For the conventional computational methods for structural reliability analysis, the common limitations are long computational time, large number of iteration and low accuracy. Thus, a new novel method for structural reliability analysis has been proposed in this paper based on response surface method incorporated with an improved genetic algorithm. The genetic algorithm is first improved from the conventional genetic algorithm. Then, it is used to produce the response surface and the structural reliability is finally computed using the proposed method. The proposed method can be used to compute structural reliability easily whether the limit state function is explicit or implicit. It has been verified by two practical engineering cases that the algorithm is simple, robust, high accuracy and fast computation

Fang, Yongfeng

Tee, Kong Fah

Yongfeng Fang

Kong Fah Tee

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STRUCTURAL ENGINEERING AND MECHANICS

Structural reliability analysis using response surface method with improved genetic algorithm

Greenwich Academic Literature Archive

1. IntroductionThe response surface method (RSM) is a very important method for structural reliability analysis. The main advantages of RSM are because it is not restrained by the number of random variables and it is also not limited by the form of the limit state function whether it is implicit or explicit (Ellingwood and Galambos 1982, Mahmoodian et al. 2012, Jiang et al. 2012, Fang et al. 2014, Tee et al. 2015). In addition, the RSM is simple to perform with high accuracy (Bai et al. 2014, Fang et al. 2013, Su et al. 2015). Nevertheless, if the initial point of choice in the RSM is unreasonable and the objective function is highly non-linear, then the rate of convergence of the RSM is slow. Therefore, the RSM for structural reliability estimation in a wide range of applications has been limited.  An improved RSM has been developed for structural reliability evaluation (Li and Chen 2013). In this method, the rate of convergence of the RSM is accelerated by rotating the coordinate system to compute the structural reliability. However, due to the rotation of the coordinate system in each computation step, the total computational time apparently has been increased when there are many random variables. Another improved RSM for reliability analysis of structures has also been proposed by Kang et al. (2010). Although the precision efficiency of the proposed improved RSM method is higher, its weight function has to be determined in structural reliability estimation. The convergence may not be guaranteed if the weight function Corresponding author, Ph.D. E-mail: fangyf_9707@126.com aPh.D. E-mail: K.F.Tee@gre.ac.uk is not correctly determined. A new quadratic function has been constructed to substitute the RSM, some variables are chosen to compute the structural reliability (Basaga et al. 2012). The rate of convergence is higher, but the proposed method is complex. The precision and accuracy of the proposed method is influenced by the complexity.   The genetic algorithm (GA) has been considered in computer applications since 1960s. Its technological merits are high degree of parallelism, less dependent on the initial value and superior robustness in the computation of extremum (Xuan and Chen 2000, Michhalewicz 1994, Yang et al. 2002). The GA has been used to study structural optimization (Shao et al. 2001, Tee et al. 2014, Khan and Tee 2016). It has been shown that soft computing is a superior method for structural reliability computation, but the study is relatively simple. An improved genetic algorithm has been proposed by He and Liang (2001). The method has accelerated convergence and improved the computational efficiency by changing trends of search points based on the improved gradient. The structural optimization has been studied by using cuckoo search algorithm and GA (Ponnambalam 2014). It has been shown that the proposed approach is superior when it is applied to discrete variables. Based on the above literature review, the RSM for structural reliability computation is studied in this paper using the improved gradient of GA. The method is suitable to be incorporated with the response surface function for both explicit and implicit form. It has been verified by two examples that the proposed method can be used to compute structural reliability with fast convergence and high computational efficiency. 2. Response Surface Method (RSM)Structural reliability analysis using response surface method with improved genetic algorithm Yongfeng Fang1 and Kong Fah Tee2a 1School of Mechanical Engineering, Guizhou University of Science Engineering, Bijie 551700, China 2Department of Engineering Science, University of Greenwich, Kent ME4 4TB, United Kingdom Abstract.  For the conventional computational methods for structural reliability analysis, the common limitations are long computational time, large number of iteration and low accuracy. Thus, a new novel method for structural reliability analysis has been proposed in this paper based on response surface method incorporated with an improved genetic algorithm. The genetic algorithm is first improved from the conventional genetic algorithm. Then, it is used to produce the response surface and the structural reliability is finally computed using the proposed method. The proposed method can be used to compute structural reliability easily whether the limit state function is explicit or implicit. It has been verified by two practical engineering cases that the algorithm is simple, robust, high accuracy and fast computation. Keywords:  structural reliability; response surface method; improved genetic algorithm  The structural limit state function (LSF) G(X)=g(x1,x2,…,xn) in the RSM can be formulated using a quadratic polynomial function as follows.  21 1( )in ni i ii iZ g a b x c x     x    (1) where xi (i=1,2,…,n)is space variable of the LSF and a, bi, ci (i=1,2,…,n) are the coefficients of Eq. (1). The computation procedure of structural reliability using the RSM is given as follows. The mean point of the random variables is chosen as the initial point 1 1 1 11 2[ , , , ]nX x x x1 21 1 1( , , , )ng x x x and 1 21 1 1 1( , , , , , )i i ng x x x f x will be determined, f is assigned to 3 in the first step computation and 1 in the next step. The 2n+1 value will be obtained and the value will be used in the coefficients of the Eq. (1) to obtain the RSM. The reliability index βk and its design testpoint kDX  of the structure will be determined by using the first order second moment method (Fang et al. 2015) based on the response surface function (k is the number of the iteration). If Eq. (2) can be established, then it is possible to evaluate structural reliability.1k k     (2) where βk is the reliability index of the structure. On theother hand, if Eq. (2) cannot be established, then the new design testing point kMX  will be obtained as follows. ( )( )( ) ( )kk k k kM D k kDg XX X X Xg X g X     (3) where Xk is the initial point where the response surfacefunction can be computed by kth iteration. kDX  and kMXare the design testing point and the interpolation point at the kth response surface function, respectively. g(Xk) and( )kDg X  are the values of the LSF which are corresponding to Xk andkDX , respectively. A huge amount of computational time and bigger error will be produced by using the above method if the LSF is a non-linear performance function. In addition, when the higher accuracy is required, the number of interpolation and the order of the approximation function will be increased. It has been shown that the computation becomes more difficult and complex. Thus, a new method described in Section 3 has been proposed to overcome the problem. 3. Improved GA by fitness functionIn structural reliability estimation, the LSF can be transformed into the fitness function (FF) as follows.  otherwise   0)(       )( maxmax cXGifXGcF(X） (4)where cmax is a constant which is the maximum value of G(X) in the evolutionary process. If the LSF is explicit, differentiable and continuous function, the new FF can be established by using the gradient of the LSF for accelerating the convergence in the evolutionary process. The new FF is given as follows. )()()()()1()()()()(minmaxminminmaxmin'ZFYFZFXFZFYFZFXF(X）F(5) where α[0,1] is power factor and it is determined according to one’s experience. Fmax(Y) and Fmin(Z) are the maximum value of the individual fitness in the current generation and the minimum value of the individual fitness in the previous generation, F(X) is the gradient of the LSF, Fmax(Y) and Fmin(Z) can be determined by using Eq. (6) and Eq. (7), respectively, ||•|| is 2-norm. Eq. (6) is given as follows 321311211111max)(,)(,)(min,,)(,)(,)(min)(nnnnnnyyFyyFyyFyyFyyFyyFYF(6) where Y=[y1,…y2],  yi1 (i=1，2，3) is ith component of y1.If the structure is plane structure, then the third component can be omitted. Eq. (7) is given as follows 321311211111min)(,)(,)(min,,)(,)(,)(min)(nnnnnnzzFzzFzzFzzFzzFzzFZF   (7) where Z=[z1,…z2], zi1 (i=1，2，3) is ith component of z1.On the other hand, if the LSF is the implicit function, the new FF is shown in Eq. (4). However, the gradient is difficult to be determined. Therefore, the gradient is substituted by one order difference as follows.)()()( ab XFXFXF   (8) )}()(,),()(max{)(11maxanbnabyFyFyFyFYF(9) )}()(,),()({min)(11minanbnabzFzFzFzFZF  (10) where •b is the paternal chromosome and •a is the progenychromosome. The procedure for the proposed method with incorporation of improved fitness GA into the RSM for structural reliability estimation is given as follows.  Table 1 Comparison of results for Example 1. RSM-MLS RSM IGA-RSM The structural reliability index 2.7100 2.7112 2.7111 Error 0.04 0.004 The largest failure point *1u -2.5411 -2.5725 -2.5722 *2u 0.9417 0.8562 0.8958 Computational time (s) 50 39 10 Step 1. The LSF is determined. Step 2. The n+1 design testing points are selected in the variable space, while f=3 (Li and Chen 2013).  Step 3. The coefficients of Eq. (1) are computed. Step 4. The maximum failure point of the response surface function g(x) is computed by using the first order second moment method. Step 5. The maximum failure point is given as X, the new response surface function is established by using Eq. (1) and Eq. (3) whereas the new FF is established by using Eq. (4).  Step 6. Fmax(Y) and Fmin(Z) are computed, if 'minmax )(-)(  ZFYF (11) Then X is located as the new central point. The 2n+1 design testing points are selected in the influence domain of the point while f=1 (Li and Chen 2013). Step 7. The 2n+1 design testing points are substituted into Eq. (1). The coefficients of the second order response surface function g(x) are determined. Step 8. The new surface function is determined. The maximum failure point kDX  and the structural reliability index βk are computed by using the first order secondmoment method, where k is the number of the iteration. Step 9. The Step 5 to Step 8 are repeated until Eq. (12) is satisfied. 11kDkDkDXXX (12)where ε is the required accuracy. 4. ExamplesExample 1. The example in Rajashekhar and Ellingwood (1993) is used in this paper to verify the proposed method. The LSF is given as follows. 200)53.0exp(]2.6)2(4.0exp[)(21uug uwhere u1 and u2 are two independent random variables which are considered to obey the standard normal distribution.  Fig. 1 10-bar truss structure Table 2 Comparison of results for Example 2 RSM-MLS RSM IGA-RSM Reliability index 4.8083 4.8089 4.8085 The largest failure points 7.6658, 9.9949 7.6636, 9.9949 7.6651, 9.9949 9.7302, 10.0075 9.7362, 10.0075 9.7300, 10.0075 10.0350, 9.9949 10.0346, 9.9949 10.0354, 9.9949 9.5916, 10.2997 9.5986, 10.2946 9.5928, 10.5981 10.0212, 9.9855 10.0209, 9.9857 10.0212, 909855 Computational time (s) 300 240 109 The comparison of the results based on RSM using moving least squares (RSM-MLS) (Kang et al. 2010), classical RSM and improved GA for RSM (IGA-RSM) is shown in Table 1. The classical RSM is used as a benchmark method for comparison. It is shown from Table 1 that the computed result obtained by IGA-RSM is more accurate than RSM-MLS, while the number of iteration for IGA-RSM is the lowest among the 3 methods. The size of chromosome population is 10, cross rate is 0.3 and variation rate is 0.1. Table 1 gives the method. It is clear that the proposed method is superior in the calculation of structural reliability.  Example 2. A ten-bar truss structure is shown in Figure 1. It is widely used as an example to illustrate structuraloptimization design and reliability estimation. Its LSF is an implicit function with random variables as shown in Eq. (13).  ( ) ( )ag   A A   (13) where Ai~N(64.52,1.27)(cm2) and σa=172.4 MPa.Similarly, the classical RSM is used as a benchmark method for comparison in this example. It is shown from Table 2 that the reliability index and the largest failure point obtained by IGA-RSM is more accurate than RSM-MLS, while the numbers of iteration of IGA-RSM, classical RSM and RSM-MLS are 34, 84 and 90, respectively. It is clear that the number iteration of IGA-RAM method is the lowest. It can be summarized that the IAG-RSM is superior in structural reliability estimation.  5. ConclusionsIn this paper, the improved fitness function of response surface function is established by using the improved GA with explicit and implicit limit state functions. The structural reliability index and the largest failure point can be determined using the improved GA to produce the new response surface function based on the required accuracy. It has been shown by two examples that the proposed method is high precision and needs fewer iterations. The method can be easily accessed in the traditional GA. It has been verified that the algorithm is simple, robust and fast computation. 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Structural reliability analysis using response surface method with improved genetic algorithm

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