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STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION USING MESHFREE METHOD K.K. Choi, N.H. Kim, and K.Y. Yi Center for Computer-Aided Design College of Engineering The University of Iowa Workshop on Meshfree Methods November 4, 2000

STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

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Page 1: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND

OPTIMIZATION USING MESHFREE METHOD

K.K. Choi, N.H. Kim, and K.Y. Yi

Center for Computer-Aided Design

College of Engineering

The University of Iowa

Workshop on Meshfree Methods

November 4, 2000

Page 2: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

CONTENTS• Introduction

• Meshfree Method– Reproducing Kernel Particle Method (RKPM)

• Nonlinear Structural Analysis

– Finite Deformation Elastoplasticity

– Frictional Contact Analysis – Penalty Method

• Design Sensitivity Analysis

– Shape Design Sensitivity Analysis for Elastoplasticity (Large Deformation)

– Die Shape Design Sensitivity Analysis for Contact Problem

• Numerical Examples of Shape Design Optimization

– Torque Arm Design Problem (Linear Elastic)

– Oil Pan Gasket Design Problem (Contact)

– Deepdrawing Design Problem (Springback)

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INTRODUCTION

Frictional Contact Analysis• Continuum-Based Contact Formulation

• Penalty Regularization of Variational Inequality

• Regularized Coulomb Friction Model

DSA of Frictional Contact Problem• DSA Variational Inequality Is Approximated Using

the Same Penalty Method

• Die Shape Change Is Considered by Perturbing Rigid Surface

• Path Dependent Sensitivity Results for Frictional Problem

Meshfree Discretization• Reproducing Kernel Particle Method Is Used

• Stable Solution for Large Shape Changing Problem

• Accurate Result for Finite Deformation Problem

• Direct Transformation/Mixed Transformation/BoundarySingular Kernel Methods for Essential B. C.

Page 4: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

INTRODUCTION cont.

Structural Analysis of Elastoplasticity• Finite Deformation Elastoplasticity Using Multiplicative

Decomposition of Deformation Gradient• Return Mapping Algorithm in Principal Stress Space• Stress Is Computed Using Hyper-Elasticity w.r.t. Stress-Free

Intermediate Configuration• Exact Linearization Is Required for Quadratic Convergence

of Analysis and Accuracy of DSA

Structural Design Sensitivity Analysis (DSA)• Material Derivative Approach Is Used for Shape DSA• Updated Lagrangian Formulation Is Used for Elastoplasticity• Shape Function of RKPM Depends on Shape Design• Direct Differentiation Method Is Used to Solve

Displacement Sensitivity• DSA Equation Is Solved at Each Converged Configuration

without Iteration• Material Derivative of Intermediate Configuration Is Updated

at Each Load Step Instead of Stress in Conventional Method

Page 5: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

REPRODUCING KERNEL PARTICLE METHOD

Reproduced Displacement Function

( ) ( ; ) ( ) ( )Raz x C x y x y x z y dyφ

Ω

= − −∫

( ) ( ) as 0 Dirac Delta MeasureRz x z x a→ →

( ; ) ( ) ( )TC x y x x y x− = −q H 2( ) [1, ( ), ( ) , , ( ) ]T ny x y x y x y x− = − − ⋅⋅⋅ −H

0 1( ) [ ( ), ( ), , ( )]Tnx q x q x q x= ⋅⋅ ⋅q

01

( ) ( ; ) ( ) ( )

( 1) ( )( ) ( ) ( )

!

Ra

n n

n nn

z x C x y x y x z y dy

d z xm x z x m x

n dx

φΩ

=

= − −

−= +

0 ( ) 1 ( ) 0 1,...,km x m x k n= = =

Correction Function

n-th Order Completeness Requirement (Reproducing Condition)

( ) 0 if

( ) 0 otherwisea

a

y x y x a

y x

φφ − > − <

− =

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RKPM cont.

Reproducing Condition

( ) ( ) (0)x x =M q H (0) [1,0,...,0]T =H

0 1

1 2 1

1 2

( ) ( ) ... ( )

( ) ( ) ... ( )( )

. . ... .

( ) ( ) ... ( )

n

n

n n n

m x m x m x

m x m x m xx

m x m x m x

+

+

=

M

1( ; ) (0) ( ) ( )TC x y x x y x−− = −H M H

1( ) (0) ( ) ( ) ( ) ( )R Taz x x y x y x z y dyφ−

Ω

= − −∫H M H

1 1

( ) ( ; ) ( ) ( )NP NP

RI a I I I I I

I I

z x C x x x x x z x x dφ= =

= − − ∆ = Φ∑ ∑

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RKPM cont.

• Shape Function ΦI(xI) Depends on Current Coordinate

Whereas FEA Shape Functions Depend on Coordinate of

the Reference Geometry

• Does Not Satisfy Kronecker Delta Property: ΦI(xJ) ≠ δIJ

• Lagrange Multiplier Method for Essential B.C.

– First-order variation is

• Direct Transformation Method, Mixed Transformation

Method, and Singular Kernel Methods Are Available.

( )D

TU z dλ ζΓ

Π = − − Γ∫

( )D D

T TU z d z dλ λ ζΓ Γ

Π = − Γ − − Γ∫ ∫

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DOMAIN DISCRETIZATION

Meshfree Shape Function

Meshfree Discretization

Page 9: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

MESHFREE METHOD

Larger Bandwidth of Stiffness Matrix Than FEM

Expensive Computational Cost

Difficulties in Imposing Essential Boundary Conditions

A Remedy to Mesh Distortion in Shape Optimization

Accurate Solution to Large Deformation Problem

Versatile hp-Adaptivity

Mesh Independent Solution Accuracy Control

Construction of Shape/Interpolation Function in Global Level

Disadvantages

Advantages

Page 10: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

SHAPE DESIGN SENSITIVITY ANALYSIS

0Ωτ

nΩτ

zτV

Initial Design

Perturbed Design zτ

ż

V : Design velocity vector, direction of design perturbationτ : Design perturbation parameterz : Displacement vectorż : Material Derivative of displacement

−+=

+=

→ ττ

ττ

ττ

τ

)()(lim

)(

0

xzVxz

Vxzzd

d

Ω∈→ xxxxT ),(: τmapping

...)(),( TOH++= xVxxT ττ

Page 11: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

NONLINEAR STRUCTURAL ANALYSIS

• Nonlinear Variational Equation (Updated Lagrangian Formulation)

• Linearization

Structural Energy Form

Contact Variational Form

Load Linear Form

Z),,(),()(

),;(),;( 1*1*

∈∀−−=

∆+∆

ΓΩΩ

zzzzzz

zzzzzzknkn

kknkkn

ba

ba

Z),(),(),( ∈∀=+ ΩΓΩ zzzzzz nn ba

( , ) :a dΩ Ω= Ω∫z z τ ε

( , )bΓ z z

∫∫ ΓΩΩ Γ+Ω=S

dd STBT fzfzz)(

* ( ; , ) [ : : ( ) : ( , )]a dΩ Ω∆ = ∆ + ∆ Ω∫z z z ε c ε z τ η z z

3 3 3a lg i j p iij i

i 1 j 1 i 1ˆc 2

= = =

∂= = ⊗ + τ∑∑ ∑∂τ

c m m cε

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FRICTIONAL CONTACT PROBLEM

Contact Penalty Function

Impenetration Condition

Tangential Slip Function

Contact Consistency Condition

gt

xcet

xc0

xen

Ω1

Ω2

Γc2Γc

1

ξ

gn

t00 0( )t c cg ξ ξ≡ −t

1 2( ( )) ( ) 0, ,Tn c c n c c c cg ξ ξ≡ − ≥ ∈Γ ∈Γx x e x x

( ) ( ( )) ( ) 0Tc c c t cϕ ξ ξ ξ= − =x x e

2 21 1

2 2C C

n n t tP g d g dω ωΓ Γ

= Γ + Γ∫ ∫

Contact Variational Form

( , )C C

n n n t t tb g g d g g d Sω ωΓΓ Γ

= Γ + Γ∫ ∫z z

-µωngn

gt

ωt

Modified Coulomb Friction Model

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DESIGN SENSITIVITY ANALYSIS

Undeformed Configuration

(Design Reference)

Intermediate Configuration

(Analysis Reference)

Current Configuration

)( pdd Fτ

)(dd zτ

V(X)

F

FeFp

• Updated Lagrangian Formulation

• Finite Deformation Elastoplasticity

• No Need to Update Velocity Fields

• Updating Sensitivity Information of Intermediate

Configuration and Plastic Variables

Page 14: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

FINITE DEFORMATION DSA

Remarks:– The same tangent operator is used as analysis -- Need accurate

computation of tangent operator– Direct Differentiation -- DSA needs to be carried out at each

converged load step– Update sensitivity information: intermediate configuration (analysis

reference), plastic internal variables, and frictional effect– Total form of sensitivity equation– No iteration is required to solve the sensitivity equation

n n

0 0 0

d d da ( , ) b ( , ) ( ) , Z

d d dτ τ τΩ τ τ Γ τ τ Ω τ τ ττ= τ= τ= + = ∀ ∈ τ τ τ

z z z z z z

* n n * n n n na ( ; , ) b ( ; , ) ( ) a ( , ) b ( , )ΓΩ ′ ′ ′+ = − −V V Vz z z z z z z z z z z

Design Sensitivity Equation

Material Derivative of Variational Equation

Page 15: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

FINITE DEFORMATION DSA cont.

( )( )

( , ) : ( ) : ( )

( , ) ( , )

ficV V P

V P

a d

div d

Ω

Ω

′ = + + Ω

+ + + Ω

z z c z c z

z z z z V

ε : ε ε : ε τ : εε : ε ε : ε τ : εε : ε ε : ε τ : εε : ε ε : ε τ : ε

τ : η τ : η τ : ετ : η τ : η τ : ετ : η τ : η τ : ετ : η τ : η τ : ε

)()( 0 Vzz nV sym ∇∇−=εεεε

( ) )(),( 00 VzVzzzz nnT

nV symsym ∇∇−∇∇∇−=ηηηη

1)( −= FFFG pddeτ

)()( Gz symP −=εεεε

( , ) ( )TP nsym= − ∇z z z Gηηηη

3

1

( ) ( )ˆ

p pfic p ii id d

n nd dpi

eeτ τ

τ τ=

∂ ∂= + ∂ ∂ ∑ mτ ατ ατ ατ α

αααα

Fictitious load

( )21 3( ) ( ) ( ) ( )d d d d

n nd d d dH H Hα α ατ τ τ τγ γ γ+ ′= + + +N Nα αα αα αα α2

1 3( ) ( ) ( )p pd d dn nd d de eτ τ τ γ+ = +

1 1

1 1 1 1 1( ) ( ) ( )p e ed d dn n n n nd d dτ τ τ

− −+ + + + += +F F F F F

1 1 1( ) ( ) ( )tr tre p e p ed d d

n n nd d dτ τ τ+ + += +F f F f F3

1

exp( )p jj

j

Nγ=

= −∑f m Incremental Plastic Deformation Gradient

Updating path-dependent terms

Page 16: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

CONTACT DSA

• Normal Contact Fictitious Load Form for DSA

*( , ) ( ; , ) ( , )dVd b b b

τ τ ττ Γ Γ ′ = + z z z z z z z

( )

( ) ( )

( ), ,

2, ,

( , ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

C C

C C

C C

T TT TN n c n n c n n c t t c

T T TTn n c t n c n n c n t c

T T T Tn n c n n c n n c n

b d g c d

g c d g c d

g c d g d

ξ ξ

ξ ξ

ω ω α

ω ω

ω ω κ

Γ Γ

Γ Γ

Γ Γ

′ = − − Γ − − − Γ

− − Γ − − Γ

− Γ + − Γ

∫ ∫

∫ ∫

∫ ∫

z z z z e e V V z z e e V V

t z z e e V t z e e V V

z e e V z z e V n

( , ) ( , ) ( , )V N Tb b b′ ′ ′= +z z z z z z

• Material Derivative of Contact Variational Form

Page 17: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

CONTACT DSA cont.

• Tangential Stick Fictitious Load Form for DSA

( )

( )( )

*

1 1, ,

21 1 1 1 1

1 1 1 1 1, ,

1

( , ) ( ; , )

2 ( ) ( )

(2 ( ) ( )

( ) ( )

C

C

C

n nT T

n TT n n nt t t t c c

Tn n n n T n n n nt t t t c c

Tn n n n n n n T n n nt n t t n c c

n n nt n

b b

g c d

g d

g g c c d

g

ξ ξ

ξ ξ

ω

ω ν β

ω β

ω

− −

Γ

− − − − −

Γ

− − − − −

Γ

′ =

+ − + Γ

+ − − + − − Γ

+ + − + Γ

z z z V z

t z z e e V z

t z z e e V z V z

t t z z e e V z

t

( )( )

2 1 1, ,

21 1 1 1 1 1 1,

21 1 1 1 1 1, , ,

( ) ( )

2 ( )

2 ( )

C

C

C

Tn n T n t t nt n c c

T Tn n n n n n n n n n nt n t c n t c c

T Tn n n n n n n n n nt n n t c n n c c

nt

c c d

g g c c d

g g g c c d

ξ ξ

ξ

ξ ξ ξ

ω β

ω β

ω κ

− −∆ −

Γ

− − − − − − −

Γ

− − − − − −

Γ

− + Γ

+ − + − − Γ

+ − + Γ

+

t z z e e V z

t t z e e V z V z

t z e e V z

( ) 1( ) ( ) ( )C

n T n n n n T n Tt t n t tg g g c dν −

Γ

− + − Γ∫ z z e t z z e V n

Page 18: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

u5

u6u7

u8

u1u2 u3

u4

2789 N

5066 NSymmetric Design

MESHFREE METHOD WITH LARGE SHAPE CHANGING PROBLEM

Initial Design Optimum Design

Analysis Result after Remeshing at Optimum Design

FEM with remodeling : 45 iterations Meshfree w/o remodeling : 20 iterations

Page 19: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

u1u1

u2u2

u3

u4u3

u4

u5

u6

u7u8

u9

GASKET SHAPE DESIGN

– 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

2( )gap∑

dEngine Block

Oil Pan Block

Page 20: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

DESIGN OPTIMIZATION

Optimization Problem 1 Optimization History

Optimum Design

2

min

. . 300

1700

1.0

0.5 0.5

C

i

d

s t F kN

kPa

gap mm

u

σ≥

≤≤

− ≤ ≤∑ d

2( )gap∑

Pressure Distribution

Page 21: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

DESIGN OPTIMIZATION OF DEEPDRAWING

E = 206.9 GPaν = 0.29σy = 167 MPaH = 129 MPaµf = 0.144Isotropic Hardening

Die

Punch Blank Holder

Blank

25 mm

26 mm

u1

u2

u3

u4

u6

u5

Performance(Ψ) ∆Ψ Ψ`∆τ RATIO(%)u1ep31 .189554+1 -.147351-6 -.147312-6 100.03ep32 .160840+1 -.458731-6 -.458733-6 100.00ep33 .113010+1 -.268919-6 -.268949-6 99.99ep34 .812794+0 -.217743-6 -.217764-6 99.99ep35 .568991+0 -.144977-6 -.144996-6 99.99ep36 .383581+0 -.802032-7 -.802123-7 99.99G .510092+2 -.159478-4 -.159503-4 99.98u2ep31 .189554+1 -.727109-7 -.726800-7 100.04ep32 .160840+1 .764579-7 .764732-7 99.98ep33 .113010+1 .118855-6 .118849-6 100.01ep34 .812794+0 .829857-7 .829822-7 100.00ep35 .568991+0 .700201-7 .700157-7 100.01ep36 .383581+0 .345176-7 .345177-7 100.00G .510092+2 .494067-4 .494016-4 100.01

G gapII

N

==∑

2

1

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DESIGN OPTIMIZATION OF DEEPDRAWING (CONT.)

Cost Function History

8.0

9.0

10.0

11.0

12.0

0 1 2 3 4Iteration

Design Parameter History

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 1 2 3 4

Iteration

u1

u2

u3

u4

u5

u6

min

s. t. .

gap

ep

2

0 2

∑≤

Optimization Problem

DOT : Sequential Quadratic Programming

Page 23: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

DESIGN OPTIMIZATION OF DEEPDRAWING (CONT.)

Plastic Strain (Initial Design)

Plastic Strain (Optimum Design)

Page 24: STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION …web.mae.ufl.edu/nkim/Papers/conf5.pdf ·  · 2002-01-28STRUCTURAL SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION

SUMMARY

• Meshfree Method Is Effective in Shape Optimization

• Very Accurate DSA Results Made Optimization Problem Converged in a Small Number of Iterations

• Deepdrawing Optimization Is Successfully Carried out to Reduce the Springback Amount

• Developed Accurate and Efficient Shape DSA Method for Finite Deformation Elastoplasticity Using Multiplicative Decomposition of Deformation Gradient

• Die Shape Design Sensitivity Formulation Is Developedfor the Frictional Contact Problem