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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. Inverse ProblemInverse Problem

3.3. Optimization MethodOptimization Method

4.4. Application to Glass BlowingApplication to Glass Blowing

5.5. Conclusions & future workConclusions & future work

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Inverse Problem Glass Blowing ConclusionsBlow Molding Optimization Method

Blow MoldingBlow Molding

glass bottles/jars

plastic/rubber containers

mould

pre-form

container

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Example: JarExample: Jar

Inverse Problem Glass Blowing ConclusionsBlow Molding Optimization Method

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ProblemProblem

Forward problem

Inverse problem

pre-form container

Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method

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Forward ProblemForward Problem

1

2

i

m

•Surfaces 1 and 2 given•Surface m fixed (mould wall)•Surface i unknown

Forward problem

Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method

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1

Inverse ProblemInverse Problem

•Surfaces i and m given

•Either 1 or 2 unknown

Inverse problem

2

i

m

Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method

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Construction of Pre-Form by PressingConstruction of Pre-Form by Pressing

1

2

Blow Molding Glass Blowing ConclusionsInverse Problem Optimization Method

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OptimizationOptimization

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

mould wallmodel container

approximate container

Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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OptimizationOptimizationmould wallmodel containerapproximate container

Minimize objective function2

2

2

i

dd d

idOptimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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Computation of Objective FunctionComputation of Objective Function Objective Function:

Composite Gaussian quadrature:

• m+1 control points (•) → m intervals•n weights wi per interval (ˣ)

2

i

dd

2 2

i

d ( )m n

i nj ij i

d w d

x

Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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Parameterization of Pre-FormParameterization of Pre-Form

P1

P5P4

P3

P2

P0

OR,φ1. Describe surface by

parametric curve• e.g. spline, Bezier curve

2. Define parameters as radii of control points:

3. Optimization problem: Find p as to minimize

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

)(p

Optimisation Method Glass BlowingBlow 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

Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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Function Evaluations per Iteration

Distance function d:o one function evaluation

Jacobian matrix:

1. Finite difference approximation:

o p function evaluations (p: number of parameters)

2. Broyden’s method:o no function evaluations, but less accurate

function evaluation = solve forward problem

Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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Neglect mass flow in azimuthal direction (uf≈0)

Given R1(f), determine R2(f)

Volume conservation:

•R(f) radius of interface

Approximation for InitialApproximation for Initial GuessGuess

streamlines

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

f r

Optimisation Method Glass BlowingBlow Molding Inverse Problem Conclusions

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InitialInitial GuessGuess

approximate inverse problem

initial guess of pre-form

model container

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Glass BlowingGlass Blowing

Blow Molding Inverse Problem ConclusionsGlass BlowingOptimization Method

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Forward ProblemForward Problem

1)Flow of glass and air Stokes flow problem

2)Energy exchange in glass and air Convection diffusion

problem

3)Evolution of glass-air interfaces Convection problem

Blow Molding Inverse Problem ConclusionsGlass BlowingOptimization Method

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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• interfaces implicitly defined• level sets maintained as signed

distances

Blow Molding Inverse Problem ConclusionsGlass BlowingOptimization Method

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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 glass air is allowed to “flow

out”

Blow Molding Inverse Problem ConclusionsGlass BlowingOptimization Method

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Comparison Approximation with Comparison Approximation with Simulation ModelSimulation Model

forward problem

pre-form container

simulation

approximation (uf≈0)

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Optimization of Pre-Form

inverse problem

initial guess

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Optimization of Pre-Form

initial guess

inverse problem

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Optimization of Pre-Form

optimal pre-form

inverse problem

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Signed Distance between Approximate and Model Container

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Summary Shape optimization method for pre-

form in blow molding• describe either pre-form surface by

parametric curve• minimize distance from approximate

container to model container• find optimal radii of control points• use approximation for initial guess

Application to glass blowing average distance < 1% of radius

moldBlow Molding Inverse Problem Glass Blowing ConclusionsOptimization Method

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Short Term Plans

Extend simulation model• improve switch free-stress to no-slip

boundary conditions

• one level set problem vs. two level set problems

Well-posedness of inverse problem

Sensitivity analysis of inverse problem

Blow Molding Inverse Problem Glass Blowing ConclusionsOptimization Method

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Parison Optimization for Ellipse

model container optimal containerinitial guess

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Blow MoldingBlow Molding

mould

ring

parison

container

e.g. glass bottles/jars

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ApproximationApproximation

Initial guess

pre-form model container

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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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Example (p = 13)

Optimisation ResultsIntroduction Simulation Model Conclusions

Method # function evaluations

# iterations

Hybrid Broyden 32 8 1.75

Finite Differences 98 9 1.36

Conclusions:

• similar number of iterations

• similar objective function value

• Finite Differences takes approx. 3 times longer

than Hybrid Broyden

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Optimal preform

Preform Optimisation for Jar

Model jar Initial guess

ResultsLevel Set MethodIntroduction Simulation Model Conclusions

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Preform Optimisation for Jar

Model jar

ResultsLevel Set MethodIntroduction Simulation Model Conclusions

Approximate jar

Radius: 1.0

Mean distance: 0.019Max. distance: 0.104

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Conclusions

ConclusionsOptimisationIntroduction Simulation Model Results

Glass Blow Simulation Model• finite element method• level set techniques for interface tracking• 2D axi-symmetric problems

Optimisation method for preform in glass blowing• preform described by parametric curves• control points optimised by nonlinear least

squares Application to blowing of jar

mean distance < 2% of radius jar

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Thank you for your attention

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ComparisonComparisonInverse problem Forward problem

two unknown intefaces one unknown interface

Inverse problem

Forward problem1

2

i

m

Inverse problem under-determined or forward problem over-determined?

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Inverse Problem

Optimisation ResultsIntroduction Simulation Model Conclusions

preform container

Unknown surfaces

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Forward Problem

Optimisation ResultsIntroduction Simulation Model Conclusions

preform container

Rm knownRi unknown

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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

Volume ConservationVolume Conservation

3 3 3 32 1 m iR R R R

R(f)

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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

Volume ConservationVolume Conservation

3 3 3 32 1 m iR R R R

R(f)