Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. Hybrid reliability analysis for structures with implicit limit state functions Wen-Cai Sun1,2, Zi-Chun Yang1,2, Lei Wang1,2 of High Temperature Structural Composite Materials for Naval Ship, 2College of Power Engineering, Naval University of Engineering, Wuhan, Hubei, China E-mail: [email protected] 1Institute For the implicit limit state equation usually encountered in structural hybrid reliability analysis, the interval parameters have been assumed to obey conservative distribution and the SVM is used to construct the hyper surface. A series of state points covering around the design point can be obtained by Markov Chain simulation. Then, combined with the structural response values, the training samples for SVM can be obtained. The structural hybrid reliability can be calculated by the combination of the approximate limit state equation and Monte Carlo method. Two numerical examples verify the feasibility and effectiveness of the proposed method. Keywords: Structural Hybrid Reliability; Implicit Limit State Equation; Support Vector Machine (SVM); Markov Chain. 1. Introduction The limit state functions of the complex structures are usually implicit or difficult to write out the analytic expressions. In addition, these structures commonly contain many types of uncertain information. At present, the Response Surface (RS) [1], Artificial Neural Network (ANN) [2] and Support Vector Machine (SVM) [3] are mostly used for approximately fitting of the implicit limit state functions. The former two are based on the empirical risk minimization principle, and have many insurmountable defects in practical applications. Hurtado explained that the root of the problems of Response Surface method is rigid non-adaptive regression technique [4]. The topology structure of ANN is difficult to choose, the generalization ability is poor, the over-learning and the local optimization of ANN are the main problem of ANN applied in reliability analysis. SVM is one of the latest research results in machine learning domain, which is based on structural risk minimization principle and can avoid the overlearning, local optimization and dimension disaster [5]. Rocco and Moreno used 86 Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 SVM to assess the reliability of network systems [6]. Hurtado and Alvarez considered structural reliability problem as pattern recognition problem and analyzed the structural reliability using SVM and stochastic finite element [3]. Support Vector Regression machine (SVR) is developed by Vapnik based on SVM theory, and is a new algorithm for nonlinear regression estimation [7]. So far, hybrid reliability researches for implicit limit state equations are rarely reported. A new type of probabilistic-interval hybrid reliability model was established in literature [8], and the solving method was derived based on failure integral formula. The problem of implicit limit state is studied in this paper based on the reliability model in literature [8]. Firstly, the sample extraction strategy is researched based on Markov chain technique [9–11]. The interval variables are dealt as conservative distribution assumption, and the optimal importance sampling function is used as the stationary distribution of the Markov chain. The state points around the design point and the alternative points are used together with the structural response values constituting the training sample set. The SVR algorithms for probabilistic-interval hybrid reliability analysis are proposed, and the limit state equation with structural risk minimization can be obtained. The hybrid reliability can be obtained through random sampling of probabilistic and interval variables. Two examples are given to verify the methods in this paper. 2. The Structural Hybrid Reliability Method Assume the structural limit state equation is M G( X ,Y ) 0 (1) where X=[X1,X2,…,Xn] and Y=[Y1,Y2,…,Ym] are probabilistic variable vector and interval variable vector, respectively. Assume the joint probability density function of X1,X2,…,Xn and Y1,Y2,…,Ym is fX,Y(x,y) and the integral formula of structural failure probability can be written as Pf f X ,Y ( x , y )dx1 dxn dy1 dym G ( x , y )0 f X ( x ) fY ( y )dx1 dxn dy1 dym (2) G ( x , y ) 0 fY ( y ) f X ( x )dx1 dxn dy1 dym G ( x , y ) 0 Pf ( y ) fY ( y )dy1 dym E[ Pf ( y )] where Ω is the variable region of the interval vector Y, G(x,y)≤0 is the failure region of the structure, Pf(y) is the structural failure probability under the 87 Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 realization value of Y. Eq. (2) shows that the structural failure degree is the mean value of the failure function with Y as its independent variable in the region of interval variables. For the case of independent interval variables, the failure integral formula is equivalent to Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. Pf n 1 V 1 n lim Pf ( yi ) lim Pf ( yi ) V n i 1 n n n i 1 (3) where yi denotes the ith sample point of the interval variable vector according to uniform distribution. For the case of relevant interval variables, the failure integral can be written as Pf Pf ( y ) fY ( y )dy1 dym Pf ( y ) fY1 ( y1 ) fY2 |Y1 ( y2 | y1 ) fYm |Y1 ,Y2 ,,Ym1 ( ym | y1 , y2 , , ym 1 )dy1 dym (4) E[ Pf ( y )] lim n 1 n Pf ( y j ) n j 1 where yj denotes the jth sample point of the interval variable vector according to conditional probability. If yk denotes the kth sample point of the interval variable vector, the Monte Carlo calculation formula of structural reliability is Pr 1 Pf 1 1 N N P (y f k ) (5) k 1 where N denote the simulation times. 3. Analysis Method for Implicit Limit State Problems 3.1. The sample extraction strategy based on Markov Chain To ensure the accuracy of structural reliability analysis, the limit state surface should have high accuracy near the design point. So, the samples that support the surface should be distributed around the design point, where it is the region of greater contribution to structural failure. If the optimal importance sampling function is used as the stationary distribution of Markov chains, the Markov chain state points can be well covered around the structural design point. Then, a good approximation of the limit state equation can be obtained through SVM training. 88 Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 When probability variables and interval variables all exist in the structure, the method in section 1 is equivalent to probability reliability analysis with the interval variables according to uniform distribution. The assumption for interval variables is proved to be conservative in many studies. So, this method is reasonable in theory, and partial to safety in application. In order to facilitate the representation, the random variables and interval variables are uniformly denoted as X=[X1,X2,…,Xn](n is the dimension of structural variables). The joint probability density function of X is f(x) and the conditional probability density function of the variables in failure region F is f x | F I F ( x ) f ( x ) Pf (6) where IF(x) is the indicator function of F. If x∈F, IF(x)=1, otherwise, IF(x)=0. Pf denotes the structural failure degree. Eq. (6) is the optimal importance sampling function in reliability calculation. Take it as the stationary distribution of Markov chain, and the obtained state points can be well distributed around the structural design point. For convenience, make q( x ) I F ( x ) f ( x ) (7) First, take one point in the failure region as the initial point according to engineering experience or numerical method. Then, on the basic of the j-1th state point x(j-1), generate the alternative point ε according to the n-dimensional proposed distribution f | x j 1 and calculate the ratio as r q q x ( j 1) (8) Then, the jth state point denoted as x(j)=[x1(j),x2(j),…,xn(j)]T of Markov chain can be obtained according to Metropolis criterion, i.e. make x(j)=ε with probability min{1,r} and x(j)=x(j-1) with probability 1-min{1,r}. Repeat the above steps, and the state points denoted as x(1),x(2),---,x(M) can be obtained whose stationary distribution is the optimal importance sampling function f(x|F). The proposed distribution f | x j 1 has a great influence on the robustness of Metropolis algorithm. f | x j 1 must be a density function with symmetric distribution, and the algorithm is more robust for bell shaped probability density function than super cube shaped one. In this paper, n dimension Gauss distribution is chosen as the proposed distribution as follows f * ( |x (j -1) ) 1 2 n/2 1/ 2 89 e 1 x j 1 2 T 1 x j 1 (9) Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 where is n n dimension covariance matrix. The standard deviation of the proposed distribution has influence on the distance of the next sample and the current sample, and the convergence rate of Markov chain is affected too. The reduced variance strategy is applied in this paper written as Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. ij j i0 ( 1, i 1, 2,, n, j 1, 2,, M ) (10) where ij is the variance of the ith component generating the jth state point, is the initial variance of the ith component and can be the original variance. Thus, the region of greater contribution to structural failure can be quickly explored in the early stage of Markov chain simulation. Then, more samples can be centered around the design point or the limit state surface in later stage. To make full use of the information generated in Markov chain simulation, the alternative sample point and the state point will be used together, and form the training samples of SVM with the corresponding output index (functional values). Thus, the fitting accuracy of the limit state equation will be improved. 0 i 3.2. Algorithm of SVM prediction a) In Markov chain simulation, the state samples, alternative samples and their corresponding output value are saved. b) Eliminate repeat samples of the state points, and merge the other state points and the alternative samples. Form the training samples T=(X×Y)l={(x1,y1),…,(xl,yl)}T together with the output index, where xi ∈Rn, Y={y1,y2,…,yl} is the output values. In addition, it can be known from the Markov chain simulation that l=M. c) Choose the insensitive coefficient, penalty parameter and the kernel function K(·,·). d) Construct the following convex quadratic programming problem: l ( i i )( j j )( ( xi )γ ( x j )) min 2l R i , j 1 l l ( i i ) yi ( i i ) 1 1 i i l () s.t. ( i i ) 0;0 i C , i 1,Λ, l i 1 (11) e) Solve the convex quadratic programming problem in d), and the Lagrange multiplier and offset b can be obtained such as 90 Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 () (1 , 1 , 2 , 2 , , l , l )T (12) l b y j ( i i ) K ( xi , x j ) , j (0, C ) (13) i 1 or l (14) Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. b y k ( i i ) K ( xi , xk ) , k (0, C ) i 1 f) Construct the regression forecasting model of SVM as l y g ( x ) ( i i ) K ( xi , x) b (15) i 1 g) Test the accuracy of training using test samples, and adjust the precision parameter, penalty parameter and the kernel function parameter. h) For the interval variables, extract N samples according to the principle of uniform distribution. For probability variables, extract N samples according to their probability distribution respectively. Then, N group forecasting samples can be obtained. i) Take the output function of the SVM as the explicit approximation of the implicit limit state function. The structural hybrid reliability can be obtained through predicting the response values of samples and Monte Carlo simulation. 4. Example 4.1. Example 1 The limit state equation of a cylindrical pressure vessel for strength failure is M K Ic 0.975 PD a / Q / t 0 where the physical meanings and the statistic features of all variables are listed in Table 1. Table 1 The physical meanings and statistical features of all variables in Example 1. Physical quantity mean value standard deviation Fracture toughness KIc /MPam1/2 Working pressure P /MPa 124 5.88 18.6 0.3933 91 Internal diameter D/mm Wall thickness t/mm Crack depth a/mm 1500±30 5±0.6 3±0.5 9in x 6in b2904-ch13 Take the results of direct Monte Carlo method as the baseline of comparison with the method proposed in this paper. The failure degree is 9.31×10-3 through ten million sampling. Use Markov chain method in this paper and extract 100 training samples. Then, use SVM to make regression training, search optimal parameters and then verify the training accuracy under different number of samples. In this example, radial basis kernel function is used. The penalty factor C and insensitive coefficient ε is +∞ and 0.01, respectively. The relative error of the reliability with σ is shown in Fig.1. After many times parameter testing, the error reaches the minimum value when σ=56. Fix the model parameters and reduce the training samples to verify the accuracy. The variation of the relative error with sample size is shown in Fig. 2. 2 0 relative error(%) -2 -4 * N =100 C=+∞ ε=0.01 -6 -8 -10 -12 -14 -16 10 20 30 40 50 60 kernel function parameterσ Fig. 1 The variation of relative error with σ. 0.10 0.05 σ=56 C=+∞ ε=0.01 0.00 relative error(%) Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. Mechatronics and Manufacturing Technologies (MMT 2016) -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 -0.35 50 60 70 80 sample size N 90 100 * Fig. 2 The variation of relative error with sample size. It is shown in Fig. 1 that the relative error can be controlled lower than 0.1% after parameter adjustment, and the result can keep stable when σ is within a certain range. 92 9in x 6in b2904-ch13 From Fig. 2, the accuracy is improved with the increasing of the training sample size. In addition, when the number of training sample is 50, the relative is -0.28%, which has enough accuracy. So, the excellent properties of SVM for small sample size problem are improved. Moreover, the sample extraction strategy can provide excellent training samples for SVM, and these samples can be well centered around the structural design point or the limit state surface. The excellent training samples play an importance role in the performance and accuracy of SVM. 4.2. Example 2 The limit state function for fracture failure of the turbine disc of a certain type of aircraft engine [12] is M sS C 2 2 2 J 2 where the variables’ physical meanings are as follows σs–strength limit; S–section area; J–moment of inertia of cross section; ρ–mass density; w–angular velocity; C–coefficient. The statistical properties are listed in Table 3. Table 3 The statistical properties of the basic variables. σs /MPa n/(r/min) C mean value 960 12150 5.3682 standard deviation 65.1 134 0.1134 Variable 3 variable ρ/(kg/ m ) mean value 8140 standard deviation 165 2 -3 S/m (10 ) J/m4(10-4) [6.283 2,6.726 4] [1.144 7,1.284 7] 0.00 -0.04 * relative error(%) Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. Mechatronics and Manufacturing Technologies (MMT 2016) N =100 C=+∞ ε =0.01 -0.08 -0.12 -0.16 -0.20 20 25 30 35 40 kernel function parameterσ Fig. 3 The variation of relative error with σ. 93 45 Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 -0.015 σ=44 C=+∞ ε=0.01 Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. relative error(%) -0.020 -0.025 -0.030 -0.035 -0.040 40 50 60 70 80 90 100 * sample size N Fig. 4 The relative errors under different number of samples. The structural failure degree simulated by direct Monte Carlo is 1.3× 10 (ten million times). The methods presented in this paper including sample extraction strategy based on Markov chain and SVM are used to reconstitute the limit state equation. Extract 100 training samples according to the above method, choose radial basis kernel function and take the penalty factor C and insensitive coefficient as +∞ and 0.01, respectively. The variation of the reliability’s relative error with the kernel function parameter σ is shown in Fig. 3. Through many debugging, the relative error reached the minimum value –0.017% when σ=44. Fix the model parameters, and the relative error under different number of samples is shown in Fig. 4. It can be shown in Fig. 3 that the relative error of reliability can keep stable with kernel function parameter σ in a large range. The absolute value of relative error can be controlled lower than 0.1%. Fig. 4 shows that high accuracy can be obtained when the sample size is 40 and the excellent properties of SVM for small sample size problem and the progressiveness of the sample extraction strategy proposed in this paper are improved. The reasons of the high accuracy of the proposed method can be summed as follows: ① SVM technology is based on structural risk minimization principle and has excellent learning ability for small samples and generalization performance. ② The training samples simulated by Markov chain can be well distributed nearing the structural design point and the limit state surface, so there are more information including in these samples about the structural failure or the limit state. ③ The alternative samples and the state samples are used together, so the information generated in sample simulation are fully used. ④ Different from the classification technology, the SVM regression technology applied the information besides the classification mark, i.e. the corresponding function value of these samples, so the information can be used more fully. -4 94 Mechatronics and Manufacturing Technologies (MMT 2016) 9in x 6in b2904-ch13 Mechatronics and Manufacturing Technologies Downloaded from www.worldscientific.com by NATIONAL UNIVERSITY OF SINGAPORE on 10/28/17. For personal use only. 5. Conclusion The limit state equations of complex engineering structures are commonly implicit. In addition, complex structures mostly have many variables with different types of uncertainty due to the lack of history and test information for partial variables. In view of the implicit limit state function in structural hybrid reliability analysis, a method based on Markov chain sample simulation strategy and SVM regression is proposed in this paper. A serial of state points distributed around the design point can be obtained through Markov chain simulation. Together with the alternative samples and the corresponding response values, the training sample set of SVM can be formed. Then, the effective approximation of the limit state equation can be got through SVM training. The structural hybrid reliability can be calculated combining with the Monte Carlo method. Acknowledgments This research is supported by the National Natural Science Foundation of China (51509254) and the Youth Science Funds of Naval University of Engineering (HGDQNJJ13013/HGDQNEQJJ15009). References 1. 2. 3. 4. 5. 6. 7. Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety, 1990, 7(1): 57–66. Deng J, Gu D S, Li S B, et al. Structural reliability analysis for implicit performance functions using artificial neural network. Structural Safety, 2005, 27(1): 25–48. Hurtado J E, Alvarez D A. Classification approach for reliability analysis with stochastic finite element moeling. Journal of Structural Engineering, 2003, 129(8): 1141–1149. Hurtado J E. An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26(3): 271–293. Deng Naiyang, Tian Yingjie. Support Vector Machine–Theory, Algorithm and Expansion. 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