Single-loop System Reliability-based Topology Optimization Accounting for Statistical Dependence between Limit-states

Transcription

1 11 h US Naional Congress on Compuaional Mechanics Minneapolis, Minnesoa Single-loop Sysem Reliabiliy-based Topology Opimizaion Accouning for Saisical Dependence beween imi-saes Tam Nguyen, Norheasern Universiy Junho Song, Glaucio Paulino, Universiy of Illinois a Urbana-Champaign 07/25/ Deparmen of Civil and Environmenal Engineering Norheasern Universiy

2 Topology Opimizaion Classical srucural design opimizaion: he opimal sizes or shapes for a given layou and conneciviy size opimizaion shape opimizaion Topology opimizaion: free-form opimizaion 2

3 Topology Opimizaion Applicaions Airbus Wing box rib 500 kg reducion/wing Skidmore, Owings & Merrill, P (SOM) ( 3

4 Topology Opimizaion Procedure Problem formulaion min C(ρ, u ) ρ s..: K(ρ) u f d d min T f ud V (ρ) ρ( ψ) dv V 0 ρ ρ( ψ) 1 Solid and Isoropic Maerial wih Penalizaion (SIMP) E( ψ) ρ( ψ) p E 0 s Iniial guess Finie Elemen Analysis Objecive Funcion & Consrains Sensiiviies Analysis Filering (Projecion) Technique Updae Maerial Disribuion compuaionally expensive P Opimizers Opimaliy Crieria (OC) Mehod of Moving Asympoes (MMA) Converged? Resul Yes No 4

5 High Resoluion Topology Opimizaion arge-scale (high resoluion) TOP arge number of finie elemens Compuaionally expensive: FEA cos Borrvall and Peersson, (2000), IJNME 96x96x96 Exising high resoluion TOP Parallel compuing (Borrvall and Peersson, 2000) Fas ieraive solvers (Wang e al. 2007) Approximae reanalysis (Amir e al. 2009) Adapive mesh refinemen (de Suler e al. 2008) A sool (884,736 B8/U) Same discreizaion for analysis and design 5

6 Muliresoluion Topology Opimizaion (MTOP) Convenional elemen-based approach (Q4/U) Same discreizaion for displacemen and densiy Displacemen Densiy/design variable Proposed MTOP approach (Q4/n25) Differen discreizaions for displacemen and densiy/design variables Displacemen Densiy Design variable 6

7 MTOP: 2D Canilever Beam Objecive: minimum compliance Consrain: volfrac = 0.5 engh scale: r min = 1.2 Coarse FE mesh Configuraion Fine FE mesh MTOP Q4/n25 FE mesh 48x16 (C=208.23) Coarse FE mesh Q4/U FE mesh 240x80 (C=210.68) Q4/U FE mesh 48x16 (C=205.57) Convergence hisory Nguyen e al. (2010), JSMO 7

8 MTOP: 3D Examples 3D cross-shaped secion 3D bridge design q Non-designable layer non-designable layer 10x120x30 6 2/3 Borrvall & Peersson 320,000 B8/U MTOP: 36,000 B8/n125 V i i V i MTOP: 5,000 B8/n125 i An exising design 8 (hp://

9 Sysem Reliabiliy-Based Topology Opimizaion Componen RBTO Sysem RBTO d, μ min f ( d, μ ) d, μ X s.. P g ( dx, ) 0 P i=1,..., n min f ( d, μ ) X i X d d d, μ μ μ U U X X X X s.. P( Esys )= P g i( dx, ) 0 P k ic k d d d, μ μ μ U U X X X i? sys 1. How o handle dependen limi-saes in SRBTO? 9 2. How o compue probabiliy accuraely in RBTO?

10 Exising SRBTO Approaches Discree srucures Mogami e al. (2006) Truss examples Ground srucure Opimal srucure Mogami e al. (2006), JSMO Coninuum srucures Silvia e al. (2010) imi-saes: saisically independen P E P E P E P E E P E E E n n n1 n n1 ( sys ) i ( i ) ( i j ) ( 1) ( 1 2 n) i1 i1 i1 ji1 P( E E E ) P( E ) P( E ) P( E ) DTO Silvia e al. (2010), JSMO SRBTO How o handle he dependence beween limi-saes? 10

11 Marix-based Sysem Reliabiliy (MSR) Mehod Song and Kang, (2009); Srucural Safey E 1 e 1 e 2 E 2 e 3 e 6 e 5 e 4 e 7 e 8 E 3 Convenien: marix-based procedures for c and p; easy SRA calculaion (inner produc) General: uniform applicaion o series, parallel, and any general sysems Flexible: inequaliy-ype informaion; incomplee informaion ( P bounds mehod) Efficien: no need o re-compue p ; replace c for SRA of a new even muually exclusive and collecively exhausive evens (MECE) Common Source Effec: can accoun for saisical dependence beween componens Decision Suppor: parameer sensiiviies, componen imporance measure; inferences 11

12 Proposed approach: SRBTO using MSR Adop a single-loop RBTO Double-loop RBTO Objecive funcion iang e al. (2007), McDonald & Mahadevan (2008) Single-loop RBTO Objecive funcion Reliabiliy eval. 1 s consrain Reliabiliy eval. n h consrain Approx. MPP 1 s consrain Approx. MPP n h consrain Karush Kuhn Tucker (KKT) opimaliy condiions Use MSR mehod o compue P sys and is gradiens min f ( d, μ ) 1 d, μx, P,..., P n s.. g ( d, X( U )) 0 i=1,..., n P i sys X i cp T c p s S s s s T Psys ( ) f ( ) d P dependen sys indepdenden Single-loop PMA MSR mehod 12

13 SORM-based Single-loop CRBTO & SRBTO Enhance he accuracy in Single-loop RBTO Firs-Order Reliabiliy Mehod (FORM) inaccurae for nonlinear limi-saes Propose o use Second-Order Reliabiliy Mehod (SORM) o improve he accuracy SORM-based Single-loop CRBTO FORM-based A he k-h sep SORM-based ( k ) ( k ) i ( k 1) i i i ( k1)( SORM ) i SORM-based Single-loop SRBTO i u u ˆ u ˆ G G p G 0 FORM-based PE ( ; P ) cp c T p ( s ) ( s ) s s S P sys T f d P A he k-h sep 13 sys sys P sys P E = cp ( SORM ) ( sys; P ) SORM-based T ( SORM ) c p ( s) s S( s) s T ( SORM ) Psys f d P sys

14 SRBTO of a Cube F 3 F 2 F 1 F 2 F 3 Objecive: minimize volume V() imi-saes: g ( ρ, F ) 120 C ( ρ, F ), i 1,2 i i i i F 1 =12 Random loads: : F ~ (F 1,F 2,F 3 ) ~ N(100,10), N(0,30), N(0,40) oad cases: F1 ( F1, F2 ), F2 F1 F3 (, ) Consrains Deerminisic TO (DTO): Componen RBTO (CRBTO): Sysem RBTO (SRBTO): gi ( ρf, ) 0, i1,2 P( g ( ρf, ) 0) P, i 1,2 i i i P( g ( ρf, ) 0) P i i sys 14

15 Opimal Topologies volfrac = 6.3% volfrac = 24.4% ( F =10) volfrac =22.3% ( F1 =10) volfrac =23.9% ( F1 =20) DTO CRBTO SRBTO SRBTO 15 Differen opologies were obained

16 componen probabiliy componen probabiliy opimal volume fracion Improve Accuracy by Second-Order Reliabiliy Mehod Componen RBTO SORM-based provides more accurae resuls han FORM-based FORM-based CRBTO SORM-based CRBTO sandard deviaion (F 1 ) MCS on FORM-based MCS on SORM-based consrain on P 1 (C 1 ) MCS on FORM-based MCS on SORM-based consrain on P 2 (C 2 ) sandard deviaion (F 1 ) sandard deviaion (F 1 )

17 volume fracion sysem probabiliy Improve Accuracy by Second-Order Reliabiliy Mehod Sysem RBTO SORM-based provides more accurae resuls han FORM-based 0.5 FORM-based SRBTO SORM-based SRBTO MCS on FORM-based MCS on SORM-based consrain on P sys sandard deviaion (F 1 ) sandard deviaion (F 1 ) 17

18 SRBTO of a Building Core P P 1,q 1 P 1,q 1 Objecive: minimize volume V() P q oad case 1 P 1,q 1 P 1,q 1 imi-saes: g C C 0 i( ρ, Fi ) i i( ρ, Fi ) Random loads: F ~ (P 1,P 2,P 3,q 1,q 2,q 3 ) 5 P 2,q 2 oad case 2 4 symmery axes P 2,q 2 P 2,q 2 P 2,q 2 oad cases: F ( Pq, ) i i i P 3,q 3 oad case 3 P 3,q 3 oad Cases P q (a op) mean c.o.v mean c.o.v C i q/2 /12 10/12 /12 P 3,q 3 P 3,q 3 Case Case Case

19 opimal volume fracion Opimal Topologies of he Building Core SRBTO ( same =0.50, diff =0.25) DTO arge sysem failure probabiliy volfrac v.s P sys Probabiliies: SRBTO/MSR v.s MCS P 1 P 2 P 3 P sys same = 0.5 diff = 0.25 SRBTO MCS volfrac = 21.93% volfrac =28.15% (P sys =0.05) volfrac =22.25% (P sys =0.85) same = 0.5 diff = 0.25 SRBTO MCS DTO SRBTO SRBTO same = 0.9 diff = 0.45 SRBTO MCS

20 opimal volume fracion Building Core wih Paern Repeiion m DTO m=3 m=6 m=10 m= DTO SRBTO ( same =0.50, diff =0.25) paern symmery SRBTO ( same =0.90, diff =0.45) paern repeiion SRBTO hp:// number of paern repeiions, m 20

21 Muliresoluion opology opimizaion (MTOP) Uses hree disinc displacemen, densiy, and design variable fields Improves efficiency, apply o large-scale problems Sysem Reliabiliy-based Topology Opimizaion (SRBTO/MSR) Uses Marix-based sysem reliabiliy (MSR) mehod Enables uniform applicaions o general sysem evens Accouns for saisical dependence SORM-based Single-loop CRBTO & SRBTO Employs second-order reliabiliy mehod (SORM): SORM-based Single-loop CRBTO & SRBTO Improves accuracy Numerical Examples Summary & Conclusions 21 Includes paern repeiion consrains Compares wih Mone Carlo simulaions