STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION USING MESHFREE METHOD

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1 STRUTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION USING MESHFREE METHOD Workshop o Meshfree Methods November 4, 000 K.K. hoi, N.H. Kim, ad K.Y. Yi eter for omputer-aided Desig ollege of Egieerig The Uiversity of Iowa

2 ONTENTS Itroductio Meshfree Method Reproducig Kerel Particle Method (RKPM) Noliear Structural Aalysis Fiite Deformatio Elastoplasticity Frictioal otact Aalysis Pealty Method Desig Sesitivity Aalysis Shape Desig Sesitivity Aalysis for Elastoplasticity (Large Deformatio) Die Shape Desig Sesitivity Aalysis for otact Problem Numerical Examples of Shape Desig Optimizatio Torque Arm Desig Problem (Liear Elastic) Oil Pa Gasket Desig Problem (otact) Deepdrawig Desig Problem (Sprigback)

3 INTRODUTION Meshfree Discretizatio Reproducig Kerel Particle Method Is Used Stable Solutio for Large Shape hagig Problem Accurate Result for Fiite Deformatio Problem Direct Trasformatio/Mixed Trasformatio/Boudary Sigular Kerel Methods for Essetial B.. Frictioal otact Aalysis otiuum-based otact Formulatio Pealty Regularizatio of Variatioal Iequality Regularized oulomb Frictio Model DSA of Frictioal otact Problem DSA Variatioal Iequality Is Approximated Usig the Same Pealty Method Die Shape hage Is osidered by Perturbig Rigid Surface Path Depedet Sesitivity Results for Frictioal Problem

4 INTRODUTION cot. Structural Aalysis of Elastoplasticity Fiite Deformatio Elastoplasticity Usig Multiplicative Decompositio of Deformatio Gradiet Retur Mappig Algorithm i Pricipal Stress Space Stress Is omputed Usig Hyper-Elasticity w.r.t. Stress-Free Itermediate ofiguratio Exact Liearizatio Is Required for Quadratic overgece of Aalysis ad Accuracy of DSA Structural Desig Sesitivity Aalysis (DSA) Material Derivative Approach Is Used for Shape DSA Updated Lagragia Formulatio Is Used for Elastoplasticity Shape Fuctio of RKPM Depeds o Shape Desig Direct Differetiatio Method Is Used to Solve Displacemet Sesitivity DSA Equatio Is Solved at Each overged ofiguratio without Iteratio Material Derivative of Itermediate ofiguratio Is Updated at Each Load Step Istead of Stress i ovetioal Method

5 REPRODUING KERNEL PARTILE METHOD Reproduced Displacemet Fuctio z R ( x) = ( x; y x) φ ( y x) z( y) dy a R z ( x) z( x) as a 0 Dirac Delta Measure φ ( y x) > 0 if y x < a a φ ( y x) = 0 otherwise a orrectio Fuctio ( xy ; x) = q( x) T H( y x) H ( ) T y x = [1,( y x),( y x),,( y x) ] q( x) T = [ q ( x), q ( x),, q ( x)] 0 1 -th Order ompleteess Requiremet (Reproducig oditio) R z ( x) = ( x; y x) φ ( y x) z( y) dy a ( 1) d zx ( ) = m0 ( x) z( x) + m ( x)! dx = 1 m ( ) 1 ( ) 0 1,..., 0 x = mk x = k =

6 RKPM cot. Reproducig oditio M( x) q( x ) = H (0) H(0) T = [1,0,...,0] M( x) m ( x) m ( x)... m ( x) 0 1 m1( x) m( x)... m+ 1( x) = m ( x) m ( x)... m ( x) + 1 xy x x y x T 1 ( ; ) = H(0) M( ) H( ) z x x y x y x z y dy R T 1 ( ) = H(0) M( ) H( ) φa ( ) ( ) NP R z ( x) = ( x; x x) φ ( x x) z x = Φ ( x) d I a I I I I I I= 1 I= 1 NP

7 RKPM cot. Shape Fuctio Φ I (x I ) Depeds o urret oordiate Whereas FEA Shape Fuctios Deped o oordiate of the Referece Geometry Does Not Satisfy Kroecker Delta Property: Φ I (x J ) δ IJ Lagrage Multiplier Method for Essetial B.. T Π= U λ ( z ζ) d D First-order variatio is T T Π= U λ zd λ ( z ζ) d D D Direct Trasformatio Method, Mixed Trasformatio Method, ad Sigular Kerel Methods Are Available.

8 DOMAIN DISRETIZATION Meshfree Discretizatio Meshfree Shape Fuctio

9 MESHFREE METHOD Advatages ostructio of Shape/Iterpolatio Fuctio i Global Level Mesh Idepedet Solutio Accuracy otrol Versatile hp-adaptivity A Remedy to Mesh Distortio i Shape Optimizatio Accurate Solutio to Large Deformatio Problem Disadvatages Difficulties i Imposig Essetial Boudary oditios Expesive omputatioal ost Larger Badwidth of Stiffess Matrix Tha FEM

10 SHAPE DESIGN SENSITIVITY ANALYSIS Perturbed Desig z τ τ τv z ż 0 mappig T x x ( x), x d z = zτ ( x + τv) dτ zτ ( x + τv) z( x) = lim τ τ 0 : τ T( x, τ ) = x + τv( x) + H. O. T. Iitial Desig V : Desig velocity vector, directio of desig perturbatio τ : Desig perturbatio parameter z : Displacemet vector ż : Material Derivative of displacemet

11 NONLINEAR STRUTURAL ANALYSIS Noliear Variatioal Equatio (Updated Lagragia Formulatio) a ( z, z) b ( z, z) = l ( z), z Z a + (, zz) = τ: εd b ( z, z) l T B T S (z) = z f d + z f d S Structural Eergy Form otact Variatioal Form Load Liear Form Liearizatio a * ( z k ; z = l k + 1, z) + b ( z) a * ( ( z z k k ; z k+ 1, z) b, z) ( z k, z), z Z a * ( z; z, z) = [ ε : c : ε( z) + τ: η( zz, )] d τ c = = m m + τ c ε a lg i j p i cij ˆ i i= 1 j= 1 i= 1

12 FRITIONAL ONTAT PROBLEM 1 x g t x c Impeetratio oditio Tagetial Slip Fuctio g t ( ξ ξ ) 1 e e c g c t x 0 t c t c c ξ g ( x x ( ξ )) e ( ξ ) 0, x, x T 1 c c c c c c otact osistecy oditio T ϕξ ( ) = ( x x ( ξ)) e ( ξ) = 0 c c c t c otact Pealty Fuctio 1 1 P = ω g d ωt gt d + otact Variatioal Form b ( zz, ) = ω gg d+ ω ggds t t t -µω g ω t g t Modified oulomb Frictio Model

13 DESIGN SENSITIVITY ANALYSIS V(X) F d dτ ( z) Udeformed ofiguratio (Desig Referece) F p d dτ ( F p ) F e urret ofiguratio Itermediate ofiguratio (Aalysis Referece) Updated Lagragia Formulatio Fiite Deformatio Elastoplasticity No Need to Update Velocity Fields Updatig Sesitivity Iformatio of Itermediate ofiguratio ad Plastic Variables

14 FINITE DEFORMATION DSA Material Derivative of Variatioal Equatio d d d dτ dτ dτ a ( z, ) b (, ) ( ), Z τ τ z τ + 0 z τ τ z τ= τ = l τ= 0 z τ τ z τ= 0 τ τ Desig Sesitivity Equatio + = * * a ( z; zz, ) b ( z; zz, ) l V( z) a V( zz, ) b V( zz, ) Remarks: The same taget operator is used as aalysis -- Need accurate computatio of taget operator Direct Differetiatio -- DSA eeds to be carried out at each coverged load step Update sesitivity iformatio: itermediate cofiguratio (aalysis referece), plastic iteral variables, ad frictioal effect Total form of sesitivity equatio No iteratio is required to solve the sesitivity equatio

15 FINITE DEFORMATION DSA cot. Fictitious load fic ( ε: ε ε: ε τ :ε ) a (, zz) = c: () z+ c: () z+ d V V P ( τ:η (, zz) τ:η (, zz) τ:ε divv) d εv ( z) = sym( 0z V) η ε η V P P V P T ( z z V) sym( z ) ( z, z) = sym 0 ( z) = sym( G) T ( z, z) = sym( z G) 0 V Updatig path-depedet terms ( ) τ 3 p p fic i i p d d i = dτ ( α ) + d ( e ) p τ m i= 1 α eˆ e d p G = F ( F ) F dτ τ ( α + ) = ( α ) + H + H γ ( γ) N + H γ ( N) d d d d dτ 1 dτ α 3 α dτ α dτ d p d p d ( e 1) ( e ) ( ) dτ + = + γ dτ 3 dτ p e 1 e 1 d d d dτ F+ 1 = dτ F+ 1 F+ 1+ F+ 1 dτ F+ 1 ( ) ( ) ( ) tr tr d e d p e p d e dτ F+ 1 = dτ f F+ 1 + f dτ F+ 1 3 p j = exp( γ N j ) j= 1 ( ) ( ) ( ) f m Icremetal Plastic Deformatio Gradiet 1 τ

16 ONTAT DSA Material Derivative of otact Variatioal Form d * τ ( z, ) ( ;, ) (, ) τ τ z τ = z z z + z z d b b b V b ( z, z) = b ( z, z) + b ( z, z) V N T Normal otact Fictitious Load Form for DSA ( ) ( ) T T T T b (, zz) = ω ( z z) ee ( V V) d ω αg c ( z z) ee ( V V) d N c c c t t c T ( ) ξ ω ( ) T T T t z zc ee t Vc, t zc, ξ eet V Vc ω g c ( ) d g c ( ) d T T T T c, ξ c, ξ c ω g c z e e V d+ ω κg ( z z ) e ( V ) d

17 ONTAT DSA cot. Tagetial Stick Fictitious Load Form for DSA b = * T( zz, ) bt( zvz ;, ) ( ) T 1 T 1 t t t z z et et Vc, ξ z c, ξ + ω g c ( ) ( + ) d ( g ) 1 1 T 1 T 1 1 t ( t t ( z z) et et ( V z Vc z c) + ω ν β + d ( ) T T 1 t β t t t z z et e Vc, ξ z c, ξ + ω g g + c c ( ) ( + ) d ω t 1 T t t 1 g t t c c ( z z) e e ( V + z ) d 1 ( β ) ( β ) T t c, ξ c, ξ T 1 T 1 1 t t t t zc, ξ e et V z Vc z c + ω g g c c ( + ) d T 1 T 1 t t t zc, ξ e e Vc, ξ z c, ξ T t κ{ ( ) ( ) } 1 T T ν gt z z et + g gt t c ( z z) et ( V ) d + ω g g g c c ( + ) d + ω

18 MESHFREE METHOD WITH LARGE SHAPE HANGING PROBLEM u u 1 u 8 u 6 u 7 u 4 u3 u N Symmetric Desig 5066 N Iitial Desig FEM with remodelig : 45 iteratios Optimum Desig Meshfree w/o remodelig : 0 iteratios Aalysis Result after Remeshig at Optimum Desig

19 GASKET SHAPE DESIGN Oil Pa Gasket to Reduce Leakage Mooey-Rivli Rubber Material Flexible-Rigid Body otact ad Self-otact oditios Sigificat Distortio i Self-otact Regios Oil u u 5 Oil Pa Block 4 u 4 u 3 u u u 3 u 1 u 1 u 9 u 6 u 7 u 8 l gap ( ) Egie Block d

20 DESIGN OPTIMIZATION Optimizatio Problem 1 Optimizatio History mi d st.. F 300kN σ 1700 kpa gap 1.0 mm 0.5 u 0.5 i l gap ( ) d Optimum Desig Pressure Distributio

21 DESIGN OPTIMIZATION OF 5 mm u u 3 Puch u 1 Blak 6 mm u 4 u 5 Die DEEPDRAWING u 6 Blak Holder E = 06.9 GPa ν = 0.9 σ y = 167 MPa H = 19 MPa µ f = Isotropic Hardeig Performace(Ψ) Ψ Ψ` τ RATIO(%) u 1 e p e p e p e p e p e p G u e p e p e p e p e p e p G G = N I = 1 gap I

22 DESIGN OPTIMIZATION OF DEEPDRAWING (ONT.) Optimizatio Problem mi gap s.t. 0. e p DOT : Sequetial Quadratic Programmig ost Fuctio History Iteratio Desig Parameter History u1 u u3 u4 u5 u Iteratio

23 DESIGN OPTIMIZATION OF DEEPDRAWING (ONT.) Plastic Strai (Iitial Desig) Plastic Strai (Optimum Desig)

24 SUMMARY Meshfree Method Is Effective i Shape Optimizatio Developed Accurate ad Efficiet Shape DSA Method for Fiite Deformatio Elastoplasticity Usig Multiplicative Decompositio of Deformatio Gradiet Die Shape Desig Sesitivity Formulatio Is Developed for the Frictioal otact Problem Deepdrawig Optimizatio Is Successfully arried out to Reduce the Sprigback Amout Very Accurate DSA Results Made Optimizatio Problem overged i a Small Number of Iteratios