SHAPE SENSITIVITY ANALYSIS AND OPTIMIZATION FOR A CONTACT PROBLEM IN THE MANUFACTURING PROCESS DESIGN

Transcription

1 SHAPE SENSITIVITY ANALYSIS AND OPTIMIZATION FOR A CONTACT PROBLEM IN THE MANUFACTURING PROCESS DESIGN 4 th World Congress of Structural and Multidisciplinary Optimization June 4-8, Dalian, China Nam H. Kim, Ki-young Yi, and K.K. Choi Center for Computer-Aided Design and Department of Mechanical Engineering The University of Iowa

2 OUTLINE DSA for Contact Problem with Frictional Return-Mapping Material Derivative of Frictional Return-Mapping Scheme Die Shape Design Parameters Smooth Contact Surface Meshfree Interpolation of C 2 -Continuous Contact Surface Numerical Examples Gasket Design Problem Metal Punch DSA Metal Extrusion Problem

3 3D CONTACT FORMULATION Penalty-Based Contact Formulation Contact Form b (, ) ˆ N zz = ω N gnz dγ Γ c X Gap Function c g = n ( x x ) 0 g Consistency Condition e c α ( x x ) = 0 Variational Equation a (, zz) + b (, zz) = l (), z z Z Ω N Ω x 1 Γ x n x c τv Slave Surface e 2 e 1 Master Surface 2 Γ x Initial Design Perturbed Design τv c

4 3D CONTACT FORMULATION cont. Material Derivative Formulas d d ( x ) = V( X) + z ( X) τ τ τ = 0 d ( ) = ( ) + + ξ dτ x V X z e c c c τ α α τ = 0 : Slave Particle : Contact Point on Master Surface Material Derivative of the Contact Form d b b b d * [ N( zτ, zτ)] N( zzz ;, ) + N( zz, ) τ τ = 0 b (, ) = b (;, ) + g V dγ c X * ˆ N zz N zvz ωn κ z n n Γ : Contact Fictitious Load b * N ( z;,) ii is same as the tangent stiffness operator that appears in contact analysis. Thus, the contact fictitious load can be calculated readily.

5 FRICTIONAL CONTACT DSA Elastoplasticity-Type Friction Model Trial Frictional Force f tr = f tr α e α f = f ω M ( ξ ξ ) tr n 1 n 1 α α T αγ γ γ Relative Slip Amount Frictional Consistency Condition tr h = f N g α µω If h 0, Stick Condition f α = f tr α Or else, Slip Condition Return-mapping algorithm in Elastoplasticity f α = µω g p N α p α = f f tr α tr

6 FRICTIONAL CONTACT DSA cont. f tr Frictional Return-mapping µω n g f n 1 ξ n 1 ξ n ξ (Slip in Parametric Domain) Frictional Form bt (, zz) = f d c αξα Γ Γ X Material derivative of the frictional form that is consistent with the frictional return-mapping algorithm has to be taken.

7 FRICTIONAL CONTACT DSA cont. Material Derivative of the Stick Condition n 1 n 1 fα = ωtφ αβξβ( z ) + ωtφ αβξβ( V) + fα + ωtmαβξβ ( z ) Φ = + n 1 αβ Mαβ Mαγ, β ( ξγ ξ γ ) d b b dτ [ * (, )] = ( ; T zz T zzz, ) : Implicit Term τ = 0 + b * T Γ ( zv ;, z) : Explicit Term + ξ + ω ξ ξ Γ c X n 1 n 1 ( fα α T Mαβ α β ) d : Path- Depenent Term The expressions of implicit and explicit terms are same. The contact stiffness matrix from the response analysis can be used for DSA.

8 FRICTIONAL CONTACT DSA cont. Material Derivative of the Slip Condition ( ˆ ˆ µω N g tr tr tr f β β α = µω N pαn z + V) + [ f p p f f p ] tr α α β α α p e f d b b dτ * [ T( zz, )] = T( zzz ;, ) τ = 0 + b * T (; zv, z) gξ β β β n 1 n 1 + µωn ( δ p p )( f ) c tr α α α + ωtmαγξγ dγ Γ X f b ( zz, ) T tr has the same expression as in the stick condition. f α The sensitivities of the frictional force and displacement contribute to the path-dependency of frictional contact DSA. f α

9 FRICTIONAL CONTAT DSA cont. Design Sensitivity Equation * * a (;, zzz ) + b (;, zzz ) = lv( z) av(, zz) bv(, zz), z Z Ω Γ Contact Fictitious Load b (, ) b N(, ) b V zz = zz + T( zz, ) Normal Contact Frictional Slip Path-dependency comes from the tangential friction. (Frictional force and contact particle displacement) The same tangent stiffness matrix from response analysis is used for DSA.

10 SMOOTH CONTACT SURFACE Construction of C 2 - continuous surface from a scattered set of particles. A design independent parametric plane is generated using surrounding particles. Meshfree shape function is independent of die shape design parameters. Slave Particle Master Surface r Projection Perturbed Surface τv c Meshfree Interpolation NP x( ξ, ξ ) = Ψ ( ξ, ξ ) x x 1 2 I 1 2 I = 1 NP dψ I ( ξ1, ξ2) ( ξ, ξ ) = dξ, α 1 2 I = 1 α I x I ξ 2 ξ 1 Parametric Plane Meshfree Shape Function Ψ(ξ 1,ξ 2 )

11 SMOOTH CONTACT SURFACE cont. Meshfree shape function in the local parametric domain Ψ ( ξ, ξ ) = H (0,0) M ( ξ, ξ ) H( ξ ξ, ξ ξ ) Φ ( ξ ξ, ξ ξ ) I T 1 I I I I a Moment Matrix Kernel Function The continuum-based design sensitivity formulation is independent of the surface generation algorithm. Only information that is already available from response analysis is necessary.

12 GASKET SHAPE OPTIMIZATION Oil Pan Gasket to Reduce Leakage Mooney-Rivlin Rubber Material Flexible-Rigid Body Contact and Self-Contact Conditions Significant Distortion in Self-Contact Regions Oil u u 5 Oil Pan Block 4 u 4 u 3 u u u 3 2 u 2 1 u 1 u 9 u 6 u 7 u 8 l gap 2 ( ) Engine Block d

13 DESIGN OPTIMIZATION Optimization Problem 1 Optimization History min d st.. F 300kN C σ 1700 kpa gap mm 0.5 u 0.5 i l gap 2 ( ) d Optimum Design Pressure Distribution

14 METAL PUNCH PROBLEM 558 Meshfree Particles (1,674 DOF) 306 Rigid Surface Particles (Smooth Surface) E = 207 GPa,ν = 0.29 Yield Stress = 167 MPa u 2 Rigid Surface Metal Plate u 1 Isotropic Hardening = 77.2 MPa Penalty Parameter = Frictional Coefficient = 0.1 Rigid Surface u 3

15 MESHFREE ANALYSIS RESULT Effective Plastic Strain Plot

16 DESIGN SENSITIVITY RESULTS Design u 1 u 2 u 3 Performance ψ ψ ψ τ ψ/ψ τ 100 z E E z E E z E E z E E z E E z E E z E E z E E z E E z E E z E E z E E z E E z E E E E z 312

17 METAL EXTRUSION PROBLEM

18 METAL EXTRUSION PROBLEM cont. Area Reduction Ratio 2.65 Maximum Plastic Strain 2.68 Length Extension Ratio 2.76

19 DESIGN PARAMETERIZATION CAD-Based Design Parameterization Die Shape Design Parameters.4073 Billet Extrusion Die Initial Billet Length = 0.6 Final Billet Length = % Extension 1.1 u 2 =.1 u 1 = 30 u 1 : Die Angle.45 u 2 : Fillet Radius u 3 =.1 u 3 : Fillet Radius 2.25

20 DESIGN VELOCITY FIELDS Design: u 1 u 2 u 3

21 DESIGN SENSITIVITY RESULTS Design Parameter u 1 u 2 u 3 Process Work Sensitivity 5.50E E E 5 Plastic Strain Sensitivity 5.32E E E 4 DSA Cost Finite Difference Cost = 0.2 Design Optimization Problem Minimize Process Work ( W ) Subject to Effective Plastic Strain ( e ) 2.68 p

22 DESIGN IMPROVEMENT Process Force Plot Initial Design New Design Process Work New Design Initial Design = 0.945

23 CONCLUSIONS DSA and optimization of the frictional contact problem is presented by using the continuum approach. The material derivative that is consistent with the frictional return mapping algorithm is derived. The smooth contact surface is used in the sensitivity formulation with design independent meshfree interpolation function. Numerical examples show the efficiency and accuracy of the proposed sensitivity calculation method. The current solid-based design approach will be further extended to the nonlinear shell structure.