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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
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)
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.
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
SHAPE DESIGN SENSITIVITY ANALYSIS
0Ω
0Ωτ
nΩτ
nΩ
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
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 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
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
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
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
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
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
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
DESIGN OPTIMIZATION OF DEEPDRAWING (CONT.)
Plastic Strain (Initial Design)
Plastic Strain (Optimum Design)
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