1 Numerical Shape Optimisation in Blow Moulding Hans Groot.

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Numerical Shape Optimisation Numerical Shape Optimisation in Blow Mouldingin Blow Moulding

Hans GrootHans Groot

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OverviewOverview

1.1. Blow moldingBlow molding

2.2. Forward ProblemForward Problem

3.3. Inverse ProblemInverse Problem

4.4. Optimisation MethodOptimisation Method

5.5. Conclusions & Future WorkConclusions & Future Work

Inverse Problem Optimization Method ConclusionsBlow Molding Inverse Problem

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Blow Molding/FormingBlow Molding/Forming

glass bottles/jars

polymer containers

mould

pre-form

container

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Glass Bottle Forming Glass Bottle Forming MachineMachine

Inverse Problem Optimization Method ConclusionsBlow Molding Inverse Problem

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ProblemProblem

Forward problem

Inverse problem

pre-form container

Blow Molding Optimization Method ConclusionsForward Problem Inverse Problem

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Forward ProblemForward Problem

R1

R2

Ri

Rm

•Surfaces R1 and R2 given•Surface Rm fixed (mould wall)•Surface Ri unknown

Forward problem

Blow Molding ConclusionsForward Problem Optimization MethodInverse Problem

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Constitutive EquationsConstitutive Equations

1)Mechanics Stokes flow problem

2)ThermodynamicsConvection diffusion

problem

3)Evolution of surfacesConvection problem


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Level Set MethodLevel Set Method

glass

airair

θ > 0

θ < 0θ < 0

θ = 0

motivation:

• fixed finite element mesh• topological changes are

naturally dealt with• surfaces implicitly defined• level sets maintained as signed

distances


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Computer Simulation ModelComputer Simulation Model Finite element method One fixed mesh for

entire flow domain 2D axi-symmetric At equipment

boundaries: no-slip of material air is allowed to “flow

out”


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Glass BlowingGlass Blowing


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R1

Inverse ProblemInverse Problem

Inverse problem

R2

Ri

Rm

•Surfaces Ri and Rm given•Surface R1 fixed•Surface R2 unknown

Blow Molding Forward Problem ConclusionsInverse Problem Optimization Method

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Blowing corner•No surface tension:infinite time

•Surface tensionnot possible: equilibrium

Blowing round cavity•Possible if:

radius of mould cavity < radius of curvature

Mould RequirementsMould Requirements


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•Surface in equilibrium:Glass Surface in EquilibriumGlass Surface in Equilibrium

pp

g0p gz

•Contact angle φ between surface and mould

•Unknowns:•curve z = f(r)•contact radius rc

•contact depth zc


zc

rc

-HφLr

z

φ

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pp

g0p gz

3/22

''( )( )

1 '( )

z rr

z r

•2D curve z(r) :

•Boundary conditions:

c cd d

cd d

( ) , ( )( ) tan , ( ) cotz z

r r

z r H z L zr L


zc

rc

-HφLr

z

φ

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pp

g0p gz

0,p

g g

3/22

''

1 ( ')

zz

z

•Second order ODE:


zc

rc

-HφLr

z

φ

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pp

g0p gz

0,p

g g

3/22

''( )

1 ( ')

''

z zzz

z

•Multiplication by z’:


zc

rc

-HφLr

z

φ

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pp

g0p gz

0,p

g g

1/22 22 1 ( ') ( )z z E

•Integration:


zc

rc

-HφLr

z

φ

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Glass Surface in EquilibriumGlass Surface in Equilibrium•First order ODE:

•Boundary conditions:

c cd d

cd d

( ) , ( )( ) tan , ( ) cotz z

r r

z r H z L zr L

2

22

4' 1

( )z

z E

•Constants:

1c

2 2

22

2

21c 4

4c

2 cos ( )

2 sin ( )

1 dz

z EH

E H

z H

r L z


zc

rc

-HφLr

z

φ

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corner

Example

corner in (1,-10)

0

0.2

20

9

gp

g

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Time ScaleTime Scale•Equilibrium (no gravity):

•Time scale:L

tV

pL LV

•Typical values:5

2 1

10 Pa s1 Pa m

10 m

310 st

•Typical blow process takes ~1s

zc

rc

-HLr

z


210 s

10s3s1s

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OptimizationOptimization

Find pre-form for approximate container with minimal distance from container design

mould wallcontainer design

approximate container

Optimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

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OptimizationOptimizationmould wallcontainer designapproximate container

Minimize objective function2

2

2

i

dd d

RidOptimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

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Computation of Objective FunctionComputation of Objective Function Objective function:

Line integral:

Composite Gaussian quadrature:

• m+1 control points (•) → m intervals•

n weights wi per interval (ˣ)

2

i

dd

2'( ) ( ( ))m n

i nj i nj ij i

w s d s x x


1 2

0'( ) ( ( )) ds d s s x x

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Parameterization of Pre-FormParameterization of Pre-Form

P1

P5P4

P3

P2

P0

OR,φ1. Describe unknown surface

by parametric curve• e.g. spline, Bezier curve

2. Define parameters as spherical radii of control points:

3. Optimization problem: Find p as to minimize

1 2 5P P P( , ,..., )R R Rp

)(pOptimization MethodInverse ProblemBlow Molding Inverse Problem Conclusions

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iterative method to minimize objective function

J: Jacobian matrix

: Levenberg-Marquardt parameter

H: Hessian of penalty functions:

iwi /ci , wi : weight, ci >0: geometric

constraint

g: gradient of penalty functions

p: parameter increment

d: distance between containers

Modified Levenberg-Marquardt Method

T T

i i i i i i i i J J I H p J d g


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( ) ( ) p 1

Tolerance

Tolerance should not be smaller than

total error of optimisation method

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Numerical Errors


Types:

Model error: εL

Truncation error: εT

Rounding error: εR

Measurement error: εM

Interpolation error: εI

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Numerical Errors


Types:

Model error: εL

Model simplifications

Discretisation of forward problem


Rounding error: εR



L( ) ( ) r p r p ε

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Numerical Errors


Types:

Model error: εL



Numerical differentiation of residual

Numerical integration (objective function)

Rounding error: εR



T

( ) ( )( )

r p e r pJ p e ε

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Numerical Errors


Types:

Model error: εL




Rounding error: εR



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Numerical Errors


Types:

Model error: εL




Rounding error: εR



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Numerical Errors


Types:

Model error: εL




Rounding error: εR



Interpolation of known surfaces (through data) Interpolation of unknown surface (parametrisation)

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Numerical Errors


Types:

Model error: εL






Interpolation of unknown surface (parametrisation)

Total Error: ε = εL + εT + εM + εI

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Restrictions on Errors


Mesh size : h• Linear elements: εL =O(h2)

• h2 εM

Distance between control points of unknown surface: ξ• Cubic spline interpolation : εI =O(ξ4)

• ξ ~ C l/(m-1)o C : constanto l : curve lengtho m : number of control points

• ξ4 h2 m C l h-1/2+1

ξ

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Truncation Error


Forward Difference Approximation: Error:

Funtion evaluations: p = number of parameters

Central Difference Approximation: Error:

Funtion evaluations: 2p = number of parameters

Broyden Update: Error: ? No function evaluations

Tε O( )

2

Tε O( )

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Forward Difference Approximation


Error bounded by

Minimum:

Conclusion:

( ) ( )( ) ( )O

r p e r p

J p e

L 12

2εM

L

4ε

M

T Lε ε O( )h

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Broyden Update


Jacobian Lipschitz continuous:

Error bound:

Conclusion:

T

11 T

i i i i ii i

i i

r r A s sA A

s s

1( ) ( )i i i J p J p s

31 1 2

( ) ( )i i i i i A J p A J p s

1 1

( ) O( ) ( ) O( ) O( )

i ii i

i

A J pA J p

s

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Error Analysis


Types:

Model error: εL= O(h2)

Truncation error: εT = O(h)

Measurement error: εM = O(h2) (by choice of h)

Interpolation error: εI = O(h2)

Total Error: ε ~εT = O(h)

Tolerance:

h

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Assumptions:Negligible mass flow in azimuthal direction (uφ ≈0)Constant viscosity

Given R1(φ), determine R2(φ)

Volume conservation:

•R(φ) radius of interface

Approximation for InitialApproximation for Initial GuessGuess

streamlines

3 3 3 32 1 m i( ) ( ) ( ) ( )R R R R

φr


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InitialInitial Guess for BottleGuess for Bottle


Rφ

φR

streamlines

r

φr

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InitialInitial GuessGuess

approximate inverse problem

initial guess of pre-form model container

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forward problem

pre-form container

simulation

approximation (uφ≈0)

Comparison Approximation with Comparison Approximation with SimulationSimulation

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Optimization of Pre-Form (no sagging)

inverse problem initial guess

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inverse problem initial guess


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inverse problem optimal preform


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ErrorSigned Distance between Approximate and Model Container

top bottom

Absolute Tollerance:h≈ 0.059

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Summary Inverse problem:

• find preform corresponding to container Shape optimization method for pre-

form in blow molding• Pre-from surface described by parametric

curve• Approximation for initial guess• Error in approximation of Jacobian is

dominant Application to glass blowing

Blow Molding Forward Problem Optimization Method ConclusionsInverse Problem

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Future Work Initial guess with sagging Sensitivity analysis (w.r.t. perturbations in

thickness)

Comparison finite difference with derivative-free optimisation

Adaptive optimisation strategy• T-splines• Adaptive mesh

Volume constraint Application to polymersBlow Molding Forward Problem Optimization Method ConclusionsInverse Problem

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Blow Molding Forward Problem Optimization Method ConclusionsInverse Problem

Thank you for your attention

Questions?

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Incompressible medium:

•R(f) radius of interface G

Simple example → axial symmetry:

•If R1 is known, R2 is uniquely determined and vice versa

1 12 2

3 3 3 32 1 m i

0 0

( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R

Initial GuessInitial Guess

3 3 3 32 1 m iR R R R

R(f)

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Inverse ProblemInverse Problem

1 given (e.g. plunger)

m, i given

•determine 2 2

1

•Optimization:•Find pre-form for container with minimal difference in glass distribution with respect to desired container

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Inverse Problem

1

2i

m

i and m given

1 and 2 unknown

Inverse problem

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1 12 2

3 3 3 32 1 m i

0 0

( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R

Volume Conservation (incompressibility)

R1

R2Ri

Rm

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1 12 2

3 3 3 32 1 m i

0 0

( ) ( ) sin( )d ( ) ( ) sin( )d) )( (R R R R

Volume Conservation (incompressibility)

•Rm fixed

•Ri variable

with R1 and R2

•R1, R2??

Ri

Rm

R1

R2

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Blow Moulding

preform container

Forward problem

Inverse problem

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Hybrid Broyden Method

Optimisation ResultsIntroduction Simulation Model Conclusions

iii

ii

ii

iii

iiii

iii

ii

ii

ii

ii

iiii

rrr

JJ

JJ

pJr

rpJr

pJr

rr

pp

pp

pp

ppJr

1

1

111

with

otherwise ,

:method bad sBroyden'

if ,

:method good sBroyden'

[Martinez, Ochi]

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Error Analysis


Mesh size : h• Linear elements: εL =O(h2)

• h2 εM

Distance between control points of unknown surface: ξ• Cubic spline interpolation : εI =O(ξ4)

• ξ ~ C l/(m-1)o C : constanto l : curve lengtho m : number of control points

• ξ4 h2 m C l h-1/2+1

1 Numerical Shape Optimisation in Blow Moulding Hans Groot.

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