10 h US Naonal Congress on Compuaonal Mechancs Columbus, Oho 16-19, 2009 Sngle-loop Sysem Relably-Based Desgn & Topology Opmzaon (SRBDO/SRBTO): A Marx-based Sysem Relably (MSR) Mehod Tam Nguyen, Junho Song, Glauco H. Paulno Deparmen of Cvl and Envronmenal Engneerng Unversy of Illnos a Urbana-Champagn 1
Deermnsc Opmzaon vs. RBDO A. Deermnsc Opmzaon mn f ( d, μ ) d, μ s.. g ( d, ) 0 =1,..., n d d d, μ μ μ L U L U g 1 (x 1,x 2 )=0 x 2 B Feasble regon g 2 (x 1,x 2 )=0 Objecve funcon ncrease f ( d, μ ) Low probably of falure Unsafe B. Relably-Based Desgn Opmzaon (RBDO) 2 A mn f ( d, μ ) d, μ s.. P g ( d, ) 0 P =1,..., n 1 x 1 d d d, μ μ μ L U L U Hgh probably of falure 2
Relably Index Approach U 2 mn f ( d, μ ) d, μ RBDO Approaches (1): RIA vs. PMA Enevoldsen and Sorensen (1994) Tu e al (1999) s. G( U) 0 - - - Frs Order Relably Mehod (FORM) MPP s where β mn U 1. β FG 0 β If consrans G < are 0 acve RIA & PMA yeld he same resuls PMA Us generally more effcen and sable han RIA mn G = 0 U cons G > 0 mn G = 0 U 1 U 1 Performance Measure Approach U 2 mn f ( d, μ ) d, μ s where G mn G( U) 1. G p FG ( β ) 0 p s. U =β MPP Decrease G Enevoldsen, I., and Sorensen, J. D., 1994, Relably-Based Opmzaon n Srucural Engneerng, Sruc. Safey, 15(3), pp 169 196. Tu, J., Cho, K. K., and Park, Y. H., 1999, A New Sudy on Relably-Based Desgn Opmzaon, ASME J. Mech. Des., 121(4), pp 557 564. 3
RBDO (2): Double- vs. Sngle-loop Double-loop RBDO Sngle-loop RBDO Objecve funcon Objecve funcon Relably Evaluaon For 1 s consran Relably Evaluaon For n h consran Approx. MPP For 1 s consran Approx. MPP For n h consran Karush Kuhn Tucker (KKT) Opmaly condons Sngle-loop s generally more effcen han double-loop 4
RBDO (3): Componen vs. Sysem mn f ( d, μ ) d, μ Componen RBDO s.. P g ( d, ) 0 P =1,..., n n consrans on componens Targe componen probables: pre-assgned mn f ( d, μ ) d, μ Sysem RBDO s.. P = P g ( d, ) 0 P k C k ONE consran on em probably Targe em probably: pre-assgned Comp. Prob.: ndrecly consraned Sysem performance s beer descrbed by SRBDO 5
Exsng SRBDO Mehods Seres Sysem Ba-abbad e al (2006): Sys. probably sum of comp. probables (upper bound) n n P =P g ( d, ) 0mn 1, P 1 1 Lang e al (2007) : Sys. probably: heorecal bound formulas (Dlevsen, 1979) (upper bound) n n max j j 1 2 P P P Slva e al (2009) : Sysem relably-based opology opmzaon (SRBTO) (ndependen componen evens) General Sysem (cu-se) MacDonald and Mahadevan (2008): Usng cu-ses n Eem E1 E2 E3 E4 E3 E5 C1 C2 C P mn 1, 3 PC (upper bound) 1 We propose usng he marx-based em relably (MSR) mehod o overcome he lmaons Song, J., and Kang, W.-H., 2009, Sysem Relably and Sensvy under Sascal Dependence by Marx-Based Sysem Relably Mehod, Sruc. Safey, 31(2), pp 148 156. 6
Marx-based Sysem Relably (MSR) Mehod Marx-based formulaon of em falure probably Example E 1 e 1 e 2 e 3 PE ( em ) E 2 T cp P( E E E ) p + p + p + p + p 1 2 3 2 4 5 6 7 [0 1 0 1 1 1 1 0 ] [p p p p p p p p ] 1 2 3 4 5 6 7 8 T e 8 e 6 e 5 e 7 E 3 e 4 muually exclusve and collecvely exhausve evens (MECE) c: even vecor descrbe he em even of neres p: probably vecor lkelhood of componen jon falures Applcable o any em even (seres, parallel, general) wh dependen componens Song, J., and Kang, W.-H., 2009, Sysem Relably and Sensvy under Sascal Dependence by Marx-Based Sysem Relably Mehod, Sruc. Safey, 31(2), pp 148 156. 7
Idenfcaon of Even Vecor c Consruc even vecor c E E c 1c 1... n E1 E2 c c.* c.*...* c E E E 1... n E1 E2 c 1 1 c 1 c 1 c n n ( ).*( ).*...*( ) E E E Effcen and easy o mplemen by marx-based language Sofware (Malab ) Can consruc drecly from even vecors of componens and em evens Can develop specfc algorhms o denfy even vecor 8
p Compuaon of Probably Vecor p Ierave marx-based procedure for sascally ndependen (s.) componens Example : n =3 T P1 P P P 1 [1] 1 1 p [2] p p p p [1] 1 1 [] P p p. P. P [ 1] [ 1] T P. P. P PP 1 2 PP [1] 2 1 2 PP PP 1 2 [1] 2 1 2 p [3] PP 1 2P3 PP 1 2P 3 PP 1 2P 3 p[2]. P3 PP 1 2P3 p. P PP P [2] 3 1 2 3 PP 1 2P3 PP 1 2P 3 PP 1 2P3 9
MSR for Sysem wh Dependen Componens Sysem probably wh dependen componens P( E ) P( E s) f ( s) ds s T T c p( s) fs( s) ds c p( s) fs( s) ds s s T cp S Ulze condonal S.I of componens gven an oucome of random varables s causng componen dependence, named Common Source Random Varables (CSRV). Even vecor c s ndependen of hs consderaon No need o consruc probably vecor for new em even 10
Idenfcaon of CSRV s Implc Approxmaon by Dunne-Sobel (DS) correlaon marx Z N(0, R), ρ ( r. r ) m j k 1 k jk m 0.5 m 2 1 k k k k1 k1 Z r U r S Z, =1,..,n are condonal s. gven S=s F he gven correlaon marx wh DS class correlaon marx wh leas square error 11
SRBDO/MSR: Algorhm Curren SRBDO algorhm Usng bound formulas (naccurae) NOT applcable o general em Sensves NOT avalable. MSR mehod Compue em falure probably Applcable o general em Independen evens and dependen evens Propose: Sngle-loop PMA SRBDO/MSR algorhm Based on Sngle-loop SRBDO-PMA procedure Sysem probably and s sensves by MSR 12
Sngle-loop PMA SRBDO/MSR: Algorhm m unknowns random varables n componen evens 1 em consran (n+m) unknowns (n+1) consrans Inal Desgn Equvalen SRBDO/PMA KKT approxmae MPP pon a each eraon for each consran based on, I of each componen Fnd MPPs of componens by PMA Fnd P and s sensves by MSR Updae d, P by an opmzaon algorhm Converged? Yes Resul No MSR npu oupu P P P P 13
SRBDO/MSR: Parameer Sensves Parameer sensves P = cp T P T p P = c Independen θ θ θ p 1 2 n =... P P p p p P θ θ θ T P [ P P P ] 1 2 n T ps () = c fs( s) ds θ s dependen Sysem probably sensves w.r. componen probables (DS class) P() s β 1 m rs k k k 1 m 0.5 2 rk k 1 P ( s) P ( s) β P ( s) 1 P β P β φ( β ) 14
Example Indeermnae Truss Srucure Obj. func: Mn. he wegh of russ Op parameer: bar areas: A Consran: Sysem. Prob. Falure: Sysem fals f wo members fal em even (cu-se) {C k } ={(1,2), (1,3), (1,4), (1,5), (1,6), (2,3),(2,4), (2,5), (2,6), (3,4), (3,5), (3,6), (4,5), (4,6), (5,6)} mn f ( d) 2( A A ) A A A A d{ A,..., A } 1 6 1 2 3 4 5 6 1 2 3 4 5 6 15 s.. P = P g ( d, ) 0 P 0.001 k 1 C k g ( d, ) A F 0.707 F 1,2 A A, A, A, A, A, A A F 0.500 F 3,...,6 A 0 6 1 2 F A Means and Sandard devaons of he random varables L 3 4 5 Random Varables (Gaussan dsrbuon) Mean Sd dev Member srengh F, =1,..6 (Mpa) 745 62 L Appled load F A (kn) 4450 45 MacDonald M., and Mahadevan S., 2008, Desgn Opmzaon wh Sysem-Level Relably, ASME J. Mech. Des., 130(2), p. 021403 15
Indeermnae Truss Srucure MacDonald and Mahadevan (*) vs. SRBDO/MSR Members Area: A ( 10 3 mm 2 ) Relably Index: SRBDO by (*) SRBDO/MSR SRBDO by (*) SRBDO/MSR 6 1 2 F A 1 18.43 17.89 2.89 2.67 L 3 5 2 18.27 17.89 2.83 2.67 3 13.51 13.20 3.16 2.99 4 4 13.44 13.20 3.12 2.99 5 13.33 13.20 3.06 2.99 L 6 13.09 13.20 2.92 2.99 f(x) 105.24 103.36 SRBDO/MSR: beer opmal desgn + symmerc resuls MCS: P =0.00107 (10 6 mes, c.o.v. = 0.03) ~ SRBDO/MSR: 0.001 16
CIM Indeermnae Truss Srucure Condonal probably mporan Measure (CIM) CIM = P( E E )= P( E E ) ' c PE ( ) T p T cp Only addonal ask s o denfy he even vecor of he new em 1.00 6 0.75 1 2 F A 0.50 L 3 5 4 0.25 0.00 1 2 3 4 5 6 Componens L Componen mporan rankng: (1,2) -> (3,4,5,6) 17
Topology Opmzaon: DTO, CRBTO, SRBTO ρ, F ρ, F g d d =1,2,3 F 1 F 2 50 Random Varables F 1 ~ N(100 10) F 2 ~ N(100 10) F 3 F 3 ~ N(100 10) DTO Mn V( ) 1 s. g (,F) 0, =1,2,3 g (,F) d (,F) d CRBTO Mn V( ) s. P g (, F) 0 1 1 P g (,F) 0 2 2 P g (,F) 0 3 3 1 2 3 max 1 1 1 max 2 2 2 max 3 3 3 P P P max where : P P P 0.02275 g (,F) d d (, F) 0 g (,F) d d (, F) 0 g (,F) d d (, F) 0 2 Mn s. V( ) d max = 1.5 SRBTO SRBTO/PMA+MSR P =P g (,F) g (,F) g (,F) 0 1 2 3 where g (, F) d d (,F) 0 max 1 1 1 g (, F) d d (,F) 0 max 2 2 2 g (, F) d d (,F) 0 max 3 3 3 140 P Mn 1 2 3, P, P, P 1 2 3 V( ) s. g (,F) 0 g (,F) 0 g (,F) 0 P P ( P, P, P ) P F=(F 1, F 2, F 3 ) 3 4 MSR 1 2 3 18
Topology Opmzaon: DTO, CRBTO, SRBTO F 1 F 2 50 (P = 0.0434 volfrac = 0.58) 140 F 3 E E Seres (P = 0.005 volfrac = 0.44) DTO (volfrac = 0.40) E E E E 1 2 3 Parallel (P = 0.005 volfrac = 0.47) CRBTO (P = 0.02275 P = 0.0434 volfrac = 0.60) E E E E 1 2 3 General SRBTO/MSR 19
Summary & Concluson Sngle-loop PMA SRBDO/MSR algorhm s proposed. The SRBDO/MSR s applcable o general em n a unform manner. Sensves for graden-based opmzaon Handle dependence beween componen evens Numercal examples: SRBDO & SRBTO examples Fuure work: 3D opology opmzaon problem 20
Fuure work: Large-scale SRBTO Problem confguraon Deermnsc Opmal Topology (half) Sze (full doman): 180x60x60 elemens Number of unknowns (half doman: 1.0 ml.) Run me: ~ 45.7 hours Flerng radus: 6 elemens (full) Hgh resoluon SRBTO s compuaonally expensve Wang, S., Surler, E., and Paulno, G.H. 2007, Large-scale Topology Opmzaon usng Precondoned Krylov Subspace Mehods wh Recyclng, In. J. Numer. Meh. Engrg., 69, pp 2441 2468. 21