Author: YehLiang Hsu, TzuChi Liu, Francis Thibault, Benoit Lanctot (20030904);
recommended: YehLiang Hsu (20040324).
Note: This paper is published in Proceedings of the I MECH E Part B
Journal of Engineering Manufacture, Vol. 218, No. 2, 1 February 2004, p.
197212(16).
Design optimization of the blow moulding process using fuzzy
optimization algorithm
Abstract
Blow moulding is
the forming of a hollow part by “blowing” a mouldcavityshaped parison that is
made by thermoplastic molten tube. Blow moulded parts often require a strict control of the thickness
distribution in order to achieve the required mechanical performance and final
weight. A fuzzy optimization algorithm for determining
the optimal die gap openings and die geometry for the required thickness
distribution in the blow moulding process is presented. The idea of the fuzzy
optimization algorithm is that, instead of using purely numerical information
to obtain the new design point in the next iteration, engineering knowledge and
human supervision process can be modeled in the optimization algorithm using
fuzzy rules. The structure of an optimization algorithm is still maintained to
guide the engineering decision process and to ensure that an optimal solution
rather than a trial and error solution can be obtained. It is shown that how a
single fuzzy engine can be used in various cases and types of optimizations of
the blow moulding process.
Keywords: blow moulding, fuzzy
optimization algorithm, computer simulation
Notation
D_{im}

mandrel diameter at the neck point

D_{ob}

diameter of bushing

D_{om}

mandrel diameter at the die exit

f

optimization objective function

m

the total number of nodes in the finite element model

n

the total number of discrete die gap opening programming points

x_{i}

die gap opening at ith programming point

Dx_{i}

the vector of change of die gap openings at ith
programming point

x^{q}

vector of x_{i} in the qth iteration

Dx^{q}

the change in x^{q}

x_{i,}_{max}

maximum allowable die gap opening for ith programming
point

x_{i,}_{min}

minimum allowable die gap opening for ith programming
point

Y

target thickness

y_{j}

the thickness at the jth node

y^{q}

vector of y_{j} in the qth iteration

Dy^{q}

the change in y^{q}

_{}

the average weighted thickness _{} of all nodes
affected by x_{i}

_{}

maximum value of average weighted thickness for ith
programming point

_{}

minimum value of average weighted thickness for ith
programming point

a^{q}

step size in the qth iteration

1.
Introduction
Blow moulding is
the forming of a hollow part by “blowing” a mouldcavityshaped parison that is
made by thermoplastic molten tube. It is the most popular and efficient process for manufacturing commodity
hollow plastic parts such as bottles, containers, and toys. More recently, this
forming process has been applied to the manufacture of complex automotive parts
such as fuel tanks, seat backs, air ducts, windshield washer and cooling
reservoirs.
The blow moulding process consists of three
phases: parison extrusion, parison inflation and part solidification. As shown
in Figure 1, the extrusion phase involves the extrusion of a polymer melt
through an annular die to form a hollow cylindrical parison with a nonuniform
material distribution and consequently nonuniform parison thickness along its
length. Once the parison is extruded to the desired length, it is inflated to
take the shape of an enclosing mould. The part then solidifies as a consequence
of heat transfer to the cooling mould. The parison thickness distribution is
modified significantly by the inflation and the solidification stages to yield
the final part thickness distribution.
Figure 1. Blow
moulding process
Blow moulded parts often require a strict
control of the thickness distribution in order to achieve the required mechanical
performance and final weight. Figure 2
shows the forming of an axisymmetric bottle. As illustrated in Figure 2(a), by
moving the mandrel up and down, the die gap can be adjusted as a function of
time. The movement of the mandrel can be programmed by the percentage of gap
openings at discrete time. When gap opening is 0%, the mandrel is at the upper
limit, which results in the minimum die gap; when gap opening is 100%, the
mandrel is at the lower limit, which results in the maximum die gap. Manipulation
of the programming of gap openings can lead to an optimal part thickness
distribution. For example, in order to obtain
uniform thickness distribution of the hollow part, the thickness of a
programmed parison must be vary along its length. As shown in Figure 2(b), the
parison thickness for the largest expansion area must be thicker than those of the
other areas.
(a)
(b)
Figure 2. Illustration of parison programming
BlowSim is a
finite element software package designed to simulate the extrusion blow moulding,
injection stretch blow moulding, and thermoforming processes. It is developed
by the Industrial Materials Institute (IMI) of the National Research Council (NRC),
Canada.
The blow moulding
process simulation consists of modelling the successive process stages in order
to predict the final part quality as a function of the operating conditions,
the mould geometry and the material properties. Predictions of final part
thickness were made by integrated simulation of the parison formation, clamping
and inflation, and part cooling and solidification stages. Programming points,
die dimensions, extrusion temperature, parting plane shape, and mould
temperature were among the operating conditions considered [1]. BlowSim
can be used to model the process phases: parison formation, clamping and
inflation, part cooling and shrinkage, and part mechanical performance. The
process modelling is based on a large displacement finite element formulation [2].
The parison deformation is modelled using a multilayer membrane element type
and a nonisothermal viscoelastic material model. The mechanical performance
is modelled with the predicted thickness distribution, and the appropriate
applied load. The simulation results of BlowSim have been validated with many
industrial cases and show good agreement.
In many industrial applications, combining simulation
tools with optimization methodologies allows the designers to treat complex
design criteria via simulation to pursue maximum part quality and minimum
manufacturing costs. In the
blow moulding process, it is desirable to manipulate the percentage
of die gap openings to obtain a
final part of constant thickness or a predefined thickness profile. It is,
therefore, an optimization problem on how to control the die gap openings to
minimize the deviation in the thickness of the final part from the target
thickness.
Figure 3 shows
the finite element model of the bottle case in Figure 2 by BlowSim. Given a set
of die gap openings at n programming points x_{i}, i=1,
2, …, n, we are able to extract the thickness of all nodes from the
simulation results by BlowSim, and apply them to the following equation to get
the objective function value:
min. _{} (1)
where y_{j} is the thickness at the jth node
in the simulation model, and Y is the corresponding target thickness,
and m is the total number of nodes in the BlowSim finite element model.
The die gap openings at discrete programming points x_{i}, i=1,
2, …, n, are the variables to be determined in the optimization process.
Obviously, the thickness at a node is a function of the corresponding die gap
openings.
Figure 3. The finite element model of the bottle
DiRaddo and GarciaRejon [3] proposed an
iterative optimization loop which combined a blow moulding process predictor
and an updating technique to search for the parison thickness profile that
results in the minimum overall difference between the specified final part
thickness distribution and the individual iteration’s output from the
predictor. Lee and Soh [4] determined the optimal thickness profiles of
a preform for a blowmoulded part having required wall thickness distribution.
A finite element model is formulated to relate the preform wall thickness
distribution to the wall thickness distribution in the blowmoulded part. The
feasible direction method is used for optimization, and the design variables
are the thickness of finite elements.
Gradient type numerical
optimization algorithms can certainly be used to solve for the optimal die gap
openings. On the other hand, manufacturing engineers usually adjust the die gap
openings empirically: reduce the die gap opening if the corresponding portion
of the final part is too thick, and vice versa.
When solving an
engineering optimization problem using numerical optimization algorithms, we
basically view the problem as a pure mathematical optimization model. Design
modifications in the optimization process rely on numerical information rather
than engineering heuristics, experience, and knowledge. This paper develops a “fuzzy
optimization algorithm” for engineering optimization problems, which enables
the use of engineering heuristics to generate the new design point of the next
iteration. The structure of an optimization algorithm is still maintained to
guide the engineering decision process and to ensure that an optimal solution
rather than a trial and error solution can be obtained. Currently this fuzzy
optimization algorithm is developed specifically for engineering optimization
problems whose objective functions are in the form of Equation (1).
This paper first
explains the concept of fuzzy optimization algorithms. The blow moulding
process optimization results are presented to demonstrate the generality of this
approach to various optimization cases in different application domains.
2.
The concept of “fuzzy
optimization algorithms”
As shown in
Figure 4, the optimization process can be viewed as a closedloop control
system. In the case of blow moulding process optimization, BlowSim is analogous
to the system process to be controlled, whereas an optimization algorithm is
analogous to the controller. In the qth iteration, BlowSim simulation
results (thickness distribution y^{q}) are input
to the optimization algorithm, which in turn generates the change in die gap
openings (Dx^{q}) according to its search rules. Die gap openings for the next
iteration are updated (x^{q}^{+1}= x^{q}+Dx^{q}) and simulated again using
BlowSim to continue the iteration. Finally, a control system attempts to
achieve a stable, predefined output. The optimization process pursues a
converging objective function value.
Figure 4. General block diagram of a design
optimization process
When we apply traditional
numerical optimization algorithms to an engineering problem, we treat the
engineering problem as a pure mathematical problem. Engineering heuristics are
totally ignored. This motivates the idea that, in addition to crisp numerical
rules, the engineering heuristics such as “reduction in the die gap opening if
the corresponding portion of the final part is too thick, and vice versa”
should also be modeled in the optimization algorithm using fuzzy rules. As
suggested in Figure 4, the “controllers” in the optimization process may as
well be fuzzy controllers!
A fuzzy system is characterized by a collection
of linguistic statements based on expert knowledge. The linguistic statements
are usually in the form of IFTHEN rules. As shown in
Figure 4, if the relations between the system process input x^{q} (die gap openings) and system process output y^{q}^{ }(thickness distribution) and Dy^{q} are known empirically (reduce the die gap opening will reduce the
thickness of the corresponding portion of the final part, and vice versa), a
fuzzy logic engine instead of a numerical optimization algorithm can be used to
generate the system process input change rate Dx^{q} according to a set of domain parameters given by the users.
Arakawa and
Yamakawa [5] demonstrated an optimization method using qualitative
reasoning, which makes use of the qualitative information that gives an
approximate direction of the optimum search. Hsu et al. [6] proposed a
fuzzy optimization algorithm and applied it for determining the “move limit”,
which is an important optimization process parameter in the sequential linear
programming algorithm. Mulkay and Rao [7] also proposed a modified
sequential linear programming algorithm using fuzzy heuristics to control the
optimization parameters. Arabshahi et al. [8] pointed out that many
optimization techniques involve parameters that are often adapted by the user
through trial and error, experience, and other insight. Instead, they applied
neural and fuzzy ideas to adaptively select these parameters.
In these papers,
fuzzy heuristics were used to control the parameters of the optimization
algorithm to improve its performance. The following sections demonstrate how
engineering heuristics can also be modeled into the fuzzy optimization
algorithm for the optimization of the blow moulding process.
3.
Die gap opening optimization
for constant part thickness
The bottle case
study in Figure 3 was first used to illustrate the fuzzy optimization
algorithm. In this example, we hope to manipulate the die gap openings at 7 discrete
programming points (x_{i},
i=1,.., 7) to obtain a uniform wall thickness
part of 2 mm.
Therefore, in the objective function Equation (1), Y = 2. Note that the
die gap opening at a discrete time point x_{i} may
affect the thickness of many nodes. BlowSim provides the “average weighted
thickness” _{} of all nodes
affected by x_{i}. As discussed earlier, designers usually adjust the die gap
openings empirically: reduce the die gap opening if the corresponding portion
of the final part is too thick, and vice versa. This engineering heuristic indicates that the average weighted
thickness of a certain portion (_{}) is
a monotonic increasing function with respect to the corresponding die gap
opening (x_{j}), and can be expressed by
5 rules:
(1)
IF
_{} is PB THEN Dx_{i}
is NB;
(2)
IF
_{} is PS THEN Dx_{i}
is NS;
(3)
IF
_{} is ZE THEN Dx_{i}
is ZE;
(4)
IF
_{} is NS THEN Dx_{i}
is PS;
(5)
IF
_{} is NB THEN Dx_{i}
is PB.
The quantization table (Table 1) gives quantitative
definitions for PB (positive big), PS (positive small), ZE (zero), NS (negative
small) and NB (negative big). There are 5 “domain parameters” in Table 1 to be determined
by the user according to the application problem. The definition of the 5 domain parameters
and their numerical values for the bottle case example are
Y: target
thickness (2mm);
_{}: minimum value of
average weighted thickness (0mm
for all programming points);
_{}: maximum value of average weighted thickness (4mm for all programming points);
x_{i,}_{min}: minimum allowable die gap opening (5% for
all programming points);
x_{i,}_{max}:
maximum allowable die gap opening (95% for all programming points).
Table 1. The quantization table
Boundaries of fuzzy input, _{}

Boundaries of fuzzy output, Dx_{i}

Quantized Level

_{}

_{}

2

_{}

_{}

1

_{}

0

0

_{}

_{}

1

_{}

_{}

2

Table 2 shows
the results for the first two iterations of the bottle example. Initially, the
die gap openings are set at 50% for all 7 programming points. From BlowSim
simulation, the average weighted thickness varies from 1.341mm to 5.119mm, and the objective
function value is 0.82. Then the fuzzy engine generates the change in die gap openings
Dx_{i} and
the die gap openings at the 7 programming points are updated. From BlowSim
simulation, the average weighted thickness now varies from 1.374mm to 3.262mm, and the objective
function value is reduced to 0.78. The fuzzy engine then generates Dx_{i} for this
iteration, and the die gap openings are updated again.
Table 2. Results for the first two iterations of
the bottle example
Programming points

1

2

3

4

5

6

7

Objective
function value

Initial values

x_{i}(%)

50

50

50

50

50

50

50

0.82263

_{}(mm)

5.119

2.671

1.362

1.341

1.858

3.957

4.828

Dx_{i}(%)

34.604

15.623

14.336

14.791

2.787

33.998

34.604

1^{st} iteration

x_{i}(%)

15.396

34.377

64.336

64.791

52.787

16.002

15.396

0.77913

_{}(mm)

3.258

1.374

1.938

2.29

2.015

2.846

3.262

Dx_{i}(%)

5.151

18.928

0.835

18.767

0.3119

4.475

5.159

2^{nd} iteration

x_{i}(%)

10.256

53.305

65.171

46.025

52.475

11.528

10.237

0.61265

_{}(mm)

3.453

2.292

1.978

1.212

1.652

2.696

3.042

Dx_{i}(%)

2.842

6.330

0.2905

18.902

6.788

2.263

2.398

Referring to
Figure 4, in the optimization iterations, we expect the objective function to
flatten out when approaching convergence. However, in reality, the objective
function value might “overshoot” when approaching convergence. In many
numerical optimization algorithm, a scalar multiplier a^{q} (often called “step size”) determining the amount of change for
this iteration is introduced, and x^{q}^{+1}= x^{q}+a^{q}Dx^{q}[9]. Usually a is adjusted dynamically throughout the iteration process. The heuristics
for adjusting a is simply, reduce if
the change in objective function value is big, and vice versa. Obviously this
can also be expressed by the same 5 rules previously discussed. In the examples
in this paper, initially a^{0}=1, and in the iteration process, a is adjusted
using the same fuzzy engine. If the change in objective function is big
(objective function increases rather than decreases), a in the next
iteration will be reduced to 0.5~1.0 times of that of the current iteration.
The current iteration will be given up if the increase in the objective function
is larger than 10%.
Finally, Figure
5(a) shows the iteration history for the bottle case, including the history of
the objective function value and the step size to show the effect of the step
size control. Figure 5(b) compares the initial (50% die gap openings for all
programming points) and final die gap openings, and Figure 5(c) compares the average
weighted thickness of the initial and final design on the 7 programming points.
(a) Iteration history
(b) Initial and final die gap openings
(c) Initial and final average weighted thickness
Figure 5. Die gap opening optimization results for
the bottle case study using 7 programming points
In this example,
the optimization process terminated after 17 iterations, when the change in
objective function value was less than 0.1%. Only 18 BlowSim simulations were
needed (one simulation was given up between iterations 3 and 4 because the
overshoot was too large), and no sensitivity calculation was required. Ideally
the objective function should converge to zero upon obtaining a part with
uniform thickness. However, the objective function at the end of the iteration
is 0.47, and as shown in Figure 5(c), the average weighted thickness of the top
and bottom portions of the bottle are still higher than the target value. As
shown in Figure 5(b), the average weighted thickness at these two portions
cannot be further reduced because the corresponding die gap openings are
already close to the lower bound 5%. Figure 6 shows the optimization result of
the bottle example with the same domain parameters using 31 programming points.
The fuzzy optimization algorithm terminated after 18 iterations, and 20 simulations
were needed in this case. Increasing the number of programming points further
reduces the objective function value. However, too many programming points in a
short parison extrusion time is sometimes not practical since the pneumatic
mandrel movement into the die head is limited by its response time. Note that
the computation cost in each iteration of the fuzzy optimization algorithm is
independent of the number of design variables.
(a) Iteration history
(b) Initial and final die gap openings
(c) Initial and final average weighted thickness
Figure 6. Die gap opening optimization results of
the bottle case study using 31 programming points
The fuzzy
optimization algorithm was then applied to the process optimization of a fluid reservoir shown in Figure 7,
which is a more complex automotive part. In this case the target thickness was 5mm, and 23
programming points were used. The values of the domain parameters in this case
were assigned as: Y =5mm,
_{}=0mm, _{}=10mm, x_{i,}_{min}=5%
and x_{i,}_{max}=95%.
Figure 7. Geometry of the windshield washer fluid
reservoir
The optimization
process terminated after 15 iterations, and Figure 8 shows the optimization
results. In Figure 8(c), the average weighted thickness of all programming
points after optimization are close to 5mm, but the objective function value in Figure 8(a)
at the end of optimization is still high (1.69). This is because the fluid
reservoir is not symmetric. For unsymmetrical parts whose cross sections are not circular, it is not possible to obtain a part with uniform thickness using a
circular die. When the parison is inflated to take the shape of an enclosing mould, the thickness
varies along the crosssection of the part. The die
geometry has to be manipulated before the die gap opening optimization in order
to obtain the desired part thickness. This will be discussed in the next
section.
(a) Iteration history
(b) Initial and final die gap openings
(c) Initial and final average weighted thickness
Figure 8. Die gap opening optimization results for
the fluid reservoir case
4.
Die geometry optimization
In BlowSim, the geometry
of the die in the closed and open positions is defined by the minimum and
maximum die gap (GapMin and GapMax) at a number of die sections or “die points”
at different angular positions. Figure 9 shows the geometry of a typical bushing
and mandrel die head components. GapMin is defined as the effective die gap for
0% die gap opening
GapMin
= (D_{ob } D_{om})/2, (2)
and GapMax is defined as the effective die gap for 100% die gap opening
GapMax
= (D_{ob }– D_{im})/2, (3)
Figure 9. Illustration of the die geometry
For symmetrical
parts with circular crosssections, a uniform thickness can be obtained using
circular die geometry. For unsymmetrical parts, GapMin and GapMax at each die
point should be optimized first to obtain a die geometry that is suitable for
the shape of the unsymmetrical part. Then the die gap opening optimization is
carried out to obtain a part with constant thickness using this die geometry.
BlowSim also provides the average weighted thickness of all nodes affected by a
die point.
Here the die
geometry optimization manipulates only GapMax while keeping GapMin fixed. The
objective is to obtain constant average weighted thickness for all die points.
The engineering heuristics for adjusting GapMax is the same as those for
adjusting die gap openings: reduce GapMax if the corresponding average weighted
thickness is too large, and vice versa. Obviously this can also be expressed by
the same 5 fuzzy rules previously discussed. Note that the objective function
used here is to minimize the deviation of the average weighted thickness of the
die points from the target thickness. Moreover, in die geometry optimization, x_{i}
becomes the GapMax at die point i.
The bottle case
in Figure 3 was used again to verify the results of die geometry optimization
using the fuzzy optimization algorithm. As shown in Figure 10(b), we
deliberately created a noncircular initial die geometry for validation
purpose. The domain parameters are: Y =2mm, _{}=0mm, _{}=4mm, x_{i,}_{min}=3mm and x_{i,}_{max}=13mm. Note that the definitions of some of
the parameters have been changed, though the same fuzzy engine is used. In this
case, the die gap openings were kept at 50% during the die geometry
optimization, and GapMin was fixed at 3mm. Figure 10(a) shows that after 11 iterations, the
fuzzy optimization algorithm converged to the expected circular die geometry
shown in Figure 10(b) because the bottle is a symmetrical part. A total of 11
BlowSim simulations were needed. Figure 10(c) shows that the final average
weighted thickness of all die points are close to the target thickness of 2mm.
(a) Iteration history
(b) Initial and final die geometry
(c) Initial and final average weighted thickness
Figure 10. Die geometry optimization results of the
bottle case
The die geometry
optimization process was then applied to the windshield washer fluid reservoir in Figure 7. In this
case the die gap openings were kept at 50% during the
die geometry optimization, and GapMin was fixed at 0.17mm. The domain parameters in the die geometry
optimization of the fluid reservoir are assigned as follow: Y =5mm, _{}=0mm, _{}=10mm, x_{i,}_{min}=0.17mm and x_{i,}_{max}=20mm. The die geometry optimization process
terminated after 15 iterations. Figure 11(a) compares the initial and final
GapMax. In this example (and the following examples), the diameter of the die
is relatively large compared to the die gap, and therefore the die geometry is
not shown here. Figure 11(b) shows that the average weighted thickness of all
die points are close to the target thickness of 5mm.
To get a better
design, die gap opening optimization was then applied to the fluid reservoir
example using the final die geometry in Figure 11(a). The domain parameters are
Y =5mm, _{}=0mm, _{}=10mm, x_{i,}_{max}=95%,
and x_{i,}_{min}=5%.
Figure 11(c) shows the
iteration history of both die geometry optimization and die gap opening
optimization. The objective function in Figure 11(c) is the deviation of the thickness of the final part from the target
thickness (Equation
(1)). Compared to the final die gap opening objective function value of 1.69
obtained from the gap opening optimization (Figure 8(a)), the die geometry
optimization objective function drops from 2.26 to 1.60, and further drops to
1.27 after die gap opening optimization. A total of 23 BlowSim simulations were
needed, 15 for die geometry optimization and 8 for die gap opening
optimization. Figure 11(d) shows the initial and final die gap openings.
(a) Initial and final GapMax
(b) Initial and final average weighted thickness
of the die geometry optimization
(c) Iteration history of the die geometry and die
gap opening optimization
(d) Initial and final die gap openings
Figure 11. Optimization results of the fluid
reservoir case
5.
Application Examples
In this section,
the process of die geometry optimization followed by die gap opening
optimization is applied to the other two unsymmetrical blow moulded parts shown
in Figure 12 and 13, the jerry can and the gas tank, respectively. Again, the
same fuzzy engine can be applied to both examples for die geometry optimization
and die gap opening optimization after a simple domain parameter mapping shown
in Table 3 and Table 4.
Figure 12. Geometry of the jerry can
Figure 13. Geometry of the fuel gas tank
Table 3. Domain parameter mapping for the jerry can
example
Domain Parameters

Die Geometry Optimization

Die gap opening Optimization

Y

2mm

2mm

_{}

4mm

4mm

_{}

0mm

0mm

x_{i,}_{max}

20mm

95%

x_{i,}_{min}

2mm

5%

Table 4. Domain parameter mapping for the gas tank
example
Domain Parameters

Die Geometry Optimization

Die gap opening Optimization

Y

5mm

5mm

_{}

10mm

10mm

_{}

0mm

0mm

x_{i,}_{max}

40mm

95%

x_{i,}_{min}

2mm

5%

In the jerry can example, 17 die points were
used, GapMin was fixed at 2mm, and the die gap openings were kept at 50% during
the die geometry optimization. The die geometry optimization process terminated
after 5 iterations. Die
gap opening optimization is then applied to the jerry can example using the
final die geometry shown in Figure 14(a). Eleven programming points were used.
The die gap opening optimization terminated after 20 iterations. Figure 14(b)
shows the final die gap openings. The objective function value drops from 0.84
of the initial design to 0.62 of the final design, and a total of 5+20=25
BlowSim simulations were needed.
(a) Initial and final GapMax
(b) Initial and final die gap openings
Figure 14. Die geometry and die gap opening
optimization results for the jerry can case
In the gas tank example, 8 die points were
used, GapMin was fixed at 2mm, and the die gap openings were kept at 50% during
the die geometry optimization. The die geometry optimization process terminated
after 11 iterations. Figure 15(a) shows the initial and final die geometry. Die gap opening optimization is then
applied to the gas tank example using the final die geometry shown in Figure
15(a). Twenty programming points were used. The die gap opening optimization
terminated after 13 iterations. Figure 15(b) shows the final die gap openings.
The objective function value drops from 1.89 of the initial design to 1.32 of
the final design, and a total of 11+13=24 BlowSim simulations were needed.
(a) Initial and final GapMax
(b) Initial and final die gap openings
Figure 15. Optimization results for the gas tank
case
6.
Conclusions
This paper
presents a fuzzy optimization algorithm for determining the optimal die gap
openings and die geometry in the blow moulding process. This fuzzy optimization
algorithm has been integrated with the computer simulation software BlowSim,
and has been tested on a number of blow moulding examples. The fuzzy
optimization algorithm does not require sensitivity information and are
completely external to BlowSim. Using a set of userdefined parameters, it is
shown that a single fuzzy engine can perform die gap opening optimization and
die geometry optimization for various cases. This characteristic makes the fuzzy
optimization algorithm easily expandable to the integration with other
simulation software in other application domains.
Comparing to
traditional numerical optimization process, the fuzzy optimization algorithm
tries to utilize engineering heuristics and is closer to the engineering
decision process. The structure of an optimization algorithm is still
maintained to guide the engineering decision process and to ensure that an
optimal solution rather than a trialand error solution can be obtained.
7.
Acknowledgement
This research
was supported by the National Science Council, Taiwan under grant No. NSC91
2212E155007, and by National Research Council, Canada under grant No. NRC
7505K01.
References
1.
Laroche, D., DiRaddo, R. W., and Aubert, R., “Process modeling of
complex blowmoulded parts,” Plastics Engineering, 1996, 52(12),
p. 3739.
2.
Laroche D., Kabanemi K., Pecora L., and DiRaddo R, “Integrate
numerical modeling of the blow moulding process,” Polymer Engineering &
Science, 1999, 39(7), p.12231233.
3.
Diraddo, R.W. and GarciaRejon, A., “Profile optimization for the
prediction of initial parison dimensions from final blow moulded part
specifications” Computers & Chemical Engineering, 1993, 17(8),
p751764.
4.
Lee, D. K. and Soh, S. K., “Prediction of optimal perform thickness
distribution in blow moulding,” Polymer Engineering and Science, 1996, 36(11),
p. 1513 1520.
5.
Arakawa, M. and Yamakawa, H., “Study on the optimum design applying
qualitative reasoning,” Transactions of the Japan Society of Mechanical Engineers,
Part C, 1990,56(522), Mar, p.398403.
6.
Hsu, Y. L., Lin, Y. F. and Guo, Y. S., “A Fuzzy Sequential
Linear Programming Algorithm for Engineering Design Optimization,” Design
Engineering Technical Conferences, ASME 1995, 1, p. 455462.
7.
Mulkay, E. L. and Rao, S. S., “Fuzzy Heuristics for Sequential Linear
Programming,” Journal of Mechanical Design, 1998, 120, p. 1723.
8.
Arabshahi, P., Choi, J. J., Marks, R. J. II, and Caudell, T. P.,
“Fuzzy parameter adaptation in optimization: some neural net training examples,”
IEEE Computational Science & Engineering, 1996, 3(1), p
5767.
9.
Vanderplaats, G. N., Numerical Optimization Techniques for Engineering Design,
McGrawHill, Inc., 1993.