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Profile optimization for the prediction of initial parison dimensions from final blow moulded part specifications
Computers them. Engng, Vol. 17. No. 8, pp. 751-764, 1993 Printi in Great Britain. All rights teservcd
0098-I 354/93 $6.00 + 0.00 Copyright Q 1993 Pergamon Press Ltd
PROFILE OPTIMIZATION FOR THE PREDICTION OF INITIAL PARISON DIMENSIONS FROM FINAL BLOW MOULDED PART SPECIFICATIONS R. W. DIRADDO and A. GARCIA-REJON National Research Council of Canada, Industrial Materials Institute, 75 boul de Mortagne, Boucherville, Quebec, Canada J4B 6Y4 (Received 10 August
1992; final revision received 29 October 1992; received for publication 25 November 1992)
Abstraet-The critical stage of the extrusion blow moulding pr-
involves the inflation of a molten parison into the final part. Modelling of the inflation stage generally entAils predicting the final part thickness distribution, given the initial parison thickness profile. We refer to this formulation of the process as the “forward formulation” or as the “forward predictor”. It would be more desirable to model the process using the inverse formulation, that is, given a tinal part thickness distribution, obtain the required initial parison thickness profile. We refer to this formulation of the process as the “inverse formulation”. Tooling costs and machine downtimes can be minimized with the information obtained from the simulation. Computational times and man-hours involved with trial-and-error runs with the forward predictor can also be minimized. The approach chosen involves placing a forward predictor in an iterative optimization loop. The loop searches for the parison thickness profile that results in the minimum overall difference between the
specifiedfinal part thicknessdistributionand the individualiteration’s output from the forward predictor. The forward predictor employs a discretized Newton-Raphscm technique in the solution. The profiled
optimization methodology utilizes updating techniques analogous to classical process control equations.
INTRODUCTION Blow
moulding process
Plastic hollow parts, such as bottles and drums, are manufactured by a process known as blow moulding. Extrusion blow moulding is employed in the production of larger containers, as well as automotive parts, such as car seats, gas tanks, bumpers and ducts. Injection blow moulding is utilized in the production of smaller, fine detail containers prevalent in the pharmaceutical and cosmetic industries. Stretchinjection blow moulding is employed in the manufacture of parts such as soda bottles, where increased strength and gas impermeability are required (Rosato and Rosato, 1989). The blow moulding process can be separated into three stages: (i) parison or preform formation; (ii) inflation; and (iii) final part cooling. The initial stage of the extrusion blow moulding process involves the extrusion of a polymer melt through an annular die. Resin is mixed, melted and transported forward by a rotating screw in conjunction with heaters located on the barrel wall. The melt is extruded through an annular die of adjustable gap, to form a hollow cylindrical membrane known as a parison. The die gap is the distance between the inner mandrel and the outer bushing. The gap can be varied 751
during the extrusion, due to the tapered nature of the die, by moving either the mandrel or the bushing in a vertical direction. The process of variable die gap extrusion is referred to as parison programming and is utilized to manipulate the thickness distribution in the final part. The viscoelastic nature of polymeric melts, during the extrusion, result in two phenomena known as swell and sag. These phenomena can result in complex diameter and thickness profiles along the length of the parison (DiRaddo and Garcia-Rejon, 1992). Swell is caused by the relaxation of the polymeric melt as it exits the die. Sag is caused by the drawdown of the parison under its own weight. Injection blow moulding involves the injection of the melt into a cavity of specified dimensions. This results in the formation of a hollow cylindrical part known as a preform. The preform dimensions are governed by the injection mould cavity dimensions. This differs from extrusion blow moulding, where the parison dimensions are determined by parison programming. The inflation and cooling stages are similar for both extrusion and injection blow moulding. In extrusion blow moulding, the parison is vertically suspended in the air during which time two mould halves enclose it by the action of a pneumatic or hydraulic mechanism. Internally applied air pressure causes the
752
R. W.
DIRADDO and
parison to inflate and take the shape of the final mould. The final phase involves the cooling of the part by the contact with the cooling lines. The mould halves then open, the final part is removed and the cycle begins again. The injection and stretch-injection blow moulding processes both employ a core rod in conjunction with moving hydraulic and pneumatic mechanisms for transferring of the preform from station to station. The preform is first transported to the inflation station, and once inflated is then transported to the cooling station. The inflation stage of the stretch-injection blow moulding process entails an additional preform stretching step. The preform is, stretched to about twice its original length, prior to the actual inflation. The stretching of the preform orients the molecules of the material in the axial direction. This orientation results in a stronger and gas impermeable finished product. Final part considerations necessitate obtaining an optimal part thickness distribution. Excessive resin usage decreases productivity by increasing cooling requirements and consequently cycle times. Costs are also increased by the use of unnecessary material. Insufficient material or a poor resin distribution can cause inadequate mechanical strength as well as inflation failures in part areas where the blow ratio is high. The mechanical strength considerations of importance can be structural (impact and stiffness), bending and deflection resistance. Ail of these properties are related primarily to the material and the final part thickness distribution. Therefore, for a given material, the most critical parameters in part specification is the final part thickness distribution (DiRaddo et al., 1991). As mentioned, the parison thickness distribution in extrusion blow moulding is dependent, to a large degree, on the die tooling and parison programming employed. An elaborate trial-and-error experimental procedure is normally required for obtaining the die gap programming profile that gives the optimal thickness distribution. The parison programming set-up procedure entails extruding a parison at an initial die gap profile, forming the final part and measuring the thickness distribution. The die gap profile is then adjusted accordingly, in a trial-and-error fashion, until the desired final part thickness distribution is obtained. This set-up procedure is very time consuming and can result in long downtimes. The preform thickness distribution obtained in injection blow moutding is dependent on the injection mould cavity design. Mould designs are usually based on heuristic knowledge of mould mak-
A.
GARCIA-REJON
ers. However, this approach usually requires a trialand-error procedure whereby several process trials and subsequent mould modifications are performed. Therefore, tooling costs and design times are increased. Modelhng of the inflation stage, whereby the final part thickness distribution is predicted from the process conditions, such as the parison thickness profile, allows for some optimization of trial-anderror procedure, by alleviating the need for actual process trials (Garcia-Rejon et al., 1991). This formulation of the process is referred to as the “forward formulation” or as the “forward predictor”. However, a trial-and-error procedure must still be performed with the simulation software to obtain the desired parison or preform thickness distribution. This can result in undesirably long computation times, as well as excessive man-hours running the individual simulations. Prior knowledge of the required parison or preform thickness distribution would eliminate the need for the trial-and-error procedures required in both extrusion and injection blow moulding. Therefore, an approach that would allow for prediction of the initial parison or preform thickness distribution from the specified final part thickness distribution would be very useful. This formulation of the process is referred as the “inverse formulation”. However, obtaining a formulation that allows for the direct prediction the parison dimensions is not trivial. In the absence of such a direct modelling scheme, one can resort to automation of the trial-and-error procedure that iteratively employs the forward formulation or forward predictor. BACKGROUND
Modelling the process with the forward formulation involves prediction of the final part thickness distribution given the initial parison thickness distribution. Two approaches, with regard to material behaviour during inflation, have been utilized for the prediction. The first method assumes that the polymer melt behaves as a viscous or a viscoelastic fluid, whereas the second method assumes that the melt behaves as an elastic solid. The assumption that the parison behaves as a viscous or a viscoelastic fluid results in a very complex computational formulation. Poslinski and Tsamopoulos (1990) treat the parison as a Newtonian fluid subject to a non-isothermal inflation. Parison position and cooling during the inflation are predicted as a function of time. Some experimental final part thickness distributions are obtained and compared to simulation results for a simple mould geometry and
Profile optimization for initial parison dimensions a constant initial thickness preform. The bottom horizontal section of the part and therefore, the effect of the pinch on the entire parison is not considered. The maximum blow-up ratio is two, although most industrial applications entail blow-up ratios up to five. Also, the inherent elastic nature of the polymer melt is not considered, since the formulation assumes a Newtonian fluid. Ryan and Dutta (1982) as well as Khayat and Garcia-Rejon (1992) assume a viseoelastic behaviour of the polymer melt. The inflation is modelled as a dynamic process, predicting the parison inflation as a function of time. A free inflation was considered. Attempts with confined inflation, that is employing a mould geometry, have not been handled to date. The constitutive equation used is the modified corotational ZFD (MZFD) model. Bellet et al. (1992) model the viscoelastic 3-D inflation of a Norton power law material. Kouba and Vlachopoulos (1992) discuss the modelling of the 3-D deformation of a K-BKZ material. The approaches discussed in Poslinski and Tsamopoulos (1990), Ryan and Dutta (1982) Khayat and Garcia-Rejon (1992), Beget et al. (1992) and Kouba and Vlachopoulos (1992) are still at a development stage and also lack comparison to a broad range of experimental data, considering the effect of process parameters such as parison thickness, parison thickness distribution and large blow-up ratios. The assumption that the material behaves as an elastic solid, during the inflation, results in a much simpler formulation and is also at a further stage of development than the cases of viscous and viscoelastic material behaviour. The inflation times in blow moulding are very short (OS-l.0 s), in comparison to extrusion times (10 s). Therefore, dynamic viscous effects, during inflation, are of less importance because of the relatively short duration of the deformation. Haessly (1989) has shown that the elastic material behaviour assumption, during the inflation, is a valid assumption, by demonstrating that high density polyethylene melts subject to free inflation exhibited perfect elastic memory after rupture. The elastic solid approach uses two different basic assumptions: continuum mechanics (Petrie and Ito, 1992) and virtual work 1980; Khayat et al., approaches (Garcia-Rejon et al., 199 1; Haessly, 1989; Green and Adkins, 1960; DeLorenzi and Nied, 1991; Harms and Michaeli, 1992; Cohen and Seitz, 1991; Kouba et al., 1992). Finite difference and finite element techniques have been employed to solve the highly nonlinear sets of equilibrium equations. Derdouri and Khayat (1992) have extended the elastic solid approach so as to predict the inflation
753
of a stretching preform in injection-stretch blow moulding. For both the continuum mechanics and virtual work approaches, the inflation steps are assumed to be a series of static equilibrium conditions, whereby the outward directed pressure force is balanced with the inward directed elastic force. The inflation can result in an experimentally observed and theoretically predicted pressure drop for some materials, such as PET (1983). A high initial pressute is required to begin the blowing. Once the blowing becomes unstable, the pressure will drop until the entire parison has been blown to a larger diameter. The pressure will then continue to rise. The new membrane locations, for each individual increment in inflating pressure (0.1 psi), are calculated. These approaches (Garcia-Rejon et al., 1991; Derdouri and Khayat, 1992) predict that the critical pressure to just fill the mould is in the 5-1.5 psi range. Experimental studies (Garcia-Rejon et al., 1991; Derdouri and Khayat, 1992) have confirmed the prediction by showing that this critical pressure is about 10 psi. The no-slip condition applies for an element touching the mould surface. The molten element is assumed to freeze and remain stationary at contact. Partial freezing of the element will still result in minimal slip at the mould surface. The parison or preform and the final part are assumed to be symmetric about the central vertical axis. Inflation is assumed to be so rapid that temperature change before mould contact is negligible. The thin shell approximation is employed and the forces in the thickness direction are neglected. As the membrane increases in thickness, the validity of the approximation suffers. A variety of elastic constitutive equations are utilized in the formulation. Treloar (1976) gives a detailed review of these models. The constitutive equations relate the elastic strain energy per unit volume W to the strain invariants. One of the simplest models capable of predicting material behaviour at high deformations, is the Mooney-Rivlin equation: w = c, (II, - 3) + C,(II* - 3),
(1)
where the first and second strain invariants, II, and II,, are related to the principal stretch ratios by: z1,=n:+n:+n:,
(2)
Iz~=R,*+;1,2+j1;2.
(3)
The parameters, 1,. 1, and L, represent the radial, longitudinal and thickness stretch ratios, respectively. The Mooney coefficients are obtained from a uniaxial tensile test (Finney and Kumar, 1988). Formulations
754
R. W. DIRADIMJ and A.
employing the Mooney-Rivlin equation have generally employed an average value of the Mooney coefficients, over the entire intlation period. DiRaddo and Garcia-Rejon (1993a) have employed an entirely different methodology as a forward predictor of the inflation process. Neural networks are utilized for the on-line prediction of the final part thickness distribution from the initial process conditions, such as parison thickness profile, parison temperature profile, mould geometry and the stress growth function. The neural network is trained by employing a gradient descent optimizationregression approach and mapping a broad range of output and input data. Once trained, the algorithm is capable of predicting outputs based on new inputs. Several advantages of neural networks over simulations based on first principles include faster response and the network’s ability to update a model to account for process shifts. Neural networks do not allow for an understanding of process fundamentals, they require a great deal of experimental data for the training procedure and problems can arise with extrapolation beyond the range defined by the experimental data. Therefore, the methodology is better suited for on-line applications, where fast response and following of process shifts is crucial. The objective of this work is to model the inflation of a parison with the inverse formulation. This involves the prediction of the initial parison thickness profile from the final part specifications. The approach should be capable of handling various parison programming situations such as constant and variable die gap profiles as well as various die temperatures. The elastic solid and viscous fluid assumptions have not been used for direct modelling of the process with the inverse formulation. The direct approach involves beginning with the membrane at the mould surface and employing the appropriate constitutive equations to deflate the membrane back to its initial shape. For the case of the elastic solid assumption with direct modelling of the process with the inverse formulation, the inflating pressure required to fill the mould is not known beforehand, and is required in the solution of the formulation. The formulation would begin with the pressure required to fill the mould and the inflated membrane shape at the mould surface. The deflating membrane shape would then be calculated for each negative increment in pressure. The method of detachment of the membrane from the mould surface is not evident. In the forward formulation, the confined mould imposed completion of the inflation for the respective segments of the
GARCIA-REJON
parison. However, for the inverse formulation, it is not possible to know, at what pressure increment, a specific part of the membrane detaches from the mould surface. Therefore, several deflating membrane shapes can occur and the possibility of a nonunique solution exists. One possible deflating shape is the actual solution. A second possible detlating shape is a parison deflating shape with the same form as the mould geometry. DiRaddo and Garcia-Rejon (1993b) have employed neural networks for the modelling of the process with the inverse formulation However, extrapolation problems were evident with test runs that were beyond the data range selected for the training. A large amount of experimental data was required for the training of the network. A third alternative for modelling of the process with the inverse formulation involves automating the trial-and-error procedure utilized with the forward predictor. As mentioned, the trial-and-error procedure entails beginning with an initial guess parison thickness profile and employing the simulation software for the prediction of the final part thickness profile. The input parison thickness profile is then adjusted after every simulation until an acceptable final part thickness distribution is obtained_ Automation of the trial-and-error procedure can minimize the computing time required for the solution. COMPUTATIONAL
APPROACH
The algorithm employed for the modelling of the process with the inverse formulation incorporates a combination of: (i) a Newton-Raphson routine for the prediction of the forward formulation; and (ii) a profiled optimization routine whereby the forward predictor is placed in an iteration loop. The approach is a classic black box optimization. The goal of the loop is to search for the parison thickness profile condition that results in the minimum error between the individual iteration final part thickness distribution and the specified final part thickness distribution (Fig. 1). Newron-Raphson routine The Newton-Raphson solution of the forward formulation, involves discretizing the parison and final part into a series of i linear elements along the length. The top of the parison and final part correspond to segment 1 and the bottom of parison or final part correspond to segment N. The approach employed involves the principle of virtual work discussed previously (Garcia-Rejon et al., 1991; Haessly, 1989; Green and Adkins, 1960; DeLorenzi and Nied, 1991; Harms and Michaeli, 1992; Cohen and Seitz,
Profile optimization for initial parison dimensions
j=
is applied in increments of 0.1 psi until mould contact is complete. The residual forces in the radial and axial directions are Frk and Fzk, respectively. The derivatives of W with respect to u, and V, can be found by employing the chain rule:
I
Input Tunrng hp(i,l)andh,(O
755
Constants,
(6)
Run Forward Process Sequence: Obtain h Ci,])
I and
E(i,j)
= hs(i)
Calculate
-h
dW
(1.j)
1
Total
Error
Yes Solution
=
hp(i,j)
-
Minlf’Wm
Error
ReaChed
?
NO
hp(i,j+
I) = hdl,j)
+ f
j-j+,
dWd&
dWdl,
dv,=dldo+dl,du,. L t
I
I E(i.j),
E(i,j-1)
1
-
Fig. 1. Flowchart of profiled optimization for a discretized model.
1991; Kouba et al., 1992). This method has been selected so as to reduce the size of the computational scheme and because it is the best developed of the alternatives. In the undeformed state, a node k of element i has the coordinates (r*, z~). After deformation, the node is displaced to coordinate (Q + o,, zk + Us), where vk and u, are the unknown components of the k node displacement vector. The other node of element i can be treated in a similar manner. The nodal displacements, uk and v, can be related to the individual stretch ratios by employing geometric considerations (Haessly, 1989). The new equilibrium position is determined by the balance between the externally applied pressure force and the internally directed elastic force. The radial force balance yields:
The elastic strain energy per unit volume W is related to the individual stretch ratios by equations (l-3). A linear 20°C temperature drop, from 200 to 18O”C, along the length of the parison has been observed (2). This drop in temperature is due to cooling of the parison as it is extruded in ambient air. The temperature dependence of the Mooney coefficients is incorporated along the length of the parison, according to the temperature profile. The stretch ratios and the derivatives of the stretch ratios with respect to the nodal displacements are functions solely of the nodal displacements (Haessly, 1989). The set of nonlinear equations is solved by using a Newton-Raphson routine, which aims at finding the unknown nodal displacements for a series of air pressures, that minimize the value of the force residuals at each node. The elements continue to deform, with increasing pressure, until they contact the mould. The element thickness distribution at mould contact is the final part thickness distribution. ProBled optimization The optimization routine involves running the forward predictor for J iterations, whereby the individual iterations can be represented by i. Therefore, at length location i and for iteration i the parison thickness is h,(i,j), the predicted final part thickness is h(i,j) and the specified final part thickness distribution is h,(i). The forward predictor can be depicted by the following relationship: h(i,j) = G(i) x hp(i,j),
and the axial force balance yields:
(5) where r is the segment radius, r,, is the segment thickness, P pressure and R, S and T are nodal displacements (Haessly,
I is the segment length, is the applied inflation functions solely of the 1989). The air pressure
(7)
(8)
where the function G(i) is dependent on process parameters such as mould geometry, melt temperature profile, segment location as well as interaction effects caused by adjacent segments along the length of the parison. The function G(i) is incorporated when running the forward predictor. The error at parison segment i for iteration j is: E&j)
= h,(i) - h(i,j),
(9)
R. W. Dr&ow
756
and A. GARCEA-REJON
and therefore substituting equation (8) into equation (9) yields: E(i,j)
= h,(i) - G(i)h,(i,j).
(IO)
Summation of equations (9) or (10) over the entire length of the parison yields: ET(j)
= C [h,(i) -
i
h(W)1
(11)
or
Ah,& j) = aE(i,j) ET(j)
= 1 [h,(i) -
i
W)~,G,Al.
(12)
The goal of the optimization is to find the parison thickness distribution h,(i,j) that gives the minimum overall error ET(j). Ideally, one would like to obtain the roots of equation (12). Another important criterion for the algorithm is the speed at which a solution is attained. Placing a numerical technique, in which each iteration can take 10 min, into an optimization search loop can result in very long computational times. Therefore it is important that convergence be. reached in a reasonable time frame. The optimization search therefore becomes a dual problem of minimizing the total overall error in the quickest possible time. This is analogous to the goal in traditional process control (Stephanopoulos, 1984). Therefore, the optimization methodology chosen uses classical control equations. Other optimization algorithms, such as gradient descent (Jenson and Jeffreys, 1977) and a root finding Newton-Raphson technique (Carnahan et al., 1969) were also studied, but the speed of convergence was found to be best for the methodology based on the process control equations (Schoebe, 1991). The additional time for the gradient descent and NewtonRaphson techniques was required to estimate the derivatives used in the respective techniques. The optimization routine involves updating the input parison thickness profile by an amount, Ah,(i,j), dependent on the present and past error profiles. The new parison thickness profile becomes: h,(i,j
+ I) = h,(i,j)
+ A&(&j).
For a proportional (P) optimizer parison thickness profile is: Ah,(i,j)
(13)
the change
= aE(i,j),
in
(14)
where a is the user-defined optimizer gain. A proportional-integral (PI) optimizer is defined by: A&(&j)
that may exist with the proportional optimizer (Stephanopoulos, 1984). A traditional P1 system integrates the past error over the total number of iterations. However, only one past error value is looked at in this case, in an effort to simplify the problem and minimize computational times. The user-defined integral term is fi. A proportional-integral-derivative optimizer (PID) is defined by:
= aE(i,j)
+ 0.5/3[E(i,j)
+ E(i,j
-
l)].
(15)
This convergence scheme employs knowledge of past error values in an effort to decrease any offset
+ 0.5 jS[E(i,j) + y[E(i,j)
- 1)
+ E(i,j - E(i,j
- l)].
(16)
The user-defined derivative term y is employed to help dampen oscillations that can result from the use of the integral term (Stephanopoulos, 1984). Equation (16) can be rearranged to yield: Ah,(i, j) = [a + OSfl + y]E(i,j) + [0.5/3 - y]E(i,j
-
1).
(17)
This equation is the standard form for an acceleration strategy used in recycle blocks for process simulations (Jenson and Jeffreys, 1977). The optimization routine continues in an iterative fashion, whereby the parison thickness profile is continuously updated, until the overall error in the final part thickness distribution is below a prespecified tolerance_ The parison thickness profile at this point is the desired solution. OF-TIMIZER
TUNING
The first phase of the methodology involves determining optimal values for the user-defined constants, a, /3and y. The criteria for selection of these “tuning constants” are minimum number of iterations for convergence and minimum offset error at the point of convergence. Once the optimization routine is “tuned” at the optimal values of a, B and y. it can be employed for prediction of various processing situations. Figure 2 shows the dependence of the error with the number of iterations for several values of updating gain a. The routine diverges for a gain of 5.0. This behaviour is analogous to process control situations where instability arises at high gains. An updating gain of 1.0 has the fastest convergence, but yields oscillations at the end of the routine. This is also analogous with process control situations, where high gains can result in an oscillatory response. Figure 3 portrays the relationship between the error and the number of iterations for two different integral constants, /3= 0 and 0.5. The updating gain for these situations is set at 0.5. The use of the integral constant induces oscillatory behaviour and decreases
---__ ...__ ---% \c Profile optimization for initial parison dimensions
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:
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Updating gain 0.1
‘?
0.5 1.0 5.0
0.100
z
s
a ::
w
0.010
0.010
0.00 I
0.001 100
10
1000
Iterations Fig. 2. Optimization search as a function of gain term. the offset error
slightly.
to the effect of integral eliminated
The behaviour control
and oscillatory
behaviour
The effect of the derivative Fig.
4. Introducing
results in damping integral both
term.
integral
and
which
the minimum
yields
of
induced
updating
the fastest
oscillatory
y is shown
constant
set to 0.5 for these situations.
situation, PID
constant
the derivative
The
in 0.5
by the
gain
are
best tuned
convergence
behaviour,
8000 parison through
can occur.
of the oscillations
The
sion blow moulding
is analogous
action, where offset is
and
is therefore
a
gap
of
Experimental
data
for
comparison
The flush gap
The bottle
to the simuextru-
is defined
and mould
are at the
programmer
geometry
employed
ratios,
from
profiles
micrometer.
This
in this entials
of the part.
were measured and measuring
8.0 mm of bottle length, with a high-precision held
of
one at the top and
of the part, to four at the middle thickness
was
(10 Hz).
in Fig. 5. The bottle geometry
of blow-up
bottle
as the gap
in the die gap at an interval
ting the bottle into segments
with a Battenfeld-Fischer
was extruded
and outer bushing
0.1 s, for a 10s extrusion work are shown
with a MAC0
1.7 cm and a flush
The parison
able to input changes
bottom
lation was obtained
1.O mm.
equipped
The parison
die of o.d.
same vertical position.
The
EXPERIMENTAL
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when the inner mandrel
a variety
optimizer.
machine
programmer.
corresponds
by cutat every hand-
to 40 sampling
R. W.
758
DIRADDO
and A. GARCIA-REJON
1.000
1.000
0.100
0.100
; : : : : : : : : : :
0.010
0.010
0.001
100
10
Iterations
Fig. 3. Optimization search as a function of integral term.
points along the length of the bottle. Point zero is at the bottom of the neck and point 40 is at the bottom of the bottle. The neck itself is neglected in the study, as its thickness profile is defined by the threading of the blow pin exterior and not the inflation. A high-density poIyethylene blow moulding grade resin (DuPont of Canada, !Sclair 56B) was employed in the study. The parison thickness profiles were measured by the technique described by DiRaddo and Garcia-Rejon (1993b). The methodology is able to measure the effects of both swell and sag on the parison thickness profile. The magnitude of the ratio of the Mooney constantc C,/C, is a direct measure of the deviation of
material behaviour from elastic theory (Treloar, 1976; Finney and Kumar, 1988). The value of 5.0 is obtained by utilizing the approach described in Finney and Kumar (1988) and employing extensional stress-strain data. The stress-strain data were obtained by employing a Rheometrics Extensional Rheometer and are presented elsewhere (GarciaRejon and DiRaddo, 1992). RESULTS
AND
DEX.XJSSION
The results of modelling of the process with the forward formulation, involving prediction of final part thickness distribution from initial parison di-
Profileoptimizationfor initial parison dimensions
759
1 .ooo
1 .ooo
-0 ----
0.5
0.100
0.100
0.010
0.010
?3 s
s
rEi
0.001
0.001 1
10
100
1000
Iterations Fig. 4. Optimizationsearch as a function of derivativeterm. mensions, are shown in Fig. 6. The results are compared to experimental data obtained off-line. The algorithm based on virtual work is employed. Results are presented for simulations run at a Mooney ratio, C,/C, of 5.0. When considering the forward predictor, some of&et is obtained between the experimental and theoretical results. The error is about 10% in the center of the part, but increases to as high as 150% towards the bottom of the part. Many factors can cause the offset, such as cooling and curvature effects introduced at the mould pinch. However, the predictions are acceptable in an engineering sense. The parison melt near the mould, at the top and bottom pinch locations, will tend to cool to a much lower temperature than the melt temperature, since these sections are exposed to conductive cooling for a brief period of time. This will result in a much
cooler and therefore stiffer material at these part sections. The inflation wilt subsequently be affected. The pinching of the parison by the mould also introduces curvature effects. The formlation assumes that the parison is of constant diameter, for the entire length of the parison, including the bottom part near the mould. However, in actuality, the bottom portion (3 cm length) takes on an elliptical shape. Therefore, for this bottom part, the parison diameter is no longer constant but is a function of the parison length. Simulations run with these nonconstant diameter parison diameter shapes did not converge, when run with the forward predictor (Derdouri and Khayat, 1992). Work is continuing to employ more elaborate numerical techniques to aid in the convergence. However, for the meantime, a constant diameter parison
R. W. DIGw
and A. GARCJA-REJON
Profile optimization
for initial parison dimeusions
761
0.24 0.22
0
2
6
10
14
18
Profile
22
26
30
34
38
point
8. Prediction of parison thickness distribution for a
bottle formed at an extrusion gap of 0.11cm.
0 -
5~
1’0
15
2’0
Profile
25
30
j5
40
45
point
Fig. 6. Prediction of bottle thickness distribution from parison thickness distribution-forward proces s sequence.
shape will bc employed and the bottom portion of the parison (profile points 3640) will not be considered. Once the optimization routine has determined the optimal PID parameters, it is considered tuned. Therefore, the PID parameters, a, fi and y are at their respective values, that render a minimum prediction error in the minimum number of iterations. These PID parameters can now be used for future predictions of the required parison dimensions, for a specified final part thickness distribution. The final part thickness distributions chosen as the set points were those obtained experimentally from the stated
size extrusions and are available in (DiRaddo Garcia-Rejon, 1!393a). Optimization routines are usually concerned with whether a global or local minimum has been obtained. Figure 7 shows the comparison between the set point part thickness distribution h,(i) for the case of a part formed at an extrusion gap of 0.11 cm, and the thickness distribution h(i,j) for this set point input, obtained in the final iteration J. The difference between the two curves is the overall error of equation (12). The two curves coincide, confirming that a global minimum condition has been obtained. Therefore, any offset that results, when comparing the predicted and experimental initial parison dimension profile, is a result only of the forward predictor’s limitations and not a result of the attainment of a local minimum in the optimization routine. The use of optimization routines also brings about concern with obtaining a unique solution. Several solutions can be found depending on the space domain where the search is taking place. Therefore,
0.14 0.13
0.32
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0.30
0.11
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8 2
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0.07
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0.01
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0.14
0.06
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0
0.16
7. Comparison
25
30
35
40
c 45
0
2
6
IO
14
I8
Profile
point
of set point distribution to the final iteration distribution.
+:::::::::::::::::::c
Fig.
22
26
30
34
38
point
Prediction of parison thickness distribution for a bottle formed at an extrusion gap of 0.13 cm.
9.
R. W. DIRADDO
and A. GARCIA-REJON 0.34 0.32 0.30
7
0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14
t
~~;;;;:‘i”‘“::;::::. 0
2
6
10
14
18
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22
26
30
34
38
0
pain!
Predictionof parison thicknessdistributionfor a bottle formed at an extrusiongap of 0.15 cm.
10.
employing different initial guesses allows for consideration of a broad space domain. A range of initial guesses was tried for the parison thickness distribution (0.50, 0.25, 0.10 and 0.05 cm). All these initial guesses, when individually input into the optimization routine, with the same set point thickness distribution, eventually converged to the same solution of parison thickness profile. Therefore, uniqueness of solution can be assumed, regardless of the initial guess of parison thickness distribution employed, as long as the guess is within the range 0.05-0.50 cm. The predictions of final part thickness distribution, after the first iteration, for the individual gaps can be found elsewhere (Schoebe, 1991). The thickness in the centre of the part, obtained after the first iteration, ranges from approx. 0.02-0.27 cm depending on the initial parison thick0.30 0.28
0.20 0.18 0.16
10
14
18
Profile
22
26
30
34
38
point
Fig. 12. Prediction of parison thickness distribution for a bottle formed with a step in the gap of magnitude -0.02 cm.
ness employed. The thickness in the centre of the part, obtained after the last iteration, is about 0.05 cm for all the initial guesses. Therefore considerable movement has occurred during the optimization search. Figures 8-10 show the required parison thickness profile for the thickness distribution of a part formed at constant extrusion gaps of 0. 1 1, 0.13 and 0.15 cm, respectively. The top of the parison is represented by Profile Point = 1. The final part thickness distribution chosen as the set point and the corresponding parison thickness profile were again obtained experimentally. The experimental parison thickness profile is constant, since the effects of sag and time-dependent swell were found to be minimal, for these cases (2). The instantaneous swell is the ratio of the experimen0.34 0.32 0.30
r
0.16 o.i4 0.12
t
0.04
l
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0.02 0
6
0.28 0.26 0.24 0.22 0.20 0.18
0.26 0.24 0.22
0.14
2
2
6
10
14
18
22
26
30
34
38
+
Profile point Fig. 11. Prediction of parison thickness distribution for a bottle formed at a die temperature of 180°C and an extrusion gap of 0. I1 cm.
0
2
6
10
14
18
Profile
22
26
30
34
38
point
Fig. 13. Prediction of parison thickness distribution for a bottle formed with steps in the gap of magnitude +/ - 0.02 cm.
Profile optimization for initial parison dimensions
tal parison thickness indicated on the respective figure, to the corresponding constant extrusion die gap. The prediction is very good in the centre of the parison. However, as larger gaps are considered, the simulation begins to present difficulties at the ends of the parison. As mentioned, the offset is caused by limitations in the forward predictor, because of the cooling ati curvature effects introduced by mould pinching. Figure 11 shows the effect of employing a lower melt temperature of 180°C and an extrusion gap of 0.11 cm. The simulation over-predictsat the bottom of the parison to a greater extent than for the case of 200°C. The swell is slightly greater at the lower temperature causing higher prediction of the parison dimension profile throughout the length. Figures 12 and 13 show the effects of variable die gap extrusions, indicative of parison programming. Figure 12 gives the prediction for the case of a final part produced at an extrusion, where a step in the die gap from 0.15 to 0.13 cm occurs. The simulation is able to predict the presence as well as the location of the step along the length of the parison. Nonlinear effects (Haessly, 1989; DiRaddo et al., 1993) are responsible for the overprediction that occurs before the step and the underprediction that occurs after the step. The forward predictor is unable to model these non-linear effects that are caused by oscillatory infiation patterns and compressive forces. Figure 13 gives the prediction for the case of a final part produced at an extrusion, where two steps in the die gap, from 0.134 11 cm and back to 0.13 cm, occur. The simulation is again able to predict the presence and locations of the steps, for this parison programming situation. The prediction is better, in an overall sense, than for the case of Fig. 12. A possible explanation for this is that the compressive forces that caused the oscillatory inffation pattern and nonlinear effects present for the case of Fig. 12, cancel each other because of the presence of equal and opposite steps (DiRaddo et al., 1993). CONCLUSION
The use of a combined Newton-Raphson and profiled optimization routine has been employed for the prediction of the parison thickness profile required for a specified final part thickness distribution. This
approach
involves
modelling
of
the inflation
process with the inverse formulation. A simulation based on modelling of the inflation process with the forward formulation is placed in an iteration loop, with the goal of determining the parison thickness
763
profile that results in the minimum difference between the specified final part thickness distribution and the output from the forward predictor. The modelling of the process with the forward formulation is based on the principle of virtual work. The parison behaviour during inflation, is considered to be an elastic membrane and the inflation is treated as a series of static deformations. The validity of the elastic material behaviour is justified because of the relatively short duration of the inflation time and by elastic recovery experiments performed in another work. The membrane is discretized into a series of elements and the membrane movement is solved by employing a Newton-Raphson routine. The optimization loop employs a methodology analogous to classical control techniques. Updating of the parison thickness profile after each iteration is performed based on the present error profile, as well as the integral and derivative of the error profile over the previous iterations. The results obtained indicate that the performance of the optimization loop is directly dependent on the performance of the forward predictor. Future work entails the improvement of the forward predictor in order to incorporate the effects of cooling and curvature introduced by the mould. Acknowledgements-The authors would like to express their gratitude to S. Schoebe and S. Lacoursiere for their contributions in this work. NOMENCLATURE C, = First Mooney coefficient C, = Second Mooney coefficient E = Elementerror ET = Summation of element errors Fd = Radial nodal force residual FTk= Axial nodal force residual G = Forward process sequence predictor function h = Final part element thickness output h, = Parison element thickness h, = Specified final part element thickness 11, = First strain invariant 11s= Second strain invariant J = Last iteration I = Element length k=Node P = Pressure R = Function of nodal displacement r = Element radius S = Function of nodal displacement T= Function of nodal displacement t,, = Undeformed element thickness z+ = Radial nodal displacement uk= Axial nodal displacement W = Strain energy per unit volume a = Updating gain p = Integral term y = Derivative term A = Updating amount i, = First stretch ratio 1, = Second stretch ratio 1, = Third stretch ratio
R. W. DIRADW
764
and A. GARCLVREJON
REFERENCES
Ballet M.,
C.
Wouters.
Camahan
Corsini,
SPE
B.,
Numerical
H. A. Methoak.
DeLorenzi H. G. and of thermoforming Polymer
P. Mercier and P. 38. 197 11992). Luther and -J. 0. Wilkes, AZ&lied Wiley, New York (1969). H. F. Nied. Finite element simulation and blow. moulding. MoaXling of Chap. 5 (A. I. Isayev, Ed.). Hanser,
ANTEC
Processing,
D.
Assaker,
Tech. Paners
NY (1991). Derdouri A. and R. E. Khayat, IMI Internal Report (1992). DiRaddo R. W. and A. Garcia-Rejon, Polymer Engng Sci. 32, 1401 (1992). DiRaddo R. W. and A. Garcia-Rejon, Polymer Engng Sci. (1993a). DiRaddo R. W. and A. Garcia-Rejon, A&. Polymer Tech. (1993b). DiRaddo, R. W, W. I. Patterson and M. R. Kamal, Znt. Polymer Process. 6, 217 (1991). DiRaddo R. W., J. Sam Wang and A. Garcia-Rejon, Polymer Engng Sci. (1993). Erwin L., M. A. Pollock ,and H. Gonzalez, Polymer Engng Sci. 23, 826 (1983).
Finney R. H. and A. Kumar, Rubber. (1988).
Chem.
Tech. 61, 884
Garcia-Rejon A. and R. W. DiRaddo. Znt. Conf. Rheology Vol. 11, (1992). Garcia-Rejon A., A. Derdouri, R. E. Khayat, M. E. Ryan and W. P. Haessly SPE ANTEC Tech. Papers 37, 836 (1991).
Green A. D. and J. E. Adkins, Large Deformations. Oxford University Press (1960). Haessly W. P., Finite element modeling and experimental study of injection blow moulding. Ph.D. Thesis, SUNY at Buffalo (1989). Harms R. and W. Michaeli, SPE ANTEC Tech. Papers 38. 641 (19921. Jenson‘ V. 6. and G. V. Jeffreys, Mathematical Metho& in Chemical Engineering. Academic Press, New York (1977). Khayat R. E. and A. Garcia-Rejon, J. Non Newt. Fluid Mech.
(1992).
l
Khayat R. E., A. Derdouri and A. Garcia-Rejon, Znt. J. Sofia% Struct. 29, 69 (1992). Kouba K. and J. Vlachopoulos, SPE ANTEC Tech. Papers 38, 114 (1992). Kouba K., 0. Bartos and 3. Vlachopoulos, Polymer Engng Sci. 32, 699 (1992). Petrie C. J. S. and K. Ito, Plastics Rubber 6, 68 (1980). Poslinski A. J. and J. A. Tsamopoulos, AZChE JI 36, 1837 (1990).
Rosato D. and D. Rosato. Biow Molding Handbook. Hanser, Munich (1989). Ryan M. E. and A. Dutta, Pofymer Engg Sci. 22, 569 (1982).
Schoebe S.. IMI Internal Report (1991). Stephanopoulos G., Chemical Process Control. PrenticeHall, Englewood Cliffs, NJ (1984). Treloar L. R. G., Proc. R. Sot. Land. A 351, 301 (1976).
0098-I 354/93 $6.00 + 0.00 Copyright Q 1993 Pergamon Press Ltd
PROFILE OPTIMIZATION FOR THE PREDICTION OF INITIAL PARISON DIMENSIONS FROM FINAL BLOW MOULDED PART SPECIFICATIONS R. W. DIRADDO and A. GARCIA-REJON National Research Council of Canada, Industrial Materials Institute, 75 boul de Mortagne, Boucherville, Quebec, Canada J4B 6Y4 (Received 10 August
1992; final revision received 29 October 1992; received for publication 25 November 1992)
Abstraet-The critical stage of the extrusion blow moulding pr-
involves the inflation of a molten parison into the final part. Modelling of the inflation stage generally entAils predicting the final part thickness distribution, given the initial parison thickness profile. We refer to this formulation of the process as the “forward formulation” or as the “forward predictor”. It would be more desirable to model the process using the inverse formulation, that is, given a tinal part thickness distribution, obtain the required initial parison thickness profile. We refer to this formulation of the process as the “inverse formulation”. Tooling costs and machine downtimes can be minimized with the information obtained from the simulation. Computational times and man-hours involved with trial-and-error runs with the forward predictor can also be minimized. The approach chosen involves placing a forward predictor in an iterative optimization loop. The loop searches for the parison thickness profile that results in the minimum overall difference between the
specifiedfinal part thicknessdistributionand the individualiteration’s output from the forward predictor. The forward predictor employs a discretized Newton-Raphscm technique in the solution. The profiled
optimization methodology utilizes updating techniques analogous to classical process control equations.
INTRODUCTION Blow
moulding process
Plastic hollow parts, such as bottles and drums, are manufactured by a process known as blow moulding. Extrusion blow moulding is employed in the production of larger containers, as well as automotive parts, such as car seats, gas tanks, bumpers and ducts. Injection blow moulding is utilized in the production of smaller, fine detail containers prevalent in the pharmaceutical and cosmetic industries. Stretchinjection blow moulding is employed in the manufacture of parts such as soda bottles, where increased strength and gas impermeability are required (Rosato and Rosato, 1989). The blow moulding process can be separated into three stages: (i) parison or preform formation; (ii) inflation; and (iii) final part cooling. The initial stage of the extrusion blow moulding process involves the extrusion of a polymer melt through an annular die. Resin is mixed, melted and transported forward by a rotating screw in conjunction with heaters located on the barrel wall. The melt is extruded through an annular die of adjustable gap, to form a hollow cylindrical membrane known as a parison. The die gap is the distance between the inner mandrel and the outer bushing. The gap can be varied 751
during the extrusion, due to the tapered nature of the die, by moving either the mandrel or the bushing in a vertical direction. The process of variable die gap extrusion is referred to as parison programming and is utilized to manipulate the thickness distribution in the final part. The viscoelastic nature of polymeric melts, during the extrusion, result in two phenomena known as swell and sag. These phenomena can result in complex diameter and thickness profiles along the length of the parison (DiRaddo and Garcia-Rejon, 1992). Swell is caused by the relaxation of the polymeric melt as it exits the die. Sag is caused by the drawdown of the parison under its own weight. Injection blow moulding involves the injection of the melt into a cavity of specified dimensions. This results in the formation of a hollow cylindrical part known as a preform. The preform dimensions are governed by the injection mould cavity dimensions. This differs from extrusion blow moulding, where the parison dimensions are determined by parison programming. The inflation and cooling stages are similar for both extrusion and injection blow moulding. In extrusion blow moulding, the parison is vertically suspended in the air during which time two mould halves enclose it by the action of a pneumatic or hydraulic mechanism. Internally applied air pressure causes the
752
R. W.
DIRADDO and
parison to inflate and take the shape of the final mould. The final phase involves the cooling of the part by the contact with the cooling lines. The mould halves then open, the final part is removed and the cycle begins again. The injection and stretch-injection blow moulding processes both employ a core rod in conjunction with moving hydraulic and pneumatic mechanisms for transferring of the preform from station to station. The preform is first transported to the inflation station, and once inflated is then transported to the cooling station. The inflation stage of the stretch-injection blow moulding process entails an additional preform stretching step. The preform is, stretched to about twice its original length, prior to the actual inflation. The stretching of the preform orients the molecules of the material in the axial direction. This orientation results in a stronger and gas impermeable finished product. Final part considerations necessitate obtaining an optimal part thickness distribution. Excessive resin usage decreases productivity by increasing cooling requirements and consequently cycle times. Costs are also increased by the use of unnecessary material. Insufficient material or a poor resin distribution can cause inadequate mechanical strength as well as inflation failures in part areas where the blow ratio is high. The mechanical strength considerations of importance can be structural (impact and stiffness), bending and deflection resistance. Ail of these properties are related primarily to the material and the final part thickness distribution. Therefore, for a given material, the most critical parameters in part specification is the final part thickness distribution (DiRaddo et al., 1991). As mentioned, the parison thickness distribution in extrusion blow moulding is dependent, to a large degree, on the die tooling and parison programming employed. An elaborate trial-and-error experimental procedure is normally required for obtaining the die gap programming profile that gives the optimal thickness distribution. The parison programming set-up procedure entails extruding a parison at an initial die gap profile, forming the final part and measuring the thickness distribution. The die gap profile is then adjusted accordingly, in a trial-and-error fashion, until the desired final part thickness distribution is obtained. This set-up procedure is very time consuming and can result in long downtimes. The preform thickness distribution obtained in injection blow moutding is dependent on the injection mould cavity design. Mould designs are usually based on heuristic knowledge of mould mak-
A.
GARCIA-REJON
ers. However, this approach usually requires a trialand-error procedure whereby several process trials and subsequent mould modifications are performed. Therefore, tooling costs and design times are increased. Modelhng of the inflation stage, whereby the final part thickness distribution is predicted from the process conditions, such as the parison thickness profile, allows for some optimization of trial-anderror procedure, by alleviating the need for actual process trials (Garcia-Rejon et al., 1991). This formulation of the process is referred to as the “forward formulation” or as the “forward predictor”. However, a trial-and-error procedure must still be performed with the simulation software to obtain the desired parison or preform thickness distribution. This can result in undesirably long computation times, as well as excessive man-hours running the individual simulations. Prior knowledge of the required parison or preform thickness distribution would eliminate the need for the trial-and-error procedures required in both extrusion and injection blow moulding. Therefore, an approach that would allow for prediction of the initial parison or preform thickness distribution from the specified final part thickness distribution would be very useful. This formulation of the process is referred as the “inverse formulation”. However, obtaining a formulation that allows for the direct prediction the parison dimensions is not trivial. In the absence of such a direct modelling scheme, one can resort to automation of the trial-and-error procedure that iteratively employs the forward formulation or forward predictor. BACKGROUND
Modelling the process with the forward formulation involves prediction of the final part thickness distribution given the initial parison thickness distribution. Two approaches, with regard to material behaviour during inflation, have been utilized for the prediction. The first method assumes that the polymer melt behaves as a viscous or a viscoelastic fluid, whereas the second method assumes that the melt behaves as an elastic solid. The assumption that the parison behaves as a viscous or a viscoelastic fluid results in a very complex computational formulation. Poslinski and Tsamopoulos (1990) treat the parison as a Newtonian fluid subject to a non-isothermal inflation. Parison position and cooling during the inflation are predicted as a function of time. Some experimental final part thickness distributions are obtained and compared to simulation results for a simple mould geometry and
Profile optimization for initial parison dimensions a constant initial thickness preform. The bottom horizontal section of the part and therefore, the effect of the pinch on the entire parison is not considered. The maximum blow-up ratio is two, although most industrial applications entail blow-up ratios up to five. Also, the inherent elastic nature of the polymer melt is not considered, since the formulation assumes a Newtonian fluid. Ryan and Dutta (1982) as well as Khayat and Garcia-Rejon (1992) assume a viseoelastic behaviour of the polymer melt. The inflation is modelled as a dynamic process, predicting the parison inflation as a function of time. A free inflation was considered. Attempts with confined inflation, that is employing a mould geometry, have not been handled to date. The constitutive equation used is the modified corotational ZFD (MZFD) model. Bellet et al. (1992) model the viscoelastic 3-D inflation of a Norton power law material. Kouba and Vlachopoulos (1992) discuss the modelling of the 3-D deformation of a K-BKZ material. The approaches discussed in Poslinski and Tsamopoulos (1990), Ryan and Dutta (1982) Khayat and Garcia-Rejon (1992), Beget et al. (1992) and Kouba and Vlachopoulos (1992) are still at a development stage and also lack comparison to a broad range of experimental data, considering the effect of process parameters such as parison thickness, parison thickness distribution and large blow-up ratios. The assumption that the material behaves as an elastic solid, during the inflation, results in a much simpler formulation and is also at a further stage of development than the cases of viscous and viscoelastic material behaviour. The inflation times in blow moulding are very short (OS-l.0 s), in comparison to extrusion times (10 s). Therefore, dynamic viscous effects, during inflation, are of less importance because of the relatively short duration of the deformation. Haessly (1989) has shown that the elastic material behaviour assumption, during the inflation, is a valid assumption, by demonstrating that high density polyethylene melts subject to free inflation exhibited perfect elastic memory after rupture. The elastic solid approach uses two different basic assumptions: continuum mechanics (Petrie and Ito, 1992) and virtual work 1980; Khayat et al., approaches (Garcia-Rejon et al., 199 1; Haessly, 1989; Green and Adkins, 1960; DeLorenzi and Nied, 1991; Harms and Michaeli, 1992; Cohen and Seitz, 1991; Kouba et al., 1992). Finite difference and finite element techniques have been employed to solve the highly nonlinear sets of equilibrium equations. Derdouri and Khayat (1992) have extended the elastic solid approach so as to predict the inflation
753
of a stretching preform in injection-stretch blow moulding. For both the continuum mechanics and virtual work approaches, the inflation steps are assumed to be a series of static equilibrium conditions, whereby the outward directed pressure force is balanced with the inward directed elastic force. The inflation can result in an experimentally observed and theoretically predicted pressure drop for some materials, such as PET (1983). A high initial pressute is required to begin the blowing. Once the blowing becomes unstable, the pressure will drop until the entire parison has been blown to a larger diameter. The pressure will then continue to rise. The new membrane locations, for each individual increment in inflating pressure (0.1 psi), are calculated. These approaches (Garcia-Rejon et al., 1991; Derdouri and Khayat, 1992) predict that the critical pressure to just fill the mould is in the 5-1.5 psi range. Experimental studies (Garcia-Rejon et al., 1991; Derdouri and Khayat, 1992) have confirmed the prediction by showing that this critical pressure is about 10 psi. The no-slip condition applies for an element touching the mould surface. The molten element is assumed to freeze and remain stationary at contact. Partial freezing of the element will still result in minimal slip at the mould surface. The parison or preform and the final part are assumed to be symmetric about the central vertical axis. Inflation is assumed to be so rapid that temperature change before mould contact is negligible. The thin shell approximation is employed and the forces in the thickness direction are neglected. As the membrane increases in thickness, the validity of the approximation suffers. A variety of elastic constitutive equations are utilized in the formulation. Treloar (1976) gives a detailed review of these models. The constitutive equations relate the elastic strain energy per unit volume W to the strain invariants. One of the simplest models capable of predicting material behaviour at high deformations, is the Mooney-Rivlin equation: w = c, (II, - 3) + C,(II* - 3),
(1)
where the first and second strain invariants, II, and II,, are related to the principal stretch ratios by: z1,=n:+n:+n:,
(2)
Iz~=R,*+;1,2+j1;2.
(3)
The parameters, 1,. 1, and L, represent the radial, longitudinal and thickness stretch ratios, respectively. The Mooney coefficients are obtained from a uniaxial tensile test (Finney and Kumar, 1988). Formulations
754
R. W. DIRADIMJ and A.
employing the Mooney-Rivlin equation have generally employed an average value of the Mooney coefficients, over the entire intlation period. DiRaddo and Garcia-Rejon (1993a) have employed an entirely different methodology as a forward predictor of the inflation process. Neural networks are utilized for the on-line prediction of the final part thickness distribution from the initial process conditions, such as parison thickness profile, parison temperature profile, mould geometry and the stress growth function. The neural network is trained by employing a gradient descent optimizationregression approach and mapping a broad range of output and input data. Once trained, the algorithm is capable of predicting outputs based on new inputs. Several advantages of neural networks over simulations based on first principles include faster response and the network’s ability to update a model to account for process shifts. Neural networks do not allow for an understanding of process fundamentals, they require a great deal of experimental data for the training procedure and problems can arise with extrapolation beyond the range defined by the experimental data. Therefore, the methodology is better suited for on-line applications, where fast response and following of process shifts is crucial. The objective of this work is to model the inflation of a parison with the inverse formulation. This involves the prediction of the initial parison thickness profile from the final part specifications. The approach should be capable of handling various parison programming situations such as constant and variable die gap profiles as well as various die temperatures. The elastic solid and viscous fluid assumptions have not been used for direct modelling of the process with the inverse formulation. The direct approach involves beginning with the membrane at the mould surface and employing the appropriate constitutive equations to deflate the membrane back to its initial shape. For the case of the elastic solid assumption with direct modelling of the process with the inverse formulation, the inflating pressure required to fill the mould is not known beforehand, and is required in the solution of the formulation. The formulation would begin with the pressure required to fill the mould and the inflated membrane shape at the mould surface. The deflating membrane shape would then be calculated for each negative increment in pressure. The method of detachment of the membrane from the mould surface is not evident. In the forward formulation, the confined mould imposed completion of the inflation for the respective segments of the
GARCIA-REJON
parison. However, for the inverse formulation, it is not possible to know, at what pressure increment, a specific part of the membrane detaches from the mould surface. Therefore, several deflating membrane shapes can occur and the possibility of a nonunique solution exists. One possible deflating shape is the actual solution. A second possible detlating shape is a parison deflating shape with the same form as the mould geometry. DiRaddo and Garcia-Rejon (1993b) have employed neural networks for the modelling of the process with the inverse formulation However, extrapolation problems were evident with test runs that were beyond the data range selected for the training. A large amount of experimental data was required for the training of the network. A third alternative for modelling of the process with the inverse formulation involves automating the trial-and-error procedure utilized with the forward predictor. As mentioned, the trial-and-error procedure entails beginning with an initial guess parison thickness profile and employing the simulation software for the prediction of the final part thickness profile. The input parison thickness profile is then adjusted after every simulation until an acceptable final part thickness distribution is obtained_ Automation of the trial-and-error procedure can minimize the computing time required for the solution. COMPUTATIONAL
APPROACH
The algorithm employed for the modelling of the process with the inverse formulation incorporates a combination of: (i) a Newton-Raphson routine for the prediction of the forward formulation; and (ii) a profiled optimization routine whereby the forward predictor is placed in an iteration loop. The approach is a classic black box optimization. The goal of the loop is to search for the parison thickness profile condition that results in the minimum error between the individual iteration final part thickness distribution and the specified final part thickness distribution (Fig. 1). Newron-Raphson routine The Newton-Raphson solution of the forward formulation, involves discretizing the parison and final part into a series of i linear elements along the length. The top of the parison and final part correspond to segment 1 and the bottom of parison or final part correspond to segment N. The approach employed involves the principle of virtual work discussed previously (Garcia-Rejon et al., 1991; Haessly, 1989; Green and Adkins, 1960; DeLorenzi and Nied, 1991; Harms and Michaeli, 1992; Cohen and Seitz,
Profile optimization for initial parison dimensions
j=
is applied in increments of 0.1 psi until mould contact is complete. The residual forces in the radial and axial directions are Frk and Fzk, respectively. The derivatives of W with respect to u, and V, can be found by employing the chain rule:
I
Input Tunrng hp(i,l)andh,(O
755
Constants,
(6)
Run Forward Process Sequence: Obtain h Ci,])
I and
E(i,j)
= hs(i)
Calculate
-h
dW
(1.j)
1
Total
Error
Yes Solution
=
hp(i,j)
-
Minlf’Wm
Error
ReaChed
?
NO
hp(i,j+
I) = hdl,j)
+ f
j-j+,
dWd&
dWdl,
dv,=dldo+dl,du,. L t
I
I E(i.j),
E(i,j-1)
1
-
Fig. 1. Flowchart of profiled optimization for a discretized model.
1991; Kouba et al., 1992). This method has been selected so as to reduce the size of the computational scheme and because it is the best developed of the alternatives. In the undeformed state, a node k of element i has the coordinates (r*, z~). After deformation, the node is displaced to coordinate (Q + o,, zk + Us), where vk and u, are the unknown components of the k node displacement vector. The other node of element i can be treated in a similar manner. The nodal displacements, uk and v, can be related to the individual stretch ratios by employing geometric considerations (Haessly, 1989). The new equilibrium position is determined by the balance between the externally applied pressure force and the internally directed elastic force. The radial force balance yields:
The elastic strain energy per unit volume W is related to the individual stretch ratios by equations (l-3). A linear 20°C temperature drop, from 200 to 18O”C, along the length of the parison has been observed (2). This drop in temperature is due to cooling of the parison as it is extruded in ambient air. The temperature dependence of the Mooney coefficients is incorporated along the length of the parison, according to the temperature profile. The stretch ratios and the derivatives of the stretch ratios with respect to the nodal displacements are functions solely of the nodal displacements (Haessly, 1989). The set of nonlinear equations is solved by using a Newton-Raphson routine, which aims at finding the unknown nodal displacements for a series of air pressures, that minimize the value of the force residuals at each node. The elements continue to deform, with increasing pressure, until they contact the mould. The element thickness distribution at mould contact is the final part thickness distribution. ProBled optimization The optimization routine involves running the forward predictor for J iterations, whereby the individual iterations can be represented by i. Therefore, at length location i and for iteration i the parison thickness is h,(i,j), the predicted final part thickness is h(i,j) and the specified final part thickness distribution is h,(i). The forward predictor can be depicted by the following relationship: h(i,j) = G(i) x hp(i,j),
and the axial force balance yields:
(5) where r is the segment radius, r,, is the segment thickness, P pressure and R, S and T are nodal displacements (Haessly,
I is the segment length, is the applied inflation functions solely of the 1989). The air pressure
(7)
(8)
where the function G(i) is dependent on process parameters such as mould geometry, melt temperature profile, segment location as well as interaction effects caused by adjacent segments along the length of the parison. The function G(i) is incorporated when running the forward predictor. The error at parison segment i for iteration j is: E&j)
= h,(i) - h(i,j),
(9)
R. W. Dr&ow
756
and A. GARCEA-REJON
and therefore substituting equation (8) into equation (9) yields: E(i,j)
= h,(i) - G(i)h,(i,j).
(IO)
Summation of equations (9) or (10) over the entire length of the parison yields: ET(j)
= C [h,(i) -
i
h(W)1
(11)
or
Ah,& j) = aE(i,j) ET(j)
= 1 [h,(i) -
i
W)~,G,Al.
(12)
The goal of the optimization is to find the parison thickness distribution h,(i,j) that gives the minimum overall error ET(j). Ideally, one would like to obtain the roots of equation (12). Another important criterion for the algorithm is the speed at which a solution is attained. Placing a numerical technique, in which each iteration can take 10 min, into an optimization search loop can result in very long computational times. Therefore it is important that convergence be. reached in a reasonable time frame. The optimization search therefore becomes a dual problem of minimizing the total overall error in the quickest possible time. This is analogous to the goal in traditional process control (Stephanopoulos, 1984). Therefore, the optimization methodology chosen uses classical control equations. Other optimization algorithms, such as gradient descent (Jenson and Jeffreys, 1977) and a root finding Newton-Raphson technique (Carnahan et al., 1969) were also studied, but the speed of convergence was found to be best for the methodology based on the process control equations (Schoebe, 1991). The additional time for the gradient descent and NewtonRaphson techniques was required to estimate the derivatives used in the respective techniques. The optimization routine involves updating the input parison thickness profile by an amount, Ah,(i,j), dependent on the present and past error profiles. The new parison thickness profile becomes: h,(i,j
+ I) = h,(i,j)
+ A&(&j).
For a proportional (P) optimizer parison thickness profile is: Ah,(i,j)
(13)
the change
= aE(i,j),
in
(14)
where a is the user-defined optimizer gain. A proportional-integral (PI) optimizer is defined by: A&(&j)
that may exist with the proportional optimizer (Stephanopoulos, 1984). A traditional P1 system integrates the past error over the total number of iterations. However, only one past error value is looked at in this case, in an effort to simplify the problem and minimize computational times. The user-defined integral term is fi. A proportional-integral-derivative optimizer (PID) is defined by:
= aE(i,j)
+ 0.5/3[E(i,j)
+ E(i,j
-
l)].
(15)
This convergence scheme employs knowledge of past error values in an effort to decrease any offset
+ 0.5 jS[E(i,j) + y[E(i,j)
- 1)
+ E(i,j - E(i,j
- l)].
(16)
The user-defined derivative term y is employed to help dampen oscillations that can result from the use of the integral term (Stephanopoulos, 1984). Equation (16) can be rearranged to yield: Ah,(i, j) = [a + OSfl + y]E(i,j) + [0.5/3 - y]E(i,j
-
1).
(17)
This equation is the standard form for an acceleration strategy used in recycle blocks for process simulations (Jenson and Jeffreys, 1977). The optimization routine continues in an iterative fashion, whereby the parison thickness profile is continuously updated, until the overall error in the final part thickness distribution is below a prespecified tolerance_ The parison thickness profile at this point is the desired solution. OF-TIMIZER
TUNING
The first phase of the methodology involves determining optimal values for the user-defined constants, a, /3and y. The criteria for selection of these “tuning constants” are minimum number of iterations for convergence and minimum offset error at the point of convergence. Once the optimization routine is “tuned” at the optimal values of a, B and y. it can be employed for prediction of various processing situations. Figure 2 shows the dependence of the error with the number of iterations for several values of updating gain a. The routine diverges for a gain of 5.0. This behaviour is analogous to process control situations where instability arises at high gains. An updating gain of 1.0 has the fastest convergence, but yields oscillations at the end of the routine. This is also analogous with process control situations, where high gains can result in an oscillatory response. Figure 3 portrays the relationship between the error and the number of iterations for two different integral constants, /3= 0 and 0.5. The updating gain for these situations is set at 0.5. The use of the integral constant induces oscillatory behaviour and decreases
---__ ...__ ---% \c Profile optimization for initial parison dimensions
1
.OQO
r’
757
-
1.000
/’
‘-------_______
‘.
“‘1
.._.
‘.
. .
._
-_
1
----
‘\\
. . . .. --
‘.
‘r
:
:
0.100
Updating gain 0.1
‘?
0.5 1.0 5.0
0.100
z
s
a ::
w
0.010
0.010
0.00 I
0.001 100
10
1000
Iterations Fig. 2. Optimization search as a function of gain term. the offset error
slightly.
to the effect of integral eliminated
The behaviour control
and oscillatory
behaviour
The effect of the derivative Fig.
4. Introducing
results in damping integral both
term.
integral
and
which
the minimum
yields
of
induced
updating
the fastest
oscillatory
y is shown
constant
set to 0.5 for these situations.
situation, PID
constant
the derivative
The
in 0.5
by the
gain
are
best tuned
convergence
behaviour,
8000 parison through
can occur.
of the oscillations
The
sion blow moulding
is analogous
action, where offset is
and
is therefore
a
gap
of
Experimental
data
for
comparison
The flush gap
The bottle
to the simuextru-
is defined
and mould
are at the
programmer
geometry
employed
ratios,
from
profiles
micrometer.
This
in this entials
of the part.
were measured and measuring
8.0 mm of bottle length, with a high-precision held
of
one at the top and
of the part, to four at the middle thickness
was
(10 Hz).
in Fig. 5. The bottle geometry
of blow-up
bottle
as the gap
in the die gap at an interval
ting the bottle into segments
with a Battenfeld-Fischer
was extruded
and outer bushing
0.1 s, for a 10s extrusion work are shown
with a MAC0
1.7 cm and a flush
The parison
able to input changes
bottom
lation was obtained
1.O mm.
equipped
The parison
die of o.d.
same vertical position.
The
EXPERIMENTAL
a converging
when the inner mandrel
a variety
optimizer.
machine
programmer.
corresponds
by cutat every hand-
to 40 sampling
R. W.
758
DIRADDO
and A. GARCIA-REJON
1.000
1.000
0.100
0.100
; : : : : : : : : : :
0.010
0.010
0.001
100
10
Iterations
Fig. 3. Optimization search as a function of integral term.
points along the length of the bottle. Point zero is at the bottom of the neck and point 40 is at the bottom of the bottle. The neck itself is neglected in the study, as its thickness profile is defined by the threading of the blow pin exterior and not the inflation. A high-density poIyethylene blow moulding grade resin (DuPont of Canada, !Sclair 56B) was employed in the study. The parison thickness profiles were measured by the technique described by DiRaddo and Garcia-Rejon (1993b). The methodology is able to measure the effects of both swell and sag on the parison thickness profile. The magnitude of the ratio of the Mooney constantc C,/C, is a direct measure of the deviation of
material behaviour from elastic theory (Treloar, 1976; Finney and Kumar, 1988). The value of 5.0 is obtained by utilizing the approach described in Finney and Kumar (1988) and employing extensional stress-strain data. The stress-strain data were obtained by employing a Rheometrics Extensional Rheometer and are presented elsewhere (GarciaRejon and DiRaddo, 1992). RESULTS
AND
DEX.XJSSION
The results of modelling of the process with the forward formulation, involving prediction of final part thickness distribution from initial parison di-
Profileoptimizationfor initial parison dimensions
759
1 .ooo
1 .ooo
-0 ----
0.5
0.100
0.100
0.010
0.010
?3 s
s
rEi
0.001
0.001 1
10
100
1000
Iterations Fig. 4. Optimizationsearch as a function of derivativeterm. mensions, are shown in Fig. 6. The results are compared to experimental data obtained off-line. The algorithm based on virtual work is employed. Results are presented for simulations run at a Mooney ratio, C,/C, of 5.0. When considering the forward predictor, some of&et is obtained between the experimental and theoretical results. The error is about 10% in the center of the part, but increases to as high as 150% towards the bottom of the part. Many factors can cause the offset, such as cooling and curvature effects introduced at the mould pinch. However, the predictions are acceptable in an engineering sense. The parison melt near the mould, at the top and bottom pinch locations, will tend to cool to a much lower temperature than the melt temperature, since these sections are exposed to conductive cooling for a brief period of time. This will result in a much
cooler and therefore stiffer material at these part sections. The inflation wilt subsequently be affected. The pinching of the parison by the mould also introduces curvature effects. The formlation assumes that the parison is of constant diameter, for the entire length of the parison, including the bottom part near the mould. However, in actuality, the bottom portion (3 cm length) takes on an elliptical shape. Therefore, for this bottom part, the parison diameter is no longer constant but is a function of the parison length. Simulations run with these nonconstant diameter parison diameter shapes did not converge, when run with the forward predictor (Derdouri and Khayat, 1992). Work is continuing to employ more elaborate numerical techniques to aid in the convergence. However, for the meantime, a constant diameter parison
R. W. DIGw
and A. GARCJA-REJON
Profile optimization
for initial parison dimeusions
761
0.24 0.22
0
2
6
10
14
18
Profile
22
26
30
34
38
point
8. Prediction of parison thickness distribution for a
bottle formed at an extrusion gap of 0.11cm.
0 -
5~
1’0
15
2’0
Profile
25
30
j5
40
45
point
Fig. 6. Prediction of bottle thickness distribution from parison thickness distribution-forward proces s sequence.
shape will bc employed and the bottom portion of the parison (profile points 3640) will not be considered. Once the optimization routine has determined the optimal PID parameters, it is considered tuned. Therefore, the PID parameters, a, fi and y are at their respective values, that render a minimum prediction error in the minimum number of iterations. These PID parameters can now be used for future predictions of the required parison dimensions, for a specified final part thickness distribution. The final part thickness distributions chosen as the set points were those obtained experimentally from the stated
size extrusions and are available in (DiRaddo Garcia-Rejon, 1!393a). Optimization routines are usually concerned with whether a global or local minimum has been obtained. Figure 7 shows the comparison between the set point part thickness distribution h,(i) for the case of a part formed at an extrusion gap of 0.11 cm, and the thickness distribution h(i,j) for this set point input, obtained in the final iteration J. The difference between the two curves is the overall error of equation (12). The two curves coincide, confirming that a global minimum condition has been obtained. Therefore, any offset that results, when comparing the predicted and experimental initial parison dimension profile, is a result only of the forward predictor’s limitations and not a result of the attainment of a local minimum in the optimization routine. The use of optimization routines also brings about concern with obtaining a unique solution. Several solutions can be found depending on the space domain where the search is taking place. Therefore,
0.14 0.13
0.32
0.12
0.30
0.11
s
0.09
2
0.08
8 2
z
0.28 1
Gap
= 0.
0.26
I1 cm
0 Set point
0.07
z_
0.10 0.06
3 ii
0.05 0.04
I\
t
0.03
z
0.24
2
0.22
z
0.20
t Y .o f _
0.18
E .z z
0.12
p.
0.0s
0.10
0.04
0.01
0.02
1 5
LO
I5
20
Profile Fig.
0.14
0.06
0.02
0
0.16
7. Comparison
25
30
35
40
c 45
0
2
6
IO
14
I8
Profile
point
of set point distribution to the final iteration distribution.
+:::::::::::::::::::c
Fig.
22
26
30
34
38
point
Prediction of parison thickness distribution for a bottle formed at an extrusion gap of 0.13 cm.
9.
R. W. DIRADDO
and A. GARCIA-REJON 0.34 0.32 0.30
7
0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14
t
~~;;;;:‘i”‘“::;::::. 0
2
6
10
14
18
Profile
22
26
30
34
38
0
pain!
Predictionof parison thicknessdistributionfor a bottle formed at an extrusiongap of 0.15 cm.
10.
employing different initial guesses allows for consideration of a broad space domain. A range of initial guesses was tried for the parison thickness distribution (0.50, 0.25, 0.10 and 0.05 cm). All these initial guesses, when individually input into the optimization routine, with the same set point thickness distribution, eventually converged to the same solution of parison thickness profile. Therefore, uniqueness of solution can be assumed, regardless of the initial guess of parison thickness distribution employed, as long as the guess is within the range 0.05-0.50 cm. The predictions of final part thickness distribution, after the first iteration, for the individual gaps can be found elsewhere (Schoebe, 1991). The thickness in the centre of the part, obtained after the first iteration, ranges from approx. 0.02-0.27 cm depending on the initial parison thick0.30 0.28
0.20 0.18 0.16
10
14
18
Profile
22
26
30
34
38
point
Fig. 12. Prediction of parison thickness distribution for a bottle formed with a step in the gap of magnitude -0.02 cm.
ness employed. The thickness in the centre of the part, obtained after the last iteration, is about 0.05 cm for all the initial guesses. Therefore considerable movement has occurred during the optimization search. Figures 8-10 show the required parison thickness profile for the thickness distribution of a part formed at constant extrusion gaps of 0. 1 1, 0.13 and 0.15 cm, respectively. The top of the parison is represented by Profile Point = 1. The final part thickness distribution chosen as the set point and the corresponding parison thickness profile were again obtained experimentally. The experimental parison thickness profile is constant, since the effects of sag and time-dependent swell were found to be minimal, for these cases (2). The instantaneous swell is the ratio of the experimen0.34 0.32 0.30
r
0.16 o.i4 0.12
t
0.04
l
Simulation
0.02 0
6
0.28 0.26 0.24 0.22 0.20 0.18
0.26 0.24 0.22
0.14
2
2
6
10
14
18
22
26
30
34
38
+
Profile point Fig. 11. Prediction of parison thickness distribution for a bottle formed at a die temperature of 180°C and an extrusion gap of 0. I1 cm.
0
2
6
10
14
18
Profile
22
26
30
34
38
point
Fig. 13. Prediction of parison thickness distribution for a bottle formed with steps in the gap of magnitude +/ - 0.02 cm.
Profile optimization for initial parison dimensions
tal parison thickness indicated on the respective figure, to the corresponding constant extrusion die gap. The prediction is very good in the centre of the parison. However, as larger gaps are considered, the simulation begins to present difficulties at the ends of the parison. As mentioned, the offset is caused by limitations in the forward predictor, because of the cooling ati curvature effects introduced by mould pinching. Figure 11 shows the effect of employing a lower melt temperature of 180°C and an extrusion gap of 0.11 cm. The simulation over-predictsat the bottom of the parison to a greater extent than for the case of 200°C. The swell is slightly greater at the lower temperature causing higher prediction of the parison dimension profile throughout the length. Figures 12 and 13 show the effects of variable die gap extrusions, indicative of parison programming. Figure 12 gives the prediction for the case of a final part produced at an extrusion, where a step in the die gap from 0.15 to 0.13 cm occurs. The simulation is able to predict the presence as well as the location of the step along the length of the parison. Nonlinear effects (Haessly, 1989; DiRaddo et al., 1993) are responsible for the overprediction that occurs before the step and the underprediction that occurs after the step. The forward predictor is unable to model these non-linear effects that are caused by oscillatory infiation patterns and compressive forces. Figure 13 gives the prediction for the case of a final part produced at an extrusion, where two steps in the die gap, from 0.134 11 cm and back to 0.13 cm, occur. The simulation is again able to predict the presence and locations of the steps, for this parison programming situation. The prediction is better, in an overall sense, than for the case of Fig. 12. A possible explanation for this is that the compressive forces that caused the oscillatory inffation pattern and nonlinear effects present for the case of Fig. 12, cancel each other because of the presence of equal and opposite steps (DiRaddo et al., 1993). CONCLUSION
The use of a combined Newton-Raphson and profiled optimization routine has been employed for the prediction of the parison thickness profile required for a specified final part thickness distribution. This
approach
involves
modelling
of
the inflation
process with the inverse formulation. A simulation based on modelling of the inflation process with the forward formulation is placed in an iteration loop, with the goal of determining the parison thickness
763
profile that results in the minimum difference between the specified final part thickness distribution and the output from the forward predictor. The modelling of the process with the forward formulation is based on the principle of virtual work. The parison behaviour during inflation, is considered to be an elastic membrane and the inflation is treated as a series of static deformations. The validity of the elastic material behaviour is justified because of the relatively short duration of the inflation time and by elastic recovery experiments performed in another work. The membrane is discretized into a series of elements and the membrane movement is solved by employing a Newton-Raphson routine. The optimization loop employs a methodology analogous to classical control techniques. Updating of the parison thickness profile after each iteration is performed based on the present error profile, as well as the integral and derivative of the error profile over the previous iterations. The results obtained indicate that the performance of the optimization loop is directly dependent on the performance of the forward predictor. Future work entails the improvement of the forward predictor in order to incorporate the effects of cooling and curvature introduced by the mould. Acknowledgements-The authors would like to express their gratitude to S. Schoebe and S. Lacoursiere for their contributions in this work. NOMENCLATURE C, = First Mooney coefficient C, = Second Mooney coefficient E = Elementerror ET = Summation of element errors Fd = Radial nodal force residual FTk= Axial nodal force residual G = Forward process sequence predictor function h = Final part element thickness output h, = Parison element thickness h, = Specified final part element thickness 11, = First strain invariant 11s= Second strain invariant J = Last iteration I = Element length k=Node P = Pressure R = Function of nodal displacement r = Element radius S = Function of nodal displacement T= Function of nodal displacement t,, = Undeformed element thickness z+ = Radial nodal displacement uk= Axial nodal displacement W = Strain energy per unit volume a = Updating gain p = Integral term y = Derivative term A = Updating amount i, = First stretch ratio 1, = Second stretch ratio 1, = Third stretch ratio
R. W. DIRADW
764
and A. GARCLVREJON
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