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 à 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 ARTICLE IN PRESS www.elsevier.com/locate/ijmecsci 0020-7403/$ -see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.11.001 à Tel.: +1 44 29 20876047. E-mail address: qiaoh@cf.ac.uk. 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 à T dG ¼ Z G T à 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 à ¼ 1 2p ln 1 r , (7) where r is the distance between a source point and a field point. In Eq. (6), q à ¼ qT à =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 à T ðeÞ dG e ¼ X N e¼1 Z G e T à 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) ARTICLE IN PRESS 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 References [1] Kown TH. Mold cooling system design using boundary element method. Journal of Engineering for Industry 1988;110:384–94. [2] Barone MR, Caulk DA. Optimal arrangement of holes in a two- dimensional heat conductor by a special boundary integral method. International Journal for Numerical Methods in Engineering 1982;18:675–85. [3] Himasekhar K, Lottey J, Wang KK. 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