design optimization using meshfree methodweb.mae.ufl.edu/nkim/papers/conf9.pdf · 2002. 1. 28. ·...

The University of FloridaCollege of Engineering

DESIGN OPTIMIZATION USING MESHFREE METHOD

Nam Ho Kim and Kyung Kook Choi

Center for Computer-Aided DesignCollege of EngineeringThe University of Iowa

2000 U.S.-Korea Conference on Science and Technology,Entrepreneurship, and Leadership


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)

– Deepdrawing Design Problem (Springback)


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

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

• Stable Solution for Large Shape Changing Problem

• Accurate Result for Finite Deformation Problem


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


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 − > − <

− =


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∑ ∑


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∫ ∫


DOMAIN DISCRETIZATION

Meshfree Shape Function

Meshfree Discretization


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


NONLINEAR STRUCTURAL ANALYSIS• Nonlinear Variational Equation

(Updated Lagrangian Formulation)

• Linearization

Structural Energy Form

Contact Variational Form

Load Linear Form

Z),,(),()(

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

∈∀−−=

∆+∆

ΓΩΩ

+Γ

+Ω

zzzzzz

zzzzzzknkn

kknkkn

ba

ba

Z),(),(),( ∈∀=+ ΩΓΩ zzzzzz nn ba

( , ) :a dΩ Ω= Ω∫z z τ ετ ετ ετ ε

( , )bΓ z z

∫∫ ΓΩΩ Γ+Ω=S

dd STBT fzfzz)(

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

3 3 3a lg i j p iij i

i 1 j 1 i 1ˆc 2

= = =

∂= = ⊗ + τ∑∑ ∑∂

c m m cττττεεεε


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


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


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


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


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


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


TORQUE ARM OPTIMIZATION

u5

u6u7

u8

u1

u2u3

u4

2789 N

5066 NSymmetric Design

Initial Design & Design Parameters Stress Contour

Minimize massSubject to σmax ≤ 800 MPa

• Automatic Node Generation Capability Is Used• 239 RKPM Particles (478 DOF)• Thickness = 0.3 cm• Steel with E = 207 GPa and ν = 0.3• Used Design Sensitivity and Optimization (DSO) Tool for

Design Parameterization and Design Velocity Computation• Meshfree Analysis = 5.19 sec.; DSA per DV = 0.57 sec on

HP Exemplar S-Class


DESIGN SENSITIVITY RESULTSPerformance(ψ) ∆ψ ψ’ ∆ψ/ψ’×100u1

Area .374606E+03 .103614E-05 .103615E-05 100.00

S82 .305009E+01 -.628917E-07 -.628906E-07 100.00

S85 .295060E+01 -.177360E-08 -.177217E-08 100.08

S88 .295177E+01 -.888293E-07 -.888278E-07 100.00

S91 .276433E+01 -.112452E-06 -.112451E-06 100.00

S97 .236361E+01 -.777825E-07 -.777813E-07 100.00

S136 .210658E+01 -.159904E-06 -.159907E-06 100.00

S133 .218336E+01 .336654E-07 .336665E-07 100.00

S100 .216967E+01 -.676240E-07 -.676229E-07 100.00

u3

Area .374606E+03 .101184E-05 .101185E-05 100.00

S82 .305009E+01 -.780835E-09 -.781769E-09 99.88

S85 .295060E+01 .146740E-09 .146784E-09 99.97

S88 .295177E+01 -.587522E-08 -.587479E-08 100.01

S91 .276433E+01 -.193873E-07 -.193869E-07 100.00

S97 .236361E+01 -.393579E-07 -.393573E-07 100.00

S136 .210658E+01 -.388212E-09 -.388864E-09 99.83

S133 .218336E+01 .415960E-09 .415245E-09 100.17

S100 .216967E+01 -.597876E-07 -.597868E-07 100.00

u5

Area .374606E+03 .200000E-05 .200000E-05 100.00

S82 .305009E+01 -.257664E-07 -.257638E-07 100.01

S85 .295060E+01 -.775315E-07 -.775303E-07 100.00

S88 .295177E+01 .894471E-08 .894603E-08 99.99

S91 .276433E+01 .385595E-07 .385607E-07 100.00

S97 .236361E+01 .108496E-07 .108508E-07 99.99

S136 .210658E+01 .353638E-07 .353624E-07 100.00

S133 .218336E+01 -.565799E-07 -.565810E-07 100.00

S100 .216967E+01 .974523E-08 .974644E-08 99.99

von Mises Stress Sensitivity w.r.t. u3

Accuracy of DSA Compared with FDM


OPTIMIZATION HISTORY

MASS HISTORY

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20ITER

Design Parameters History

-5

-3

-1

1

3

5

0 5 10 15 20

ITER

u1u2u3u4u5u6u7

Cost Function Design Parameters


Weight Reduction: 0.878kg → 0.421 kg (48%)

FEA Took 45 Iterations with 8 Remodelings (AIAA Paper by J. Bennett & M. Botkin), whereas Meshfree w/o Any Remodeling Took 20 Iterations

OPTIMIZATION RESULTS

Confirmed Analysis Results After Meshfree Remodeling at the Optimum Design


DEEPDRAWING PROCESS OPTIMIZATION

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


DSA AND OPTIMIZATION

Optimization Problem

G gapII

N

==∑

2

1

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

2

1

2

3

4

5

6

minimize ( ( ) )

subject to 0.2

0.6

0.1 0.1

0.01 0.1

1.1 1.1

1.1 1.1

0.01 0.01

0.02 0.1

i i

p

i

G P

e

t

u

u

u

u

u

u

= −

≤≥

− ≤ ≤− ≤ ≤− ≤ ≤− ≤ ≤

− ≤ ≤− ≤ ≤

∑ x x


OPTIMIZATION HISTORY

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

No. of Design Iteration : 4No. of Structural Analysis : 9


OPTIMIZATION RESULT

Plastic Strain (Initial Design)Maximum = 0.126

Plastic Strain (Optimum Design)Maximum = 0.111


OPTIMIZATION RESULTS cont.

Smaller Thickness Variation Due to Smaller Plastic Strain


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


FUTURE PLANS

• Develop Configuration Design Sensitivity Formulationfor Meshfree Shell Structure

• Current Gauss Integration Method Will Be Replaced byStabilized Conforming Nodal Integration for DSA

• Develop DSA of Contact Formulation Using Meshfree Smooth Contact Surface Representation

• Integrate Automatic HP-Adaptivity Into Shape DesignOptimization Process

design optimization using meshfree methodweb.mae.ufl.edu/nkim/papers/conf9.pdf · 2002. 1. 28. ·...

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