!PPLICATIONOF-ONTE#ARLO3IMULATIONFOR&REE0ISTON%NGINE#YLINDER"LOCK$ESIGN
.URAINI!BDUL!ZIZ!HMAD+AMAL!RIFlN-OHD)HSAN-OHD*AILANI-OHD.OOR AND.ORDIN*AMALUDDIN
$EPARTMENTOF-ECHANICALAND-ATERIALS%NGINEERING
&ACULTYOF%NGINEERING"UILT%NVIRONMENT 5NIVERSITI+EBANGSAAN-ALAYSIA
5+-"ANGI3ELANGOR -ALAYSIA
0HONE
%MAILANNAZIZA GMAILCOM
2ECEIVED$ATETH!UGUST !CCEPTED$ATETH-ARCH
!"342!#4
4HISPAPERPRESENTSASTOCHASTICSIMULATIONSTUDYTOTAKEACCOUNTTHESEUNCERTAINTIESINACYLINDERBLOCK STRUCTUREOFFREEPISTONENGINE4HECOMPUTATIONALSTOCHASTICSTRUCTURALMECHANICSANDANALYSISALLOWS A RATIONAL TREATMENT OF STATISTICAL UNCERTAINTIES INVOLVED IN STRUCTURAL ANALYSIS AND DESIGN )T CONSISTS ASTOCHASTICSIMULATIONOFTHESTATICANALYSISOFTHEMODELSTRUCTUREWHERETHEANALYSISWASEXECUTED FROMlNITEELEMENTSOFTWARE4HESIMULATIONSTUDYPRODUCESADATATHATIDENTIFYTHEDESIGNINPUTAND OUTPUTPARAMETERTHATNEEDTOBEOPTIMISEDBASEDON-ONTE#ARLOMETHOD"YEXECUTINGTHEDESIGN IMPROVEMENTANALYSISINTHESIMULATIONPROCESSMULTIPLESOLUTIONSARRIVEDFROMTHETARGETBEHAVIOUR WHICHHAVEBEENSPECIlEDFORTHESTRUCTURE&ORTHISCYLINDERMODELITISFOUNDTHATFROMIMPROVEMENT DESIGNTHEREISASIGNIlCANTREDUCEINSTRESSVALUEOF-0AMODULUSOFELASTICITYFORMATERIAL%OF -0AANDINCREASINGINMODULUSOFELASTICITYFORMATERIAL%OF-0ABYCONTROLLINGTHE TARGETBEHAVIOURMAXIMUMDISPLACEMENTOFTHECYLINDER,MAX4HECORRELATIONSBETWEENTHESEVARIABLES WERESHOWNINTHESTUDYWHICHALLOWDESIGNERSTOIDENTIFYTHESTRENGTHOFTHERELATIONSHIPBETWEEN VARIABLESANDTHEUNCERTAINTIESINTHEDESIGN
+EYWORD3TOCHASTICSIMULATIONUNCERTAINTIESCYLINDERBLOCK-ONTE#ARLO
!"342!+
+ERTASKERJAINIMEMBENTANGKANKAJIANMENGGUNAKANKAEDAHSIMULASISTOKASTIKDALAMMENGAMBILKIRA FAKTORKETIDAKTENTUANPADASTRUKTURBLOKSILINDERBAGIENJINOMBOHBEBAS!NALISISDANKAEDAHPERKOMPUTERAN STOKASTIKBAGISTRUKTURMEKANIKMEMBERIKANPENDEKATANYANGRASIONALDALAMMELIBATKANKETIDAKTENTUAN SECARASTATISTIKDALAMMENGANALISISDANMEREKABENTUKSESUATUSTRUKTUR+AJIANINIMENGANDUNGISIMULASI STOKASTIKBAGIANALISISSTATIKMODELSTRUKTURYANGTELAHDILAKSANAKANMENGGUNAKANPERISIANKAEDAHTERHINGGA +AJIANSIMULASIINIMENGHASILKANDATAYANGDIKENALPASTISEBAGAIMASUKANBAGIREKABENTUKJUGAPARAMETER KELUARAN YANG PERLU DIOPTIMUMKAN BERASASKAN KAEDAH -ONTE #ARLO $ENGAN MELAKSANAKAN ANALISIS PEMBAIKANREKABENTUKDALAMPROSESSIMULASIBEBERAPAPENYELESAIANTELAHDIPEROLEHIHASILDARISASARAN
KELAKUANYANGTELAHDIKENALPASTIUNTUKSTRUKTURTERSEBUT$ARIPADASIMULASIPEMBAIKANREKABENTUK DIDAPATIBAHAWADENGANMENGAWALSASARANKELAKUANIAITUANJAKANMAKSIMUMSILINDER,MAXTERDAPAT PENGURANGANDALAMTEGASANYANGMEMBERINILAISEBANYAK-0AJUGANILAIMODULUSKEKENYALANBAHAN
%IAITU-0ANAMUNTERDAPATPENINGKATANDALAMMODULUSKEKENYALANBAHAN%IAITU -0A+AJIANINIMEMBENTANGKANPERKAITANANTARAPEMBOLEHUBAHYANGMANAIANYADAPATMEMBANTU PEREKABENTUKDALAMMENGENALPASTIKEKUATANPERHUBUNGANANTARAPEMBOLEHUBAHPEMBOLEHUBAH JUGAKETIDAKTENTUANYANGUJUDDALAMREKABENTUK
+ATAKUNCI3IMULASISTOKASTIKKETIDAKTENTUANBLOKSILINDER-ONTE#ARLO
).42/$5#4)/.
4HE ANALYSIS OF STRUCTURAL BEHAVIOUR IS USUALLY PERFORMED FOR SPECIlED STRUCTURAL PARAMETERS ANDLOADINGCONDITIONS!CONVENTIONALNUMERICAL ANALYSISLIKElNITEELEMENTUTILIZESAMINIMUMAND MAXIMUM VALUES OF THE STRUCTURAL PARAMETERS
"UTINREALLIFESTRUCTURALSYSTEMSTHEPARAMETERS ARECHARACTERIZEDBYINHERENTRANDOMNESS-OST MECHANICAL AND STRUCTURAL SYSTEMS OPERATE IN RANDOMENVIRONMENTS-ARCZYK )NMOST PRACTICAL SITUATIONS THE MAJOR SOURCES OF SUCH RANDOMNESSCANBEIDENTIlEDINUNCERTAINTYOF MATERIAL PROPERTIES LOADING CONDITIONS ETC FOR EXAMPLETHEREMAYBEMEASUREMENTINACCURACY IN THE MANUFACTURE PROCESS 4HEREFORE THE CONCEPTOFUNCERTAINTYPLAYSANIMPORTANTROLE IN THE ENGINEERING DESIGN PROBLEMS4HE MOST COMMONAPPROACHTOPROBLEMSOFUNCERTAINTY ISTOMODELTHESTRUCTURALPARAMETERSASRANDOM VARIABLES
3TOCHASTIC IS SYNONYMOUS WITHhRANDOMv )T IS USED TO INDICATE THAT A PARTICULAR SUBJECT IS SEEN FROM POINT OF VIEW OF RANDOMNESS 3TOCHASTIC IS OFTEN USED AS COUNTERPART OF THE WORDhDETERMINISTICv WHICH MEANS THAT RANDOMPHENOMENAARENOTINVOLVED4HEREFORE STOCHASTIC MODELS ARE BASED ON RANDOM TRIALS WHILE DETERMINISTIC MODELS ALWAYS PRODUCE THESAMEOUTPUTFORAGIVENSTARTINGCONDITION /RIGLIO 4HESYSTEMNEEDSAPROBABILISTIC APPROACHASAREALISTICANDRATIONALPLATFORMFOR BOTHDESIGNANDANALYSIS3TOCHASTICPROCEDURES HAVE PROVIDES THE RATIONAL TREATMENT OF THE UNCERTAINTIES IN THE ASSESSMENT OF LOADING MATERIAL AND GEOMETRIC PROPERTIES OF THE STRUCTURE USING THE PROBABILITY AND STATISTIC APPROACH
0ARAMETERUNCERTAINTIESARETYPICALLYSPECIlED IN TERMS OF INTERVAL ANALYSIS MEMBERSHIP FUNCTIONS AND PROBABILITY DENSITY FUNCTIONS 0$&S 4HE MAIN CHARACTERISTICS OF THE INTERVAL ANALYSIS ARE THE VARIABLES REPRESENT LOWER AND
UPPER BOUNDS WAS THE LEAST ACCURATE METHOD FORUNCERTAINTYMODELLING#RESPO !SFOR MEMBERSHIP FUNCTIONS FUZZY LOGIC IS THE BASIS FOR ASSESSING THE UNCERTAINTIES IN THE SYSTEM OUTPUTWHICHPROVIDESANINTERMEDIATELEVELOF DETAIL4HEUNCERTAINTIESUSING0$&SAREREFERRED AS THE PROBABILISTIC METHODS THAT PROVIDE THE BEST DESCRIPTION OF THE UNCERTAIN PARAMETERS BY TREATING THEM AS RANDOM VARIABLES -ONTE
#ARLO 3IMULATION -#3 )MPORTANCE 3AMPLING ,ATIN(YPERCUBE3AMPLINGAND'ENERALIZED#ELL -APPING ARE AMONG THE NUMERICAL METHODS COMMONLY USED TO ESTIMATE 0$&S 3CHUÑLLER )N THIS STUDY THE STOCHASTIC METHOD INCORPORATES UNCERTAINTY AND VARIABILITY BASED ON-#34HESTUDYINVOLVEDTHESTATICANALYSISOF CYLINDERBLOCKMODELOFAFREEPISTONENGINETHAT USEDAPREDICTEDMAXIMUMCOMBUSTIONPRESSURE AROUND-0A4HEMODELLINGOFTHECOMPONENT WASPERFORMEDINlNITEELEMENTSOFTWAREINORDER TO ASSIGN ITS MAXIMUM BOUNDARY CONDITIONS MATERIALSPROPERTIESANDITSCONSTRAINTSPOSITION 4HESTOCHASTICSIMULATIONWASCONDUCTEDUSING DIFFERENT SOFTWARE WHICH USED -#3 BASED METHOD4HERESULTSFROMTHESIMULATIONPROCESS WILL GIVE THE DESIRED OUTPUT AND IDENTIFY THE UNCERTAIN AND CORRELATION AMONG THE DESIGN VARIABLESOFTHECOMPONENTS
34/#(!34)#-/$%,).'!.$02/#%33 3TOCHASTICMODELLINGAIMSONTHECHARACTERIZING THEDIFFERENTRANDOMVARIABLESTHATDESCRIBETHE UNCERTAINTIES INVOLVED IN THE SYSTEM "ASICALLY THE STRUCTURAL PROBLEMS INVOLVED LOADING AND RESISTANCEVARIABLESSUCHASPRESSUREAND9OUNGS MODULUS !S MENTIONED BY (ERMAN ET AL THE LOAD EFFECTS IN A STRUCTURE DEPEND ON A SET OF VARIABLESTHATARESTRUCTUREINDEPENDENTWHICH CHARACTERIZETHELOADINGANDSEVERALSTRUCTURE DEPENDENT PARAMETERS &OR EXAMPLE ON THE
STRENGTHSIDEPLASTICYIELDINGOFTHESTRUCTUREIS DEPENDSONTHESTATEOFSTRESSDISTRIBUTIONASWELL ASONTHEGEOMETRYANDMATERIALPROPERTIES )N STOCHASTIC MODELS OF STRENGTH SEVERAL GEOMETRICVARIABLESAREINVOLVESWHICHCANBE MODELLED AS RANDOM TO TAKE INTO ACCOUNT THE UNCERTAINTIESINFABRICATIONORINMEASUREMENT PROCEDURES INCLUDING THE MATERIAL PROPERTIES
3TOCHASTIC ANALYSIS IS AN ANALYSIS THAT RELATED TO A PROCESS INVOLVING A RANDOMLY DETERMINED SEQUENCE OF OBSERVATIONS WHERE EACH OF IS CONSIDERED AS A SAMPLE OF ONE ELEMENT FROM A PROBABILITY DISTRIBUTION 4HE UNCERTAINTY PARAMETERS IN THE DISTRIBUTION ARE TREATED AS CONTINUOUS RANDOM VARIABLES 4HE RANDOM VARIABLE WAS DISTRIBUTED ACCORDING TO THE PRESCRIBED DENSITY FUNCTION 0$& DESCRIBED A PROBABILISTICFUNCTIONWHERETHERESULTSWEREAN ESTIMATEDVALUE)NORDERTOlNDMOREACCURATE RESULTS STATISTICAL SIMULATION METHOD CAN BE USED TO GENERATE AN ARTIlCIAL DATA THAT EXHIBIT THESAMEMECHANICALBEHAVIOURASEXPERIMENTAL DATA-AHNKEN
3TOCHASTIC SIMULATION SHOWS THE UNCERTAIN VARIABLE WAS SIMULATED USING -#3 TECHNIQUE AS ILLUSTRATED IN &IGURE 4HE lGURE ILLUSTRATES THEIDEAOF-ONTE#ARLOORSTATISTICALSIMULATION
Random numbers on [0,1]
x1, x2,x3,...
Probabability Density Function (PDF) which describe the system
Results of simulation:
Material Properties, thickness, pressure, displacement, f(x)
x
&)'52%-ONTE#ARLO3IMULATIONOFASYSTEM
AS APPLIED TO AN ARBITRARY PHYSICAL SYSTEM BY ASSUMING THE EVOLUTION OF THE SYSTEM CAN BE DESCRIBEDBY0$&4HENTHE-#3WILLPROCEEDBY SAMPLINGFROMTHIS0$&WHICHNECESSITATESAFAST ANDEFFECTIVEWAYTOGENERATERANDOMNUMBERS UNIFORMLYDISTRIBUTEDONTHEINTERVAL
"YINTRODUCING-ONTE#ARLOMETHOD6ERENA INTHEANALYSISFORACONTINUOUSRANDOM
VARIABLESXTHE0$&DElNESTHEPROBABILITYTHAT WHENTHEVARIABLEISSAMPLEDAVALUELYINGINTHE RANGEXTOXDXISFX DXASIN&IGUREANDITCAN BEWRITTENAS
PROBX≤ Xg≤ XDX ≡0X≤ X|≤XDX FX DX
ANDIFFX ≥ 0 –∞ < X< ∞ THEN∞∫
–∞FXg DXg
4HE MOST IMPORTANT DISTRIBUTION IS THE
#UMULATIVE $ISTRIBUTION &UNCTION #$& WHERE IT HAS BECOME THE BASIC DISTRIBUTION IN -#3 4HE#$&GIVESTHEPROBABILITYTHATTHERANDOM FUNCTIONXISLESSTHANOREQUALTOXASILLUSTRATE IN&IGUREANDCANBEWRITTENAS
#$&≡ PROBXg≤ x) ≡&X
∞∫
–∞FXg DXg
f(x)dx = probability that the r.v.x’
is in dx about x f(x’)
x x‘
dx
&)'52%0ROBABILITYDISTRIBUTIONFUNCTION
&)'52%#UMULATIVEDISTRIBUTIONFUNCTION 1
F(x)
x
WHERE &X PROPERTIES OBEYS THE FOLLOWING CONDITIONS &X NONDECREASING &X ∈;=
AND &n∞ AND&∞
&ROM THE CUMULATIVE DISTRIBUTION -#3 WILL PROCEED THE SAMPLING PROCESS BY READING A QUANTILESQOFTHEDISTRIBUTION!QUANTILESISAN INVERSEFORMOFTHE#$&BYGENERATINGASAMPLE PFROMTHE#$&&X 4HEQUANTILETRANSFORMATION PROCEDURESARE-ARCZYK
s 3AMPLE A UNIFORM RANDOM NUMBER P IN
;=WHICHKNOWNAS &XP P
s 2EADTHEQUANTILEFUNCTIONQP FROMTHE DISTRIBUTION
s )NVERTTHEQUANTILEFUNCTIONANDSOLVEFOR THESIMULATEDVALUEXP
QP &nP XP
&OREACHDESIGNVARIABLESWITHUNCERTAINTIES THE POSSIBLE VALUES ARE DEFINED BY MEANS OF PROBABILITYDISTRIBUTIONSUCHAS'AUSSIAN.ORMAL 7EIBULL 5NIFORM ,OGARITHMIC AND $ISCRETE 'AUSSIAN DISTRIBUTIONS GX ARE MAINLY USED IN THE SIMULATION PROCESS IN ORDER TO KNOW HOW THE STRUCTURE BEHAVES4HE DISTRIBUTION IS CHARACTERIZEDBYITSTWOPARAMETERSTHEMEAN MANDTHEVARIANCEσ
g x x m
( ) exp – ¥
§¦
´
¶µ
¨ ª
©©
·
¹
¸¸ 1
2
1 2
2
S P S
34/#(!34)#3)-5,!4)/.
3TOCHASTICSIMULATIONWITHlNITEELEMENTMODEL STARTSBYSPECIFYINGTHETOLERANCESANDSCATTEROF ALLTHEINPUTVARIABLESUSEDINTHEMODEL)NTHE STATICANALYSISOFTHREEDIMENSIONALMODELOFTHE
CYLINDER BLOCK COMPONENT THE STIFFNESS MATRIX MIGHTBERANDOMDUETOUNPREDICTABLEVARIATION OF SOME MATERIAL PROPERTIES RANDOM COUPLING STRENGTH BETWEEN COMPONENTS UNCERTAIN BOUNDARY CONDITIONS ETC4HE BLOCK STRUCTURE WHICH IS COMPOUNDED WITH LINER AS SHOWN IN &IGURE WAS GIVEN A MAXIMUM PREDICTED COMBUSTIONPRESSURE0OF-0AATTHEUPPER EDGEOFTHELINER4HEREWEREALSOTWOPRELOADED PRESSURE0AND0OCCURATTHEUPPERANDINNER HOLEATTHEBLOCKRESPECTIVELY3INCETHEMATERIAL ISCOMPOUNDEDTHEREARETWOTYPEOFMATERIAL PROPERTIES USED IN THE ANALYSIS WHICH ARE IRON MATERIAL ANDALUMINIUMMATERIAL FORLINER ANDBLOCKRESPECTIVELY
&OR THIS PARTICULAR COMPOUNDED STRUCTURE GENERALLY THE INNER TUBE OR LINER IS PUT INTO COMPRESSIONANDTHEOUTERPARTWILLBEINTENSION 7HENANINTERNALPRESSUREISAPPLIEDITWILLCAUSES ATENSILEHOOPSTRESSATTHEINSIDEOFTHEOUTER TUBEWHICHWILLMAKETHELINERDIAMETERDECREASE ANDTHEOUTERLINERDIAMETERINCREASEDASSHOWN INEQUATION AND 2YDER &ORINNER LINERDIAMETERITISDECREASEDBY
1
E vP xd1
¥
§¦
´
¶µ
S
WHERE D IS THE DIAMETER &OR THE OUTER LINER DIAMETERISINCREASEDBY
1
E vP xd1
¥
§¦
´
¶µ
S
4HEMODELISEXECUTEDFORNUMBEROFTIMES n USING'AUSSIANDISTRIBUTIONTOIDENTIFY HOW THE MODEL PERFORMED AND PROVIDE THE INFORMATIONONTHEMODELOUTPUTS4HEOUTPUTS INDICATE THE CORRELATION BETWEEN THE VARIABLES AND ITS PERFORMANCE SCATTER ALSO IDENTIFIES VARIABLE THAT INFLUENCE THE OUTPUT MOST4HE CORRELATIONSBETWEENTWOVARIABLESEXPRESSTHE STRENGTHANDRELATIONSHIPBETWEENTHEVARIABLES ALSO TAKE INTO ACCOUNT THE SCATTER IN OTHER VARIABLES IN THE SYSTEM4HE CORRELATION VALUES
RANGEFROMnTOWHEREVALUECLOSETOEITHER nORINDICATESASTRONGCORRELATION4WOTYPE OFPOSSIBLECORRELATIONCOEFlCIENTWEREUSEDIN THESIMULATIONPROCESSWHICHARE-3#2OBUST
$ESIGN
• 0EARSONSCORRELATIONCOEFlCIENTORLINEAR CORRELATIONCOEFlCIENT RWHICHMEASURES THE LINEAR CORRELATION BETWEEN VARIABLES
&ORTWOSTOCHASTICVARIABLESXANDYWITH Ix.RIS
r
x m y m x m y m
i x i y
i
i x i y
i
¨ ª©
·
¹¸
¤
¤
2 2
P3 P1
P2
&)'52%&INITEELEMENTMODELOFENGINECYLINDERBLOCK
• 4HE3PEARMANRANKCOEFlCIENTORNONLINEAR CORRELATIONCOEFlCIENT RSISMORERELIABLE IN DETERMINING SIGNIFICANT RELATIONSHIP BETWEENTHEVARIABLES4HECOMPUTATION OFTHECORRELATIONISPERFORMEDBYRANKING THEVARIABLESFROMTHEHIGHESTTOLOWEST ASSIGNINGFROMTO.4HERANKINGISUSED TOCREATE0IE#HARTSTOSHOWTHEINmUENCE OFINPUTSONOUTPUTS4HERANKISCOMPUTED ASTHELINEARCORRELATIONBETWEENTHERANKS OFXI2IANDTHERANKSOFYI3IWHICHIS
r
R m S m
R m S m
s
i x i y
i
i R
i i i S
¨ ª©
·
¹¸¨ª
¤
¤
2 ©©¤
2·¹¸
&ROMTHESIMULATIONDECISIONMAPWILLSHOWS THE RELATION BETWEEN THE INPUT VARIABLES AND THEOUTPUT4HEINFORMATIONWILLASSISTFORBETTER
Finite Element Model
Stochastic Design Improvement Stochastic Simulation - Model Health Checking
&)'52%&LOWCHARTOFSTOCHASTICSIMULATIONPROCESS
Pressure 1, P1
Max Displacement Magnitude, Lmax
Max von Mises stress, Smax 1
11
3
&)'52%$ECISIONMAPOFSTOCHASTICSIMULATION
RESULTS BY SIMULATING THE DESIGN IMPROVEMENT PROCESS )N THIS PROCESS THE SPECIFIED TARGET VALUES WHICH INCORPORATE TOLERANCE WERE ASSIGNEDINTHEOUTPUTVARIABLES&IGURESHOWS THEmOWCHARTOFTHEOVERALLSIMULATIONPROCESS 2%35,43!.$$)3#533)/.
3TOCHASTIC SIMULATION ENABLE TO IDENTIFY THE RELATIONSHIP BETWEEN THE INPUT AND OUTPUT VARIABLES OF THE CYLINDER BLOCK STATIC ANALYSIS AS ILLUSTRATED IN THE DECISION MAP IN &IGURE 4HIS MAP INDICATED THE RELATION BETWEEN THE STRONGEST INPUT VARIABLES AND OUTPUT VARIABLES BASED ON THE STOCHASTIC PROCESS &ROM THE MODEL HEALTH SIMULATION RESULTS FOR TIMES EXECUTIONSTHEDECISIONMAPSHOWSTHERELATION BETWEEN0ANDTWOOUTPUTVARIABLESWHICHARE MAXIMUM DISPLACEMENT MAGNITUDE ,MAX AND MAXIMUMVON-ISESSTRESS3MAX
)NTHEPIECHARTIN&IGUREITSHOWSTHAT0 CONTRIBUTEPERCENTOFOVERALLRESULTFOR3MAX SINCESTRESSANDPRESSUREHASALINEARRELATIONSHIP
%VENTHOUGHTHEPIECHARTOFTHE3MAXSHOWSTHAT 0INDICATEDASTHEMOSTINmUENTIALVARIABLETHERE WEREOTHERVARIABLESTHATGAVECERTAINEFFECTTO THEOUTPUTSWHICHARETHE.UORPOISONRATIOν AND9OUNGSMODULUS%OFMATERIALWHICHIS ALUMINIUM FOR BLOCK MATERIAL )T IS ALSO KNOWN THAT STRESS HAVE LINEAR RELATIONSHIP WITH% AND STRAINWHERESTRAINISRELATEWITHν&ROMTHEPIE CHARTITSHOWSTHATTHEMOSTINmUENTIALMATERIAL AFFECTING THE STRESS VALUE OF THE COMPOUNDED BLOCKSTRUCTUREISALUMINIUM
4HE GRAPH IN &IGURE INDICATES A LINEAR POSITIVE CORRELATION OF BETWEEN3MAX AND 0WITHTHEMOSTLIKELYVALUEOFTHESTRESSIS -0A AT 0EQUAL TO -0A WHICH IS SLIGHTLY HIGHERTHANTHEMAXIMUMPREDICTEDCOMBUSTION PRESSUREOF-0A4HISPHENOMENONEXPLAINED THEUNCERTAINTYOFTHEANALYSISWHERETOLERANCE WASTAKENASUNCERTAINTYFACTORWHICHOCCURRED INTHEINPUTANDOUTPUTVALUEOFTHEVARIABLES
&ORMAXIMUMDISPLACEMENT,MAXTHEDECISION MAP IN &IGURE SHOWS THAT THE RELATION WITH 0 AND 3MAX !S SHOWN IN &IGURE 0 INDICATES ABOUTPERCENTFROMOVERALLVARIABLESTHAT INFLUENTIAL ,MAX4WO OTHER VARIABLES THAT ALSO CONTRIBUTEINDISPLACEMENTOFTHESTRUCTUREARE%
AND%WITHANDPERCENTRESPECTIVELY
!SSTATEDTHISISDUETOTHETENSILEHOOPSTRESS THATOCCURREDATTHEINSIDEOFTHEOUTERLINERTUBE )T ALSO SHOWS STRONG POSITIVE LINEAR CORRELATION OFBETWEEN,MAXAND0ASSHOWNIN&IGURE WITHTHEMOSTLIKELYVALUESOFDISPLACEMENTIS
XMMAT0EQUALTO-0A
&ROMTHECORRELATIONGRAPHSIN&IGUREAND BOTH3MAXAND,MAXSHOWSTHATTHEMOSTLIKELY VALUEFOR0IS-0A"YUSING,MAXASTHETARGET VARIABLEANIMPROVEMENTHASBEENMADEINORDER TO GET BETTER RESULTS FOR ,MAX )N THE STOCHASTIC DESIGN IMPROVEMENT 3$) THE SOFT TARGET VARIABLEISSETFOR,MAXANDTHE9OUNGSMODULUS%
FORBOTHMATERIALSWERESETASDESIGNVARIABLE)N STATISTICALPOINTOFVIEWTHEMOSTLIKELYVALUEFROM THEPREVIOUSSIMULATIONWASTAKENASTHEMEAN
&)'52%0IECHARTOF-AXVON-ISESSTRESS
Pressure 1, P1: 51.86%
Material 2, Nu: 18.26 Material 2, E: 7.32%
Material 1, E: 4.77%
Pressure 2, P2: 2.17%
Material 2, Rho: 1.72%
Material 2, G: 1.50%
Material 1, Nu: 1.14%
Material 1, G: 0.73%
Others: 0.54%
Max von Mises stress, Smax
&)'52%0IECHARTOFDISPLACEMENTMAGNITUDE
Pressure 1 - P1: 44.49%
Material 1 - E: 26.37%
Material 2 - E: 16.97%
Pressure 2 - P2: 2.73%
Material 2 - G:2.40%
Material 1 - Nu: 2.26%
Material 1 - G: 1.64%
Pressure 3 - P3: 1.58%
Material 2 - Nu: 0.76%
Others: 0.80%
Max Displacement Magnitude, Lmax
&)'52%#ORRELATIONGRAPHOFVON-ISESSTRESSAGAINSTPRESSURE
&)'52%#ORRELATIONGRAPHOFDISPLACEMENTAGAINSTPRESSURE
most likely value = 0.283
0.256 m = 0.286 0.317
0.258 m = 0.283 0.314
(×10–2)
(×10–2)
&)'52%'AUSSIANDISTRIBUTIONOF,MAXFORA -ODELHEALTHSIMULATIONB 3TOCHASTICDESIGNIMPROVEMENT
2%&%2%.#%3
#RESPO , ' /PTIMIZATION OF 3YSTEM WITH 5NCER TAINT Y )NITIAL $EVELOPMENTS FOR 0ERFORMANCE2OBUSTNESSAND2ELIABILITY"ASED
$ESIGNS )#!3% 2EPORT .O .!3!#2
(ERMAN ' - #HRISTOPH % " #HRISTIAN ' " AND 3OARES#'5NCERTAINTIESIN0ROBABILISTIC .UMERICAL !NALYSIS OF 3TRUCTURES AND 3OLIDS n 3TOCHASTIC &INITE %LEMENTS 3TRUCTURAL3AFETY n
-AHNKEN2)DENTIlCATIONOF-ATERIAL0ARAMETERS FOR#ONSTITUTIVE%QUATIONS
%RWIN3TEIN2ENEDE"ORSTAND4HOMAS*2(UGHES
%DS %NCYCLOPEDIAOF#OMPUTATIONAL-ECHANICS
-ARCZYK * 0RINCIPLES OF 3IMULATION"ASED
#OMPUTER!IDED%NGINEERING-ADRID
-ARCZYK * 3TOCHASTIC 3IMULATION 5SING -3#
2OBUST $ESIGN n "ASIC4HEORY -3# 3OFTWARE
#ORPORATION53!
4!",%/BSERVABLESIMULATIONDATA /BSERVABLE
DATA
,MAX X MM
3MAX -0A
%-0A MATERIAL
%-0A MATERIAL
0 -0A
!CTUAL
CONDITION
-OSTLIKELY
VALUE
3$)
FORNEWGAUSSIANDISTRIBUTIONINTHE3$)&IGURE A ANDB SHOWSTHEDISTRIBUTIONBEFOREAND AFTERTHE3$)SIMULATIONRESPECTIVELY
4HEAREAOFACCEPTABLEDATAINTHEGAUSSIAN DISTRIBUTION OF 3$) HAS BEEN CHANGE COMPARE TOTHEEXISTINGAREAIN&IGUREA 4HEEXISTING DISTRIBUTIONSHOWSTHATTHELOWERLIMITOF WAS INCREASED TO IN THE 3$) DISTRIBUTION 4HE UPPER LIMIT OF THE EXISTENCE DISTRIBUTION DECREASED FROM TO IN THE 3$) DISTRIBUTION4HISSHOWSTHATTHE3$)DISTRIBUTION WASSHIFTINGTOTHELEFTSINCETHETARGETVARIABLE ISTOREDUCETHEMAXIMUMDISPLACEMENTASSMALL ASPOSSIBLE4ABLESHOWSTHEOBSERVABLEDATAOF INPUTANDOUTPUTVARIABLESOFBOTHSIMULATIONS 4HETABLESHOWSTHREEDIFFERENTTYPEOFDESIGN WHERE DESIGNERS CAN CHOOSE ACCORDING TO THE PRIORITY OF THE DESIGN4HE ACTUAL CONDITION GAVEHIGHER,MAXCOMPAREDTOOTHERSGIVENTHE SAMEVALUEOF3MAXWITH3$))NTERMSOFMATERIAL PROPERTIES 3$) HAS THE LOWEST VALUE OF %AND HIGHESTVALUEOF%COMPAREDTOOTHERS"UTFOR 03$)GAVETHELOWESTVALUE
#/.#,53)/.
3UMMARIZEFROMTHISSTUDYEXPLAINEDTHEEFFECT OF DESIGN VARIABLES WHICH WERE TREATED AS CONTINUOUS RANDOM VARIABLE USING STOCHASTIC SIMULATION &ROM THE STATIC ANALYSIS OF FINITE ELEMENT MODEL THE MODEL HEALTH SIMULATION HASSHOWNTHERELATIONSHIPAMONGTHEOUTPUT
ANDINPUTVARIABLESOFTHEDESIGN4HESTOCHASTIC PRODUCED SOME INDICATOR DIAGRAM SUCH AS THE DECISION MAP SCATTER OF CORRELATION GRAPH AND PIECHARTWHICHCANBEUSEDTOINDICATETHEMOST INmUENTIALVARIABLE!NIMPROVEMENTALSOCANBE MADEBYIDENTIFYINGWHICHVARIABLESINmUENTTHE OVERALLPERFORMANCEOFTHEDESIGN
&ROMTHESIMULATIONRESULTSOFCYLINDERBLOCK ANALYSISEVENTHOUGHTHESTRESSRESULTSDOESNOT EFFECTMUCHONTHEDESIGNSINCETHEMAXIMUM RESULTS IS FAR BELOW THE YIELD STRENGTH BUT THE INmUENCEOFUNCERTAINTYIN0AND%GAVECERTAIN AFFECTONTHEDESIGNINTHEDATASHOWNIN4ABLE 4HEREAREMANYCOMBINATIONSOFRESULTSTHAT CANBEUSEDFORINTHEDESIGNDEPENDINGONTHE END USER SPECIlCATIONS $ESIGNER CAN CHOOSE EITHERTOSPECIlEDCERTAINPRESSURETOTHEBLOCK ORTARGETINGCERTAINALLOWABLEDISPLACEMENTTHAT THESTRUCTUREMAYEXPERIENCE!SFORTHEANALYSIS SINCETHERUNNINGENGINEGAVEAMAXIMUM0OF -0AITISBETTERTOCHOOSETHE3$)COMBINATIONS EVEN THOUGH THE % VALUE SLIGHTLY HIGHER THAN ACTUALINTERMSOFCOSTBUTITENABLETOREDUCE THE%VALUEWHICHCANBALANCETHEOVERALLCOST OFTHEDESIGN
!#+./7,%$'%-%.43
4HISPROJECTWASSPONSOREDBY-ALAYSIAN-INISTRY OF 3CIENCE4ECHNOLOGY AND )NNOVATION UNDER 0ROJECT)20!02
-#32OBUST$ESIGN6ERSIONR5SERS-ANUAL -3#3OFTWARE#ORPORATION53!
2YDER ' ( 3TRENGTH OF -ATERIALS -ACMILLAN
%DUCATION,TD5+
3CHUÑLLER ') #OMPUTATIONAL 3TOCHASTIC -ECHANICSn2ECENT!DVANCES#OMPUTERSAND 3TRUCTURESn
/RIGLIO6INCENZO3TOCHASTIC&ROM-ATH7ORLD
! 7OLFRAM 7EB 2ESOURCE ONLINE HTTP MATHWORLDWOLFRAMCOM3TOCHASTICHTML TH -AY
6ERENA -5 )NTRODUCTION TO -ONTE #ARLO -ETHODS #OMPUTATIONAL 3CIENCE AND
%DUCATIONAL 0ROJECT 5NITED 3TATES ONLINE HTTPWWWPHYORNLGOVCSEPTH-AY
