Other Uncertainty Analysis Methods

Transcription

1 Chapter 9 Other Uncertant Analss Methods Chapter 9 Other Uncertant Analss Methods 9. Introducton In Chapter 7, we have dscussed relablt analss methods FORM and SORM. Both of the methods can provde reasonabl accurate relablt analss results. But the ma be neffcent when the number of random varables s lare and when the dervatve of a performance functon s evaluated numercall (for eample, throuh the fnte dfference method). The reason s that both of the methods requre searchn the MPP and that ths search process nvolves a number of determnstc analses. Furthermore, the dstrbutons of the nput random varables have to be known. In some enneern applcatons, a precse dstrbuton of a random varable ma not be avalable because of lmted nformaton. For eample, onl a small sample sze of a random varable s avalable, onl the mean and standard devaton can be obtaned, or onl the nterval where a random varable resdes s known. For these stuatons, one ma consder usn the moment matchn method or the worst case analss method that we wll dscuss below. If the evaluaton of the performance functon s computatonall epensve, a smplfed model must be created to replace the ornal performance functon for the uncertant analss purpose. Response surface method (RSM) s one of such methods and wll also be dscussed n ths chapter. 9. Moment Matchn Method If the frst two moments (mean and standard devaton) of a random varable are known, the moment matchn method can be used to estmate the mean and standard devaton of a performance functon. Then the mean and standard devaton of the performance functon ma be used to estmate the probablt of falure. Assume that the random varables = (,,, n ) the means and standard devatons of = (,,, n ) are = (,,, n ) = ( σ, σ,, σ n ) functon = ( ) at the means = ( ) are mutuall ndependent and that and s, respectvel. The frst order Talor epanson of the performance follown lnearzaton ( ),,, n,,, provdes the n n ( ) T ( ) L( ) = ( ) + ( ) = ( ) + ( )( ), (9.) = where ( ) s the radent of ( ) U at = ( ),,,. ( ) s ven b n

2 Probablstc Enneern Desn ( ) ( ) ( ) ( ) =,,, (9.) n Then, the mean of ( ) s appromated b the mean of the lnearzed functon L( ) and s ven b The standard devaton of ( ) s ven b ( ). (9.) σ n ( ) σ =. (9.4) If the nput random varables = ( ),,, are assumed to be normall n dstrbuted, the lnearzed performance functon L( ) s also normall dstrbuted snce L( ) s a lnear combnaton of normal varables. The probablt of falure s therefore computed b n ( ) pf PL { ( ) < 0} =Φ ( ) =Φ ( ) σ σ =. (9.5) The moment matchn method s also called the frst order and the second moment (FOSM) method snce t nvolves the frst order dervatve and second moment. Two appromatons are used n the moment matchn method the dstrbutons of random varables = (,,, n ) are assumed normall dstrbuted, and the performance functon ( ) s lnearzed at the means of = (,,, n ). FOSM does not need the MPP search and t s more effcent than both FORM and SORM. If the dervatve of ( ) s evaluated numercall b the fnte dfference method, the total number of functon evaluaton s n +. n evaluatons are for the dervatve calculaton, and one evaluaton s for the calculaton of the performance at the means of = (,,, n ). In man cases, FOSM can provde satsfactorl accurate results. Because of ts easness and effcenc, FOSM has been wdel used, especall n probablstc mechancal component desn and robust desn. However, FOSM s not accurate as FORM or SORM n eneral. As shown n F. 9., the probablt of falure p s estmated b the probablt nteraton over the hatched f

3 Chapter 9 Other Uncertant Analss Methods T reon, whch s on the upper rht sde of the straht lne ( ) + ( )( ) = 0. The actual probablt nteraton reon s the shaded reon that s on the upper rht sde of the curve ( ) = 0. Because of the appromatons, FORM ma result n a snfcant error for hhl nonlnear functons wth random varables that follow nonnormal dstrbutons. Appromated probablt nteraton reon Actual probablt nteraton reon (falure reon) ( ) = 0 ( ) > 0 ( ) ( ) ( ) < 0 T ( ) + ( )( ) = 0 ( ) = ( ) Appromated dstrbuton o Ornal dstrbutons Appromated dstrbuton Fure 9. Appromatons n Moment Matchn Method Eample 9. relablt analss of the cantlever beam Use FORM to solve Eample 7.. The performance functon s ven b 4L P P 0 = D Ewt + t w where P ~ N(500,00) lb and P ~ N(000,00) lb, and the other parameters n the above equaton are constants (see Eample 7.). The radent of the performance functon s ven b,

4 4 Probablstc Enneern Desn 4L P P ( ) =, =,. P P Ewt P 4 P P 4 P w + t + t w t w The mean of ( ) s computed b 4L P P ( ) = D + = + Ewt t w = and the standard devaton of ( ) s computed b σ = σ + σ P P P P P, P P, P 4 P P 4 P P P w P t P 4L = + σ + + σ = 0.74 Ewt t w t w The probablt of falure s then calculated b p f PL { ( U ) < 0} =Φ ( ) =Φ = σ 0.74 The soluton of the probablt of falure from Monte Carlo Smulaton wth 0 5 smulatons s p f = (see Eample 8.), and the soluton from FORM s p = (see Eample 7.). If we consder the Monte Carlo Smulaton soluton as f the accurate soluton, the soluton from FOSM s not as accurate as that of FORM. However, FOSM s much more effcent than FORM snce the former does not need to search the MPP. As the eample demonstrates, FOSM onl requres the functon value and ts radent at the means of random varables. It s effcent and s therefore wdel used n man enneern applcatons, especall n robust desn. It should be noted that one potental problem of FOSM s that f a performance functon epressed b another equvalent formulaton, the soluton of the probablt of falure ma var. For eample, f the above performance functon s rewrtten wth an equvalent formulaton 4

5 5 Chapter 9 Other Uncertant Analss Methods 4L P P = D0 +, Ewt t w the probablt of falure from FOSM wll become p f = 0.005, whch s qute dfferent from the soluton based on the ornal formulaton. FOSM s also commonl appled n probablstc mechancal component desn. The follown eample demonstrates such an applcaton. Eample 9. probablstc shaft desn A shaft subjected to an aal forces Q s shown n F. 9.. The mean and standard devaton of Q are = 40 kn and σ =. kn, respectvel. The mean and standard Q Q devaton of the eld strenth S are = 667 MPa and σ = 5. MPa, respectvel. The standard devaton of the shaft radus r s radus Q r such that the relablt of the component s R = S S σ r = 0 m. Determne the mean of the Q r Fure 9. A Shaft Subjected to Aal Forces In ths desn problem, there s onl one desn varable, whch s the radus of the shaft. For comparson, the conventonal (determnstc) desn s also ven below. Determnstc desn In the determnstc desn, the means (nomnal values) are used. To make sure that the desn s safe enouh, a lare safet factor S F = s chosen. The normal stress of the shaft s calculated b From the defnton of the safet factor S F Q S =. π r S F S S π r = =, S Q 5

6 6 Probablstc Enneern Desn The radus s then determned b SQ F 40 0 r = = = m= 7.6mm. 6 πs π Probablstc desn The performance functon s defned as Q = S S = S. π r The mean of s ven b = S. π The standard devaton of s calculated b Q r = Q + r + S Q r S σ σ σ σ Q Q σq σr ( ) σs σ σ σ Q r S r r r r = + + = π π π π Then, the probablt of falure s computed b Therefore, p ( 0) f = P< =Φ = R σ. or, σ =Φ ( R), +Φ ( R) σ = 0. Substtutn the equatons of and σ nto the above equaton elds 6

7 7 Chapter 9 Other Uncertant Analss Methods whch s Q Q S +Φ ( R) 4 0 σ Q + σ r + σs =, π r π r π r ( ) ( ) ( ) π r π r π r Φ ( 0.999) = 0 The soluton to the above equaton s - r = 6. 0 m= 6. mm. Comparn the result from the probablstc desn and that of the determnstc desn, t s seen that the latter s much more conservatve because t requres a larer radus. Wth the requred relablt of 0.999, the determnstc approach over desned the component and resulted n a hher relablt than requred. (Interested readers ma want to calculate the relablt for the determnstc desn.) On the other hand, f a smaller factor of safet (e...5) were used, the determnstc desn would be rsk snce the relablt would be less than the requred value. As dscussed prevousl, FOSM ma not result n accurate relablt estmaton. If hher accurac s desred for the desn, one should use the advanced probablstc desn methods that wll be dscussed n Part III n ths book. 9. Worst Case Analss Some tmes the nformaton about random varables ma be lmted. We onl know the nterval over whch a random varable ma resde. Some of the stuatons where the uncertantes are characterzed wth ntervals are as follows. () Sometmes a quantt ma not have been studed at all, and the onl real nformaton about t comes from theoretcal constrants. Phscal lmts ma be used to crcumscrbe possble ranes of quanttes even when no emprcal nformaton about them s avalable. [] () Condton montorn calls for perodc nspecton of components. A component ma be n workn condton at one nspecton, but n falure condton at the net. The tme to falure s therefore n a wndow of tme between the last two nspectons durn whch the component faled. () A measurement from a devce s assocated wth an nterval based on the number of reported dts. For eample, the value 9. ma be assocated wth the nterval [9.0, 9.5] where the two endponts are the two closest landmark values. 7

8 8 Probablstc Enneern Desn (4) Desn enneers often specf ther desn varables n the form of nomnal values plus-or-mnus tolerances. For a completel new desn, t s hard to know how the desn varables could be dstrbuted over the tolerance ranes before phscal deploment []. (5) Man enneern formulatons have ther applcaton lmts. Intervals are used to dentf the choces of formulatons. For eample, f the rato of thckness to nternal dameter of a clnder s between. and., the clnder s consdered as a thn-wall clnder; f the rato s reater than., t s consdered as a thck-wall clnder []. (6) Analss enneers often estmate the analss error b the percentae of the nomnal analss results. For eample, for a partcular applcaton, a fnte element analss ma be reported to have a 0% error []. For stuatons where onl ntervals of random varables are known, we can use the worst case analss to fnd the nterval of a performance functon. Assume that the nterval for random varable s [ a, b ], then ts averae s ven b a + b =. (9.7) Obvousl, the dstance between the averae and one of the endponts s half of the rane b a, namel, b a = b = a =. (9.8) The procedure of worst case analss s descrbed as follows. Frst, the performance functon s lnearzed b the frst order Talor epanson at the averae of the nput random varables = (,,, n) b n ( ) ( ) ( ) + ( ). (9.9) = The averae of the performance functon s evaluated at the averae (,,, ) n =, namel, ( ) = ( ). (9.0) The devaton of the performance functon from ts averae s then ven b n ( ) = ( ) ( ) = ( ). (9.) = 8

9 9 Chapter 9 Other Uncertant Analss Methods Because we are nterested n the worst case, we take the absoluton values of the dervatves n the above equaton and use the mamum chanes of random varables. Then, the worst case dfference (the mamum dfference) of the performance functon s ven b n n ( ) ( ) = = b a ( ) = = Therefore, the performance wll var n the follow rane,. (9.) n n ( ) ( ), + = ( ), ( ) +. (9.) = = If the safe reon s defned b ( ) > 0, the worst case performance functon wll be, and ths worst case value should be reater than 0, namel, > 0. It should be noted that the above method for dentfn the worst case of the performance functon s an appromaton. The appromaton comes from usn the frst order Talor epanson and takn the absolute value of the dervatves. Therefore, the soluton from Eq. 9. has some error. To accuratel dentf the worst case value of a performance functon, the mamum (or mnmum) value of the performance has to be searched over the ranes of all the nput varables. Optmzaton technques can be used for ths purpose. It should also be noted that the results from the worst case scenaro ma be too conservatve. Eample 9. worst case analss of the cantlever beam The same beam problem as n Eample 9. s used to demonstrate the worst case analss method. The tp dsplacement of the cantlever beam s ven b where the eternal forces 4L P P = Ewt + t w P and, P are known n the ntervals [400, 600] and [800, 00], respectvel. Therefore, P = 00, and = 00. The averaes of P and P are P = 500 and P = 000, respectvel. The averae of s computed b P 9

10 0 Probablstc Enneern Desn 6 4L P P = + = + = n Ewt t w The radent of at the averaes of P and P s ven b 4L P P =, Ewt 4 P P 4 P P w t + + t w t w =, , = ( ) Then, = P+ P = =0.496 n P P Thus, the rane of the performance functon s ven b [ ] = + [ ] mn The larest (worst case) dsplacement s, ma, =0.55,.09 n. = + =.09 n. worst If the allowable tp dsplacement s.5, the desn wll be consdered safe. However, f the allowable tp dsplacement s.0, the desn wll be consdered as a falure; n ths case, the desn has to be modfed. Eample 9.4 shaft desn b worst case analss The shaft desn problem n Eample 9. s solved aan b the worst case analss. The ranes of the random varables are set to standard devatons, namel, r = σ r = 0 m, Q = σ Q =.=.6 kn, S = σ S = 5.= 75.9 MPa. 0

11 Chapter 9 Other Uncertant Analss Methods The averaes of the uncertan varables are r, averae radus r s to be determned. Q = 40 kn, and S = 66.7 MPa. The The averae performance functon s ven b The rane of s ven b Q = S. π r = + r+ = + r+ Q Q S Q S Q r S π r π r. The worst case s and should be reater than zero; therefore, Q Q = S 0 Q + r+ S =, π r π r π r whch elds the follown eqauton =. πr πr πr Solvn the above equaton elds the desn varable, r = 6.6 mm, whch s reater than the radus obtaned from moment matchn method. Ths ndcates that the worst case analss ma result n a conservatve desn. 9.4 Response Surface Method In man enneern applcatons, the evaluaton of a performance functon s computatonall epensve. Uncertant analss needs a number of such evaluatons. One soluton to ths problem s to create a surroate model to replace the ornal epensve performance functon. The evaluaton of a surroate model s much cheaper than that of the ornal performance functon. The basc dea s to perform a number of eperments (numercall or phscall) at dfferent desn ponts (or nputs), and then the performance functon values and correspondn nputs are used to ft the smplfed surroate model. Ths process s called Desn of Eperments (DOE), or more precsel, Computer Desn of Eperments f the eperments are conducted numercall. Once a surroate model s establshed, the uncertant analss methods such as Monte Carlo Smulaton, FORM, and SORM can be appled for uncertant analss.

12 Probablstc Enneern Desn There are several tasks n DOE, ncludn selectn the tpe of surroate model, dentfn desn ponts where the eperments wll be performed, and solvn the unknown coeffcents of the surroate model. Generall, the functons whch can accurate represent the ornal functon and need a small number of eperments are favorable. Response Surface Model (RSM) s one of those functons. RSM s a polnomal tpe of functon. Net we wll use a smple eample to dscuss RSM wth the follown procedure. The eneral procedure of RSM method s as follows. Step : Determne the desn (nput) varables and response varables. Step : Determne the desn varable bounds. Step : Plan the eperment, ncludn the number of eperments, levels of desn varables, and the tpe of response surface. Step 4: Perform eperment to obtan the response varables at the desn ponts determned n Step. Step 5: Determne the unknown coeffcents of the response surface model and perform other analses such as senstvt analss. Step 6: Use the response surface model for uncertant analss. Net, a smple eample wll be used to demonstrate a -level full factoral desn, where two ponts (levels) of each desn varable and all the combnatons of desn varable levels are used. In ths eample, the response Y s calculated a some computer smulaton proram, whch s ver tme consumn. To et a cheaper model of the response n terms of desn varables, the DOE s performed. The desn varables nclude two contnuous varables and, and a dscrete varable that takes values of ether A or B. Step : Determne the desn varables and response varables The desn varables are the dmensonal varables and, and the materal tpe. The frst two dmensonal desn varables are consdered contnuous, and the last desn varable s a dscrete varable snce t represents the tpe of materals. Step : Determne the desn varable bounds The bounds of are [ mn, ma ] = [ 60, 70] mm and [ mn, ma ] = [ 0, 40] mm, respectvel. There are two materal tpes avalable; therefore can be ether Tpe A or Tpe B. Step : Plan the eperment Two levels for each desn varable are consdered for the eperment. For smplct, all the desn varables are transformed at the scale of [-, +], where - stands for the lower bound mn, and + stands for the upper bound ma. A contnuous desn varable at the scale of [-, +] s then epressed b ( ) mn = ma mn. (9.4)

13 Chapter 9 Other Uncertant Analss Methods A two-level full factoral desn s used for ths problem. All the combnatons (desn ponts) of two levels of the desn varables are consdered, and the two levels are selected on the lower bounds and upper bounds of the desn varables. The follown table ves the desn ponts. (The table s called DOE matr). There are 8 eperments n the table. For, - represents Tpe A materal, and + represents Tpe B materal. Table 9. DOE Matr Eperment A lnear functon of the stress Y s selected for the RSM, whch s ven b Y = β + β + β + β, (9.5) 0 0 where β ( = 0,,,) are unknown coeffcents. The DOE matr s vsualzed n F. 9., where the crcles represent the desn ponts. Fure 9. The vsualzaton of the DOE Matr Snce 8 values of the response Y are to be obtaned from the 8 eperments, mamall, 8 unknown coeffcents can be ncluded n the RSM. Therefore, the follown hher order polnomal wth 8 unknown coeffcents and nteracton terms s also an alternatve RSM.

14 4 Probablstc Enneern Desn Y = β + β + β + β + β + β + β + β, (9.6) 0 0 n whch β, β, β, and β are addtonal undetermned coeffcents. Step 4: Perform eperment to obtan the response The FEA s performed at the 8 desn ponts ven n Table 9. and F. 9.. The ep calculated stresses Y are lsted n Table 9.. Table 9. DOE Matr Eperment ep Y The epermental results are also plotted n F. 9., where the quanttes wthn the crcles are the response varable Y calculated from the FEA smulaton. Step 5: Determne the unknown coeffcents of the RSM The undetermned coeffcents are solved out wth the least square dfference between the ep predcted response from Eqs. 9.5 or 9.6 and the epermental results Y. The model for solvn the undetermned coeffcents s ven b for the leaner model n Eq. 9.5, and ( ) β0 β 0 β β 8 ep (9.7) = Mn Y ( ) β0 β 0 β β β β β β 8 ep = Mn Y for the quadratc model n Eq. 9.6., (9.8) Eqs. 9.7 and 9.8 can be solved b optmzaton. Alternatvel, based on the above models, the coeffcents can also be calculated as follows. The frst coeffcent s the averae of the responses from the eperment, 4

15 5 Chapter 9 Other Uncertant Analss Methods β 0 8 ep = Y = = The coeffcents of the frst order terms β, β, and β are calculated from the man effects. The man effect E of varable s the averae chane when chane from - to + whle other varables reman unchaned at ther averaes. Therefore the man effect E s computed as the dfference between the averae value P + of the response at the hh level (+) and the averae value P of the response at the low level (-). For, β s computed as follows. The averae response at hh level (+) (see F. 9.4) s, ep ep ep ep P+ = ( Y + Y4 + Y6 + Y8 ) = ( ) = 75.75, and the averae response at low level (-) s, ep ep ep ep P = ( Y + Y + Y5 + Y7 ) = ( ) = The man effect of s ven b E= P+ P = ( ) = Fure 9.4 The Man Effect of 5

16 6 Probablstc Enneern Desn The coeffcent of s half of the man effect E, namel, β = E. (9.9) Therefore, β = E/=.5. Smlarl, β and β are calculated as follows. ep ep ep ep P+ = ( Y + Y4 + Y7 + Y8 ) = ( ) = 6.75 (see F. 9.5), ep ep ep ep P = ( Y + Y + Y5 + Y6 ) = ( ) = 66.75, E= P+ P = = 5. Hence, β = E/= Fure 9.5 The Man Effect of ep ep ep ep P+ = ( Y4 + Y5 + Y6 + Y7 ) = ( ) = 65 (see F. 9.5), 6

17 7 Chapter 9 Other Uncertant Analss Methods ep ep ep ep P = ( Y + Y + Y + Y4 ) = ( ) = 6.5, E = P+ P = =.5. Hence, β = E/=.5. (9.) Fure 9.6 The Man Effect of Therefore, the lnear RSM s obtaned, whch s ven b Y = (9.0) Usn Eq. 9.4,,, and are transformed nto ornal varables, and then the RSM s rewrtten n terms of the ornal varables as below. Y ( + 60) ( + 0) = Or Y = (9.) In the quadratc model n Eq. 9.6, the nteracton terms are ncluded. If treatn,,, and as three new ndvdual varables, we can use the same approach as we dd above to calculate the coeffcents of the nteracton terms. For ths 7

18 8 Probablstc Enneern Desn purpose, Table 9. s then rewrtten n order to nclude the new varables (the nteractons). The table s ven below. Table 9.4 Epermental Results Eperment ep Y The calculatons are ven below. P+ = ( Y+ Y4+ Y5+ Y8) = ( ) = 65. P = ( Y + Y + Y6 + Y7) = ( ) = 6.5. E = P+ P = (65 6.5) =.5. β = E /= P+ = ( Y + Y+ Y6+ Y8) = ( ) = P = ( Y + Y4 + Y5 + Y7) = ( ) = E= P+ P = ( ) = 0. β = E /= 5. P+ = ( Y + Y + Y7 + Y8) = ( ) = P = ( Y + Y4 + Y5 + Y6) = ( ) = E= P P = ( ) =

19 9 Chapter 9 Other Uncertant Analss Methods β = E /= 0. P+ = ( Y + Y+ Y5+ Y8) = ( ) = P = ( Y+ Y4+ Y6+ Y7) = ( ) = 64. E = P+ P = ( ) = 0.5. β = E/= 0.5. Therefore, the nonlnear RSM s ven b Y = (9.) Usn Eq. 9.4, we can obtan the RSM n terms of the ornal varables below. Or, ( + 60) ( + 0) Y = ( + 60) ( + 0) ( + 60) ( + 60) ( + 0) Y = (9.) Step 6: Use the response surface model for uncertant analss. After the RSM s obtaned, the ornal epensve FEA model wll be replaced for uncertant analss and probablstc desn. Step 6 wll be demonstrated n the follown eample. Eample 9.5 RSM based relablt analss of the cantlever beam We wll use RSM to solve the relablt analss for the cantlever beam ven n Eamples 7. and 9. In ths problem, n addton to the random varables P ~ N(500,40) lb and P ~ N(000,80) lb, the Youn s modulus E s also consdered 6 5 normall dstrbuted wth E ~ N(0 0,0 0 ) ps. Net, we wll create a RSM n terms of the three random varables b usn the -level full factoral desn. 9

20 0 Probablstc Enneern Desn Step : Determne the desn varables and response varables The desn varables are the three random varables P, P, and E. The response varable s the tp deflecton of the beam, Y, whch s ven b Y 4L P P = + Ewt t w, n whch all the constants are the same as n Eample 7.. Step : Determne the desn varable bounds The lower and upper bounds are determned b σ prncple. The bounds are ven b and [ P, P ] = σ, + σ = [ 80, 60], mn ma P P P P mn ma P P P P [ ] P, P = σ, + σ = 760, 40, 6 6 [, ] = [, + ] = 7 0, 0 Emn E ma E σ E E σ E. Step : Plan the eperment Two levels for each desn varable are consdered for the eperments. The ntended RSM s ven b Y = β + β P + β P + β E + β PP + β PE + β PE + β PPE. 0 The DOE matr s ven n Table 9.5, and the values of the random varables are shown n brackets. Step 4: Perform eperment to obtan the response The tp deflectons at the 8 desn ponts are calculated and are lsted n Table 9.5. Table 9.5 DOE Matr and Epermental Results Eperment P (lb) P ( lb) E (ps) ep Y (n) - (80) - (760) - (7 0 6 ).9669 (60) - (760) - (7 0 6 ).00 - (80) (40) - (7 0 6 ) (60) (40) - (7 0 6 ).09 0

21 Chapter 9 Other Uncertant Analss Methods 5 - (80) - (760) ( 0 6 ) (60) - (760) ( 0 6 ) (80) (40) ( 0 6 ) (60) (40) ( 0 6 ).657 Step 5: Determne the unknown coeffcents of the response surface model The unknown coeffcents are computed usn the equatons dscussed above, and the RSM s then ven b Y = P P 0.75E 0.09PP 0.06PE PE 0.00 PPE. + Ths model s for the normalzed random varables at the scale of [-, ]. Eq. 9.4 can be used to convert the model nto the ornal random varables. Step 6: Use the response surface model for uncertant analss Snce the RSM s cheap to compute now, Monte Carlo smulaton wth a lare sample p = P = D Y <, where sze of 0 6 s used to compute the probablt of falure { } 0 f 0 0 D s the allowable deflecton and D0 = n. The calculated probablt of falure based on the RSM s p f = To confrm the result, the same sze of Monte Carlo smulaton s also performed wth the ornal performance functon and the result s p = It s noted that wth onl 8 functon evaluatons (for constructn the f RSM), the RSM based relablt analss method produces a ver accurate relablt result. Heren, onl a smple two-level full factoral desn s dscussed. To make a RSM more accuratel represent the ornal functon, hher levels can be used. For eample, n a - level desn full factoral desn, for each desn varable, n addton to the two end ponts, the center pont s also used n constructn the RSM. Wth hher level DOE, a hh order RSM, for eample, a cubc polnomal, can be used to acheve accurate results. However, more levels and a hher order RSM need more eperments. To make reasonable ood trade-off between accurac and effcenc, a fractonal factoral desn ma be consdered, where not all the combnatons of the desn varables levels wll be consdered. The complete methodolo of RSM can be found n man statstcal eperment desn books. To better capture the hh nonlneart and acheve hher accurac, other DOE methods such as Krn method, MARS (Adaptve Reresson Splnes), and radal bass functons have been developed and have been ncreasnl used n enneern applcatons. A vast amount of lterature s avalable n ths subject.

22 Probablstc Enneern Desn 9.5 Concluson The task of uncertant analss s to dentf the probablstc characterstcs of performance functons. The probablstc characterstcs of performance functons wll be used n the desn stae for manan and mtatn the effects of nput uncertant on the performance. The probablstc characterstcs of performance nclude the moments (mean, standard devaton, etc.), percentle values, the probablt of falure, relablt, and probablt dstrbutons. Dependn on dfferent applcatons, dfferent probablstc characterstcs wll be needed. For eample, robust desn needs mean and standard devaton, relablt-based desn needs relablt, and rsk analss needs the probablt of falure. In Chapters 7, 8 and 9 (ths chapter), we dscussed the commonl used uncertant analss methods. It should be noted that there s no unversal uncertant analss method that suts all the stuatons n enneern analss and desn. For dfferent problems, we ma choose dfferent uncertant analss methods. When we consder choosn uncertant analss methods, accurac and effcenc are the major concerns. There alwas ests a conflct between accurac and effcenc. The robustness, whch measures f a method can successfull dentf the uncertant analss solutons, s also a factor of consderaton. A comparson amon the methods we have dscussed s ven n Table 9.6 [].

23 Chapter 9 Other Uncertant Analss Methods Table 9.6 Comparson of Uncertant Analss Methods MCS FORM SORM* FOSM* Worst Case Analss RSM Requres nput dstrbutons Yes Yes Yes Dstrbutons or the frst two moments Intervals Ma or ma not Deals wth correlaton Yes Yes Yes Yes No Yes Requres dervatve of performance functon Effcenc Capablt and accurac Robustness Reference No Yes Yes Yes Yes Needs a lare number of functon evaluatons, especall when the probablt s hh. Gves accurate solutons when enouh samples are used; can enerate the complete dstrbuton. Ver robust (can alwas fnd the soluton.) Effcent for small or moderate number of random varables; effcent than SOME; effcenc decreases wth a lare number of random varables. The accurac depends on the performance functon and nput dstrbutons; enerall, more accurate than moment matchn and RSM. The MPP search ma not convere. Effcent for small number of random varables; needs nd dervatves; effcenc decreases wth a lare number of random varables. The accurac depends on the performance functon and nput dstrbutons; enerall, more accurate than FORM. The MPP search ma not convere. Ver effcent. Smple to use, but enerall not accurate. Ver effcent. Appromaton on the bunds of a performance functon. Ma or ma not Effcent wth small or moderate number of desn varables. Hue computaton al demand when the number of random varables s lare. The accurac depends on how accuratel the RSM represents the performance functon; ma result n errors. Robust Robust Robust f MCS s used; the MPP search ma not convere f FORM or SORM s used. * If a radent-free optmzaton alorthm s used, no dervatve s needed. [] Ferson S., Josln, C.A., Helton, J.C., Oberkampf, W.L., and Sentz, K., 004, Summar from the Epstemc Uncertant Workshop: Consensus amd Dverst, Relablt Enneern and Sstem Safet, 85 (-), pp

24 4 Probablstc Enneern Desn [] Du,., Sudjanto, A., and Huan, B., "Relablt-Based Desn under the Mture of Random and Interval Varables," n press, ASME Journal of Mechancal Desn, 006. [] Du,. and Chen, W., "Towards a Better Understandn of Modeln Feasblt Robustness n Enneern," ASME Journal of Mechancal Desn, Vol., No. 4, pp ,