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).
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.
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
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
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
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 0. 4. Repeat steps to 3 tmes. 5. Determne the number o alures,, based on the counter ( I) value. 6. Compute and = ( ) ( ) = 7. Compute ( ) ( ) COV = 6
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 3 0.050 ormal.3x0 5 -m 0.00 ormal CHATER 7b. SIULATIO Slde o. 3 Example (DC) DC_5000 Cycles 0.00 0.35 0.090 0.30 0.080 0.070 0.5 robablty o Falure 0.060 0.050 0.040 0.030 COV () 0.0 0.5 0.0 Coecent o aton () 0.00 0.00 0.05 0.00 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 Smulaton Cycles 7
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 0. 5. Repeat steps 3 to 4 tmes. 6. Compute X ( X ) ( ) = I[ g( X ) 0] hx ( X ) ( ) = = ( ) 7. Compute ( ) ( ) COV = ( ) 8
CHATER 7b. SIULATIO Slde o. 6 Example (IS) Z = F y S w IS_5000 Cycles L 0.40 0.40 0.35 robablty o Falure 0.0 0.00 0.080 0.060 0.040 0.00 COV () 0.30 0.5 0.0 0.5 0.0 0.05 Coecent o aton ( ) 0.00 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 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
CHATER 7b. SIULATIO Slde o. 8 Example (CE) Z = F y S w CE_5000 Cycles L 0.00.60E-0 robablty o Falure 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.00 0.00.40E-0.0E-0.00E-0 8.00E-0 6.00E-0 4.00E-0.00E-0 Coecent o aton ( ) COV () 0.00E+00 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 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
CHATER 7b. SIULATIO Slde o. 0 Example (GCE) Z = F y S GCE_5000 Cycles L w 0.00 5 0.090 0.080 5 4 robablty o Falure 0.070 0.060 0.050 0.040 0.030 4 3 3 Coecent o aton ( ) 0.00 COV () 0.00 0 0 000 000 3000 4000 5000 6000 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 = ( )
CHATER 7b. SIULATIO Slde o. Example (AV) Z = F y S AV_5000 Cycles L w 0.00 0.030 0.090 0.080 0.05 robablty o Falure 0.070 0.060 0.050 0.040 0.030 0.00 0.05 0.00 Coecent o aton ( ) 0.00 0.00 COV () 0.005 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 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 ( )
CHATER 7b. SIULATIO Slde o. 4 Example (SS) Z = F y S SS_5000 Cycles L w 0.00 0.400 robablty o Falure 0.80 0.60 0.40 0.0 0.00 0.080 0.060 0.040 0.350 0.300 0.50 0.00 0.50 0.00 Coecent o aton ( ) 0.050 0.00 COV () 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 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
CHATER 7b. SIULATIO Slde o. 6 Example (CV) Z = F y S CV_5000 Cycles L w 0.03 robablty o Falure 0.40 0.0 0.00 0.080 0.060 0.040 0.00 COV () 0.0 0.0 0.00 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.00 0.00 Coecent o aton ( ) 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 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
CHATER 7b. SIULATIO Slde o. 8 Example (RR) Z = F y S RR_5000 Cycles L w 0.0 0.30 0.00 0.5 robablty o Falure 0.080 0.060 0.040 COV () 0.0 0.5 0.0 Coecent o aton ( ) 0.00 0.05 0.00 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 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
CHATER 7b. SIULATIO Slde o. 30 Example (RR&S) Z = F y S RR&S_5000 Cycles L w 0.0 0.30 0.00 0.5 robablty o Falure 0.080 0.060 0.040 0.0 0.5 0.0 Coecent o aton ( ) COV () 0.00 0.05 0.00 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 Smulaton Cycles CHATER 7b. SIULATIO Slde o. 3 Example (Results or All VRT s) VRT 000 3000 5000 COV() Tme COV() Tme COV() S.D. VAR Tme Relatve Ecency Rato DC 0.0750 0..7 0.0747 0.0643 6.0 0.0768 4.90E-0 3.77E-03.4E-05 98.0 CE 0.0798 5. 0.083 3 63.4 0.0834.09E-04.75E-05 3.05E-0 05.5 4360 GCE 0.088.8 0.08 38.8 0.088 6.04E-05 5.00E-06.50E- 66.7 83689 IS 0.077 0.00 3.0 0.0957 0.065 94.0 0.0986 4.77E-0 4.70E-03.E-05 55.9 0.4 CV 0.094 0.0037 8.8 0.096 0.008 0.7 0.00.E-03.4E-04 4.60E-08 98.7 5 SS 0.0770 0.033 5.9 0.0760 0.0074 48.0 0.0858 8.3E-03 7.3E-04 5.08E-07 80. 34 AV 0.09 0.0030 40. 0.083 0.003 37.3 0.084 8.4E-04 6.94E-05 4.8E-09 47.6 65 RR 0.0936 0.00 9. 0.0944 7 56.4 0.0968 5.5E-0 5.08E-03.58E-05 93.7 RR&S 0.0935 0.00 3. 0.0898 7 69.3 0.0880 5.9E-0 4.57E-03.08E-05 4.7 AS 0.080 0.080 0.080 6
CHATER 7b. SIULATIO Slde o. 3 Example (Results or All VRT s) 5000 Smulaton Cycles Estmated probablty o alure 0.300 0.50 0.00 0.50 0.00 AS (0 08) DC CE GCE IS CV SS AV RR RR&S AS 0.050 0 000 000 3000 4000 5000 6000 umber o smulaton cycles CHATER 7b. SIULATIO Slde o. 33 Example (Results or All VRT s) 5000 Smulaton Cycles ance ( ) 5.00E-04 4.50E-04 4.00E-04 3.50E-04 3.00E-04.50E-04.00E-04 DC CE GCE IS CV SS AV RR RR&S.50E-04.00E-04 5.00E-05 0.00E+00 0 000 000 3000 4000 5000 6000 umber o Smulaton Cycles 7
CHATER 7b. SIULATIO Slde o. 34 Example (Results or All VRT s) 5000 Smulaton Cycles Coecent o aton ( ) 5.00E-0 4.50E-0 4.00E-0 3.50E-0 3.00E-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 0 000 000 3000 4000 5000 6000 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) = 0.083 Estmated probablty o alure 0.0800 0.0600 0.0400 0 DC CE GCE IS CV SS AV RR RR&S AS Seres 0.0768 0.0834 0.088 0.0986 0.00 0.0858 0.084 0.0968 0.0880 0.080 Technques 8
CHATER 7b. SIULATIO Slde o. 36 Example (Results or All VRT s) 5000 Smulaton Cycles 0.0090 0.0080 0.0070 COV( ) 0.0060 0.0050 0.0040 DC CE GCE IS CV SS AV RR RR&S 0.0030 0 DC CE GCE IS CV SS AV RR RR&S Seres 0.0490 0.0477 0.00 0.0083 8 5 5 Techques CHATER 7b. SIULATIO Slde o. 37 Example (Results or All VRT s) 5000 Smulaton Cycles 300 50 Tme (seconds) 00 50 00 50 0 DC CE GCE IS CV SS AV RR RR&S Seres 98.0 05.5 66.7 55.9 98.7 80. 47.6 93.7 4.7 Technques 9
CHATER 7b. SIULATIO Slde o. 38 Example (Results or All VRT s) 5000 Smulaton Cycles 0000 9000 8000 Relatve Ecency Rato 7000 6000 5000 4000 3000 DC CE GCE IS CV SS AV RR RR&S 000 000 0 DC CE GCE IS CV SS AV RR RR&S Seres 4360 83689 0.4 5 34 65 Technques 0