AIAA Optimal Sampling Techniques for Zone- Based Probabilistic Fatigue Life Prediction

Transcription

1 AIAA Optmal Samplng Technques or Zone- Based Probablstc Fatgue Le Predcton M. P. Enrght Southwest Research Insttute San Antono, TX H. R. Mllwater Unversty o Texas at San Antono San Antono, TX 43 rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conerence and Exhbt Non-Determnstc Approaches Forum Aprl 22-25, 2002 Denver, Colorado

2 AIAA OPTIMAL SAMPLING TECHNIQUES FOR ZONE-BASED PROBABILISTIC FATIGUE LIFE PREDICTION Mchael P. Enrght Southwest Research Insttute San Antono, Texas Harry R. Mllwater Unversty o Texas at San Antono San Antono, Texas Abstract When computatonal accuracy must be assured, the eort assocated wth smulaton-based probablstc le predcton can be sgncant. DARWIN TM (Desgn Assessment o Relablty Wth INspecton) s a smulaton-based computer program or probablstc atgue le predcton o rotors and dsks n commercal arcrat jet engnes. DARWIN uses a zone-based probablstc approach n whch the dsk s dvded nto zones o approxmately equal rsk, and the number o smulatons (samples) s unquely dened or each zone. Snce atgue alure probabltes are typcally very small or these dsks (.e., on the order o 0-9 per lght cycle or 0-5 cumulatve), a large number o Monte Carlo samples may be requred to satsy computatonal accuracy requrements. In ths paper, technques or reducng the total varance assocated wth probablstc le predcton are compared or a dsk wth a xed number o zones. It s shown that allocaton o samples based on rsk contrbuton actors or an optmal approach can sgncantly reduce the total number o samples requred or a speced accuracy. In addton, a hybrd approach s presented n whch optmal samplng s combned wth a semautomated zone renement procedure. Ths approach s llustrated or an arcrat rotor dsk n whch t s shown that zone renement prmarly nluences mean dsk rsk, whereas optmal samplng nluences dsk rsk varance. These strateges can be used to mprove the ecency o zone-based probablstc atgue le predctons where computatonal accuracy must be assured. Introducton The need or probablstc damage tolerance methods or le management o arcrat turbne rotors s well recognzed. In act, a recent FAA advsory crcular (AC33.4-) speces that a probablstc approach be used to account or rare metallurgcal anomales (reerred to as hard alpha (HA)) that may be present n ttanum rotor dsks. Although rare, these anomales have led to uncontaned engne alures such as the ncdent n Soux Cty n (Fgure ). Fgure. Uncontaned engne alure at Soux Cty, Iowa was drectly lnked to a rare metallurgcal anomaly. Under the drecton o the Federal Avaton Admnstraton, a probablstc damage tolerance computer program, DARWIN TM, has been developed at Senor Research Engneer, Member AIAA Assstant Proessor, Member AIAA Copyrght 2002 Southwest Research Insttute. Publshed by the Amercan Insttute o Aeronautcs and Astronautcs, Inc. wth permsson. Amercan Insttute o Aeronautcs and Astronautcs

3 Southwest Research Insttute or atgue le predcton o ttanum rotors & dsks contanng hard alpha deects 3. It was developed n collaboraton wth a Steerng Commttee consstng o our major U.S. arcrat engne manuacturers (General Electrc, Honeywell, Pratt & Whtney, and Rolls-Royce). The FAA has stated that use o DARWIN s an acceptable method or complyng wth AC DARWIN uses a zone-based probablstc approach 4 n whch the dsk s dvded nto zones o approxmately equal rsk, and the number o smulatons (samples) s unquely dened or each zone. Snce atgue alure probabltes are typcally very small or these dsks (.e., on the order o 0-9 per lght cycle or 0-5 cumulatve), a large number o Monte Carlo samples may be requred to satsy computatonal accuracy requrements. Snce metallurgcal anomales (deects) can occur at any locaton, the dsk s dvded nto a number o zones o approxmately equal rsk (Fgure 2). The probablty o alure p wthn a sngle zone s p = δ p (2) where p = PF ( ) = probablty o racture n zone gven an ntal deect, and zone. The dsk alure probablty the m zone F values: 2 PF ( ) = Pgx [ ( ) 0] P n s the unon o P = P[ F F F ] (3) m Over the past several years, a number o enhancements have been added to DARWIN to mprove computatonal ecency (e.g., le approxmaton uncton or determnstc atgue crack growth computatons 5, mportance samplng or zone-based varance reducton 6-8, sem-automated zone renement 9 or mproved zone dscretzaton, among others). However, none o these methods address the allocaton o samples to specc zones. In ths paper, technques or reducng the total varance assocated wth probablstc le predcton are compared or a dsk wth a xed number o zones. It s shown that reallocaton o samples based on rsk contrbuton actors or an optmal approach can sgncantly reduce the total number o samples requred or a speced accuracy. In addton, a hybrd approach s presented n whch optmal samplng s combned wth a sem-automated zone renement procedure. Ths approach s llustrated or an arcrat rotor dsk n whch t s shown that zone renement nluences mean dsk rsk, whereas optmal samplng nluences dsk rsk varance. Zone-Based Probablstc Fatgue Le Predcton The alure lmt state g( X ) n DARWIN s based on low cycle atgue alure: where gx ( ) = K K( X ) () c K = racture toughness, K( X ) = stress c ntensty actor, and X = nput varable vector. Fgure 2. Zone-based rsk assessment model used n DARWIN. The deect occurrence rate δ s modeled as a Posson dstrbuton. It s assumed that the probablty o havng two or more sgncant deects n a dsk s neglgble, so Eqn. 3 can be smpled as or P m P( F ) (4) = 2 Amercan Insttute o Aeronautcs and Astronautcs

4 P m δp (5) σ pˆ = pˆ ( ˆ p) = (6) n Opportuntes or Computatonal Ecency When perormng probablstc atgue le predctons, a number o opportuntes are avalable or mprovng the overall ecency o the computatons. Over the past several years, the ollowng enhancements have been added to DARWIN to address code ecency 0 : Determnstc Le Predcton I Monte Carlo Smulaton (MC) s used, the alure lmt state must be evaluated or each random sample usng a cycle-based atgue crack growth algorthm (Flght_Le 0 ). In DARWIN., the LAF (Le Approxmaton Functon) was ntroduced to reduce the number o calls to the Flght_Le algorthm. The LAF creates an array o determnstc le and assocated crack area values or a amly o ntal deects. Durng MC smulaton, crack growth s evaluated by nterpolaton usng values n the LAF arrays, whch substantally reduces computaton tme. Zone-Based Varance Reducton Importance Samplng was ntroduced n DARWIN , 0. The zone alure probablty s rst computed wthout nspecton usng an ecent numercal ntegraton scheme, ollowed by samplng wth nspecton ocused n the alure regon. Ths technque s sgncantly aster than standard MC, partcularly the zone alure probablty s very small. Zone Dscretzaton/Renement The dsk alure probablty s dependent on the level o dscretzaton o the zones, and decreases wth ncreasng dscretzaton. Snce computaton tme ncreases as zones are added, an eort should be made to dscretze only the zones that have a sgncant contrbuton to rsk. In DARWIN 3.5, a semautomated zone renement algorthm 9-0 was added allowng the user to denty and subdvde hgh rsk zones. Dsk-Based Varance Reducton where p ˆ = probablty o racture estmate n zone gven an ntal deect, and zone. n = number o samples n The mean and varance o the dsk alure probablty are as ollows: µ = δ pˆ (7) σ Pˆ m = m ˆ δ σ P pˆ = = (8) where p ˆ s the alure probablty estmate or the dsk. The condence bounds or the dsk, Fgure 3, are lower bound = µ ˆ k α σ ˆ (9) P /2 P upper bound = µ ˆ + k α σ ˆ (0) P /2 P where k α /2 = standard normal varate or a ( α ) condence nterval. Probablty Densty samplng error -k α/2 µ p ˆ (-α) condence nterval Target rsk k α/2 Fgure 3. Condence ntervals assocated wth smulaton-based probablstc atgue le predcton. The varance o the alure probablty n each zone σ ˆp s dependent on the number o samples : Dsk-based varance reducton ocuses on decreasng the overall dsk varance by reallocatng samples to zones that have the most sgncant contrbuton to rsk. Dsk rsk condence bounds can be used to ensure that 3 Amercan Insttute o Aeronautcs and Astronautcs

5 the alure probablty o a dsk does not exceed the target value (.0 E-9 as speced n AC33.4- ) or a gven condence (e.g., 95%). As shown schematcally n Fgure 4, dsk varance can be reduced, then the mean dsk rsk can be ncreased yet satsy the rsk target or a gven condence. Probablty Densty Change n dsk mean Target rsk Decreasng dsk varance Dsk Falure Probablty P Fgure 4. Dsk Varance reducton allows hgher mean dsk alure probablty or a gven target rsk. Two approaches or reallocatng samples are consdered or reducng dsk varance: RCF (rsk contrbuton actor) approach, and an optmal approach. For the RCF approach, the number o samples n each zone s based on an estmate o the zone n RCF : where 2 m kα /2 2 2 γ P = N = δ p( p) (5) Further detals regardng the development o these relatonshps can be ound n Res Applcaton o Dsk-Based Varance Reducton or a Ttanum Rotor Dsk wth a Fxed Number o Zones Consder the arcrat rotor dsk shown n Fgures 5 and 6. The desgn le o the dsk s 20,000 lght cycles. Internal stresses and temperatures are dented usng nte element analyss based on operatonal loadng condtons. Fve prmary random varables are consdered or probablstc analyss. The man descrptors or three o these varables (stress scatter, le scatter, and nspecton tme) are ndcated n Table. For the remanng two varables (deect area, probablty o detecton (POD)), emprcal dstrbutons (AIA POST95-3FBH-3FBH deect dstrbuton, #3 FBH : Reject Calbraton POD Curve) ound n AC33.4- were used or probablstc atgue le predctons. 44 zones were used to model the dsk. Addtonal detals regardng the selecton o random varables and assocated dstrbutons can be ound n Res. 0 and 2. 2 n = RCF N () where RCF δ ˆ p = Pˆ (2) k N = γ and 2 α /2 2 ( P ) P = total number o samples n dsk γ P P P ˆ = = relatve samplng error For the optmal approach 8, the number o samples s: n = δ m p( p) N δ p ( p ) = (3) (4) Fgure 5. Arcrat rotor dsk applcaton cross secton. Table. Ttanum Arcrat Rotor Applcaton Example Random Varables Random Medan COV(%) Dstrbuton Varable Stress.0 20 Lognormal Scatter Le Scatter.0 40 Lognormal Inspecton Tme 0,000 cycles 20 Normal 4 Amercan Insttute o Aeronautcs and Astronautcs

6 4.0E+0 3.0E+0 TARGET RISK (DTR) FAA AC samples per zone 000 0,000 00,000 PROBABILITY DENSITY 2.0E+0.0E+0 UPPER 95% CONFIDENCE BOUND MEETS TARGET RISK AT 00,000 SAMPLES PER ZONE 0.0E E E-0.00E-09.05E-09.0E-09.5E-09.20E-09 PROBABILITY OF FAILURE (WITHOUT INSPECTION) Fgure 8. Increasng the number o zone samples reduces varance but s computatonally expensve. Fgure 6. Arcrat rotor dsk applcaton 3D vew. Falure probablty results are shown n Fgure 7 (00 samples per zone). It can be observed that the mean dsk alure probablty s below the target rsk value (.0 E-9 ). However, the upper condence bound result (no nspecton) s well above the target rsk, so varance reducton s needed. As shown n Fgure 8, the desred varance reducton can be acheved by ncreasng the number o samples n each zone to 00,000. However, ths requres a total o over 4 mllon samples or the dsk. In Fgure 9, a comparson o the condence bounds versus number o dsk samples s shown or three sample allocaton approaches: unorm (.e., same number o samples n all zones), RCF, and optmal. For a speced number o samples t can be observed that, compared to the unorm approach, the condence bounds are narrower or the RCF and optmal approaches. The target rsk can be acheved wth the RCF and optmal approaches wth approxmately 40,000 dsk samples (over 4 mllon dsk samples are requred or the unorm approach). It s nterestng to note that the optmal method converges only slghtly aster than the RCF approach. 2.0E+09 WITHOUT INSPECTION Hybrd Approach: Zone Renement Combned wth Optmal Samplng PROBABILITY DENSITY.5E+09.0E E+08 WITH INSPECTION LOWER (95%) CONFIDENCE BOUND TARGET RISK (DTR) FAA AC UPPER (95%) CONFIDENCE BOUND 0.0E E E-0.0E-09.5E E-09 PROBABILITY OF FAILURE (WITHOUT INSPECTION) Fgure 7. Upper condence bounds on dsk rsk results or xed number o zones ntally do not satsy FAA target rsk. As stated prevously, the dsk alure probablty decreases wth ncreasng zone dscretzaton. As shown schematcally n Fgure 0, the total number o samples wthn a dsk remans xed, then the prmary eect o zone renement s a decrease n mean dsk alure probablty. On the other hand, dsk varance reducton prmarly nluences the dsk rsk COV (Fgure ). A hybrd approach s proposed n whch zone renement s used wth relatvely ew samples per zone to establsh a mean dsk alure probablty value that s below the target rsk. Dsk varance reducton methods are then appled to ensure that the dsk rsk upper condence bound satses the target alure probablty constrant. 5 Amercan Insttute o Aeronautcs and Astronautcs

7 PROBABILITY OF FAILURE.6E-09.4E-09.2E-09.0E E-0 6.0E-0 Unorm - lower bound (95%) Unorm - mean Unorm - upper bound (95%) RCF - lower bound (95%) RCF - mean RCF - upper bound (95%) Optmal - lower bound (95%) Optmal - mean Optmal - upper bound (95%) TARGET RISK (DTR) 4.0E-0.0E+03.0E+04.0E+05.0E+06.0E+07 NUMBER OF SAMPLES IN DISK Fgure 9. Comparson o three dsk varance reducton technques (unorm, RCF, optmal). Probablty Densty ZONE REFINEMENT Increasng zone renement Dsk Falure Probablty P Fgure 0. For a xed number o dsk samples, zone renement prmarly nluences mean dsk rsk. Applcaton Example: Hybrd Approach Ths example llustrates the hybrd approach whch combnes zone renement wth optmal samplng dscussed n the prevous secton. Consder agan the arcrat rotor dsk shown n Fgures 5 and 6. The dsk s subjected to ve successve zone renements usng the sem-automated algorthm n DARWIN 3.5 9,0 (Fgure 2). The number o dsk samples s held constant durng zone renement (3800 total dsk samples). A nal teraton s perormed n whch the samples are reallocated to rsk crtcal zones usng the optmal samplng approach. Probablty Densty OPTIMAL SAMPLING Increasng number o samples Dsk Falure Probablty P Fgure. Optmal samplng prmarly nluences dsk rsk COV. 6 Amercan Insttute o Aeronautcs and Astronautcs

8 Renement 38 Zones (a) Renement 4 36 Zones (d) Renement 2 53 Zones (b) Renement 5 22 Zones (e) Renement 3 82 Zones (c) Fgure 2. Hybrd approach applcaton zone renement sequence. 7 Amercan Insttute o Aeronautcs and Astronautcs

9 Normalzed dsk rsk mean and COV values are shown n Fgure 3 or each teraton. It can be observed that durng teratons -5 (zone renement only), the dsk rsk mean decreases sgncantly, but dsk rsk COV remans relatvely hgh. Durng teraton 6 (optmal samplng only) the dsk rsk COV decreases sgncantly (note: the total number o samples s ncreased to 4700 or the nal teraton). In Fgure 4, the 95% condence bounds are shown or zone renement teratons -5 ( unorm ndcates equal number o samples n each zone wth a constant number o dsk samples). For comparson purposes, the condence bounds assocated wth the optmal approach are also shown (note: n Fgure 4, the optmal approach s appled at each zone renement teraton). For ths example, the target rsk could have been acheved by applyng optmal samplng ater zone renement teraton number 2. On the other hand, the target could also be acheved usng zone renement only (target satsed at zone renement 3). However, snce the human tme assocated wth zone renement s generally sgncantly greater than or dsk varance reducton, optmal samplng should be appled as soon as t s practcal to do so (.e., apply as soon as mean dsk alure probablty s sucently lower than target rsk). NORMALIZED FAILURE PROBABILITY (P / P, ) ZONE REFINEMENT ONLY MEAN (NO INSP) COV (NO INSP) MEAN (WITH INSP) COV (WITH INSP) ZONE REFINEMENT ITERATION OPTIMAL SAMPLING ONLY Fgure 3. Hybrd approach results: Iteratons -5 - zone renement, teraton 6 optmal samplng. 8 Amercan Insttute o Aeronautcs and Astronautcs

10 PROBABILITY OF FAILURE 3.0E E E-09.5E-09.0E-09 Unorm - lower bound (95%) Unorm - mean Unorm - upper bound (95%) Optmal - lower bound (95%) Optmal - mean Optmal - upper bound (95%) TARGET RISK (DTR) FAA AC E-0 0.0E ZONE REFINEMENT ITERATION Fgure 4. Comparson o 95% condence bounds or unorm samplng and optmal samplng versus renement teraton. Conclusons and Recommendatons Ecent technques or reducng dsk varance were presented and llustrated or a dsk wth a xed number o zones. It was shown that, compared to the unorm approach, the RCF and optmal dsk varance reducton technques can be used to satsy target rsk requrements wth sgncantly ewer samples. For the example problem consdered, the condence bounds assocated wth the optmal approach converge only slghtly aster than those assocated wth the RCF approach. A hybrd approach was also presented n whch zone renement s combned wth optmal samplng. It was shown that the number o dsk samples remans xed durng renement, zone renement prmarly nluences the mean dsk rsk, whereas optmal samplng prmarly nluences dsk rsk COV. Ths hybrd approach appears to be a good strategy or meetng target rsk requrements usng a mnmum o human and computer resources. Acknowledgements Ths work was supported by the Federal Avaton Admnstraton under Cooperatve Agreement 95-G- 04 and Grant 99-G-06. The authors wsh to thank the FAA Techncal Center project managers, Bruce Fenton and Joe Wlson or ther contnued dlgence and encouragement and Tm Mouzaks o the FAA Engne and Propeller Drectorate or hs contnued support. The ongong contrbutons o the TRMD Steerng Commttee (Darryl Lehmann, Pratt & Whtney; Sandeep Muju, Honeywell; Jon Tschopp, General Electrc; Geo Ward, Rolls-Royce) are also grateully acknowledged. 9 Amercan Insttute o Aeronautcs and Astronautcs

11 Reerences. "Advsory Crcular - Damage Tolerance or Hgh Energy Turbne Engne Rotors," U.S. Department o Transportaton, Federal Avaton Admnstraton, AC 33.4-, Washngton, DC, Jan "Arcrat Accdent Report - Unted Arlnes Flght 232 McDonnell Douglas DC-0-0 Soux Gateway Arport, Soux Cty, Iowa, July 9, 989," Natonal Transportaton Saety Board, NTSB/AAR-90/06, Washngton, DC, Nov Southwest Research Insttute, Alled Sgnal, General Electrc, Pratt and Whtney, Rolls-Royce Allson, Scentc Formng Technologes, "Turbne Rotor Materal Desgn - Fnal Report," Federal Avaton Admnstraton, DOT/ FAA/ AR-00/64, Washngton, DC, Dec Mllwater, H.R., Ftch, S., Wu, Y-T., Rha, D.S., Enrght, M.P., Leverant, G.R., McClung, R.C., Kuhlman, C.J., Chell, G.G., Lee, Y.-D., A probablstcally-based damage tolerance analyss computer program or hard alpha anomales n ttanum rotors, Proceedngs, 45 th ASME Internatonal Gas Turbne & Aeroengne Techncal Congress, Munch, Germany, May 8-, Wu, Y-T., Enrght, M.P., McClung, R.C., Mllwater, H., Leverant, G.R., Probablstc methods or desgn assessment o relablty wth nspecton (DARWIN TM ), Proceedngs, 4 st Structures, Structural Dynamcs, and Materals Conerence, Atlanta, GA, Aprl 3-6, Enrght, M.P. and Wu, Y-T. Probablstc atgue le senstvty analyss o ttanum rotors, Proceedngs, 4 st Structures, Structural Dynamcs, and Materals Conerence, Atlanta, GA, Aprl 3-6, Wu, Y-T., Mllwater, H.R., and Enrght, M.P., Ecent and accurate methods or probablstc analyss o ttanum rotors, Proceedngs, 8 th ASCE Specalty Conerence on Probablstc Mechancs and Structural Relablty, South Bend, IN, July 24-26, Wu, Y.T., Enrght, M.P., and Mllwater, H.R. Probablstc Methods or Desgn Assessment o Relablty Wth INspecton, AIAA Journal, AIAA, 40(5), 2002, Mllwater, H.R., Enrght, M.P., Ftch, S., A convergent probablstc technque or rsk assessment o gas turbne dsks subject to metallurgcal deects, Proceedngs, 43 rd Structures, Structural Dynamcs, and Materals Conerence, Denver, CO, Aprl 22-25, Southwest Research Insttute, DARWIN User s Gude, Verson 3.5, Ang, A. H-S., and Tang, W.H., Probablty Concepts n Engneerng Plannng and Desgn, Wley, New York, 975, pp Aerospace Industres Assocaton Rotor Integrty Subcommttee, The development o anomaly dstrbutons or arcrat engne ttanum dsk alloys, Proceedngs, 38 th Structures, Structural Dynamcs, and Materals Conerence, Kssmmee, FL, Aprl 7-0, 997, pp Amercan Insttute o Aeronautcs and Astronautcs