Tài liệu A systematic computer-aided approach to cooling system optimal design in plastic injection molding docx

International Journal of Mechanical Sciences 48 (2006) 430–439
A systematic computer-aided approach to cooling system optimal
design in plastic injection molding
H. Qiao
Ã
Cardiff School of Engineering, Queen’s Buildings, The Parade, PO Box 925, Cardiff, CF24 0YF, UK
Received 10 December 2003; received in revised form 10 October 2005; accepted 10 November 2005
Available online 27 December 2005
Abstract
Cooling system design is of great importance for plastic injection molding because it significantly affects the productivity and quality of
the final products. In this paper, a systematic computer-aided approach is developed to achieve the cooling system optimal design. This
approach expiates the trial and error process normally practiced in conventional cooling system design based on the designer’s experience
and intuition. Various aspects of the optimization process for cooling system design are investigated including cooling analysis using
boundary element method (BEM), a perturbation-based approach to design sensitivity analysis, optimization problem formulation, and
a novel hybrid optimizer based on Davidon–Fletcher–Powell (DFP) method and simulated annealing (SA). A case study shows that the
proposed methodology for cooling system optimal design is efficient, robust and practical.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: Boundary element method; Cooling system design; Perturbation-based approach; Simulated annealing
1. Introduction
The injection molding cycle consists of several stages,
such as filling, packing, and cooling. The cooling system
design is of considerable importance since about 80% of
the cycle time is taken up by the cooling phase. The cooling
system must be able to remove the heat at the required rate
so that the plastic part can be ejected without distortion. At
the same time, the cooling of the part should be kept as
uniform and balanced as possible so that undesired defects
such as sink marks, differential shrinkage, internal thermal
residual stress and warpag e can be reduced.
In cooling system design, design variables typically include
the size, location and layout of cooling c hannels, and the

thermal properties, temperature and flow rate of the coolant.
With so many design param eters involved, design work to
determine t he optimum c ooling s ystem is e xtremely difficult.
Traditionally, the designer uses his experience and intuition to
guide the design process. This manual design process has the
advantage that the designer’s judgement can be utilized for the
design. However, as the design problem becomes more
complex, the manual d esign process based only on t he
designer’s judgement becomes inadequate and even impossi-
ble. For an optimum design, the designer needs a more
powerful tool integrating the cooling analysis and optimization
programs into the design process. Using such a tool, a design
can be improved systematically, automatically and efficiently.
The analysis of heat transfer within the mold and the
part plays a crucial role in the optimum cooling system
design. Since the mold usually has a complicated structure
with runners and cooling channels, more efforts have been
made on the mold cooling analysis. There are mainly two
approaches considered for the mold cooling analysis: cycle-
averaged approach and transient approach.
In cycle-averaged approach, the mold thermal analysis is
based on the steady-state heat transfer approximation for
the cycle-averaged temperature field during the continuous
cyclic transient cooling analysis [1]. Barone and Caulk [2]
developed, as an approximation, a special boundary
integral equation for two-dimensional (2D) large regions
with small circular holes by analyzing stead-state heat
conduction. Kwon [1] demonstrated the fundamentals of
the cycle-averaged cooling modeling and some features of
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the boundary element method (BEM) mold cooling de sign
system in both two and three dimensions. In his research, a
cycle-averaged heat flux coefficient for the polymer part is
obtained from the typical solution of the one-dimensional
(1D) heat diffusion equation. Himasekhar et al. [3]
performed a comparative study for the 3D mold cooling
analysis based on cycle-averaged approach. A local 1D
transient analysis is performed for the plastic part using an
implicit finite difference method with a variable mesh. An
iterative procedure is developed for coupling the cycle-
averaged mold analysis and the transient part analysis. The
same method was also applied by Chen and Hu [4] . Some
researchers [5,6] developed an alternative formulation by
separating the mold temperature into two components: a
steady component and a time-varying component. The
steady component is first calculated in the same way. The
time-varying component is then evaluated by the difference
between instantaneous heat flux and cycle-averaged heat
flux on the mold-part interface. Rezayat and Burton [7]
developed a special boundary integral equation for 3D
complex geometry. In the proposed formulation, the mold-
part interface is replaced by the mid-surface of the part, the
numerical quadrature over the surface of the cooling
channels is reduced to integration over the axis of

cylindrical segments and modeling and meshing of the
exterior surface may be avoided. Some other researchers
[8–10] have applied this formulation in their BEM
numerical implementation.
Although the cycle-averaged approach can predict well
the overall performance of the cooling system, a more
accurate transient analysis can enhance the unde rstanding
of shrinkage and warpage of the plastic part. Hu et al. [11]
adopted the dual reciprocity boundary element method
(DRBEM) to calculate the transient temperature distribu-
tions during the cooling process. The instantaneous heat
flux from the polymer part to the mold is obtained using
the finite difference method. Time stepping is achieved by
means of finite difference schemes. Tang et al. [12,13]
presented another approach to simulate the transient heat
transfer within a mold during the injection molding
process. The transient temperature distributions in the
mold and the polymer part are simultaneously computed
using the Galerkin finite element method. Similar to the
DRBEM, finite difference time stepping is also adopted for
the temporal discretization.
Using the information provided by the cooling analysis,
the design can be further improved by the optimization
algorithm. Till now only few researchers applied numeri cal
optimization techniques to automate the search for an
optimal cooling system design. Barone and Caulk [2] used
the CONMIN program, which employs a conjugate
gradient search algorithm and the method of feasible
directions, to search for the optimal layout of the cooling
channels based on the proposed special boundary integra-

tion so that the uniformity of the cavity surface tempera-
ture can be achieved. Parang et al. [14] used the same
optimization scheme for optimal positioning of holes in
arbitrary 2D regions. Tang et al. [13,15] presented an
approach for the optimal cooling system design based on
the transient cooling analysis. The design constraints and
objective function are evaluated using finite element
analysis. The constrained optimization problem is solved
using Powell’s conjugate direction method with an interior
penalty function.
Design sensitivity analysis (DSA) has gained importance
as a major component of the optimal design pr ocess in
recent years. Considerable economy can be achieved by
calculating the accurate sensitivities in order to reach the
optimum in a reasonable number of solutions of the
analysis model. Matsumoto and Tanaka [16] presented a
DSA formulation based on the direct differentiation of the
regularized boundary integral equation with respect to the
design variables for 2D steady-state heat conduction
problems and applied it to the optimal design of cooling
channels in injection molds. Park and Kwon [9] developed
a DSA for the injection mold cooling system using the
direct differentiation approach based on the modified
boundary integral equation presented by Rezayat and
Burton [7]. Based on this DSA, an optimal arrangement of
circular holes is found to make the temperature distribu-
tion over the part surface as uniform as possible by
employing the augmented Lagrangian multiplier method
and Davidon–Fletcher–Powell method [10].
In this paper, a new systematic computer-aided ap-

proach is developed for cooling system optimal design.
Compared with the work done by other researchers
mentioned above, this approach has the advantage of
being simple in concept and easy in implementation while
showing high computational efficiency and quality.
2. Numerical formulations
2.1. Mold cooling analysis
The purpose of mold cooling analysis is to analyze the
temperature profile along the mold cavity wall by
numerical simulation to improve the cooling system design.
In this paper, the cycle-averaged approach is used since it is
simple, computationally efficient and yet sufficiently
accurate for mold design purposes [1].
The cycle-averaged mold temperature distribution T can
be obtained by solving the steady-state heat conduction
problem and hen ce the governing equation is Laplac e’s
equation, which can be written as
r
2
T ¼ 0, (1)
over the mold so lid zone bounded by the boundary G.Three
kinds of boundary conditions must also be satisfied including:
 Mold cavity surface: The cycle-averaged heat flux on the
mold cavity surface is given by
Àk
qT
qn
¼
¯
q, (2)

ARTICLE IN PRESS
H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439 431
where n is the normal to the surface, k is the thermal
conductivity of the mold and
¯
q is the cycle-averaged heat
flux given by Ref. [3]
¯
q ¼
1
t
f
þ t
c
þ t
o
Z
t
f
0
q
1
ðtÞ dt

þ
Z
t
f
þt
c

t
f
q
2
ðtÞ dt þ
Z
t
f
þt
c
þt
o
t
f
þt
c
q
3
ðtÞ dt
#
, ð3Þ
where t
f
, t
c
, t
o
are filling, cooling, and mold opening
time, respectively, and q
1

, q
2
, q
3
are instantaneous heat
flux values during filling, cooling, and mold opening
time, respectively.
The value of heat flux q
1
during filling time t
f
can be
calculated by performing a filling stage analysis using
any commercial software, but it is not considered in this
paper. The flux q
3
during the mold opening time is
typically very small and hence is neglected [3]. The value
of the heat flux q
2
during cooling stage can be calculated
by performing a transient part cooling analysis using
either the finite difference method or the finite element
method (FEM ).
 Cooling channel surface: On the cooling channel surfa ce,
a convective boundary condition is defined as [17]
Àk
qT
qn
¼ h

c
ðT À T
c
Þ, (4)
where h
c
represents the heat transfer coefficient between
the mold and the coolant at a temperature of T
c
.
 Mold exterior surface: On the exterior surface, a
convective boundary condition is imposed, i.e.
Àk
qT
qn
¼ h
a
ðT À T
a
Þ, (5)
where h
a
represents the heat transfer coefficient between
the mold and the ambient environment air at a
temperature of T
a
.
The mold temperatur e is calculated using BEM. A major
advantage of the BEM is that there is no need to mesh the
whole solid zone but only the boundary surfaces of the

mold and cooling ch annels. Hence mesh da ta preparation
is greatly simplified and less computing time and storage is
needed for the same level of accuracy than in the case of
FEM. Moreover, the main interest in the mold cooling
analysis is to calculate the temperature and flux on the
mold cavity wall. In using BEM, less unwanted informa-
tion about internal points is obtained. Therefore, the BEM
is particularly well suited for mold cooling analysis.
The boundary integral equation (BIE) for Eq. (1) can be
written as [18]
C
P
T
P
þ
Z
G
q
Ã
T dG ¼
Z
G
T
Ã
q dG, (6)
where C
P
and T
P
are the coefficient and temperature at the

boundary point P, respectively. T
*
is the fundamental
solution of the Laplace’s equation and can be easily verified
in two dimensions as
T
Ã
¼
1
2p
ln
1
r
, (7)
where r is the distance between a source point and a field
point. In Eq. (6), q
Ã
¼ qT
Ã
=qn, q ¼ qT=qn, and n is the unit
outward normal to the boundary G. Dividing the boundary
G into N boundary elements, with boundary G
e
for element
e, the following discretized form of equation is obtained
from Eq. (6) for node i:
C
i
T
i

þ
X
N
e¼1
Z
G
e
q
Ã
T
ðeÞ
dG
e
¼
X
N
e¼1
Z
G
e
T
Ã
q
ðeÞ
dG
e
; (8)
where T
(e)
, q

(e)
are the temperature and flux distribution
over the element e. C
i
, T
i
are the coefficient and
temperature at node i.
By introducing shape functions and evaluating the
integration constants in Eq. (8), one obtains the follo wing
system of equations in matrix form after taking all the
nodes into consideration [18]:
HT ¼ Gq. (9)
T is a vector containing temperatures at all the nodes and q
is a vector containing heat fluxes at all the nodes. H and G
are two matrices containing the integration constants.
To solve system (9), boundary conditions must be
specified and system (9) needs to be reordered in such a
way that all unknowns are taken to the left side, i.e.
Au ¼ Bv . (10)
u is a vector of unknowns, and v is a known vector; A and
B are matrices whose columns are a combination of
columns of H and G.
2.2. Design sensitivity analysis
Design sensitivity analysis (DSA) is to investigate the
rates of change of response quantities with respect to
design variables. Thes e rates of change are the gradient (or
derivative) information essential for coupling optimization
methods and analysis procedures. During the optimization
process, considerable economy can be achieved by calcu-

lating the gradients as part of the analysis procedure.
Therefore, accurate and efficient sensitivity analysis has
been an important topic in the field of design optimization.
In the following, a perturbation-based approach is derived
using BEM.
Given an initial design variables vector x
0
, the corre-
sponding overall system of equations of BEM can be
written similarly to Eq. (10) as
A
0
u
0
¼ B
0
v
0
, (11)
where u
0
is the column vector of u nknown boundary
response, and v
0
is the vector of specified boundary
conditions. A
0
and B
0
are sq uare and recta ngular matrices,

respectively. The matrix A
0
can be decomposed into the form
A
0
¼ LU,(12)
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H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439432
where L and U are a lower and an upper triangular matrix,
respectively. Given a change Dx in the design variables so that
the modified design is
x ¼ x
0
þ Dx,(13)
the corresponding system of equations becomes
Au ¼ Bv,(14)
where
A ¼ A
0
þ D A, (15a)
B ¼ B
0
þ DB,(15b)
u ¼ u
0
þ D u ,(15c)
v ¼ v
0
þ D v ,(15d)
where D(

) denotes the corresponding changes involved in
matrices, response and boundary conditions.
The object is to find efficient and high-quality approx-
imations of the response u due to changes in the design
variables Dx without solving the modified analysis Eq. (14).
Eq. (14) can be rewritten as
ðA
0
þ D AÞðu
0
þ DuÞ¼ðB
0
þ DB Þðv
0
þ D v Þ. (16)
Substituting Eq. (11) yields
ðA
0
þ D AÞDu ¼ B
0
Dv þ DBv
0
þ D BDv À DAu
0
. (17)
Premultiplying A
À1
0
, Eq. (17) can be expressed as
ðI þ A

À1
0
DAÞDu ¼ s
0
, (18)
where
s
0
¼ A
À1
0
ðB
0
Dv þ DBv
0
þ DBDv À DAu
0
Þ. (19)
Defining
E ¼ A
À1
0
DA, (20)
Eq. (18) can be rewritten as
Du ¼ðI þ EÞ
À1
s
0
. (21)
If kk is matrix norm, and if kEko1, then applying the

binomial series expansion of Ref. [19], the following
expression is obtained:
Du ¼ðI À E þ E
2
ÀÁÁÁÞs
0
. (22)
Defining
s
1
¼ÀEs
0
,
s
2
¼ÀEs
1
,
.
.
.
ð23Þ
Du can be expressed as
Du ¼ s
0
þ s
1
þ s
2
þÁÁÁ, (24)

and the sensitivity for the ith design variable can be
obtained as
qu
qx
i
¼
Du
Dx
i
. (25)
The efficiency of this approach lies in the fact that for the
given decomposed form of Eq. (12), the calculati on of the
vectors s
0
; s
1
; s
2
; involves only forward and back
substitution. Compared with the traditional finite differ-
ence method, no additional matrix factorization is required
to calculate the sensitivity for the given design x
0
.
Compared with the methods present ed in Refs. [9,16], this
approach avoids the use of numerical integration of a new
class of fundamental solution sensitivity kernels for the
computations of the sensitivity matrices and the sensitivity
of boundary conditions. Therefore, this approach has the
advantages of the conventional finite difference method,

being simple in concept and easy in implementation, while
overcomes the drawback of low efficiency. It should be
noted that its validity and reasonable level of accuracy are
generally limited to cases where only small changes have
been introduced into a design.
2.3. Optimization algorithm
Various numerical optimization algorithms have been
developed to solve optimization problems. Unfortunately
no universal algorithm exists which works well for all
problems. This is because the convergence and the
efficiency of a particular algorithm are dependent on the
problem to be solved. In this paper, a hybrid optimizer for
the cooling system optimal design is developed based on
the Davidon–Fletcher–Powell (DFP) method and the
simulated annealing (SA) algorithm.
2.3.1. DFP method
The DFP method is one of the best general-purpose
unconstrained optimization techniques making use of the
derivatives. This method is very powerful and quadratically
convergent. It is very stable even if the objective function is
highly distorted and eccentric [20]. In order to use the DFP
method to solve constrained problems, the interior penalty
function method is applied so that the subsequent points
generated will always lie within the feasible domain during
the minimization process. The iterative procedure of the
DFP method can be found in Ref. [20].
2.3.2. SA algorithm
Simulated anneal ing is a stochastic optimization technique
for non-linear programming (NLP) problems. The basic idea
of this met hod is to generate a random point in order to

avoid getting trapped a t a local minimum. The new worse
trial point can either be accepted or rejected . The decision is
based on a probability, which is computed by using a
parameter called temperature. The procedure of implementa-
tion [21] and some modifications are stated as follows:
(i) Start with an initial feasible point x
0
and evaluate the
objective function e( x
0
). Set the initial temperature y,
iteration numbers K ¼ 0 and k ¼ 1.
(ii) Generate a new feasible point x
k
randomly. Evaluate
e(x
k
) and Df ¼ f ðx
k
ÞÀf ðx
0
Þ.
ARTICLE IN PRESS
H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439 433
(iii) If Deo0, then take x
k
as the new best point x
0
, and
set f ðx

0
Þ¼f ðx
k
Þ and go to step (iv). Otherwise,
generate a random number bA[0,1] and calculate the
probability as
p ¼ e
ÀDf =y
.
If bop, then take the x
k
as the new best point x
0
.
(iv) If koN (N is the allowed maximum number of trial
points generated within one iteration), then set k ¼
k þ 1 and go to step (ii). Otherwise, go to step (v).
(v) If there is no successful acceptance after N trials, then
stop. Otherwise, go to step (vi).
(vi) If KoM (M is the iteration limit), reduce the
temperature y ¼ ty (t is a constant less than 1). Then
set K ¼ K þ 1, k ¼ 1, and go to step (ii). Otherwise,
stop.
2.3.3. Hybrid SA–DFP optimizer
The DFP method is efficient for a search for a local
optimum. Since most engineering optimization problems
are non-convex in the search space, this method may
obtain only a local optimum solution. The simulated
annealing algorithm, on the other hand, is a stochastic
optimizer that can handle non-convex problems and look

for the global optimum. The principal drawback of this
method is its computational cost when dealing with
numerically expensive problems, like the cooling system
design analyze d via boundary element models. In this
paper, a fast quasi-global optimization approach based on
a hybrid combining the simulated annealing and the DFP
method, namely SA–DFP, is developed. This approach can
reduce the CPU time of SA while retaining the main
characteristics of SA. The flow chart for SA–DFP is shown
in Fig. 1.
The basic idea of SA–DFP is that a local optimum can
be found qui ckly by using the DFP method, and by using
SA the search point can escape from the valley of the local
optimum in order to reach the global optimum. In the
SA–DFP search, the DFP method can be considered as a
sub-search in the SA search. The DFP search will be
performed after step (iii) of the SA search using the new
generated point x
k
as the start point if x
k
is accepted by the
SA algorithm. The overall SA–DFP search process can
also be considered as a sequence of DFP, SA, DFP,
SA; , DFP, SA. The decision when the switch occu rs
from one search to the other is based on the following
guidelines:
DFP-SA, if the local optimum has be en found by DFP;
SA-DFP, if the new trial point has been accepted by
SA.

In step (ii) of SA algorithm, although x
k
¼
½x
ðkÞ
1
x
ðkÞ
2
ÁÁÁ x
ðkÞ
n

T
is generated randomly, in practice it
is not generated arbitrarily. Normally it is within a certain
neighborhood of the current point. Thus SA is not a pure
random search within the entire design space. In the
present implementation, if x
iL
and x
iU
are lower and upper
bounds for the ith design variable, then x
k
is calculated as
follows:
For i ¼ 1ton
Calculate d ¼ min ðx
iU

À x
ðkÀ1Þ
i
; x
ðkÀ1Þ
i
À x
iL
Þ;
Generate a random number z, zA[0, 1];
Generate a new x
i
randomly as follows:
x
ðkÞ
i
2½x
ðkÀ1Þ
i
À d; x
ðkÀ1Þ
i
; if zp0:5
x
ðkÞ
i
2½x
ðkÀ1Þ
i
; x

ðkÀ1Þ
i
þ d; if z40:5
End
2.4. Implementation for cooling system optimal design
Fig. 2 shows a conceptual illustration of the optimal
cooling system design methodology. Such a computer-
aided optimal design consists of two main modules:
analysis and optimization. An iterative design loop is
formed based on these two modules to find the optimal
design. As the whole design loop is done by computer, the
design process will be more econo mical and reliable
compared with the manual design appro ach.
ARTICLE IN PRESS
Set start point and
control parameters
DFP search for local
optimum
Accept the new
point ?
Generate new random
point
Finished ?
END
Yes
No
Yes
No
Set the new point as
the start point

SA Search
START
Fig. 1. SA–DFP optimizer.
H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439434
Analysis techniques including cooling analysis
and sensitivity analysis using BEM have been discussed
in previous sections. The cooling system optimal design
can be considered as a constrained NLP problem.
Design goals are implemented into the optimization
formulation in terms of objective function, constraints
and design variables. In the following, these aspects
will be described for the implementation of the optimal
design.
2.4.1. Objective function
The objective function is chosen to mini mize the
difference between temperatures distributed over the mold
cavity surface as
f ðxÞ¼
1
S
Z
S
ðTðxÞÀ
¯
TðxÞÞ
2
dS, (26)
where x is a design v ector, T is the temperature
distribution, S denotes the cavity surface area, and the
average surface temperature

¯
T is defined as
¯
TðxÞ¼
1
S
Z
S
TðxÞ dS. (27)
In the boundary element model, the temperature distribu-
tion can be represented by the nodal temperatures.
Therefore, Eq. (26) can be rewritten as
f ðxÞ¼
1
N
w
X
N
w
i¼1
ðT
i
ðxÞÀ
¯
TðxÞÞ
2
, (28)
where N
w
is the total number of nodes over the cavity wall,

T
i
ði ¼ 1; 2; ; N
w
Þ are the temperatures at the corre-
sponding nodes, and the average temperature is
¯
TðxÞ¼
1
N
w
X
N
w
i¼1
T
i
ðxÞ. (29)
2.4.2. Design variables
In this paper, location s of cooling channels are selected
as design variables, which will be determined through the
optimization process. Let N
c
be the number of cooling
channels. In the 2D case, the location of a circular cooling
channel can be specified by the x and y coordinates of its
center. For example, the location of the ith cooling channel
is denoted as the center C
i
¼ðx

i
; y
i
Þ. Therefore, the total
number of design variables is 2N
c
.
2.4.3. Constraints
Constraints on cooling channels include specified lower
limits for the spacing between the cooling channels, and the
spacing between each channel and the outer boundary C.
If the ith cooling channel located at C
i
has the diameter
D
i
, the constraints can be expressed in the following:
jC
i
À C
j
jX
D
i
2
þ
D
j
2
þ d

1
; i ¼ 1; 2; ; N
c
À 1,
j ¼ i þ 1; i þ 2; ; N
c
, ð30aÞ
jC
i
À CjX
D
i
2
þ d
2
; i ¼ 1; 2; ; N
c
, (30b)
where 9aÀb9 denotes the distance between a and b. d
1
40
and d
2
40 are clearan ces specified to avoid interference of
the cooling channels with each other and with the outer
boundary, respectively.
2.4.4. Optimization method
The optimization method applied to solve this con-
strained NLP problem is SA–DFP method developed in
the previous section. The DFP method is a first-order

method that uses the gradie nt of the objective function to
determine the search direction. The gradient of f with
respect to the design variables can be expressed from
Eq. (28) as follows
rf ðxÞ¼
2
N
w
X
N
w
i¼1
½ðT
i
ðxÞÀ
¯
TðxÞÞðrT
i
ðxÞÀr
¯
TðxÞÞ, (31)
where r
¯
TðxÞ can be obtained from Eq. (29) as
r
¯
TðxÞ¼
1
N
w

X
N
w
i¼1
rT
i
ðxÞ. (32)
Therefore, in order to use SA–DFP method to find the
minimum point of f, we must calculate both the tempera-
tures and their gradients over the cavity surface, which are
obtained through cooling analysis and sensitivity analys is,
respectively. The interior penalty function is used to handle
the constraints. The gradients of constraints can be
obtained via the finite difference method with the same
ARTICLE IN PRESS
Set design goals and
control parameters
OptimizerNew Design
Sensitivity Analysis
Initial Design
Cooling Analysis
Finished ?
END
Mesh Generation
Yes
No
Iterative
Design Loop
Analysis Optimization
START

Fig. 2. Optimization methodology for cooling system design.
H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439 435
step size as the perturbation-based approach to sensitivity
analysis.
3. Numerical results
3.1. Design sensitivity analysis
In this example, the perturbation-based sensitivity
analysis approach (PSA) is applied to calculate the
sensitivity for the injection mold shown in Fig. 3.
Parameters used for cooling analysis are listed in
Tables 1 and 2. Two BEM mesh models with linear
elements, as shown in Fig. 4, are used. The cooling channel
is located at (50, 50). Mold cavity surface temperature
sensitivity with respect to the x-coordinate of the cooling
channel center is investigated. Second-order approximation
is used in Eq. (24). Comparison with the forward finite-
difference method (FDM) is also given to demonstrate the
efficiency of the proposed approach. The simulation is
carried out on a PC with PII 350 MHz CPU and
256 Mbytes RAM. The sensitivity distributions are shown
in Fig. 5 and the CPU time results are listed in Table 3.It
can be seen that the PSA approach has the same level of
accuracy as the FDM for the same step size but has a
higher efficiency. It is also seen from Table 3 that 31.8%
CPU time can be saved for Model 1, whereas 50.9% CPU
time can be saved for Model 2 compared with a full
analysis for the same model. The lower the percentage of
elements that vary according to the design variable, the
more the CPU time that can be saved using PSA approach.
3.2. Optimization of cooling system design

To justify the method presented above for the cooling
system optimal design, a simple mold with four cooling
channels shown in Fig. 3 is used to demonstrate the
optimization. Here suppose all the cooling channel sizes are
the same, D ¼ 10 mm. Other cooling parameters are shown
in Tables 1 and 2. Due to the symmetry, only one channel
location needs to be determined during optimization
process. Thus in 2D case the number of design variables
is 2, i.e. x and y coordinates of the center. The objective
function defined by Eq. (28) is minimized over the cavity
surface to achieve the uniformity of the temperature
distribution. The constraint is defined by Eq. (30b), in
which d
2
¼ 5 mm. The initial design and the optimal design
obtained by SA–DFP search are shown in Fig. 6. The
results of the temperature distribution over the cavity
surface for the initial design and the optimal design are
shown in Fig. 7. It can be found that the temperatur e
ARTICLE IN PRESS
400
200
10
4
40
100
200
50
y
x

Fig. 3. Injection mold with cooling channel layout (unit: mm).
Table 1
Cooling operation conditions
Coolant temperature, T
c
(1C) 30
Ambient air temperature, T
a
(1C) 30
Polymer injection temperature, T
inj
(1C) 220
Heat transfer coefficient of coolant, h
c
(W/m
2
1C) 3650
Heat transfer coefficient of air, h
a
(W/m
2
1C) 77
Mold opening time, t
o
(s) 4
Mold cooling time, t
c
(s) 16
Table 2
Material properties

Material Density, r
(kg/m
3
)
Specific heat, c
(J/kg K)
Conductivity, k
(W/m K)
Mold 7670 426 36.5
Polymer 938 1800 0.25
The number of elements = 44
The number of elements = 69
(a)
(b)
Fig. 4. BE mesh models for sensitivity analysis: (a) Model 1, and (b)
Model 2.
H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439436
distribution is improved after optimization. The final
result, however, is still not very satisfactory. This is mainly
because the number of cooling channels is not enough to
maintain a more uniform temperature distribution.
To improve the uneven temperatures present in the 4-
channel case, four additional same-sized cooling channels
are added. In this 8-channel case, the locations of two
cooling channels need to be determined during the
optimization process due to the symmetry. Therefore in
the 2D case, the number of design variables is 4 and the
objective function is the same as that in the 4-channel case.
The constraints are defined by Eqs. (30) with
d

1
¼ d
2
¼ 5 mm. The initial design and the optimal design
achieved by SA–DFP method are shown in Fig. 8.
In order to demonstrate the effectiveness and efficiency of
the SA–DFP approach to cooling system optimal design,
comparison with the results obtained using the DFP method
alone and the SA algorithm alone is given. The optimal
values are listed in Table 4. The results of temperature
distribution for initial design and optimal designs by DFP,
SA and SA–DFP search are shown in Fig. 9.
From Table 4, it is found that in the optimal design
obtained by the DFP method, the location of the second
cooling channel is very far away from the cavity surface.
Thus the cooling performance is similar to that of the 4-
channel case. This can also be verified from its temperature
distribution curve. After optimization by SA–DFP and SA,
we see a pronounced improvement in surface temperature.
This is because the cooling system optimal design is
basically a non-convex optimization problem and the
DFP method is very likely to be trapped in a valley with
a local optimum, which may not be good enough.
Therefore, global optimization methods such as SA–DFP
and SA are more robust for cooling system optimal design.
Compared with the SA search, the SA–DFP search takes
only about
1
3
CPU time to arrive at the comparative

optimal design for this particular case. Therefore, the
SA–DFP is a more efficient optimization method than SA
for cooling system optimal design. Moreover, the more
design variables are involved, the more CPU time can be
saved by employing SA–DFP.
4. Conclusions
An optimization methodology for computer-aided cool-
ing system design in injection molding is presented. Various
aspects of optimization including cooling analysis and
sensitivity analysis using the boundary element method,
optimization problem formulation and a novel hybrid
optimizer are investigated.
Cooling analysis, which is performed to evaluate the
objective function, plays an impor tant role during the
optimization process. More efforts are devoted to mold
ARTICLE IN PRESS
Table 3
CPU time (s) for sensitivity analysis
Finite-
difference
method
Perturbation-
based approach
Time saving
(%)
Model 1 0.22 0.15 31.8
Model 2 0.57 0.28 50.9
Model 1: step size = 0.05mm
-0.2
-0.1

0
0.1
0.2
0 20406080100
x (mm)
Sensitivity
Model 1: step size = 0.1mm
-0.2
-0.1
0
0.1
0.2
0 20406080100
x (mm)
Sensitivity
FDM
PSA
FDM
PSA
FDM
PSA
FDM
PSA
Model 2: step size = 0.05mm
-0.2
-0.1
0
0.1
0.2
0 20406080100

x (mm)
Sensitivity
Model 2: step size = 0.1mm
-0.2
-0.1
0
0.1
0.2
0 20406080100
x (mm)
Sensitivity
Fig. 5. Sensitivity distributions along the cavity surface.
H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439 437
cooling analysis because of its complexity in computation.
The cycle-averaged approach is used to perform cooling
analysis because of its efficiency, especially suited for the
repeated analysis during the iterative optimization process.
A pert urbation-b ased approach using the boundary
element method is developed to perform design sensitivity
analysis. This approach has the same accuracy as the
conventional finite difference approac h, but at a higher
efficiency. Moreo ver, significant programming efforts re-
quired using direct differen tiation method or adjoint variable
method for design sensitivity analysis could be avoided.
Although there are many numerical optimization meth-
ods, none of them is perfect in both the computational
effort involved (efficiency) and the accuracy of the
calculations (quality). The DFP method is efficient to find
a local optimum, but the optimum obtained is normally
dependent on the initial design. The simulated annealing is

a stochastic optimizer which is capable of finding the global
optimum, but more evaluat ions of the objective function
are required. Thus, a novel hy brid optim izer is developed
by combining the DFP method with the simulated
annealing. This is a global-like optimization method, which
at the same time has a high efficiency.
Optimization of cooling system design is implemented by
a systematic computer-aided methodology. Cycle-averaged
cooling analysis, perturbation-based sensitivity analysis,
and the hybrid SA–DFP optimizer are applied to search for
the optimal design. Significant uniformity of the tempera-
ture distribution along the cavity surface is obtained as a
result of the optimization process. The numerical results
show that the methodology proposed for cooling system
optimal design is efficient, robust and practical.
Acknowledgements
Most results reported in this paper were obtained in the
CAD/CAM Lab, Nanyang Technological University,
Singapore.
ARTICLE IN PRESS
400
200
y
x
Initial Design
(50, 50)
Optimal Design
(37.02, 32.03)
Fig. 6. Initial and optimal design for 4-channel case.
50

55
60
65
70
-100 -80 -60 -40 -20 0 20 40 60 80 100
x (m)
Temperature (°C)
Initial
SA-DFP
Fig. 7. Temperature distribution along the cavity surface for 4-channel
case.
400
200
y
x
Initial Design
(50, 50)
(100, 50)
Optimal Design
(19.88, 24.78)
(62.14, 22.73)
Fig. 8. Initial and optimal design for 8-channel case.
Table 4
Initial and optimal designs for 8-channel case
Initial design Optimal design
DFP SA SA–DFP
(50, 50) (32.67, 24.37) (20.31, 19.62) (19.88, 24.78)
(100, 50) (95.22, 74.38) (65.58, 18.45) (62.14, 22.73)
Fig. 9. Temperature distribution along the cavity surface for 8-channel
case.

H. Qiao / International Journal of Mechanical Sciences 48 (2006) 430–439438
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