Design Sensitivity Analysis and Optimization of Nonlinear Transient Dynamics

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1 Desig Sesitivity Aalysis ad Otimizatio of Noliear Trasiet Dyamics 8th AIAA/USAF/NASA/ISSOMO Symosium o Multidisciliary Aalysis ad Otimizatio Nam Ho Kim ad Kyug Kook Choi Ceter for Comuter-Aided Desig The Uiversity of Iowa

2 Itroductio Cotets Structural Dyamics Multilicative Elastolasticity Liearizatio ad Taget Stiffess Imlicit Time Itegratio Shae Desig Sesitivity Aalysis Fiite Deformatio Elastolasticity Time Itegratio of Desig Sesitivity Equatio Udate Path-Deedet Variables Numerical Examles Bumer Imact Problems Pressurized Sheet Metal Stamig Problems

3 Itroductio Shae Desig Sesitivity Aalysis of Noliear Structure Fiite Deformatio Elastolasticity Usig Multilicative Decomositio of the Deformatio Gradiet Classical Retur Maig Algorithm Is Preserved Usig Pricial Kirchhoff Stress ad Logarithmic Strai Desig Sesitivity Equatio Is Obtaied at the Iitial Domai ad The Trasformed ito the Curret Domai to Recover the Udated Lagragia Form Path-Deedecy Comes from the Itermediate Cofiguratio ad Plastic Evolutio Variables Exact Taget Oerator Yields Iteratio-Free DSA

4 Itroductio cot. Desig Sesitivity Aalysis of Structural Dyamics Newmark Family Imlicit Time Itegratio Is Used to Solve 2d-Order Desig Sesitivity Differetial Equatio with Homogeeous Iitial Coditios Desig Sesitivity Equatio Solves for the Material Derivative of Acceleratio at Coverged Time Stes Time Itegratio Material Derivative of Dislacemet Sesitivity Equatio Is More Efficiet for the Imlicit Time Itegratio Method Tha the Exlicit Method Comared to the Cost of Resose Aalysis

5 Multilicative Elastolasticity Kiematics (Deformatio Gradiet) FX ( ) = F e ( XF ) ( X) Pricial Logarithmic Strai e = e 3 T e 2 = ν i i i= 1 b FF m e log( ν ) 1 1 e2 log( ν 2) e = = e log( ν ) 3 3 Kirchhoff Stress 3 i τ = τ m i= 1 i i i i m = i : Pricial Vector τ i : Pricial Stress Isotroic Material Assumtio

6 Costitutive Relatio Trial Elastic Pricial Stress (τ ) tr = c e e tr e 2 c = ( λ + µ ) µ I 3 dev All icremetal deformatio is assumed to be elastic Yield Fuctio f ( η, e ) = η 2 3 κ ( e ) η = dev( τ ) α = η tr (2µ + H α ( e )) γ 2 3 κ ( e ) = 0 Retur Maig Algorithm tr τ + 1 = ( τ ) 2µγ N α = α γ Hα ( e ) N e = + e + γ 2 1 3

7 Liearizatio ad Taget Stiffess Taget Stiffess Tesor alg i j i cij 2 τ i trial i= 1 j= 1 i= 1 c = m m + c 2 alg τ e 2 4µγ c = = c 4 µ AN N [ Idev N N] tr tr e η Liearizatio c alg is the same as classical elastolasticity i ricial stress/strai sace a(, zz) = τε() zdω Structural Eergy Form Ω ij ij ( εij ijklεkl τijηij ) a * (; z z, z) () z c ( z) + ( z, z) dω Ω Udated Lagragia Formulatio

8 Itermediate Cofiguratio Udate Itermediate Cofiguratio f F 3 tr 1 e tr i = 1 = ex( ei ) i= 1 3 i = ex( γ Ni ) m i= 1 F FF m e = f F e tr e F = F F 1 Icremetal Plastic Deformatio Gradiet It is assumed that the icremetal lastic si is arbitrary Cotiuous itermediate cofiguratio

9 Variatioal Formulatio of Structural Dyamics Weak Form d( z, z) + a( z, z) = l( z), z Z T d( z, z) z z, ttd Kietic Eergy Form z Ω z T b T h l( z) = z f dω+ z f dγ Load Liear Form h Ω Γ Variatioal Equatio i Structural Domai 2d-Order Differetial Equatio i Time Domai Iitial Coditios 0 zx (, 0) = z( x) x Ω 0 z ( x, 0) = z ( x) x Ω, t, t =z ρ Ω

10 Imlicit Time Itegratio Newmark Method Predictor z = z + ( 1 γ ) t z r 1 1, t, t z = z+ t z + ( β) t z r , t 2 Corrector z = z +γ t z r, t, t z= z +β t 2 z, r tt β, γ : Newmark Parameters

11 Imlicit Time Itegratio cot. Liearizatio d [ ( z, z) Is Noliear w.r.t. z ] a Ω z z z z z k+ 1 * k k+ 1 (, ) + a ( ;, ) = l( z) a( z, z) d( z, z), z Z k k : Time t k : Iteratio Couter Acceleratio Form k z = t z k + 1 β d( z, z) + β t a ( z ; z, z) k 1 2 * k k 1 l k k = ( z) a( z, z) d( z, z), z Z Udate Kiematic Variables z = z + z k+ 1 k k+ 1 r z 0 = z

12 Fiite Deformatio DSA V(X) F d dτ ( z) Udeformed Cofiguratio (Desig Referece) F d dτ ( F ) F e Curret Cofiguratio Itermediate Cofiguratio (Aalysis Referece) Udated Lagragia Formulatio Fiite Deformatio Elastolasticity No Need to Udate Velocity Fields Udatig Sesitivity Iformatio of Itermediate Cofiguratio ad Plastic Variables

13 Fiite Deformatio DSA cot. Material Derivative z = d dτ ( z) 1 = lim τ ( τ 0 τ + ) ( ) = z + zv [ z X τ V z X ] 0 V(X) : Desig Velocity Field Material Derivative of Structural Eergy Form d dτ * [ a( zz, )] = a ( zzz ;, ) + a ( zz, ) V Exlicitly Deedet Terms o V(X) Path-Deedet Terms Imlicitly Deedet Terms Same As Structural Liearizatio

14 Fiite Deformatio DSA cot. Fictitious load P P fic ( ε ε τ η τ ε ) V V ( εij ( z) ijklεkl ( z) τijηij ( z, z) τijεij ( z) V) a (, zz) = () zc () z+ (, zz) + () z dω V ij ijkl kl ij ij ij ij Ω ε V + c + + div dω Ω Exlicitly Deedet Terms () z = sym( 0 z V) sym η V T (,) zz = ( z 0z V) sym( z V) 0 Path-Deedet Terms ε P () z = sym( G) η τ T ( z, z) = sym( z G) τ τ 3 fic i d i d i = ( ) ( e ) dτ α + dτ m i= 1 α eˆ e d G = F ( F ) F dτ 1

15 DSA for Structural Dyamics Kietic Eergy [ d( z, z )] = τ ρ z z d Ω+ ρ z z div V d Ω d d T T Ω Ω d( z, z) + d ( z, z) V Desig Sesitivity Equatio [ d( z, z)] + [ a( z, z)] = [ l( z)], z Z d d d dτ dτ dτ * d( z, z) + a ( z; z, z) = l ( z) a ( z, z) d ( z, z), z Z V V V Iitial Coditios (Homogeeous) z ( x, 0 ) = 0 x Ω z, t ( x, 0 ) = 0 x Ω

16 DSA for Structural Dyamics cot. Predictor z = z + ( 1 γ ) t z r 1 1, t, t z = z+ t z + β t z r , t ( ) 1 2 Corrector z = z +γ t z r, t, t z= z +β t 2 z r Acceleratio Form DSA * d( z, z) + β t 2 a ( z; z, z) = l ( z) a ( z, z) V V * r d ( z, z) a ( z; z, z), z Z V Sesitivity Equatio Is Liear ad Solves for Total Acceleratio

17 Udate Path-Deedet Variables Udatig Plastic Variables d dτ 2 ( ) ( α + ) = ( α ) + H + H γ ( γ) N + H γ ( N) d d d d dτ 1 dτ α 3 α dτ α dτ d d 2 d dτ e+ 1 = dτ e + 3 dτ γ ( ) ( ) ( ) 1 tr ( N) = [ Idev N N] 2 µ ( e ) ( α) tr η d d d dτ dτ dτ d dτ ( F + ) = ( I+ z) = z z V d 1 dτ e 1 e 1 d d d dτ F+ 1 = dτ F+ 1 F+ 1+ F+ 1 dτ F+ 1 ( ) ( ) ( ) ( F ) = ( f ) F + f ( F ) d e d e d e dτ + 1 dτ + 1 dτ + 1 d tr d d [ 2µ ( e ) ( )] Aκ ( e ) T ( γ ) = AN α Udatig Itermediate Cofiguratio tr dτ dτ tr dτ d dτ e 1 e 1 d e e = + 1 dτ ( F ) F ( F ) F

18 Bumer Imact Problem u 6 u 8u10 u 4 u 2 u3 u 5 u 1 u 7u9 u u u 11 u u u 15 Aalysis Meshfree Method Desity ρ = 7,800 kg/m 3 Iitial Velocity v 0 = 8.05 km/hr Aalysis Time t = 0 ~ 10 msec Time Icremet t = 0.1 msec Moutig Disl. d = 2.8 cm Thickess h = 0.5 cm Cotact Pealty No. w = 1,000 Frictio Coeff. µ f = 0.4 Lame s Costats λ = GPa µ = 80.2 GPa Plastic Hardeig H = 1.1 GPa Isotroic Hardeig Iitial Yield Stress σ Y = 500 MPa Newmark Parameters γ = 0.26 β = 0.5

19 Aalysis Results Resose Aalysis 1,600 sec Sesitivity Aalysis 853 / 16 sec Effective Plastic Strai Vo Mises Stress

20 Time History 8.0 BUMPER IMPACT PROBLEM 6.0 Dis Vel Acc Time (msec) Time History of Node 24

21 Time History cot BUMPER IMPACT PROBLEM 8.0 Dis Vel Acc Time (msec) Time History of Node 39

22 Sesitivity Results ad Otimizatio Problem Performace(Ψ) Ψ Ψ ( Ψ / Ψ ) 100% u 2 e E E E e E E E e E E E z x e e e FCx E E E u 4 e E E E e E E E e E E E z x e E E FCx E E E u 6 e E E E e E E E e E E E zx e E E F Cx E E E u 8 e E E E e E E E e E E E z x e E E FCx E E E u 10 e E E E e E E E e E E E z x e E E F Cx E E E u 12 e E E E e E E E e E E E z x e E E F Cx E E E Desig Otimizatio Problem Defiitio MIN Area ST e e.. 6(0.07) (0.02) 0.04 e 12(0.02) 0.04 e 13(0.02) 0.04 e 14(0.02) 0.04 e 15(0.07) 0.04 e 16(0.09) 0.04 e 17(0.05) 0.04 e 28(0.04) 0.04 e 29(0.05) 0.04 e 45(0.01) 0.04 e 46(0.01) 0.04 e 65(0.06) 0.04 e 66(0.04) 0.04 e 67(0.02) 0.04 FCx (2.0) u 1.0 i= 1,16 i

23 Otimizatio Results Iitial Desig Otimized Desig Effective Plastic Strai Vo Mises Stress

24 Otimizatio History BUMP IMPACT PROBLEM COST*.001 Eff6 Eff12 Eff13 Eff14 Eff15 Eff16 Eff17 Eff28 Eff29 Eff65 Eff66 Eff67 Fcx* ITER 12 Resose Aalysis : 30 Sesitivity Aalysis : 12

25 Pressurized Sheet Metal Stamig Iitial Geometry ad Desig Parameters Pressure u 17 u 18 Sheet Metal u 9 u 11 u 12 u 15 u 13 u 14 u16 u 1 u 5 u 6 u 3 u 4 u 7 u 8 u 10 Rigid Die u 2 Die Shae DSA ad Otimizatio Kim et al. Com. Mech. 25 (2000)

26 Pressure Load Time History 14.0 TIME HISTORY OF PRESSURE(MPa) Time(msec)

27 Time History of Deformatio SHEET METAL STAMPING Dis. Vel. Acc Time(msec)

28 Aalysis Results Effective Plastic Strai Srig-Back (1.75) Vo Mises Stress

29 Desig Otimizatio 2 MIN G = π ( x) x dγ Γ ST.. e i 0.16 i= 22, 28, 49,55, 68,70,72, u 3.0 j = 1,,18 j Resose Aalysis : 12,018 sec Sesitivity Aalysis : 3,215/18 sec Iitial Shae Otimum Shae

30 Otimizatio History 0.40 Otimizatio of Sheet Metal Stamig COST*.01 EffSt22 EffSt28 Effst49 EffSt55 EffSt68 EffSt70 EffSt72 EffSt ITER Resose Aalysis : 41 Sesitivity Aalysis : 14

31 Otimizatio Results Effective Plastic Strai Predetermied Shae Vo Mises Stress

32 Coclusios A Accurate ad Efficiet Shae DSA ad Otimizatio of Structural Trasiet Dyamics is Proosed. Fiite Deformatio Elastolastic Material ad Frictioal Cotact Coditio Are Cosidered i DSA Desig Sesitivity Equatio Is Solved at Each Coverged Time Ste without Iteratio Usig the Same Taget Stiffess Matrix from Aalysis Sesitivity Equatio Is More Efficiet for the Imlicit Time Itegratio Method Tha the Exlicit Method Comared to the Cost of Resose Aalysis