Probability, Statistics, and Reliability for Engineers and Scientists SIMULATION
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1 CHATER robablty, Statstcs, and Relablty or Engneers and Scentsts Second Edton SIULATIO A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng 7b robablty and Statstcs or Cvl Engneers Department o Cvl and Envronmental Engneerng Unversty o aryland, College ark CHAA HALL/CRC CHATER 7b. SIULATIO Slde o. Smulaton ethods Smulaton s the process conductng experments on a model. A model s a representaton or the real system or the real component or the purpose o studyng the perormance. Hgh cost. Dculty (mpossblty).
2 CHATER 7b. SIULATIO Slde o. Smulaton ethods onte Carlo technques are technques or testng engneerng systems by mtatng ther real behavor. the accuracy o the smulaton estmator ncreases as the smulaton cycles ncrease. CHATER 7b. SIULATIO Slde o. 3 Smulaton ethods The perormance uncton s dened as Z = R L = g( X, X, X 3, K, X n ) X,, X, X 3, K X n n = random varables I I I Z > 0 Z < 0 Z = 0 survval. alure. lmt state.
3 CHATER 7b. SIULATIO Slde o. 4 Smulaton ethods The relablty o each component n the system s the probablty that the strength o the component exceeds the appled loadngs on the same component. The probablty o alure o the component s the probablty that the strength o the component s less than the appled loadngs on the component. CHATER 7b. SIULATIO Slde o. 5 Smulaton ethods The onte Carlo Smulaton ethods: Drect onte Carlo smulaton ance reducton technques Improve the smulaton accuracy: reduce the varance o the estmated probablty o alure, Improve the smulaton ecency: reduce the number o smulaton cycles 3
4 CHATER 7b. SIULATIO Slde o. 6 Smulaton ethods Steps or smulatons based varance reducton technques: Select g(x) tmes Compute () COV() Generate (u) Draw sample (R.V.) Evaluate I[g(x)] Estmate CHATER 7b. SIULATIO Slde o. 7 Smulaton ethods In the smulaton technques, compute: The estmated probablty o alure. The varance o the estmated probablty o alure. The coecent o varaton o the estmated probablty o alure. The computatonal tme (CU). σ The relatve ecency rato = σ T T 4
5 CHATER 7b. SIULATIO Slde o. 8 Smulaton ethods The VRT s are classed based on ther common characterstcs:. The mportance samplng category: ore samples are taken rom the regon o nterest.. The correlated samplng category: Lnear correlaton among the randomly generated varables. 3. The condtonal expectaton category: Condtonng on one or more o the generated random varables. 4. The general technques category: Indvdual characterstcs. CHATER 7b. SIULATIO Slde o. 9 Smulaton ethods Table. Classcaton o ance Reducton Technques. Importance samplng category: Importance samplng technque. Adaptve samplng technque. Strated samplng technque. oststrated samplng technque. Latn hypercube samplng technque. Updated Latn hypercube samplng technque. Sphercal samplng technque. Truncated samplng technque. Correlated samplng category: Antthetc ate technque. Common Random umbers technque. Control ate technque. Rotaton Samplng technque. Condtonal expectaton category: Condtonal expectaton technque. Generalzed condtonal expectaton technque. Adaptve hybrd condtonal expectaton technque. General technques category: Response surace technque. Adaptve response surace technque. Russan roulette technque. Russan roulette and splttng technque. Jackkne technque. 5
6 CHATER 7b. SIULATIO Slde o. 0 VRT: Drect onte Carlo Technque (DC) Draw samples o the basc random varables based on ther probablstc characterstcs and eedng them n the perormance uncton. = ( ) ( ) = ( ) ( ) COV = ( X ) = g( X, X, ) Z = g, K X n CHATER 7b. SIULATIO Slde o. VRT: Drect onte Carlo Technque (DC) Smulaton steps or DC:. Select a perormance uncton and denty ts random varables and ther probablstc characterstcs.. Generate random numbers ( u) and then the random varables values by usng the nverse transormaton method. 3. Evaluate the perormance uncton (lmt state uncton), g( X ), add to the alure counter,( I(). ), g < 0 and add 0 g Repeat steps to 3 tmes. 5. Determne the number o alures,, based on the counter ( I) value. 6. Compute and = ( ) ( ) = 7. Compute ( ) ( ) COV = 6
7 CHATER 7b. SIULATIO Slde o. Example (DC) w oment alure mode o a steel beam subjected to unormly dstrbute loadng. Z = F y S L Where F y S Random able ean Value COV Dstrbuton Type 90 a 0.5 ormal Yeld stress ( F y ) Secton odulus (S) Load moment, ( ) = materal yeld stress. = elastc secton modulus. = moment eect due to appled loadng. 8.9x0-4 m ormal.3x0 5 -m 0.00 ormal CHATER 7b. SIULATIO Slde o. 3 Example (DC) DC_5000 Cycles robablty o Falure COV () Coecent o aton () Smulaton Cycles 7
8 CHATER 7b. SIULATIO Slde o. 4 VRT: Importance Samplng Technque (IS) The smulaton samples are concentrated n the alure regon. The random varables are generated accordng to selected probablty dstrbutons wth mean values closer to the desgn pont. X ( X ) = X ( X ) dx = hx ( X ) dx h ( X ) D D X ( ) = = ( = ( ) ( X) X I[ g( X) 0] = hx ( X) ) ( ) COV = ( ) CHATER 7b. SIULATIO Slde o. 5 VRT: Importance Samplng Technque (IS) Smulaton steps or IS:. Select a perormance uncton and denty ts random varables and ther probablstc characterstcs.. Select the mportance densty uncton, h X x, and dene the orgnal densty uncton, X ( x). 3. Generate random numbers ( u) and then the random varables values by usng the nverse transormaton method. 4. Evaluate the perormance uncton (lmt state uncton), g( X ), add to the alure counter,( I(). ), g < 0 and add 0 g Repeat steps 3 to 4 tmes. 6. Compute X ( X ) ( ) = I[ g( X ) 0] hx ( X ) ( ) = = ( ) 7. Compute ( ) ( ) COV = ( ) 8
9 CHATER 7b. SIULATIO Slde o. 6 Example (IS) Z = F y S w IS_5000 Cycles L robablty o Falure COV () Coecent o aton ( ) Smulaton Cycles CHATER 7b. SIULATIO Slde o. 7 VRT: Condtonal Expectaton Technque (CE) Randomly generatng all the basc random varables except one varable. Select the control varable X k wth the hghest varablty. [ g ( X : =,, K, n k) ] = FX k k ; = ( = ( ) = = ( ) ( ) COV = ( ) ) 9
10 CHATER 7b. SIULATIO Slde o. 8 Example (CE) Z = F y S w CE_5000 Cycles L E-0 robablty o Falure E-0.0E-0.00E E E E-0.00E-0 Coecent o aton ( ) COV () 0.00E Smulaton Cycles CHATER 7b. SIULATIO Slde o. 9 VRT: Generalzed Condtonal Expectaton Technque (GCE) The number o control varables are consdered more than one. = F µ S Fy S µ σ Fy + σ = = ( ) = = ( ( ) ) ( ) COV = ( ) 0
11 CHATER 7b. SIULATIO Slde o. 0 Example (GCE) Z = F y S GCE_5000 Cycles L w robablty o Falure Coecent o aton ( ) 0.00 COV () Smulaton Cycles CHATER 7b. SIULATIO Slde o. VRT: Antthetc ate Technque (AV) egatve correlaton between derent cycles o smulaton s nduced n order to decrease the varance o the estmated mean value. u and -u are used. = () + ( ) () ( ) () ( ) ( ) = [ ( ) ( ) (, )] + Cov + 4 ( ) COV = ( )
12 CHATER 7b. SIULATIO Slde o. Example (AV) Z = F y S AV_5000 Cycles L w robablty o Falure Coecent o aton ( ) COV () Smulaton Cycles CHATER 7b. SIULATIO Slde o. 3 VRT: Strated Samplng Technque (SS) The alure doman s dvded nto several dsjont subdomans. ore samples are then taken rom the most mportant subdomans. = 5 = ( ) ( ) COV = = = 5 = I [ g( X ) < 0] σ ( )
13 CHATER 7b. SIULATIO Slde o. 4 Example (SS) Z = F y S SS_5000 Cycles L w robablty o Falure Coecent o aton ( ) COV () Smulaton Cycles CHATER 7b. SIULATIO Slde o. 5 VRT: Control ate Technque (CV) Takes advantage o correlaton between certan varables. Another random varable wth known mean s selected to adjust the. Generate an ntal run to estmate the Cov( g( X ), g o ( X )) adjustment constant, a = ( g o ( X )) [ I [ g ( x ) < 0] a( I ( g o ( x ) < 0) )] + aµ c = = [ ] = ( g( X )) + a g o ( X ) acov g X, ( ) COV = ( ) ( ) ( ( ) g ( X )) o 3
14 CHATER 7b. SIULATIO Slde o. 6 Example (CV) Z = F y S CV_5000 Cycles L w 0.03 robablty o Falure COV () Coecent o aton ( ) Smulaton Cycles CHATER 7b. SIULATIO Slde o. 7 VRT: Russan Roulette Technque (RR) Some smulaton cycles are klled (ceased to exst) by chance wth a certan probablty. The survval probablty s determned and the the survval weght s then adjusted as w = ( J n s the Russan roulette counter or survved smulaton cycles. ( ) = = I[ g( X ) 0]. J. w ( ) = = n ( ) COV = ( ) ( ) survval ) 4
15 CHATER 7b. SIULATIO Slde o. 8 Example (RR) Z = F y S RR_5000 Cycles L w robablty o Falure COV () Coecent o aton ( ) Smulaton Cycles CHATER 7b. SIULATIO Slde o. 9 VRT: Russan Roulette & Splttng Technque (RR&S) The klled samples n the Russan Roulette are compensated n the splttng process n ths technque. re-determne the number o splts, k. Wght the samples as ( / k). = = I[ g( X ) 0]. J n where J n ( = Russan roulette counter ) = = ( ( ( ) ) ) COV = ( ) 5
16 CHATER 7b. SIULATIO Slde o. 30 Example (RR&S) Z = F y S RR&S_5000 Cycles L w robablty o Falure Coecent o aton ( ) COV () Smulaton Cycles CHATER 7b. SIULATIO Slde o. 3 Example (Results or All VRT s) VRT COV() Tme COV() Tme COV() S.D. VAR Tme Relatve Ecency Rato DC E E-03.4E CE E-04.75E E GCE E E-06.50E IS E E-03.E CV E-03.4E E SS E E E AV E E E RR E E-03.58E RR&S E E-03.08E AS
17 CHATER 7b. SIULATIO Slde o. 3 Example (Results or All VRT s) 5000 Smulaton Cycles Estmated probablty o alure AS (0 08) DC CE GCE IS CV SS AV RR RR&S AS umber o smulaton cycles CHATER 7b. SIULATIO Slde o. 33 Example (Results or All VRT s) 5000 Smulaton Cycles ance ( ) 5.00E E E E E-04.50E-04.00E-04 DC CE GCE IS CV SS AV RR RR&S.50E-04.00E E E umber o Smulaton Cycles 7
18 CHATER 7b. SIULATIO Slde o. 34 Example (Results or All VRT s) 5000 Smulaton Cycles Coecent o aton ( ) 5.00E E E E E-0.50E-0.00E-0.50E-0.00E-0 DC CE GCE IS CV SS AV RR RR&S 5.00E-0 AV SS CE 0.00E+00 CV umber o smulaton cycles CHATER 7b. SIULATIO Slde o. 35 Example (Results or All VRT s) 5000 Smulaton Cycles 0.00 DC (500'000 cycles) = Estmated probablty o alure DC CE GCE IS CV SS AV RR RR&S AS Seres Technques 8
19 CHATER 7b. SIULATIO Slde o. 36 Example (Results or All VRT s) 5000 Smulaton Cycles COV( ) DC CE GCE IS CV SS AV RR RR&S DC CE GCE IS CV SS AV RR RR&S Seres Techques CHATER 7b. SIULATIO Slde o. 37 Example (Results or All VRT s) 5000 Smulaton Cycles Tme (seconds) DC CE GCE IS CV SS AV RR RR&S Seres Technques 9
20 CHATER 7b. SIULATIO Slde o. 38 Example (Results or All VRT s) 5000 Smulaton Cycles Relatve Ecency Rato DC CE GCE IS CV SS AV RR RR&S DC CE GCE IS CV SS AV RR RR&S Seres Technques 0
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