Numerical shape optimisation in blow moulding
Citation for published version (APA):Groot, J. A. W. M. (2011). Numerical shape optimisation in blow moulding. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR709254
DOI:
10.6100/IR709254
Document status and date: Published: 01/01/2011
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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
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
prof.dr.ir. R.M.M. Mattheij
Copromotor: dr. C.G. Giannopapa
Dimensionless Numbers Br Brinkman number , page 39
Br= ¯ ¯µV2
λ T0−Tm
!
De Deborah number , page 46
De= τµp Fr Froude number , page 39 Fr= VgL2 Nu Nusselt number , page 40
Nu=αD¯ ¯ λ
Pe P´eclet number , page 39
Pe=ρ¯cpVD
¯ λ
Re Reynolds number , page 39
Re= ρVD¯µ
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
I identity matrix, page 121
J Jacobian matrix of least squares residual, page 119
q heat flux , page 24 [W m−2] r least squares resiudal, page 117 t unit tangent, page 36
u flow velocity , page 24 [m s−1]
x position , page 24 [m]
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 [s−1] I identity tensor, page 29
T Cauchy stress tensor , page 24 [Pa]
Greek
α heat transfer coefficient , page 37 [W m−2K−1] ¯
α typical heat transfer coefficient ,
page 40 [W m−2K−1]
¯
λ typical conductivity , page 38 [W m−1K−1] ¯µ typical viscosity , page 38 [Pa s] β friction coefficient , page 36
[N m−3s] δ step size of difference formula,
page 122
˙ε strain rate , page 31 [s−1] total error of optimisation method,
page 125
C error in computation of the least
squares residual, page 124
F error in numerical solution of
for-ward problem, page 123 I interpolation error, page 125
R rounding error, page 125
T truncation error, page 124 Γ boundary/surface, page 22
γ surface tension , page 34 [N m−1] Γ1 inner melt surface, page 22
Γ2 outer melt surface, page 22
Γf melt-air boundary, page 22
Γi inner container surface, page 90
γLip Lipschitz constant, page 124 Γm mould boundary/surface, page 22
Γo outer boundary, page 22
Γs symmetry axis, page 22
κ curvature , page 34 [m−1]
λ effective conductivity , page 26 [W m−1K−1] λLM Levenberg-Marquardt
parameter, page 121
µ dynamic viscosity , page 29 [Pa s] µ0 zero-shear-rate viscosity , page 29
[Pa s] Ω fluid domain, page 22
ωk kth weight of Gaussian quadrature rule, page 117
Φ heat source density , page 24 [W m−3] Φ objective function, page 116 ρ density , page 24 [kg m−3] Σ forming machine domain, page 22
σ equivalent stress , page 31 [Pa] σy yield strength , page 46 [Pa]
θ level set function, page 57
θ1 level set function corresponding to inner melt-air interface, page 58 θ2 level set function corresponding to
outer melt-air interface, page 58 ϕ polar coordinate/angle, page 97 ϕc expansion angle, page 104 e
Φ weighted objective function, page 130
ζ sum of weighted penalty functions, page 130
Latin
¯cp typical specific heat , page 38 [J kg−1K−1] Cn space of n times continuously di
ffer-entiable functions, page 52
Cn0 space of functions in Cn that van-ish on the boundary of the domain, page 59
H1 Sobolev space, page 52
Lp set of Lebesgue p-integrable func-tions, page 52
c geometric constraint function, page 130
cp specific heat , page 27 [J kg −1K−1]
D typical diameter , page 38 [m] d signed distance function, page 58 e specific internal energy , page 24
g gravitational acceleration in axial direction , page 38 [m s−2] h typical mesh size, page 126 k tensile modulus , page 32 [Pa s] L typical length scale , page 38 [m] m reciprocal of power-law index,
page 32
nG number of points for Gaussian
quadrature rule, page 116
nint number of subintervals for
compos-ite quadrature rule, page 117 np number of parameters, page 115
p pressure , page 29 [Pa]
pin inlet pressure , page 36 [Pa] R spherically radial coordinate,
page 97
r cylindrically radial coordinate, page 77
R1 spherical radius of inner melt
sur-face, page 101
R2 spherical radius of outer melt sur-face, page 101
Ri spherical radius of inner container surface, page 102
Rm spherical radius of mould surface,
page 102
t time , page 24 [s]
Tg glass transition temperature ,
page 25 [K]
V typical flow velocity , page 38 [m s−1] Vr stretch rod speed , page 36 [m s−1] z axial coordinate, page 77
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 24v
·
A= vTAT:
double dot product , page 24A
:
B= tr AB dev deviatoric part , page 29
dev T= T −1 3tr T
~. double square brackets for jump conditions , page 35
~A = A2− A1
D
Dt material derivative , page 24
D
Dt =
∂
∂t+ u
·
∇ ⊗
∇ differential operator , page 24
∇= (∂x
1, . . . , ∂xn)
T
⊗ matrix/tensor product , page 24
A ⊗ B= ABT ∂ boundary of domain , page 22
∂S = S \S◦
!
tr trace , page 29 tr A= P
jAj j
Superscripts
∗ dimensionless quantity (omitted from Chapter 3), page 38
T transpose, page 29
Subscripts
* end of process, page 90 a air, page 22
b baffle, page 22 c contact point, page 94 F forward problem, page 123 f free surface, page 22
G Gaussian quadrature rule, page 116 I interpolation, page 125
int interval, page 117
l forming material/melt, page 22 Lip Lipschitz, page 124
LM Levenberg-Marquardt, page 121 M measurement, page 125
m mould, page 22
o outer/enclosing, page 22 opt optimum, page 117 p parameters, page 115 q equipment, page 22 R rounding, page 125 r stretch rod, page 22 res residual, page 119 s symmetry (axis), page 22 T truncation, page 124
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 Kreji´c, 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 implementa-tion of the blow moulding model; without his support the numerical simulaimplementa-tions would probably not have been possible.
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 L¨opker, 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 Mar`ese Wolfs-van 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. Dur-ing this period many social events were organised within CASA, varyDur-ing from daily lunches to weekly game evenings, regular sport events or nights out in town to the half-yearly CASA PhD days, the half-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 in-creasing 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
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, Lu-cia Scardia, Olga Shchetnikava, Berkan Sesen, Antonino Simone, Sudhir Srivastava, J¨urgen 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.
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 devel-oped. 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 prob-lem 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 distri-bution 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
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, Bezi´er curve, and computes the optimal positions of the control points of the curve as to minimise the objective function. The objective function represents the dis-tance 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. There-fore, 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 analyt-ical 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.
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 vari¨eren 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 ontwor-pen. 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 initi¨ele bedrijfscondities gezocht, zoals de voorvorm en de initi¨ele temperatuurverdeling, om een houder met precies deze wand-dikte 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.
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 voorvormopper-vlak door middel van een parametrische kromme, bijv. een spline of een Bezi´er 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 berekenin-gen en kan het dominante deel van de rekenkracht vormen. Daarom worden eindige dif-ferentie benaderingen van de Jacobiaan matrix gecombineerd met secant updates. Een foutanalyse wordt uitgevoerd om een optimale fouttolerantie voor het optimalisatie algo-ritme 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 bolco¨ordinaten te verwaarlozen, zodat het inverse probleem analytisch kan worden opgelost.
1 Introduction 1
1.1 Blow Moulding Manufacturing Processes . . . 1
1.1.1 History . . . 1
1.1.2 Hollow Container Manufacturing . . . 5
1.2 Process Simulation and Optimisation . . . 11
1.2.1 Simulation . . . 11
1.2.2 Optimisation . . . 12
1.3 Objectives . . . 15
1.4 Thesis Outline . . . 17
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
2.7.1 Glass blow moulding . . . 41
2.7.2 PET Stretch Blow Moulding . . . 44
2.7.3 General Blow Moulding . . . 47
3 Numerical Methods 49 3.1 Discretisation Procedure . . . 49
3.2 Variational Formulation . . . 52
3.3 Interface Capturing . . . 56
3.4 Solution Methods . . . 64
4 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
5 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
6 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
7 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
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 ap-plications has rapidly increased over the last century, traditional blow moulding is a technique that has been invented thousands of years ago.
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 cave-dwellers 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 be-tween 2750 and 2625 B.C. [184]. However, it is thought that the Egyptians and Baby-lonians 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 Thout-mosis III (1504-1450 BC), who brought glassmakers to Egypt as prisoners.
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 blow-ing 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 Ro-mans 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, Eng-land, around 1821 [18, 67]. Still, until the end of the eighteenth century, glass manufac-turing 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
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 pro-duction 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 keep-ing liquids in householdkeep-ing [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
America [115]. Nowadays the range and variety of blow moulded products is still in-creasing.
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
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 form-ing machine. In the formform-ing 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. Con-secutive 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 pari-son is formed. Two widely used techniques are blow-blow moulding and press-blow moulding.
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 sig-nificantly. 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 hy-drogen, 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.
gob mould plunger baffle (a)press preform (b)blow
Figure 1.5: Schematic drawing of a press-blow process
mould air baffle
glass
ring
(a)settle blow
mould glass air baffle 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 non-alcoholic beverages c Pascal Artur/VOA; photo used courtesy of the Phototheque of Saint-Gobain.
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 plasti-cator. 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 tem-perature 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 pres-surised 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 em-ploying rotating knives are used to trim the top of a bottle. Other trimming opera-tions 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].
distinguished: injection blow moulding, extrusion blow moulding and stretch blow mould-ing.
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 (a) mould preform (b) container (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
mould around the parison when it leaves the die (Fig. 1.8(a)-1.8(b)) and pres-surised 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 mould-ing process. First a parison is formed by injection mouldmould-ing. Usually, the parison is packaged for later use. Then in the stretch blow moulding process it is re-heated 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
Process Simulation and Optimisation
1.2.1 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. Un-fortunately, measurements are often complicated, considering that blow moulding pro-cesses take place at high rates in closed constructions under complicated circumstances, such as high temperatures. Furthermore, trial-and-error experiments with blow mould-ing equipment are usually expensive and time consummould-ing. 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 infor-mation 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 sim-ulations can also be used for comparison with measurements.
Process optimisation: to optimise an existing blow moulding process. Process simu-lations 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 pro-cess, 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 Tech-niques, 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.
The difficulty in blow moulding a container with a prescribed wall thickness distri-bution 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 com-pute the corresponding container. Then the initial operating settings of the blow mould-ing 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 exper-tise 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 de-sired 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 distribu-tion 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 rheo-logical 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 distri-bution 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 pre-form. Several other optimisation methods have been considered in literature to optimise the wall thickness distribution in stretch blow moulding [190] and extrusion blow mould-ing [73, 95, 213] of polymers in three dimensions. An engineermould-ing 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
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 dis-tribution 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 pre-form 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 rela-tively 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 Bezi´er 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 Bezi´er 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 thick-ness, 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 ini-tial guessshould 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 al-ways the best choice, for example if measurements are complicated and knowledge about
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, pro-vided 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 con-tainer 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 for-ward 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
physical quantities, which are not included in the general problem formulation. Further-more, 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 pro-cess, 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 ad-dressed. Inverse problems, particularly in engineering, are often ill-posed. Blow mould-ing processes involve heat transfer by convection and conduction and shape transforma-tion 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 sur-face of the blow moulded container coincides with the mould sursur-face. 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.
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 op-timisation strategies to solve the inverse opop-timisation 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 dif-ferences 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 optimi-sation 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 analyt-ical 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
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 pro-cess. Firstly, in § 2.1 the geometry of the forming machine is defined and the compu-tational 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 appli-cable 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 con-ditions 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 dimen-sionless form based on specific data for the various materials in order to simplify the problem formulation. This section is concluded by formulating a dimensionless, simpli-fied 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 varia-tional 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
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 re-initialisation 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 for-mulations 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 solv-able. 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 dura-tion 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.
Chapter 6 deals with efficient shape optimisation strategies to solve the inverse op-timisation problem. In § 6.1 the opop-timisation problem is formulated and the 2D axial-symmetrical unknown surface is parameterised by means of a parametric curve. The pa-rameters 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 cor-responds 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 al-gorithms 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 prob-lem 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.
Mathematical Modelling of Blow
Moulding
In this chapter a general and complete mathematical model for blow moulding is pre-sented. 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
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 byΣ. The equipment domain can be sub-divided into subdomains for different parts, such as a mould domain Σm, a stretch rod
domainΣrand a baffle domain Σb. The ‘fluid’ domain Ω consists of the open forming material or melt domainΩl, the open air domainΩa, the inner melt-air interfaceΓ1and the outer melt-air interfaceΓ2. The melt-air boundary, which is also referred to as the melt surfaces, is given byΓf = Γ1∪Γ2. For glass and extrusion blow moulding the fluid domainΩ := Ωl∪Ωa∪Γfis fixed, while for stretch blow mouldingΩ changes in time due to the motion of the stretch rod. Furthermore,ΩlandΩaare 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. DomainΣ is enclosed byΓo∪Γs. DomainΩ 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,
Figure 2.1 illustrates the domain decomposition of forming machines for 2D axial-symmetrical 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.
baffle Γs Ω Ωa air Γ2 Γ Γb Γo Γo n t Γa,o Γg,o Ωg glass Γm Γ1 mould r z
(a) counter blow moulding machine
t Ωa air Ωa air Γo Γm Γ1 Γ2 Γs Γa,oΓg,o Σr stretch rod Γr,o t Ωg material mould r z Γr stretch rod n
(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 mod-elling it can be assumed that the equipment has constant temperature, so that the mathe-matical model can be restricted to the forming material and air.
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 Dt + ρ∇
·
u= 0, in Ω \ Γf× T, (2.2.1a) ρDu Dt = ∇·
T + ρg, in Ω \ Γf× T, (2.2.1b) ρDe Dt = T:
∇ ⊗ u − ∇·
q+ Φ, in Ω \ Γf× T, (2.2.1c) where T is the time interval of the forming process. The computational domain is re-stricted 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 veloc-ity 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 mate-rial and air and depend on the solution, the gravitational acceleration g [m s−2] is con-stant and the heat source densityΦ [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 equationdx
dt = u 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 consid-ered, nl. thermodynamical and rheological relationships. Thermodynamical constitutive relations are important, as blow moulding involves high temperatures; the typical tem-perature 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 ma-terial. The rheological constitutive equations describe the deformation that the forming material undergoes; they capture the stress-strain constitutive relation.
