Numerical Modeling of Viscoelastic Stress Relaxation During GlassLens Forming Process

Anurag Jain and Allen Y. Yiw

Department of Industrial, Welding and Systems Engineering, The Ohio State University, Columbus, Ohio 43210

With the recent advances in numerical simulation capabilitiesand computing technology, finite element method (FEM) can beapplied to predict the performance of a precision aspherical lensmolding process. In this paper, various stages of the lens moldingprocess have been modeled using a commercial FEM code MSCMARC. Stress relaxation effect during the forming stage hasbeen incorporated into the numerical model by using a general-ized Maxwell model. Successful comparison of the predictedresults has been made with the experimental data. The variousaspects of the simulation that would enable a more realisticmodeling of the process have been identified for future research.

I. Introduction

THE conventional production of lenses is a rather complicatedprocess. Glass raw material that has been pressed to a rough

form has to be processed on both sides in order to obtain a fin-ished precision optical component. The lens blank is cleaned andmounted on a grinding block, and the first side is then groundand polished. The same procedure is then repeated for the sec-ond side. The lens is then centered and cleaned. Although therehave been recent advances in the optical fabrication techniquessuch as magnetorheological polishing, ion beam polishing, andsingle point diamond turning, the overall complexity and costinvolved in these processes are very high for medium to highvolume production of aspherical glass optics. In addition, thereare environmental concerns because of the use of a cutting fluidand health hazards associated with glasses having high leadconcentrations.

The conventional fabrication methods are more suitable formanufacturing spherical glass lenses. However, these lenses re-sult in errors in imaging such as spherical aberration, coma, andastigmatism. Aspherical lenses, on the other hand, have one orboth surfaces that do not conform to a sphere and providegreater advantage over spherical lenses because of reduced lightlosses and aberrations, better image quality, and compact lensassemblies. Some of the important fields of application ofaspherical optical components are medical equipment, cameras,CD-ROM technology, and military equipment. In order tofulfill the high demands on refractive and reflective opticalcomponents, it is necessary to develop accurate fabrication tech-niques that generate aspherical optical surfaces in glass with afigure accuracy of l/10 (l5 633 nm), and a surface roughness of1–5 nm rms.

For over 30 years, work has been carried out on developing aprocess for precision molding of aspherical glass lenses.1–4 In alens molding process2 a glass blank or gob is initially heated toa temperature above its transition and subsequently pressed be-tween the two mold halves into a lens shape. Controlled cooling(annealing) of the formed lens is then carried out with the molds

in the closed position in order to remove forming stresses andavoid heat sink marks on the lens surface. The lens is eventuallyreleased at a temperature close to room temperature and finallyallowed to cool to ambient temperature. If designed correctly,this process can be easily adapted for high volume production ofprecision aspherical glass lenses. Additionally, this process is anet shape and an environmentally friendly process.

Although the precision molding process saves productionsteps in comparison with the traditional production methods,there are difficulties in processing that must be overcome. Themolds must have a very precise form and a smooth surface thatcan resist attack from the hot glass during repeated pressingcycles. Careful temperature control, thermal expansion of themolds, curve conformance, mold life, and adherence of glassmaterial to the mold surface are some of the other technicalchallenges that are associated with this process. This paper anal-yzes the various stages of the lens molding operation using finiteelement method (FEM) simulation program by considering theviscoelastic nature of glass. It is aimed to develop a reliableFEM model that can be utilized to gain a fundamental under-standing of the process and address some of the aforementionedissues related to it.

II. Background

Because of the industrial relevance of the process, considerableresearch has been carried out by various researchers in the areaof glass molding.5–8 Most of this research deals with thermalaspects of the molding process like heat transfer across theglass–mold interface, contact behavior of hot glass with the mo-lds, and temperature distribution in glass during forming oper-ations. Research has usually been performed using experimentaltechniques and one- and two-dimensional mathematical modelshave been formulated.

A limited amount of work has been published in the area ofglass forming simulation. Most of this work is based on the as-sumption that glass is a Newtonian fluid, and viscoelastic effectsand structural relaxation have been neglected.9–11 Soules et al.12

used the finite element program MARC to determine the ther-mal stresses induced during uniform cooling, reheating, and iso-thermal hold in case of different glass plate configurations. Theviscoelastic theory describing stress and structural relaxation ef-fects was incorporated into the FEM program to determine theresidual stress distribution inside glass. A good comparison ofthe predicted results was obtained with the analytical results andexperimental stress measurements. Hyre13 used the FEM pro-gram POLYFLOW for the numerical simulation of differentstages of glass container manufacturing process. By implement-ing FEM, the author was able to gain a better insight into theimpact of forming and conditioning stage process parameters onthe final container quality, which can help in further improvingthe process.

The above-mentioned research activities were conducted withgreat detail but did not address the issues concerning precisionglass molding process and its numerical modeling. Our imme-diate effort therefore is to expand their results to molding ofprecision glass optics and accurately predict the performance of

JournalJ. Am. Ceram. Soc., 88 [3] 530–535 (2005)

DOI: 10.1111/j.1551-2916.2005.00114.x

530

W.-Y. Ching—contributing editor

wAuthor to whom correspondence should be addressed. email: yi.71@osu.eduManuscript No. 11061. Received May 25, 2004; approved September 7, 2004.

the process using numerical simulation. In an earlier researchcarried out by the authors, lens molding experiments were per-formed to determine the feasibility of the molding process tomake glass lenses for high-precision optical applications.2 In thisstudy FEM simulations have been conducted, corresponding tothe experimental conditions in reference2, and a comparison ofexperimental results with the predicted data has been made.

III. Research Objective

The overall objective of this study was the numerical simulationof the lens molding process using FEM. The specific tasks in thisstudy were to:

(i) include a mathematical model of viscoelasticity (stressrelaxation) of glass material into the numerical simulation andobserve its effect on the molding loads and residual stresses in-side the formed lens.

(ii) predict mold distortion because of thermal loading andlens surface deflection at different stages of the molding process.

(iii) determine the feasibility of applying FEM for predict-ing lens molding process performance and outlining other issuesthat may help in better modeling and process prediction.

IV. Numerical Model

In the present study, Two-dimensional-axisymmetric simu-lations of the lens molding process were conducted using acommercial general-purpose, non-linear FEM programMARC,which is suitable for viscoelastic modeling of materials.MARC assumes that the deviatoric and volumetric behaviorof a viscoelastic material can be fully uncoupled and that thebehavior can be described by a time-dependent shear and bulkmodulus. Glass can be modeled as a thermorheologically simplematerial by means of the Narayanaswamy14,15 model that ena-bles the user to study the time dependence of physical propertiesof glass when subjected to temperature change. The governingand constitutive equations were solved using the appropriateboundary conditions and have been described in the followingsection.

(1) Governing Equations

When modeling incompressible materials, the stress can be di-vided according to the Cauchy stress tensor16 given by

r ¼ �pIþ s (1)

where the first term on the right-hand side is the hydrostaticstress corresponding to pressure p, I is the unit tensor, and s isthe extra stress tensor, that basically constitutes the shear stresscomponents of r and is responsible for the fluid deformation.

The conservation of linear momentum for the system gives16

rDv

Dt¼ H � sþ rb (2)

where the operator D/Dt is the material time derivative, v is thevelocity vector, r is the density, and b is a body force per unitmass acting at a given location in a body.

Assuming that the density of glass during the entire formingprocess remains constant, the mass balance equation reducesto16

H � v ¼ 0 (3)

Finally, the energy balance equation for the incompressiblefluid is given by16

rCpqTqt

þ v � HT� �

¼ kH2Tþ s : Dþ r _R (4)

where Cp is the specific heat, k is the thermal conductivity, Dand s are the rate of deformation and extra stress tensor, re-spectively, and _R is the rate of internal heat generation.

Glass was modeled as a Newtonian fluid in which case theextra stress tensor is given by16

s ¼ 2ZD (5)

where Z is the shear viscosity, which, for Newtonian fluids, var-ies only with temperature.

(2) Viscoelastic Stress Analysis

Viscoelastic behavior is the time-dependent response of a mate-rial to stress or strain. In the class of viscoelastic materials suchas glass, the application of a constant load is followed by a de-formation, which can be made up of instantaneous deformation(elastic effect) followed by continual deformation with time (vis-cous effect), which results in the decay of the applied load and istermed as relaxation. Stress relaxation in high-temperature glasshas been studied extensively by glass researchers.17–21

Viscoelastic stress relaxation can best be illustrated with thehelp of mechanical models—combinations of springs and dash-pots, where the spring represents the elastic behavior and dash-pot represents viscous behavior. The generalized Maxwellmodel, shown in Fig. 1, was used to represent the stress relax-ation behavior of glass in this study. In the figure, Gi representsthe elastic shear modulus of the springs and Zi represents theviscosity of the dashpot.

The stress relaxation modulus (G1(t)) and the stress relaxationfunction (c1(t)) for this model can be represented by22

G1ðtÞ ¼ 2GXmi¼1

wi exp �t=tið Þ (6)

c1ðtÞ �G1ðtÞG1ð0Þ

¼Xmi¼1

wi expð�t=tiÞ (7)

where G1(t) is the shear stress relaxation modulus, a time-de-pendent analog of shear modulus G (at t5 0, G1(0)5 2G), ti isthe shear stress relaxation time (s) given by Zi/Gi, and wi is theweighing factor such that

Xmi¼1

wi51:

A viscoelastic counterpart of the elastic-thermal analysisequation that is implemented in FEM is given by Eq. (8)12

rðtÞ ¼Z t

0

DEðt� t0Þ d

dt0eðt0Þ � ethðt0Þ� �

dt0 (8)

where DE is the elastic modulus matrix, r is the stress, and e andeth are the actual and thermal strain tensors, respectively. Theshear modulus G in matrix DE is then replaced by G1 (t). Both

G Gm

Gi

1

i

m

e12

1

η η η

Fig. 1. Generalized Maxwell model for modeling viscoelastic stressbehavior.

March 2005 Numerical Modeling of Viscoelastic Stress Relaxation 531

the thermal strain and the elastic modulus would relax with timein the glass transition zone and therefore the change in strainshould be calculated in increments between t�Dt and t. Thetotal change in stress is then given by

Dr ¼ DE½De� Deth� �Xmi¼1

ð1� e�Dt=ti Þriðt� DtÞ (9)

where Dt is a small time step and ri is a partial stress component.The portion of the stress that relaxed during the time step Dt andthe change in the thermally induced stress are converted to nodalforces and subtracted from external forces in the model. Chang-es in displacements, strains, and stresses can be calculated in thesame way as in a simple elastic analysis.12

(3) Material Properties

Table I summarizes the room temperature thermal and mechan-ical properties of the glass blank and the molds that were used inthe simulation. The properties of BK-7 glass23,24 were used tomodel the glass disk, and the properties of Tungsten carbide(Fuji alloy, J05 grade)25 were used to model the upper and lowermolds and dies (see Fig. 3). Stress relaxation times (ti) andweighing factors (wi)

12,26 (used in Eq. (7)) for a similar glass at6001C have been tabulated in Table II.

The relaxation times at 7001C, which is the molding temper-ature in our study, was estimated using the Arrhenius equationgiven by19

t ¼ t0 exp DH=RTð Þ (10)

where s0 is a constant, DH is the activation energy (5.91� 105 J/mol)12, and R is the ideal gas constant.

The instantaneous value of the shear modulus (G1(0) in Eq.(7)) at forming temperature was obtained from Spinner27 andGoldblatt et al.28 It is expected that at temperatures higher thanthe transition temperature (B5571C for BK7 glass in our case),the structure of glass breaks down and there is a rapid decreasein the elastic characteristics of glass. The relaxation times arevery small because of very low viscosities and therefore viscousflow is a dominant mode of energy dissipation. Because of thedependence of physical properties on glass composition, the val-ue of the shear modulus was selected for a glass having a similarcomposition as BK7 glass.

(4) Boundary Conditions and Contact

Figure 2 shows the two-dimensional-axisymmetric model of alens molding process with the displacement and thermal bound-ary conditions applied to it. The lower mold and dies were con-strained in the y direction. Mold movement used in the actualexperiments was applied to the numerical model, as a displace-ment boundary condition to the upper mold and dies. Addi-tionally, a gravitation boundary condition was applied to theglass blank in order to take into account the flow of glass be-cause of gravity loads at molding temperature. For the glassblank it was assumed that conduction and convection are theprimary modes of heat transfer mechanism. A convective heattransfer coefficient of 20 W � (m2 �K)�1 and a constant value ofthe inter-body heat transfer coefficient of 5000 W � (m2 �K)�1

was used to simulate the heat exchange between the glass andmolds.29,30 The thermal boundary conditions applied to theglass blank are

�kqTqn

¼ hintðT � TcÞonSc (11)

�kqTqn

¼ h1ðT � T1Þ onSh (12)

where k is the thermal conductivity of glass, Tc is the contacttemperature of the opposite object at the glass–mold interface,TN is the sink/source temperature, n is the unit vector normal tothe boundary surface, and hint and hN are the inter-body heattransfer coefficient and convective coefficient, respectively. Ther-mal conductivity of glass varies with temperature in a complexmanner because of the radiation heat transfer effects at highertemperatures. Many investigators have treated the heat transferin glass at higher temperatures by a term called ‘‘effective’’ or‘‘apparent’’ thermal conductivity, which is a characteristic of thegiven experimental conditions, and hence varies with each set-up.8 Because of the unavailability of data a constant value ofthermal conductivity was used in our study. Additionally, theresults from measurement and computer calculation performedby different researchers have shown that for the molding con-ditions applied in this study (i.e. molding temperature and glassselection), radiation will have a negligible contribution to theheat transfer at the glass–mold interface and also withinglass.8,31 Therefore, for simplicity the radiation heat transfermechanism was not taken into account in this study.

Mold temperature history obtained from the experiments wasapplied to the numerical model through fixed temperatureboundary condition at the nodes. Because of the high thermaldiffusivity of tungsten carbide molds, it was assumed that theentire mold object would be at a uniform constant temperature,i.e. temperature at the measurement location is the same as thetemperature of the entire mold object.

Upper mold Upper mold die

Lower mold Lower mold die

Glass blank

X

Y

Z

Constrainedboundary conditionDisplacementboundary conditionConvective boundarycondition

Interface heat transfer

Sc

Sc

Sc

Sc

ShSh

Sh

Fig. 2. Two-dimensional-axisymmetric model of lens molding.

Table I. Mechanical and Thermal Properties of Glass Blankw

and Moldsz

BK 7 glass

Tungsten carbide

(Fuji alloy, J05

grade)

Material type Viscoelastic ElasticElastic modulus (MPa) 81 000 570 000Density (kg/m3) 2500 14 650Thermal conductivity(W � (m � 1C)�1)

1.1 63

Specific heat (J � (kg � 1C)�1) 800 450Coefficient of thermal expansion(� 10�6/1C)

7.1 4.9

wSchott Glass Inc.23 and Bansal and Doremus.24 zFujidie Co.25

Table II. Weighing Factors and Relaxation Times Used in theNumerical Model for Viscoelastic Analysis

w

Term no. Weighing factor Relaxation time (s)

1 0.422 0.09972 0.423 0.00943 0.154 0.0003

wSoules et al.12 and MSC MARC.26

532 Journal of the American Ceramic Society—Jain and Yi Vol. 88, No. 3

A simplified shear friction model represented by Eq. (13) wasused to model glass–mold interface friction

t ¼ m � km (13)

where t is the shear strength of the interface (MPa), m is theshear factor (0rmr1), and km is shear yield stress of the softermaterial (MPa). A shear friction coefficient of 1.0 was used thatassumes that complete sticking takes place between the glass andthe molds.11

Four-node axisymmetric, thermo-mechanical coupled, quad-rilateral elements were used to model the objects. Approximate-ly 2500 elements were used to model the glass blank, 500elements were used to model the upper and lower molds, and200 elements were used to model the mold dies. No remeshingcriterion was used during the simulation. The complete simula-tion was performed in five stages: (1) Heating of molds and glassblank to a molding temperature of 7001C, (2) Lens forming at7001C (adiabatic) at a mold velocity of 0.1 mm/s, (3) Initial slowcooling of closed mold assembly and lens (at the rate of 0.81C/s),(4) Rapid cooling of the closed mold assembly and lens to thelens release temperature of 2001C (at the rate of 1.61C/s), and (5)Cooling the lens independently to 201C.

V. Results and Discussion

(1) Molding Simulation

Figure 3 shows the different stages of the FEM simulation of alens molding process. Figure 3(a) shows the initial glass andmold configuration and corresponds to the heating stage, duringwhich the temperature of all the objects is raised from 201C, tothe molding temperature of 7001C. Figure 3(b) shows the mol-ding stage during which the glass blank is pressed between theupper and lower mold halves into the shape of a lens. Figure 3(c)shows the annealing stage of the simulation during which theformed lens was cooled from 7001C to the release temperature of2001C. As in the experiments the annealing in the simulationwas performed at two different cooling rates: (a) initial slowcooling at the rate of 0.81C/s and (b) fast cooling of 1.61C/s.

From the standpoint of process performance and quality ofthe lens, annealing stage is one of the most critical stages of thelens molding process. As the formed lens begins to lose heat tothe molds, a time-dependent change in the physical propertiesoccurs and the phenomenon is called structural relaxation, therate of which is inversely proportional to viscosity. In the liquidstate, glass is so hot and viscosity is so low that its structurechanges in step with temperature. However, at lower tempera-tures and higher viscosity structural changes lag behind temper-ature changes and therefore the structure begins to deviate fromthe equilibrium line. Therefore, when glass is cooled rapidly

from a temperature above the transition region, it retains someproperties of higher temperature, and these properties ‘‘relax’’ tothat characteristic of the lower temperature as a function oftime. This behavior is shown by most mechanical, thermal, andeven optical properties of glass-like refractive index. A mathe-matical model describing structural relaxation was not includedin our current simulation model and the authors therefore be-lieve that the predicted residual stress and lens geometry atdifferent temperature (discussed in Section V(3)) may not com-pletely agree with those observed in the actual process.

(2) Comparison of Molding Loads

Figure 4 shows the comparison between the experimental andpredicted molding loads. A very good agreement between theforce magnitudes is observed in the figure. It has to be noted thatbecause of the unavailability of the force data for blank formingexperiments, comparison has been made with the force resultsfrom the gob forming tests. Because of the same gob and blankmaterial (i.e. BK7 glass) and molding conditions used in the ex-periments (i.e. temperature, molding speed, mold geometry,etc.), it is expected that the forming load required to formboth blank and gob would approximately be the same. Hence,the force data are only shifted along the time axis because of thedifferent instants at which the molding stage was initiated in goband blank tests; the magnitude and the shape of the formingloads are similar.

Forces begin to increase at the instant the upper mold comesin contact with the glass material and reach a maximum value atthe end of the forming stage. A slight decrease in the magnitudeof the load is observed at the start of the annealing stage. Duringthis stage a small amount of deformation of the glass material

Fig. 3. (a) Heating of the glass blank and the molds to the molding temperature of 7001C. (b) Molding of the glass blank at constant temperature. (c)Annealing of the molded lens to the release temperature of 2001C.

0

100

200

300

400

500

600

700

0 200 400 600 1000800Time [Sec]

Predicted load Experimental load

Mo

ldin

g lo

ad [

N]

ReleaseAnnealing stageMolding

stageHeating

stage

Fig. 4. Comparison of experimental and predicted molding loads.

March 2005 Numerical Modeling of Viscoelastic Stress Relaxation 533

takes place (of the order of a few microns) because of the closingof the increasing gap between the upper and lower mold halvesbecause of the shrink in mold size at lower temperatures. Asteady load of 500 N is required to maintain the glass deforma-tion till the end of the annealing cycle. Finally, the force dropsdown to zero at the time of mold release.

These force results validate the stress relaxation model thatwas used as an input to the simulation. The relaxation timesdetermine the rate at which the forming stresses would decayinside the glass lens while the elastic and shear modulus char-acterize the resistance of the viscoelastic material to permanentdeformation. These data inputs would therefore determine theforming stresses inside glass during the molding stage and hencethe amount of load that would be required for deformation.

(3) Mold and Lens Distortion

Figure 5 shows the comparison between the mold geometry atroom temperature (shown with a mesh) and the molding tem-perature of 7001C. A uniform expansion of approximately 45mm on the aspherical surface of the upper mold is observed be-cause of the thermal expansion of the molds at high tempera-tures. A similar expansion would also be observed in the lowermold and the mold dies. The amount of mold expansion is det-rimental to the lens molding process, because of the accuracyrequirements of a precision lens.

Therefore, in order to mold high-precision optical elements, itis desirable to compensate for mold expansion and shrinkage bydesigning the molds with a small amount of offset. This offset inmold geometry is usually achieved by the mold iteration processin which molding tests are carried out repeatedly with differentmold geometries until the lens meets the design specifications.32

It may be possible to implement FEM simulation of mold dis-tortion for improving the compensation technology of opticalmolds.

Figure 6(a) shows the predicted deflection of the lens geom-etry during the cooling phase from the release temperature of2001C to the room temperature. The deflection of the lens is lessthan a micrometer and remains approximately constant for adistance of 4 mm from the lens center. On moving further awayfrom the center, the deflection begins to increase uniformly andreaches a maximum value at the lens periphery. Figure 6(b)shows the contour plot of the equivalent stress distribution in-side a formed lens at the time of lens release (i.e. at 2001C) andFig. 6(c) shows the equivalent stress distribution inside the lenswhen it has cooled down to room temperature. It can be ob-served from the figures that there is a higher stress concentrationat the lens periphery as compared with the lens center. Theseresidual stresses of approximately 2 MPa in the outside region

possibly cause slight distortion of the lens during the coolingphase to room temperature. At room temperature not only is themagnitude of stress lower (o1 MPa) but also the region overwhich these stresses are distributed is reduced.

(4) Contact and Heat Transfer

Figure 7 shows the FEM results of the contact between the glassand molds at different stages of the molding cycle. The shadedregion in the figure depicts the glass and mold contact, with thethickness of the contact zone being proportional to the magni-tude of interface pressure.

Figure 7(a) shows the glass mold contact at 7001C. At thistemperature glass is soft and conforms very well to the moldsurfaces, resulting in a higher real area of contact. Figure 7(b)shows the glass mold contact at the end of the annealing cyclewhen the molds have cooled down to 2001C. The area of contactis much lower as compared with that at higher temperatures andit is mostly confined to the lens center and the ends. This low-ering of contact is mostly because of two reasons: (a) solidifica-tion of the glass material because of which it no longer conformsto the mold surfaces and (b) thermal shrinkage of the moldsbecause of lowering of temperatures.

These results highlight the importance of the transient heattransfer phenomenon at the glass–mold interface during themolding cycle. During annealing, the primary mode of heattransfer between glass and molds is conduction. The heat trans-fer between the two bodies would depend upon the interfacepressure, mold surface roughness, and the difference in glass–mold temperatures. Additionally, a radiation field exists inside

Fig. 5. Comparison of upper mold geometry at room temperature withthe predicted geometry at molding temperature.

Fig. 6. (a) Predicted lens deflection on cooling from the release tem-perature of 2001–201C. (b) Equivalent stress distribution in the moldedlens at (b) 2001C and (c) at 201C.

Fig. 7. Numerical results showing the contact between the formed glasslens and the upper and lower mold halves at the end of (a) forming cycle(7001C) (b) annealing cycle (2001C).

534 Journal of the American Ceramic Society—Jain and Yi Vol. 88, No. 3

glass that contributes to the heat transfer between the glass andmold. In the semi-transparent wavelength range (2.5–5 mm),heat can be transported by radiation within glass. If this radi-ation escapes from the glass and reaches the mold surface, it iseither absorbed or reflected back into glass. By absorbing thisradiation, the mold surface removes heat from glass, which istransferred into the mold. In order to accurately model the lensmolding process it is therefore important to input the time-de-pendent value of the heat transfer coefficient, which is a functionof instantaneous molding conditions, and include a numericalmodel of the radiation heat transfer mechanism. Because of thecurrent limitations with the available FEM programs the radi-ation heat transfer mechanism could not be incorporated intothe simulation model.

VI. Conclusions

Precision lens molding operation has been simulated by takinginto account viscoelastic stress relaxation behavior of glass. Thisresearch is one of the first attempts to apply FEM to predict theperformance of a lens molding process and to gain an insightinto the complex mechanisms observed in any glass formingoperation. Some of the main findings of the work are summa-rized below:

1. FEM simulation can be used to predict the performanceof a lens molding process. But a more sophisticated model isneeded to make a more accurate process prediction.

2. Structural relaxation needs to be taken into account inthe numerical model for precise prediction of the final lens shapeand residual stresses. Knowledge of high-temperature glassproperties including viscosity and elastic properties is neededfor accurate process modeling. These issues are being dealt in theongoing research being performed by the authors.

3. The heat transfer model should also incorporate the ra-diation heat transfer mechanism both within glass and at theglass–mold interface. Because of the limitation of the availableFEM code this heat transport mechanism could not be takeninto account in the numerical model.

4. Thermal expansion of the mold takes place at the mol-ding temperature, which is significant from the standpoint ofaccuracy demands on an optical component. The molded lensalso changes shape when it is cooled down from molding tem-perature to room temperature. These changes in lens and moldgeometries can be compensated for in the initial mold surfacegeometry by making use of FEM.

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