Mixed Efficient Global Optimization for Time- Dependent Reliability Analysis

Transcription

1 Journal of Mechanical Deign Volume 137 Iue 5 Mixed Efficient Global Optimization for Time- Dependent Reliability Analyi Zhen Hu and Xiaoping Du 1 Department of Mechanical and Aeropace Engineering Miouri Univerity of Science and Technology Wet 13th Street, Toomey Hall 272,Rolla, MO 65409, U.S.A.; Tel: ; dux@mt.edu 1

2 AUTHORS INFORMATION: Zhen Hu, M.S. Reearch Aitant Department of Mechanical and Aeropace Engineering Miouri Univerity of Science and Technology 290D Toomey Hall 400 Wet 13th Street Rolla, MO Xiaoping Du, Ph.D. Profeor Department of Mechanical and Aeropace Engineering Miouri Univerity of Science and Technology 272 Toomey Hall 400 Wet 13th Street Rolla, MO (voice) (fax) 2

3 Abtract Time-dependent reliability analyi require the ue of the extreme value of a repone. The extreme value function i uually highly nonlinear, and traditional reliability method, uch a the Firt Order Reliability method, may produce large error. The olution to thi problem i uing a urrogate model for the extreme repone. The objective of thi work i to improve the efficiency of building uch a urrogate model. A mixed efficient global optimization (m-ego) method i propoed. Different from the current EGO method which draw ample of random variable and time independently, the m-ego method draw ample for the two type of ample imultaneouly. The m- EGO method employ the Adaptive Kriging - Monte Carlo Simulation (AK-MCS) o that high accuracy i alo achieved. Then Monte Carlo imulation i applied to calculate the time-dependent reliability baed on the urrogate model. Good accuracy and efficiency of the m-ego method are demontrated by three example. Keyword: Kriging, Reliability analyi, Mixed EGO, Time-dependent 3

4 1 Introduction Reliability i defined within a period of time when a limit-tate function involve time. For thi cae, time-independent reliability analyi methodologie [1, 2] are not applicable, and time-dependent reliability method hould be ued. Even though other method [3-5] exit for time-dependent reliability problem, the mot widely ued method are firt paage method and extreme value method. The former method are eaier to implement and are therefore more popular, but may not be a accurate a the latter method. The two type of method are briefly reviewed below. The firt-paage method calculate the probability that the repone exceed it failure threhold (limit tate) for the firt time in a predefined period of time. The event that the repone reache it limit tate i called an upcroing, and the upcroing rate i the rate of change in the upcroing probability with repect to time. If the firt-time upcroing rate i available, the time-dependent probability of failure can be eaily computed. But it i difficult to obtain the firt-time upcroing rate. For thi reaon, approximation method are widely ued. The mot commonly ued method i the Rice formula [6], which ue upcroing rate throughout the entire period of time with the aumption that all the upcroing are independent. Many method have been developed baed on the Rice formula. For intance, an aymptotic outcroing rate for tationary Gauian procee wa derived by Lindgren [7] and Breitung [8, 9]. The bound of the upcroing rate of a non-tationary Gauian proce were given by Ditleven [10]. To olve general time-dependent reliability problem, Hagen and Tvedt [11, 12] propoed a parallel ytem approach. A PHI2 method wa developed by Sudret [13]. Hu and Du alo developed a time-dependent 4

5 reliability analyi method baed on the Rice formula [14]. Even if ome modification have been made [15-18], the upcroing method may till produce large error when upcroing are trongly dependent. The extreme value method approximate the time-dependent reliability from another apect by uing the extreme value of the repone with repect to time. If the ditribution of the extreme value can be accurately etimated, the accuracy of the reliability analyi will be higher than the upcroing rate method ince the independent upcroing aumption i eliminated. Accurately and efficiently etimating the ditribution of the extreme value, however, i a challenge ince global optimization with repect to time hould be performed repeatedly. In general, the extreme value of the repone i much more nonlinear than the repone itelf with repect to the input random variable. For ome problem, the ditribution of the extreme repone i multimodal with different mode (peak of probability denity) even though the repone itelf follow a unimodal ditribution [19]. For thi reaon, uing Deign of Experiment (DOE) to obtain a urrogate model of the extreme repone become promiing and practical. For example, Wang and Wang [20] propoed an extreme repone method uing the Efficient Global Optimization (EGO) approach [21]; Chen and Li [22] tudied how to evaluate the ditribution of the extreme repone uing the probability denity evolution method [22]. The efficiency of the exiting extreme value method with DOE, uch a the approach reported in [20], can be improved. To how the feaibility of the improvement, we firt define the repone function a Y = g( X, t), where X = [ X1, X2,, X n ] i a vector of random variable, t i time, and Y i a repone. In many application, the repone 5

6 function may be a black box, uch a CAE (computer-aided engineering) model. To obtain the extreme value of Y, current method draw ample of X firt. Then at each ample point of X, ample of t are drawn through EGO [21], which produce the extreme repone with repect to time at each ample point of X. Thu, the value of the extreme repone are available at all the ample point of X, and a urrogate model of the extreme repone i then built. Sampling on X and t i performed at two neted and independent level, and we therefore call the method the independent EGO method. The interaction effect of X and t are not conidered at the two eparate ampling level. The efficiency could be improved if X and t are ampled imultaneouly. Thi motivated u to develop a new method with higher efficiency. Thi work develop a new time-dependent reliability method baed on EGO and the active learning trategy [23]. The new method i named the mixed-ego method ince X and t are ampled imultaneouly. The ignificance of thi work conit of the following element: A mixed EGO method: the new method generate ample of X and t imultaneouly o that the interaction effect of X and t i conidered. Thi require fewer training point for building urrogate model and therefore increae the efficiency of the EGO method. The integration of the mixed-ego method with the Adaptive-Kriging Monte Carlo imulation (AK-MCS) [23]. The integration make the urrogate model of the extreme repone accurate near or at the limit tate and hence improve both accuracy and efficiency of the time-dependent reliability analyi. 6

7 The remainder of thi paper tart from Section 2 where EGO and time-dependent reliability are reviewed. The new method i dicued in Section 3. Three example are preented in Section 4, and concluion are given in Section 5. 2 Background EGO i ued in thi work. We at firt review EGO and dicu the definition of timedependent reliability. We then review the current independent EGO method for timedependent reliability analyi. 2.1 Efficient Global Optimization (EGO) Since being propoed by Jone in 1998 [21], EGO ha been widely ued in variou area [26-29]. EGO i baed on the DACE model [30] or the Kriging model. Both of the model are updated by adding training point gradually, but they ue different criteria for model updating. The EGO model i updated with a new training point that imize the expected improvement function (EIF) while the DACE model i updated with a new training point that minimize the mean quare error. A imum EIF help find a point with the highet probability to produce a better extreme value of the repone. Many tudie have demontrated that EGO can ignificantly reduce the number of training point for global optimization. EGO at firt contruct a Kriging model uing initial training point, and the expected improvement (EI) i calculated uing the mean and covariance of the Kriging model. The model i then updated by adding a new point with the imum EI. The procedure continue until convergence i achieved. The Kriging model gˆ( x ) i given by 7

8 yˆ = gˆ( x) = hx ( ) T β + Z( x) (1) in which h() i called the trend of the model, β i the vector of the trend coefficient, and Z() i a tationary Gauian proce with a mean of zero and a covariance given by Cov Z a Z b = σ R ab (2) 2 [ ( ), ( )] Z (, ) 2 where σ Z i the variance of the proce, and R(, ab ) i the correlation function. The commonly ued correlation function include the quared-exponential and Gauian type [30]. At a training point x, ŷ i a Gauian random variable denoted by 2 yˆ gˆ( x)~ N( µ ( x), σ ( x)) = (3) in which N(, ) tand for a normal ditribution; µ () and σ () are the mean and tandard deviation of ŷ, repectively. At a training point x, µ ( x) = g( x) and σ ( x) = 0, and gˆ( x ) therefore pae all the point that have been ampled. For the global imum of g( x ), the improvement i defined by I y y * = (, 0), where * y i the current bet olution (the imum repone) obtained from all the ampled training point. It expectation or EI i then computed by [21] * * * µ ( x) y µ ( x) y EI( x) = ( µ ( x) y ) Φ + σ( x) φ σ( x) σ( x) (4) where Φ ( ) and φ ( ) are the cumulative ditribution function (CDF) and probability denity function (PDF) of a tandard Gauian variable, repectively, and * y i y = { g( x )} * () i i= 1, 2,, k (5) in which k i the number of current training point. 8

9 By imizing EI, we find a new training point. ( k) 1) x = arg EI( x ) (6) xîx Algorithm 1 below decribe the procedure of EGO. More detail can be found in Ref. [21] and [30]. Place Table 1 here In Step 3, e EI (a mall poitive number) i ued a a convergence criterion. The imum EI i caled in Line 7 a uggeted in [19]. 2.2 Time-dependent reliability For a general limit-tate function Y = g( X, t), a failure occur if in which e i the failure threhold. Y = g( X, t) ³ e (7) For a time interval [ t0, t ], the time-dependent reliability i defined by [5] { X t) < t t t } Rt (, t) = Pr Y= g(, e, [, ] (8) 0 0 where Pr{} tand for a probability, and t [ t0, t ] mean all time intant on [ t0, t ]. The time-dependent probability of failure i defined by where tand for there exit. { t) t, } p ( t, t ) = Pr Y = g( X, e, t [ t ] (9) f 0 0 pf ( t0, t ) i a non-decreaing function of the length of 0 [ t, t ]. The longer i the period of time, the higher i pf ( t0, t ) in general. 9

10 2.3 Time-dependent reliability analyi with urrogate model The failure event in Eq. (7) i equivalent to Y imum repone on [ t0, t ] and i given by > e, where Y i the global Y = arg ma x{ g( X, t) } (10) tî[ t0, t ] Then pf ( t0, t ) i rewritten a 0 { X } p ( t, t ) = Pr Y ( ) > e (11) f For many problem, Y i highly nonlinear with repect to X and may follow a multimodal ditribution. Uing exiting reliability method, uch a the Firt and Second Order Reliability Method (FORM and SORM), may reult in large error. Monte Carlo imulation become a choice if a urrogate model, Y ˆ = g ( X), of Y, can be built. A dicued previouly, the direct EGO method i employed to olve time-dependent reliability problem by Wang and Wang [20]. The urrogate model of extreme repone Y = gˆ ( X) wa built with a neted procedure. The outer loop generate ample of X while the inner loop i executed to find the time t when the repone i imum. Sample of t are generated by EGO in the inner loop. A more direct and general independent EGO procedure imilar to the aforementioned neted procedure [20] i ummarized below. Outer loop: Sampling on X for building Y ˆ = g ( X). Inner loop: EGO for yma x = { g( x, t) } at x, which i a ample of X. tî[ t0, t ] The aociated algorithm or Algorithm 2 i hown a follow. 10

11 Place Table 2 here In Step 3, e MSE i a mall poitive number ued a the convergence criterion for the mean quare error MSE. The independent EGO method may not be efficient for two reaon. Firt, the onedimenional EGO with repect to t i performed repeatedly at each ample point of X. A mentioned previouly, X and t are treated independently at two eparate level, and the interaction of X and t i therefore ignored. Not conidering the interaction effect of X and t may reult in low computational efficiency. Second, a mall MSE i expected for an accurate urrogate model for reliability analyi. Contructing a urrogate model with a low MSE, however, i computationally expenive becaue Y i in general highly nonlinear and poibly multimodal. 3 A mixed-ego baed method In thi ection, we dicu the mixed-ego baed method that overcome the drawback of the independent EGO method. The new method build a urrogate model Y = gˆ ( X) for the global extreme repone through another urrogate model Y = gˆ( X, t). It i efficient becaue of the following reaon: Uing Y = gˆ( X, t) can the effectively account for the joint effect of X and t and will reduce the number of ample of both X and t. 11

12 The mixed EGO along with the AK-MCS method [23] can efficiently and accurately approximate the extreme repone at or near the limit tate. High accuracy at or near the limit tate will reduce the number of ample of X. 3.1 Overview A dicued in [19], the accuracy of reliability analyi i determined by only the accuracy of the urrogate model at the limit tate or Y = g ( X ) = e. For thi reaon, we focu on achieving high accuracy of Y ˆ = g ( X) at or near the limit tate. By doing o, the number of ample can be reduced. Since the limit-tate Y = g ( X ) = e i of the greatet concern, the ample updating criterion need to be modified. In thi work, we integrate the AK-MCS method [23] with the propoed mixed-ego method. The overall procedure of the mixed-ego baed method i provided in Table 3, and the detailed algorithm will be dicued in Subection 3.3 and 3.4 and will be ummarized in Section Place Table 3 here The major difference between the independent EGO method and the mixed-ego method i that X and t are ampled at two eparate level in the former method while X and t are ampled imultaneouly in the latter method. 3.2 Initial ampling The initial ample x and t are generated to create an initial urrogate model for Y. The commonly ued ampling approache include the Random Sampling (RS), Latin 12

13 Hypercube Sampling (LHS), and Hammerley Sampling (HS) [31]. In thi work, the HS method i ued a it perform better in providing uniformity propertie over a multidimenional pace [32]. Sample are generated by the HS method in the [0, 1] domain. They are then tranformed into ample of X and t according to their probability ditribution uing the invere probability method. t i treated a if it wa uniformly ditributed. Suppoe that the dimenion of X i n and that k initial ample are generated. The ample x are é (1) (1) (1) x1 x2 x ù n (2) (2) (2) (1) (2) ( k) x1 x2 x é n ; ; ; ù x ; êë x x x úû ; ( k) ( k) ( k) x1 x2 x ëê n ûú (12) in which x = [ x, x,, x n ] i the i-th ample point. () i () i () i () i 1 2 k initial ample of t are alo generated along with thoe of X. We then have the following combined initial ample. é (1) (1) (1) (1) x1 x2 xn, t ù (2) (2) (2) (2) x1 x2 xn, t [, ] x t = ( k) ( k) ( k) ( k) x1 x2 xn, t êë úû (13) We then call the limit-tate function to obtain repone at the above ample point and build a mixed EGO model Y = gˆ( X,) t with repect to X and t. Y = gˆ( X,) t i called a mixed model becaue it i a function of X and t. Then, the extreme value repone y at x are identified baed on the mixed EGO model that will be dicued in the following ection. 13

14 3.3 Contruct initial Y ˆ = g ( X) with the mixed EGO model Thi i Step 2 of the mixed-ego method in Table 3. With t, the EI in Eq. (4) i rewritten a () i * () i * () i () i * µ ( x, t) y () i i µ ( x, t) y i EI( x, t) = ( µ ( x, t) yi ) Φ ( x, t) () i + σ φ () i σ( x, t) σ( x, t) (14) where * y i i the current bet olution (imum repone), and () i µ ( x, t) and σ () i ( x, t) () i are the mean and tandard deviation at [ x, t], repectively. The expreion of EI are the ame a thoe for the independent EGO method and the mixed EGO model. The difference lie in the way of computing () i () i µ ( x, t) and σ ( x, t). For the independent EGO method, () i () i µ ( x, t) and σ ( x, t) are obtained from the onedimenional Kriging model Y= gt ˆ( ), which i contructed in the inner loop for t when X i fixed. For the mixed EGO model, they are computed from the Kriging model Y = gˆ( X, t), which i contructed when X and t vary imultaneouly. Once convergence i reached, the imum repone with repect to x will be available. Then the initial model Y ˆ = g ( X) can be built. The algorithm (Algorithm 3) for the initial Y ˆ = g ( X) i given a follow. Place Table 4 here In Line 2, x contain initial ample ued to contruct Y = gˆ ( X), and x t contain x and added ample of X for model ˆ(, ) Y = g X t. In Line 3, e EI i ued a a 14

15 convergence criterion for the imum EI. In Line 5, x t i computed by plugging () i EI(, ) () () i y i, µ Y ( x, t) and µ, which are obtained from Y = gˆ( X, t), into Eq. (14). () i Y ( x, t) Note that in the mixed EGO model, all the ampled point of both X and t are ued to identify the new training point of t. But in the independent EGO model, only the ampled point of t are ued to update training point of t. From the output of the mixed EGO model, we obtain the extreme value y correponding to the ample () i x, i 1, 2,, = k. In the following ection, we dicu how to identify a new training point ( k 1) x + and the aociated g x +. ( ( k 1) ) 3.4 Update Y ˆ = g ( X) with AK-MCS and the mixed EGO model The initial model of the extreme repone Y = gˆ ( X) obtained above in general i not accurate. New training point of X hould be added. A dicued in Sec. 3.1, it i deirable to generate more training point near or at the limit tate. Several approache are propoed for thi purpoe. For intance, the efficient global reliability analyi (EGRA) method [33] generate more training point adaptively near the limit tate. Baed on EGRA, an active learning approach called the AK-MCS method [23] i developed to further ue the joint probability denity of random variable for generating training point without uing global optimization. Uing the principle of AK-MCS, Wang and Wang later propoed a confidence enhanced equential ampling approach [34]. Dubourg and Sudret integrated the importance ampling approach with the AK-MCS method [35] and further improved the efficiency. All of the above approache are baed on the Kriging model. Approache baed on other urrogate model technique are alo available. For example, upport vector machine are ued to generate explicit limit-tate boundarie [36], 15

16 and the ame technique i alo applied to identify dijoint failure domain and limit tate boundarie for dicontinuou repone [37]. The two method are further improved by Baudhar and Mioum [38]. In thi work, the AK-MCS method i employed to identify new training point x ( k + 1) near the limit tate of the extreme repone Y = gˆ ( X). It ha two advantage over the EGRA method: the joint probability denity of random variable i conidered during the ampling proce, and it avoid global optimization in earching for new training point. A brief review of the AK-MCS method i given in Appendix A. Note that other ampling method mentioned above can be ued a well. With the new training point ( k 1) x + identified from AK-MCS, we can find the extreme repone g x + to update the urrogate model for Y. Obtaining ( ( k 1) ) g x + i ( ( k 1) ) equivalent to olving the following one dimenional global optimization problem: t = arg { y = g( x, t)} (15) ( k + 1) ( k + 1) t [ t0, t ] To reduce the number of function call, we till ue the mixed EGO model preented in the lat ubection, and we alo ue the data et of [ x, t ] and t y obtained in Sec Algorithm 4 below how the detail of finding Place Table 5 here ( k 1) x + and g x +. ( ( k 1) ) 16

17 In Line 13, EI( new new new new x, t) i computed by plugging y, µ ( x, t), and σ ( x, t), which are obtained from Y = gˆ( X, t), into Eq. (14). When the convergence criterion i atified, we obtain the urrogate model Y = gˆ ( X). We then ue MCS to calculate reliability. A Y = gˆ ( X) i accurate, o i the reliability calculated by MCS with a ufficiently large ample ize. Note that MCS will no longer call the original limit-tate tate function. We now have all the algorithm for the new method. Next we put everything together and give the complete algorithm. 3.5 Summary of the mixed EGO-baed method Combining Algorithm 3 and 4 yield the complete algorithm of the mixed EGO baed method, or Algorithm 5, given below. Place Table 6 here 4 Numerical example In thi ection, three numerical example are ued to demontrate the effectivene of the propoed approach. Each of the example i olved uing the following four method. Rice: The outcroing rate method baed on the Rice formula and Firt Order Reliability Method (FORM) [14, 39]. Independent EGO: The independent EGO method or the neted EGO [20] Mixed EGO: The propoed mixed EGO-baed method MCS: The direct MCS method uing the original limit-tate function 17 Y Y

18 The other three method are ued to compare the accuracy and efficiency of the mixed EGO method. 4.1 A nonlinear mathematical model A function of X and t i given in Eq. (16), where X i a random variable following a normal ditribution 2 X ~ N (10, 0.5 ). 1 yxt (, ) = in(2.5 X) co( t) 0.4) 2 X ) 4 2 (16) The time-dependent probability of failure i given by p( t, t) = Pr{ yx (, r) > 0.014, $ rî [1, 2.5]} (17) f 0 According to Eq. (8), p ( t0, t ) i equivalent to the following probability: f p ( t, t ) = Pr{ Y > 0.014} (18) f 0 Before calculating reliability, we at firt evaluate the mixed EGO model (or Algorithm 3) becaue it i the core component of the mixed EGO baed method. We generate different number of initial ample of X and t. We then identify repect to y with x uing the exiting independent EGO method and the mixed EGO method, repectively. The convergence criterion of the two method i e EI -5 = 10. The number of ample of X are et to 10, 15, 18, and 20. The number of function evaluation (NOF) required for identifying y for different number of initial ample of X are given in Table 7. The reult how that the independent EGO call the limit-tate function 127 time to identify the extreme value when the ample ize i 15 while the mixed EGO method only need 59 function evaluation. The reult for ample ize of 10, 18, and 20 18

19 are imilar. Fig. 1 how the value of Y (i.e. y ) obtained from the two method, a well a the true Y, for the ample ize of ten. Place Table 7 here Place Figure 1 here The reult how that both model are accurate to extract the extreme repone. The number of function evaluation by the mixed EGO model i le than that by the independent EGO method. The former i therefore more efficient. Thi become more apparent when the number of ample of X become larger. We now examine the performance of the mixed EGO baed method for the timedependent reliability analyi. We ue the MCS olution a a benchmark for accuracy comparion. The percentage of error i computed by MCS pf - pf e% = 100% (19) MCS p f where MCS p f i from MCS that call the original limit-tate function, and p f i from a non-mcs method. The reult of reliability analyi are hown in Table 8. Place Table 8 here 19

20 The reult how that the accuracy and efficiency of the mixed EGO baed method are much better than the outcroing rate method (Rice formula) and the independent EGO method. 4.2 A vibration problem A vibration problem a hown in Fig. 2 i modified from Ref. [40] by treating the tiffne of pring k 2, damping coefficient c 2, ma m 2, the tiffne of pring k 1, and ma m 1 a random variable. There are totally five random variable. The random variable are given in Table 9. Place Table 9 here Place Figure 2 here The amplitude of the vibration of ma m 1 ubjected to force f in( t) 0 W i given by q æ c W )( k -m W ) ö = ç ø f èç c2w( k1-m1w-m2w ) )( k2m2w-( k1-m1w)( k2-m2w)) 1/2 (20) where W i the excitation frequency, which i conidered a time, or t =W. Eq. (20) can be nondimenionalized uing a tatic deflection of the main ytem. The non-dimenional diplacement of m 1 i given by [40] ( ( 2 )) 1/2 Y = g( X, W= ) k K / K ) K (21)

21 where X = [ k1, m1], and K i, i = 1, 2, 3, are given by ( 2 2 ( 2 ) 2 ) K = c W ) k -m W (22) K = c W( k -mw-m W ) (23) ( 2 ( 2 )( 2 )) K = kmw-k-mw k -mw (24) Y i conidered over a wide excitation frequency band, 8 W 28 (rad/). Since W i treated a t, the period of time i [8, 28] rad/. A failure i defined a the event when Y i larger than 35. The probability of failure on [8, 28] rad/ i given by p (8, 28) = Pr{ g( X, W ) > 35, $W Î[8, 28]} (25) f Fig. 3 how one repone of Y at the mean of random variable. It i highly nonlinear. Place Figure 3 here We ue the independent EGO and the mixed EGO baed method to calculate the time-dependent probability of failure. Table 10 how the reult from different method. Note that the Rice formula baed method i not applicable for thi example a the repone i highly nonlinear. The reult how that the propoed mixed-ego method i much more efficient than the independent EGO method. It hould be noted that the vibration problem i employed a an example to verify the ability of the propoed method in olving highly nonlinear problem. Since the limit-tate function i treated a a black box, the propoed method can be applied to other vibratory problem a well. 21

22 Place Table 10 here 4.3 A beam ubjected to time-variant loading A corroded beam ubjected to tochatic load a hown in Fig. 4 i ued a the third example. Thi example i modified from [41]. Place Figure 4 here A failure occur when the tre of the beam i larger than the ultimate trength of the material. The time-dependent probability of failure i given by in which p = Pr{ g( X, Y( t), t) > 0, $Î t [0, 35]} (26) f 2 ( ) ( )( ) 2 t g( X, Y ( t), t) = F( t) L /4) r a b L /8 - a -2kt b - 2 kt /4 (27) u where X = [ u, a0, b0], Yt () = [ Ft ()], r = 4 t N, k = m / year, and L = 5m. Here, Ft () i a tochatic loading preented by the pectral repreentation method [42] a follow: where 7 æ 7 ö F( t) = 6500 ) x i ( aij in( bt ij cij )) ç ) i= 1 çè j= 1 ø å å (28) 22

23 é ù a = ú ëê úû é ù b = ú ëê úû é ù c = êë úû There are totally ten random variable, which are defined in Table 11. Place Table 11 here The time-dependent probability of failure over [0, 35] year i calculated by aforementioned four method, and the reult are given in Table

24 Place Table 12 here The reult how that the mixed EGO-baed method i much more efficient and accurate than the independent EGO method and the Rice formula baed method. 5 Concluion The ditribution of the extreme value of a time-dependent limit-tate function i required to evaluate the reliability defined within a period of time. The extreme value may be highly nonlinear with a multimodal ditribution with repect to random input variable. For thi reaon, exiting approximation method, uch a FORM, SORM, and the upcroing method, may produce large error. Uing Monte Carlo imulation baed on the urrogate model of the extreme repone become more practical. Thi work develop a new reliability method that can efficiently and accurately contruct urrogate model of extreme repone. The Efficient Global Optimization (EGO) i employed, and the ample point of input random variable and time are imultaneouly generated. With thi treatment, the new method i much more efficient than the exiting method where the two et of ample are generated independently in two neted loop. The urrogate model from the new method i accurate near or at the limit tate, and it accuracy in other area i not important for the reliability aement. Thi i another reaon for the high efficiency. After the urrogate model i available, the 24

25 reliability can then be eaily etimated by Monte Carlo imulation, which will not call the original limit-tate function any more. A indicated in Sec. 4.2, where t i the frequency, intead of a time factor, the propoed method can be ued for limit-tate function with random input variable X and a general interval variable t. The latter may not necearily be a time factor. The reliability produced by the propoed method become the wort-cae reliability with repect to the interval variable. The new method i baed on the Kriging model, and during the ampling and model updating proce, the Kriging model i called repeatedly. The computational cot of calling the Kriging model i minor or moderate compared to that of calling a limit-tate function whoe evaluation may be computationally expenive. Acknowledgement Thi material i baed upon work upported by the National Science Foundation through grant CMMI The upport from the Intelligent Sytem Center (ISC) at the Miouri Univerity of Science and Technology i alo acknowledged. Appendix A. Review of the AK-MCS method The AK-MCS method [23] i developed baed on the EGRA method [33]. In the AK- MCS method, the Kriging model i combined with MCS to adaptively update training point near or at the limit tate. A Kriging model G = f ˆ( x ) i contructed with initial 25

26 training point { xg, }. In order to identify a potential dangerou point (i.e. the new training point), which may caue the change of the ign of the repone variable, one can ue a learning function defined by [23] MCS U ( x ) = MCS µ ( x ) G MCS σ ( x ) G (A1) where MCS x i a group of ample drawn from the ditribution of random input variable, MCS µ ( x ) i the prediction form the Kriging model G = f ˆ( x ), and 2 MCS σ ( x ) i the G variance of the prediction. Note that the population of MCS ample i ued to conider the joint probability denity information of random variable and avoid the complicated global optimization ued in the EGRA method. MCS Then, a new training point i identified by minimizing U ( x ). A topping criterion G ε i defined a min{ U ( x )} ε [23]. After the convergence i achieved, the probability U U of failure p f i etimated by plugging current available ample x total into the urrogate total total MCS model, where x = [ x ; x ] i ued to tore total ample for the current iteration. Then the coefficient of variation of probability of failure where N MCS i the total number of ample in Cov pf i checked by Cov = (1 p ) ( p N ) (A2) pf f f MCS total x. If Cov pf > 0.05, continue above procedure. Otherwie, top and obtain the approximated p f. The general algorithm of the AK-MIS ummarized below. Place Table 13 here 26

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30 Lit of Table Caption Table 1 Detailed procedure of algorithm 1 Table 2 Detailed procedure of algorithm 2 Table 3 Major Procedure of the mixed-ego baed method Table 4 Detailed procedure of algorithm 3 Table 5 Detailed procedure of algorithm 4 Table 6 Detailed procedure of algorithm 5 Table 7 NOF required for different number of ample of X Table 8 Reult of example 1 Table 9 Variable and parameter of Example 2 Table 10 Variable and parameter of Example 2 Table 11 Random variable of Example 3 Table 12 Reult of Example 3 Table 13 Detailed procedure of algorithm 6 Lit of Figure Caption Figure 1 Figure 2 Figure 3 Figure 4 Y from independent EGO and mixed EGO and the true value A vibration problem One repone Y at the mean value point of random variable Corroded beam ubjected to tochatic loading 30

31 Table 1 Detailed procedure of algorithm 1 Algorithm 1 Efficient Global Optimization (EGO) (1) (2) ( k) 1 Generate initial ample x ; [ x ; x ; ; x ] (1) (2) ( k) 2 Compute y = [ g( x ), g( x ),, g( x )]; et m = 1 3 While { m = 1} or { EI( x ) < eei } do xîx 4 Contruct a Kriging model yˆ = gˆ( X) uing { x, y } 5 Find y = 6 Search for { g( x )} * () i i= 1, 2,, k+ m 1 ( k) m) x = arg EI( x ), where EI( ) xîx x i computed by Eq. (4) 7 Scale EI( x)= EI( x ) / b(1), where β (1) i the firt element of the xîx xîx trend coefficient β given in Eq. (1) ( k m) 8 Compute g( x + ); update 9 m= m End While ( k) m) y = [ y, g( x )] and x x x ( ) [ ; k ) ; m ] 31

32 Table 2 Detailed procedure of algorithm 2 Algorithm 2 Independent EGO method (1) (2) ( k) 1 Generate initial ample x ; [ x ; x ; ; x ] 2 (1) (2) ( k) Solve for y = [ g ( x ), g ( x ),, g ( x )], where () i () i ( x ) { ( x, )} t [ t0, t ] g = g t, uing EGO; et m = 1 3 While { m = 1} or { MSE( x ) < emse } do xîx 4 Contruct a Kriging model Y ˆ = g ( X) uing { x, y } 5 Find 6 Search for ( k) m) x = arg {MSE( x )} xîx ( k m) ( k m) g ( x + ) { g( x +, t)} t [ t0, t ] ( ) [ ; k ) m ( k m) x ; x x ] and y = y g x + = uing EGO 7 Update [, ( )] 8 m= m+ 1 9 End While 10 Reliability analyi uing Y ˆ = g ( X) 32

33 Table 3 Major Procedure of the mixed-ego baed method Step 1: Initial ampling 1. Generate initial ample x and t Step 2: Build initial extreme repone model (Algorithm 3) 2. Build time-dependent urrogate model Y = gˆ( X,) t 3. Solve for the imum repone Y at x baed on Y = gˆ( X,) t uing the mixed EGO method 4. Build initial extreme repone model Y ˆ = g ( X) Step 3: Update extreme repone model (Algorithm 4) 5. Adding new training point of X by the AK-MCS method [23] 6. Identify extreme value aociated with the new training point uing the mixed EGO method 7. Obtain final model Y ˆ = g ( X) Step 4: Reliability analyi 8. Monte Carlo imulation baed on Y ˆ = g ( X). 33

34 Table 4 Detailed procedure of algorithm 3 Algorithm 3 Mixed EGO model for initial Y ˆ = g ( X) 1 At initial ample point, compute y = [ y ] = [ g( x, t )] () i () i () i i= 1,, k i= 1,, k 2 Set xt = x, m = 1, and the initial current bet olution vector y = y 3 While { m = 1} or { I < e EI } do 4 Contruct Kriging model Y = gˆ( X, t) uing {[ x, t ], y } ( i 5 EI ) EI ( i) Find a point with imum EI: [ x, t ] = arg { {EI( x, t)}}, where t i= 1, 2,, k t [ t0, t ] () i iei [1,, k] and EI( x, t) i computed baed on Y = gˆ( X, t) ; calculate ( iei I ) EI = EI( x, t ) / β( x, t) (1). EI ( EI ) EI 6 Compute y = g( x i, t ) EI EI y if y > y ( iei) 7 Update current bet olution y ( iei) = yma x ( iei) otherwie ( EI ) 8 Update data point x ; [ x ; x i EI EI t t ], t ; [ t ; t ], and y = [ y, y ] 9 m= m End While 11 Record y, [ xt, t ], and y 12 Contruct Y ˆ = g ( X) uing { x, y } 34

35 Table 5 Detailed procedure of algorithm 4 Algorithm 4 Sampling update total 1: Set r = 1 and x = [] 2: While { r = 1} or { Cov > 0.05} do pf 3: Set p = 1 4: MCS Generate n MCS ample of X, x, i= 1, 2,, n ; let x total = [ x total ; x MCS ] i 5: While { p = 1} or { U min < ε U }, where ε U i the convergence criterion, do 6: Contruct a Kriging model Y ˆ = g ( X) uing { x, y } and predict MCS repone and their variance at x i uing Y ˆ = g ( X) 7: MCS Compute U ( x ) uing Eq. (A1); identify a new training point by x new i new = arg min {U( x)} and U min = U( x ) MCS xîx 8: Generate a new time intant t r from uniform ditribution on [ t0, t ] 9: MCS new new Compute y = g( x, t r ); update x ; [ x ; x ], t ; [ t ; t r ], and y = [ y, y ] MCS new MCS 10: Set y = y and q = 1 new 11: While { q = 1} or { EI( x, t) > eei } do 12: 13: Find 14: Scale tî[ t0, t ] 35 t Contruct n + 1 dimenional Kriging model Y = gˆ( X, t) uing {[ x, t ], y } EI t uch that baed on Y = gˆ( X, t) t EI t [ t0, t ] t MCS new = {EI( x, t)}, where EI( x new, t) i computed x x, where β (1) (, t) i the firt new EI new EI EI(, t ) = EI(, t ) / β( x, t) (1) element of the trend coefficient of Y = gˆ( X, t) model 15: EI Compute y new EI = g( x, t ) 16: EI EI new, if new y y > y Update current bet olution y = new y, otherwie 17: new EI EI Update data point xt ; [ xt; x ], t ; [ t ; t ], y = [ y, y ] 18: q= q+ 1 19: End While 20: new new Record y ; [ y; y ], x ; [ x ; x ], x t, t, and y 21: p= p+ 1 22: End While 23: Contruct Kriging model of Y ˆ = g ( X) uing { x, y } and compute p ˆ f x t

36 by plugging Y = gˆ ( X ) total x into 24: Compute Cov = (1 pˆ ) ( pˆ rn ) 25: r= r+ 1 26: End While pf f f MCS 36

37 Table 6 Detailed procedure of algorithm 5 Algorithm 5 Complete algorithm 1) Step 1: Initialization (1) (2) ( k) (1) (2) ( k) Generate initial ample x ; [ x ; x ; ; x ] and t ; [ t ; t ; ; t ] uing the Harmmerley ampling method. 2) Step 2: Build initial model Y ˆ = g ( X) (Algorithm 3) a) Compute y = [ y ] = [ g( x, t )] ( i) (1) (1) i= 1,, k i= 1,, k b) Set xt = x, m = 1, and the initial current bet olution vector y = y c) While { m = 1} or { I < e EI } do i) Contruct an n + 1 dimenional Kriging model Y = gˆ( X, t) uing{[ x, t ], y } ( iei ) EI ( i) ii) Find a point with imum EI: [ x, t ] = arg { { EI( x, t)}}, i= 1, 2,, k t [ t0, t ] ( iei ) EI where i EI [1,, k] ; calculate I = EI( x, t ) / β( x, t) (1). EI ( iei ) EI iii) Compute y = g( x, t ) EI EI, if ( EI) y y > y i iv) Update current bet olution y ( iei ) = y ( iei ), otherwie ( i ) EI EI v) Update data point xt ; [ xt; x ], t ; [ t ; t ], y = [ y, y ] vi) m= m+ 1 End While d) Record y, [ xt, t ] and y ; Set p = 1. 3) Step 3: Update Y ˆ = g ( X) with AK-MCS and mixed EGO (Algorithm 4) Algorithm 4 4) Step 4: Reliability Analyi Reliability analyi uing Y ˆ = g ( X) t 37

38 Table 7 NOF required for different number of ample of X Number of NOF ample of X Independent EGO Mixed EGO

39 Table 8 Reult of example 1 Method NOF pf ( t0, t ) ( 10-4 ) Error (%) Rice Independent EGO Mixed-EGO baed MCS N/A 39

40 Table 9 Variable and parameter of Example 2 Variable Mean Standard deviation Ditribution k 1 (N/m) Normal m 1 (kg) Normal k 2 (N/m) Normal m (kg) Normal 2 c (N/m) Normal 2 40

41 Table 10 Variable and parameter of Example 2 Method NOF pf ( t0, t ) ( 10-2 ) Error (%) Rice N/A N/A Independent EGO Mixed-EGO baed MCS N/A 41

42 Table 11 Random variable of Example 3 Variable Mean Standard deviation Ditribution u (Pa) Normal a 0 (m) Normal b 0 (m) Normal x Normal 1 x 0 50 Normal 2 x 0 98 Normal 3 x Normal 4 x Normal 5 x 0 98 Normal 6 x Normal 7 42

43 Table 12 Reult of Example 3 Method NOF pf ( t0, t ) ( 10-2 ) Error (%) Rice Independent EGO Mixed-EGO baed MCS N/A 43

44 Table 13 Detailed procedure of algorithm 6 Algorithm 6 Algorithm of AK-MCS total 1: Set q = 1 and x = [] 2: While { q = 1} or { Cov > 0.05} do pf 3: Set p = 1 4: MCS Generate n MCS ample of X, x, i= 1, 2,, n ; let x total = [ x total ; x MCS ] i 5: While { p = 1} or { U min < ε U } do 6: Contruct a Kriging model of G = f ˆ( X ) uing initial training point{ xg, } MCS and obtain the prediction and variance by plugging x into G = f ˆ( X ) 7: new new Identify new training point by x = arg min {U( x)} and U min = U( x ) MCS xîx 8: new new new new Compute y = f( x ); Update x; [ xx ; ] and g= [ g, y ] 9: End While 10: total Compute p ˆ f by plugging x into G = f ˆ( X ) 11: Compute Cov _ p f with Cov _ p = (1 p ) ( p qn ) 12: q= q+ 1 13: End While f f f MCS MCS i 44

45 Y from the mixed EGO model Y from the independent EGO method True Y True Y Fig. 1 Y from independent EGO and mixed EGO and the true value 45

46 f in( W t) 0 k 2 k 1 m 1 c 2 m 2 q 1 Fig. 2 A vibration problem 46

47 Y Ω Fig. 3 One repone Y at the mean value point of random variable 47

48 L L/2 A F kt A-A b 0 kt a 0 Fig. 4 Corroded beam ubjected to tochatic loading 48