information associated with each random variable are provided in Table 7. Table 7. Statistical information of the considered random variables of the 3D FGM frame Random variables

Distribution type

Mean

Standard deviation

P1

Normal [72]

100 kN

2.5 kN

P2

Normal [72]

200 kN

5 kN

EU (Zr2O) EL (Al)

Lognormal [46, 72] 151 GPa

7.55 GPa

Lognormal [46, 72]

3.5 GPa

70 GPa



k

b h Case A Case B

Beta Lognormal [46, 72] Lognormal [46, 72] Lognormal Beta

0.3 20 mm 40 mm 2 0.2

0.0065 0.4 mm 0.8 mm 0.04 0.004

After conducting the convergence study, the frame tower has been discretized into 690 nodes and 1008 finite elements. The displacements in three directions (i.e., u, v, and w) at two observation points, A and B, with coordination of (0.5, -1.866, 5) and (-1, 0, 4), respectively, are selected as observation points. In this numerical investigation, the size of the training dataset is selected as 100. The reason for this selection is that at this training sample size, a convergence of estimation was captured when the proposed X-SVR approach was utilized. Furthermore, the exhaustive MCS approach with 1 million simulations has also been adopted for the purpose of result verification. In Case A, the statistical information, including the PDFs, CDFs, and the structural reliabilities of the concerned structural displacements have been investigated by both the proposed approach and the MCS method. The estimated PDFs, CDFs, and REs of CDF of

u A and wB are presented in Figures 10 and 11, and the estimated probability of these considered responses are summarized in Tables 8 and 9.

(a) (b) (c) Figure 10. Estimated (a) PDF and (b) CDF of uA in Case A; (c) RE of CDF

(a) (b) (c) Figure 11. Estimated (a) PDF and (b) CDF of wB in Case A; (c) RE of CDF

Table 8. Estimated probability of uA in Case A Location

MCS

X-SVR

P(uA  uMCS  3 uMCS )

0.000640 0.000548

P(uA  uMCS  2 uMCS )

0.019081 0.018872

P(uA  uMCS   uMCS )

0.158625 0.159434

P(uA  uMCS )

0.509543 0.509474

A

A

A

A

A

A

A

P(uA  uMCS   uMCS )

0.840961 0.840236

P(uA  uMCS  2 uMCS )

0.973246 0.973386

P(uA  uMCS  3 uMCS )

0.997686 0.997824

A

A

A

A

A

u

MCS A

u

MCS A

A

= 3.1876×10-4 m = 1.5516×10-5 m

Table 9. Estimated probability of wB in Case A Location

MCS

X-SVR

P(wB  wMCS  3 wMCS )

0.000622 0.000612

P(wB  wMCS  2 wMCS )

0.018914 0.018899

P(wB  wMCS   wMCS )

0.158747 0.158937

P(wB  wMCS )

0.509970 0.509905

B

B

B

B

B

B

B

P(wB  wMCS   wMCS )

0.841143 0.840908

P(wB  wMCS  2 wMCS )

0.973083 0.973147

P(wB  wMCS  3 wMCS )

0.997644 0.997708

B

B

B

B

B

B

 wMCS = 9.2125×10-3 m B



MCS wB

= 5.0357×10-4 m

From Figures 10 and 11, the estimations of the PDFs and CDFs for uA and wB can be effectively achieved by the proposed approach. Moreover, by utilizing 100 training samples, the proposed approach has been competent to deliver adequate estimations for all the concerned displacements by comparing with the MCS method. The accuracy of the regression model established by the X-SVR has been thoroughly verified by the comparisons of results between X-SVR and MCS approaches, scatter plots, R2-values, and RSD-values.

In addition to the PDFs and CDFs, it is evidently demonstrated in Tables 8 and 9 that the estimated reliabilities of the concerned displacements at point A by the proposed technique agree well with the MCS method, regardless of the selected locations changing from   3 to   3 . By further closely examining Tables 8 and 9, the effectiveness of the proposed XSVR approach in structural reliability analysis can once again be confirmed by having the maximum absolute relative error of the estimated probability for the concerned displacements less than 0.1%. Furthermore, in order to demonstrate the applicability of the proposed approach for situations where the statistical information (i.e., mean, standard deviation, and distribution type) of a specific considered system uncertain parameter changes, the power-law index k is adjusted to the random variable in Beta distribution with distinctive mean and standard deviation in Case B. A similar investigation is conducted in Case B to obtain the statistical information about wB . The estimated PDFs and CDFs of the concerned displacements are presented in Figure 12. In addition, the static structural reliability analysis of wB is studied and summarized in Table 10.

(a) (b) (c) Figure 12. Estimated (a) PDF and (b) CDF of wB in Case B; (c) RE of CDF Table 10. Estimated probability of wB in Case B Location

P(wB  

MCS wB

 3

MCS MCS wB

X-SVR

)

0.000541 0.000510

P(wB  wMCS  2 wMCS )

0.018335 0.018244

P(wB  wMCS   wMCS )

0.159011 0.159297

P(wB  wMCS )

0.511099 0.511131

B

B

B

B

B

P(wB  wMCS   wMCS )

0.841362 0.841096

P(wB  wMCS  2 wMCS )

0.972549 0.972574

P(wB  wMCS  3 wMCS )

0.997410 0.997464

B

B

B

B

B

B

 wMCS = 6.2945×10-3 m B



MCS wB

= 3.8681×10-4 m

As illustrated in Figure 12, though there is a distinctive variation in the statistical information of the power-law index k, the proposed approach still provides effective estimations by using the identical size of the training set. Moreover, by examining the R2values, RSD-values, maximum RE-values, as well as the scatter subplots, it is strongly indicated that the established regression model through the proposed X-SVR is capable of providing high-quality computational results. It is further confirmed in Table 10 that the proposed approach is effective in estimating the structural reliabilities for the concerned displacements at point B with a maximum absolute relative error of 0.034%. Therefore, for both Cases A and B in this example, the proposed semi-simulative approach is competent to effectively provide not only the reliabilities of any concerned structural displacements, but also the sufficient statistical moments, such as means, standard deviations, PDFs, and CDFs etc., by using 100 training samples. By further closely comparing the estimated statistical moments of wB in Cases A and B, it is also noticed that, for this example, the mean of wB has been slashed for approximately 31.7% and the associated standard deviation for 23.2%, by only adjusting the statistical information of the power-law index k. Therefore, in practical engineering, through controlling the power-law index in FGM structures, not only the material general properties can be modified, but also the coefficients of variation within the probabilities of the structural responses might be adjusted. Furthermore, the last highlight worth to mention here is that by utilising the identical computational environment, the proposed approach only took 146.36 seconds on average to establish the effective regression model when the size of the training set is 100, whereas the MCS method with 1 million simulations consumed more than 21.75 hours. This tremendous gap in computational costs further outstands the advantages of the proposed approach. 4.5. An extension of Example 4.4 To further demonstrate the applicability of the proposed approach for problems involving large numbers of uncertain parameters, an extension of the 3D FGM frame tower, which has been studied in the previous section, is re-investigated here. The geometry, boundary

conditions, FEM information, and material model are inherited from the previous investigation. Unlike Example 4.4, the material properties of each layer of the structure considered in this example are modelled as independent random variables with distinctive statistical information. A full list of incorporated uncertain parameters for the re-investigation is listed in Table 11. In total, there are 32 independent random parameters involved in this investigation. Table 11. Statistical information of the considered random variables for the extension of the 3D FGM frame tower

Layer 1

Layer 2

Layer 3

Random variables

Distribution type

Mean

Standard deviation

P1

Normal [72]

100 kN

2.5 kN

P2

Normal [72]

200 kN

5 kN

EU1

Lognormal [46, 72] 151 GPa

1.51 GPa

EL1

Lognormal [46, 72]

70 GPa

0.7 GPa

1 b1 h1 k1

Beta

0.294

0.0065

Lognormal [46, 72]

20 mm

0.2 mm

Lognormal [46, 72]

40 mm

0.4 mm

Lognormal

2

0.02

EU2

Lognormal [46, 72] 151 GPa

3.02 GPa

EL2

Lognormal [46, 72]

70 GPa

1.4 GPa

2 b2 h2 k2

Beta

0.297

0.0065

Lognormal [46, 72]

20 mm

0.4 mm

Lognormal [46, 72]

40 mm

0.8 mm

Lognormal

2

0.04

EU3

E

3 L

3 b3 h3 k3 Layer 4

Lognormal [46, 72] 151 GPa

4.53 GPa

Lognormal [46, 72]

70 GPa

2.1 GPa

Beta

0.3

0.0065

Lognormal [46, 72]

20 mm

0.6 mm

Lognormal [46, 72]

40 mm

1.2 mm

Lognormal

2

0.06

EU4

Lognormal [46, 72] 151 GPa

6.04 Gpa

EL4

Lognormal [46, 72]

70 GPa

2.8 GPa

4 b4 h4

Beta

0.303

0.0065

Lognormal [46, 72]

20 mm

0.8 mm

Lognormal [46, 72]

40 mm

1.6 mm

k4 Layer 5

Lognormal

2

0.08

EU5

Lognormal [46, 72] 151 GPa

7.55 GPa

EL5

Lognormal [46, 72]

70 GPa

3.5 GPa

5 b5 h5 k5

Beta

0.306

0.0065

Lognormal [46, 72]

20 mm

1.0 mm

Lognormal [46, 72]

40 mm

2.0 mm

Lognormal

2

0.10

where EU , for =1,...,5 denotes the Young’s modulus of the upper surface of the FGM for the  th layer; EL , for =1,...,5 denotes the Young’s modulus of the lower surface of the FGM for the  th layer;   , for =1,...,5 denotes the Poisson’s ratio of the FGM for the  th layer; b , for =1,...,5 denotes the width of the cross-sectional area of the FGM for the

 th layer; h , for =1,...,5 denotes the height of the cross-sectional area of the FGM for the  th layer; k  , for =1,...,5 denotes the power-law exponent of the FGM for the  th layer. To highlight the superior performance of the proposed X-SVR approach in term of computational accuracy, the traditional support vector regression (SVR), Gaussian process regression (GPR), and neural network (NN) are implemented for comparison. Furthermore, the results of MCS with 1 million simulations are also adopted as reference. Identical to the previous investigations, 100 training samples are utilized for four adopted machine learning techniques to estimate the vertical deflection at point B, i.e., wB . The corresponding PDF and CDF are presented in Figure 13(a) and (b), respectively. To more intuitively illustrate the accuracy of various techniques, the REs of CDF are calculated, as shown in Figure 13(c).

(a) (b) (c) Figure 13. Estimated (a) PDF and (b) CDF of wB ; (c) RE of CDF

From Figure 13, it is well-demonstrated that the implemented machine learning techniques are capable of estimating wB , by simultaneously considering 32 systematic uncertain parameters. Noticeable deviations in estimated PDF and CDF can be observed in Figure 13, especially when comparing the estimations of the X-SVR with the traditional SVR technique. By further referring to the detailed information, i.e., RE of CDF, it is obvious that the accuracy of the proposed X-SVR approach is superior to the other incorporated techniques, by possessing the highest R2-value of 0.9977 and lowest absolute max RE of 1.1703%. Moreover, the static structural reliability analysis of wB at some concerned locations, i.e., wMCS , wMCS , wMCS , and wMCS , has been investigated   wMCS  2 wMCS  3 wMCS B B B B B B B through these machine learning techniques, and the corresponding results are summarized in Table 12. Table 12. Estimated probability of wB Location

MCS

X-SVR

SVR

GPR

NN

P(wB  wMCS  2 wMCS )

0.019076 0.020533 0.010900 0.022060 0.018940

P(wB  wMCS   wMCS )

0.158899 0.157010 0.146078 0.157669 0.151903

P(wB  wMCS )

0.509393 0.503606 0.534042 0.500892 0.496027

B

B

B

B

B

P(wB  wMCS  2 wMCS )

0.973252 0.977992 0.996474 0.977974 0.985802

P(wB  wMCS  3 wMCS )

0.997617 0.998738 0.999995 0.998651 0.999815

B

B

B

B

 wMCS = 9.2214×10-3 m B



MCS wB

= 4.5956×10-4 m

After comparing the estimated probability in Table 12, it is evidently demonstrated that the proposed X-SVR approach is competent to provide relatively lower deviation against the MCS results in general. Therefore, the effectiveness of the proposed X-SVR approach in engineering applications with relatively large number of uncertain parameters is clearly illustrated. Moreover, basing on the regression model, the computational costs of the considered structural reliability analysis have been greatly reduced by all machine learning aided approaches. The computational efficiency through the established regression model is highlighted by comparing the costs against the brutal completion of MCS with 1 million simulations. It is worth to mention that the proposed X-SVR consumed approximately 14 seconds to complete 1 million estimations; whereas the exhaustive MCS spent nearly 22

hours. Consequently, the applicability, accuracy, as well as the efficiency of the proposed machine learning aided structural reliability analysis for this extension of the 3D FGM frame tower example have been comprehensively demonstrated. 4.6. Remarks After numerical investigations, several advantages of the proposed X-SVR approach can be summarized as: (1) The optimization problem in the proposed X-SVR approach is a strictly proven quadratic programming (QP) problem with simple non-negative constraints. This means that the global optimum solution can be effectively obtained through any available QP solver; (2) The relationship between systematic uncertainties and concerned response can be expressed with an explicit formulation; (3) The newly developed generalized Gegenbauer polynomial kernel (GGK) has been integrated within the proposed X-SVR approach; (4) An auto-tuning scheme for the involved hyperparameters has been fulfilled through the application of Bayesian optimization methods. Despite of these innovations, the application of the proposed approach is still limited by the computational capability of the workstation, especially in dealing with high dimensional problems. 5. Conclusion In this paper, a novel machine learning aided static structural reliability analysis approach is proposed for functionally graded frame structures involving random variables. A new machine learning based approach, namely the extended Support Vector Regression (XSVR) method, is freshly developed herein. Applied with newly proposed generalized Gegenbauer Kernel and Bayesian optimization method, the X-SVR approach has improved performance in regression through automatically selecting the hyperparameters. Distinctive to the implicit constitutive relationship, the regression model generated by the X-SVR approach can be explicitly expressed as a kernelized function. Based on the generated regression model, the structural reliability analysis can be fulfilled with effectiveness and robustness. To demonstrate the applicability, stability, and accuracy of the proposed computational approach, the Borehole function and 50-D function have been investigated in comparison to

the traditional SVR with Gaussian kernel first. In addition, two functionally graded frame structures with the consideration of practical stimulated uncertainties are thoroughly investigated to demonstrate the applicability, high precision, as well as the computational efficiency of this proposed approach. Consequently, it will draw more attention as a new semi-simulative method for structural reliability analysis and provide more guidance value to engineers in both structural design and safety assessment stages. 6. Acknowledgement The work presented in this paper has been supported by Australian Research Council projects DP160103919 and IH150100006. 7. Reference [1]

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Appendix A The detailed expression of each component within k e is explicitly formulated as the following:

k  k  k e 1,1

k

e 7,1

e 7,7

Axxe  le

(A1)

Axye  y2,e 12Cxxe  3  le (1   y ,e )2 ksye le (1   y ,e )2

e 2,2

A  12 Dxxe  3  e xz z ,e 2 2 le (1  z ,e ) ksz le (1  z ,e ) e

e 3,3

k

 k

e 9,3

k

e 9,9

i e e e k5,3  k9,5  k11,3  k11,9 

k



e 11,11

k

Axze z2,e 6 Dxxe  le2 (1  z ,e )2 2ksze (1  z ,e )2

(A5)

le (1  z ,e )2

k  k

k

 k

e 8,6

k

e 12,2

e e k6,6  k12,12 

e 8,2

k

e 11,5

k

e 12,6

k

where

 k



 k

Axze le z2,e



4ksze (1  z ,e )2

(A6)

(A7)

Axye  y2,e 6Cxxe  2  le (1   y ,e )2 2ksye (1   y ,e ) 2

(A8)

 k

e 12,8



Bxxe  le

e 12,1

k

e 12,7

Cxxe ( y2,e  2 y ,e  4)

e 8,8



e 7,6

(A3)

(A4)

ksye le

Dxxe (z2,e  2z ,e  4)

e 6,1

e 6,2

Fxye

2

Fxze ksze le

e e e k4,4  k10,4  k10,10 

e 5,5

(A2)

le (1   y ,e )2



Axye le  y2,e 4ksye (1   y ,e )2

Axye  y2,e 12Cxxe  3  le (1   y ,e )2 ksye le (1   y ,e )2

Dxxe (z2,e  2z ,e  2) le (1  z ,e )2

Cxxe ( y2,e  2 y ,e  2) le (1   y ,e )2





Axze le z2,e 4ksze (1  z ,e )2

Axye le  y2,e 4ksye (1   y ,e )2

(A9)

(A10)

(A11)

(A12)

Axxe  be 

h2

h 2

Axye  Axze  be 

E e ( y )dy h2

h 2

Bxxe  be 

h2

G e ( y )dy

(A13) (A14)

yE e ( y )dy

(A15)

y 2 E e ( y )dy

(A16)

Dxxe 

be3 h 2 e E ( y )dy 12 h 2

(A17)

Fxye 

be3 h 2 e G ( y )dy 12 h 2

(A18)

h 2

Cxxe  be 

h2

h 2

Fxze  be 

h2

h 2

 y ,e 

z ,e 

y 2G e ( y )dy

24 I ze (1   e ) Asye le2 24 I ye e 2 sz e

A l

(1   e )

(A19) (A20)

(A21)

where ksye  1/ K sye and ksze  1/ K sze ; K sye and K sze denote the shear correction factors; le , be , and he denote the length, width, and height of the beam element, respectively; Asye and Asze denote the two effective shear areas of the beam element; I ye and I ze denote the second moment of areas about the y-axis and z-axis of the beam element, respectively; E e ( y ) and

G e ( y ) denote the Young’s modulus and Shear modulus of the beam element at location y, respectively; and 

e

denotes the Poisson’s ratio of the beam element.