Authors: YehLiang Hsu, TzuChi Liu, Francis Thibault1 and Benoit Lanctot
(20020921); recommended: YehLiang Hsu (20030214).
Note: This paper is revised and presented at The 2002 NRCNSC
CanadaTaiwan Joint Workshop on Advanced Manufacturing.
A fuzzy optimization algorithm
for the blow moulding process
This paper
presents recent results of an ongoing NRCNSC joint research project on the
development of a multidisciplinary design optimization (MDO) methodology for
blow moulded automotive parts. In particular, this paper demonstrates a fuzzy
optimization algorithm for determining the optimal gap openings and die geometry
in the blow moulding process. Traditional numerical optimization algorithms
treat the optimization problem as pure mathematical problems. Engineering
knowledge about the problem is not utilized in the optimization process. 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 modelled in the
optimization algorithm using fuzzy rules. It is shown that how a single fuzzy
optimization engine can be used in various types of optimizations.
Key Words: blow
moulding, process optimization, fuzzy logic
1.
Introduction
Blow moulding is
the forming of a hollow part by “blowing” a mouldcavityshaped parison that is
made by thermoplastic molten tube. Blow moulding is the most popular and efficient process for
manufacturing commodity hollow plastic parts such as bottles, containers, toys,
etc. 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, part inflation and part solidification. 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 the 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.
Blow moulded parts often require a strict
control of the thickness distribution in order to achieve the required
mechanical performance and final weight. Manipulation
of the die gap programming points can lead to an optimal part thickness
distribution. Figure 1 shows the forming of an axis symmetric bottle. As
illustrated in Figure 1(a), the die gap can be adjusted as a function of time
in order to obtain the desired thickness profile along the extruded parison,
which determines the thickness of the blown
hollow part. For example, in order to obtain uniform thickness distribution of
the hollow part, the thickness of programmed parison must be nonuniform. As
shown in Figure 1(b), the parison thickness for the greatest expansion area
must be thicker than those of the other areas.
(a) (b)
Figure 1. 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 Material Institute (IMI) of National Research Council (NRC), Canada. The blow moulding process simulation
consists of the modelling of 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. 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 [Laroche et al., 1999]. 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 the maximum part quality and minimum
manufacturing costs. In the blow moulding process, it
is desirable to manipulate the die gap programming to obtain a final part of constant
thickness or a predefined thickness profile. It is, therefore, an optimization
problem on how to control the gap openings to minimize the deviation of the
thickness of the final part from the target thickness. Given a set of gap
openings, we are able to extract the thickness of all nodes from the simulation
results by BlowSim, and apply them into the following equation to get the
objective function value:
min. _{} (1)
where y_{i} is the thickness at the ith node
in the simulation model, Y_{i} is the corresponding target
thickness, and n is the total number of nodes. The gapopenings at
discrete time points are the design variables. Obviously, y_{i} is
a function of the gap openings.
Gradient type numerical
optimization algorithms provide a numerical tool to solve for the optimal gap openings.
On the other hand, manufacturing engineers usually adjust the gap openings
empirically: reduce the gap opening if the corresponding portion of 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, experiences, 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 an optimal 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 that this approach can
be general to various optimization cases in different application domains.
2.
The concept of “fuzzy
optimization algorithms”
As shown in
Figure 2, the optimization process can be viewed as a closedloop control
system. The optimization model in an optimization process is analogous to the
plant in a control system; an optimization algorithm is analogous to the
controllers. Initial parameters are input to the optimization algorithm, which
in turn generates a trial design point according to its search rules. The
optimization model is then evaluated at this trial design point, and the
information such as objective and constraint function values and sensitivity is
fed back. The optimization algorithm is triggered again to generate the next
design point, using the information from previous iterations. Finally, a
control system attempt to achieve a stable, predefined output. The optimization
process pursues a converging objective function value.
Figure 2. General block diagram of a design
optimization process
Traditional
numerical optimization algorithms are analogous to direct digital controllers.
The algorithms are usually “crisply” designed for well defined mathematical
models. Their numerical rules for generating the next design point are exact
and definite, and they can usually be proved to have nice converging behavior
when applying to well defined mathematical models. However, in engineering
optimization problems, we seldom have well defined mathematical models. The functions
in an engineering optimization problem often do not have exact algebraic forms
in terms of the design variables, and they can only be evaluated through
experiments or computer simulations, which are expensive and imprecise in
nature. Sensitivity required in most numerical optimization algorithms is often
obtained from finite difference methods. Very often the cost of the number of
function evaluations required to meet the “crisp” definition of the numerical
algorithms is simply too high to be affordable.
When we apply
numerical optimization algorithms on 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 human supervision process should also be modelled in an optimization
algorithm using fuzzy rules. As suggested in Figure 2, 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 2, if the relations between the system process input x^{q} (gap openings) and system process output y^{q}^{ }(thickness in the blow moulding example) and Dy^{q} are known empirically
(reduce the gap opening will reduce the thickness of the corresponding portion
of the final part, and vice versa), the fuzzy logic
optimization engine will generate the system process input change rate Dx^{q} according to a set of domain parameters given by the
users. The step size a^{q} in Dx^{q} is set to be 1 initially, but is also
controlled by the same fuzzy optimization engine. The system process input is
then updated (x^{q}^{+1} = x^{q} + Dx^{q}) and the new
system process outputs are fed back to compare with set point Y. This
iterative process continues until the predefined convergence criterion is met.
Hsu et al [1995]
proposed this concept of a fuzzy optimization algorithm, and Mulkay and Rao
[1998] also presented the same idea. In both work, fuzzy heuristics are used to
control the parameters of the optimization algorithm to improve its
performance. The following sections demonstrate how engineering heuristics are modelled
into the fuzzy optimization algorithm for the optimization of the blow moulding
process.
3.
The blow moulding examples for
constant thickness
3.1
The bottle example
The bottle
example in Figure 1 is first used to illustrate the fuzzy optimization process.
In this example, we hope to manipulate the gap openings at 7 control points to
obtain a constant thickness part at 2mm.
Therefore, in the objective function Equation (1), Y_{i} = 2,
and n = 7. As discuss earlier, manufacturing engineers usually adjust
the gap openings empirically: reduce the gap opening if the corresponding
portion of final part is too thick, and vice versa. This engineering heuristic indicates that the thickness of a
certain node (y_{i}) is a monotonic increasing function with
respect to the corresponding gap opening (x_{j}), and can be expressed by 5 fuzzy rules:
(1)
IF
y_{i} is PB THEN Dx_{i} is NB;
(2)
IF
y_{i} is PS THEN Dx_{i} is NS;
(3)
IF
y_{i} is ZE THEN Dx_{i} is ZE;
(4)
IF
y_{i} is NS THEN Dx_{i} is PS;
(5)
IF
y_{i} is NB THEN Dx_{i} is PB.
The quantization table (Table 1) gives quantitative
definitions for PB (positive big), PS (positive small), ZE (zero), NB (negative
small) and NB (negative big). There are 5 “domain parameters” in Table 1 to be
decided by the user according to the application problem. From BlowSim simulation, when the gap openings at the 7 control
points are all set to be 75%, the maximum thickness of the part is 6.05mm, and the minimum thickness of the
part is 1.36mm. Therefore, the definitions of the 5 domain
parameters and their values for the bottle example are
Y_{i}: Target value of system process output (target thickness, 2mm);
y_{i,}_{max}: Maximum
value of system process output y_{i} (maximum thickness in the
initial design, 6mm);
y_{i,}_{min}: Minimum
value of system process output y_{i} (minimum thickness in the
initial design, 1mm);
x_{i,}_{max}: Maximum
value of system process input x_{i} (maximum allowable gap
opening, 95%);
x_{i,}_{min}: Minimum
value of system process input x_{i} (minimum allowable gap
opening, 5%).
Table 1. The quantization table
Boundaries of fuzzy input, y_{i}

Boundaries of fuzzy output, Dx_{i}

Quantized Level

_{}

_{}

2

_{}

_{}

1

_{}

0

0

_{}

_{}

1

_{}

_{}

2

In this research, the process of assigning the
values of the 5 domain parameters to the current application problem is called “domain
parameters mapping.” There are two process parameters Q_{is} and
Q_{os} in Table 1, which define the linearity of quantized level
with respect to the fuzzy input and fuzzy output. This relation is linear when Q_{is}=2,
and both Q_{is} and Q_{os} are set to be 2
through out this paper.
In the blow
moulding process simulation, BlowSim provides the “average weighted thickness”
of all nodes affected by the gap opening of a certain control points. Given a
set of gap openings, we are able to extract the average weighted thickness of
all control points from the simulation results by BlowSim. The fuzzy
optimization engine will then generate a set of change in gap openings Dx_{i} for
the next iteration according to the current average weighted thickness and the
domain parameters defined by the user.
Referring to
Figure 2, the step size a
is set to be 1 initially, but is also controlled by
the same fuzzy optimization engine. In the optimization iterations, we expect a
converging behavior in the objective function. However, if the step size a is too big,
the objective function value might “overshoots.” On the other hand, if the step
size a is too small, the convergence will be slow. Ideally, step size a should be
adjusted dynamically through out the iteration process, and the heuristic rule
for adjusting a is simply, reduce a if the change in objective function value
is big, and vice versa. Obviously this can also be expressed by the same 5
fuzzy rules previously discussed, only now fuzzy input _{}, where f_{k} is
the objective function at kth iteration, and fuzzy output Dx_{i}
becomes change in step size Dr. The step size for the (k+1)th
iteration is, _{}.
Same domain
parameter mapping is also required here: Y_{i}=0 (we
expect no change in objective function value when converging), y_{i,}_{max}=10%,
y_{i,}_{min}=10%,
x_{i,}_{max}=1.0,
and x_{i,}_{min}=0.5
(step length in the next iteration will be 05~1.0 times of that of the previous
iteration). These parameters are used for step size control through out this
paper.
Finally, Figure
4(a) shows the iteration history of the bottle example, and Figure 4(b)
compares the gap openings of the initial and final design. Figure 4(c) compares
the average weighted thickness of the initial and final design on the 7 control
points, and Figure 4(d) compares the thickness distribution of the initial and
final parts.
In this example,
the optimization process terminates after 18 iterations, when the change in
objective function value is less than 0.1%. Only 18 BlowSim simulations are
needed, and sensitivity calculation is not required. Ideally the objective
function should converge to zero if a constant thickness part is obtained.
However, as shown in Figure 4(c), the thickness of the top and bottom portions
of the bottle are higher than the target value. As shown in Figure 4(b), the
thickness at these two portions cannot be further reduced because the corresponding
gap openings are already close to the lower bound 5%. Figure 5 shows the
optimization result using 31 control points. Increasing the resolution of the
control points further reduces the objective function value.
(a) (b)
(c) (d)
Figure 4. Optimization results of the bottle
example with 7 control points
(a) (b)
(c) (d)
Figure 5. Optimization results of the bottle
example with 31 control points
3.2
Fluid reservoir and gas tank
example
The fuzzy
optimization algorithm is then used to for process optimization of two complex automotive parts, the fluid
reservoir and the gas tank. In the fluid reservoir example, the target
thickness is 5mm. In
the initial design with 23 control points, the maximum
thickness of the part is 6.7009mm,
and the minimum thickness of the part is 2.3885mm.
With these information, the values of the domain parameters in this case are
assigned as follow: Y_{i} =5mm,
y_{i,}_{max}=7mm, y_{i,}_{min}=2mm, x_{i,}_{max}=95%,
and x_{i,}_{min}=5%.
The optimization
process terminates after 15 iterations. Figure 6 shows the optimization
results. In Figure 6(c), the weighted average thickness of all control points
are close to 5mm,
but the objective function value is still higher than 0. This is because the
fluid reservoir is not symmetric. It is not possible to obtain a constant
thickness part using a circular die. Die geometry optimization is needed here,
and will be discussed in the later sections.
In the gas tank example, the target thickness
is also 5mm, and 20 control
points are used. With the information from the simulation
of the initial design, the values of the domain parameters in this case are
assigned as follow: Y_{i} =5mm,
y_{i,}_{max}=13mm, y_{i,}_{min}=4mm, x_{i,}_{max}=95%,
and x_{i,}_{min}=5%.
The optimization process terminates after 27 iterations, though a better
objective function value has been obtained in the 3^{rd} iteration.
Figure 7 shows the optimization results. Again, in Figure 7(c), the weighted
average thickness of all control points are close to 5mm, but the objective function value is still high.
And when the average weighted thickness approaches 5mm the thickness of all nodes do not necessarily
approach 5mm. This is also because
the gas tank is not symmetric. Figure 7(b) shows that the gap openings of the
first 8 nodes have reached the lower bound, which also prevents the objective
function value to go down further.
(a) (b)
(c) (d)
Figure 6. Optimization results of the fluid
reservoir example
(a) (b)
(c) (d)
Figure 7. Optimization results of the fluid
reservoir example
4.
Performance optimization
Another major requirement of blow moulded parts
is its mechanical performance. A common practice for achieving this
goal is to minimize the part weight subject to mechanical performance
constraints such as maximum stress and part deflection under top load, internal
pressure, etc. Figure 8(a) shows the bottle example again. Two loading
conditions are considered separately, top load and internal pressure. The
maximum stress (from finite element software) under these two loads Max. (s_{topload}, s_{pressure}) shall not exceed a predefined value.
In order to minimize the part weight, it is
desirable to find an optimal thickness profile that is fully stressed under
these two loads, that is, Max. (s_{topload}, s_{pressure}) equals the maximum allowable stress at all nodes. This is again an optimization problem
whose objective function is in the form of Equation (1). The engineering
heuristic is very similar too: increase the thickness if the stress is too high
and vice versa. The same fuzzy optimization algorithm can be used to find the
optimal thickness profile, which is also described by control points. Now fuzzy input y_{i} is the Max. (s_{topload}, s_{pressure}) on the ith control point, and fuzzy output Dx_{i} becomes change in thickness on the ith control
point. The initial design in this case is 2mm
constant thickness. The values of the domain parameters
in this case are assigned as follow: Y_{i}=16.5MPa (target
stress, yield stress/2); y_{i,}_{max}=33MPa (yield stress); y_{i,}_{min}=10MPa
(low stress in the initial design), x_{i,}_{max}=4mm (thickness upper bound), and x_{i,}_{min}=1mm (thickness lower bound). The
optimization process terminates after 8 iterations (Figure 8(b)). When we
superimposes the stress distributions of two loads for the final design in
Figure 8(c), we can see that constant stress is achieved in almost all control
points. Figure 8(d) shows the final thickness profile.
(a) (b)
(c) (d)
Figure 8. Performance optimization results of the
bottle example
This optimal
thickness profile can be used as target thickness to obtain the gap openings
that achieves this thickness profile. As shown in, Figure 9(a) shows the
resulting thickness profile obtained using 7 control points, which do not agree
very well with the optimal thickness profile. Figure 9(b) shows the stress
distribution of the final design. An alternative is to directly use gap
openings at the control points as design variables, that is, to find the
optimal gap openings that achieve the constant stress boundary. The engineering
heuristic is, reduce the gap opening if the stress is low, and vice versa. The
values of the domain parameters in this approach are: Y_{i}=16.5MPa;
y_{i,}_{max}=33MPa;
y_{i,}_{min}=10MPa,
x_{i,}_{max}=95%,
and x_{i,}_{min}=5%.
Figure 10 shows the results of this approach.
At this point,
both results are not good. The resolution using 7 control points is not enough
to represent a complex shape in Figure 8(d) seems to be the major problem. This
example will be redone with more control points.
(a) (b)
Figure 9. Combining performance optimization with
process optimisation
(a) (b)
Figure 10. An alternative approach for combining
performance optimization with process optimization
5.
Die geometry optimization
As discussed
earlier, in some cases, especially for unsymmetrical parts, die geometry has to
be manipulated in order to obtain desired part thickness. The geometry of the
die in the closed and open positions is defined by the minimum and maximum die
gap at a number of "die points" in different of angular positions.
The minimum and maximum die gap at each die point (and of course gap openings)
can be manipulated to obtain a constant thickness part. BlowSim also provides
the average weighted die point thickness (same philosophy as average weighted
thickness previously used for gap openings).
The first trial
for die geometry optimization will be to manipulate only the maximum die gap,
while keeping the gap openings at the "optimal levels" obtained
previously. The domain parameters mappings for the gas tank example in this
case are: Y_{i} =5mm,
y_{i,}_{max}=50mm, y_{i,}_{min}=[2.8 8.4 11.2
9.8 2.8 5.6 11.2 7.0 2.8]mm, x_{i,}_{max}=10mm, and x_{i,}_{min}=0.5mm. Figure 11(a) shows the iteration
history of the bottle example, and Figure 11(b) compares the maximum die gaps
of the initial and final design. Figure 11(c) compares the average weighted die
point thickness of the initial and final design on the 9 die points, and Figure
11(d) compares the thickness distribution of the initial and final parts. The
results are not satisfactory at this point. Figure 11(a) does not show any sign
of convergence, though Figure 11(a) shows the average weighted thickness is
closer to target 5mm
than the initial design.
(a) (b)
(c) (d)
Figure 11. Optimization results of the fluid
reservoir example
6.
Author Biography
Dr. YehLiang
Hsu received his PhD from Stanford University in
1992. He is professor and chairman of ME Department, Yuan Ze University,
Taiwan. He specializes in design optimization.
TzuChi Liu is a 2^{nd} year PhD student.
7.
References
Hsu, Y. L., Lin,
Y. F., and Guo, Y. S., “A Fuzzy Sequential Linear Programming Algorithm for
Engineering Design Optimization,” 1995 Design Engineering Technical Conferences,
v1, p. 455462, ASME.
Laroche D.,
Kabanemi K., Pecora L., and DiRaddo R, Polymer Engineering & Science,
v39, 7, p.12231233, 1999.
Mulkay, E. L.,
and Rao, S. S., “Fuzzy Heuristics for Sequential Linear Programming,” Journal
of Mechanical Design, v120, p. 1723, 1998.