How to maximise Value and the S133 Issue Carol Matthews Partner, Wright Hassall.
DESIGN OPTIMIZATION USING MESHFREE METHODweb.mae.ufl.edu/nkim/Papers/conf9.pdf · 2002. 1. 28. ·...
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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
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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)
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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
• Direct Transformation/Mixed Transformation/BoundarySingular Kernel Methods for Essential B. C.
• Stable Solution for Large Shape Changing Problem
• Accurate Result for Finite Deformation Problem
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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OPTIMIZATION RESULT
Plastic Strain (Initial Design)Maximum = 0.126
Plastic Strain (Optimum Design)Maximum = 0.111
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OPTIMIZATION RESULTS cont.
Smaller Thickness Variation Due to Smaller Plastic Strain
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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
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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