Numerical shape optimisation in blow moulding

Numerical shape optimisation in blow moulding Groot, J.A.W.M.

DOI: 10.6100/IR709254 Published: 01/01/2011

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Citation for published version (APA): Groot, J. A. W. M. (2011). Numerical shape optimisation in blow moulding Eindhoven: Technische Universiteit Eindhoven DOI: 10.6100/IR709254

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Numerical Shape Optimisation in Blow Moulding

c 2011 by J.A.W.M. Groot, Eindhoven, The Netherlands. Copyright All rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author. c Pascal Artur/VOA, with courtesy of Right image on front cover and Figure 1.7 used copyright the Phototheque of Saint-Gobain. Figure 1.1 used courtesy of Wilson Museum in Castine, ME, USA. c Trustees of the British Museum, with premission from British muFigure 1.2 used copyright seum in London, UK. Printed by Printservice Technische Universiteit Eindhoven Cover design by Paul Verspaget Grafische Vormgeving-Communicatie

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Groot, Johannes A.W.M. Numerical Shape Optimisation in Blow Moulding by Johannes A.W.M. Groot. Eindhoven: Technische Universiteit Eindhoven, 2011. Proefschrift. A catalogue record is available from the Eindhoven University of Technology Library ISBN 978-90-386-2460-0 NUR 919

Numerical Shape Optimisation in Blow Moulding

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op dinsdag 19 april 2011 om 16.00 uur

door

Johannes Alphonsus Wilhelmus Marius Groot geboren te Hengelo

Dit proefschrift is goedgekeurd door de promotor: prof.dr.ir. R.M.M. Mattheij Copromotor: dr. C.G. Giannopapa

Nomenclature x

position , page 24

N

set of integers, page 60

R

set of real numbers, page 52

T

time interval of forming process, page 24

E˙

strain rate tensor , page 29

I

identity tensor, page 29

T

Cauchy stress tensor , page 24 [Pa]

[m]

Dimensionless Numbers Br

Brinkman number , page 39 Br =

De

Fr Nu

Pe

Re

!

µV ¯ 2 λ¯ T0 −Tm

Deborah number , page 46 De = τ µp 2 Froude number , page 39 Fr = VgL Nusselt number , page 40

αD ¯ λ¯

ρ¯cp VD λ¯

Nu =

Péclet number , page 39 Pe =

[s−1 ]

Greek

Reynolds number , page 39 Re =

ρVD µ¯

α

heat transfer coefficient , page 37 [W m−2 K−1 ]

α¯

typical heat transfer coefficient , page 40 [W m−2 K−1 ]

λ¯

typical conductivity , page 38 [W m−1 K−1 ]

µ¯

typical viscosity , page 38

β

friction coefficient , page 36 [N m−3 s]

δ

step size of difference formula, page 122

Bold Latin e

unit vector, page 38

n

unit normal, page 34

A

secant approximation of Jacobian matrix, page 120

g

gravitational acceleration , page 24 [m s−2 ]

Hf

Hessian of f , page 117

ε˙

strain rate , page 31

I

identity matrix, page 121

total error of optimisation method, page 125

J

Jacobian matrix of least squares residual, page 119

C

error in computation of the least squares residual, page 124

F

error in numerical solution of forward problem, page 123

I

interpolation error, page 125

M

measurement error, page 125

[W m−2 ]

q

heat flux , page 24

r

least squares resiudal, page 117

t

unit tangent, page 36

u

flow velocity , page 24

[m s−1 ]

[Pa s]

[s−1 ]

vi

Nomenclature

R

rounding error, page 125

σ

equivalent stress , page 31

[Pa]

T

truncation error, page 124

σy

yield strength , page 46

[Pa]

Γ

boundary/surface, page 22

θ

level set function, page 57

γ

surface tension , page 34

θ1

Γ1

level set function corresponding to inner melt-air interface, page 58

inner melt surface, page 22

Γ2

θ2

outer melt surface, page 22

level set function corresponding to outer melt-air interface, page 58

Γf

melt-air boundary, page 22

ϕ

polar coordinate/angle, page 97

Γi

inner container surface, page 90

ϕc

expansion angle, page 104

γLip

Lipschitz constant, page 124

e Φ

Γm

weighted page 130

mould boundary/surface, page 22

Γo

ζ

outer boundary, page 22

sum of weighted penalty functions, page 130

Γs

symmetry axis, page 22

Latin

κ

curvature , page 34

λ

effective conductivity , page 26 [W m−1 K−1 ]

λLM

[N m−1 ]

[m−1 ]

µ

dynamic viscosity , page 29 [Pa s]

µ0

zero-shear-rate viscosity , page 29 [Pa s]

function,

c¯ p

typical specific heat , page 38 [J kg−1 K−1 ]

Cn

space of n times continuously differentiable functions, page 52

Cn0

space of functions in C n that vanish on the boundary of the domain, page 59

H1

Sobolev space, page 52

Lp

set of Lebesgue p-integrable functions, page 52

Levenberg-Marquardt parameter, page 121

objective

Ω

fluid domain, page 22

ωk

kth weight of Gaussian quadrature rule, page 117

c

geometric page 130

Φ

heat source density , page 24 [W m−3 ]

cp

specific heat , page 27 [J kg−1 K−1 ]

D

typical diameter , page 38

Φ

objective function, page 116

d

signed distance function, page 58

e

specific internal energy , page 24 [J kg−1 ]

ρ

density , page 24

Σ

forming machine domain, page 22

constraint

function,

[m]

−3

[kg m ]

Nomenclature

vii

g

gravitational acceleration in axial direction , page 38 [m s−2 ]

V

typical flow velocity , page 38 [m s−1 ]

h

typical mesh size, page 126

Vr

stretch rod speed , page 36 [m s−1 ]

k

tensile modulus , page 32

z

axial coordinate, page 77

L

typical length scale , page 38

m

reciprocal of power-law index, page 32

nG

number of points for Gaussian quadrature rule, page 116

nint

number of subintervals for composite quadrature rule, page 117

[Pa s] [m]

np

number of parameters, page 115

p

pressure , page 29

[Pa]

pin

inlet pressure , page 36

[Pa]

R

spherically page 97

r

cylindrically page 77

R1

spherical radius of inner melt surface, page 101

R2

Notations and Conventions v

vector, page 24

S

set/interval, page 24

S

(vector) space, page 52

T

tensor, page 24

s, S

scalar, page 24 Operators

·

inner product , page 24 v A = vT A T

·

:

double dot product , page 24 A B = tr AB

dev

deviatoric part , page 29 dev T = T − 13 tr T

~.

double square brackets for jump conditions , page 35 ~A = A2 − A1

spherical radius of outer melt surface, page 101

D Dt

material derivative , page 24 D ∂ = + u ∇ ⊗ Dt ∂t

Ri

spherical radius of inner container surface, page 102

∇

differential operator , page 24 ∇ = (∂ x1 , . . . , ∂ xn )T

Rm

spherical radius of mould surface, page 102

⊗

matrix/tensor product , page 24 A ⊗ B = ABT

t

time , page 24

∂

boundary of domain , page 22

Tg

glass transition page 25

radial

radial

coordinate,

coordinate,

[s] temperature

, [K]

:

·

◦

!

∂S = S \ S tr

trace , page 29

P tr A = j A j j

viii

∗

T

Nomenclature Superscripts

l

forming material/melt, page 22

dimensionless quantity (omitted from Chapter 3), page 38

Lip

Lipschitz, page 124

LM

Levenberg-Marquardt, page 121

transpose, page 29

M

measurement, page 125

Subscripts

m

mould, page 22

o

outer/enclosing, page 22

opt

optimum, page 117

p

parameters, page 115

q

equipment, page 22

*

end of process, page 90

a

air, page 22

b

baffle, page 22

c

contact point, page 94

F

forward problem, page 123

R

rounding, page 125

f

free surface, page 22

r

stretch rod, page 22

G

Gaussian quadrature rule, page 116

res

residual, page 119

I

interpolation, page 125

s

symmetry (axis), page 22

int

interval, page 117

T

truncation, page 124

Acknowledgements This thesis presents the main results of the work done for my PhD project on Numerical Shape Optimisation in Blow Moulding at Eindhoven University of Technology. It is the result of more than four years of experience in analysis, numerical simulation and shape optimisation of blow moulding. As for today I am sincerely grateful for the opportunity to work on this PhD project. Therefore, I would like to express my cordial gratitude to everyone who has by any means contributed to it. Although I have worked on this project independently, I could not have come this far without the support of many. First and foremost my gratitude goes to my direct supervisors prof. dr. Bob Mattheij and dr. Christina Giannopapa. I am immensely thankful to Bob for his invaluable advice and the freedom that he gave me. His trust and wisdom were indispensable assets for the success of the project. I would like to thank Christina for her intensive supervision. Her passion and enthusiasm greatly stimulated in my work and her result-orientated guidance has considerably contributed to my work. Cordial thanks are addressed to the other members of my core committee, prof. dr. Nataˇsa Krejić, prof. dr. Han Slot and prof. dr. Kees Vuik, for being part of my committee, reading my thesis and providing valuable comments. My gratitude also goes to the additional members of the extended committee, dr. Sjoerd Rienstra and dr. Jacques Dam. I would like to extend words of thanks to dr. Bas van der Linden, dr. Ronald Rook and Sam Moussa for their help with difficulties regarding operating systems and software. I am much indebted to Bas for his active and unconditional help with the implementation of the blow moulding model; without his support the numerical simulations would probably not have been possible.

x

Nomenclature In addition, I would like to express my appreciation to the applied analysis group, who helped me a lot with several analysis questions during the Wednesday morning meetings. I would like to thank Yves van Gennip and Kundan Kumar for organising these fruitful meetings. Special thanks go to dr. George Prokert for the many useful advices he gave me on various matters. I am also sincerely grateful to dr. Martijn Anthonissen, dr. Andreas Löpker, dr. Frans Martens, Dipl.-Math. Christiane Peters, dr. Arris Tijsseling and dr. Jan ten Thije Boonkkamp for their great help and support with my teaching activities. Furthermore, I would like to express my thanks to Enna van Dijk and Marèse Wolfsvan de Hurk for their help with administrative matters and their active participation in the organisation of many social events. I would like to thank Wilson Museum, the British Museum and Saint Gobain for their permission for use of materials in this thesis. It were my colleagues at CASA and other departments, who made the four years of my PhD programm so enjoyable and working at CASA so special. The working environment at CASA was pleasantly informal and it was a pleasure to meet so many colleagues who were amazingly enthusiastic about research as well as socialising. During this period many social events were organised within CASA, varying from daily lunches to weekly game evenings, regular sport events or nights out in town to the halfyearly CASA PhD days, the yearly CASA outing and CASA cooking event, as well as occasional travels within and outside the Netherlands. I am happy that I could be part of these informal get-together-events, or even help organising some of these, and it was delightful to see that many of the social events organised from within CASA were increasing in frequency and popularity, so much that they were also attended by colleagues from other departments. The unforgettable times I have experienced at CASA could not have been possible without my dear colleagues and with many of them I have created a special bond. Thank you, dear Nico van der Aa, Steffen Arnrich, Laura Astola, Evgeniya Balmashnova, Mayla Bruso, Nicodemus Banagaaya, Gaetan Bisson, Dion Boesten, David Bourne, John Businge, Mirela Darau, Willem Dijkstra, Remco Duits, Ali Etaati, Yabin Fan, Tasnim Fatima, Malik Furqan, Yves van Gennip, Shruti Gumaste, Andriy Hlod, Davit Harutyunyan, Michiel Hochstenbach, Qingzhi Hou, Zoran Ilievski, Roxana Ionutiu, Bart Janssen, Godwin Kakuba, Badr Kaoui, Sinatra Kho, Evelyne Knapp, Jan Willem

Nomenclature

xi

Knopper, Kundan Kumar, Agnieszka Lutowska, Kamyar Malakpoor, Temesgen Markos, Oleg Matveichuk, Jos Maubach, Martien Oppeneer, Jos en Peter in ’t panhuis, Miguel Patr´ıcio, Prasad Perlekar, Maxim Pisarenco, Rostyslav Polyuga, Corien Prins, Michiel Renger, Eloy Romero, Patricio Rosen Esquivel, Maria Rudnaya, Valeriu Savcenco, Lucia Scardia, Olga Shchetnikava, Berkan Sesen, Antonino Simone, Sudhir Srivastava, Jürgen Tas, Maria Ugryumova, Marco Veneroni, Arie Verhoeven, Erwin Vondenhoff, Nata Voynarovskaya, Niels Willems, Yeneneh Yimer Yalew, Shona Yu and everyone else who made my life at CASA so much more enjoyable. I would also like to thank the organisers of the PhDays ’07-’09, as well as all the participants in the PhDays, who each year managed to turn this social event into a tremendous success. I am thankful to my training partners and teachers at the sport centre of Eindhoven University of Technology. During my PhD I have spent much of my spare time in the sport centre, which at all times gave me fresh courage, energy and motivation to continue my PhD project. It was a great honour to perform various martial arts under Senzei Huub Meijer’s supervision. Finally, I would like to thank my family and friends for their continuous love and support, and their patience and understanding when my PhD project was suppressing my social life. I am particularly grateful to my parents; without their support this thesis might not have been possible. Lastly, but not least, I thank the beautiful Neda Sepasian for bringing me joy and spending wonderful times together during the last year of my PhD program.

Summary Blow moulding is a popular manufacturing process for the production of plastic and glass containers, e.g. bottles, jars, jerrycans. In a blow moulding process a so-called preform of molten material is brought into a mould and subsequently inflated with air as to take the mould shape. Blow moulding processes typically vary in the way the preform is produced and brought into the mould. The stretch blow moulding process is a variation of the blow moulding process in which the preform is simultaneously inflated with air and stretched with a stretch rod. A two-dimensional axial-symmetrical blow moulding simulation model is developed. The numerical simulation model is based on Finite Element Methods and uses Level Set Methods to track the moving interfaces between the melt and air. Level Set Methods mark the location of the interfaces implicitly by a so-called level set function and therefore do not require re-meshing of the finite element mesh. The efficiency of the simulation model is illustrated by applying it to the stretch blow moulding of a plastic water bottle and the blow moulding of a glass beer bottle. The model is validated by means of volume conservation and comparison with data provided by industry. Two mathematical problems are considered in blow moulding. The forward problem is to find the final container that is blow moulded from a given preform under certain operating conditions. In practice often a container with a certain wall thickness distribution is desired. Then the corresponding initial operating conditions, such as the shape of the preform and the initial temperature distribution, are sought in order to produce a container with exactly this thickness distribution. In this case the inverse problem is considered, to find the shape of the preform, given a designed container, such that the container can be blow moulded from the preform. The solvability and sensitivity of the inverse problem are analysed. It is shown that under some circumstances the melt-air interfaces can reach a force equilibrium state

xiv

Summary

during blow moulding. Consequently, constraints on the mould surface and process time are necessary so that the inverse problem is solvable and not excessively sensitive to perturbations in the shape. The sensitivity of the inverse problem with respect to perturbations in the shape can be estimated by means of an approximation of the melt flow. Numerical shape optimisation is used to find a solution of the inverse problem. The optimisation method describes the unknown preform surface by a parametric curve, e.g. spline, Beziér curve, and computes the optimal positions of the control points of the curve as to minimise the objective function. The objective function represents the distance between the inner surface of the computed container, which is the solution of the forward problem for the approximate preform, and the inner surface of the designed container. Gradient-based optimisation algorithms are discussed to find the optimal positions of the control points. In gradient-based optimisation information about the gradient of the objective function with respect to changes in the parameters, i.e. the positions of the control points, is used to find the optimum. However, computing the gradient is extremely computationally expensive and can form the computational overhead. Therefore, finite difference approximations of the Jacobian are combined with secant updates. An error analysis is performed to choose an optimal error tolerance for the optimisation algorithm. The optimisation methods are applied to glass blow moulding and results are compared with each other. An initial guess for the iterative optimisation algorithms is constructed by an analytical approximation of the optimum. The approximation is derived by omitting the mass flow in polar direction in spherical coordinates, so that the inverse problem can be solved analytically.

Samenvatting Blaasvormen is een veelgebruikt fabricageproces voor de productie van plastic en glazen houders, bijv. flesjes, potjes of kannen. In een blaasvormproces wordt een zogenoemde voorvorm van vloeibaar materiaal in een mal gebracht en vervolgens opgeblazen met samengeperste lucht, zodat het de malvorm aanneemt. Blaasvormprocessen variëren typerend in de manier waarop de voorvorm wordt geproduceerd en in de mal wordt gebracht. Het rek-blaasvormproces is een variant op het blaasvormproces waarin de voorvorm gelijktijdig wordt opgeblazen met lucht en gerekt met een rekstaaf. Een twee-dimensionaal axiaal-symmetrisch blaasvorm simulatiemodel is ontworpen. Het numerieke simulatiemodel is gebaseerd op Eindige Elementen Methoden en gebruikt Level Set Methoden om de bewegende randen tussen de smelt en de lucht te traceren. Level Set Methoden markeren de locatie van de randen impliciet met een zogenoemde level-set functie en werken daarom op een vast eindige elementen rooster. De doeltreffendheid van het simulatiemodel wordt ge¨ıllustreerd door het toe te passen voor het rek-blaasvormen van een plastic waterfles en het blaasvormen van een glazen bierfles. Het model wordt gevalideerd door middel van volumebehoud en vergelijking met data vanuit de industrie. Twee wiskundige problemen in blaasvormen worden beschouwd. Het voorwaartse probleem is het bepalen van de uiteindelijke houder door een gegeven voorvorm onder zekere bedrijfscondities te blaasvormen. In de praktijk wordt vaak een houder met zekere wanddikte gevraagd en worden de bijbehorende initiële bedrijfscondities gezocht, zoals de voorvorm en de initiële temperatuurverdeling, om een houder met precies deze wanddikte te produceren. In dit geval wordt het inverse probleem beschouwd: het bepalen van de voorvorm, gegeven de ontworpen houder, zodanig dat de houder kan worden geblaasvormd uit de voorvorm.

xvi

Samenvatting De oplosbaarheid en gevoeligheid van het inverse probleem worden geanalyseerd.

Hierbij wordt aangetoond dat onder zekere omstandigheden, de smelt-lucht randen een krachtevenwichtstoestand kunnen bereiken gedurende het blaasvormen. Dientengevolge zijn beperkingen op het maloppervlak en de procesduur noodzakelijk, zodanig dat het inverse probleem oplosbaar is en niet overmatig gevoelig voor verstoringen in de vorm. De gevoeligheid van het inverse problem met betrekking tot verstoringen in de vorm kan worden geschat door middel van een benadering van de smeltstroming. Numerieke vormoptimalisatie wordt toegepast om een oplossing van het inverse probleem te vinden. De optimialisatiemethode beschrijft het onbekende voorvormoppervlak door middel van een parametrische kromme, bijv. een spline of een Beziér kromme, en berekent de optimale posities van de controlepunten van de kromme, zodanig dat de objectieve functie minimaal is. De objectieve functie representeert de afstand tussen de binnenste oppervlakken van de berekende houder en de ontworpen houder. Gradient-gebaseerde optimalisatiealgorithmen worden besproken om de optimale posities van de controlepunten te vinden. In gradient-gebaseerde optimalisatie wordt informatie over de gradient van de objectieve functie met betrekking tot veranderingen in de parameters, d.w.z. de posities van de controlepunten, gebruikt om het optimum te vinden. Het puntsgewijs berekenen van de gradient kost echter uitermate veel berekeningen en kan het dominante deel van de rekenkracht vormen. Daarom worden eindige differentie benaderingen van de Jacobiaan matrix gecombineerd met secant updates. Een foutanalyse wordt uitgevoerd om een optimale fouttolerantie voor het optimalisatie algoritme vast te stellen. De optimalisatiemethoden worden toegepast voor glasblaasvormen. Een begingok voor de iteratieve optimalisatiealgoritmen wordt geconstrueerd door een analytische benadering van het optimum. De benadering wordt afgeleid door de massastroming in polaire richting in bolcoördinaten te verwaarlozen, zodat het inverse probleem analytisch kan worden opgelost.

Contents 1

Introduction

1

1.1

Blow Moulding Manufacturing Processes . . . . . . . . . . . . . . . .

1

1.1.1

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Hollow Container Manufacturing . . . . . . . . . . . . . . . .

5

Process Simulation and Optimisation . . . . . . . . . . . . . . . . . . .

11

1.2.1

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.2.2

Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.4

Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.2

2

Mathematical Modelling of Blow Moulding

21

2.1

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2

Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3

Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.1

Compressibility and Thermal Expansion . . . . . . . . . . . . .

25

2.3.2

Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.3.3

Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.3.4

Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.3.5

Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.3.6

Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.4

Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.5

Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . .

35

2.6

Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.7

Model Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

xviii

3

4

5

6

7

8

Contents 2.7.1

Glass blow moulding . . . . . . . . . . . . . . . . . . . . . . .

41

2.7.2

PET Stretch Blow Moulding . . . . . . . . . . . . . . . . . . .

44

2.7.3

General Blow Moulding . . . . . . . . . . . . . . . . . . . . .

47

Numerical Methods

49

3.1

Discretisation Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.2

Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.3

Interface Capturing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.4

Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Blow Moulding Results

69

4.1

Level Set Methods and Fast Marching Methods . . . . . . . . . . . . .

69

4.2

Glass Blow-Blow Moulding . . . . . . . . . . . . . . . . . . . . . . .

76

4.3

PET Stretch Blow Moulding . . . . . . . . . . . . . . . . . . . . . . .

83

Mathematical Analysis of the Inverse Problem

89

5.1

Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . .

89

5.2

Restrictions on the Mould Surface . . . . . . . . . . . . . . . . . . . .

92

5.3

Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.4

Approximate Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Shape Optimisation Strategy

113

6.1

Parametrisation of the Inverse Optimisation Problem . . . . . . . . . . 113

6.2

Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4

Error Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.5

Initial Guess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.6

Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Shape Optimisation Results

131

7.1

Initial Guess for Blow Moulding an Ellipsoidal Glass Container . . . . 131

7.2

Optimisation of the Preform Shape for Blow Moulding a Glass Bottle . 135

Conclusions and Recommendations

143

Chapter 1

Introduction

1.1

Blow Moulding Manufacturing Processes

Blow moulding is a manufacturing process for the production of hollow containers, such as bottles, jars and jerrycans. In a blow moulding process a molten material is brought into a mould and inflated with gas to force it in the mould shape. Various materials can be blow moulded, but the process is mainly used for the production of plastic and glass containers. Blow moulding is advantageous because of its fast production rate at relatively low cost. Blow moulded containers are widely used all over the world to contain liquids from soft drinks, milk, beer and juice to shampoo, gel and liquid soap to oil and petrol.

1.1.1

History

Although nowadays blow moulding is a fully automated process and the range of applications has rapidly increased over the last century, traditional blow moulding is a technique that has been invented thousands of years ago.

2

1 Introduction The earliest glass objects used by men were found in nature. These glass objects involved chipped pieces of obsidian, a natural volcanic glass, which were used by cavedwellers for the production of primitive tools and weapons [177].

Figure 1.1: Examples of volcanic glass formed from acid lava: on the top left obsidian and to the right the same volcanic material which cooled and hardened before the gas bubbles escaped; photo used courtesy of Wilson Museum, Castine, ME, USA

According to the Roman historian Pliny the Elder (23 A.D. - 79 A.D.) the history of glass making begins with the accidental discovery by Phoenician merchants. The legend recounts shipwrecked sailors at the coast of present Lebanon who set their cooking pots on blocks of natron (soda) from their cargo and built a fire under it. The next morning they found that the fire had molten a mixture of sand and soda, which had cooled and hardened into glass. Nowadays, the scientific reliability of Pliny’s tale is called into question [113, 184]. On the other hand, it is believed that the Egyptians and Mesopotamians discovered glass making. The earliest known manmade glass are Egyptian beads dating from between 2750 and 2625 B.C. [184]. However, it is thought that the Egyptians and Babylonians started making their own glass objects in the form of beads and jewelry much earlier [67, 177]. Yet it took until at least 1500 B.C. before the first glass containers were produced by Egyptian craftsmen. The first bottles were produced by winding pieces of glass around moulds of concentrated sand and scraping the inside of the bottle [177]. The earliest examples of Egyptian glassware are three vases bearing the name of the Pharaoh Thoutmosis III (1504-1450 BC), who brought glassmakers to Egypt as prisoners.

1.1 Blow Moulding Manufacturing Processes

Figure 1.2: The Felix bottle dating from the 3rd century A.D. found at Faversham, Kent c Trustees of the British Museum

A revolutionary turn occurred when the Syrian craftsmen discovered the glass blowing pipe between 27 B.C. and 14 A.D. However, the Romans were the first to experiment with blowing glass inside moulds, which eventually led to an improvement of glass jars and bottles and the production of glass drinking vessels. It is also believed that the Romans were the first to use glass for architectural purposes, after the discovery of clear glass in Alexandria around 100 A.D. Not much later the invention of glass blowing in combination with colourants, led to the invention of stained glass windows [177].

Figure 1.3: Roman glassblow pipes

The first complete blow moulding of bottles seems to have originated in Bristol, England, around 1821 [18, 67]. Still, until the end of the eighteenth century, glass manufacturing techniques were based on skill and empirical knowledge rather than science [27]. Rapid advances in chemistry and physics toward the end of the eighteenth century led to prosperous progress, such as the development of optical instruments, the invention of

3

4

1 Introduction the tank furnace around 1816, the introduction of the iron mould in 1847 and improved chemical durability of glass [27].

Figure 1.4: Glass blowing as illustrated in Ref. [105]

In 1851 a U.S. patent for blow moulding a plastic material other than glass was issued to Samuel Armstrong [115]. It concerned unique novelty items made of natural latex [116]. The earliest attempts to blow mould thermoplastics were performed by blowing steam by means of a nozzle between two sheets of cellulose nitrate clamped between a split mould to soften the sheets and blow them in the mould shape. The steam temperature and pressure exerted by the clamping of the split mould parts caused the sheets to fuse together into one plastic container [67]. In the thirties efforts were made to apply the technique to other plastic materials such as cellulose acetate and polystyrene. With the introduction of low density polyethylene after the second world war the production of low density polyethylene squeeze bottles caused a rapid expansion of the blow moulding industry and plastic bottles gradually replaced glass bottles for e.g. shampoo and liquid soap. Furthermore, the development of polyethylene terephthalate (PET) in 1941 led to the application of reheat stretch blow moulding, which is nowadays a popular technique for the production of plastic bottles for e.g. soft drinks. During the fifties there was an enormous demand for plastic containers for keeping liquids in householding [67]. Together with the mass production of high density polyethylene and polypropylene and the appearance of more advanced blow moulding equipment, a virtual explosion of blow moulded products was seen in Europe and North

1.1 Blow Moulding Manufacturing Processes America [115]. Nowadays the range and variety of blow moulded products is still increasing.

1.1.2

Hollow Container Manufacturing

A typical hollow container manufacturing process passes through the following essential stages: 1. compounding and melting the forming material, 2. producing the so-called preform or parison, 3. blow moulding the preform into the mould shape, 4. applying ancillary treatments to finish the product. To get a better understanding of blow moulding, the process stages are described in more detail. A distinction is made between the blow moulding of glass and polymers, because of the basic differences in material properties. • Glass Glasses have some characteristic physical properties that make them suitable for blow moulding. Hot glass is sufficiently fluid to flow by gravity and be blown by relatively low air pressure. It gradually stiffens as it cools down as a result of the rapidly increasing viscosity. Furthermore, it has suitable heat transfer properties for rapid processing [67]. On the other hand, glasses have high melting points and therefore have to be processed under extreme temperatures. In addition, solid glass can easily break when subject to high stresses. Therefore, additional process stages may be required to reduce the stresses in the glass. Essential manufacturing processes for hollow glass containers are described below in the order of their application.

Melting In industry the vast majority of glass products is manufactured by melting raw materials and recycled glass in tank furnaces at an elevated temperature [109,177]. Examples of raw materials include silica, boric oxide, phosphoric oxide, soda and

5

6

1 Introduction lead oxide. The temperature of the molten glass in the furnace ranges between 1200 and 1600 ◦ C. A slow formation of the liquid is required to avoid bubble forming [177]. Parison forming The glass melt is cut into uniform gobs, which are gathered in a forming machine. In the forming machine the glass gobs slightly cool down to below 1200 ◦ C. Subsequently, the individual molten gobs are formed into a preform or parison. Different types of products require different parison forming techniques. Widely used techniques are press forming and blow forming. They are explained further on. Container forming The glass parison is inverted by means of a robotic arm and two mould halves are closed around the parison just below the neck. In the mould the parison is first left to sag due to gravity for a short period. Then pressurised air is blown in the mould by means of a blow head to force the glass in a mould shape. Finally, the mould shape is left in the mould to cool down before it is ejected from the mould. After the formation the glass objects are rapidly cooled down as to take a solid form. Annealing Development of stresses during the formation of glass may lead to static fatigue of the product, or even to dimensional changes due to relaxation or optical refraction. The process of reduction and removal of stresses due to relaxation is called annealing [177]. In an annealing process the glass objects are positioned in a so-called Annealing Lehr, where they are reheated to a uniform temperature region, and again gradually cooled down. The rate of cooling is determined by the allowable final permanent stresses and property variations throughout the glass [177]. Surface treatment An exterior surface treatment is applied to reduce surface defects. Flaws in the glass surface are removed by chemical etching or polishing. Consecutive flaw formation may be prevented by applying a lubricating coating to the glass surface. Crack growth is prevented by chemical tempering (ion exchange strengthening), thermal tempering or formation of a compressive coating [177]. The basic difference between glass blow moulding techniques is the way the parison is formed. Two widely used techniques are blow-blow moulding and press-blow moulding.

1.1 Blow Moulding Manufacturing Processes In a press-blow process first a parison is constructed by a press stage. Figure 1.5 shows a schematic drawing of a press-blow process. In the press stage a glass gob is dropped down into a mould, called the blank mould, and then pressed from below by a plunger (Fig. 1.5(a)). Once the gob is inside the blank mould, the baffle (upper part of the mould) closes and the plunger moves up. When the plunger is at its highest position, the ring closes itself around the plunger, so that the mould-plunger construction is closed from below. When the plunger is finally lowered, the ring is decoupled from the blank mould and the parison is carried by means of a robotic arm to another mould for the final blow stage, in which the container is formed (Fig. 1.5(b)). In a blow-blow process the parison is formed by a blow stage. Figure 1.6 shows a schematic drawing of the first blow stage of a blow-blow process. First in the settle blow the glass gob is blown from above to form the neck of the container (Fig. 1.6(a)), then in the counter blow from below to form the parison (Fig. 1.6(b)). After the counter blow the preform is carried to the mould for the final blow stage. Figure 1.7 shows a photo of a blow-blow machine. The press-blow technique was originally limited to the forming of wide mouthed containers, such as jars. By using the blow-blow technique it is easier to produce narrow mouthed containers. On the other hand, the press-blow process is easier to be controlled. It is interesting to note that the temperature of the glass gob is typically around 1, 000 ◦ C, while the temperature of the material of the forming machine is typically around 500 ◦ C. Consequently, the surface temperature of the material will increase significantly. To keep the temperature of the material within acceptable bounds the mould and plunger are heat insulated by means of water-cooled channels.

• Polymers Polymers processed by blow moulding are mainly thermoplastics, such as polyethylene, polypropylene and polystyrene. Plastics are composed of polymers of carbon and hydrogen, often together with other components, such as oxygen, nitrogen, chlorine or sulfur. Plastics can be stretched without sacrificing strength, which is a property that is regularly employed in blow moulding. Plastics can be processed under relatively low temperatures, but have poor heat transfer properties.

7

8

1 Introduction preform

gob

baffle

mould

plunger

(a) press

(b) blow

Figure 1.5: Schematic drawing of a press-blow process baffle

baffle glass air

mould

mould air glass

ring

(a) settle blow

ring

(b) counter blow

Figure 1.6: Schematic drawing of the first blow stage of a blow-blow process

Figure 1.7: Glass blow-blow machine for manufacturing bottles for wine, spirits and nonc Pascal Artur/VOA; photo used courtesy of the Phototheque of Saintalcoholic beverages Gobain.

1.1 Blow Moulding Manufacturing Processes Essential manufacturing processes for hollow plastic containers are described below in the order of their application.

Plasticising To mould plastic products the plastic is plasticised by means of a plasticator. In a plasticator a polymer resin is conveyed and mixed through a rotating screw, while it is kept at an elevated, uniform temperature. Controlling the temperature is important for plastic homogenisation [159]. Parison forming The parison is formed either by injection moulding or by extrusion. In injection moulding the parison is formed by injecting a polymer melt into a tube-shaped mould around a core rod, which forms the inner shape of the parison. In an extrusion process the polymer melt is extruded in the parison shape. Two extrusion methods in use are continuous extrusion and intermittent extrusion. In continuous extrusion the parison is extruded continuously and individual parts of fixed length are severed. In intermittent extrusion the extruder does not rotate while the parison is blown [67, 191]. Injection moulding is often used for the production of relatively small or wide-mouthed containers. Extrusion can be used for a wide variety of containers. Moreover, continuous extrusion provides a fast production rate and can be used with unplasticised PVC and polyethylene [67]. As a rule of thumb the melt temperature for parison forming is about 50 ◦ C above the melting point. If the melt temperature is higher, the polymer may degrade [156]. Most thermoplastics have melting points between 100 and 300 ◦ C. Container forming The parison is delivered into a mould, where it is inflated with pressurised air to expand it to the mould shape. The temperature of the parison is usually just above the glass transition temperature. Finally, the mould shape is left in the mould to cool down before it is ejected. Trimming and reaming Trimming is the most common ancillary operation. Tools employing rotating knives are used to trim the top of a bottle. Other trimming operations are carried out manually using knives or automatically by mechanical cutters. The inside of the neck is usually reamed to the desired diameter. Resulting swarf is removed by blasting pressurised air in the container [67]. In polymer container manufacture three main variations of blow moulding can be

9

10

1 Introduction

distinguished: injection blow moulding, extrusion blow moulding and stretch blow moulding. Injection blow moulding First a parison is injection moulded and subsequently inverted on the stretch rod and clamped in the mould between the mould halves. Then the rod opens and pressurised air is blown through the rod to form the container.

extruder barrel die

preform

(a)

mould

container

(b)

(c)

Figure 1.8: Schematic drawing of an extrusion blow moulding process

preform

mould

(a)

stretch rod

pressurised air

(b)

container

(c)

Figure 1.9: Schematic drawing of a stretch blow moulding process

Extrusion blow moulding Figure 1.8 shows a schematic drawing of the extrusion blow moulding process. An extruded parison is captured from above by closing the

1.2 Process Simulation and Optimisation

11

mould around the parison when it leaves the die (Fig. 1.8(a)-1.8(b)) and pressurised air is blown in the mould from below to form the parison into the container (Fig. 1.8(c)). Stretch blow moulding Figure 1.9 shows a schematic drawing of a stretch blow moulding process. First a parison is formed by injection moulding. Usually, the parison is packaged for later use. Then in the stretch blow moulding process it is reheated to above the glass transition temperature and transferred on the stretch rod into a mould (Fig. 1.9(a)). Subsequently, the preform is simultaneously stretched with the stretch rod and inflated with pressurised air (Fig. 1.9(b)-1.9(c)). Stretch blow moulding is a popular manufacturing technique for the production of PET (polyethylene terephtalate) bottles.

1.2 1.2.1

Process Simulation and Optimisation Simulation

Blow moulding processes take place at high production rates. At the same time quality factors of the products, such as smoothness, strength, weight and cooling conditions, are optimised. To optimise and control the process, thorough knowledge is required. Unfortunately, measurements are often complicated, considering that blow moulding processes take place at high rates in closed constructions under complicated circumstances, such as high temperatures. Furthermore, trial-and-error experiments with blow moulding equipment are usually expensive and time consuming. Process simulation offers a good alternative. Blow moulding processes can be simulated by means of a mathematical model, which is solved numerically to visualise the process at discrete times. The input information for the simulation includes a preform shape, an initial temperature distribution and an inlet air pressure. A representative numerical simulation should give as output the container’s final shape and wall thickness as well as the stress and thermal deformations the forming material (e.g. glass, polymer) and equipment undergo during the process. Process simulation can be used for several purposes. Process analysis: to analyse and comprehend the blow moulding process. Process simulations can also be used for comparison with measurements.

12

1 Introduction

Process optimisation: to optimise an existing blow moulding process. Process simulations can help minimise undesired variations in the wall thickness and reduce the weight while maintaining the strength. They can also help optimise cooling conditions and increase the production speed [75]. Process innovation: to analyse a completely new process. Prior to setting up a new process, it can be analysed and optimised in advance by means of process simulation. Over the last few decades mathematical models for process simulation have become increasingly important in understanding, controlling and optimising the process [109, 192]. The growing interest of blow moulding industry for process simulation has been a motivation for a fair number of publications on this subject. The earliest papers that deal with process simulation date from the eighties [30, 31, 38, 42, 49, 203, 205]. Numerous papers and theses on the subject appeared afterward for glass blow moulding [32, 75, 76, 97], thermoforming [50,107], extrusion blow moulding [48,112,124,189] and (injection) stretch blow moulding [84, 124, 129, 130, 150, 166, 201, 208]. Generally, Finite Element Methods (FEMs) are employed for numerical simulation of blow moulding processes. FEMs are usually coupled to Interface Tracking Techniques, which attempt to track the melt-air interfaces explicitly and update the finite element mesh as the interfaces evolve. The procedure of updating the mesh can become increasingly computationally expensive as the mesh size decreases or the mesh has to be updated more frequently. Many algorithms for efficient mesh updating have been proposed [30, 32, 97, 107, 124, 166]. Surprisingly, hardly any attempt has been made to completely avoid re-meshing, for example by an Eulerian approach1 .

1.2.2

Optimisation

A key issue in blow moulding is the wall thickness distribution of the final container. The thicker the container, the stronger it is and the less easily it breaks. The thinner the container, the lighter it is and the less costs are spent on material. The optimal wall thickness distribution follows from a comparative assessment between these aspects. The ideal thickness distribution is not necessarily uniform, as some parts of the container are more vital than others, e.g. corners, and should be thicker for optimal strength. 1

In an Eulerian approach the melt-air interfaces are described implicitly by interface functions on a fixed mesh.

1.2 Process Simulation and Optimisation

13

The difficulty in blow moulding a container with a prescribed wall thickness distribution is that the corresponding initial operating settings, such as the preform shape, the initial temperature distribution or the inlet pressure, are not known beforehand. There are essentially two ways to deal with this. The first approach is by means of numerical simulation. If the initial operating conditions can be estimated, for example by empirical expertise of the process, a mathematical model can be developed that can directly compute the corresponding container. Then the initial operating settings of the blow moulding process can be adjusted to improve the wall thickness of the computed container. Finding the desired wall thickness distribution is a trial-and-error procedure, which is often based on the manufacturer’s knowledge of the process. However, empirical expertise is often insufficient to find the initial operating settings, particularly when innovative processes are involved. A more efficient approach is to solve an inverse mathematical problem to find the initial operating settings corresponding to the container with the desired wall thickness distribution. This class of inverse problems is quite challenging, as in general coupled, highly nonlinear physical systems and complicated geometries are involved. Usually, numerical optimisation methods are employed to find a solution to the inverse problem. Over the last few decades numerical optimisation has become increasingly popular in blow moulding. An early attempt to optimise the container wall thickness distribution in blow moulding dates from the early nineties and was based on a Neural Network approach to find the preform thickness, the mould geometry and a representative rheological parameter corresponding to the optimal wall thickness distribution [55]. In the same year a combined Newton-Raphson and profiled optimisation routine to predict the parison thickness distribution required for a specified wall thickness distribution was presented [54]. In both papers the area of application was focussed on extrusion blow moulding. Few years after an attempt was made to optimise the wall thickness distribution in stretch blow moulding [114]. The authors used an optimisation method based on a method of feasible directions that attempts to find the optimal thickness of the preform. Several other optimisation methods have been considered in literature to optimise the wall thickness distribution in stretch blow moulding [190] and extrusion blow moulding [73, 95, 213] of polymers in three dimensions. An engineering approach to find the optimal parison shape for glass blow moulding was presented in Ref. [122, 135]. The authors combined a computer aided design (CAD) model, a 3D thermomechanical finite element model with adaptive mesh techniques and an optimisation technique based on

14

1 Introduction

the Levenberg-Marquardt method. The algorithm attempted to optimise the geometry of the mould for the first blow stage of the blow-blow process, given the wall thickness distribution at the end of the second blow stage. Optimisation methods have also been used to estimate the heat transfer coefficient or the initial temperature distribution in glass blow moulding [35, 134] and polymer injection stretch blow moulding [14, 139, 155]. It may have become clear from the foregoing that various initial operating settings can be optimised to obtain a container with the desired wall thickness, such as the preform shape, initial temperature distribution, inlet air pressure, etc. This thesis focusses on finding the optimal shape of the preform. The shape of the preform is often relatively easy to control in the parison forming process (e.g. injection moulding, extrusion, pressing) and can be directly and intuitively related to the container shape. To optimise the preform shape numerical shape optimisation is used. In numerical shape optimisation it is common practice to discretise the shape by approximating its boundary by a parametric surface. For an axial-symmetrical shape the parametric surface can be represented by a parametric curve, e.g. a spline or Beziér curve. This technique is a well-known for shape optimisation in e.g. metal forming [68, 69, 153, 214], but not so widely integrated in blow moulding. An example in blow moulding in which the geometry of the blank mould for the first blow stage of a glass blow-blow moulding process is described by a Beziér curve is presented in Ref. [122]. In blow moulding various approaches have been proposed to optimise the initial operating settings. For example, in Ref. [55] several settings, such as the parison thickness, temperature and mould diameter were used as parameters for optimisation. In Ref. [54, 114] a method was presented that attempts to find the optimal thickness of the finite elements of the parison. Finally, in extrusion blow moulding it is common practice to optimise the die gap opening at different points in time [73, 95, 213]. An advantage of approximating the unknown preform surface by a parametric curve is that the shape can be relatively easily controlled by a finite set of control points. In this way generally high accuracy can be reached depending on the interpolation method. For iterative optimisation an initial guess of the optimum is required. A suitable initial guess should be close enough to the optimum to ensure that the optimisation method converges towards it. In blow moulding literature the initial guess is usually constructed from measurements or the manufacturers knowledge about the process, which is not always the best choice, for example if measurements are complicated and knowledge about

1.3 Objectives

15

the process is limited. In Ref. [54] several initial guesses were tried and it was verified that the optimisation algorithm (a combined Newton-Raphson and profiled optimisation routine) converged to the optimal parison thickness for extrusion blow moulding, provided that the initial guess was within a certain range from the optimum. To the author’s best knowledge no attempt has been made to construct an initial guess of the preform shape by means of an analytical approximation of the optimal shape.

1.3

Objectives

The main objective of this thesis can be stated as: “find a preform shape, such that a container with a prescribed wall thickness distribution can be produced by blow moulding the container from the preform under certain operating conditions”. Before entering into further detail, two central problems are highlighted. The forward problem is to find the shape of the container, given the shape of the preform. The inverse problem is to find the shape of the preform, given the shape of the container. The forward problem needs to be solved in order to solve the inverse problem. The following goals are pursued in this thesis. 1. Give a general and complete problem formulation for forward and inverse blow moulding. 2. Present a class of efficient numerical methods to solve the forward problem, based on Level Set Methods. 3. Analyse the solvability and sensitivity of the inverse problem. 4. Find an efficient shape optimisation strategy to solve the inverse optimisation problem. The general and complete mathematical problem for blow moulding consists of a complete system of governing equations with jump conditions on the interfaces between materials, boundary conditions and initial conditions. It can be solved in an essentially identical way for the extrusion, injection, stretch blow moulding process for polymers and final blow moulding stage for glass. The various blow moulding processes and material properties are distinguished by defining constitutive relationships between the

16

1 Introduction

physical quantities, which are not included in the general problem formulation. Furthermore, it is described how a complete mathematical model covering physical phenomena involved in the process, such as surface tension, heat radiation and stress relaxation, can be simplified to a model that merely describes the dominant phenomena driving the process, hence significantly reducing the computational effort required to solve the problem. In order to solve the forward problem numerically a Galerkin Finite Element Method is used. Re-meshing is avoided by using a Level Set Method, which mark the location of the interfaces implicitly by a so-called level set function. Level Set Methods also have other attractive properties: they automatically deal with topological changes, can be used with high order of accuracy in general [144] and easily extend to three dimensions [185]. A so-called triangulated Fast Marching Method is used for re-initialisation of the level set function. The efficiency of the numerical method is illustrated by applying it to the stretch blow moulding of a PET bottle and the blow moulding of a glass bottle. All results presented are 2D axial-symmetrical. The model is validated by verifying conservation properties and, if data is available, by comparison with measurements. An interesting characteristic of blow moulding is that multiple moving interfaces are involved. The Level Set Method can either mark the location of each single moving interface by separate level set functions or mark all moving interfaces by one level set function. It is proven that both level set problems give the same solution. The second method is computationally cheaper, but can lead to numerical difficulties, because the gradient of the level set function is not everywhere defined. Both Level Set Methods are compared with each other. An analysis of the inverse problem with respect to solvability and sensitivity is addressed. Inverse problems, particularly in engineering, are often ill-posed. Blow moulding processes involve heat transfer by convection and conduction and shape transformation by surface tension, which are well-known sources of ill-posed inverse problems. This thesis aims at finding a way to deal with this within an acceptable tolerance by establishing conditions for the solvability of the inverse problem. Inverse problems in blow moulding are characterised by the fact that the outer surface of the blow moulded container coincides with the mould surface. This puts some constraints on the mould shape, since the inverse problem can only be solvable if the mould shape can be blown in finite time under given operating conditions. By means of a quantitative analysis and logical deduction the nature of such constraints is studied.

1.4 Thesis Outline

17

The inverse problem is sensitive to changes as the melt is stretched and becomes thinner during forming. The inverse problem can also be sensitive to changes because the melt surfaces converge towards an equilibrium in which a force balance occurs, for example because of surface tension. A sensitivity analysis is performed to investigate the extent of the sensitivity. The inverse problem is formulated as an optimisation problem. Efficient shape optimisation strategies to solve the inverse optimisation problem are discussed. In these strategies the unknown, axial-symmetrical preform surface is described by a parametric curve. The shape of the parametric curve is controlled by a set of control points. Then the positions of the control points are searched for as to optimise the container wall thickness distribution. To this end a finite set of parameters subject to optimisation is defined as a subset of the coordinates of the control points. The number of parameters is restricted to a minimum to limit the computational time, while providing sufficient accuracy for the approximation of the unknown surface. The computational time of the optimisation method is roughly proportional to the number of function evaluations, i.e. the number of times a forward problem is solved. Approximating the gradient of the residual with respect to the parameters by finite differences costs several function evaluations and can form the computational overhead. An alternative is to combine finite difference optimisation with Broyden’s method. The efficiency of the method is illustrated by applying it to the blow moulding of a glass bottle with prescribed wall thickness. An error analysis is performed to choose an optimal error tolerance for the optimisation algorithm. In this way a numerical solution of the inverse problem with optimal accuracy can be obtained with respect to given errors in the input and the model. An initial guess for the iterative optimisation algorithm is constructed by an analytical approximation of the optimum. The approximation is derived by omitting the mass flow in polar direction in spherical coordinates, so that the inverse problem can be solved analytically.

1.4

Thesis Outline

This thesis is structured as follows. In order to solve the inverse problem it is formulated as an optimisation problem and an iterative method is used to solve the optimisation

18

1 Introduction

problem. The forward problem needs to be solved at least each iteration of the method in order to calculate the container shape. Therefore, first a mathematical model for the forward problem is presented in Chapter 2, followed by numerical methods to solve the forward problem in Chapter 3. Results are shown in Chapter 4. Then the more complicated inverse problem is analysed in Chapter 5. Optimisation methods to solve the optimisation problem are discussed in Chapter 6. Solutions of the inverse problem are presented in Chapter 7. Finally, the work is concluded in Chapter 8, where also recommendations for future work are given. In the remainder this section the structure of these chapters is discussed into more detail. Chapter 2 presents a general and complete mathematical model for industrial blow moulding. The following successive steps are followed to model a blow moulding process. Firstly, in § 2.1 the geometry of the forming machine is defined and the computational domain split up into subdomains for the various parts. Secondly, in § 2.2 the governing equations, which are based on the conservation laws for mass, momentum and energy, are formulated in the open subdomains. The governing equations are applicable to all continuous media. The set of governing equations is completed by adding an equation for the propagation of the moving boundaries between the domains. Thirdly, constitutive relationships between the physical material properties are defined for each component and for the melt in particular. Constitutive equations for forming materials are discussed in detail in § 2.3. Fourthly, in § 2.4-2.5 jump, boundary and initial conditions are described. Fifthly, in § 2.6 the governing equations are reformulated based on the constitutive relationships and the resulting mathematical problem is brought into dimensionless form. Finally, in § 2.7 a quantitative analysis is performed for the dimensionless form based on specific data for the various materials in order to simplify the problem formulation. This section is concluded by formulating a dimensionless, simplified mathematical problem for blow moulding processes in general, which is assumed in the remainder of this thesis. Chapter 3 discusses numerical methods for solving the forward problem for blow moulding. First in § 3.1 the procedure for the spatial and temporal discretisation of the problem is described. For the spatial discretisation Finite Element Methods are used, which are based on the variational problem formulation. Therefore, in § 3.2 the variational problem is formulated. The moving interfaces are implicitly incorporated in the weak formulation. Next in § 3.3 Interface Capturing Techniques are discussed to track the moving interfaces. Interface Tracking Techniques are also briefly addressed. Then

1.4 Thesis Outline

19

the Level Set Method used in this thesis is described. The Level Set Method can either mark the location of each moving interface by a level set function or mark all moving interfaces by one level set function. These Level Set Methods are compared to each other and it is proven that both level set problems give the same solution. Different reinitialisation techniques to keep the desired shape of the level set function are discussed, in particular the triangulated Fast Marching Method. Finally, in § 3.4 solution methods to solve the discretised system of equations are described. Chapter 4 presents results of the numerical methods presented in Chapter 3 by means of several 2D axial-symmetrical applications. In § 4.1 a relatively simple example of blow moulding a glass container in an ellipsoidal mould is presented. The comparison between solving two level set problems, one for each glass-air interface, and solving one level set problem for both interfaces is emphasised. In addition, results of re-initialisation by the Fast Marching Method are assessed. In § 4.2 the blow-blow moulding of a glass beer bottle is simulated. The results are validated by verifying conservation properties and by comparison with data obtained from measurements. In § 4.3 the stretch blow moulding of a PET water bottle is simulated using the viscoplastic rheological model described in Chapter 2. Chapter 5 formulates and analyses the inverse problem. In § 5.1 mathematical formulations for the forward problem and inverse problem are given. In § 5.2 constraints on the mould surface are prescribed, which should hold for the inverse problem to be solvable. The case in which the outer melt surface reaches a force equilibrium state before it reaches the mould is studied and a time scale until equilibrium occurs is estimated. In § 5.3 the sensitivity of the inverse problem is analysed by:

• studying the case in which a container is blow moulded for an infinite time duration under the ideal circumstances in which the mathematical model is valid, • by approximating the sensitivity with respect to perturbations in the shape.

In § 5.4 an analytical approximation of the inverse problem is derived, by omitting the mass flow in polar direction in spherical coordinates, so that the inverse problem can be solved analytically. An example in which the approximation is compared with the numerical simulation is presented.

20

1 Introduction Chapter 6 deals with efficient shape optimisation strategies to solve the inverse op-

timisation problem. In § 6.1 the optimisation problem is formulated and the 2D axialsymmetrical unknown surface is parameterised by means of a parametric curve. The parameters subject to optimisation of the container wall thickness distribution are defined as the spherical radii of the control points. To optimise the wall thickness distribution the method aims at minimising a scalar objective function of the parameters. In § 6.2 the objective function is chosen as the L2 -norm over the distance from the designed inner container surface to the computed inner container surface. Here the designed surface corresponds to the optimal wall thickness distribution and the computed surface is obtained by solving the forward problem for the approximate preform. The objective function is computed numerically by a composite Gaussian quadrature formula. In § 6.3 several algorithms to solve the parameterised optimisation problem are described. The emphasis is on computationally cheap ways to approximate the derivatives. The algorithms stop if a suitable stopping criterium is satisfied. In § 6.4 an optimal error tolerance for the stop criterium is derived by means of an error analysis. For reliable convergence it is desired that the initial guess for the iterative optimisation methods is near the optimum. In § 6.5 an initial guess is constructed by means of the analytical approximation of the inverse problem presented in Chapter 5. The control points are chosen equidistantly along the initial guess of the unknown surface. Finally, geometric constraints are imposed on the parametric curve. In § 6.6 the optimisation method is modified as to account for the inequality constraints. Chapter 7 presents results of the optimisation methods for solving the inverse problem by means of several 2D axial-symmetrical applications. In § 7.1 a relatively simple example of blow moulding a glass preform in an ellipsoidal mould is presented. In § 7.2 the wall thickness distribution is optimised for blow moulding a glass bottle. The convergence results of the optimisation method are assessed. Finally, Chapter 8 states the conclusions and presents recommendations for future work.

Chapter 2

Mathematical Modelling of Blow Moulding

In this chapter a general and complete mathematical model for blow moulding is presented. The model covers the extrusion, injection, stretch blow moulding process for polymers and final blow moulding stage for glass. The model can be used calculate the location of the melt-air interfaces at any point in time during the process, as well as the stress and thermal deformation the materials undergo during the process. The model is used in the following chapters to solve the forward problem for blow moulding.

2.1

Geometry

In order to formulate a mathematical problem for blow moulding, the various mechanical parts of the forming machine are distinguished. Each part is characterised by its own physical properties and behaviour, all of which are combined into one governing model

22

2 Mathematical Modelling of Blow Moulding

for the forming process. Therefore, each part of the forming machine induces a problem domain. In this context the forming machine is defined as the whole of forming equipment, as well as the space enclosed by the equipment. The domain for the forming equipment is subdivided into subdomains for the different components, e.g. mould, ring, stretch rod. The domain for the space enclosed by the equipment is subdivided into subdomains for the forming material and air. The following subdomains and boundaries are defined with corresponding symbolic notations. The entire open domain of the forming machine, consisting of equipment, forming material (melt) and air, is denoted by Σ. The equipment domain can be subdivided into subdomains for different parts, such as a mould domain Σm , a stretch rod domain Σr and a baffle domain Σb . The ‘fluid’ domain Ω consists of the open forming material or melt domain Ωl , the open air domain Ωa , the inner melt-air interface Γ1 and the outer melt-air interface Γ2 . The melt-air boundary, which is also referred to as the melt surfaces, is given by Γf = Γ1 ∪ Γ2 . For glass and extrusion blow moulding the fluid domain Ω := Ωl ∪ Ωa ∪ Γf is fixed, while for stretch blow moulding Ω changes in time due to the motion of the stretch rod. Furthermore, Ωl and Ωa are variable in time for any forming process. The boundaries of the domains are: Γb

:

baffle boundary

Γ1

:

inner melt surface

Γm

:

mould boundary

Γ2

:

outer melt surface

Γo

:

outer boundary

Γs

:

symmetry axis

Γf

:

melt-air boundary

Γr

:

stretch rod surface

Note that not necessarily all boundaries exist for each forming machine. Domain Σ is enclosed by Γo ∪ Γs . Domain Ω is enclosed by ∂Ω := Ω ∩ Γq ∪ Γo ∪ Γs , with Γq := Γb ∪ Γm ∪ Γr

(equipment boundary).

(2.1.1)

In addition, define ∂Ωa := Ωa ∩ ∂Ω and ∂Ωl := Ωl ∩ ∂Ω. Finally, the boundaries for the stretch rod, melt and air domain are distinguished: Γa,q

=

∂Ωa ∩ Γq \ Γr ,

Γl,q

=

∂Ωl ∩ Γq \ Γr ,

Γa,o

=

∂Ωa ∩ Γo ,

Γl,o

=

∂Ωl ∩ Γo ,

Γr,o

=

∂Ωr ∩ Γo .

2.1 Geometry

23

Figure 2.1 illustrates the domain decomposition of forming machines for 2D axialsymmetrical counter blow moulding (Fig. 2.1(a)) and stretch blow moulding (Fig. 2.1(b)). For completeness also subdomains and boundaries of the forming machine are given. The domains for final blow moulding of glass containers are the same as for stretch blow moulding, if the air domain is extended to the stretch rod domain.

Γo Γb

baffle

Ωa

t

air

n

Γr,o Γa,o Γg,o Ωa air Γ2 Γs Γ1

Γs Γ2

Γo

Ωg Γm mould

glass

Σr

Ωa air

stretch rod

Γm

Γo

t n

Γr

mould

Ωg

material z

Γ1

Γg,o

Γa,o

r

(a) counter blow moulding machine

z r

(b) stretch blow moulding machine

Figure 2.1: 2D axial-symmetrical problem domain and subdomains of forming machines

Subdomains of the equipment can be of interest when modelling the heat exchange between the forming material, air and equipment. For less advanced heat transfer modelling it can be assumed that the equipment has constant temperature, so that the mathematical model can be restricted to the forming material and air.

24

2.2


Governing Equations

The mathematical model is based on the conservation of mass, momentum and energy for both the forming material and air: Dρ + ρ∇ · u = 0, in Ω \ Γf × T, Dt Du ρ = ∇ · T + ρg, in Ω \ Γf × T, Dt De ρ = T : ∇ ⊗ u − ∇ · q + Φ, in Ω \ Γf × T, Dt

(2.2.1a) (2.2.1b) (2.2.1c)

where T is the time interval of the forming process. The computational domain is restricted to the fluid domain Ω, i.e. the heat transfer in the equipment is omitted in the model. The solution of system of equations (2.2.1) is {u, ρ, e}, with flow velocity u [m s−1 ], density ρ [kg m−3 ] and internal energy e [J kg−1 ]. The stress tensor T [Pa] and heat flux q [W m−2 ] follow from the constitutive equations for the forming material and air and depend on the solution, the gravitational acceleration g [m s−2 ] is constant and the heat source density Φ [m s−2 ] can depend on reaction heat or external heat sources. In addition, the location of the melt-air interfaces is of interest, which follows from the ordinary differential equation dx =u dt

in T,

(2.2.2)

for all x(t) ∈ Γ(t), for any moving boundary Γ(t).

2.3

Constitutive Equations

The constitutive relationships characterise the idealised physical behaviour of a forming material. In forming processes two essential types of constitutive relations can be considered, nl. thermodynamical and rheological relationships. Thermodynamical constitutive relations are important, as blow moulding involves high temperatures; the typical temperature range for blow moulding of polymers is within 100 ◦ C − 300 ◦ C and for glass blow moulding this range can extend to 800 ◦ C−1400 ◦ C. Temperature variations within these ranges may cause significant changes in the physical properties of the forming material. The rheological constitutive equations describe the deformation that the forming material undergoes; they capture the stress-strain constitutive relation.

2.3 Constitutive Equations

2.3.1

25

Compressibility and Thermal Expansion

The compressibility measures the relative change in volume with respect to a change in pressure. The compressibility of glass in the forming temperature range is typically of the order 10−2 to 10−1 GPa−1 , which is sufficiently small to assume incompressibility. For example, for silica glass the compressibility is 2.7 · 10−2 GPa−1 [24]. The compressibility of polymers is generally somewhat larger, but nonetheless under low pressure polymers can be considered incompressible. For example, graphs of the volume-pressure equations of state for polyethylene show that the compressibility at 300 K does not exceed 0.25 GPa−1 [110]. For comparison the inlet pressure for blow moulding is generally not higher than 1 MPa. Because of the large temperature variations, thermal expansion of the forming material is an important property in hollow container manufacture. The following densitytemperature relation for thermal expansion can be deduced: ρ(T ) = ρ0 1 − αV T − T ref .

(2.3.1)

Here • αV [K−1 ] is the volumetric thermal expansion coefficient, • T ref [K] is a reference temperature, • ρ0 [kg m−3 ] is the density at the reference temperature. The volumetric thermal expansion coefficient is often assumed constant. For molten glass it typically ranges from 5 · 10−5 to 8 · 10−5 K−1 [197]. For molten polymers the volumetric thermal expansion coefficient is usually higher (generally in the range from 5 · 10−4 to 15 · 10−4 K−1 [62]), but temperature variations during the forming of polymers are typically lower. For amorphous polyethylene terephthalate (PET) above the glass transition temperature T g , αV is around 9.87 · 10−4 K−1 [106]. Thus the change in the density of PET within a temperature variation of 50 K is still much less than 10 percent (see Fig. 2.2). In general, variations in the density due to temperature variations during blow moulding are considered small enough to be ignored.

26


1.35 1.34

Density (x 103 kg m−3)

1.33 1.32 1.31 1.3 1.29 1.28 1.27 1.26 70

75

80

85

90 95 100 Temperature (oC)

105

110

115

120

Figure 2.2: Linearised thermal expansion of amorphous PET

2.3.2

Heat Flux

The heat flux q [W m−2 ] is the result of conduction and radiation. Fourier’s Law of thermal conduction states q = −λ

· ∇T,

(2.3.2)

where λ is the effective conductivity [W m−1 K−1 ], given by λ = λc + λr .

(2.3.3)

Here λc is the thermal conductivity and λr is the radiative conductivity. • Glass The thermal conductivity measures 1.0 W m−1 K−1 at room temperature for soda-lime glass and increases with approximately 0.1 W m−1 K−1 per 100 K [197]. The thermal conductivity of glass is often assumed constant. The calculation of the radiative conductivity λr is mostly a complicated process. For non-transparent glasses it can be simplified by the Rosseland approximation [59, 160, 197] λr (T ) =

16 n2 σT 3 , 3 α

(2.3.4)


27

where • σ is the Stefan Boltzmann radiation constant [W m−2 K−4 ], • n is the average refractive index [-], • α is the absorption coefficient [m−1 ]. The radiative conductivity λr given by (2.3.4) is called the Rosseland parameter. Relation (2.3.4) cannot be applied for highly transparent glasses, since in this case not all radiation is absorbed by the glass melt [197]. A more simple approach is to omit the radiative term, which is often reasonable for clear glass [111, 120]. For more information on heat transfer in glass by radiation the reader is referred to [109, 120, 121, 136, 197].

• Polymers For polymers the thermal conductivity is low and often difficult to measure [64]. It tends to be weakly dependent on temperature and composition [138]. The contribution of radiation is generally ignored.

2.3.3

Specific Heat

Since forming materials are incompressible, the internal energy can be expressed in terms of temperature using the thermodynamical identity de = c p dT.

(2.3.5)

• Glass The specific heat of glass c p [J kg−1 K−1 ] is slightly temperature dependent. In Ref. [9] an increase in the specific heat of less than ten percent in a temperature range of 900 K to 1300 K for soda-lime-silica glass is reported. For various specific heat capacity models for glass the reader is referred to Ref. [9, 176].

28


• Polymers For polymers the relation between the specific heat and temperature is usually complicated and difficult to measure. Extensive work on the relation between the heat capacity and temperature for a wide range of different polymers was performed by M. Dole and others (e.g. Ref. [46, 182, 207]). Often linear or quadratic approximations are used to estimate the relation between the specific heat and temperature within a certain temperature range [182]. For instance, the approximation c p (T ) \ [J kg−1 K−1 ] = 1500 + 4(T \ [K] − 378.15)

(2.3.6)

can be used for PET [14, 139] (see also Ref. [165] for data). A more detailed study of the heat capacity of PET is given in Ref. [182].

2.3.4

Rheology

Forming materials, such as glass and polymers, are substances with complex molecular structures. Consequently, modelling the deformation of forming materials is often not straightforward. In this section rheological models are discussed for the two representative blow moulding processes dealt with in this thesis, namely glass blow moulding and polymer stretch blow moulding. It should be kept in mind that the models should not merely capture the rheological behaviour as accurately as possible, but they should also be relatively simple in order to be computationally feasible. As will be seen in Chapter 6, the shape of the preform is optimised by means of an iterative process in which the numerical calculation of the container forms the computational overhead. Relatively complex rheological models can be used if high accuracy is required, but the computational cost for optimising the preform shape may become excessive. • Glass blow moulding In general, glass can be treated as an isotropic viscoelastic Maxwell material [9, 31, 32, 177]; that is, the strain rate tensor E˙ [s−1 ] can be split up into an elastic and a viscous part: E˙ = E˙ e + E˙ v ,

(2.3.7)


29

where the elastic and viscous strain rate tensors, E˙ e and E˙ v respectively, are given by [32] 1+ν 1 − 2ν ∂T 1 E˙ e = αV + tr(T˙ ) + dev(T˙ ) E ∂t 3 E 1 E˙ v = dev(T ). 2µ

(2.3.8) (2.3.9)

Here T˙ [Pa s] is the stress rate tensor, αV [ ◦ C−1 ] is the volumetric thermal expansion coefficient (see (2.3.1)), E [Pa] is Young’s modulus, ν [-] is Poisson’s ratio and µ [Pa s] is the dynamic viscosity. However, at relatively low viscosities the relation between shear stress and viscosity becomes approximately linear. For example, for soda-limesilica glasses the viscosity in terms of the strain rate and the temperature is approximated by [21, 181]

˙ T) = µ(E,

µ0 (T ) , 1 + 3.5 · 10−6 E˙ µ0.76 0 (T )

(2.3.10)

where µ0 is the Newtonian or zero-shear-rate viscosity. Consequently, the motion of glass is dominated by viscous flow and the influence of elastic effects can be neglected [9, 31, 177]. Moreover, since glass is practically incompressible, it follows that

˙ = 0. tr(E)

(2.3.11)

It can be concluded that glass in the forming temperature range behaves as an incompressible Newtonian fluid [9, 31, 197], i.e.

˙ T = −pI + 2µE,

(2.3.12)

where p [Pa] is the pressure. Finally, the strain rate tensor can be written as the symmetric gradient of the flow velocity, i.e. 1 E˙ = ∇ ⊗ u + (∇ ⊗ u)T . 2

(2.3.13)

30


• Polymer stretch blow moulding Much research has been conducted on the rheological behaviour of polymers. Essentially, three different kinds of models have been introduced in literature to describe the mechanics of a polymer melt subject to inflation, namely hyperelastic, viscoelastic and viscoplastic models. Regarding the short inflation time (∼ 0.5s), a hyperelastic constitutive relationship can be assumed [49, 72]. In particular, earlier papers propose isothermal, hyperelastic models, many of which were first developed for thermoforming processes [49, 50, 203]. However, hyperelastic models have difficulties with proper material characterisation and the prediction of the time-dependent behaviour of the polymer during a blow moulding process [72, 112]. Moreover, in Ref. [28] it was observed that the behaviour of PET (polyethylene terephthalate) is highly rate dependent, which is an effect that hyperelastic models generally cannot deal with. Viscoelastic models for stretch blow moulding processes have been studied extensively. For example, in Ref. [108] a K-BKZ viscoelastic model was adopted to simulate the blow moulding process. The K-BKZ model [15, 102, 123, 145] is an integral model that relates the stress to the complete history of deformation. In Ref. [166] a liquid-like visco-elastic constitutive equation of Johnson-Segalman type [100] was used to describe the mechanical behaviour of PET during a stretch blow moulding process. In this case the Cauchy stress tensor has the following form: T = −pI + 2µv E˙ + T e .

(2.3.14)

The extra-stress tensor T e can be determined by means of an integral form of the Johnson-Segalman constitutive model. This Johnson-Segalman type model was applied to the stretch blow moulding of PET. The simulation results were in good agreement with the experimental data. In Ref. [130, 131, 209] the Buckley-Jones model was used to model the constitutive behaviour of PET during the stretch blow moulding process. In the Buckley-Jones model [25, 26] the Cauchy stress is expressed as the sum of a bond stretching stress and a conformational stress. The Buckley-Jones model is attractive in that it encompasses the full range of behaviour exhibited by PET during industrial drawing processes [3, 4, 209]. In Ref. [130, 131] different constitutive models were compared for injection stretch blow moulding, namely a hyperelastic, creep and viscoelastic Buckley model. The Buckley model could predict the final bottle thickness distribution much


31

more accurately than the hyperelastic and creep model and produced results that closely matched with the experimental data. Because of the high deformation speed, also viscoplastic models can be considered, which exhibit some advantages over hyper- and viscoelastic models. Both viscoelastic and viscoplastic models take into account strain hardening and strain rate effects. Contrary to viscoplastic models, viscoelastic models can also calculate the stress relaxation during stretch blow moulding, which is a dominant deformation mode in relatively slow deformation processes, such as extrusion. However, because of the short time duration of a stretch blow moulding process, the loss of accuracy as a result of omitting stress relaxation is marginal, while the computational time can be decreased significantly [202]. This can be a motivation to employ a viscoplastic model. Another advantage of viscoplastic models is that they are generally very stable, whereas difficulties with convergence can be encountered when using hyperelastic or viscoelastic models [202]. Viscoplastic models have mainly been used for metal forming processes, but not extensively for stretch blow moulding. Viscoplastic models for stretch blow moulding were suggested in Ref. [19, 43, 200, 201]. In addition, other models, such as elasto-visco-plastic models [38] and hyperplastic models [91], have been proposed in literature. Since viscoplastic models are rather computationally cheap and numerically stable, they are highly suitable in the context of numerical shape optimisation, in which the mathematical model is invoked iteratively as a procedure to compute the container shape. Therefore, in this thesis the viscoplastic model presented in Ref. [201, 216] is used to describe the rheological behaviour of the polymer. The model is based on the LevyMises flow rule: 2σ ˙ dev T = E, 3 ε˙

(2.3.15)

where the strain rate ε˙ and the equivalent stress σ are given by r

2˙ ˙ E : E, 3 r 3 σ= dev(T ) : dev(T ). 2 ε˙ =

(2.3.16) (2.3.17)

32


In Ref. [200, 201] the material behaviour was determined by performing an uniaxial tensile test. From the test the following relation was deduced:

σ = k(T )ε˙ m .

(2.3.18)

Substitution of (2.3.18) in (2.3.15) results in the isotropic, viscous polymer flow,

˙ dev(T ) = 2µE,

(2.3.19)

with non-Newtonian viscosity 1 µ(ε, ˙ T ) = k(T )ε˙ m−1 . 3

(2.3.20)

In Ref. [19, 88] the G’Sell-WLF material model was proposed for the tensile modulus:    A1 (T − T g )    1 − exp(−A3 ζ) exp(A4 ζ A5 ), k(T ) = k0 exp  A2 + T + T g

(2.3.21)

where ζ is the cumulated strain rate.

2.3.5

Viscosity

As blow moulding processes are influenced by strong viscous forces, viscosity plays an important role. Forming materials have a large viscosity range for varying temperature. For glass it ranges from 10 Pa s at the melting temperature (about 1500 ◦ C) to 1020 Pa s at room temperature [177,197]. For polymers viscosities can range from 10−3 to 1012 Pa s, depending on the type of polymer, with a typical value of the order 103 Pa s at 230 ◦ C [17, 34, 125, 215]. The viscosity increases rapidly as the melt is cooled, so that the material will retain its shape after the forming process. Typical values for the viscosity in blow moulding lie between 102 and 105 Pa s.


33

• Glass The temperature dependence for the viscosity of glass within the forming temperature range is given by the VFT-relation, due to Vogel, Fulcher and Tamman [177, 197]: ! B µ(T ) = µL exp . T − TL

(2.3.22)

Quantities µL [Pa s], B [ ◦ C] and T L [ ◦ C] represent the Lakatos coefficients, which depend on the composition of the glass melt. Figure 2.3(a) shows how strongly the viscosity depends on temperature for soda-lime-silica glass with Lakatos coefficients µL = 2.870 · 10−2 Pa s, B = 8575 ◦ C and T L = 259 ◦ C [154]. 5.5 5 log10 viscosity

4.5 4

3.5 3

2.5 2 800

850

900


1100

1150

1200

(a) VFT-relation 5

log10 viscosity

4.5 4 3.5 3 2.5 100

120

140

160


240

260

280

300

(b) Modified WLF-relation

Figure 2.3: Viscosity-temperature relation

34


• Polymers The temperature dependence of the zero-shear-rate viscosity of polymers just above the glass transition temperature can be described by the modified WLF-equation [206]    A1 (T − T g )   , µ0 (T ) = k0 exp − (2.3.23) A2 + (T − T g ) where the coefficients k0 [Pa s], A1 [-] and A2 [K] depend on the type of polymer and are determined from experimental shear viscosity data. Note the analogy with (2.3.21). Figure 2.3(b) shows the viscosity-temperature relation for polypropylene with coefficients k0 = 4.66 · 1012 Pa s, A1 = 26.12 and A2 = 51.6K [215]. Coefficients for some other representative polymers can be found in e.g. Ref. [148].

2.3.6

Surface Tension

Surface tension allows the melt to resist the external forces that form the material. Surface tension results in a surface force f s of the melt on the surrounding air: f s = −γκn,

(2.3.24)

where normal n points in the air domain. Here γ [N m−1 ] is the surface tension and κ [m−1 ] is the curvature. The surface tension depends on the material composition, the temperature and the molecular weight. Surface tension of glass components around 1,300 ◦ C ranges from 0.29 N m−1 for SiO2 to 0.58 N m−1 for Al2 O3 [9,27]. A wide range of values for different glass compositions and temperatures can be found in Ref. [9]. Surface tension of most polymers between 100 ◦ C and 300 ◦ C ranges between 2 · 10−2 and 3 · 10−2 N m−1 [16]. Values for surface tension and its temperature gradient for various forming materials are given in Tab. 2.1.

2.4

Jump Conditions

The governing equations in § 2.2 are defined in Ω \ Γf . On the free surface Γf jump conditions are introduced. The jump conditions between two immiscible viscous fluids are the continuity of the flow velocity, ~u = 0,

on Γf × T,

(2.4.1a)

2.5 Boundary and Initial Conditions material

T [ ◦ C] 150 to 167 140 to 152 1,300 to 1,400 1,100 to 1,300

Polyethylene [16] Polystyrene [16] Lime Aluminosilicate Glass [198] Soda Silicate Glass [10, 146, 198]

35 γ [10−2 N m−1 ] 2.2 to 2.8 3.1 to 3.3 35.0 to 37.0 28.5 to 30.0

dγ/dT [10−5 N m−1 K−1 ] -7.3 to -4.0 -5.4 to -7.2 2.0 to 4.0 -6.0 to 30.0

Table 2.1: Values for surface tension and its temperature gradient for various forming materials

with ~u = ua − ul , and a dynamic jump condition stating the balance of stress across the fluid interface [101, 142], ~T n = −γκn,

on Γf × T.

(2.4.1b)

Finally, for the heat transfer between the forming material and air continuity of the heat flux is imposed, ~q · n = 0,

2.5

on Γf × T.

(2.4.2)

Boundary and Initial Conditions

To complete the set of governing equations boundary and initial conditions are imposed. Boundary conditions for the flow problem can be determined as follows. • On Γs symmetry conditions are imposed. • On Γl,o and Γa,o the normal stress should be equal to the external pressure. • On Γr the velocity is given by the stretch rod speed. • On Γl,q a suitable slip condition for the melt should be adopted. A commonly used boundary condition to describe fluid flow at an impenetrable wall [52, 70, 98, 158] is Navier’s slip condition: (T n + βu) · t = 0,

(2.5.1)

36

2 Mathematical Modelling of Blow Moulding where β [N m−3 s] is the friction coefficient. Tangent t has positive orientation along the contour around the melt domain. The order of magnitude of the friction coefficient depends on many parameters, such as the type of glass, temperature, pressure or presence of a lubricant [56,63,158]. For β → ∞ Navier’s slip condition together with the boundary condition for an impenetrable wall, u · n = 0, can be reformulated as a no-slip condition, i.e. u = 0.

(2.5.2)

• On Γa,q free-stress conditions are proposed. In practice air can escape through small cavities in (part of) the mould wall. This specific aspect is modelled by allowing air to flow freely through the mould wall. In summary, the boundary conditions for the flow problem can be formulated as: u · n = 0,

T n · t = 0,

on Γs × T,

u · n = 0,

(T n + βu) · t = 0,

on Γl,q × T,

T n · n = 0,

T n · t = 0,

on Γa,q × T,

T n · n = p0 ,

T n · t = 0,

on Γo × T,

u · ez = Vr ,

u · er = 0,

on Γr × T,

(2.5.3)

where er and ez are the unit vectors in radial and axial direction, respectively, Vr [m s−1 ] is the stretch rod speed and     pin on Γa,o × T p0 =    0 on Γl,o × T.

(2.5.4)

Here pin is the pressure at which air is blown into the mould. Alternatively, one may prefer to introduce the boundary condition u · n = 0,

T n · t = 0,

on Γl,o × T.

(2.5.5)

This boundary condition avoids outflow of glass through the mould entrance during blowing. However, this involves the definition of separate boundaries Γa,o and Γl,o , which change in time. This is not always convenient, particularly if a fixed mesh is used for the

2.6 Dimensional Analysis

37

discretisation of the fluid domain (see Chapter 3). Instead Γl,o can be conceived as the boundary between the melt and the ring, thus imposing a no-slip condition, u = 0,

on Γl,o × T.

(2.5.6)

The energy boundary conditions follow from symmetry and heat exchange with the surroundings: q · n = 0,

on Γs × T,

q · n = α T − T∞ ,

on Γo ∪ Γq × T,

(2.5.7)

where T ∞ is the temperature of the surroundings. More advanced boundary conditions can be introduced on Γl,q to compensate for the absence of a radiation model, in particular for glass. For details the reader is referred to Ref. [9, 120]. Finally, the initial conditions consist of values for u [m s−1 ] and e [J kg−1 ] at initial time t = 0s, as well as the initial position of the moving boundaries Γ1 , Γ2 and possibly Γr . Initial conditions are specified in § 2.7.

2.6

Dimensional Analysis

Based on the constitutive relationships in § 2.3 the following assumptions can be posed for forming materials. • Forming materials are incompressible and do not undergo significant thermal expansion. Consequently, ρ can be assumed constant in space and time. • The heat flux follows Fourier’s Law (2.3.2). • The internal energy is related to the temperature by the heat capacity through the thermodynamical identity (2.3.5). For simplicity pressurised air is considered an incompressible, viscous fluid. With these assumptions the governing equations (2.2.1) can be written as ∇ · u = 0,

in Ω \ Γf × T,

Du = ∇ · T − ρgez , in Ω \ Γf × T, Dt DT ρc p = T : ∇ ⊗ u + ∇ · (λ∇T ) , in Ω \ Γf × T, Dt ρ

(2.6.1a) (2.6.1b) (2.6.1c)

38


where the heat source density Φ is omitted and the gravity force is written as

g = −ρgez .

(2.6.2)

n o The solution of system of equations (2.6.1) is u [m s−1 ], p [Pa], T [K] . The pressure dependence is hidden in the stress tensor. The constitutive relations imply that (2.6.1) is a fully coupled system of equations: the viscous stress is temperature dependent and the heat transfer is partly effected by convection and diffusion.

In order to analyse the flow and energy problem quantitatively, the governing equations (2.2.1) with jump conditions (2.4.1) and boundary conditions (2.5.3) and (2.5.7) are written in dimensionless form. Define a typical: velocity V, length scale L, diameter D, viscosity µ, ¯ specific heat c¯ p , effective conductivity λ¯ and mould temperature T m . Then introduce the dimensionless variables t∗ :=

V(t − t0 ) , L

c∗p :=

cp c¯ p

,

x , L

u∗ :=

u , V

T ∗ :=

T − Tm , T0 − Tm

Dλ , Lλ¯

µ∗ =

µ , µ¯

T ∗ :=

DT , µV ¯

x∗ := λ∗ :=

(2.6.3)

where t0 is the initial time of the process and T 0 is the initial temperature distribution. For convenience all dimensionless variables, spaces and operators in this chapter with respect to dimensionless variables (2.6.3) are denoted with superscript ∗ . Substitution of the dimensionless variables (2.6.3) in the governing equations (2.6.1) leads to the dimensionless form, ∇∗ · u∗ = 0, Re Pe c∗p

in Ω∗ \ Γ∗f × T∗ ,

Du∗ Re ∗ ∗ e, ∗ = ∇ ·T − Fr z Dt

(2.6.4a)

in Ω∗ \ Γ∗f × T∗ ,

DT∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = Br T : ∇ ⊗ u + ∇ · λ ∇ T , Dt

(2.6.4b) in Ω∗ \ Γ∗f × T∗ .

(2.6.4c)

2.6 Dimensional Analysis

39

Here Re =

ρV D , µ¯

(2.6.5)

V2 , gL ρ¯c p V D Pe = , λ¯ Fr =

Br =

(2.6.6) (2.6.7)

µV ¯ 2 λ¯ T 0 − T m

(2.6.8)

are the Reynolds number, Froude number, Péclet number and Brinkman number, respectively. The Reynolds number measures the ratio of inertia to viscous forces, the Froude number the ratio of inertia to gravitational forces, the Péclet number the ratio of the advection rate to the diffusion rate and the Brinkman number the dissipation rate to the conduction rate. The dimensionless numbers are useful for assessing the order of magnitude of the different terms in (2.6.4).

For completeness also the jump and boundary conditions are written in dimensionless form. The jump conditions (2.4.1) and (2.4.2) in dimensionless form read ~u∗ = 0,

on Γ∗f × T∗ ,

~T ∗ n = −γ∗ κ∗ n,

on Γ∗f × T∗ ,

(2.6.9a) (2.6.9b)

where

γ∗ =

γ , µV ¯

κ∗ = Dκ,

(2.6.10)

and

~λ∗ n · ∇∗ T ∗ = 0,

on Γ∗f × T∗ .

(2.6.11)

40


Boundary conditions (2.5.3) retain the same form in terms of the dimensionless variables, u∗ · n = 0, u∗ · n = 0,

T ∗ n · t = 0, T ∗ n + β∗ u∗ · t = 0,

T ∗ n · n = 0,

T ∗ n · t = 0,

∗

T n· n = ∗

u · ez =

p∗0 , Vr∗ ,

on Γ∗s × T∗ , on Γ∗l,q × T∗ , on Γ∗a,q × T∗ ,

∗

on

∗

on

T n · t = 0, u · er = 0,

Γ∗o Γ∗r

(2.6.12)

∗

×T , × T∗ ,

with dimensionless friction coefficient β∗ :=

Lβ , µ¯

(2.6.13)

and dimensionless pressure  Lp in   on Γ∗a,o × T∗  µV ¯ p∗0 =   0 on Γ∗ × T∗ .

(2.6.14)

l,o

Finally, consider boundary conditions (2.5.7). Let α¯ be a typical value for the heat transfer coefficient, then the dimensionless heat transfer coefficient is defined by α∗ := α/α. ¯ The dimensionless boundary conditions for the energy problem become: λ∗ ∇∗ T ∗ · n = 0, on Γ∗s × T∗ (2.6.15) on Γ∗o ∪ Γ∗q × T∗ , λ∗ ∇∗ T ∗ · n = Nu α∗ T∗ − T∗∞ , where Nu =

αD ¯ , λ¯

(2.6.16)

is the Nusselt number. The Nusselt number measures the ratio of convective to conductive heat transfer across the boundary.

2.7

Model Simplifications

System of governing equations (2.6.4) with jump conditions (2.6.9) and boundary conditions (2.6.12) and (2.6.15) holds for all blow moulding processes under the assumptions made at the beginning of the previous section. Based on a quantitative analysis the governing equations can often be simplified considerably depending on the type of forming process. In this section a quantitative analysis is performed for the two representative forming processes in this thesis: glass blow moulding and PET stretch blow moulding.

2.7 Model Simplifications

2.7.1

41

Glass blow moulding

Glass blow moulding can refer to either the final blow stage in a press-blow or blow-blow process or to the counter blow stage in a blow-blow process. The orders of magnitude of most physical parameters are typically the same for these forming processes. The temperature of the glass melt in the final blow is usually slightly lower than in the counter blow, but this does not lead to a significant difference in the order of magnitude of the physical parameters. Therefore, the same typical values for both the counter blow and the final blow are considered. Table 2.2 shows specific data for glass blow moulding. The dimensionless numbers Quantity glass density glass viscosity gravitational acceleration surface tension flow velocity length scale of the parison diameter of the mould initial glass temperature mould temperature specific heat of glass effective conductivity of glass

: : : : : : : : : : :

Symbol [Unit] ρ [kg m−3 ] µ¯ [kg m−1 s−1 ] g [m s−2 ] γ [N m−1 ] V [m s−1 ] L [m] D [m] T 0 [ ◦ C] T m [ ◦ C] c¯ p [J kg−1 K−1 ] λ¯ [W m−1 K−1 ]

Value 2.5 · 103 104 9.8 0.5 10−1 10−1 10−2 103 5 · 102 1.5 · 103 5

Table 2.2: Specific data for glass blow moulding

for glass blow moulding are: Reg ≈ 2.5 · 10−4 ,

Fr ≈ 1.0 · 10−2 ,

Peg ≈ 7.5 · 102 ,

Brg ≈ 4.0 · 10−2 . (2.7.1)

The Reynolds number is sufficiently small to neglect the inertia terms in the momentum equations (2.6.4b). The order of magnitude of the gravity term is given by Reg Fr

≈ 2.5 · 10−2 ,

which is not extremely small. Moreover, gravity is the driving force during the sagging stage, hence it cannot be ignored. Therefore, the glass flow is described by the Stokes

42


flow equations: ∇∗ · u∗ = 0,

∇∗ · T ∗ = g∗ ,

in Ω∗l .

(2.7.2)

Here T ∗ satisfies (2.3.12) in terms of the dimensionless variables, and g∗ is the dimensionless gravity force, given by g∗ =

Re e. Fr z

(2.7.3)

The Péclet number for glass is moderately large, while the Brinkman number is small in comparison. Thus, the heat transfer is dominated by convection, conduction and radiation: ∗ ∗ ∗ DT ∗ ∗ ∗ =∇ · λ ∇ T , Dt

in Ω∗l × T∗ ,

(2.7.4)

where λ∗ is defined as λ∗ :=

λ . ρ¯c p V L

(2.7.5)

Note that definition (2.7.5) is different from the definition in (2.6.3) in § 2.6. With definition (2.7.5) the Nusselt number becomes Nu =

α¯ . ρ¯cp V

(2.7.6)

Since the heat transfer coefficient is typically small (α¯ 1), the Nusselt number is small (Nu 1), whence the boundary conditions can be simplified to λ∗ ∇∗ T ∗ · n = 0,

on ∂Ω∗ × T∗ .

(2.7.7)

In addition, the magnitude of the surface force is assessed. Consider the dimensionless dynamic jump condition (2.6.9b). The curvature κ has an order of magnitude equal to 1/D. Then definition (2.6.10) gives rise to the following dimensionless values: γ∗ ≈ 5 · 10−4 ,

κ∗ ≈ 1.

(2.7.8)

Consequently, the influence of surface tension can be neglected. However, in Chapter 5 some mathematical questions are posed in which surface tension plays an important role.


43

Therefore, the dynamic jump condition of the form (2.6.9b) including surface tension is nevertheless maintained in this thesis. Although specific data for glass is widely available, the physical properties of the hot, compressed air used to blow the preform is not so well presented in literature. Nonetheless, the typical values in Tab. 2.3, which are in the same order of magnitude as ambient air, are used for the quantitative analysis. The resulting dimensionless numbers are: Quantity initial air temperature specific heat of air thermal conductivity of air air density air viscosity

: : : : :

Symbol [Unit] T 0 [ ◦ C] c¯ p [J kg−1 K−1 ] λ¯ [W m−1 K−1 ] ρ [kg m−3 ] µ [kg m−1 s−1 ]

Value 7.5 · 102 103 10−1 1 10−4

Table 2.3: Specific data for pressurised air

Pea ≈ 10,

Bra ≈ 4 · 10−8 ,

Rea ≈ 10.

(2.7.9)

It can be concluded that the heat transfer in air is described by (2.7.4) in Ω∗a × T∗ , while the flow of air is described by the dimensionless Navier-Stokes equations: ∇∗ · u∗ = 0, Re

in Ω∗a × T∗ ,

Du∗ p¯ = ∇∗2 u∗ − Re 2 ∇∗ p∗ − g∗ , Dt∗ ρV

(2.7.10a) in Ω∗a × T∗ ,

(2.7.10b)

where p¯ is the typical pressure. For the inlet air the typical pressure can be chosen as p¯ = pin . The inlet pressure is typically of the order pin ∼ 105 Pa, so for the inlet air p¯ ρV

2

≈ 107 ,

(2.7.11)

from which the equations of motion (2.7.10b) can in principle be reduced to ∇∗ p∗ = 0.

(2.7.12)

Clearly, the pressure drop in the inlet air is negligible compared to the pressure drop in glass. However, in order to have sufficient degrees of freedom to solve the flow problem,

44


the pressure should be coupled to the flow velocity in the equations of motion (2.7.10b). Moreover, it is convenient to consider the general choice p¯ = ρV 2 ,

(2.7.13)

for all air in the mould. In this case the flow equations for air are scaled analogously to those for glass. Consequently, the full Navier-Stokes equations should be solved in the air domain. Unfortunately, this means that the flow problem for air is more complicated than for glass, while in glass blow moulding only the motion of glass is of concern. Therefore, air is replaced by a fictitious fluid with the same physical properties as air, but with a much higher viscosity, e.g. µa = 1. Then the Reynolds number of the fictitious fluid Re ≈ 10−3 is small enough to reasonably neglect the influence of the inertia forces. On the other hand, the viscosity of the fictitious fluid is still much smaller than the viscosity of glass, so that the pressure drop in the fictitious fluid is still negligible compared to the pressure drop in the glass [6, 75, 90]. As a result, the flow of the fictitious fluid can be described by the Stokes flow equations (2.7.2) in Ω∗a .

2.7.2

PET Stretch Blow Moulding

The governing equations for PET stretch blow moulding can be simplified following an analogous quantitative analysis as for glass blow moulding. Consider the specific data for PET stretch blow moulding in Tab. 2.4. The dimensionless numbers for PET stretch blow moulding are: RePET ≈ 1.3 · 10−5 ,

Fr ≈ 1.0 · 10−2 ,

PePET ≈ 8.1 · 103 ,

BrPET ≈ 46. (2.7.14)

The Reynolds number is extremely small for PET, so the influence of inertia can be neglected. The Reynolds number is also small enough to neglect the order of magnitude of the gravity term, RePET ≈ 1.3 · 10−3 . Fr This indicates that the flow problem for PET is dominated by viscous forces only: ∇∗ · u∗ = 0,

∇∗ · T ∗ = 0,

in Ω∗l ,

(2.7.15)

2.7 Model Simplifications Quantity PET density PET viscosity gravitational acceleration surface tension flow velocity length scale of the parison diameter of the mould initial PET temperature mould temperature stretch rod temperature specific heat of PET effective conductivity of PET

45

: : : : : : : : : : : :

Symbol [Unit] ρ [kg m−3 ] µ¯ [kg m−1 s−1 ] g [m s−2 ] γ [N m−1 ] V [m s−1 ] L [m] D [m] T 0 [ ◦ C] T m [ ◦ C] T r [ ◦ C] c¯ p [J kg−1 K−1 ] λ¯ [W m−1 K−1 ]

Value 1.3 · 103 105 9.8 2.5 · 10−2 10−1 10−1 10−2 102 10 40 1.5 · 103 0.24

Table 2.4: Specific data for PET stretch blow moulding

The Péclet number is large for PET, which indicates that the contribution of conduction to the heat transfer is much smaller than the contribution of convection. The Brinkman number is moderately large for PET, but still a few orders of magnitude smaller than the Péclet number. Thus, the influence of heat dissipation is marginal compared to heat convection. As a result, the heat transfer is convection dominated: DT ∗ = 0, Dt∗

in Ω∗l × T∗ ,

(2.7.16)

This means that the temperature is preserved along streamlines. Consequently, if the initial temperature of PET is uniform, it will remain uniform. It should be mentioned, however, that close to the equipment wall large temperature variations occur over a small length scale. Therefore, the conductive heat flux is somewhat larger close to the equipment wall. On the other hand, the Graetz number, which measures the ratio of thermal convection in the direction of the flow to conduction normal to the flow, is large, 1 D GzPET := π PePET ≈ 6.4 · 102 . 4 L

(2.7.17)

Hence thermal conduction towards the equipment wall is negligible [5, 156]. Yet also the viscosity in this region may increase by several orders, so that the fluid friction may not be negligible and the influence of heat generation by dissipation should be taken into account as well [158]. Although the reader should take notice of these boundary layer

46


effects, it is simply assumed that they are small enough to be ignored. The dimensionless value for the surface tension is γ∗ ≈ 2.5 · 10−6 ,

(2.7.18)

which is extremely small, hence negligible. Next a quantitative analysis of the rheological behaviour of the PET melt is performed. The Deborah number De measures the ratio of elastic forces to viscous forces. For stretch blow moulding of PET it is given by De = τ

pin ∼ 1, µ

(2.7.19)

where τ = 0.1 is the typical relaxation time of PET [166]. It can be concluded that the elastic forces are of the same order of magnitude as the viscous forces during the stretch blow process. The typical yield strength of PET at T = 373K is σy = 10 MPa [57, 119]. For comparison the typical stress during stretch blow moulding is σ ¯ = µ¯

V ≈ 1 MPa. D

(2.7.20)

Clearly, the isotropic yield criterium σ ¯ ≤ σy is barely satisfied; that is, the typical stress for stretch blow moulding is close to the elastic limit. From this analysis it can be concluded that the rheological behaviour is elasto-visco-plastic. Therefore, the choice of a viscoplastic material model is not unreasonable, particularly in view of modelling simplifications that disregard elastic effects. Note that a viscoplastic model can only be used to describe the actual stretch blow process, while the PET melt is subject to extreme stresses. It cannot be used to model the complete process, including the cooling of the container in the mould, where stress relaxation is important. Consider the viscoplastic model in § 2.3.4. Based on the assumption that the temperature distribution of PET is uniform, the energy problem is omitted in the simulation and the coefficient k in (2.3.18) is treated as a constant. Substitution of (2.3.20) in (2.3.19) yields m−1 ˙ ∗ 2 T ∗ = −p∗ I + k∗ ε˙ ∗ E, 3

(2.7.21)

where the dimensionless pressure is scaled in the same way as the Cauchy stress tensor.


47

For air the same typical values as for glass blow moulding are considered (see Tab. 2.3), except that the inlet air temperature is 20 ◦ C. Consequently, the temperature difference T 0 − T m is much lower. On the other hand, Bra ∼ 10−5 , which is still small enough to neglect the influence of heat dissipation in air. Furthermore, since the heat conduction in PET is ignored, also the heat conduction in the fictitious fluid can be reasonably omitted. As a result (2.7.16) also holds in Ω∗a . Analogously to the analysis for glass blow moulding, air can be replaced by a fictitious fluid to simplify the flow problem.

2.7.3

General Blow Moulding

In general forming materials, such as glasses and polymers, are isotropic, incompressible, viscoelastic fluids. The constitutive model for the stress-strain interaction depends on the forming material. However, since large deformations occur in a short time duration, the mechanical stress strongly depends on the strain rate. Therefore, the motion of the forming material is modelled as a viscous fluid flow, possibly with non-Newtonian viscosity or including an extra stress tensor to account for elasticity. Forming materials typically have an extremely low Reynolds number, so that the influence of inertia can be neglected. The heat transfer in forming materials is usually convection dominated. The contribution of viscous dissipation is generally relatively small, because of the small length scale (hence small flow velocity) and large temperature gradients. The influence of conduction is not always small, but highly depends on the forming material. Because of the low Reynolds number for blow moulding, air can always be replaced by a fictitious fluid with higher viscosity, so that the motion of air can be described by Stokes flow. The heat transfer in air can be described by the heat convection-diffusion equation as for the forming material. In conclusion, a blow moulding process can be described by the following simplified system of governing equations: ∇∗ · u∗ = 0,

in Ω∗ \ Γ∗f ,

(2.7.22a)

∇∗ · T ∗ = g∗ , in Ω∗ \ Γ∗f , DT ∗ c∗p ∗ = ∇∗ · λ∗ ∇∗ T ∗ , in Ω∗ \ Γ∗f × T∗ , Dt

(2.7.22b) (2.7.22c)

48


with jump conditions (2.6.9), (2.6.11), boundary conditions (2.6.12), (2.6.15) and initial condition T (x∗ , 0)∗ = T 0∗ (x∗ ),

x∗ ∈ Ω∗ ,

(2.7.23)

and the ordinary differential equation dx∗ = u∗ dt∗

in T∗ ,

(2.7.24)

with initial conditions Γ∗ (0) = Γ∗0 , for all moving boundaries Γ∗ (t∗ ), t∗ ∈ T∗ .

(2.7.25)

Chapter 3

Numerical Methods

The mathematical problem defined in the previous chapter involves a strongly coupled system of partial differential equations and complicated geometries with moving boundaries. To solve this problem, efficient numerical methods are required. The numerical algorithm should give as output the container’s final shape and wall thickness as well as the stress and thermal deformations the forming material and equipment undergo during the blow moulding process.

3.1

Discretisation Procedure

To find a numerical solution to a system of partial differential equations the system first needs to be discretised. The discretisation procedure is important for the efficiency, robustness and stability of the numerical algorithm. Partial differential equations are usually discretised in both space and time for which different techniques can be used. A Galerkin Finite Element Method (FEM) is applied for the spatial discretisation of the solution of the system of governing equations (2.7.22). FEMs have been extensively employed in literature for numerical simulation of blow moulding processes and

50

3 Numerical Methods

have proven their efficiency for solving the various underlying mathematical problems many times. The numerous examples include Ref. [30, 32, 42, 75, 97, 205] for glass blow moulding, Ref. [49, 107] for thermoforming, Ref. [38, 124, 166, 201, 208] for PET polymer stretch blow moulding and Ref. [124, 189, 212] for extrusion blow moulding. Different methods that are used less often in literature for numerical simulation of blow moulding are Finite Volume Methods [210] and Boundary Element Methods [53]. The FEM employs stable elements for the spatial discretisation of the Stokes flow problem (2.7.22a), (2.7.22b), (2.6.9), (2.6.12). Because all examples in this thesis are axial-symmetrical, only elements for two-dimensional axial-symmetrical problems are considered. Both first order and second order triangular elements are considered (see Fig. 3.1). First order stable elements for Stokes flow problems are the so-called Minielements [12, 47, 79]. Mini-elements use linear basis functions at joint nodes for both velocity and pressure at the vertices and a so-called bubble function at an additional node in the barycentre of the element (see Fig. 3.1(a)). The bubble function is the normalised product of the linear basis functions. Second order elements include Taylor-Hood elements and P2 /P0 elements. Taylor-Hood elements use joint nodes for the pressure and the velocity unknowns (see Fig. 3.1(b)). Second-order triangular Taylor-Hood elements (P2 /P1 ) use continuous quadratic basis functions for velocity and linear basis functions for pressure [79, 218]. Elements P2 /P0 use continuous quadratic basis functions for velocity and piecewise discontinuous pressure [79, 96, 218] (see Fig. 3.1(c)). Although the velocity is quadratic, the piecewise constant pressure controls the error of the approximation [39]. velocity pressure

(a) Mini element

(b) P2 /P1 Taylorhood element

(c) P2 /P0 element

Figure 3.1: Stable elements for 2D (axial-symmetrical) Stokes flow problems

For the spatial discretisation of the energy problem (2.7.22c, 2.6.11, 2.6.15, 2.7.23)

3.1 Discretisation Procedure

51

linear or quadratic Lagrange elements are used [44, 217]. An implicit Finite Difference Method (FDM) is used for the temporal discretisation. FDMs that are used in this thesis include the implicit Euler scheme or, for more accuracy, second order schemes, such as Crank-Nicholson or BDF (Backward Difference Formula). Each time step the static Stokes flow problem, the energy problem and the moving interface problem are solved in respective order using the FEM. The flow velocity obtained from the Stokes flow problem is used to solve the energy problem and the moving interface problem. Then the temperature obtained from the energy problem and the new position of the moving interfaces are used to compute the physical parameters at the next time step. The time stepping stops when the end time is reached. If desired, a method that decides whether the outer melt-air interface has coincided with the mould surface can be built in the time stepping algorithm, in which case the algorithm stops before the end time is reached. Figure 3.2 shows a flow chart of the time stepping procedure. Start process time loop t := t + ∆t

cp, λ, u

u

Energy problem

Interface problem

T

Γ1,2 Parameters (e.g. viscosity)

cp, λ, µ Flow problem

u, p yes

End time reached? no Mould wall covered?

no

yes End process

Figure 3.2: Flow chart of numerical simulation of a blow moulding process

52

3 Numerical Methods

3.2

Variational Formulation

FEMs discretise the variational formulation of the mathematical problem. The variational formulation combines the system of equations (2.7.22), (2.7.24) with jump conditions (2.6.9), (2.6.11), boundary conditions (2.6.12), (2.6.15) and initial conditions (2.7.23), (2.7.25) into one statement. A variational formulation of the Stokes flow problem is given in Theorem 3.2.1. In this variational formulation no-slip conditions are imposed on Γl,q and the stretch rod domain is omitted, but it can be easily extended to other cases. Theorem 3.2.1. Define vector spaces Q := L2 (Ω; R), (3.2.1) U := u ∈ H 1 Ω \ Γf ; Rd ∪ L2 Γf ; Rd u · n = 0 on Γs , u = 0 on Γl,q , ,(3.2.2) where superscript d denotes the dimension of the flow. The variational formulation of Stokes flow problem (2.7.22a), (2.7.22b), (2.6.9), (2.6.12) is to find (u, p) ∈ U × Q, such that for all (v, q) ∈ U × Q,     a(v, u) + b(v, p) = c(v),    b(u, q) = 0.

(3.2.3)

where a : U × U 7→ R and b : U × Q 7→ R are bilinear forms and c : U 7→ R is a linear form, defined by a(v, u) =

Z Ω\Γf

b(v, q) = −

Z

c(v) = −

Z

∇⊗v

Ω\Γf Ω\Γf

: dev

T dΩ,

(3.2.4)

q∇ · v dΩ, g · v dΩ +

(3.2.5) Z Γo

p0 n · v dΓ −

Z Γf

γκn · v dΓ.

Proof. Suppose (u, p) ∈ C2 (Ω \ Γf ; Rd ) × C1 (Ω). Let v ∈ u ∈ C1 Ω \ Γf ; Rd ∪ C0 Γf ; Rd u · n = 0 on Γs , q ∈ C 0 (Ω)

u = 0 on Γl,q ,

(3.2.6)

3.2 Variational Formulation

53

(note that C1 is dense in H 1 and C0 is dense in L2 ). First note that a(v, u) − b(v, p) =

Z Ω\Γf

(∇ ⊗ v) : dev T − pI dΩ =

Z

∇ ⊗ v : T dΩ.(3.2.7)

Ω\Γf

By means of (3.2.7) the first equation in (3.2.3) can be written as Z

Ω\Γf

Z ∇ ⊗ v : T + g · v dΩ =

Γo

p0 n · v dΓ −

Z Γf

γκn · v dΓ.

(3.2.8)

Application of identity ∇ · (T v) = v · (∇ · T ) + ∇ ⊗ v : T ,

(3.2.9)

yields Z

Ω\Γf

Z ∇ ⊗ v : T + g · v dΩ = −

Ω\Γf

v · (∇ · T + g) dΩ +

Z Ω\Γf

∇ · (T v) dΩ,(3.2.10)

By Gauss’ divergence theorem the second term on the right hand side of (3.2.10) can be written as Z Ω\Γf

∇ · (T v) dΩ =

Z

=

Z

Γs ∪Γq∪Γo Γs

n · T vdΓ +

Z Γf

(T n · t) (v · t) dΓ + +

Z Γf

~n · T vdΓ

Z Γa,q ∪Γo

~T n · vdΓ.

Substitution of (3.2.10) and (3.2.11) in (3.2.8) results in Z Z Z (T n · t) (v · t) dΓ + v · (∇ · T + g) dΩ = Ω\Γf

T n · vdΓ

Γs

+

Z

+

Z

Γo Γf

(3.2.11)

Γa,q

T n · vdΓ

T n − p0 n

· vdΓ

(~T n + γκn) · vdΓ

(3.2.12)

54

3 Numerical Methods

Variational problem (3.2.12) holds for all v ∈ C1 (Ω \ Γf ; Rd ) if and only if (2.7.22b) is satisfied and T n· t = 0

on Γs ∪ Γo ,

T n · n = p0

on Γo ,

Tn=0

on Γa,q ,

~T n = −γκn on Γf . Furthermore, b(u, q) = 0 for all q ∈ Q if and only if (2.7.22a) is satisfied.

For completeness, also a variational formulation of the energy problem is given in Theorem 3.2.2. Without loss of generality the time domain can be extended to infinity. The specific heat is assumed to be independent of temperature. Theorem 3.2.2. Let T = [0, ∞). Define vector spaces n o V := T ∈ H 1 Ω \ Γf × [0, ∞); R T (x, 0) = T 0 (x) for x ∈ Ω , (3.2.13) n 1 2 W := ω ∈ H Ω \ Γf × [0, ∞); R ∪ L Γf × [0, ∞); R lim ω(x, t) = 0 t→∞ o for x ∈ Ω . (3.2.14) Then the variational formulation of energy problem (2.7.22c), (2.6.11), (2.6.15), (2.7.23) is to find T ∈ V, such that for all φ ∈ W, Z∞ Z 0

Ω\Γf

! Z∞ Z ∂φ − φu · ∇T − λ∇T · ∇φ dΩdt = cp T αφ T ∞ − T dΓdt + ∂t Γo 0 Z T 0 φ0 dΩ, (3.2.15) Ω\Γf

where φ0 (x) = φ(x, 0) for x ∈ Ω and (u, p) ∈ U × Q is a solution of (3.2.3) for all (v, q) ∈ U × Q.

Proof. Suppose T ∈ C2 (Ω \ Γf × [0, ∞); R). Let φ ∈ C1 (Ω \ Γf × [0, ∞); R) ∪ C0 (Γf × [0, ∞); R). Successive application of the product rule for differentiation and Gauss’ di-

3.2 Variational Formulation

55

vergence theorem yields Z∞ Z 0

Ω\Γf

=

! ∂φ cp T − φu · ∇T − λ∇T · ∇φ dΩdt ∂t

Z∞ Z

! ∂ c p T φ − ∇ · λφ∇T dΩdt ∂t

Ω\Γf

0

Z∞ Z −

=−

0 Z∞ Z

∂Ω

0

+

Ω\Γf

Z Ω\Γf

cp

∂T ∂t

! + u · ∇T φ − ∇ · λ∇T φ dΩ

λφ∇T · n dΩdt −

Z∞ Z Γf

0

c p T 0 φ0 dΩ −

Z∞ Z 0

Ω\Γf

~λφ∇T · ndΩdt ! DT − ∇ · λ∇T φ dΩdt cp Dt

(3.2.16)

Substitution of (3.2.16) in (3.2.15) results in Z∞ Z 0

Ω\Γf

! Z∞ Z DT cp − ∇ · λ∇T φ dΩdt = − λ∇T · nφ dΩdt Dt Γs 0

Z∞ Z − 0

Γo ∪Γq

λ∇T · n − Nu α T − T∞ φ dΩdt Z − Γf

~λ∇T · nφ dΩdt.

(3.2.17)

Variational problem (3.2.17) holds for all φ ∈ C1 (Ω × [0, ∞); R) if and only if (2.7.22c) is satisfied and λ∇T · n = 0

on Γs ,

λ∇T · n = Nu α T − T∞ ~λ∇T · n = 0

on Γo ∪ Γq ,

on Γf .

56

3 Numerical Methods

3.3

Interface Capturing

Problem (2.7.22), (2.7.24) with jump conditions (2.6.9), (2.6.11), boundary conditions (2.6.12), (2.6.15) and initial conditions (2.7.23), (2.7.25) is a two-phase fluid flow problem, involving the flow of both melt and air. In order to model the two-phase fluid flow problem, the glass-air interfaces Γf , separating the air domain Ωa and the melt domain Ωl , have to be captured. There are various numerical techniques to deal with the moving interfaces in two-phase fluid flow problems. They can be classified in two main categories [75, 78]: Interface-Tracking Techniques (ITT) and Interface-Capturing Techniques (ICT). Interface-Tracking Techniques (ITT) attempt to find the moving interfaces explicitly. ITT involve separate discretisations of domains Ωa and Ωg ; the meshes of both domains are updated as the flow evolves by following the velocity of the interfaces. The major challenge of ITT is the mesh update. The procedure of updating the mesh can become increasingly computationally expensive as the mesh size decreases or the mesh has to be updated more frequently. ITT have been extensively used in the modelling of blow moulding processes, including glass blow moulding [30, 42, 97], thermoforming [50, 107], PET stretch blow moulding [124, 166, 199] and extrusion blow moulding [48, 112, 124]. Interface-Capturing Techniques (ICT) are based on an implicit formulation of the interfaces by means of interface functions, which allows them to function on a fixed mesh. An interface function marks the location of the corresponding interface by a level set. Two widely used ICT are Volume-Of-Fluid Methods (VOFMs) [94, 161] and Level Set Methods (LSMs) [2, 33, 174, 188]. In VOFMs the interface function denotes the fraction of volume of a fluid within each element. VOFMs are conservative and can deal with topological changes of the interface. However, they are often rather inaccurate; high order of accuracy is hard to achieve because of the discontinuity of the interface function [144]. Furthermore, they can suffer from small remnants of mixed-fluid zones [141, 151]. Still VOFMs are attractive because of their rigorous conservation properties. In LSMs the interface is generally represented by the zero contour of the interface function. LSMs automatically deal with topological changes and it is in general easy to obtain high order of accuracy [144]. In addition, properties of the interfaces, such as the normal and the curvature, are straightforward to calculate. LSMs also generalise

3.3 Interface Capturing

57

easily to three dimensions [185]. A drawback of LSMs is that they are not conservative. Poor mass conservation of LSMs for incompressible two-phase fluid flow problems is addressed in Ref. [61, 144, 193]. A major concern in LSMs is the re-initialisation of the interface function in order to avoid numerical problems. ICT have rarely been used in blow moulding. For examples of LSMs for the modelling of glass blow moulding the reader is referred to Ref. [6, 75]. By the best of the author’s knowledge no literature is found regarding the forming of hollow polymer containers using LSMs, although pseudo-concentration methods have been used for the modelling of polymer injection moulding [34, 90]. Of the aforementioned methods LSMs seem to be most attractive for the applications in this thesis. The interfaces are accurately captured, topological changes are naturally dealt with, a generalisation to three dimensions is relatively easy and complicated re-meshing algorithms are avoided. In order to compensate for the mass loss or gain coupled LSMs and VOFMs have been developed [151,178,187]. However, Ref. [75] reports a change in mass of less than 1% during the glass blow process simulations using LSMs, which can be further improved by using higher order time integration schemes or by taking smaller time steps. This indicates that in this case LSMs are suited to be used as ICT. The basic idea of LSMs is to embed the moving interfaces as the zero level set of the interface function θ, the so-called level set function (see Fig. 3.3): θ x, t = 0,

x ∈ Γf (t), t ∈ T.

(3.3.1)

The equation of motion of the interfaces follows from the chain rule and ordinary differential equation (2.7.24), ∂θ x, t + u · ∇θ x, t = 0, ∂t

x ∈ Γf (t), t ∈ T.

(3.3.2)

Initially, the level set function is defined as the signed Euclidean distance function to the interfaces Γf (0). If furthermore the level set equation (3.3.2) is extended to the fluid domain the corresponding level set problem becomes  ∂θ    + u · ∇θ = 0, in Ω × T,      ∂t        −d x, Γf (0) , x ∈ Ωa (0)   θ(x, 0) :=       d x, Γf (0), x ∈ Ωg (0),

(3.3.3)

58

3 Numerical Methods

where d x, Γ := inf kx − yk2 .

(3.3.4)

y∈Γ

Figure 3.3: Level set function for a glass preform

For the discretisation of the level set problem the same procedure can be followed as for the energy problem. Note that since two interfaces are involved θ is not everywhere differentiable. This problem can be overcome by defining two level set functions θ1 and θ2 , one for each melt-air interface, and subsequently solving the two corresponding level set problems. Then the level set function for both interfaces is θ = min{θ1 , θ2 }. In this thesis this problem is referred to as the minimum level set problem. It can be proven that the solution θ of the single level set problem and the solution θ = min{θ1 , θ2 } of the minimum level set problem are the same in the weak sense Theorem 3.3.1. Let u ∈ C1 (Ω × [0, ∞); R). Let the single level set problem be formulated as  Dθ   in Ω × [0, ∞)  Dt = 0, n o    θ(x, 0) = min θ1,0 (x), θ2,0 (x) ,

for x ∈ Ω,

(3.3.5)


59

and let the minimum level set problem be formulated as  Dθ  1  in Ω × [0, ∞)   Dt = 0,   Dθ  2   in Ω × [0, ∞)  Dt = 0,    θ1 (x, 0) = θ1,0 (x), for x ∈ Ω        θ2 (x, 0) = θ2,0 (x), for x ∈ Ω       θ(x, t) = min θ1 (x, t), θ2 (x, t) ,

(3.3.6) for (x, t) ∈ Ω × [0, ∞),

with classical solution θ ∈ C1 Ω\Γmax ×[0, ∞); R ∪C0 Γmax ×[0, ∞); R , where Γmax (t) = n o x ∈ Ω θ1 (x, t) = θ2 (x, t) . Then weak solutions in L2 (Ω × [0, ∞); R) of the single level set problem and the minimum level set problem are equal almost everywhere. Proof. Consider the weak formulation of the single level set problem: Z ∞Z Z Dφ θ dΩdt = − θ φ t=0 dΩ. Dt 0 Ω Ω

(3.3.7)

for all test functions φ ∈ C10 (Ω × [0, ∞); R). Define subspaces n o Ω1 (t) = x ∈ Ω θ1 (x, t) < θ2 (x, t) n o Ω2 (t) = x ∈ Ω θ2 (x, t) < θ1 (x, t) .

(3.3.8) (3.3.9)

n o Using θ(x, 0) = min θ1,0 (x), θ2,0 (x) , x ∈ Ω and the weak formulations for θ1 and θ2 , it follows that Z ∞Z 0

Dφ θ dΩdt = − Ω Dt

Z

min θ1 , θ2 φ t=0 dΩ ZΩ Z =− θ1 φ t=0 dΩ −

θ2 φ t=0 dΩ Ω1 (0) Ω2 (0) ! Z ∞ Z Z Dφ Dφ θ1 = dΩ + θ2 dΩ dt. Dt Dt 0 Ω1 (t) Ω2 (t)

Thus, ∞

Z 0

Z Ω1 (t)

θ − θ1

Dφ dΩ + Dt

Z Ω2 (t)

θ − θ2

! Dφ dΩ dt = 0. Dt

(3.3.10)

Since u and φ are smooth functions, there exists a function ψ ∈ C01 (Ω × R), such that ψ=

Dφ . Dt

(3.3.11)

60

3 Numerical Methods

As a result, Z ∞Z Ω

0

θ − min θ1 , θ2 ψdΩdt = 0,

(3.3.12)

which implies θ = min θ1 , θ2 almost everywhere in Ω × [0, ∞).

In the following theorem u depends on θ rather than on time. Theorem 3.3.2. Let u ∈ C1 Ω × L2 Ω × [0, ∞); R ; R be a function of x ∈ Ω and a functional of θ ∈ L2 Ω × [0, ∞); R . Then weak solutions in L2 (Ω × [0, ∞); R) of the single level set problem and the minimum level set problem are equal almost everywhere. In the proof the flow velocity in equation (3.3.10) should be replaced by u˜ θ, min{θ1 , θ2 }; x , where u˜ is given by ˜ f, g; x) = u(

f u( f ; x) − gu(g; x) . f −g

(3.3.13)

Since u˜ is not smooth in general, it cannot be expected that a smooth solution of the adjoint equation (3.3.11) exists. However, the proof can be extended by approximating θ1 , θ2 and θ by sequences of smooth functions θ1,n n∈N , θ2,n n∈N and θ,n n∈N , respectively, and then solve the linear partial differential equation ∂φn + u˜ θ,n , θ j,n · ∇φ j,n = ψ, ∂t

(3.3.14)

for φ j,n ∈ C01 (Ω j × R), for j = 1, 2. If this problem has a solution, then ∞Z

Z 0

Ωj

Z θ − θ j ψdΩdt =

∞Z

Ωj

0

Z Ωj

=

∞

Z 0

θ − θj

∂φn ∂t

dΩ +

θ − θ j u˜ θ,n , θ j,n · ∇φn dΩ dt

Z

− θu(θ) − θ j u θ j · ∇φn dΩ + Z Ω θ − θ j u˜ θ,n , θ j,n · ∇φn dΩ dt Ω

=

Z 0

∞Z θ − θ j u˜ θ,n , θ j,n − u˜ θ, θ j · ∇φn dΩdt. (3.3.15) Ω


61

Thus θ = θ j almost everywhere in Ω j × [0, ∞), if the sequences θ1,n n∈N , θ2,n n∈N and θ,n n∈N converge to the solutions θ1 , θ2 and θ, respectively, locally in L1 Ω × [0, ∞) as n → ∞, provided k∇φn k and supp φn are uniformly bounded in n ∈ N. The details of the proof are rather lengthy and are omitted here. Instead the reader is referred to Ref. [58, 143]. To completely avoid re-meshing algorithms in the numerical model for stretch blow moulding, also the surface of the stretch rod is described by a level set function. In this case the spatial domain for the level set problems is extended to Ω ∪ Σr . The level set problem for the stretch rod moving with constant velocity ur = Vr ez is trivial, as θr (x, t) = θr,0 (x + ur min{t − tstart , tstretch − tstart }).

(3.3.16)

Here θr (x, t) is the level set function corresponding to the stretch rod at position x and time t, tstretch is the stretch time, tstart is the time at which the stretch rod starts moving and θr,0 (x) := θr (x, 0).

(3.3.17)

The motion of the stretch rod is also described by the Stokes flow equations (2.7.15) in Σr . For the stretch rod an artificial viscosity µr is imposed, which is much larger than the viscosity of PET, so that the strain rate in the rod is negligible. A disadvantage of modelling the motion of the stretch rod by a level set function is that automatically a no-slip condition is imposed on the stretch rod boundary, which is not always realistic or desirable. One way to allow some slip is to define a gradual transition of the viscosity over the stretch rod boundary. An example is given in § 4.3. One of the difficulties encountered in LSMs is maintaining the desired shape of the level set function. The flow velocity does not preserve the signed distance property, but may instead considerably distort and stretch the shape of the function, which eventually leads to additional numerical difficulties [36,174]. To avoid this the evolution of the level set function is stopped at a certain point in time to rebuild the signed distance function. This process is referred to as re-initialisation. There are several ways to accomplish this. One approach is to solve the partial differential equation [149, 174, 188] ∂θ˜ ˜ − 1 = 0, + sign(θ˜0 ) k∇θk 2 ∂τ

in Ω × Υ,

(3.3.18)

62

3 Numerical Methods

˜ 0) = θ˜0 := θ(x, t), to steady state. If properly implemented the function θ˜ conwith θ(x, verges rapidly to the signed distance function around the interface [149, 188]. However, the technique does not properly preserve the location of the interface [174, 186]. This problem was fixed in Ref. [186] by adding a constraint to (3.3.18) that enforces mass conservation within the grid cells. This re-initialisation technique has appeared to be quite successful in practice. Another approach is to solve the Eikonal equation, k∇θk2 = 1,

(3.3.19)

given θ = 0 on Γf , using Fast Marching Methods (FMMs) [37, 173, 174]. FMMs build the solution outward starting from a narrow band around the interface and subsequently marching along the grid points. Depending on the implementation FMMs can be extremely computationally efficient. For the results in this thesis a triangulated FMM is used as a re-initialisation algorithm [104, 173]. The method assumes that the level set values at the nodes within a narrow band around the interface are the correct signed distance values and then builds a signed distance function outward by marching along the nodes. Figure 3.4 shows the algorithmic representation of the FMM for a two-dimensional mesh. The curve represents a two-fluid interface. The algorithm is initialised by computing initial values for distance function d in all nodes adjacent to the interface φ−1 (0). A second order accurate initialisation procedure is discussed in Ref. [37]. The nodes adjacent to the interface, to which the initial values are assigned, are tagged as proximate and are colour marked as dark grey in Fig. 3.4(a). The proximate nodes are the first nodes to be added to the set of accepted nodes, which contains all nodes of which the corresponding values are accepted as a distance value. The remaining nodes are initially tagged as distant, which are colour marked as black in Fig. 3.4, and are not (yet) of interest. The nodes adjacent to the set of accepted nodes are candidates to be added to either the set of accepted nodes or the set of distant nodes and are the trial nodes, which are colour marked as medium grey in Fig. 3.4(b). These nodes have a trial value assigned to them that might not yet be the correct distance value. When the values of all trial nodes have been updated, the trial nodes are removed from the set of trial nodes and added to the set of accepted nodes (see Fig. 3.4(c)). If multiple values are assigned to a trial node, the smallest one holds. This procedure repeats itself until the set of accepted nodes contains all nodes. Consider an element in Fig. 3.4(c) with one trial node C and two accepted nodes A and B. If the mesh is structured as in Fig. 3.4 and the angle between edges AB and AC


63

Interface

Interface

(a) Proximate and initial distant nodes

proximate: distant: trial: accepted:

B

A

(b) Trial nodes in first iteration

C

correct value by assumption not yet considered value might not be correct value updated by FMM

Interface

(c) Trial and accepted nodes in second iteration

(d) Legend

Figure 3.4: Fast Marching Methods

is right, then the distance value dC at node C is the solution of the quadratic equation

dC − dA

2

+ dB − dA

2

= dC − dB 2 .

(3.3.20)

Suppose the mesh is not structured, but the triangles have different edges. Figure 3.5 shows a triangle with angles α, β, γ and edge lengths a, b, c. The interface is approximated by a line l such that the distance from nodes A and B to l is equal to the approximate distance to the interface, that is dA and dB , respectively. Suppose dB ≤ dA . Then the angle δ between AB and l is determined by sin(δ) =

dA −dB c .

As a result distance value

dC can be computed as dC = a sin(δ + β) + dB .

(3.3.21)

64

3 Numerical Methods

l

dB

dA

δ

dA-dB

β

B

c

A α a b γ

C

Figure 3.5: Triangulated Fast Marching Method for an unstructured mesh

The line through C with the shortest distance to l from C should intersect the triangle. Therefore the following requirement has to be satisfied: 0≤a

cos(δ + β) ≤ c. cos(δ)

(3.3.22)

If (3.3.22) is not fulfilled, the update is performed by taking dC = min dA + b, dB + a .

(3.3.23)

It is also possible that only one node is accepted. If for example only A is accepted, then the values at B and C can be computed as dB = dA + c and dC = dA + b.

3.4

Solution Methods

For sufficient accuracy the finite element mesh should consist of an adequate number of elements. Consequently, a large-scale discretised system of equations may have to be solved. Therefore, efficient solvers are required to save computer memory and computational time. If the discretised system of equations is not extremely large-scale an efficient direct solver is used, e.g. PARDISO [162–164]. On the other hand, in the simulation of blow moulding processes often rather fine meshes have to be used to obtain sufficient accuracy, so that one has to resort to iterative solvers. In this case BiCGStab

3.4 Solution Methods

65

is used [196], with geometric multi-grid as a pre-conditioner. The multi-grid method uses V-cycles, incomplete LU factorisation as pre- and post-smoothers and a robust direct solver as a coarse grid solver. Initially, a drop tolerance of 10−4 is used for the incomplete LU factorisation. If no convergence is achieved for the given drop tolerance, the drop tolerance is decreased, until convergence is reached. If the solver still does not converge after decreasing the drop tolerance several times, an out-of-core direct solver is used. If no multi-grid algorithm is available, an incomplete LU factorisation can be used as a pre-conditioner, but one has to take care to find a suitable drop tolerance and an efficient minimum fill reordering algorithm, such as the minimum degree algorithm, Sloan algorithm or (reverse) Cuthill Mc Kee algorithm (see e.g. Ref. [60, 117, 157]). A flow chart of the solution process is given in Fig. 3.6. The solver performance for flow problems in forming processes (in particular pressing) is discussed in detail in Ref. [80]. Finite element discretisation of the Stokes flow equations is generally known as one of the most prominent sources of indefinite linear systems [66]. They are characterised by a large zero-block on the diagonal in the right-bottom of the stiffness matrix, which may lead to complications in solving the system. Essentially, there are three well-known methods to deal with this type of problem (see also Ref. [47, 80]). Penalty Function Methods The continuity equation is perturbed by a small factor ε of the pressure: ∇ · u + εp = 0.

(3.4.1)

It can be proven that the solution of the perturbed system approximates the original solution, provided that ε is small enough [47]. Advantages of the penalty function method are that the size of the discretised system of equations can be reduced by decoupling the linear systems for the continuity equation and the momentum equation and no partial pivoting is required [47,170,172]. However, some drawbacks are that an unknown error εp is introduced and the inverse parameter 1 ε

outweighs other entries in the stiffness matrix, causing a significant increase

in the condition number. Consequently, the penalty function is not suitable for large-scaled problems. Methods of Divergence-Free Basis Functions The basis functions are chosen, such that the continuity equation is automatically satisfied by imposing an additional incom-

66

3 Numerical Methods

Start solving

yes

Large-scale system?

Iterative solver (BiCGstab) Multigrid available?

yes (Geometric) Multigrid preconditioning

Direct solver (PARDISO)

no

ILU factorisation preconditioning

ILU factorisation preand post-smoothing

no

decrease drop tolerance for ILU factorisation

no Convergence reached?

no

yes

Min. drop tol. reached?

yes Out-of-core direct solver

End solving

Figure 3.6: Flow chart for solving the discretised system of equations

pressibility condition. In this way the number of unknowns is strongly reduced. However, the method can only be applied for specific types of elements [47], such as Crouzeix-Raviart [45] and Argyris [7, 8]. Furthermore, some complications may arise in the application of this method, because basis functions that satisfy the incompressibility condition have to be constructed, boundary conditions must be transformed and the solution is to be transformed back to its original degrees of freedom [47]. Integrated Methods The integrated method does not uncouple the velocity and pressure unknowns, which is the case in the penalty function method and the method of divergence-free basis functions. Often, partial pivoting is applied to deal with the zero diagonal elements. However, this can change the profile or bandwidth

3.4 Solution Methods

67

of the coefficient matrix. The alternative is to reorder the unknowns, such that the velocity unknowns are calculated before the pressure unknowns in the same nodes. For the reordering one of the aforementioned minimum fill reordering algorithms can be used. A disadvantage of integrated methods compared to the other two methods is the larger computational time, because of the coupling between the velocity and pressure unknowns. In this thesis integrated methods are used, either with partial pivoting or reordering (depending on the software used). Methods of divergence-free basis functions are also interesting for the applications in this thesis, because of the uncoupling of velocity and pressure unknowns, and may be tested in the future. The convection-diffusion equation (2.7.22c) is stabilised by means of a streamlineupwind Petrov-Galerkin (SUPG) method [20, 99]. The upwind parameter is defined by [75] hξ u · ∇bi ξ˜ = , 2 kuk2

in ei .

(3.4.2)

Here h is the width of the element in flow direction, bi is the ith basis function, ei is the ith element and ξ is a tuning parameter. Usually, the tuning parameter is not too different from the classical case ξ = 1 [71]. In Ref. [75] a modified tuning parameter is used for modelling the heat transfer in glass blow moulding (see also Ref. [175]).

Chapter 4

Blow Moulding Results

The efficiency of the numerical methods presented in Chapter 3 is illustrated by means of several examples in glass blow moulding and PET stretch blow moulding. To this end the corresponding numerical model is implemented in a FEM library. Then the mathematical problem is solved numerically to visualise the solution at successive points in time. The input information for the model typically includes a preform shape, a mould geometry, an initial temperature distribution and an inlet air pressure. The numerical results are compared with each other or with available data provided by industry and validated by verifying volume conservation. All results presented are 2D axial-symmetrical.

4.1

Level Set Methods and Fast Marching Methods

This section discusses numerical results of LSMs and FMMs for a simple glass blow moulding process in a 2D axial-symmetrical ellipsoidal mould. The model has been implemented in COMSOL 3.5 with MATLAB [40]. Figure 4.1 shows the unstructured mesh. Figure 4.2 shows the spherical preform. Here glass is purple and air is blue. An

70

4 Blow Moulding Results

inlet air pressure is applied on the upper boundary to blow the preform into the mould shape. For simplicity a constant glass viscosity with value µg = 103 is considered in this example, i.e. the flow is adiabatic. Figure 4.3 shows the level sets of the preform.

Figure 4.1: Unstructured triangular mesh.

Figure 4.2: Preform.

Figure 4.3: Level sets of preform.

Let both interfaces be represented by one level set function (see § 3.3). Solving one level set problem for two interfaces can lead to difficulties in the re-initialisation process, since some of the trial nodes are located right between the proximate elements. Moreover, as the mesh is unstructured and the mesh density is not uniform, FMMs do not march everywhere over the same distance per iteration. As a result the algorithm may cross the centre line between the interfaces while marching from one interface, while it has not reached the centre line from the other interface. Consequently, in some trial nodes the distance to the farther interface is computed. To overcome this, the old level set values between the interfaces are considered as distance values to the nearest interface in order to indicate whether the centre line is reached. If the old distance values are decreasing farther away from one interface, the corresponding trial values are not updated and the algorithm does not march any further from here. These trial values are instead updated in the Fast Marching process that starts from the other interface. Figure 4.4 illustrates the procedure. The Fast Marching algorithm first attempts to update the level set value in trial node C given the level set values in accepted nodes A and B, while marching from interface I. In this example the level set value in node A is θA = 1.4 and the old level set value in node C is θC = 1.3, which represents the distance to the nearest interface in the previous time step. Since the level set value decreases, the Fast Marching Method does not update the trial value in node C and does not add C to

4.1 Level Set Methods and Fast Marching Methods

71

the set of accepted nodes. In this way it is avoided that the distance values of P and Q are updated from node C. The triangle is labelled as visited, so the algorithm does not attempt to update the node C again from node A and B. In the next iteration the value in C is updated from nodes P and Q, while marching from interface J. In this example the distance from node C to interface J is dC = 1.5, which is smaller than the distance from C to interface I, dC = 1.8.

θB=1.3 B

dC=1.8 A

dC=1.5 C

θA=1.4 θC=1.3

Q

P

Interface I

Interface J

Figure 4.4: Updating a trial node C centred between interfaces I and J from accepted nodes A and B or from accepted nodes P and Q by FMMs

Figure 4.5, 4.6 and 4.7 show results for the simple glass blow moulding process at time t = 0.55 with and without re-initialisation by means of FMMs. Figure 4.5 shows the mould shape computed without re-initialisation. Compared to the signed distance levels of the preform in Fig. 4.3, the level sets of the mould shape in Fig. 4.5(b) have been significantly distorted and lost their physical meaning. Figure 4.6 shows the mould shape computed with re-initialisation over the whole domain, so that the level sets represent the signed distance to the interfaces. Finally, Figure 4.7 shows the mould shape computed with re-initialisation over a distance of 0.25 from the interfaces. In this procedure the absolute distance function is initially set to a constant value of 0.25 in all trial and distant nodes. Subsequently, the FMM is applied until trial values greater than the distance upperbound of 0.25 are computed. The algorithm does not update these trial values and stops marching any farther away from these nodes. Since the elements that are farther away from the interfaces are not updated, a significant amount of computational time is

72


(a) Glass-air interfaces

(b) Level sets

Figure 4.5: Mould shape at t = 0.55 computed without re-initialisation


(b) Signed distance levels

Figure 4.6: Mould shape at t = 0.55 computed with re-initialisation over the whole domain


(b) Signed distance levels

Figure 4.7: Mould shape at t = 0.55 computed with re-initialisation over a distance of 0.25 from the interfaces


73

saved. The maximum distance value should be large enough to have an accurate signed distance representation close to the interfaces, which depends on the flow velocity, mesh size and time step. Typically, a few times the mesh size is sufficient. Table 4.1 shows the difference in CPU time of one time step with re-initialisation by FMMs over different distances from the interfaces. It appears that the savings in CPU time are remarkable. From these results it can be concluded that FMMs are successful for practical use. Table 4.1: CPU times of one time step with re-initialisation by FMMs over different distances from the interfaces

Max distance for FMMs ∞ 0.5 0.33 0.25 0.1 0

CPU time 9.0152 7.7996 7.1994 6.9770 5.7039 4.1414

Next the computational performance of the single level set problem and the minimum level set problem is compared for the simple glass blow moulding process. The single level set problem is preferable since it saves computational time. Figure 4.8 shows the ellipsoidal mould shape at t = 0.55 computed by solving one level set problem and Figure 4.9 shows the same mould shape computed by solving two level set problems. The results are quite similar. Although it does not immediately appear from the figures, the difference between the signed distance functions is largest at the centre between the interfaces. This can be explained by the discontinuity in the derivative on the centre line and the fix for the re-initialisation algorithm for the single level set problem discussed previously. For comparison a CPU time of 806.0 seconds and a mass gain of 0.87 percent was obtained for the minimum level set problem, while a CPU time of 587.0 seconds and a mass gain of 0.20 percent was obtained for the single level set problem. Since the level set function is re-initialised over a relatively small distance from the interfaces, there is no significant CPU time difference for re-initialisation between the problems. Although solving the single level set problem seems to be beneficial, solving the minimum level set problem becomes advantageous as the melt becomes thinner. Table 4.2 shows CPU times and mass gains for the numerical simulation of the simple glass blow

74


moulding process with the single and minimum level set problem for different preform thicknesses and time steps. For the time integration an IDA scheme has been used [92], which employs a variable-order variable-step-size BDF method. While solving the single level set problem is more efficient if the melt is sufficiently thick, the mass difference between the preform and container rapidly increases as the preform becomes thinner and eventually the glass breaks. On the other hand, no complications arise if the minimum level set problem is solved, until the preform thickness is decreased further to 0.050. If the single level set problem is solved and the melt is too thin, the level set function can even become unstable, which manifests itself in the form of rapidly growing oscillations in the simulation. Obviously, the breaking is the result of a lack of accuracy. Since the gradient of the single level set function is undefined in the centre of the glass layer, the computation of the gradient of the single level set function around the centre requires more accuracy than for the minimum level set function. Consequently, the glass breaks more easily for the single level set function. The breaking can be avoided by decreasing the time step or the mesh size. However, then solving the single level set problem becomes computationally more expensive than solving the minimum level set problem with the original time step and mesh size, whereas the mass difference is still larger. Table 4.2 shows results for different time steps. In all the cases for the smaller time steps solving two level set problems gives good mass conservation. The significant mass difference, breaking and resulting unstable behaviour are severe drawbacks of the single level set problem, since blow moulding generally produces thin containers. Therefore, the minimum level set problem is preferable in view of the application.


Figure 4.8: Mould shape at t = 0.55 computed by solving the single level set problem

75

Figure 4.9: Mould shape at t = 0.55 computed by solving the minimum level set problem

Table 4.2: CPU time and mass gain for solving the single and minimum level set problems for two moving interfaces for different preform thicknesses and time steps.

preform thickness

time step

0.250

5 · 10−3

0.225

5 · 10−3

0.200

5 · 10−3

0.175

5 · 10−3

0.150

5 · 10−3

0.125

5 · 10−3

0.100

5 · 10−3

0.075

5 · 10−3

0.050

5 · 10−3

0.125 0.100 0.125

2.5 · 10−3 2.5 · 10−3 3.75 · 10−3

# level set problems one two one two one two one two one two one two one two one two one two one one one

CPU time 593.8 806.0 572.3 833.5 549.0 808.8 566.2 674.0 534.9 696.1 532.1 696.1 500.1 706.5 463.0 712.5 476.8 787.2 1184.3 1146.5 791.7

% mass gain 0.88 0.87 0.17 0.05 -0.23 -0.05 0.02 -0.02 -1.28 -0.94 -16.6 -0.54 -46.2 -2.18 -31.0 -3.70 320.8 -60.25 1.67 -10.00 -3.51

(breaks!) (breaks!) (breaks!) (unstable!) (breaks!)

76


4.2

Glass Blow-Blow Moulding

This section shows numerical results for the blow-blow moulding process for an axialsymmetrical beer bottle [76, 77, 87]. The outer surface of the preform in the final blow stage is used as the mould shape for the counter blow stage. The dimensional height of the bottle is 21.0 cm, the dimensional radius of the bottle is 1.27 cm at the neck to 2.65 cm at the widest part. Both process stages stop at a given end time. The propagation of the interfaces and the temperature distribution during the blow stages are visualised. Two level set functions are used to capture the moving interfaces; one for the outer interface and one for the inner interface. The SEPRAN finite element package has been used for computing the results in this section [171]. Mini-elements and first order Lagrange elements have been used for the spatial discretisation of the flow problem and the energy/level set problem, respectively (see § 3.1). Graphs of the finite element mesh can be found in Fig. 4.10. The Euler Implicit scheme has been used for the time integration. The resulting equations have been solved using the BiCGstab method with incomplete LU factorisation.

2

1

(a)

(b)

Figure 4.10: Finite element mesh and interface representation: (a) preform (b) bottle.

It is common practice in glass manufacturing that there is a temperature variation in the glass preform, which results in wall thickness variations [76–78]. Therefore, a nonuniform initial temperature distribution was considered for the final blow stage. The

4.2 Glass Blow-Blow Moulding

77

temperature data was provided by industry and approximated by a function by means of least squares fitting, yielding 4 X T 0 (r, z) = T 0 1 − λz e−κz z + (−1)k T k,r rk + T k,z zk .

(4.2.1)

k=1

Here r and z are the radial and axial coordinates, respectively, with z = 0m corresponding to the lowest point of the mould. The coefficients are given in Tab. 4.3. Figure 4.11 compares the temperature data with the least squares approximation in (4.2.1). The root mean square error of the approximation is 15.08 ◦ C. Coefficient [Unit] T 0 [K] T 1,r [Km−1 ] T 2,r [Km−2 ] T 3,r [Km−3 ] T 4,r [Km−4 ] T 1,z [Km−1 ] T 2,z [Km−2 ] T 3,z [Km−3 ] T 4,z [Km−4 ] λz [−] κz [m−1 ]

value 3.2104 · 103 6.9036 · 104 1.7209 · 107 1.3748 · 109 3.2284 · 1010 4.4453 · 104 3.0587 · 105 6.9616 · 105 0.0 3.3887 · 105 2.0 · 102

Table 4.3: Coefficients in interpolation formula for the non-uniform temperature distribution.

First consider the simulation of the counter blow stage. The initial temperature of the mould and the air in the counter blow stage is 500 ◦ C. The gob has a uniform initial temperature of 1, 175 ◦ C. The inlet air pressure is 138kPa. The time duration of the counter blow stage is 1.75s. Figure 4.12 visualises the evolution of the glass domain (red) and the air domain (blue). Figure 4.13 shows the temperature distribution. Next consider the simulation of the final blow stage. First the preform is left to sag due to gravity long enough that it just does not touch the bottom of the mould. The sag time used for the simulations is 0.3s. Then pressurised air is blown into the mould with an inlet air pressure of 3kPa. The pressure is much lower than for the counter blow stage, because the preform is relatively thin and therefore can easily break. The time duration of the second blow stage is 1.025s. Figure 4.14 visualises the evolution of the glass

78


0 1100

1050 −50 1000

950 −100 900

850 −150 0

2

4

6

8

10

12

14

16

18

20

◦

(a) Data [ C]

0

1150

1100

1050

−50

1000

950 −100 900

850 −150 0

2

4

6

8

10

12

14

16

18

20

◦

(b) Approximation in data points [ C]

Figure 4.11: Temperature distribution of preform.

domain (red) and the air domain (blue). Figure 4.15 shows the temperature distribution.

Figure 4.16 compares the thickness of the final product in the blow-blow simulation with the thickness data provided by industry. The signed distance representation of the level set function is used to measure the glass thickness in the simulation.


t=0.0s

0.25s

79

0.75s

1.75s

Figure 4.12: Evolution of the glass domain during the counter blow [76]. Glass is red, air is blue.

Figure 4.13: Temperature distribution during the counter blow [76].

In the comparison it should be taken into account that errors in the wall thickness are prone to both modelling and measurement inaccuracies. In order to assess the results, the error in the measurement of the initial glass temperature distribution should be estimated. The glass temperature data was provided by industry, but it is unclear how it was obtained. The most common methods to measure the temperature are by means of thermocouples and single-wavelength pyrometers (e.g. Ref. [137]). In the forming of glass containers thermocouples are impractical [136, 194]. With a pyrometer it is

80


t=0.0s

0.3s

0.725s

1.025s

Figure 4.14: Evolution of the glass domain during the final blow [76]. Glass is red, air is blue.

Figure 4.15: Temperature distribution during the final blow [76].

only possible to determine the surface temperature. Literature reports several attempts to determine the temperature inside the glass, but this is usually coupled to significant errors [13,194,195]. In Ref. [194] it is stated that the accuracy in the measurement of the surface temperature by a pyrometer is typically around 5 ◦ C. Reconstruction of the temperature distribution in the glass from the measured spectral intensity typically results in


81

−3

x 10

−3

x 10

4

0.2

0.2

0.18

0.18

3.5

0.16

4

0.16

0.14

3.5

0.14

3

0.12

0.12

0.1

3

0.1

2.5

0.08

2.5

0.08

0.06

2

0.06

0.04

2

0.04 1.5

0.02

0.02

0 0 0.02

0 0 0.02

(a) Simulation

1.5

(b) Measurements

Figure 4.16: Final product thickness distribution. −3

−3

x 10 0.2

x 10 0.2

4

4

0.18

0.18

0.16

0.16

3.5

3.5

0.14

0.14

0.12

0.12

3

3

0.1

0.1 2.5

0.08

0.08 2.5

0.06

0.06 2

0.04

0.04 2

0.02

0.02

0 0 0.02 ◦

1.5

0 0 0.02 ◦

(a) Decrease of 5 C in the initial glass

(b) Increase of 5 C in the initial glass tem-

temperature

perature

Figure 4.17: Final thickness distribution in the simulation for different temperature distributions.

82


an error of the order 10 ◦ C. Figure 4.17 shows that a temperature difference of 10 ◦ C results in a wall thickness variation of nearly the same order as the difference in wall thickness between Fig. 4.16(a) and Fig. 4.16(b). Note that the error of the least squares approximation (4.2.1) is of the same order of magnitude as the expected error due to measurements and reconstruction of the temperature distribution in the glass. A more accurate approximation of the temperature data does not give any significant improvement. From this it can be concluded that the computed thickness in the simulation is as near to the measured thickness as can be expected. Thus, taking the expected error into account, the wall thickness distribution in the simulation is in good agreement with the measured thickness distribution. In order to further assess the accuracy of the numerical results, the glass volume conservation is used. Figure 4.18 shows the percentage volume change in time. The volume change has a maximum of 1.5%. The volume conservation can be further improved using smaller time steps, higher mesh quality and a second order accurate discretisation schemes such as Crank-Nicholson. This phenomenon was also observed in Ref. [109] and for glass pressing in Ref. [111]. Good volume conservation was also observed in Ref. [75].

% glass volume change

2.5 2 1.5 1 0.5 0 0

0.5

1 time [s]

1.5

2

Figure 4.18: Volume conservation of the consecutive first and second blow stages using the Euler Implicit time discretisation scheme with time step 10−4 .

4.3 PET Stretch Blow Moulding

4.3

83

PET Stretch Blow Moulding

This section shows numerical results for the stretch blow moulding process for an axialsymmetrical PET water bottle [83, 84]. The dimensional height of the bottle is 21.6 cm, the dimensional radius of the bottle is 1.5 cm at the neck to 3.15 cm at the widest part. The propagation of the interfaces during the process is visualised. The 2D axial-symmetrical finite element model has been implemented in COMSOL 3.5 with MATLAB. Each time step the system of equations is solved using COMSOL 3.5, while the re-initialisation of the level set functions is done in MATLAB between successive time steps. The level set problems are solved using the convection-diffusion application mode in COMSOL 3.5. First order elements have been used for the spatial discretisation: Mini-elements for the flow problem and Lagrange elements for the level set problems. Figure 4.19 shows a typical mesh for the stretch blow moulding simulation. For the temporal discretisation the IDA scheme has been used (see § 4.1). The discretised system of equations is solved as described in § 3.4.

21.6cm

1.5cm

3.15cm

Figure 4.19: Typical mesh for PET bottle

Three level set functions are used to describe the evolution of the moving interfaces: θ1 for the inner polymer-air interface, θ2 for the outer polymer-air interface and θr for the stretch rod surface. Level set functions θ1 and θ2 are found by solving level set

84


problem (3.3.3) and θr is given by (3.3.16). Figure 4.20 gives an overview of the material domains and the corresponding signs of the level set functions. The continuum domains are defined as follows: air

:

max{min{θ1 , θ2 }, θr } < 0,

polymer

:

min{θ1 , θ2 } > 0,

stretch rod

:

θr > 0.

θr = 0

stretch rod

The viscosity value µr = 1012 has been used for the stretch rod.

air

air

θ1 < 0 θ1 > 0 θ2 > 0

θ2 < 0

θr>0 θ r < 0

θ1 = 0

polymer θ2 = 0

Figure 4.20: Sign of the level set functions

Figure 4.21 shows the preform and initial position of the rod. The stretch rod is initially positioned at a height H0 of 9.72 cm from the bottle mouth. The radius r0 of the stretch rod is 0.375 cm. At t = tstart the rod starts moving from this position with constant speed Vr . The level set function for the stretch rod surface in terms of 2D axial-symmetrical coordinates (r, z) is      r,q θr (r, z, t) = r0 −     r2 + z˜2 (z, t),

if z˜(z, t) ≥ 0, if z˜(z, t) < 0,

(4.3.1)

where z˜(z, t) = z + Vr min{t − tstart , tstretch − tstart } − H0 .

(4.3.2)


85

The stretch time tstretch can either be given manually or is determined by the numerical model. In the latter case the stretching is stopped when the polymer hits the bottom of the mould. This method is used for the simulations.

9.72cm

0.25cm

stretch rod PET air

Figure 4.21: Preform and initial position of the rod for the simulation

In Ref. [199, 201] the material parameters m = 0.54, k = 292529 are given for the uniaxial tensile stress test (2.3.18). Figure 4.22 shows the viscosity-strain rate relation given by (2.3.20) in log-log scale. Figure 4.23 shows results of the stretch blow moulding simulations for a PET bottle at different times for inlet pressure pin = 1 MPa. First only air is blown inside without stretching. Then at t = 0.1 s the stretch rod starts moving with speed Vr = 0.45m s−1 . µ [kg m−1 s−1 ]

10

10

6

5

4

10 −2 10

−1

10

0

10

ε¯˙ [s−1 ]

1

10

10

2

Figure 4.22: Non-newtonian viscosity for uniaxial tensile stress test

86


The white arrows depict the flow velocity vectors. As to be expected in stretch blow moulding, the polymer form first bulges at the top, while the bottom is stretched, then the bulging gradually extends to the bottom. The bottle in Fig. 4.23(f) has an almost uniform thickness, with a minimum of approximately 1 mm at the bottom part and a

(a) t = 0.1s

(b) t = 0.2s

(c) t = 0.3s

(d) t = 0.4s

(e) t = 0.5s

(f) t = 0.62s

Figure 4.23: Stretch blow moulding process for PET bottle with pin = 1 MPa and Vr = 0.45m s−1 at different times.


87

maximum of slightly more than 3 mm at the neck part. This matches with the thickness profile of a realistic water bottle, which has an almost uniform thickness around 1.5 mm. In order to assess the accuracy of the numerical results presented, the PET volume conservation is verified. It can be expected that the volume change is comparable with the glass blow-blow moulding results, except that the geometry of the PET bottle is more complex. Figure 4.24 shows the percentage volume change in time. The volume change has a maximum of 2.5%. As for the glass blow-blow moulding results, the volume conservation can be further improved using smaller time steps and higher mesh quality.

% glass volume change

1 0.5 0 −0.5 −1 −1.5 −2 0

0.1

0.2

0.3

0.4

0.5

0.6

time [s]

Figure 4.24: Volume conservation of the stretch blow moulding process for the PET bottle using the IDA time discretisation scheme with time step 5 · 10−4 .

Unfortunately, no experimental data is available for comparison. It can be expected that the accuracy of the initial temperature measurement contributes less to the error than in glass blow-blow moulding. On the other hand, the modelling error is likely to be somewhat larger because of omitting stress relaxation.

Chapter 5

Mathematical Analysis of the Inverse Problem

The previous chapters dealt with the forward problem. The forward problem is to find the final container that is formed from a given preform under certain operating conditions. In practice often a container with a certain wall thickness distribution is desired. Then the corresponding initial conditions, such as the shape of the preform and the initial temperature distribution, are sought for in order to produce a container with exactly this thickness distribution. Thus, an inverse problem is considered, that is, the initial conditions are to be determined, given the thickness distribution of the final container.

5.1

Mathematical Formulation

The following problems are considered:

90

5 Mathematical Analysis of the Inverse Problem

The forward problem Find the shape of the container, given the shape of the preform. The forward problem represents the physical process of blowing the preform into the mould shape.

The inverse problem Find the shape of the preform, given the shape of the container. Usually, a container with a certain shape is desired, but in order to form this container first the preform needs to be known.

To solve these problems mathematically, a more formal mathematical formulation of the forward and inverse problem is required. Let Γ1,0 , Γ2,0 , Γi and Γm be the inner and outer preform surfaces, the inner container surface and the mould surface, respectively, as depicted in Fig. 5.1 for a 2D axial-symmetrical jar. The dotted lines represent the unknown surfaces. Since the mould surface is fixed, also one of the preform surfaces is fixed, thus mapping one surface onto another.

Gi

G1,0

Gm G2,0

Figure 5.1: Blow moulding problem description

The forward problem is formulated as follows: find the location of inner glass surfaces Γ1 at t = t∗ , such that Γ2 (t∗ ) = Γm and Γ2 (t) , Γm for t < t∗ , given Γ1 (0) = Γ1,0 and Γ2 (0) = Γ2,0 . In Chapter 2 a mathematical model for the forward problem was presented,

5.1 Mathematical Formulation

91

consisting of the system of equations ∇ · u = 0,

in Ω \ Γf ,

(5.1.1a)

∇ · T = g,

in Ω \ Γf ,

(5.1.1b)

cp

DT = ∇ · (λ∇T ) , Dt dx = u, dt

in Ω \ Γf × T,

(5.1.1c)

in T,

(5.1.1d)

with jump conditions ~u = 0

on Γf × T,

~T n = −γκn, ~λn · ∇T = 0,

(5.1.1e)

on Γf × T,

(5.1.1f)

on Γf × T,

(5.1.1g)

boundary conditions u · n = 0,

T n · t = 0,

u · n = 0,

(T n + βu) · t = 0,

T n = 0,

λ∇T

on Γs ,

(5.1.1h) on Γl,q ,

(5.1.1i)

on Γa,q ,

(5.1.1j)

T n = p0 n,

on Γo ,

(5.1.1k)

u = Vr ez ,

on Γr ,

(5.1.1l)

on ∂Ω × T,

(5.1.1m)

· n = 0,

and initial conditions T (x, 0) = T 0 (x),

x ∈ Ω,

(5.1.1n)

Γ1 (0) = Γ1,0 ,

(5.1.1o)

Γ2 (0) = Γ2,0 .

(5.1.1p)

The forward problem is solved by the numerical methods described in Chapter 3. The inverse problem is to find the location of the initial glass surface Γ2,0 = Γ2 (0), given the surface Γ2 at time t = t∗ , such that Γ2 (t∗ ) = Γm and Γ2 (t) , Γm for t < t∗ , and

92


given Γ1 (t∗ ) = Γi and Γ1 (0) = Γ1,0 . The inverse problem is more complicated than the forward problem. One might intuitively think of solving the forward problem backward in time. However, this would mean that also the convection-conduction equation has to be solved backward in time and the backward convection-conduction equation is ill-posed. Even if the process is considered adiabatic, complications arise in the application of the boundary conditions. Because of the boundary condition for an impenetrable wall on the mould, the backward velocity cannot be determined without posing any additional assumptions. An alternative, popular method for solving the inverse problem is numerical optimisation. Optimisation methods are discussed in Chapter 6.

5.2

Restrictions on the Mould Surface

A condition for the solvability of the inverse problem is that the mould surface is defined such that a mould shape can be blown in finite time for a given inlet pressure [86]. In the case the outer melt surface converges towards an equilibrium state, it cannot take the mould shape. This is possible if a force balance occurs between the surface tension and the pressure difference across the surface before the mould shape is reached. Figure 5.2 illustrates such a situation.

mould surface

R = p/γ

Figure 5.2: The outer surface reaches an equilibrium state before it reaches the mould shape

5.2 Restrictions on the Mould Surface

93

The constraint on the mould surface is such that the outer melt surface cannot reach an equilibrium state, before in coincides with the mould surface. In its most simple form the constraint on the mould surface can be formulated as: Rm > Req ,

(5.2.1)

where Req and Rm are the radii of curvature for the outer melt surface in presumed equilibrium and the mould, respectively. However, this relation is more complicated if the radius of curvature of the melt surface in equilibrium depends on the mould surface. A differential equation for the melt surface in equilibrium is derived [86]. In equilibrium u = 0, hence T = −pI , so (5.1.1b) and (5.1.1f) give ∇ p˜ = g˜ ,

on Γ2 ,

p˜ = γκ,

(5.2.2a)

on Γ2 .

(5.2.2b)

where p˜ = pl − pa and g˜ = gl − ga ≈ gl . Let g˜ = −Gez , then from 5.2.2 it follows that the surface in equilibrium is given by γκ = p˜ 0 − Gz,

on Γ2 .

(5.2.3)

where p˜ 0 = p˜ z=0 . For a two-dimensional axial-symmetrical surface of the form (r, z(r)), the curvature can be written in terms of r as − 3 1 κ(r) = z00 (r) 1 + z0 (r) 2 2 + . r

(5.2.4)

Substitution in (5.2.3) yields a differential equation for z as a function of r, γz00 = p˜ 0 − Gz −

3 γ 1 + z0 2 2 . r

(5.2.5)

γ r

is neglected. Then successive multiplication of

Assume that p˜ 0 γr ; that is, the term 0

(5.2.5) by z and integration results in the first order ordinary differential equation 0

z =

v u t

4γ2 2 − 1, p˜ 0 − Gz 2 + E

for some constant E to be determined later.

(5.2.6)

94


j

-H

j

z

rc

r

zc

L

Figure 5.3: Outer melt surface in equilibrium near a corner in the mould wall.

Consider a situation in which the outer surface approaches a corner in the mould wall (see Fig. 5.3). The corresponding boundary conditions are z(rc ) = −H,

z(L) = zc ,

z0 (rc ) = tan ϕ,

z0 (L) = cot ϕ,

(5.2.7)

where ϕ is the contact angle between the surface and the mould (see Fig. 5.3). For the boundary conditions it is assumed that the contact angle between the surface and the mould is known, which in most cases can be determined experimentally (see e.g. Ref. [147,204] for glass). The two extra boundary conditions are necessary to determine the unknowns rc and zc . Substitution of boundary conditions (5.2.7) into (5.2.6) yields E = 2γ cos ϕ + GH + p˜ 0 2 , ! q 1 p˜ 0 − 2γ sin ϕ − 14 π + GH + p˜ 0 2 . zc = G

(5.2.8) (5.2.9)

In order to find rc separation of variables is applied to (5.2.6) and the differential equation is integrated over the interval (rc , L), which leads to Zzc    −H

4γ2 ( p˜ 0 − Gz)2 + E

− 12  2 − 1 dz = L − rc .

(5.2.10)

5.2 Restrictions on the Mould Surface

95

−9.9965 −9.997

z

−9.998

−9.999

−10 0.9965

0.9975

0.9985

0.9995

1

r

Figure 5.4: Surface in equilibrium.

Finally, differential equation (5.2.6) with initial condition z(rc ) = −H is solved numerically. Figure 5.4 shows a surface in equilibrium near a corner in the mould wall for G = 2.5 · 10−2 , γ = 5.0 · 10−3 , p˜ 0 = 0.5, H = 10, L = 1 and ϕ = π9 . The walls in Fig. 5.4 are on the left and lower axes. The surface closely approaches the corner in the mould wall, which means that with a little smoothing the mould shape could be blown. The time scale at which an equilibrium state can be expected is estimated by a dimensional analysis [86]. Near an equilibrium state the velocity is close to zero, hence the pressure difference is of the same order of magnitude as the surface force, p˜ ∼ γκ. Hence, using the scaling of the dimensionless stress tensor in (2.6.3), V.

γκD . µ

(5.2.11)

Consider specific values for blow moulding µ ∼ 104 Pa s, κD ∼ 1, γ ∼ 0.5Pa m. Then the typical velocity near the equilibrium state is V = 5 · 10−5 m s−1 . With L ∼ 10−2 m, the typical time scale can be measured as τ∼

L ∼ 200s. V

(5.2.12)

In practice the time scale can be somewhat larger, because of the increasing viscosity and influence of stress relaxation. Since this time scale is much larger than the process time,

96


t~1s

t~5s t~25s t~100s zc equilibrium mould -H

z r

rc

mould

L

Figure 5.5: Time scales of melt surface converging to equilibrium state at different locations for various time scales.

the outer melt surface may still be at some distance from the equilibrium state at the end of the process and the curvature of the surface may still be much smaller. Therefore, the mould constraint should be much stricter, such that the melt surface will coincide with the mould surface within the process time. Thus, the curvature of the mould should be much smaller than the curvature of the melt surface in equilibrium.

5.3

Sensitivity

The inverse problem is sensitive to changes, if small changes in the inner container surface give relatively large changes in the corresponding preform surface. The inverse problem is expected to be sensitive to changes as the melt is stretched and becomes thinner during forming. Another reason because of which the inverse problem can be sensitive to changes is that the inner container surface converges towards an equilibrium state, e.g. because of surface tension. It is investigated whether the inner container surface converges towards an equilibrium state during blowing and whether this significantly influences the sensitivity of the inverse problem. Suppose the outer melt surface is on the mould wall. Equations (5.2.2) hold on Γ1 . Consider the following cases.

5.3 Sensitivity

97

• If g = 0, the curvature is constant, i.e. the surface in equilibrium is spherical. No equilibrium state can be reached in general, since the geometry of the container is not spherical and the material is incompressible. • If g = 0 and γ = 0, the system is in equilibrium if pl = pa irrespective of the shape. • If g = −ρgez , the curvature is linear in z. Because of the gravity the melt flows steadily to the bottom of the mould, while the surface will be curved. Because the melt does not slip along the mould wall, this equilibrium state cannot be reached and convergence to it will be extremely slow. • If g = −ρgez and γ = 0, the melt steadily flows to the bottom of the bottle, but the surface will be flat. Thus, in general the inner container surface converges towards an equilibrium shape, but the convergence will be extremely slow because the influence of gravity and surface tension is small and the melt does not slip along the mould wall. On the other hand, if the gravity and surface tension are neglected, the pressure difference converges to zero irrespective of the shape, so not all preforms will converge to the same mould shape. Since the only force acting on the surface is caused by the pressure difference, no significant change in the mould shape is expected. As changes in the blown mould shape occur relatively slow and the end time is chosen as small as possible, it can be assumed the convergence towards an equilibrium state does not cause the inverse problem to be excessively sensitive to changes. A sensitivity analysis is applied to the 2D axial-symmetrical, adiabatic case [86]. The equations for the propagation of interface j in spherical coordinates (R, ϕ) around the center of the mould opening are h∂ i 1 sin ϕ ψ(R j , ϕ) ϕ=ϕ j dt R j sin ϕ j ∂ϕ dϕ j i 1h∂ = uϕ (R j , ϕ j ) = − R ψ(R, ϕ j ) Rj . R=R j dt R j ∂R dR j

= uR (R j , ϕ j ) =

(5.3.1) (5.3.2)

where ψ is the stream function. The thickness of a perturbation in the shape is of interest. Therefore, the propagation of the perturbation on a spherical preform surface in radial

98


direction is estimated. Assume that dϕ j dt

≈ 0.

(5.3.3)

Then from equation (5.3.2) it follows that ψ(R j , ϕ j ) ≈

˜ j) ψ(ϕ Rj

,

(5.3.4)

for some function ψ˜ : [0, π2 ] → R. Subsequently, (5.3.1) gives 1 ˜ j ) + ψ˜ 0 (ϕ j ) t + R3j,0 3 R j (ϕ j , t) ≈ 3 cot ϕ j ψ(ϕ

(5.3.5)

Finally, a small perturbation 0 < 1 in preform radius R j,0 gives a perturbation in the container radius R˜ j (t) = R j (t) +

R2j,0 R2j (t)

+ O( 2 ),

(5.3.6)

and since the mass flow in polar direction is neglected R˜ 1 (t∗ ) = R1 (t∗ ) +

R2j,0 R2j (t∗ )

Note that the coefficient

+ O( 2 ).

R2j,0 R2j (t∗ )

(5.3.7)

is usually smallest near the symmetry axis. Figure 5.6

shows the preform and resulting container for the blowing of a 2D axial-symmetrical wine glass. Figure 5.6(a)-5.6(b) show results without perturbation and Figure 5.6(a)5.6(b) show results with perturbation at the symmetry axis marked by green circles. In this example the real inner radius of the glass mouth is 3.5 cm and the real height is 7 cm. The bump has a real thickness of 0.175 cm on the symmetry axis (where uϕ = 0). The preform has real outer radius R2,0 = 4.025 cm on the symmetry axis. Then the real thickness of the perturbation in the inner container surface is 0.058 cm - a decrease by factor 3 (see (5.3.7)). It can be concluded that the inverse problem is moderately sensitive to perturbations in the preform. Small changes in the designed container lead to slightly larger changes in the corresponding preform. Reversely, errors in the preform shape result in relatively small errors in the computed containers. Therefore, the error tolerance for solving the inverse problem can be set relatively large.

5.3 Sensitivity

99

(a) Original preform

(b) Original container

◦ ◦ (c) Preform with bump

(d) Container with bump

Figure 5.6: Blowing a wine glass with and without bump.

The sensitivity also has consequences for the finite element mesh distribution. Relatively large elements should be used around the preform surfaces, while small elements should be used near the mould surface. Such a mesh distribution was used to compute the numerical results in Chapter 4 as shown in Fig. 4.10 and Fig. 4.19. Approximation 5.3.7 can be used to estimate the difference in element size and the error tolerance for the inverse problem. By scaling the mesh distribution with the sensitivity, i.e. the spatial discretisation error is roughly proportional to the size of perturbations in the shape, the typical mesh size can be used as a uniform quantity to estimate the error.

100


5.4

Approximate Problem

In this section an analytical approximation of the Stokes flow problem is derived. This approximation is used to approximate and analyse the solution of the inverse problem and as an initial guess for iterative optimisation algorithms for solving the inverse problem. The approximation is based on two assumptions [81, 85]. Firstly, the melt viscosity is assumed to be constant. For glass this is reasonable if the Péclet number is large and the initial temperature is approximately uniform (see § 2.7). Secondly, the melt tends to flow in perpendicular direction to the layer, except for the region where the melt is close to the mould. In Figure 5.7(a) the flow velocity field is depicted for a nearly spherical preform and an ellipsoidal mould (see example in § 4.1). A explanation for this phenomenon is that there is a relatively large pressure difference between the two melt surfaces. This appears from the quantitative analysis in § 2.7 and can be seen in Fig. 5.7(b). Moreover, the high viscosity causes a laminar flow behaviour. Thus, it is fair to believe that the melt tends to flow in normal direction from one surface to the other. Furthermore, assume that the melt surfaces and the mould surface are concave around an origin, which is located at the centre of the mould opening in Fig. 5.7. Then it is also reasonable to assume that the melt surfaces evolve in spherically radial direction from this origin. This gives the following assumption:

(a) Streamlines

(b) Pressure approximation

Figure 5.7: Initial streamlines and approximate pressure for blowing a glass preform. The flow seems to be perpendicular to the glass surfaces, except close to the contact surface of the glass and the mould

5.4 Approximate Problem

101

dR = uR , dt dϕ R = uϕ ≈ 0, dt

(5.4.1) (5.4.2)

where (R, ϕ) are 2D axial-symmetrical spherical coordinates. This assumption is useful for analysis purposes, in which one is primarily interested in the outward evolution of the melt surfaces towards the mould wall. For this reason the assumption was also used in (5.3.3) to estimate the sensitivity. In summary, the following assumption is made: Assumption 5.4.1. 1. the viscosity is constant in time and space, 2. the polar velocity component is zero: uϕ = 0. The origin of the spherical coordinate system is not necessarily located at the centre of the mould opening. For a bottle the mould wall, hence the container surfaces, are partially convex at the neck part. Consequently, the melt is deformed in polar direction to fit the mould during blowing. A resolution is to start at a point in time at which the neck part is already covered [81, 85]. Then the origin can be chosen at a position lower than the mould opening, such that both the mould wall and the preform are concave with respect to the origin (see Fig. 5.8). The mould wall and the preform can still be slightly convex without excessively violating Assumption 5.4.1. With Assumption 5.4.1 it is possible to find a direct relation between the preform and the container [81, 85]. Volume conservation of the melt enclosed by the surfaces R1 (ϕ) and R2 (ϕ) between any two angles ϕ1 and ϕ2 in [0, π2 ] results in    ϕ   Zϕ2 RZ2 (ϕ)  Z 2  d  2 d   2 3 3 2π R sin ϕ dRdϕ = π  R2 (ϕ) − R1 (ϕ) sin ϕ dϕ = 0.(5.4.3)   dt  3 dt    ϕ1 R1 (ϕ)

ϕ1

With Assumption 5.4.1 it follows that ϕ  Z 2  Zϕ2 d  3 ∂ 3  3 R2 (ϕ) − R1 (ϕ) sin ϕ dϕ = R2 (ϕ) − R31 (ϕ) sin ϕ dϕ.  dt  ∂t  ϕ1

ϕ1

(5.4.4)

102


R φ

R φ

(a)

(b)

Figure 5.8: Different positions of the origin for a spherical coordinate system on the symmetry axis

Since this holds for any pair of angles ϕ1 , ϕ2 , it can be concluded that ∂ 3 R2 (ϕ) − R31 (ϕ) = 0, ∂t

0≤ϕ≤

π . 2

(5.4.5)

Let R1,0 (ϕ), R2,0 (ϕ), Ri (ϕ) and Rm (ϕ) denote the spherical radii of the preform surfaces, the inner container surface and the mould surface, respectively. Then R32,0 (ϕ) − R31,0 (ϕ) = R3m (ϕ) − R3i (ϕ),

0≤ϕ≤

π . 2

(5.4.6)

Thus, under Assumption 5.4.1, if either R1,0 (ϕ) or R2,0 (ϕ) is known, the other surface of the preform can be uniquely determined, provided that a container with the outer surface on the mould can be blown. Note that at this point no use was made of the assumption of constant viscosity, so approximation (5.4.6) can also be used if the viscosity is strongly non-Newtonian or temperature dependent. The assumption of constant viscosity is made to derive an approximation of the time dependent behaviour of the melt surfaces, but the relation between the mould shape and preform is independent of the viscosity. Next an analytical expression for the evolution of the melt surfaces during blowing is derived by solving the approximate mathematical problem. In view of Assumption 5.4.1


103

the dimensionless Stokes flow equations for the melt with constant viscosity in 2D axialsymmetrical, spherical coordinates (R, ϕ) is considered [1], thereby neglecting uϕ :   ∂uR  Re ∂p  1 ∂ 2 ∂uR 2uR 1 ∂  − =  2 R − 2 + 2 sin ϕ cos ϕ, ∂r ∂R ∂ϕ Fr R ∂R R R sin ϕ ∂ϕ 1 ∂p 2 ∂u Re R = 2 R+ sin ϕ, R ∂ϕ R ∂ϕ Fr 1 ∂ 2 R uR = 0. R2 ∂R

(5.4.7a) (5.4.7b) (5.4.7c)

From the continuity equation (5.4.7c) it follows that the the flow velocity has the following form, uR (R, ϕ) = v(ϕ)R−2 .

(5.4.8)

The resulting momentum equations are 00 −2  2 0 ∂ Re   R R cos ϕ = v + v cot ϕ R , p +  Fr  ∂R     R ∂ p + Re R cos ϕ = 2v0 R−2 , ∂ϕ

(5.4.9)

Fr

Integration with respect to R and ϕ, respectively, yields 1 00 Re v (ϕ) + v0 (ϕ) cot ϕ R−3 − R cos ϕ + Cϕ (ϕ), 3 Fr Re R cos ϕ + CR (R), = 2v(ϕ)R−3 − Fr

p(R, ϕ) = −

(5.4.10)

where Cϕ and CR are integration constants with respect to the corresponding integration variable. It follows that v00 + v0 cot ϕ + 6v = −ς2 = constant.

(5.4.11)

This equation has solution v(ϕ) = AvA (ϕ) + BvB (ϕ) + C,

(5.4.12)

104


with vA (ϕ) = 3 cos2 ϕ − 1 vB (ϕ) = 6 cos ϕ + 2 1 − 3 cos2 ϕ log |cscϕ + cot ϕ| C=−

ς2 . 6

(5.4.13) (5.4.14) (5.4.15)

To specify boundary conditions the part of the outer melt surface that is in contact with the mould at time t must be determined. In the most simple case an angle ϕ = ϕc (t) can be found, such that the melt surface is in contact with the mould for ϕ ≥ ϕc (t) and with air for 0 ≤ ϕ < ϕc (t) (see Fig. 5.9). Here ϕ = 0 corresponds to the symmetry axis and ϕ =

π 2

corresponds to the mould entrance. The angle ϕc is called expansion

angle. For ϕ ≥ ϕc (t) the flow velocity is zero; the surfaces do not evolve in time, but have already taken the final mould shape. The domain of interest is then bounded by ϕ = 0 and ϕ = ϕc (t). In this case the boundary conditions follow from axial-symmetry and no-slip: v0 (0) = 0,

v(ϕc ) = 0.

(5.4.16)

As a result A=

C 1 − 3 cos2 ϕc

,

B = 0.

(5.4.17)

The constant C can be found by imposing appropriate boundary conditions for the pressure. The more realistic case is more complicated. In most practical applications part of the melt will touch the bottom of the mould before the entire mould wall is covered. In this case two expansion angles ϕc,1 (t) and ϕc,2 (t) should be introduced. The resulting boundary conditions are v(ϕc,1 ) = 0,

v(ϕc,2 ) = 0.

(5.4.18)

In this case A and B depend on ϕc,1 and ϕc,2 : vB ϕc,1 − vB A=C vB ϕc,2 vA ϕc,1 − vB vA ϕc,2 − vA B=C vB ϕc,2 vA ϕc,1 − vB

ϕc,2

, ϕc,1 vA ϕc,2 ϕc,1 . ϕc,1 vA ϕc,2

(5.4.19) (5.4.20)


105

Ri

ϕ = 12 π

Rm ϕ = ϑ(t )

R1 R2

ϕ=0

Figure 5.9: Radii of melt surfaces and expansion angle ϑ.

Cases with more than two expansion angles are also possible. Figure 5.10 shows a case of four expansion angles. Since by Assumption 5.4.1 pairs of expansion angles ϕc,1 , ϕc,2 and ϕc,3 , ϕc,4 advance independently of each other along the mould surface, only the case of two expansion angles is of interest here.

f1

R φ f2 f3 f4

Figure 5.10: Four expansion angles

First consider the simple case with one expansion angle. To solve evolution equation

106


(5.4.1) the expansion is determined as a function of time. Let R(t) be the radius of a surface with initial condition R(0; ϕ) = R0 (ϕ). Substitution of (5.4.8) into (5.4.1) yields dR = v R−2 . dt

(5.4.21)

Separation of variables gives Zt

1 3 v ϕ, ϕc (τ) dτ = R (t; ϕ) − R30 (ϕ) . 3

(5.4.22)

0

Note that the flow velocity only changes in time as a function of the expansion angle. Therefore, R can be defined as a function of ϕ and ϕc in the same way as v. Furthermore, t ≡ t(ϕc ) can be seen as the time it takes for the melt to touch the mould at angle ϕ = ϕc , i.e. t is the inverse of ϕc . For convenience the integration variable is changed into ϕc , that is Zϕ0 − ϕc

dt 1 3 dα = R (ϕ, ϕc ) − R30 (ϕ) . v ϕ, α dα 3

(5.4.23)

Integration by parts leads to −t(ϕc )v ϕ, ϕc +

Zϕ0

t(α) ϕc

∂ 1 3 v ϕ, α dα = R (ϕ, ϕc ) − R30 (ϕ) . ∂α 3

(5.4.24)

By choosing ϕ = ϕc the integral equations for R = R1 , R2 become Zϕ0 ϕc

1 3 ∂ v ϕc , α dα = Ri (ϕc ) − R31,0 (ϕc ) t(α) ∂α 3 =

1 3 Rm (ϕc ) − R32,0 (ϕc ) . 3

(5.4.25)

Note that the second relation is exactly the same as (5.4.6). For one expansion angle it ∂ is easy to find an explicit formulation for the time, since ∂α v ϕ, α is separable. Recall that the flow velocity for one expansion angle is   2   3 cos ϕ − 1  , v(ϕ) = C 1 − 3 cos2 ϕc − 1

(5.4.26)


107

where C is independent of ϕc . As a result the time can be obtained as a function of the expansion angle by dividing the integral equation by 3 cos2 ϕc − 1 and differentiating with respect to ϕc : 3 3 t ϕc = Rm ϕc − R32,0 ϕc C 3 cos2 ϕc − 1 d 3 . + Rm ϕc − R32,0 ϕc dϕc 6 sin ϕc cos ϕc

(5.4.27)

Since t(ϕ0 ) = 0, this expression induces the constraint

R0m (ϕ0 ) − R02,0 (ϕ0 ) cscϕ0 secϕ0 = 0,

(5.4.28)

which means that the tangent of the surface at the expansion angle should equal the tangent of the mould surface. If the constraint is not satisfied, it is not possible to blow a container with outer surface Γm under Assumption 5.4.1. A constraint that should be satisfied so that the container is blown in finite time is that the last term in (5.4.27) is finite at ϕc = 0, which holds because of axial-symmetry. As a final result the evolution of surface R(ϕ, ϕc ) is given by 1 R ϕ, ϕc = 3t ϕc v ϕ, ϕc + V(ϕ, ϕc ) + R30 (ϕ) 3 1 = W(ϕ, ϕc ) + R3m (ϕc ) − R32,0 (ϕc ) + R30 (ϕ) 3 ,

(5.4.29)

where 3 cos2 ϕ − 1

∂ t(α) ∂α v ϑ, α dα,

(5.4.30)

3 cos ϑ − cos ϕ 2 Rm (ϑ)R0m (ϑ) − R22,0 (ϑ)R02,0 (ϑ) . 2 sin ϑ cos ϑ

(5.4.31)

V(ϕ, ϑ) = 3

2

3 cos ϑ − 1 2

W(ϕ, ϑ) =

Zϕ0 ϑ 2

Consider the case of two expansion angles ϕc,1 and ϕc,2 . The case of one expansion angle ϕc,1 could be seen as a special case with ϕc,2 = 0. In the case of two expansion angles an additional unknown is introduced and the boundary conditions of the flow problem become more complicated. As for the case of one expansion angle the flow only changes in time as a function of the expansion angles. Let R ϕ; ϕc,1 (t), ϕc,2 (t) be a

108


surface with initial condition R ϕ; ϕ1,0 , ϕ2,0 ) = R0 (ϕ). Separation of variables of (5.4.21) gives Zt

1 3 v ϕ; ϕc,1 (τ), ϕc,2 (τ) dτ = R ϕ; ϕc,1 (t), ϕc,2 (t) − R30 (ϕ) . 3

(5.4.32)

0

The difficulty is that the expansion angles are unknown functions of time. Since expansion angle ϕc,1 decreases monotonously in time, it can be considered as a dependent variable, while t and ϕc,2 are both considered as functions of ϕc,1 . Changing the integration variable into ϑ := ϕc,1 results in Zϕ1,0 dt 1 3 R (ϕ, ϑ) − R30 (ϕ) . dα = − v ϕ; α dα 3

(5.4.33)

ϑ

Then integration by parts leads to −t(ϑ)v ϕ, ϑ +

Zϕ1,0 ∂ 1 t(α) v ϕ, α dα = R3 (ϕ, ϑ) − R30 (ϕ) . ∂α 3

(5.4.34)

ϑ

By choosing ϕ = ϑ a Volterra integral equation of the first kind is obtained, Zϑ

t(α)K ϑ, α dα = I(ϑ),

(5.4.35)

ϕ1,0

with ∂ v ϕ, ϑ ∂ϑ 1 3 I(ϑ) = R2,0 (ϑ) − R3m (ϑ) 3

K(ϕ, ϑ) =

(5.4.36) (5.4.37)

From (5.4.12) it follows that the kernel K is degenerate, K(ϕ, ϑ) = A0 (ϑ)vA (ϕ) + B0 (ϑ)vB (ϕ). As a result the solution of (5.4.35) is [152]   ! Z 1 d  A0 (ϑ)vB (ϑ)Θ(ϑ) ϑ d I(α) dα  t(ϑ) = 0   , K(ϑ, ϑ) A (ϑ) dϑ  ϕ1,0 dα v B (α) Θ(α)

(5.4.38)

(5.4.39)


109

with Z  Θ(ϑ) = exp 

ϑ ϕ1,0

 ! d B0 (α) A0 (α)vB (α)  dα . dα A0 (α) K(α, α)

(5.4.40)

Note that A0 and B0 still depend on ϕc,2 , which is an unknown function of ϑ = ϕc,1 . The expansion angles seem to be related to each other in an essentially nontrivial way, which complicates matters considerably. Furthermore, even if a relation between the expansion angles is established, the analytical solution (5.4.35) for the time as a function of the expansion angles has to be evaluated numerically. As an example the blowing of an axial-symmetrical ellipsoidal glass container is considered [81, 85] (see § 2.7 and Fig. 5.7). In this example the origin for the spherical coordinate system is chosen at the centre of the mould opening. If the shape of the preform is spherical around the origin, the glass has at most one continuous contact surface with the mould wall, i.e. the outer melt surface has only one expansion angle. The mould surface is given by the ellipse − 1 3 Rm (ϕ) = %m 1 − cos2 ϕ 2 , 4

(5.4.41)

with %m = Rm ( π2 ). Suppose the inner preform surface R1,0 is essentially circular with a horizontal tangent at the symmetry axis. This makes it relatively easy to define the initial level set function corresponding to the inner surface as a signed distance function for the simulation. In cylindrical coordinates (r, z) the surface can be defined as: q max(0, r − rO )2 + (z − zO )2 = %i ,

(5.4.42)

for some constants rO , zO and %i . In spherical coordinates this expression reads   %i − zO sec ϕ, if ϕ ≤ ϕO ,     R1,0 (ϕ) =  q     r sin ϕ − z cos ϕ + %2 − R cos ϕ + z sin ϕ2 , otherwise, O O O O i with       %i − zO  , ϕO = acos  q   2  (%i − zO )2 + rO q 2 R O = rO + z2O .

(5.4.43)

110


The outer preform surface R2,0 (ϕ) should satisfy the following constraints: R2,0 (ϕ0 ) − Rm (ϕ0 ) = 0, 0 0 R2,0 (ϕ0 ) − Rm (ϕ0 ) cscϕ0 secϕ0 = 0,

(5.4.44b)

R02,0 (0) = 0,

(5.4.44c)

R2,0 (ϕc ) < Rm (ϕc ), for ϕc < ϕ0 .

(5.4.44a)

(5.4.44d)

Considering the constraints let the derivative of the outer surface have the following form: R02,0 (ϕ) = a sin ϕc cos ϕc + b sin2 ϕc cos ϕc + c sin ϕc cos2 ϕc .

(5.4.45)

Then integration yields b c a 2 sin ϕc − cos2 ϕc + sin3 ϕc − cos3 ϕc + d. 4 3 3

R2,0 (ϕ) = Suppose ϕ0 =

π 2.

(5.4.46)

00 π π The second constraints gives R00 2,0 ( 2 ) = Rm ( 2 ). To satisfy (5.4.44d)

000 π π it is required that R000 2,0 ( 2 ) ≥ Rm ( 2 ), which leads to c ≥ 0. This constraint leaves some

freedom to choose R2,0 (0) = %m . For this choice substitution of (5.4.45) in (5.4.44) gives a = − 32 , b = 34 , c = 32 , d = %m + 81 . Finally, the function W in (5.4.31) for this example becomes ! 1 5 1 0 2 W(ϕ, ϕc ) = 3 cos ϕ − cos ϕc R ϕ + R2,0 ϕc R2,0 (ϕc )cscϕc secϕc . (5.4.47) 8 m c 6

2

2

The numerical methods presented in Chapter 3 are used to compute the container us√ 1 and rO = %m + zO − (%2m − z2O ). For this example the ing the given preform with zO = 16 values %m = 1 and %i =

3 4

are chosen. Figure 5.11 compares the results of the analytical

approximation and the numerical computation of the container. The purple area is the glass domain given by the level set functions; the blue area is the air domain. The blue lines denote the glass surfaces given by the analytical approximation. Figure 5.11(a) shows the preform and Figure 5.11(b) shows the container. Figure 5.12 plots the signed distance between the analytically approximated and numerically computed inner container surfaces relatively to the mould opening radius as a function of ϕ. Figure 5.13 plots the spherical radii of the glass surfaces on the symmetry axis, divided by the radius of the mould opening, as functions of the expansion angle ϕc , i.e. R1 (0; ϕc )/R2 π2 and


(a) preform

111

(b) resulting container

distance between surfaces

Figure 5.11: Glass surfaces of the preform and the container. The blue lines are the analytical approximations and the purple area is the glass domain in the simulations. 0.04 0.02 0 −0.02 −0.04 −0.06 0

pi/6

φ

pi/3

pi/2

Figure 5.12: The signed distance between the inner surfaces of the containers for the analytical approximation and the simulation relatively to the mould opening radius as a function of the angle ϕ.

bottom radius

2 inner bottom radius outer bottom radius 1.5

1

0.5 0

pi/6 pi/3 expansion angle

pi/2

Figure 5.13: The spherical radii of the glass surfaces on the symmetry axis relatively to the mould opening radius as functions of the expansion angle ϕc .

112 R2 (0; ϕc )/R2

5 Mathematical Analysis of the Inverse Problem π 2 .

The expansion angle advances clockwise in time, i.e. from ϕc =

π 2

to

ϕc = 0. In Fig. 5.11(b) it can be observed that the container surfaces are quite close to each other, which means that Assumption 5.4.1 is well-founded in this example. The container obtained by the approximation is slightly thicker at the top and thinner at the bottom than the numerically computed container. This could be expected, because the mass flow in polar direction, which is neglected in the approximation, is typically in counter-clockwise direction, due to the gravity and inlet pressure. The approximation is close enough to the numerically computed container to be used as an initial guess for iterative optimisation, as explained in § 6.5 of next chapter, or to validate the analysis of the inverse problem based on Assumption 5.4.1, for example the sensitivity analysis in § 5.3. Results for iterative optimisation for solving the inverse problem taking the approximation as an initial guess are shown in Chapter 7.

Chapter 6

Shape Optimisation Strategy

To solve the inverse problem, an efficient numerical shape optimisation method is used. The unknown preform surface is described by a parametric curve and an optimisation algorithm is introduced to find the positions of the control points that optimise the container wall thickness distribution. Following the analysis of the inverse problem in the previous chapter, it is assumed that the mould shape can be blown within finite time and the finite element mesh distribution is such that it accounts for the sensitivity of the inverse problem.

6.1

Parametrisation of the Inverse Optimisation Problem

The inverse problem can be formulated as an optimisation problem: find the shape of the preform, such that the difference between the inner surfaces of the blow moulded container and the designed container is minimal. Figure 6.1 illustrates the optimisation problem. If inner preform surface Γ1,0 is known, then outer preform surface Γ2,0 should be such that the difference between Γ1 (t∗ ) and Γi is minimal, where t∗ is such that Γ2 (t∗ ) =

114

6 Shape Optimisation Strategy

Γm and Γ2 (t) , Γm for t < t∗ . If the inverse problem has a solution, then this difference is zero, i.e. Γ1 (t∗ ) = Γi . mould designed container computed container

Γ1

Γ2

Γi Γm

Figure 6.1: Difference between designed and computed container

The solution of the inverse problem is a continuous surface. Conversely, numerical optimisation algorithms attempt to find the optimal values of a finite set of parameters. Thus, to find a numerical solution of the inverse problem, the surface should be represented by a finite set of parameters. For efficient optimisation it is desirable to restrict the number of parameters to a minimum, while still being able to recover the complete surface with sufficient accuracy. In shape optimisation (the cross-section of) the unknown surface is usually represented by a parametric curve using interpolation. Examples of parametric curves are Beziér curves and cubic splines. The parameters can be defined as the coordinates of the control points. Depending on the interpolation technique, the control points can be points on the actual curve (e.g. spline interpolation), or virtual points around the curve (e.g. Beziér interpolation). Not necessarily all coordinates of the control points have to be adjusted by the optimisation method. There are points of consideration when choosing parameters. Firstly, the number of parameters should be restricted to save computational time. Indeed, each

6.2 Objective Function

115

change in a parameter corresponds to a mathematical problem for a blow moulding process. Secondly, by adjusting the control points in arbitrary directions, a chaotic scattering of control points over the computational domain may occur, which might even lead to tangled curves. Thirdly, for optimal accuracy a good sampling of the control points is required. An alternative is to change the control points only in normal direction to the curve. However, then the direction in which the control points are adjusted is changing, which can bring the optimisation algorithm on the wrong track, particularly if information from previous iterations is used to compute the new direction. Moreover, it is still possible that the curve can get tangled. To avoid this the control points are adjusted in normal direction to the initial curve. As discussed in § 5.4, if the curve is convex around an origin, the normal direction can be approximated by the radial direction in spherical coordinates (R, ϕ) with respect to the origin. This choice of the parameters is used for the optimisation methods presented in this chapter. Let the outer glass surface of the preform be subject to optimisation. Let the surface be represented by a parametric curve with np + 1 control points, P0 , P1 , . . . , Pnp say, as illustrated in Fig. 6.2 for a cubic spline with np = 4. Then the parameters are the spherically radial coordinates of the control points, except for the topmost control point P0 , which is fixed to the mould. Define the vector p comprising the parameters to be optimised by !T p := RP , . . . , RP 1

.

(6.1.1)

np

The numerical optimisation problem is to find the optimal parameter vector.

6.2

Objective Function

The objective of the optimisation method is to fit the computed inner container surface Γ1 (t∗ ) to the designed surface Γi (see Fig. 6.1). The computed surface is given by the zero level set at time t∗ , i.e. −1 Γ1 (t∗ ) = θ1,∗ (0),

(6.2.1)

with θ1,∗ (x) = θ1 (x, t∗ ). The designed surface Γi is usually given by a set of points and reconstructed by interpolation.

116

6 Shape Optimisation Strategy OR,φ P0

P1

P2

P5

P4

P3

Figure 6.2: Parametrisation of a preform surface by a cubic spline with five control points. The control points are variable in radial direction.

The optimisation method minimises a scalar objective function that represents the difference between the surfaces Γ1 (t∗ ) and Γi . A suitable choice of the objective function is the average distance from Γi to Γ1 (t∗ ) with respect to the L2 -norm, 1 Φ(p) := 2 =

1 2

Z Γi Z 1 0

2 dΓ θi,∗;p

2 xi (s) x0i (s) ds, θi,∗;p

where xi : [0, 1] → Γi is the parametrisation of Γi . The subscript

(6.2.2)

indicates that the level set function is computed for parameter vector p. Recall that θ1,∗ Γi is the signed p

distance function from Γi to Γ1 (t∗ ). The choice of the objective function in (6.2.2) is strictly convex and coercive, hence the minimiser exists and is unique, provided the inverse problem is practically well-posed. Another choice of the objective function that guarantees uniqueness of the minimiser is the maximum distance from Γi to Γ1 (t∗ ). Although this choice is arguable, as the optimisation method will focuss on the location where the distance between the surfaces is largest, the objective function is not differentiable with respect to p. The integral in (6.2.2) is computed by applying a composite nG -point Gaussian quadrature rule: nG nint X 2 1X Φ(p) ωk x0 (s jk ) θi,∗;p x(s jk ) . 2 j k

(6.2.3)

6.3 Algorithms

117

Here {ωk }k=1,...,nG are the weights of the quadrature rule, {s jk }k=1,...,nG are the points of the

quadrature rule in the jth subinterval of [0, 1] for j = 1, . . . , nint , and nint is the number of subintervals. For convenience the objective function is presented as 1 Φ rT r, 2

(6.2.4)

where the residual vector r(p) comprises the elements q ri (p) := r( j−1)(nG −1)+k (p) = ωk |x0 (s jk )|θi,∗;p x(s jk ) .

(6.2.5)

The components of the q residual vector represent the distances from Γi to Γ1 (t∗ ) in points x(s jk ) with weights

6.3

ωk |x0 (s jk )|.

Algorithms

Numerical optimisation methods for nonlinear problems are iterative algorithms. They approximate the objective function in a point pk , e.g. by a Taylor series, and search for a better approximation pk+1 of the optimum in a neighbourhood of pk . The process starts with k = 0 and is repeated for k = 1, 2, . . . until the approximation is sufficiently close to the optimum. Derivative-based optimisation methods typically approximate the objective function by second-order Taylor series around pk [140], 1 Φ(pk + sk ) = Φ(pk ) + sTk ∇p Φ(pk ) + sTk HΦ (pk )sk + O ksk k3 . 2

(6.3.1)

with ksk k 1. Here HΦ (p) is the Hessian of Φ with respect to p. The quadratic model in sk is minimised by HΦ (pk )sk = −∇p Φ(pk ).

(6.3.2)

Since sk is merely an approximation, the objective function can be updated by a new quadratic model around pk+1 := pk + sk and the process is repeated. Provided p0 is sufficiently close to the optimum popt , this method will converge to the optimum. This method is known as the Newton method. The Newton method forms the basis for many popular optimisation methods. If HΦ (pk ) is nonsingular and the derivatives are exact, the Newton method has asymptotically quadratic convergence rate. However, a direct

118


Newton method is only locally convergent and therefore requires a good initial guess. Furthermore, the Hessian matrix and gradient of the objective function are not readily available, but have to be computed numerically. There are essentially two strategies to ensure global convergence of the Newton method, namely line search and trust region strategies [140]. • A line search can be used to optimise the magnitude of the search direction s. The distance by which p is adjusted along s can be found by approximating the minimiser ξopt of Φ(p + ξs). Methods to find a suitable approximate minimiser are discussed in e.g. Ref. [140]. • A trust region method can be applied to find a search direction s within a trust region. To this end the quadratic model of the objective function is minimised subject to the constraint ksk ≤ ∆ for some trust region radius ∆. Once close to an optimum, these strategies should become superfluous. In this thesis a trust region method is proposed, which is described further on. The second issue is the numerical computation of the derivatives. Computing the Hessian matrix in the Newton method requires a substantial amount of computational effort, as the direct computation of the objective function is expensive. By approximating the derivatives by a FDM (Finite Difference Method), the gradient requires at least O np objective function evaluations, while the Hessian matrix requires at least O n2p function evaluations, which is an excessive amount of computational time for one iteration. To avoid the numerical computation of the Hessian matrix, Quasi-Newton methods can be used. Quasi-Newton methods replace the Hessian matrix HΦ (pk ) by an approximation Bk . The approximation Bk can be obtained in several ways, including secant updating, neglecting some terms and finite difference approximation. One of the most effective secant methods for updating the Hessian matrix, is the so-called BFGS (Broyden, Fletcher, Goldfarb, and Shanno) method, which updates Bk as [140] Bk+1 = Bk −

Bk sk sTk Bk sT Bk sk

+

yk yTk yTk sk

,

(6.3.3)

with yk = ∇p Φ(pk+1 ) − ∇p Φ(pk ).

(6.3.4)

6.3 Algorithms

119

The BFGS method has superlinear convergence rate, despite the fact that Bk does not converge to HΦ (pk ). The Gauss-Newton method for nonlinear least squares can be seen as a special case of Quasi-Newton methods. Substitution of (6.2.4) in (6.3.2) gives   nres   X JT (p )J(p ) + r j (pk )Hr j (pk ) sk = −JT (pk )r(pk ). (6.3.5) k k  j=1

Here J is the Jacobian matrix of r with respect to p, Hr j (p) is the Hessian of the jth component of r with respect to p for j = 1, . . . , nres and nres = nG nint is the number of residuals. Since the computation of Hessian matrices of the residuals is avoided, the second term on the left-hand side is neglected. This is reasonable as the residuals are approximately zero in the optimum, provided the inverse problem has a solution. Then (6.3.5) becomes JT (pk )J(pk )sk = −JT (pk )r(pk ).

(6.3.6)

This system can be recognised as the normal equations of matrix equation J(pk )sk = −r(pk ),

(6.3.7)

hence as a Newton iteration for solving the system of nonlinear equation r(p) = 0. The Gauss-Newton method has quadratic convergence rate close to the optimum. Although several strategies have been discussed to replace the Hessian with some approximation, also computing the gradient or Jacobian matrix each iteration can become excessively expensive for practical purposes. Recall that a finite difference approxi mation of the gradient (or Jacobian matrix) requires at least O np objective function evaluations, while only one function evaluation is required per iteration for updating the parameter vector and occasionally few additional function evaluations are required for improving convergence. Therefore, if np is large, alternatives should be sought that almost completely avoid the computation of derivatives. Similarly to the case of the Hessian matrix, a secant method can be used to update the Jacobian matrix. The most popular secant method for updating the Jacobian matrix in the Newton method is Broyden’s method [22, 23, 51, 133]. In fact there are two Broyden’s methods, known as Broyden’s good method and Broyden’s bad method. As the name reveals, Broyden’s good method appears to be much more efficient in practice than Broyden’s bad method [51]. However, in few cases Broyden’s bad method

120


performs better. In Ref. [127, 128] a criterium was presented based on which it can be decided whether Broyden’s good method or Broyden’s bad method should be used. Based on this criterium a hybrid Broyden method was developed, which updates the approximation Ak of the Jacobian matrix J(pk ) after each iteration by either Broyden’s good or bad method: Ak+1 = Ak +

Ak+1 = Ak +

(rk+1 − rk − Ak sk )sTk sTk sk

,

sTk sk−1 (rk+1 − rk )T (rk − rk−1 ) , (6.3.8a) if T < sk sk (rk+1 − rk )T Ak sk

(rk+1 − rk − Ak sk )(rk+1 − rk )T (rk+1 − rk )T Ak sk

,

otherwise.

(6.3.8b)

Here formula (6.3.8a) is known as Broyden’s good method and formula (6.3.8b) is called Broyden’s bad method. Secant methods may considerably reduce the computational time, if this is dominated by the number of function evaluations. A drawback of secant methods is that they do not provide an accurate approximation of the Jacobian matrix, which is a requirement for a reliable descent direction and stopping criteria in minimisation [51]. As a result the parameter vector may fail to converge to an optimum. To overcome this secant methods can be combined with FDMs; if no convergence is reached after few iterations with the secant method, the Jacobian matrix can be approximated by finite differences. An alternative for secant methods is automatic differentiation. However, automatic differentiation requires access to the source code and the number of arithmetic operations performed to solve the forward problem is exhaustive. Furthermore, automatic differentiation is not always available, for example if commercial software is used to compute the container. Another alternative for secant methods in optimisation is derivative-free optimisation [41]. Since in derivative-free optimisation methods no gradient information is available, the convergence is usually much slower. On the other hand, typically fewer objective function evaluations per iteration are required. However, since the number of parameters is not extremely large, it is not expected that the difference in function evaluations will be considerable, as in derivative-free optimisation often few function evaluations are required to find a suitable search step. On the other hand, Broyden’s method requires only one function evaluation per iteration, unless difficulties with convergence are encountered. Moreover, Broyden’s method can be conveniently combined with fi-

6.3 Algorithms

121

nite difference approximations, following the strategy in § 6.4, which can significantly improve the efficiency of both the secant optimisation method and finite difference optimisation method in terms of function evaluations. Yet, derivative-free optimisation may be a future consideration. In this thesis the hybrid Broyden method is used in combination with the GaussNewton method. One of the difficulties encountered in the Gauss-Newton method is that the Jacobian matrix is often nearly singular [140]. This is particularly the case when the Jacobian matrix is approximated by Broyden’s method and when the Gauss-Newton method is used to solve an inverse problem, which are well known sources of ill-posed or ill-conditioned optimisation problems. This difficulty can be avoided by using a trust region method. In Ref. [140] the following Lemma is proven. Lemma 6.3.1. The search step s is a solution of the trust region subproblem min kJ(p)s + r(p)k22 ,

(6.3.9a)

subject to ksk ≤ ∆,

(6.3.9b)

s

for some trust region ∆ > 0 if and only if there is a scalar λLM ≥ 0, such that

JT (p)J(p) + λLM I s = −JT (p)r(p), λLM (∆ − ksk) = 0.

(6.3.10a) (6.3.10b)

This means that if the solution of (6.3.6) is inside the trust region, i.e. ksk ≤ ∆, then s solves the trust region subproblem (6.3.9). Otherwise, there is a scalar λLM ≥ 0, such that s solves (6.3.10a), while satisfying ksk = ∆. This gives rise to the Levenberg-Marquardt method, which in each iteration determines the search step sk by solving the linear matrix equation

JT (pk )J(pk ) + λLM, k I sk = −JT (pk )r(pk ).

(6.3.11)

The Levenberg-Marquardt method was first published in Ref. [118] and rediscovered in Ref. [126]. The original Levenberg-Marquardt method adjusted the Levenberg-Marquardt parameter λLM, k heuristically by multiplying or dividing it by a fixed factor. The connection with trust region methods was first established in Ref. [132]. The LevenbergMarquardt parameter acts as a regularisation parameter in case the JT (pk )J(pk ) is nearly

122


singular and interpolates between a steepest descent direction sk = −JT (pk )r(pk ) and a Gauss-Newton iteration. The Levenberg-Marquardt parameter λLM, k can also be used for indicating whether the Jacobian matrix is poorly approximated by a secant method, such that no reliable descent direction can be obtained [82]. For the true Jacobian matrix, the requirement Φ(pk+1 ) < Φ(pk ) is always satisfied for sufficiently large λLM,k [126]. If λLM, k becomes large and Φ does not decrease, it is likely that the Jacobian matrix is poorly approximated. Moreover, if λLM, k is large compared to kJT (pk )J(pk )k, the search step sk does not approximate the matrix equation (6.3.7), which makes it unsuitable for a Broyden update. If λLM, k is considered to be too large, the Jacobian matrix can be approximated by a FDM.

6.4

Error Tolerance

The iterative algorithm stops if ksk k < η1 or Φ(pk ) < η2 ,

(6.4.1)

for some error tolerances η1 , η2 . The error tolerances cannot be chosen arbitrarily small, since sk and Φ(pk ) are subject to errors. In order to determine the optimal tolerances an error analysis is performed. In derivative-based optimisation the approximation of the derivatives is a source of errors. For the error estimation two different types of methods for approximating the Jacobian matrix are considered: FDMs and Broyden’s method. In the case of FDMs, the Jacobian J(p) can be computed by the Forward Difference Formula: J(p)e =

r(p + δe) − r(p) + O(δ), δ

(6.4.2)

for some unit vector e in Rnp . Since (6.4.2) is computed for all np basis vectors spanning Rnp , the Forward Difference Formula requires the computation of np additional function values given r(p). The error in the finite difference approximation directly influences the error of the optimisation algorithm. Expression (6.4.2) suggests that the step size δ is chosen as

6.4 Error Tolerance

123

small as possible. However, also the error in computing the residual r should be taken into account. Let this error be denoted by F . Then the sum of errors is bounded by 2F 1 + Mδ, δ 2

(6.4.3)

where the error in computing the residual is bounded by F and M is an upperbound for Hr j (pk ), j = 1, . . . , nres . This upperbound is minimal for r δ=

4 √ F ∼ F . M

(6.4.4)

The order of accuracy can be improved by employing a Central Difference Formula to compute the Jacobian: J(p)e =

r(p + δe) − r(p − δe) + O(δ2 ), 2δ

(6.4.5)

Unfortunately, the Central Difference Formula is almost twice as computationally expensive as the Forward Difference Formula, since 2np additional function values are required. The sum of errors is bounded by F 1 2 + Kδ , δ 6

(6.4.6)

where K is an upperbound for the third order derivatives of the residual vector. The upperbound for the total error is minimal for 6 δ= K F

! 13

1

∼ F3 .

(6.4.7)

Since the error in the Central Difference Formula is O(δ2 ), the error in the solution of the 2

inverse problem is at least O(F3 ), which is not a considerable improvement considering the fact that the method is twice as expensive. Still the Central Difference Formula can be used when the optimal accuracy is reached for the Forward Difference Formula to gain a few extra digits in the last few iterations. Next suppose Broyden’s method is used to approximate J(p). Only Broyden’s good method (6.3.8a) is considered, as the theoretical properties of Broyden’s bad method

124


(6.3.8b) are similar [51]. Assume that the Jacobian matrix J is Lipschitz continuous with Lipschitz constant γLip , i.e. kJ(pk+1 ) − J(pk )k ≤ γLip ksk k.

(6.4.8)

In Ref. [51] the following upperbound for the error of the Broyden’s good update is given: 3 kAk+1 − J(pk+1 )k ≤ kAk − J(pk )k + γLip ksk k. 2

(6.4.9)

Suppose J(p0 ) is approximated by Forward Difference Formula (6.4.2). Then the initial √ error is O(δ), where δ ∼ F . Since the forward problem is not sensitive to changes in the parameters (see § 5.3), γLip is small. Assuming that r is twice differentiable, it holds that γLip =

n

max

p∈R p , j=1,...,nres

kHr j (p)k ≤ M.

(6.4.10)

The order of magnitude of ksk k is typically O(δ), which should also be the order of magnitude of error tolerance η1 in (6.4.1), since it is as close as any Newton-like method can get to the optimum. Thus, the error of the Broyden update is at least O(δ) and can have larger order of magnitude far away from the optimum. In the optimisation method there are also other sources of errors than the numerical solution of the forward problem and approximation of the Jacobian matrix. In order to estimate the total error of the optimisation method, all sources of errors should be taken into account. Finally, the error tolerances η1 and η2 in (6.4.1) should be of the same order as the total error. The following errors are considered. Model errors C The computation of the residual r is subject to errors. The error between the computed residual and the exact residual is represented by the model error C . Truncation errors T The error in the approximation of the Jacobian by finite differences with the true residual is referred to as the truncation error T . Also the error in the Jacobian update can be viewed as a truncation error and is denoted as T . Truncation errors are not only the result of approximating derivatives. The composite quadrature rule in (6.2.3) also induces a truncation error. The order of

6.4 Error Tolerance

125

magnitude of the truncation error of the composite nG -point Gaussian quadrature rule with nint subintervals of [0, 1] is  !   1 2nG   . O  nint

(6.4.11)

Clearly, not many interpolation points are required for the composite quadrature rule to match the accuracy of the numerical differentiation that approxi method − 14 mates the Jacobian. For example, if nG = 2, then nint & O T is sufficient. Thus the truncation error in (6.2.3) is negligible. Rounding errors R Performing arithmetic operations in machine precision results in a rounding error R . In general, rounding errors are negligible compared to model and truncation errors. Measurement errors M Data provided by manufacturers contains measurements errors. Since this is often the desired accuracy, it is assumed that the numerical accuracy of the optimisation method is of the order of magnitude O(M ). In practice the measurement error can be smaller if the numerical accuracy is limited by the computational effort. Interpolation errors I The interpolation of the unknown preform surface by a parametric curve gives an interpolation error. Errors in the interpolation of measured data are counted as measurement errors. The error in the forward problem F is not mentioned in this list, as it is the result of the model error C and the measurement error M . Neglecting the influence of rounding errors, the total error of the optimisation method is the sum of the contributions of all errors: = C + T + M + I .

(6.4.12)

In the following a quantitative analysis of the errors is performed. The analysis is based on the following assumptions. • The model error C is the sum of model simplifications and discretisation of the forward problem. The contribution of model simplifications can be omitted, as it is irrelevant for the choice of the error tolerance.

126

6 Shape Optimisation Strategy • The time step is small compared to the mesh size with respect to the order of accuracy of the discretisation schemes. Then for linear finite elements the model error C is O(h2 ), where h is the typical mesh size. Re-initialisation of the level set function preserves this property. • Given M , the mesh size h is chosen such that h2 & M . • A cubic spline is used for the interpolation of the unknown interface, hence the interpolation error is O(E4 ), where E is the largest distance between the control points. The number of parameters is chosen such that E4 . h2 . Thus, if ` is the length of the axial symmetrical cross-section of the surface, then E ∼ 1

np & CI `h− 2 + 1,

CI np −1 `

and

(6.4.13)

for some constant CI that is independent of h and E. Since the fourth order derivative of the surface along the cross section is typically small, it is expected that CI 1. Consequently, only few control points are required to meet the required accuracy. • The Forward Difference Formula combined with Broyden’s method is used for the approximation of the Jacobian. The truncation error T is O(δ). Furthermore, the optimal choice for the forward difference scheme is used, i.e. F = CF h2 + O(h p ), p > 2, so that the minimal upperbound (6.4.4) for δ is r δ=2

CF h. M

(6.4.14)

Unfortunately, constants M and CF are difficult to estimate without any additional computational effort, so the precise magnitude is unknown. Constant M can be estimated by approximating the diagonal elements of Hr j , e.g. by means of the BFGS method. Constant CF is independent of the mesh size, but it does depend on the mesh quality, the geometry and the problem. It can be estimated by Richardson extrapolation with different mesh sizes. On the other hand, if the problem is well scaled, it can be assumed that [140] CF ∼ 1, M

(6.4.15)

6.4 Error Tolerance

127

and the choice δ = 2h is fairly close to optimal. However, for complicated geometries it is difficult to obtain optimal mesh quality in the whole domain, so that CF will be somewhat larger and e.g. δ = 4h would be a better choice. Based on the foregoing assumptions, the error F in the forward problem is estimated by F ∼ C ∼ h2 & M + I .

(6.4.16)

Furthermore, T ∼

√ F ∼ h.

(6.4.17)

As a result, = O(h);

(6.4.18)

that is, the error of the optimisation method is dominated by the mesh size. Based on the error analysis the following modified stopping criterium is suggested for the optimisation method: n o max ksk k, Φ(pk+1 ) < 2h or ksk k
0,

j = 1, . . . , np .

(6.6.1)

130


Secondly, the preform should have a minimum thickness, RPj − R0,1 (ϕ j ) − η > 0,

j = 1, . . . , np .

(6.6.2)

Let the constraints be given by cl (p) > 0,

l = 1, . . . 2np .

(6.6.3)

The constraints are incorporated in the optimisation method by introducing a weighted objective function, given by e Φ(p) = Φ(p) + ζ(p).

(6.6.4)

Here function ζ is the sum of the so-called weighted penalty functions ζ(p) =

2np X wl , c (p) l l

(6.6.5)

with non-negative weights wl [29, 74, 82, 122, 167]. The optimum is assumed to be strictly inside the constrained domain, otherwise the constrained domain should be either expanded or the solution is physically not relevant. Under this assumption the optimality condition ∇Φ = 0 remains the same as for unconstrained optimisation [140]. In Ref. [65] it is proven that under certain topological assumptions the penalty function method with strictly decreasing weights converges to an optimum. The introduction of the weighted objective function gives rise to a modification of the Levenberg-Marquardt method by including the derivatives of the weighted penalty functions. The kth iteration of the modified Levenberg-Marquardt method is [167] T J (pk )J(pk ) + λLM, k I + Hζ (pk ) sk = −JT (pk )r(pk ) + ∇p ζ(pk ).

(6.6.6)

Each time a descent direction is found, the weights are decreased by a fixed factor.

Chapter 7

Shape Optimisation Results

The optimisation methods described in the previous chapter are applied to find a preform shape for which the container with prescribed wall thickness distribution can be blow moulded for several applications in glass blow moulding. The objective function values are computed by the numerical techniques discussed in Chapter 3. For each application the optimal preform and the corresponding container, obtained by computing the blow moulding of the optimal preform, are visualised and the convergence results of the optimisation method are plotted.

7.1

Initial Guess for Blow Moulding an Ellipsoidal Glass Container

Consider the axial-symmetrical glass container with elliptical cross section in § 2.7 and § 5.4. As before let %m = 1 and %i =

3 4.

Figure 7.1 shows the designed container.

132

7 Shape Optimisation Results

Suppose the inner container surface is given by r     9 19 2 Γi : r, z (ϕ) =  rO + − zO sin ϕ, − cos ϕ , 16 10

(7.1.1)

where the angle 0 ≤ ϕ ≤ 21 π is oriented as in Fig. 5.9. The inner preform surface is given by (5.4.43), while the outer preform surface is unknown.

Figure 7.1: The desired container.

A cubic spline with six control points is used for the parametrisation of the outer preform surface. The parameter vector, given by (6.1.1), comprises the spherically radial coordinates of the control points. Instead of solving (6.5.3), the angles of the control points are chosen as follows. First, points are chosen on the mould boundary by substituting the following values of ϕ in (7.1.1): ϕˆ j =

j(2np + j − 3) 6(np − 1)np

π,

j = 1, . . . , np .

(7.1.2)

The angles of the control points of the parametric curve are the angles of the points on the mould boundary, i.e.    r(ϕˆ j )   , ϕ j = atan − z(ϕˆ j )

j = 1, . . . , np .

(7.1.3)

Note that the difference between consecutive angles ϕˆ j − ϕˆ j−1 increases by the ratio of the semi-major axis of the elliptical cross-section of the mould to the semi-minor axis, which is a factor 2. Since the outer preform surface is approximately spherical, the control points are nearly equidistant for this choice.

7.1 Initial Guess for Blow Moulding an Ellipsoidal Glass Container

133

The initial guess for R2,0 required for the iterative optimisation method is given by (6.5.1). Figure 7.2 shows the initial guess of the preform, approximated by the cubic spline, and the resulting container by computing the blow moulding of the preform. For comparison, if the radius of the mould opening were 5 cm, the average distance between the inner surfaces of designed container in Fig. 7.1 and the computed container in Fig. 7.2(b) would be approximately 3 mm.

(a) preform


Figure 7.2: Initial guess of the preform.

(a) preform


Figure 7.3: Optimal preform for the designed container.

The outer preform surface is subject to constraints. The positions of the control points should satisfy constraints (6.6.1) and (6.6.2) with η = 0.1. These constraint are included in the objective function as weighted penalty functions, i.e. (6.6.4) and (6.6.5). Furthermore, the initial guess of the preform surface satisfies constraints (5.4.44), as described in § 6.5. Constraint (5.4.44b) does not have to be satisfied by the initial guess,

134


but can be left to the optimisation method. The Levenberg-Marquardt method with combined FDM and hybrid Broyden’s method described in § 6.3 is applied to find the optimal positions of control points corresponding to the designed container. During each iteration the control points are adjusted and the corresponding container is computed. The difference between the inner glass surfaces of the computed container and the designed container is computed to determine the new value of the objective function. The error tolerance for the objective function is 1 2h

≈ 0.01. Figure 7.3 shows the final results of the optimisation method. Figure 7.3(a)

shows the optimal preform and Figure 7.3(b) shows the resulting container. The computed container appears to be in good agreement with the designed container in Fig. 7.1. The number of iterations required for the optimisation is 13. Figure 7.4 plots the convergence behaviour of the objective function. Starting from the initial guess, the optimisation method initially rapidly converges to the optimum, but gradually slows down as the error in the solution approaches the discretisation error. Since the error tolerance is taken rather strict, the Levenberg-Marquardt algorithm is eventually considerably slowed down in its effort to find a parameter λLM for which the objective function decreases. Figure 7.5 plots the signed distance between the computed and designed inner container surfaces as a function of ϕ, i.e. the difference between the containers in Fig. 7.1 and Fig. 7.3(b).

Figure 7.4: Convergence of objective function.

7.2 Optimisation of the Preform Shape for Blow Moulding a Glass Bottle

135

signed distance

0.02 0.01 0 −0.01 −0.02 0

pi/16

pi/8

3pi/16

φ

pi/4

5pi/16 3pi/8 7pi/16

Figure 7.5: Signed distance between the desired container and the computed container as a function of the angle.

7.2

Optimisation of the Preform Shape for Blow Moulding a Glass Bottle

This section shows numerical results for optimising the preform shape for blow moulding a axial-symmetrical glass bottle with the desired wall thickness. A beer bottle with optimal wall thickness distribution is designed and one is interested in blow moulding this bottle. The dimensional height of the bottle is 20.1 cm and the dimensional radius is 1.25 cm at the neck to 2.89 cm at the widest part. Figure 7.6 shows the designed bottle. Two cases are considered: blow moulding with sagging for 0.5 s and blow moulding without sagging. The bottle is produced by blowing air in the mould with an inlet pressure of 10 kPa. The 2D axial-symmetrical finite element model has been implemented in COMSOL 3.5 with MATLAB. The details are the same as for the stretch blow moulding of the PET bottle in § 4.3. Figure 7.7 shows a typical structured mesh for the blow moulding simulation. Two level set functions are used to capture the moving interfaces. No-slip conditions are imposed on Γl,q . Let the origin of the spherical coordinate system be given at xO = (0, zO ), with zO = −5.58cm (see Fig. 5.8). An initial guess R2,0 for the iterative optimisation method with respect to origin xO is determined by (6.5.1). The radii R1,0 , Ri and Rm can be calculated as follows. Radius R1,0 is given by a piecewise linear and quadratic curve connected by a point xQ2 . This representation is chosen so that the initial level set function can easily be calculated analytically as a signed distance function to the surface. For the line piece

136


20.1cm

1.25cm

2.89cm

Figure 7.6: Designed bottle with optimal wall thickness distribution.

Figure 7.7: Typical structured mesh for glass bottle

between points xQ1 = (rQ1 , 0) and xQ2 = (rQ2 , zQ2 ) the radius is given by R1,0 (ϕ) =

rQ1 (zQ2 − zO ) + zO rQ2 (rQ2 − rQ1 ) cos(ϕ) + zQ2 sin(ϕ)

.

(7.2.1)

For the circle piece between points xQ2 and xQ3 = (0, zQ3 ) the radius is given by

R1,0 (ϕ) = (zQ2

v  21  u t 2 2   r r  Q2 Q2   2 2  . − sin (ϕ) + 2 sin (ϕ) + − zO ) 1 − 2 cos(ϕ) 2 2   z Q2 z Q2 

(7.2.2)

The mould radius Rm is given by piecewise rational Beziér curves, i.e. PnBC nBC

k nBC −k wk xk k=0 k s (1 − s) x(s) = Pn n k , nBC −k BC BC s (1 − s) w k k=0 k

(7.2.3)

given points xk and weights wk , k = 0, . . . , nBC . The mould radius is simply given by q Rm (θ(s)) = r(s)2 + z(s) − zO 2 .

(7.2.4)

7.2 Optimisation of the Preform Shape for Blow Moulding a Glass Bottle The angle θ as a function of the curve parameter s is   PnBC nBC k !  k=0 k s (1 − s)nBC −k wk rk  r(s)   θ(s) = − arctan = − arctan  Pn n k  nBC −k BC BC z(s) wk zk k=0 k s (1 − s)

137

(7.2.5)

To find s as a function of θ the following equation needs to be solved: ! nBC X nBC k s (1 − s)nBC −k wk yk (θ) = 0, k k=0

(7.2.6)

yk (θ) = rk + zk tan(θ).

(7.2.7)

with

For n = 2 the following solution can be found: q 2y0 − y1 + y21 − 4y0 y2 . s= 2(y2 − y1 + y0 )

(7.2.8)

For n = 1 this reduces to s=

y0 . y0 − y1

(7.2.9)

The inner container surface Ri is described by a cubic spline. As for the mould radius, the inner container surface is given by q Ri (θ(s)) = r(s)2 + z(s) − zO 2 .

(7.2.10)

The curve parameter s as a function of the angle θ can be found numerically by using a root finder. A cubic spline with six control points is used for the parametrisation of the outer preform surface. The angles of the control points are determined by solving (6.5.3) by an adaptive Simpson rule. Figure 7.8 shows the initial guess of the control points. The following constraints on the outer preform surface should be satisfied. The positions of the control points are subject to the thickness constraint (6.6.2) with η = 0.2. The mould constraints in (6.6.1) are replaced by rm z=0 − rPj > 0.5, zm r=0 − rPj > 0.5,

j = 1, . . . , np − 1,

(7.2.11)

j = 2, . . . , np .

(7.2.12)

138


Figure 7.9 illustrates the constrained domain. These constraints are included in the objective function as weighted penalty functions, i.e. (6.6.4) and (6.6.5). Finally, constraint (5.4.44b) is satisfied by fixing the derivative of the spline at end point P5 , such that it is perpendicular to the symmetry axis. Figure 7.10 shows the blow moulded bottles corresponding to the initial guess computed with and without sagging. Figure 7.10(a) shows the mould shape computed with sagging for 0.5 s and Figure 7.10(b) shows the mould shape computed without sagging. Obviously, the bottle computed without sagging is much closer to the designed bottle in Fig. 7.6 than the bottle computed with sagging. This could be expected as during sagging the mass flows in negative axial direction. Consequently, the mass flow in polar direction during sagging is clockwise. In the initial guess of the preform shape the mass flow in polar direction is completely omitted. Without sagging there is also mass flow in clockwise polar direction, which causes the difference between the exact and the approximated preform shape. Since the mass flow in polar clockwise direction is much larger with sagging, the error in the analytical approximation is also significantly larger. On the other hand, the approximation without sagging is quite satisfactory and suitable to be used as an initial guess. The iterative optimisation algorithms described in § 6.3 are used to find the optimal positions of the control points of the outer preform surface as to minimise the difference between the inner surfaces of the computed container and the designed container in the sense of (6.2.2). Consider a finite element mesh with 40 elements along the mould opening by 272 elements along the symmetry axis and a mesh distribution as in Fig. 7.7. The Levenberg-Marquardt method with combined FDM and secant method is used to find the optimal preform. The typical mesh size for this example is h = 0.059. Figure 7.11 shows the convergence results. From the second iteration the hybrid Broyden method is used to update the Jacobian matrix. However, in the sixth iteration no convergence is reached, so the Jacobian matrix is approximated by finite differences. In the seventh iteration another Broyden update is performed, but from the eighth iteration onwards only forward difference approximations are used, since the parameter vector is close to the optimum. Only 59 function evaluations are required until convergence. Obviously, Broyden’s method is considerably slower in convergence than finite difference optimisation. However, since only four parameters have to be optimised, an iteration of Broyden’s method is about four times faster. The benefit of secant methods

7.2 Optimisation of the Preform Shape for Blow Moulding a Glass Bottle

R

R r O

139

r O

φ

Figure 7.8: Initial guess of control points.

(a) computed with sagging

φ

Figure 7.9: Constrained domain of control points.

(b) computed without sagging

Figure 7.10: Mould shapes for initial guess.

140


Figure 7.11: Convergence of objective function.

really becomes clear when the number of parameters increases, while for fewer parameters secant methods are of no advantage. On the other hand, if the number of parameters is large, derivative-free optimisation could be considered. The authors have applied few tests for various, relatively simple problems, from which it appeared that optimisation with combined Broyden’s method and finite difference approximation with a moderate number of parameters costs fewer function evaluations than pure finite difference optimisation. Figure 7.12 shows the result of the optimisation method. Figure 7.12(a) shows the optimal shape of the preform and Fig. 7.12(b) shows the resulting container. The resulting container in is in good agreement with the designed container in Fig. 7.6. The mean distance between the inner container surfaces is 7.83 · 10−2 cm and the maximum distance is 1.78 · 10−1 cm. For comparison the radius of the bottle neck is 1.25 cm. The result is encouraging and gives confidence that the method can be used in practice.

(a) Optimal preform

(b) Optimal mould shape

Figure 7.12: Optimal preform and mould shape for beer bottle.

Chapter 8

Conclusions and Recommendations

The following objectives are achieved in this thesis. 1. A general and complete problem formulation for forward and inverse blow moulding is given. The forward problem is based on the governing equations for the conservation of mass, momentum and energy, which are applicable to all continuous media. It can be solved in an essentially identical way for the extrusion, injection, stretch blow moulding process for polymers and final blow moulding stage for glass. The problem is simplified to describe the most essential physical phenomena in blow moulding. 2. An efficient numerical method to solve the forward problem is given, based on Finite Element Methods and using Level Set Methods to track the moving interfaces. Level Set Methods do not require re-meshing, which can be a considerable saving on computational time. It is shown that for numerical stability two level set problems should be solved, one for each melt-air interface. Taking the expected measurement error into account, the wall thickness distribution obtained

144

8 Conclusions and Recommendations for blow moulding a glass bottle is in good agreement with the measured thickness distribution. Furthermore, it is verified that the numerical solutions satisfy mass conservation with good accuracy, which clearly shows the strength of the Level Set Method for this application. 3. The solvability and sensitivity of the inverse problem is analysed and conditions are established to deal with this within an acceptable tolerance. Constraints on the mould surface are imposed, such that the mould shape can be blown within a reasonable process time for given inlet pressure. An approximation is derived to estimate the sensitivity of the forward and inverse problem with respect to perturbations. 4. A practical shape optimisation method to solve the inverse optimisation problem is developed. The method uses an efficient strategy to cpmbine finite difference approximations of the Jacobian matrix with secant updates in order to save on computationally expensive function evaluations. A quantitative error analysis is performed to determine the optimal error tolerance for the optimisation method, so that the optimisation problem can be solved with optimal accuracy. For the error analysis the mesh distribution is scaled with the sensitivity. A suitable initial guess is constructed by means of an analytical approximation of the inverse problem. Although this thesis covers a wide range of aspects regarding mathematical mod-

elling and shape optimisation of blow moulding processes, there are still a few suggestions for future work. Regarding the numerical simulation method it would be interesting to perform a comparative study with different numerical models. For example, a comparison with simulation models that use Interface Tracking Techniques combined with re-meshing techniques to deal with the moving interfaces would help show the efficiency of the Level Set Method in the simulation of blow moulding processes. For the stretch blow moulding process a comparison between the viscoplastic rheological model presented and a viscoelastic model with respect to accuracy can be considered as future work. Although the mathematical model presented covers a wide range of applications in blow moulding, the number of examples presented can still be extended. It could be interesting to develop examples for other blow moulding processes, such as extrusion blow moulding, although it can be expected that results are comparable to the stretch

145 blow moulding and glass blow moulding examples. Extension of the model to fully three-dimensional cases, such as the blow moulding of decorative perfume bottles, is also a future consideration. Regarding the shape optimisation method, the computational time required by the optimisation method to obtain a solution of the inverse problem within the desired accuracy should be further reduced. In this thesis the number of function evaluations required was reduced to save computational time, e.g. by avoiding the direct computation of derivatives. Another approach is to make the function evaluations less computationally expensive. This can be achieved by means of multi-level optimisation, in which the finite element mesh is adaptively refined as the optimisation algorithm approaches the optimum [93, 93, 103, 179, 211]. The error tolerance can be set increasingly stricter by refining the mesh following an adaptive procedure, starting from a relatively loose tolerance, until the desired accuracy is reached. In this way the function evaluations are computationally cheap at first and get more expensive near the optimum. Similarly, the number of parameters can be increased, such that the interpolation error of the parametric curve matches the model error (see § 6.4). To this end T-splines can be used to approximate the unknown surface [11, 168, 169]. An advantage of T-splines is that control points can be added without altering the shape of the curve. The optimisation algorithm can be started with a coarse mesh and few control points. Each time the mesh is refined by the optimisation method, also the number of control points is increased, such that the interpolation error is kept of the same order of magnitude as the model error, without altering the shape of the parametric curve. Shape optimisation methods employing T-splines are presented in e.g. Ref. [89, 183]. In Ref. [180] a similar approach is presented for shape optimisation, in which B-splines are used to describe the unknown surface and a knot insertion algorithm is introduced to keep the shape of the spline unchanged when control points are added. Multi-level optimisation methods have appeared to be quite successful in practice. Derivative-free optimisation may also be a future consideration. Derivative-free optimisation methods are described in Ref. [41]. Another possible point of improvement of the performance is to include more information about the optimal shape of the preform in the optimisation procedure. Although the shape of the outer (or inner) preform surface is unknown, the volume of the preform is the same as for the designed container. This information is used to construct the initial guess, but the volume of the preform can be modified by the optimisation algorithm

146

8 Conclusions and Recommendations

when adjusting the parameters. To avoid this, volume conservation of the optimisation algorithm can be enforced by means of an equality constraint. To include the equality constraint an optimisation method for constrained optimisation may be introduced, e.g. quadratic programming. The choice of the objective function can be analysed further in order to improve the efficiency of the optimisation method. Various choices of the objective function are possible. For example, a weighted sum of objective functions can be optimised, including the averaged or maximum distance between the computed and designed surface and the bending energy required to bend the computed surface in the desired shape. Finally, the initial guess constructed by the analytical approximation appears to be highly suitable for solving the inverse problem by iterative optimisation. However, the initial guess should be modified to account for sagging in glass blow moulding. To this end another analytical approximation may be derived for sagging. Then first the melt surfaces at the end of the sagging stage are determined with the approximation for blowing and the unknown preform surface is determined from these melt surfaces with the approximation for sagging. However, this approach is not straightforward, as at the end of the sagging stage both melt surfaces are unknown and therefore are not uniquely determined.

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Index absorption coefficient, 27

Buckley-Jones model, 30

annealing, 6 Lehr, 6 automatic differentiation, 120

compressibility, 25, 97 condition number stiffness matrix, 65

baffle, 7

conductivity

Beziér curves, 14, 114, 136

effective, 26, 38, 41, 44

blank mould, 7, 14

radiative, 26–27

blow head, 6

thermal, 26, 27, 43, 45

blow moulding

constraints

extrusion, 10, 12–15, 31

geometric, 129–130, 133, 137, 138

glass, 3, 6, 7, 12–14, 16, 17, 23, 28–

mould surface, 16, 93, 95

29, 33, 41–44, 57, 69, 71, 74, 109–112, 133, 135, 138 blow-blow, 6, 7, 14, 23, 41, 76– 82, 87

preform surface, 107, 110, 129, 133, 138 contact angle, 94 container, 1, 3–7, 9–16, 28, 31, 49, 74,

counter blow, see blow-blow

76, 83, 85, 90, 96–98, 101, 102,

press-blow, 7, 41

107, 109–113, 115, 133, 134,

settle blow, see blow-blow

138, 140

injection, 10 stretch, 4, 10–14, 16, 23, 30–32, 44– 47, 61, 69, 83–87 boundary conditions, 35–38, 40, 42, 48, 52, 104, 107 free-stress, 36 Navier’s slip, 35, 36 no-slip, 36, 37, 52, 61, 104 Boundary Element Methods, 50

designed, 98, 113, 115, 131, 133– 135, 138, 140 wall thickness, 11–14, 17, 49, 76, 78–82, 87, 89, 113, 131, 135 control points, 14, 17, 113–115, 126, 129, 132–134, 137, 138 convection-diffusion equation, 42, 47, 55, 67 convergence rate

168

INDEX BFGS method, 119

inverse problem, see optimisation

Gauss-Newton method, 119

measurement, 125, 126

Newton method, 117

model, 124–126

creep, 30

optimisation, 123, 125

curvature, 34, 42, 56, 93, 95–97

rounding, 125 truncation, 124–126

density, 24–25, 41, 43, 44

expansion angle, 104–110

descent direction, 120, 122, 127, 130

extrusion, 9–10

steepest, 122 die, 11, 14

continuous, 9 intermittent, 9

dimensionless number Brinkman, 39, 41–45 Deborah, 46

Fast Marching Methods (FMMs), 16, 62– 64, 69–73

Froude, 39, 41, 43, 44

fictitious fluid, 44, 47

Graetz, 45

Finite Difference Methods (FDMs), 17,

Nusselt, 40, 42

51, 82, 118–120, 122–123, 127,

Péclet, 39, 41–45, 100

134, 138

Reynolds, 39, 41, 43, 44, 47 domain

BDF, 51, 74 Central Difference Formula, 123

air, 22, 23, 44, 56, 77, 78, 110

Crank-Nicholson, 51, 82

baffle, 22

Euler Explicit, 122–124, 126, 128

equipment, 22, 23

Euler Implicit, 51, 76

fluid, 22, 24, 37

Forward Difference, see Euler Ex-

forming machine, 22 forming material, see melt melt, 22, 23, 56, 77, 78

plicit IDA, 74, 83 Finite Element Methods (FEMs), 12, 14,

mould, 22

16, 49–52, 65, 69, 70, 76, 83,

stretch rod, 22, 52, 84

99, 135–138 P2 /P0 element, 50

Eikonal equation, 62

Lagrange element, 51, 76, 83

elasto-visco-plastic, 31

Mini-element, 50, 76, 83

error

Taylor-Hood element, 50

discretisation, 99, 126, 134

Finite Volume Methods, 50

forward problem, 123–125, 127

forming machine, 6, 7, 22–23

interpolation, 125, 126

Fourier’s Law, 26, 37

INDEX

169

friction coefficient, 36, 40

inlet pressure, 5, 11, 13, 14, 25, 36, 40, 43, 69, 70, 77, 85, 92, 112, 135

Gauss-Newton method, 119, 121

integrated method, 66

glass, 1–6, 25–29, 33, 34, 41, 100

Interface-Capturing Techniques (ICT), 56,

gob, 6, 7, 77 lime-aluminosilicate, 34 obsidian, 2

57 Interface-Tracking Techniques (ITT), 12, 56

recycled, 5

internal energy, 24, 27, 37

silica, 25, 34

interpolation, 14, 115

soda-lime, 26 soda-lime-silica, 27, 29, 33

spline, 114, 126 isotropic, 28, 32, 47

glass melting, 5 tank furnace, 4, 5 glass transition temperature, 9, 11, 25, 34 global convergence, 118

Johnson-Segalman model, 30 jump conditions, 34–35, 38–39, 42, 48, 52 K-BKZ model, 30

gravity, 5, 6, 38, 41, 42, 44, 97, 112 level set function, 16, 57–64, 70–71, 73, heat

74, 76, 78, 83, 84, 109, 110, capacity, 28, 37 conduction, 26, 45, 47 flux, 24, 26, 35, 37, 45 radiation, 26, 27, 37

115, 116, 126, 135 Level Set Methods (LSMs), 15, 16, 56– 64, 69–71, 73–74, 83, 84 Levenberg-Marquardt

source density, 24, 38

method, 14, 121–122, 134, 138

specific, 27–28, 38, 41, 43, 44, 54

parameter, 121–122, 127, 128, 134

transfer, 5, 7, 23, 24, 26–28, 35, 37– 40, 42, 43, 45–100 transfer coefficient, 14, 40, 42

Levy-Mises flow rule, 31 line search, 118 Lipschitz

hyperelastic, 30, 31

constant, 124

hyperplastic, 31

continuity, 124

ill-posed problems, 16, 92, 116, 121

Maxwell material, 28

initial conditions, 35, 37, 48, 52, 106,

melt surfaces, 22, 37, 56, 76, 83, 92–94,

108 injection moulding, 9–11, 57

96, 97, 100–102, 104, 110 melting point

170

INDEX glass, 5 polymers, 9

mesh size, 12, 56, 73, 74, 99, 126–128, 138 metal forming, 31 method of divergence-free basis functions, 66

error tolerance, 17, 98, 99, 122, 124, 134 initial guess, 14–15, 17, 112, 118, 128–129, 133–135, 137, 138 parameters, 14, 17, 114–116, 124, 126, 129, 132, 138 problem, 15, 17, 113–115

minimum fill reordering, 65, 67

search step, 121, 122

mould, 1, 4, 6, 7, 9, 11, 22, 41, 44, 77,

shape, 14, 15, 31, 113, 114

97, 100 surface, 16, 22, 92–96, 99–102, 104, 105, 107, 109, 132, 136–137 moving boundaries, 16, 34–35, 37, 48, 51, 56–61, 70, 76, 83, 135 Navier-Stokes equations, 43, 44 Newton method nonlinear equations, 119 optimisation, 117, 118

parametric curve, 14, 17, 113–115, 125, 129, 132, 137 parison, see preform penalty function method, 65, 66 plasticator, 9 plasticising, 9 plastics, see polymers plunger, 7 Poisson’s ratio, 29 polymers, 4, 7–9, 25, 27, 28, 34

Newtonian fluid, 29

cellulose acetate, 4

non-Newtonian fluid, 32, 47

cellulose nitrate, 4

nonlinear least squares, see Gauss-Newton

degradation, 9

method, see Gauss-Newton numerical integration

polyethylene, 4, 7, 9, 25, 34 polyethylene terephthalate (PET), 4,

adaptive simpson, 137

11, 16, 25, 28, 30, 44–47, 61,

composite Gaussian quadrature, 116,

83, 85, 87

117, 124, 125

polypropylene, 4, 7, 34 polystyrene, 4, 7, 34

objective function, 116–119, 122, 127, 128, 131, 134 weighted, 130, 133, 138 optimisation

PVC, 9 resin, 9 preform, 5–7, 9–11, 13–15, 28, 69, 74, 76, 77, 84, 90, 96–98, 100–102,

derivative-based, 117, 122

109, 110, 113, 115, 125, 129,

derivative-free, 120, 121, 140

130, 133–135, 138, 140

INDEX

171

pressurised air, 5, 6, 9–11, 36, 37, 43–44, 47, 77 pyrometer, 79

Incomplete LU factorisation, 65, 76 splines, 14, 114, 115, 126, 129, 132, 133, 137, 138

Quasi-Newton methods, 118, 119 re-initialisation, 16, 61–64, 70–73, 83, 126

Stefan Boltzmann radiation constant, 27 Stokes flow, 42, 44, 47, 50–54, 61, 65, 100, 103

re-meshing, 12, 16, 57

stopping criterium, 120, 122, 127

reaming, 9

strain rate, 29, 31, 32, 47, 61, 85

refraction index, 27

tensor, 28, 29

regularisation, 121

stream function, 97

residual, 117, 119, 123, 124

streamline-upwind Petrov-Galerkin (SUPG),

rheology, 24, 28–32, 46 Richardson extrapolation, 126

67 stress, 5, 6, 11, 49

ring, 7, 22, 37

equivalent, 31, 32

Rosseland

rate tensor, 29

approximation, 26

relaxation, 6, 31, 46, 87, 95

parameter, 27

shear, 29 tensor, 24, 30, 35, 38, 46

sagging, 6, 41, 77, 138 screw, 9

yield strength, 46 stretch rod, 11, 22, 23, 61, 84, 85

search step, 122, 127, 128

speed, 36, 84, 85

secant methods, 118–120, 122, 138

surface, 61, 83, 84

BFGS, 118, 119, 126

stretch time, 61, 85

Broyden’s method, 17, 119, 121–124,

surface tension, 17, 34, 41, 42, 44, 46,

126, 128, 134, 138 bad method, 119, 120, 124

92, 93, 95, 97 surface treatment, 6

good method, 119, 120, 123, 124 sensitivity, 15–17, 96–99, 112

tensile modulus, 32, 46, 85

solvers

thermal expansion, 25, 37

direct, 64 iterative, 64 BiCGstab, 64, 76 pre-conditioners Geometric multi-grid, 65

coefficient, 25, 29 thermocouple, 79 thermodynamical identity, 27 thermodynamics, see heat transfer thermoforming, 30

172

INDEX

thermoplastics, see polymers time step, 51, 57, 70, 73, 74, 82, 83, 87, 126 trimming, 9 trust region, 121 method, 118, 121 radius, 118 subproblem, 121 uniaxial tensile test, 32, 85 Variational Formulation, 52–55 VFT-equation, 33 viscoelastic, 28, 30, 31, 47 viscoplastic, 30, 31, 46 viscosity, 5, 29, 32–34, 38, 41, 43–45, 61, 84, 85, 95, 100, 101, 103 Newtonian, see zero-shear-rate zero-shear-rate, 29, 34 Volterra integral equation of the first kind, 108 Volume-Of-Fluid Methods (VOFMs), 56– 57 weighted penalty functions, 130, 133, 138 WLF-equation, 32, 34 Young’s modulus, 29

Curriculum Vitae Hans Groot was born on 02-09-1980 in Hengelo, The Netherlands. After finishing atheneum in 1999 at Titus Brandsma Lyceum in Oss, The Netherlands, he studied Industrial Applied Mathematics at Eindhoven University of Technology in Eindhoven, The Netherlands. In the summer of 2005 he did an internship at DLR in Berlin, Germany, as part of his Master studies. In November 2005 he worked as a researcher at the Hermann Foettinger Institute in Berlin, Germany as a continuation of his internship. In the summer of 2006 he did a Master project at TNO Science and Technology in Eindhoven, The Netherlands. In the winter of 2006 he worked as a student assistant for LIME Consultancy at Eindhoven University of Technology in Eindhoven, The Netherlands. In 2007 he graduated within CASA on Computational Science and Engineering. From 2007 he started a PhD project at Eindhoven University of Technology in Eindhoven, The Netherlands, of which the results are presented in this dissertation. During his PhD he has been active at the ASME Pressure Vessels and Piping Conference as a.o. co-chair and reviewer. In 2010 he received an Honorable Mention and Award in the PhD category of the PVP2010 Student Paper Symposium and Competition at 2010 ASME Pressure Vessels and Piping/K-PVP conference, Bellevue, Washington.

Numerical shape optimisation in blow moulding

Numerical shape optimisation in blow moulding

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