System Reliability-Based Design and Multiresolution Topology Optimization

Transcription

1 Dssertaton Defense System Relablty-Based Desgn and Multresoluton Topology Optmzaton Tam H. Nguyen 07/16/2010 Advsors: Glauco H. Paulno & Junho Song 1 Department of Cvl and Envronmental Engneerng Unversty of Illnos at Urbana-Champagn

2 Contents Introducton Multresoluton Topology Optmzaton (MTOP) Improvng Multresoluton Topology Optmzaton (MTOP) System Relablty-based Desgn Optmzaton (SRBDO) System Relablty-based Topology Optmzaton (SRBTO) Summary and Conclusons 2

3 Topology Optmzaton Classcal structural desgn optmzaton: the optmal szes or shapes for a gven layout and connectvty sze optmzaton shape optmzaton Topology optmzaton: the best topology, shape, sze under a gven doman and boundary condtons 3

4 Topology Optmzaton Applcatons Arbus Wng box rb 500 kg reducton/wng Skdmore, Owngs & Merrll, LLP (SOM) ( 4

5 Large-scale Topology Optmzaton Coarse mesh Low resoluton Fne mesh Computatonally expensve F 1 F 3 F B B8 ~ 1.0 ml. unknowns Fast solver, PC, C++ Run tme: ~ 45.7 hours An example usng Matlab code Wang, de Stuler, and Paulno, (2007), IJNME Queston 1: How to obtan hgh resoluton wth affordable computatonal cost? 5

6 Relablty-Based Desgn Optmzaton Determnstc Optmzaton Safe Low probablty of falure B Objectve functon ncrease f ( d, μ ) X Relablty-Based Desgn Optmzaton (RBDO) A Unsafe Hgh probablty of falure 6

7 System Relablty-Based Desgn Optmzaton Component RBDO System RBDO d, μ mn f ( d, μ ) d, μ X t s. t. P g ( dx, ) 0 P =1,..., n mn f ( d, μ ) X X d d d, μ μ μ L U L U X X X X s. t. P( E )= P g ( dx, ) 0 P sys k C d d d, μ μ μ L U L U X X X k? t sys Queston 2: How to handle system probablty n RBDO? 7

8 Objectves 1. To obtan hgh resoluton wth affordable computatonal cost n topology optmzaton. 2. To handle system probablty n Relablty- Based Desgn Optmzaton (RBDO). 3. To apply RBDO framework n topology optmzaton (RBTO). 8

9 Multresoluton Topology Optmzaton 9

10 Topology Optmzaton Procedure Problem formulaton mn C(ρ, u ) ρ st..: K(ρ) u f d mn T f ud V (ρ) ρ( ψ) dv V 0 ρ ρ( ψ) 1 Sold and Isotropc Materal wth Penalzaton (SIMP) E( ψ) ρ( ψ) p E d 0 s Intal guess Fnte Element Analyss Objectve Functon & Constrants Senstvtes Analyss Flterng (Projecton) Technque Update Materal Dstrbuton computatonally expensve P Optmzers Optmalty Crtera (OC) Method of Movng Asymptotes (MMA) Converged? Result Yes No 10

11 Hgh Resoluton Topology Optmzaton Large-scale (hgh resoluton) TOP Large number of fnte elements Computatonally expensve Exstng hgh resoluton TOP Parallel computng (Borrvall and Petersson, 2000) Fast solvers (Wang et al. 2007) Approxmate reanalyss (Amr et al. 2009) Adaptve mesh refnement (de Stuler et al. 2008) 11

12 TOP (1): Parallel Computng Parallel computng: Borrvall and Petersson, (2000), IJNME Doman decomposton 40x120x120 96x96x96 A cross-shaped secton (320,000 B8/U) A stool (884,736 B8/U) 12

13 Fast teratve solvers TOP (2): Fast Solvers Wang, de Stuler, and Paulno, (2007), IJNME Use precondton Krylov subspace methods wth recyclng Reduce computatonal tme for FEA Coarse mesh: 32x12x12 Confguraton p Soluton on a PC wth approx. 1 mllon unknowns Fne mesh: 180x60x60 13

14 TOP (3): Approxmate Reanalyss Reduce the number of FEA solutons FEA at an nterval of teratons Approxmate at other teratons Effcency factor: 1 ~ 5 tmes Intal guess Fnte Element Analyss or Approxmate the Dsplacement Objectve Functon & Constrants Senstvtes Analyss Flterng (Projecton) Technque Update Materal Dstrbuton MBB: 60x20 Q4/U Cantlever: 48x16x16 B8/U Converged? Yes No Amr, Bendsoe, and Sgmund, (2009), IJNME Result 14

15 TOP (4): Adaptve Mesh Refnement AMR TOP Refne the sold and surface regons de Stuler, Wang, and Paulno, (2008), IASS-IACM Reduce the total number of FEs Obtan resoluton as fne unform mesh (effcency factor 3) P Confguraton: 2:1:1 Unform mesh: 128x64x64 524,288 B8/U AMR: ntal mesh 64x32x32 ntal 65,520 B8/U fnal 228,692 B8/U 15

16 Remarks on Hgh Resoluton TOP Large-scale TOP Fne mesh: Large number of fnte elements FEA cost ncreases Exstng approaches: Powerful computng resources: many processors Reduce cost assocated wth FEA: Fast solvers Approxmate reanalyss Adaptve mesh refnement Same dscretzaton for analyss and desgn 16

17 Proposed Multresoluton TOP (MTOP) Conventonal element-based approach (Q4/U) Same dscretzaton for dsplacement and densty Dsplacement Densty/desgn varable Proposed MTOP approach (Q4/n25) Dfferent dscretzatons for dsplacement and densty/desgn varables Dsplacement Densty Desgn varable 17

18 MTOP: Integraton of Stffness Matrx Stffness matrx e e T K B DBd Numercal ntegraton K e n 1 T B DB A Q4/n25 A SIMP model V Nn Nn p T p e ( ) e 0 e A ( ) 1 1 K B D B I Senstvty N n (ρ ) I K d d d d p j j e Ke ρ j 1 ρ p 1 ρ p(ρ ) I n ρ n ρ n n B8/n125 18

19 MTOP: Projecton (flterng) Compute densty from desgn varables Mnmum length-scale (Guest et al. 2004, Almeda et al. 2009) f( d ) n n S m S d w( r ) n n wr ( ) m wr ( ) m r mn r mn r m f r m r mn 0 otherwse ρ wr ( n ) d w( r ) n m S m 19

20 MTOP Examples: 2D Cantlever Beam Objectve: mnmum complance Constrant: volfrac = 0.5 Length scale: r mn = 1.2 Nguyen, Paulno, Song, and Le, (2010), JSMO Confguraton MTOP Q4/n25 FE mesh 48x16 (C=208.23) Q4/U FE mesh 240x80 (C=210.68) Q4/U FE mesh 48x16 (C=205.57) Convergence hstory 20

21 MTOP Examples: 2D Mchell Truss Doman (3:2) Analytcal soluton Sgmund soluton (Sgmund, 2000) MTOP: 180x120 Q4/n25 elements 21

22 MTOP: 3D Cross-shaped Secton Borrvall & Petersson (2000) 320,000 B8/U elements MTOP 5,000 B8/n125 elements 22

23 MTOP: 3D Brdge Desgn L Confguraton L q Non-desgnable non-desgnable layer layer 6L 10x120x30 L 2L/3 L MTOP B8/n125 36,000 elements Exstng desgn ( 23

24 Can MTOP s effcency be mproved? MTOP approach Densty/desgn varable: same fne mesh FE mesh: coarse Reduce cost K(ρ) u f Improvng MTOP effcency? d Intal guess Fnte Element Analyss Objectve Functon & Constrants Senstvtes Analyss Flterng (Projecton) Technque MTOP??? Q4/n25 Update Materal Dstrbuton Converged? No Dfferent dscretzatons for densty & desgn varable? Result Yes 24

25 Improvng Multresoluton Topology Optmzaton (MTOP) MTOP approach (Q4/n25) or (Q4/n25/d25) Dsplacement Densty Desgn varable Proposed MTOP approach (Q4/n25/d9) and (Q4/n25/d16) Dsplacement Densty Desgn varable Q4/n25/d9 Q4/n25/d16 25

26 Improvng Multresoluton Topology Optmzaton (MTOP) MTOP MTOP Dsplacement Densty Desgn varable V V ρ ρ d d j d j B8/n125 B8/n125/d8 B8/n125/d15 ρ V d ρ V d j d j TET4/n64 TET4/n64/d8 TET4/n64/d10 26

27 MTOP: Projecton (Q4/n25/d9) Compute densty from desgn varables Mnmum length-scale (Guest et al. 2004, Almeda et al. 2009) f( d ) n Dsplacement n S m S d w( r ) n n wr ( ) m Densty Desgn varable 1 wr ( ) m r mn r mn r m f r m r mn 0 otherwse S r mn r n n r n w(r) ρ wr ( n ) d w( r ) n m S m r mn 27

28 normalzed computatonal tme Complance MTOP: MBB Beam Messerschmtt-Bolkow-Blohm (MBB) beam? 60x conventonal (60x20 Q4/U) MTOP (60x20 Q4/n25/d4) MTOP (60x20 Q4/n25/d9) MTOP (60x20 Q4/n25/d16) MTOP (60x20 Q4/n25) conventonal (300x100 Q4/U) Iteraton 300x100 Q4/U 60x20 Q4/U convergence 60x20 Q4/n25/d25 60x20 Q4/n25/d9 60x20 Q4/n25/d16 60x20 Q4/n25/d : conventonal (60x20 Q4/U) 2: MTOP (60x20 Q4/n25/d4) 3: MTOP (60x20 Q4/n25/d9) 4: MTOP (60x20 Q4/n25/d16) 5: MTOP (60x20 Q4/n25) 6: conventonal (300x100 Q4/U) Effcency 28

29 normalzed computatonal tme Complance MTOP: A Cube wth Concentrated Load Nguyen, Paulno, Song, and Le, (submtted), IJNME 24x24x24 L=24 L MTOP B8/n125/d8 MTOP B8/n125/d27) MTOP B8/n125/d64) MTOP B8/n125) L P MTOP B8/n125/d125 (C=29.04) Iteraton convergence MTOP B8/n125/d64 (C=29.06) MTOP B8/n125/d27 (C=29.08) MTOP B8/n125/d8 (C=29.33) : MTOP B8/n125/d8 2: MTOP B8/n125/d27 3: MTOP B8/n125/d64 4: MTOP B8/n Effcency 29

30 Adaptve MTOP Why Adaptve MTOP? Further mprove the effcency Reduce the number of densty elements and desgn varables durng optmzaton process? Adaptve MTOP (e.g. Q4/U & Q4/n25/d4) Q4/n25/d4 requres more computatonal cost than Q4/U Q4/n25/d4 provdes hgher resoluton Use Q4/n25/d4 where and when needed only, otherwse Q4/U Unchanged the Fnte Element Mesh durng optmzaton process 30

31 Adaptve MTOP Procedure Intal guess Fnte Element Analyss Element type array A(,j) = 1 (all Q4/U elements) Objectve Functon & Constrants Senstvtes Analyss Q4/U Q4/n25/d4 A(,j) = 1 A(,j) = 25 Flterng (Projecton) Technque Update Materal Dstrbuton Check & update element type array No Converged? Yes Result Element type Q4/U Q4/n25/d4 f L < < U A(,j) = 1 A(,j) = 25 Q4/n25/d4 Q4/U f k < L or U < k, k=1,25 A(,j) = 25 A(,j) = 1 31

32 Adaptve MTOP: 2D Cantlever Optmal topologes by MTOP and adaptve MTOP L 2L P Confguraton 160x80 Q4/U 32x16 Q4/U Adaptve MTOP optmzaton process 32x16 Q4/n25/d4 32x16 - adaptve Q4/n25/d4 & Q4/U Mesh (element types) Q4/U Q4/n25/d4 ntal teraton ntermedate teraton fnal teraton Topology 32

33 Adaptve MTOP: 3D Cantlever Beam L 2L L P Confguraton 24x12x12 B8/U 24x12x12 B8/n125/d8 Adaptve MTOP optmzaton process 24x12x12 B8/n125/d8 & B8/U Intal teraton (3,456 B8/U) Intermedate teraton (2,288 B8/U & 1,168 B8/n125/d8) Fnal teraton (2,072 B8/U & 1,384 B8/n125/d8) (FE mesh : 24x12x12 unchanged) 33

34 System Relablty-Based Desgn/Topology Optmzaton 34

35 RBDO Formulaton Component RBDO System RBDO d, μ mn f ( d, μ ) d, μ X t s. t. P g ( dx, ) 0 P =1,..., n mn f ( d, μ ) X X d d d, μ μ μ L U L U X X X X s. t. P( E )= P g ( dx, ) 0 P sys k C d d d, μ μ μ L U L U X X X k? t sys 35

36 Matrx-based System Relablty (MSR) Method Song and Kang, (2009); Structural Safety E 1 e 1 e 2 e 3 E 2 e 6 e 5 e 4 e 7 e 8 E 3 Convenent: matrx-based procedures for c and p; easy SRA calculaton (nner product) General: unform applcaton to seres, parallel, and any general systems Flexble: nequalty-type nformaton; ncomplete nformaton ( LP bounds method) Effcent: no need to re-compute p ; replace c for SRA of a new event mutually exclusve and collectvely exhaustve events (MECE) Common Source Effect: can account for statstcal dependence between components Decson Support: parameter senstvtes, component mportance measure; nferences 36

37 Proposed approach: SRBDO usng MSR Adopt a sngle-loop RBDO (Lang et al. 2007) Double-loop RBDO Objectve functon Sngle-loop RBDO Objectve functon Relablty eval. 1 st constrant Relablty eval. n th constrant Approx. MPP 1 st constrant Approx. MPP n th constrant Karush Kuhn Tucker (KKT) optmalty condtons Use MSR method to compute P sys and ts gradents mn f ( d, μ ) t 1 d, μx, P,..., P n t t s. t. g ( d, X( U )) 0 =1,..., n P sys X T c p s S s s s T t Psys cp t ( ) f ( ) d P dependent sys ndepdendent Sngle-loop PMA MSR method 37

38 Proposed approach: SRBDO usng MSR (contd.) Senstvty w.r.t. desgn varables d, x} P sys θ ps () c s ds T = fs( ) θ s p( s) 1 2 n P( s) ˆ P( s) = p( s) p( s)... p( s) P( s) θ θ θ T P( s) [ P ( s) P ( s) P ( s)] 1 2 n Use probabltes and senstvtes by component relablty analyss (FORM) Senstvty w.r.t. component falure probablty P t P ( s) P ( s) β P ( s) 1 P β P β φ( β ) 38

39 SRBDO of Truss System mn f ( d) 2( A A ) A A A A d { A,..., A } 1 6 sys t s. t. P = P g ( dx, ) 0 P k 1 C g ( dx, ) A F F 1, 2 A, A, A, A, A, A k A sys A F F 3,...,6 A 0 Mnmze total weght of the system Defnton of system falure: at least two members fal (cut-set systems): effects of load redstrbutons NOT consdered 6 L F A Random Varables (Gaussan dstrbuton) Mean Std Dev Member strength F, =1,..6 (Mpa) Appled load F A (kn) L 39

40 SRBDO of Truss System (contd.) Members Area: A ( 10 3 mm 2 ) Relablty Index: McDonald & Mahadevan SRBDO/MSR McDonald & Mahadevan SRBDO/MSR f(x) > Better optmal desgn (.e. less total weght) and symmetrc results Monte Carlo smulatons (c.o.v. = 0.03, 10 6 tmes) on the system falure probablty: P sys = (cf. MSR gves 0.001) 40

41 CIM SRBDO of Truss System (contd.) Condtonal probablty Importance Measure (CIM) CIM = P( E E )= sys P( EEsys ) c' PE ( ) sys T p T cp Relatve contrbuton of components to the system falure probablty (can be computed effcently by MSR method) Components 41

42 SRBDO of Truss System (contd.) Effects of load re-dstrbutons (sequental falures) Effects of correlaton between random varables and between components

43 Exstng SRBTO Approaches Dscrete structures Mogam et al. (2006) Truss examples Ground structure Optmal structure Mogam et al. (2006), JSMO Contnuum structures Slva et al. (2010) Lmt-states: statstcally ndependent n n n 1 n n 1 ( sys ) ( ) ( j ) ( 1) ( 1 2 n) j 1 P E P E P E P E E P E E E P( E1E 2E3) P( E1 ) P( E2) P( E3) DTO Slva et al. (2010), JSMO SRBTO Objectve: SRBTO for contnuum structures wth dependent lmt-states? 43

44 Proposed Approach: SORM-based RBTO Enhance the accuracy n RBTO Frst-Order Relablty Method (FORM) naccurate for nonlnear lmt-states Propose to use Second-Order Relablty Method (SORM) to mprove the accuracy g d, X( U) 0 SORM-based CRBTO t ( k ) t At the k-th step t t ( k ) t ( k 1) t( k 1)( SORM ) U 1 KKT U ˆ t α SORM-based SRBTO β t αˆ t β t t U * U t PE ( ; P ) c T p t ( s ) ( s ) s s S cp P sys T t t sys f d P t sys U 2 At the k-th step PE t( SORM ) ( sys; P ) c T t( SORM ) ( ) s ( S ) s T t( SORM ) t Psys cp f d P t sys 44

45 SRBTO of a Stool F 2 F 3 F F 3 2 F 1 L Objectve: mnmze volume V( ) Lmt-states: g ( ρ, F ) 120 C ( ρ, F ), 1,2 L F 1 L=12 Random loads: : F ~ (F 1,F 2,F 3 ) ~ N(100,10), N(0,30), N(0,40) Load cases: F1 ( F1, F2 ), F2 F1 F3 (, ) Constrants Determnstc TO (DTO): Component RBTO (CRBTO): System RBTO (SRBTO): g ( ρf, ) 0, 1,2 t P( g ( ρf, ) 0) P, 1,2 P( g ( ρf, ) 0) P t sys 45

46 Optmal Topologes volfrac = 6.3% volfrac = 24.4% ( F =10) volfrac =22.3% ( F1 =10) volfrac =23.9% ( F1 =20) DTO CRBTO SRBTO SRBTO 46

47 volume fracton system probablty optmal volume fracton component probablty component probablty Improve Accuracy by Second-Order Relablty Method Component RBTO Nguyen, Song, and Paulno, (submtted), JSMO FORM-based CRBTO SORM-based CRBTO MCS on FORM-based MCS on SORM-based constrant on P 1 (C 1 ) MCS on FORM-based MCS on SORM-based constrant on P 2 (C 2 ) standard devaton (F 1 ) System RBTO standard devaton (F 1 ) standard devaton (F 1 ) 0.5 FORM-based SRBTO SORM-based SRBTO MCS on FORM-based MCS on SORM-based constrant on P sys standard devaton (F 1 ) standard devaton (F 1 ) 47

48 SRBTO of a Buldng Core P P 1,q 1 P 1,q 1 Objectve: mnmze volume V( ) P q Load case 1 P1,q 1 P 1,q 1 Lmt-states: g C C 0 ( ρ, F ) ( ρ, F ) Random loads: F ~ (P 1,P 2,P 3,q 1,q 2,q 3 ) L L 5L P 2,q 2 Load case 2 4 symmetry axes P 2,q 2 P 2,q 2 P 2,q 2 Load cases: F ( Pq, ) P 3,q 3 Load case 3 P 3,q 3 Load Cases P q (at top) mean c.o.v mean c.o.v t C q/2 L L L/12 10L/12 L/12 P 3,q 3 P 3,q 3 Case Case Case

49 optmal volume fracton Optmal Topologes of the Buldng Core Nguyen, Song, and Paulno, (submtted), JSMO SRBTO ( same =0.50, DTO dff =0.25) target system falure probablty volfrac v.s P sys Probabltes: SRBTO/MSR v.s MCS P 1 P 2 P 3 P sys volfrac = 21.93% volfrac =28.15% (P sys =0.05) volfrac =22.25% (P sys =0.85) same = 0.5 dff = 0.25 same = 0.5 dff = 0.25 SRBTO MCS SRBTO MCS DTO SRBTO SRBTO same = 0.9 dff = 0.45 SRBTO MCS

50 optmal volume fracton Buldng Core wth Pattern Repetton L m L DTO m=3 m=6 m=10 m= DTO SRBTO ( same =0.50, pattern symmetry SRBTO ( same =0.90, dff =0.25) dff =0.45) pattern repetton SRBTO number of pattern repettons, m 50

51 MTOP & MTOP: Use three dstnct dsplacement, densty, and desgn varable felds Improve effcency, apply to large-scale problems Adaptve MTOP: Use MTOP and MTOP elements where and when needed Reduce the number of densty elements and desgn varables SRBDO/MSR: Apply to general system ncludng lnk-set, cut-set Address dependence between lmt-states, provde senstvty SRBTO Summary & Conclusons Propose accurate sngle-loop SORM-based CRBTO & SRBTO approaches Include pattern repetton constrants 51

52 Future Research Topcs Optmal locatons of desgn varables n MTOP MTOP approach n nonlnear and stress-based problems MTOP usng Krylov subspace methods and recyclng SRBDO wth mult-scale MSR approach SRBDO wth mxed contnuous-dscrete random varables 52

53 Contrbutons Nguyen, T. H., Paulno, G. H., Song, J., Le, C. H., (2010). "A computatonal paradgm for multresoluton topology optmzaton (MTOP)." Structural and Multdscplnary Optmzaton 41(4): Nguyen, T. H., Song, J., Paulno, G. H., (2010). "Sngle-loop system relablty-based desgn optmzaton usng matrx-based system relablty method: theory and applcatons." Journal of Mechancal Desgn 132(1): Sutradhar, A., Paulno, G. H., Mller, M. J., Nguyen, T. H., (2010). Topologcal optmzaton for desgnng patent-specfc large cranofacal segmental bone replacements. Proceedngs of the Natonal Academy of Scences 107(30) Nguyen, T. H., Paulno, G. H., Song, J., Le, C. H., "Improvng multresoluton topology optmzaton va multple dscretzatons." Internatonal Journal for Numercal Methods n Engneerng (submtted). Nguyen, T. H., Song, J., Paulno, G. H., "Sngle-loop system relablty-based topology optmzaton consderng statstcal dependence between lmt states." Structural and Multdscplnary Optmzaton (submtted). 53

54 Acknowledgements Advsors: Glauco H. Paulno & Junho Song Commttee: Glauco H. Paulno, Junho Song, Jerome F. Hajjar, C. Armando Duarte, Rav C. Penmetsa, Wllam F. Baker, Alessandro Beghn, Alok Sutradhar Fnancal Support: Vetnam Educaton Foundaton Natonal Scence Foundaton Computatonal Mechancs Group, Structural System Relablty Group and colleagues at UIUC Specal thanks to my famly 54

55 55 Thank you for your attenton!