Monte Carlo Simulation


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1 Chapter 8 Monte Carlo Smulaton Chapter 8 Monte Carlo Smulaton 8. Introducton Monte Carlo smulaton s named ater the cty o Monte Carlo n Monaco, whch s amous or gamblng such as roulette, dce, and slot machnes. Snce the smulaton process nvolves generatng chance varables and exhbts random behavors, t has been called Monte Carlo smulaton. Monte Carlo smulaton s a powerul statstcal analyss tool and wdely used n both nonengneerng elds and engneerng elds. It was ntally used to solve neutron duson problems n atomc bomb work at Alamos Scentc Laboratory n 944. Monte Carlo smulaton has been appled to dverse problems rangng rom the smulaton o complex physcal phenomena such as atom collsons to the smulaton o trac low and Dow Jones orecastng. Monte Carlo s also sutable or solvng complex engneerng problems because t can deal wth a large number o random varables, varous dstrbuton types, and hghly nonlnear engneerng models. Derent rom a physcal experme nt, Monte Carlo smulaton perorms random samplng and conducts a large number o experments on computer. Then the statstcal characterstcs o the experments (model outputs) are observed, and conclusons on the model outputs are drawn based on the statstcal experments. In each experment, the possble values o the nput random varables = (,,, n ) are sampled (generated) accordng to ther dstrbutons. Then the values o the output varable Y are calculated through the perormance uncton Y = g( ) at the samples o nput random varables. Wth a number o experments carred out n ths manner, a set o samples o output varable Y are avalable or the statstcal analyss, whch estmates the characterstcs o the output varable Y. The outlne o Monte Carlo smulaton s depcted n Fg. 9.. Three steps are requred n the smulaton process: Step samplng on random nput varables, Step evaluatng model output Y, and Step 3 statstcal analyss on model output. We wll ocus our dscussons on ndependent random varables. However, Monte Carlo smulaton s applcable or dependent varables. The three steps o Monte Carlo smulaton are dscussed n the ollowng sectons.
2 Probablstc Engneerng Desgn Dstrbutons o nput varables Step : Samplng o random varables Generatng samples o random varables Samples o nput varables Step : umercal Expermentaton Evaluatng perormance uncton Samples o output varables Analyss Model Y = g() Step 3: Statstc Analyss on model output Extractng probablstc normaton Probablstc characterstcs o output varables Fgure 9. Monte Carlo Smulaton 8. Samplng on nput random varables The purpose o samplng on the nput random varables = (,,, n ) s to generate samples that represent dstrbutons o the nput varable rom ther cds F ( x) ( =,,, n). The samples o the random varables wll then be used as nputs to the smulaton experments. Two steps are nvolved or ths purpose: Step generatng random varables that are unormly dstrbuted between and, and Step transormng the values o the unorm varable obtaned rom Step to the values o random varables that ollow the gven dstrbutons F ( x) ( =,,, n). Step Generatng random varables that are unormly dstrbuted between and The mportance o unorm numbers over the contnuous range [, ] s that they can be transormed nto real values that ollow any dstrbutons o nterest. In the early tmes o smulaton, random numbers were generated by mechancal ways, such as drawng balls, throwng dce, as the same way as many o today s lottery drawngs. ow any modern computers have the capablty to generate unormly dstrbuted random varables
3 Chapter 8 Monte Carlo Smulaton between and. There are a number o arthmetc randomgenerators developed or the computerbased random generaton. Random varables generated ths way are called pseudo random numbers. A randomgenerator produces a sequence o unorm numbers between and. The length o the sequence beore repeatng tsel s machne and algorthm dependent. The ollowng unorm random varables n the nterval o [, ] are generated wth the MATLAB random varable generator rand. Table 9. [, ] Unorm Random Varables Step Transormng [, ] unorm varables nto random varables that ollow the gven dstrbutons The task s to transorm the samples o [, ] unorm varable, z = ( z, z,, z ), where s the number o samples, generated rom Step, nto values o random varable that ollows a gven dstrbuton F ( x ). There are several methods or such a transormaton. The smple and drect transormaton s the nverse transormaton method. By ths method, the random varable s gven by x = F ( z ), =,,, (9.) where F s the nverse o the cd o the random varable. The transormaton s demonstrated n Fg. 9.. F x z x Fgure 9. The Inverse Transormaton Method 3
4 4 Probablstc Engneerng Desgn For example, s normally dstrbuted wth ( µ, σ ), snce x µ z = F ( x) =Φ σ (9.) then x = µ + σ Φ ( z) (9.3) 8.3 umercal Expermentaton Suppose that samples o each random varable are generated, then all the samples o random varables consttute sets o nputs, x = ( x, x,, xn ), =,,,, to the model Y = g( ). Solvng the problem tmes determnstcally yelds sample ponts o the output Y. y = g( x ), =,, (9.4) 8.4 Extracton o probablstc normaton o output varables Ater samples o output Y have been obtaned, statstcal analyss can be carred out to estmate the characterstcs o the output Y, such as the mean, varance, relablty, the probablty o alure, pd and cd. The assocated equatons are gven below: The mean Y y = = (9.5) The varance σ (9.6) Y = ( y Y) = I the alure s dened by the event g, the probablty o alure s then calculated by (see Eq. 6.9) { } p = P g = ( x, x,, x ) dxdx dx = ( x) dx (9.7),,, n n n g( x) g( x) where = (,,, n ) and x=( x, x,, x n ). 4
5 Chapter 8 Monte Carlo Smulaton The equaton can be rewrtten as where I () s an ndcator uncton, whch s dened by p = I( x) ( x) dx (9.8) + g( x) I( x ) = (9.9) otherwse Accordng to Eq. 3.9, the ntegral on the rghthand sde o Eq. 9.8 s smply the expected value (or average) o I( x ). Thereore, as p can be estmated by the average value o I( x ) p = I( x) = I( x ) = (9.) = where s the number o samples that have the perormance uncton less than or equal to zero,.e. g. The relablty s then estmated by R = P{ g > } = p = (9.) Smlar to the calculaton o the probablty o alure, the cd s gven by where the ndcator uncton s dened by F ( y) = P g y = I ( y ) (9.) Y ( ) ' = ( ) ' g x y I ( x ) = (9.3) otherwse The pd Y ( y) can be obtaned by the numercal derentaton o cd FY ( y ). 5
6 6 Probablstc Engneerng Desgn 8.5 Examples Two examples are used to demonstrate Monte Carlo smulaton. Example 9. The allowable stress o a mechancal component s normally dstrbuted, (,) MPa, and the maxmum stress s normally dstrbuted, (,) MPa. What s the probablty o alure p? Analytcal soluton For ths smple problem, an analytcal soluton exsts. Snce both and are normally dstrbuted, Y = g( ) = s also normally dstrbuted. Then the probablty o alure s gven by µ Y ( ) p = PY ( < ) = FY() =Φ =Φ =.855 σ Y + Monte Carlo smulaton soluton Table 9. shows the results rom Monte Carlo smulaton. samples or and are drawn rom ther respectve dstrbutons. Then the output Y s calculated wth the uncton Y = g( ) =, and the ndcaton uncton I s also calculated. Table 9. Samples rom Monte Carlo Smulaton Smulaton Y I
7 Chapter 8 Monte Carlo Smulaton Smulaton Y I From Eq. 9., the probablty o alure s computed by p = I( x ) = I( y) = =. = Other characterstcs o Y can also be calculated as ollows. The mean The standard devaton Y = y = y =.983 MPa = = S = ( y Y) = ( y.983) = MPa = = Y = g( ) = = Fgure 9.3 Samples o and 7
8 8 Probablstc Engneerng Desgn The samples o and are plotted n Fg As shown n the gure and Table 9., two samples are n the alure regon; namely, the number o alures s two. The result o the probablty o alure p has a large error compared to the analytcal soluton because the number o smulatons s small. I the number o smulatons ncreases, the soluton o p wll be more accurate. The solutons rom derent number o smulatons are dsplayed n Table 9.3. The numbers o alures and the errors o the solutons o p are also shown n the table. As the number o smulatons ncreases, the error decreases. Theoretcally, the number o smulatons approaches nnty, the soluton o p converges to the true soluton. The relatonshp between the error and the number o smulatons wll be dscussed n next secton. Table 9.3 Results rom Derent umbers o Smulatons p Error 46.9% 5.9% 4.96%.3% .5% Example 9. A cantlever beam s llustrated n Fg Ths s the same problem we have presented n Example 7.. L P y t P x w Fgure 9.4 A Cantlever Beam One o the alure modes s that the tp dsplacement exceeds the allowable value, D. The perormance uncton s the derence between D and the tp dsplacement and s gven by 3 4L Py Px g = D Ewt + t w 8
9 Chapter 8 Monte Carlo Smulaton where D =, '' 3 6 E = 3 ps s the modulus o elastcty, are wdth and heght o the cross secton, respectvely, and '' L = s the length, w and t '' w = and '' t = 4. P x and are external orces whch ollow normal dstrbutons, and Px ~ (5,) lb and P ~ (,) lb. y The probablty o alure s dened as the probablty o the allowable dsplacement less than the tp dsplacement,.e. P y 3 4 P L y Px p = P g = D + Ewt t w Ths problem nvolves only two random varables and can be easly vsualzed wthn a D random space as shown n Fg It s seen that the space s dvded by the curve o lmt state g = nto the sae regon and alure regon. At rst, samples o P x and P y are drawn and are shown n Table 9.4. The samples are also depcted n the Fg. 9.5 (a). Smulatons Table 9.4 Random Samples or the Example P x P y Y At all the sample ponts, the perormance uncton g> (see Table 9.4 and Fg. 9.5(a)). In other words, all the samples all nto the sae regon. Thereore the probablty o alure s equal to zero. Ths means that the sample sze s not large enough. I samples are used, there s one sample allng nto the alure regon (see Fg. 9.5 (b)). Thereore the probablty s /=.. I we ncrease the sample sze to,, there are 7 samples where the beam als. Ths gves an estmate o the probablty o alure o 7/ =.7 (see Fg. 9.5 (c)). As we have seen n the last example, the larger the sample sze s, the more accurate the estmate o the probablty o alure wll be. I, smulatons are perormed, the estmate o the probablty o alure s.443 as shown n Fg. 9.5 (d). Recall that the probablty o alure computed by FORM n Example 7. s.454 and by SORM n Example 7.5 s.498. I the probablty o alure obtaned rom 9
10 Probablstc Engneerng Desgn Monte Carlo smulaton wth, smulatons s consdered as an accurate soluton, and the example conrms that SORM s more accurate than FORM or ths problem. Fgs. 9.6 and 9.7 show the cd and pd o the perormance uncton g wth, smulatons. 3 3 umber o smulatons = umber o alures = p =. 5 umber o smulatons = umber o alures = p =. 5 Falure regon Sae regon Py 5 Falure regon Sae regon Py (a) = 8 9 umber o smulatons = umber o alures = 443 p = Py Falure regon Falure regon Sae regon umber o smulatons = umber o alures = 7 p = (b) = 3 Py 5 Px Px Sae regon Px Px (c) = (d) = Fgure 9.5 Monte Carlo Smulaton o the Cantlever Beam F g( g).4 g ( g) g g Fgure 9.6 cd Fgure 9.7 pd.5.5
11 Chapter 8 Monte Carlo Smulaton 8.6 Error Analyss As demonstrated n precedng examples, the accuracy o Monte Carlo Smulaton depends on the number o smulatons. The hgher the number o smulatons s, the more accurate the estmate wll be. As the number o smulatons approaches nnty, the soluton o Monte Carlo smulaton wll converge to the true probablty that s under estmaton. Snce relablty assessment normally needs hgh accuracy, t s mportant to know the error nvolved n the estmated probablty o alure. On the other hand, t s also mportant to know how many smulatons are requred to acheve the desred accuracy. The percentage error o the estmate o the probablty alure s ound to be T ( p ) = (9.4) T p ε % u α / where u α /s the (a/) quantle (percentle value) o the standard normal dstrbuton, T and p s the true value o the probablty o alure. The above equaton gves the percentage error under the (a)% condence. The error s not n an absolute (determnstc) sense. On the contrary, t ndcates that there s a (a)% chance that ε the probablty o alure wll be n the range o p ± p wth smulatons. The commonly used condence level s 95% under whch the error s approxmately gven by Snce the theoretc value o the probablty o alure probablty o alure T ( p ) ε % (9.5) T p p s used n Eqs. 9.4 and 9.5 to replace For example, p =. and the number o smulatons condence, rom Eq. 9.5 the error s calculated to be % T p s unknown, the estmated T p. 5 =, wth 95% ε=. Wth 95% lkelhood, thereore, the probablty o alure wll be p =.±.. I we desre the error to be wthn %, rom the same equaton, the number o smulaton calculated should be 399,6. For the beam problem n Example 9., the error o the probablty o alure wth, samples s (.443) ε % = 3.4%.443
12 Probablstc Engneerng Desgn Snce the pont estmate o the probablty o alure s.443, the 95% condent nterval s.443 ± %,.e. [.4,.47]. Accordng to Eq. 9.4, hgher relablty (or lower probablty o alure) requres a hgher 6 number o smulatons. For example, a probablty o alure o = ndcates that only one tem wll al out o mllon tems. The requred number o smulatons s 8 about = 4 wth % error under 95% condence. Thereore hgher relablty requres a hgher computatonal eort. For the same reason, hgher accuracy or estmatng cd n dstrbuton tal areas, a large number o smulatons must be used. From the above dscusson, the eatures o Monte Carlo smulaton are summarzed as ollows. ) Monte Carlo smulaton s easy to use or engneers who have only lmted workng knowledge o probablty and statstcs. ) Monte Carlo smulaton s easble to use or vrtually any perormance unctons and dstrbutons. 3) Monte Carlo smulaton s computatonally robust; wth sucent number o smulatons, t can always converge. 4) The problem dmenson (the number o random varables) does not aect the accuracy o Monte Carlo smulaton as ndcated n Eq Ths eature s benecal to large scale engneerng problems. 5) For relablty analyss, Monte Carlo smulaton s generally computatonally expensve. The hgher the relablty s, the larger the smulaton sze s needed. Because o the accuracy, Monte Carlo smulaton s wdely used n ) engneerng applcatons where the model evaluatons (determnstc analyses) are not computatonally expensve and ) valdatng other methods. However, due to ts computatonal necency, Monte Carlo smulaton s not commonly used or problems where determnstc analyses are expensve. p 8.7 MPP Based Importance Samplng From the precedng dscussons, t s noted that the computatonal cost s very hgh when the probablty o alure s small. The reason s that only the samples that all nto the alure regon can contrbute to the probablty estmaton. Importance samplng technque was developed wth the motvaton to mprove the computatonal ecency. The central dea o mportance samplng s to sample the random varables accordng to an alternatve set o dstrbutons such that more samples wll be n the alure regon.
13 Chapter 8 Monte Carlo Smulaton More samples wll thereore contrbute to the probablty estmaton. The dea s llustrated n Fg ew dstrbutons MPP ( x, x ) (the whte crcle) * * Lmt state g()= Sae regon g()> Orgnal dstrbutons ew dstrbutons Fgure 9.8 Importance Samplng As shown n Fg. 9.8, all the samples (the lower cloud) generated rom the orgnal dstrbutons o and are n the sae regon. They do not have any contrbuton to the probablty estmaton. I a new set o dstrbutons o and s selected such that many samples wll all nto the alure regon, then the samples (the upper cloud n the gure) wll contrbute to the probablty estmaton sgncantly. There are several mportance samplng schemes, and we wll dscuss the MPP based mportance samplng heren. Ater the MPP s obtaned, samples are pcked around the MPP to evaluate the probablty o alure through mportance samplng (see Fg. 9.8). To do so, an mportancesamplng densty, h ( x ), s ntroduced nto the Monte Carlo estmaton, Eq. 9.7, to obtan the probablty o alure ( ) x p = Ig [ ( )] ( x) dx = Ig [ ( )] h( x) dx h ( x) (9.6) where the mportancesamplng densty h ( x ) s the same as ( x ) except that the * * * * means values o are replace by the MPP x = ( x, x,, x n ). For example, the dstrbuton o s normally dstrbuted wth ( µ, σ ) and x µ ( x) = exp, the correspondng mportance samplng dstrbuton πσ σ 3
14 4 Probablstc Engneerng Desgn * * x x wll be ( x, σ ) and h ( x) = exp. It s noted that the mportance πσ σ * samplng densty h ( x ) s centered at x. Eq. 9.6 ndcates that the probablty o alure s the mean o the ntegrand ( ) x Ig [ ( )] h ( x) that s evaluated at the samples o drawn rom the mportance samplng densty ( x ). Thereore, h p ( x ) = Ig [ ( x)] h ( x ) = (9.7) As shown n Fg. 9.8, wth the sample sze, a sgncant number o alures occur when the samples are drawn rom the mportance samplng densty h ( x ). It can be approved that wth the same sample sze the error bound o mportance samplng s much smaller than the general Monte Carlo smulaton. Example 9.3 Use MPP based mportance samplng to solve Example 9.. From Example 7., the MPP n Uspace rom FORM s u * = (.7367,.6376). Ther * transormaton to space s x = (673.67, 6.38) The orgnal dstrbutons o two random varables are Px ~ (5,) and Py ~ (,), and the dstrbutons or mportance samplng are then gven by (673.67,) or P x and (6.38,) or P y. Fg. 9.9 shows the results o the general Monte Carlo smulaton where there are only 7 alures out o samples. The probablty o alure s.7 whch has a large error as compared to the accurate soluton p =.443 wth, smulatons. Fg. shows the results rom mportance sample wth only samples, out o whch 96 samples are n the alure regon. The calculated probablty o alure s.4976 and s very close to the accurate soluton rom, general Monte Carlo samples. 4
15 Chapter 8 Monte Carlo Smulaton Fgure 9.9 General Monte Carlo smulaton Fgure 9. Importance Samplng 8.8 MATLAB Random umber Generators MATLAB provdes random number generators or commonly used dstrbutons. ) normrnd random matrces rom normal dstrbuton R = normrnd(mu,sigma) returns a random number chosen rom the normal dstrbuton wth parameters MU (mean) and SIGMA (standard devaton). R = normrnd (MU,SIGMA,M,) returns an M by matrx. ) lognrnd random matrces rom the lognormal dstrbuton R = lognrnd (MU,SIGMA) returns a random number chosen rom the lognormal dstrbuton wth parameters MU (mean) and SIGMA (standard devaton). R = LOGRD(MU,SIGMA,M,) returns an M by matrx. 3) exprnd random matrces rom exponental dstrbuton R = exprnd (MU) returns a random number chosen rom the exponental dstrbuton wth parameter MU (mean). R = exprnd (MU,M,) returns an M by matrx. 4) unrnd random matrces rom contnuous unorm dstrbuton R = unrnd (A,B) returns a random number chosen rom the contnuous unorm dstrbuton on the nterval rom A to B. 5
16 6 Probablstc Engneerng Desgn R = UIFRD(A,B,M,) returns an M by matrx. In addton to the above random varable generators, MATLAB can also generate random varables or more dstrbutons. 6
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