Technical Report

Optimization of injection molding process parameters using sequential
simplex algorithm

Behrooz Farshi

*

, Siavash Gheshmi, Elyar Miandoabchi

School of Mechanical Engineering, Iran University of Science & Technology, Tehran 16846, Iran

a r t i c l e

i n f o

Article history:
Received 21 April 2010
Accepted 25 June 2010
Available online 30 June 2010

a b s t r a c t

In this study warpage and shrinkage as defects in injection molding of plastic parts have been under-
taken. MoldFlow software package has been used to simulate the molding experiments numerically. Plas-
tic part used is an automotive ventiduct grid. The process optimization to minimize the above defects is
carried out by sequential simplex method. Process design parameters are mold temperature, melt tem-
perature, pressure switch-over, pack/holding pressure, packing time, and coolant inlet temperature.
The output parameters aside from warpage and shrinkage consist of part weight, residual stresses, cycle
time, and maximum bulk temperature. Results are correlated and interpreted with recommendations to
be considered in such processes.

Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Many products in different areas such as aviation, automotive,

electronic apparatus are produced using plastic injection molding.
Having special features like capability to produce complex parts,
light weight, resistance to corrosion, ease of producing compared
to conventional materials, are the main reasons for their popular-
ity. High quality and precision can be achieved using this method
for manufacturing plastic parts. The need for lighter, more aes-
thetic and durable products necessitates manufacturing thinner
parts. Since most molten plastics cannot fill the mold cavity of thin
walled parts suitably, plastic injection molding need be used which
can result in warpage

[1]

. Therefore, reduction and control of war-

page is of importance in enhancement of the quality. Hence, war-
page minimization plays a key role in the product optimization.
As the thickness decreases, the strength is also weakened. There-
fore, ultimately the problem can be solved using the right kind of
material for the purpose of durability.

Ordinarily, production shop operators can adjust only one pro-

cess parameter at a time and this does not necessarily lead to the
real optimum combination of process parameters. This is particu-
larly true when the objective function like warpage and/or volu-
metric shrinkage is an implicit function of the control variables
and possible interaction among them.

Warpage and volumetric shrinkage as major defects in such

manufactured parts are subject to change by the shape of parts,
modifying the mold and having different sets of process parame-
ters. The design of mold and part are usually considered in the very
start of design procedure and remain unchanged. Consequently,

determination of the best set of process parameters a priori by
an optimization procedure is the best way for minimization of such
defects

[1–3]

.

In this field some researchers have focused on finding surrogate

models like support vector regression, neural network and polyno-
mial regression in lieu of expensive and time-consuming experi-
mentations.

These

surrogate

models

are

considered as

a

mathematical approximation replacing the actual simulation anal-
yses. Using response surface method and neural network model,
Erzurumlu made reduction in warpage in thin shell plastic parts

[4]

. Kurtaran et al. optimized warpage for a bus ceiling lamp casing

utilizing genetic algorithm and neural network model

[5]

. Zhou

et al. used support vector regression for optimization of injection
molding process

[6]

. Shen et al. optimized process parameters for

reducing maximum volumetric shrinkage difference using genetic
algorithm and neural networks

[7]

.

These papers apparently show that surrogate models are good

approximations of the actual ones reducing time and computa-
tional cost. However, these surrogate models are classified as
one-step optimization, without iterations. Therefore, the accuracy
of the surrogate models determines how accurate the optimum
solution is.

Since it is a time-consuming work to optimize warpage and vol-

umetric shrinkage, an efficient optimization method called
‘‘sequential simplex” optimization is used here; a zero order opti-
mization method not requiring any gradient computations. In this
paper, we firstly introduce sequential simplex optimization meth-
od, and subsequently the working models.

Huang and Tai stated that the most crucial factors that affect

warpage in injection molding of a thin shell part are packing pres-
sure, mold temperature, melt temperature and packing time

[8]

.

However, since minimization of both warpage and volumetric

0261-3069/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:

10.1016/j.matdes.2010.06.043

*

Corresponding author. Tel.: [phone]; fax: [phone].
E-mail address:

[e-mail]

(B. Farshi).

Materials and Design 32 (2011) 414–423

Contents lists available at

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Materials and Design

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shrinkage is considered in this paper, a higher number of variables
namely, mold temperature, melt temperature, pressure switch-
over, pack/holding pressure, packing time, coolant inlet tempera-
ture are considered as the variables for optimization.

Thus in this study the values of process parameters are sequen-

tially obtained by the Finite Element Analysis (FEA) software Mold-
Flow, and used in the sequential simplex algorithm for gradual
convergence to the optimum level. Many researchers have used
MoldFlow for their numerical experimentations, and have shown
that it can adequately simulate analysis of injection molding pro-
cess with a good precision and accuracy

[1,2,9–12]

.

In this paper, volumetric shrinkage is also minimized and its

corresponding warpage, cycle time, maximum bulk temperature
and part weight are obtained. I addition warpage is minimized sep-
arately, and its corresponding volumetric shrinkage, cycle time,
maximum bulk temperature and part weight are determined. Then
a comparison is made in order to find the best compromised pro-
cess parameters for highest quality commensurate with least time
as the most significant cost factor.

In this study, thin shell part which is an automotive ventiduct

grid is selected which has also been used in the paper of Sedaghat
et al.

[12]

.

2. Definition of sequential simplex optimization method

The sequential simplex algorithm uses what is known as EVOP

(EVolutionary OPeration). There are two major types of sequential
simplex algorithm which are fixed-size simplex and variable-size
simplex, the definition of which are given in reference

[13]

.

2.1. Fixed-size simplex

Spendley et al. published a paper in 1962, in which they set out

to make EVOP an automated procedure

[14]

. They introduced the

simplex geometry as a figure having one vertex more than the
number of variables of the optimization design space, bound by
the lowest number of sides. The principle of this optimization
method is to successively reject the worst vertex of the simplex
as the most undesirable design point and replace it with a better
one. This is done by projection of the rejected point through the
centroid of remaining vertices as is indicative of a path of progress
towards better points. This procedure gives a new simplex on
which another simplex operation can be done. This procedure
can go on till the optimum point is reached where the vertices coa-
lesce over that point.

Fig. 1a

shows the sequential steps involved in

this procedure.

2.2. Variable-size simplex

Nelder and Mead made two basic modifications to the fixed-

size simplex of Spendley et al.

[15]

. These modifications let the

simplex expand in favorable direction and contract in unfavorable
ones. Since the size of simplex is subject to change in this modified
method, it is called variable-size simplex.

Fig. 1b

shows the stages

of this procedure.

In fixed-size simplex, the simplex eventually rotates around the

optimum point unable to converge on it. However, in variable-size
simplex, the simplex continually changes its size to eventually col-
lapse onto the optimum point. Considering the advantages of var-
iable-size simplex method it is adopted for use in this
investigation.

3. Finite element model of automotive ventiduct grid

Geometry of the automotive ventiduct grid used in this study is

shown in

Fig. 2

a after plastic injection molding. It was designed in

CATIA without consideration of defects like warpage and volumet-
ric shrinkage and saved as a STL file to be imported to MoldFlow.
Afterwards, finite elements (FE) model of the automotive ventiduct
grid was created by MoldFlow which is a commercial software
package using hybrid finite element/finite difference method for
solving pressure, flow, and temperature field problems

[16]

. The

FE model it created is shown in

Fig. 2

b. It has length, width and

weight of 230 mm, 180 mm and about 90 g, respectively. The auto-
motive ventiduct grid is made of high density Polyethylene. The FE
model includes 3092 fusion elements.

Fig. 2

c shows the FE model

with cooling channel used in the injection molding.

Sedaghat et al.

[12]

showed that analysis with fusion elements

in a mesh of 2801 nodes is successful in predicting the experimen-
tal results of real injection molding process by comparing values
predicted from FE model and those measured on actual automotive
ventiduct grid. In our study fusion element is chosen with different
number of nodes tested to find the number of nodes that suitably
predicts the injection molding with least analysis time. Warpage is
selected to represent the injection molding process for this pur-
pose. The analysis was carried out on a Pentium 4 PC with 3 GHz
CPU, and 1 GB of RAM.

Fig. 3a

shows the warpage versus the num-

ber of nodes plot indicating that the chosen 3092 nodes can virtu-
ally be adequate for warpage computations.

Fig. 3b

shows the time

Fig. 1a. Fixed-size simplex procedure.

Fig. 1b. Variable-size simplex procedure.

B. Farshi et al. / Materials and Design 32 (2011) 414–423

415

that takes to get the results from the PC versus the number of
nodes used in the model.

It is apparent that increasing the number of nodes above 3092,

warpage shows negligible changes, while CPU time significantly in-
creases by increasing the number of nodes. Therefore, fusion ele-
ment with 3092 nodes is chosen in this study.

4. Numerical experimentation

4.1. Characterization of materials

The material selected for experimental study in this paper was

high density PolyEthylene (PE), Petrothene LS506000. This grade

Fig. 2. (a) Automotive ventiduct grid after injection molding. (b) Finite element model of automotive ventiduct grid (3092 nodes). (c) FE model with cooling.

Fig. 3a. Warpage (mm)-number of nodes.

416

B. Farshi et al. / Materials and Design 32 (2011) 414–423

offers a high stiffness with good impact strength as well as being
easy to process. Its properties are shown in

Table 1

. Dimensional

accuracy, surface finish and serial production were requirements
of manufacturing automotive ventiduct grid; therefore, tool steel
P-20 was selected as mold material. This material keeps its proper-
ties for a long time increasing the tool life. The surface hardness for
this material is about 32–35 RC

[17]

. The properties of the tool

steel P-20 are given in

Table 2

. The characteristics of cooling sys-

tem are shown also in

Table 3

.

4.2. Process parameters and their constraints

Mold temperature, melt temperature, pressure switch-over,

pack/holding pressure, packing time, and coolant inlet temperature
are considered as the variables for optimization. Their limiting con-
straints are shown in

Table 4

.

5. Volumetric shrinkage

5.1. Volumetric shrinkage optimization by sequential simplex
algorithm

Since warpage and shrinkage are considered as defects, mini-

mizing both of them is a useful task in manufacturing processes.
Warpage is a term used for warping of the part in injection mold-
ing due to the non-uniform contraction of different points and vol-
umetric shrinkage is the overall contraction of the part when it is
cooled. Minimizing these two will result in better product quality.

Volumetric shrinkage is often compensated for by a coefficient

of contraction in practical mold designs. Excessive shrinkage may
cause volumetric changes that can produce out of tolerance dimen-
sions in the final product. Consequently, one of the aims of this

Fig. 3b. Total time (s)-number of nodes.

Table 1
Properties of the Petrothene LS506000.

Property

Unit

Value, name

Family name

–

Polyethylenes

Trade name

–

Petrothene LS506000

Family abbreviation

–

HDPE

Material structure

–

Crystalline

Elastic modulus

MPa

911

Poisson’s ratio

–

0.426

Shear modulus

MPa

319

Melt density

g/cm

3

0.74921

Solid density

g/cm

3

0.92915

Table 2
Properties of the tool steel P-20.

Property

Unit

Value

Density

g/cm

3

7.8

Specific heat

J/(kg °C)

460

Thermal conductivity

W/(m °C)

29

Young’s modulus

GPa

200

Poisson’s ratio

–

0.33

Coefficient of thermal expansion

1/°C

1.2  10

5

Table 3
Characteristics of cooling system.

Characteristic

Unit

Value, name

Number of channels

–

3

Channels’ diameter

mm

10

Distance between the cooling system and the part

mm

20

Distance between channel’s centers

mm

65

Channels’ longitudinal length

mm

460

Type of channels

–

Longitudinal

Table 4
Constraints of process parameters.

Process parameter

Unit

Lower
limit

Upper
limit

Mold temperature

°C

20

60

Melt temperature

°C

165

205

Pressure switch-over

%Volume filled

90

99

Pack/holding pressure

%Maximum injection
pressure

50

75

Packing time

s

10

15

Coolant inlet

temperature

°C

20

30

B. Farshi et al. / Materials and Design 32 (2011) 414–423

417

project is to minimize the shrinkage as much as possible for above
reason.

Fig. 4a

shows the plot of the minimization procedure used for

volumetric shrinkage of the automotive ventiduct grid using
sequential simplex optimization algorithm.

Figs. 4b–4e

shows the

corresponding warpage, part weight, cycle time, maximum bulk
temperature, respectively for the volumetric shrinkage minimiza-
tion procedure.

Table 5

contains the data for the optimum point

for shrinkage minimization. Following the minimization trend of
generating sequential points one can also obtain a point of maxi-
mum shrinkage whose corresponding data can be significant.

Ta-

ble 6

contains the data related to the residual stresses

corresponding the maximum and minimum of shrinkage design
points in both the first and second directions. It is observed from
the entries in

Table 6

that minimum shrinkage design corresponds

to maximum residual stresses in both directions.

The residual stresses trapped in the final product can also be

considered as a defect, and must be controlled. Furthermore, the

shrinkage minimization trend as depicted in plots of

Figs. 4c and

4d

indicate that the part weight and cycle time both continually in-

crease towards minimum shrinkage point, while

Fig. 4b

shows

somewhat indifferent warpage response. This observation suggests
that since the cycle time, part weight, and residual stress increases
are undesirable a compromise must be made regarding the mini-
mum volumetric shrinkage.

6. Warpage

6.1. Warpage optimization by sequential simplex algorithm

Automotive ventiduct grid is considered as a thin shell plastic

part and warpage is one of the most important defects in thin shell
parts. Consequently, minimization of warpage seems reasonable
and necessary.

Fig. 5a

shows warpage minimization procedure for automotive

ventiduct grid using sequential simplex optimization algorithm.

Fig. 4a. Minimization procedure of volumetric shrinkage.

Fig. 4b. Warpage corresponding to volumetric shrinkage.

418

B. Farshi et al. / Materials and Design 32 (2011) 414–423

Figs. 5b–5e

shows the corresponding volumetric shrinkage, part

weight, cycle time, maximum bulk temperature, respectively of
the procedure.

Table 7

contains the data for optimum point for warpage mini-

mization. Following the minimization trend of generating sequen-
tial points one can also obtain a point of maximum warpage whose
corresponding data can be significant.

Table 8

contains the data re-

lated to the residual stresses corresponding the maximum and
minimum of warpage design points in both the first and second
directions. It is observed from the entries in

Table 8

that minimum

warpage design corresponds to maximum residual stresses in both
directions.

7. Discussion and results

The followings are the results obtained in this study:

 When minimizing volumetric shrinkage, corresponding war-

page was fluctuating slightly about the warpage corresponding
to the minimized volumetric shrinkage, 2.382 mm. Correspond-

Fig. 4c. Part weight corresponding to volumetric shrinkage.

Fig. 4d. Cycle time corresponding to volumetric shrinkage.

B. Farshi et al. / Materials and Design 32 (2011) 414–423

419

ing warpage was almost constant around 2.382 mm which is
close to minimized warpage, 2.086 mm. Therefore, using the
corresponding process parameters of minimized volumetric
shrinkage is reasonable when minimum volumetric shrinkage
and minimum warpage are both of concern and time is not
important because few parts are needed.

 Part weight was increasing as the volumetric shrinkage was

going towards the minimum point. When volumetric shrinkage
was minimized, the part weight reached 92.9012 g. The lowest

part weight during the minimization procedure of volumetric
shrinkage was about 90.28. This shows that part weight is of
no importance in determining the best process parameters
because the total difference between the lowest and highest
part weight is about 2.5 g or about 3%.

 When minimizing volumetric shrinkage, it followed a trend of

sharp increase in cycle time. Therefore, it is not recommended
to use its corresponding process parameters when time is of
concern as in a mass production process.

 A decreasing trend was observed in maximum bulk tempera-

ture when minimizing volumetric shrinkage.

 As shown in

Table 6

, residual stresses were increased when

volumetric shrinkage was minimized. However, if the resid-
ual stresses increase is of concern then a compromise
regarding the use of corresponding process parameters must
be made.

 When minimizing warpage, corresponding volumetric shrink-

age fluctuated about minimum warpage’s corresponding
shrinkage of 5.8076%. This value is relatively close to the mini-
mized volumetric shrinkage i.e., 2.3077%. Comparing this value
to the worst volumetric shrinkage in optimization procedure
which is about 11%; it can be concluded that using the mini-
mum warpage parameters results in reasonable minimum
shrinkage also in most cases.

 As shown in

Table 8

, residual stresses increased about 10%

when warpage was minimized. However, in most cases such
an increase is not considered alarming.

 Part weight was increasing as the warpage was being mini-

mized. Part weight corresponding to minimum warpage
reached 92.1248 g which is only 3% higher than the lowest
value of 90.28 during the process. This shows that part weight
is of no importance in determining the best process parameters
whether it is shrinkage or warpage minimization.

 When minimizing warpage, a decreasing trend of cycle time

was seen. The corresponding cycle time of minimized warpage
was about six times less than that of minimized volumetric
shrinkage. Consequently, it is reasonable to use the correspond-
ing process parameters of minimized warpage, with some com-
promise on the shrinkage when time is the most important
factor.

 Maximum bulk temperature was almost constant in the proce-

dure of minimizing warpage.

Fig. 4e. Maximum bulk temperature corresponding to volumetric shrinkage.

Table 5
Minimized volumetric shrinkage data.

Name

Unit

Value

Minimum volumetric shrinkage

–

2.3077

Corresponding warpage

mm

2.382

Corresponding part weight

g

92.9012

Corresponding cycle time

s

105.8

Corresponding maximum bulk

temperature

°C

32.471

Corresponding mold temperature

°C

30.9827646

Corresponding melt temperature

°C

165.3093513

Corresponding pressure switch-

over

% Volume filled

98.72913442

Corresponding pack/holding

pressure

% Maximum injection
pressure

72.81917651

Corresponding packing time

s

11.77099053

Corresponding coolant inlet

temperature

°C

26.44657231

Table 6
Results regarding max. and min. volumetric shrinkage.

Name

Unit

Value

Maximum volumetric shrinkage’s maximum residual stress in

1st direction

MPa

64.05

Maximum volumetric shrinkage’s maximum residual stress in

2nd direction

MPa

56.47

Minimum volumetric shrinkage’s maximum residual stress in

1st direction

MPa

77.82

Minimum volumetric shrinkage’s maximum residual stress in

2nd direction

MPa

57.03

420

B. Farshi et al. / Materials and Design 32 (2011) 414–423

It can be concluded that warpage minimization in the process of

injection molding with compromise to control shrinkage can result
in shorter cycle time and less residual stresses and can be best for
economical production process. At the optimum point for warpage

a sensitivity analysis with respect to the process parameters has
been performed. The results indicate that the three most important
parameters are pressure switch-over, mold temperature and cool-
ant inlet temperature with values of 7%, 5% and 3% respectively.

Fig. 5a. Minimization procedure of warpage.

Fig. 5b. Volumetric shrinkage corresponding to warpage.

B. Farshi et al. / Materials and Design 32 (2011) 414–423

421

The other parameters showed lower sensitivities. This seems to be
in contrast with the results of

[1,8]

indicating packing pressure and

[2]

indicating packing time as the most important factors. How-

ever, results of

[11]

seem to be in agreement with those obtained

in this study.

Furthermore, it was shown that the sequential simplex optimiza-

tion method

[13]

is an effective and useful procedure for online opti-

mization of such process problems as was also recommended in

[3]

.

8. Conclusions

In this study an optimization procedure for minimum war-

page and volumetric shrinkage of injection molding of automo-

tive ventiduct grid has been developed. It is based on sequential
simplex method which takes into consideration six process
parameters. Unlike many similar attempts, side effects on other
factors not directly included in the procedure are also investi-
gated. Consequently, it was observed that some factors such
as cycle time showed drastic increase in case of volumetric
shrinkage as compared to warpage minimization. Therefore, a
compromise recommendation can be offered for a near opti-
mum

shrinkage

in

combination

with

optimum

warpage

involving low cycle time and low residual stresses simulta-
neously. It was shown that the sequential simplex optimization
procedure is a viable and efficient method for injection molding
problems.

Fig. 5c. Part weight corresponding to warpage.

Fig. 5d. Cycle time corresponding to warpage.

422

B. Farshi et al. / Materials and Design 32 (2011) 414–423

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Fig. 5e. Maximum bulk temperature corresponding to warpage.

Table 7
Minimized warpage data.

Name

Unit

Value

Minimum warpage

mm

2.086

Corresponding volumetric

shrinkage

–

5.8076

Corresponding part weight

g

92.1248

Corresponding cycle time

s

20.7

Corresponding maximum bulk

temperature

°C

112.38

Corresponding mold temperature

°C

46.6902761

Corresponding melt temperature

°C

180.5392475

Corresponding pressure switch-

over

% Volume filled

98.35565248

Corresponding pack/holding

pressure

% Maximum injection
pressure

73.9989183

Corresponding packing time

s

13.11980438

Corresponding coolant inlet

temperature

°C

28.28933632

Table 8
Results regarding max. and min. warpage.

Name

Unit

Value

Maximum warpage’s maximum residual stress in 1st

direction

MPa

60.58

Maximum warpage’s maximum residual stress in 2nd

direction

MPa

53.57

Minimum warpage’s maximum residual stress in 1st direction

MPa

66.46

Minimum warpage’s maximum residual stress in 2nd

direction

MPa

58.74

B. Farshi et al. / Materials and Design 32 (2011) 414–423

423