1

Development of a Numerical Optimisation Method for

Blowing Glass Parison Shapes

Hans Groot

2

Overview

Introduction

Glass Blow Simulation Model

Optimisation Method

Results

Conclusions

3

Glass Melting

Glass Conditioning

Automatic Inspection

Glass Manufacturing

Glass Forming

Surface Treatment

Introduction ResultsSimulation Model Optimisation Conclusions

4

Glass Forming

• press

• press-blow

• blow-blow

Introduction ResultsSimulation Model Optimisation Conclusions

5

Blow-Blow Process

glass

mould

ring

preform

mould

ring

Introduction ResultsSimulation Model Optimisation Conclusions

6

Blow Model

1)Flow of glass and air Stokes flow problem

• Viscous forces dominate• Temperature dependent glass viscosity

2)Energy exchange in glass and air Convection diffusion

problem• No viscous dissipation

3)Evolution of glass-air interfaces Convection problem for level sets

Simulation Model ResultsIntroduction Optimisation Conclusions

7

Level 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

Simulation Model ResultsIntroduction Optimisation Conclusions

8

Computer Simulation Model Finite element

discretisation One fixed mesh for

entire flow domain 2D axi-symmetric At equipment

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

out”

Simulation Model ResultsIntroduction Optimisation Conclusions

9

Bottle Blowing Simulation

TemperatureGlass-air interfacesSimulation Model ResultsIntroduction Optimisation Conclusions

10

Glass Distribution for Jar

Preform 2: breaks!Preform 1: thickenings!

11

Given container find preform

Optimisation:• Find preform that minimises difference in glass

distribution between model container and container obtained by blow process

Inverse Problem

Optimisation ResultsIntroduction Simulation Model Conclusions

12

Least Squares Minimisation Problem Residual:

•

•

Minimise objective function:

*ˆ θθr

Optimisation ResultsIntroduction Simulation Model Conclusions

d

true interface

approximate interface

θ

*

* :

iθ withcontainer model of values distance signedθ

*:ˆ

iθ withcontainer eapproximat of values distance signedθ

2

21 r

13

Optimisation Strategy

1. Describe interfaces by parametric curves• e.g. splines, Bezier curves

2. Define parameters:

3. Compute signed distance

4. Minimise

),,...,,,,,(4411211 QQQQPPP zrzrzzrp

Optimisation ResultsIntroduction Simulation Model Conclusions

P2P1

Q3

Q2

Q1

Q0

Q4Q5

P0

zr

)(ˆ pθ)(p

14

iterative method to minimise 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

r: residual

Modified Levenberg-Marquardt Method

Optimisation ResultsIntroduction Simulation Model Conclusions

iiTiiiii

Ti grJpHIJJ

15

Computation of Jacobian

1. Finite difference approximation:

requires p function evaluations,

p: number of parameters

2. Secant method: updates Jacobian in incremental direction

no function evaluations

may fail to find descent direction

finite difference approximation

Optimisation ResultsIntroduction Simulation Model Conclusions

16

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]

17

Example

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

18

Optimal preform

Preform Optimisation for Jar

Model jar Initial guess

ResultsLevel Set MethodIntroduction Simulation Model Conclusions

19

Preform Optimisation for Jar

Model jar

ResultsLevel Set MethodIntroduction Simulation Model Conclusions

Approximate jar

Radius: 1.0

Mean distance: 0.019Max. distance: 0.104

20

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

21

Short Term Plans

ConclusionsOptimisationIntroduction Simulation Model Results

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

22

Thank you for your attention

23

Blowing Model: 2nd blow

(a) t=0.0

(b) t=0.5

(c) t=0.88

(d) t=1.17

(a)

(b)

(c)

(d)

24

Blowing Model: 1st Blow

Axi-symmetric blowing of parison

1165 oC glass

500 oC mould

m

s

i

2

1

25

ring

preform

mould

26

Re-initialisation by FMM Algorithm

CB

A

Δx

222 x )()( BCAC

Level Set Method ResultsIntroduction Simulation Model Conclusions

27

Preform Optimisation for Jar

Level Set Method ResultsIntroduction Simulation Model Conclusions

Model jar Initial guessOptimal preform