698 電器外殼注射模設計（有cad圖+文獻翻譯）,698,電器外殼注射模設計（有cad圖+文獻翻譯）,電器,外殼,注射,設計,cad,文獻,翻譯 Li et al. / J Zhejiang Univ Sci A 2007 8(7):1077-1083 1077 Single gate optimization for plastic injection mold * LI Ji-quan ? , LI De-qun, GUO Zhi-ying, LV Hai-yuan (Department of Plasticity Technology, Shanghai Jiao Tong University, Shanghai 200030, China) ? E-mail: hutli@ Received Nov. 22, 2006; revision accepted Mar. 19, 2007 Abstract: This paper deals with a methodology for single gate location optimization for plastic injection mold. The objective of the gate optimization is to minimize the warpage of injection molded parts, because warpage is a crucial quality issue for most injection molded parts while it is influenced greatly by the gate location. Feature warpage is defined as the ratio of maximum displacement on the feature surface to the projected length of the feature surface to describe part warpage. The optimization is combined with the numerical simulation technology to find the optimal gate location, in which the simulated annealing algorithm is used to search for the optimum. Finally, an example is discussed in the paper and it can be concluded that the proposed method is effective. Key words: Injection mold, Gate location, Optimization, Feature warpage doi:10.1631/jzus.2007.A1077 Document code: A CLC number: TQ320.66 INTRODUCTION Plastic injection molding is a widely used, com- plex but highly efficient technique for producing a large variety of plastic products, particularly those with high production requirement, tight tolerance, and complex shapes. The quality of injection molded parts is a function of plastic material, part geometry, mold structure and process conditions. The most important part of an injection mold basically is the following three sets of components: cavities, gates and runners, and cooling system. Lam and Seow (2000) and Jin and Lam (2002) achieved cavity balancing by varying the wall thick- ness of the part. A balance filling process within the cavity gives an evenly distributed pressure and tem- perature which can drastically reduce the warpage of the part. But the cavity balancing is only one of the important influencing factors of part qualities. Espe- cially, the part has its functional requirements, and its thicknesses should not be varied usually. From the pointview of the injection mold design, a gate is characterized by its size and location, and the runner system by the size and layout. The gate size and runner layout are usually determined as constants. Relatively, gate locations and runner sizes are more flexible, which can be varied to influence the quality of the part. As a result, they are often the design pa- rameters for optimization. Lee and Kim (1996a) optimized the sizes of runners and gates to balance runner system for mul- tiple injection cavities. The runner balancing was described as the differences of entrance pressures for a multi-cavity mold with identical cavities, and as differences of pressures at the end of the melt flow path in each cavity for a family mold with different cavity volumes and geometries. The methodology has shown uniform pressure distributions among the cavities during the entire molding cycle of multiple cavities mold. Zhai et al.(2005a) presented the two gate loca- tion optimization of one molding cavity by an effi- cient search method based on pressure gradient (PGSS), and subsequently positioned weld lines to the desired locations by varying runner sizes for Journal of Zhejiang University SCIENCE A ISSN 1673-565X (Print); ISSN 1862-1775 (Online) E-mail: jzus@ * Project (No. 50675080) supported by the National Natural Science Foundation of China Li et al. / J Zhejiang Univ Sci A 2007 8(7):1077-1083 1078 multi-gate parts (Zhai et al., 2006). As large-volume part, multiple gates are needed to shorten the maxi- mum flow path, with a corresponding decrease in injection pressure. The method is promising for de- sign of gates and runners for a single cavity with multiple gates. Many of injection molded parts are produced with one gate, whether in single cavity mold or in multiple cavities mold. Therefore, the gate location of a single gate is the most common design parameter for optimization. A shape analysis approach was pre- sented by Courbebaisse and Garcia (2002), by which the best gate location of injection molding was esti- mated. Subsequently, they developed this methodol- ogy further and applied it to single gate location op- timization of an L shape example (Courbebaisse, 2005). It is easy to use and not time-consuming, while it only serves the turning of simple flat parts with uniform thickness. Pandelidis and Zou (1990) presented the opti- mization of gate location, by indirect quality measures relevant to warpage and material degradation, which is represented as weighted sum of a temperature dif- ferential term, an over-pack term, and a frictional overheating term. Warpage is influenced by the above factors, but the relationship between them is not clear. Therefore, the optimization effect is restricted by the determination of the weighting factors. Lee and Kim (1996b) developed an automated selection method of gate location, in which a set of initial gate locations were proposed by a designer and then the optimal gate was located by the adjacent node evaluation method. The conclusion to a great extent depends much on the human designer’s intuition, because the first step of the method is based on the designer’s proposition. So the result is to a large ex- tent limited to the designer’s experience. Lam and Jin (2001) developed a gate location optimization method based on the minimization of the Standard Deviation of Flow Path Length (SD[L]) and Standard Deviation of Filling Time (SD[T]) during the molding filling process. Subsequently, Shen et al.(2004a; 2004b) optimized the gate location design by minimizing the weighted sum of filling pressure, filling time difference between different flow paths, temperature difference, and over-pack percentage. Zhai et al.(2005b) investigated optimal gate location with evaluation criteria of injection pressure at the end of filling. These researchers presented the objec- tive functions as performances of injection molding filling operation, which are correlated with product qualities. But the correlation between the perform- ances and qualities is very complicated and no clear relationship has been observed between them yet. It is also difficult to select appropriate weighting factors for each term. A new objective function is presented here to evaluate the warpage of injection molded parts to optimize gate location. To measure part quality di- rectly, this investigation defines feature warpage to evaluate part warpage, which is evaluated from the “flow plus warpage” simulation outputs of Moldflow Plastics Insight (MPI) software. The objective func- tion is minimized to achieve minimum deformation in gate location optimization. Simulated annealing al- gorithm is employed to search for the optimal gate location. An example is given to illustrate the effec- tivity of the proposed optimization procedure. QUALITY MEASURES: FEATURE WARPGE Definition of feature warpage To apply optimization theory to the gate design, quality measures of the part must be specified in the first instance. The term “quality” may be referred to many product properties, such as mechanical, thermal, electrical, optical, ergonomical or geometrical prop- erties. There are two types of part quality measures: direct and indirect. A model that predicts the proper- ties from numerical simulation results would be characterized as a direct quality measure. In contrast, an indirect measure of part quality is correlated with target quality, but it cannot provide a direct estimate of that quality. For warpage, the indirect quality measures in related works are one of performances of injection molding flowing behavior or weighted sum of those. The performances are presented as filling time dif- ferential along different flow paths, temperature dif- ferential, over-pack percentage, and so on. It is ob- vious that warpage is influenced by these perform- ances, but the relationship between warpage and these performances is not clear and the determination of these weighting factors is rather difficult. Therefore, the optimization with the above objective function Li et al. / J Zhejiang Univ Sci A 2007 8(7):1077-1083 1079 probably will not minimize part warpage even with perfect optimization technique. Sometimes, improper weighting factors will result in absolutely wrong re- sults. Some statistical quantities calculated from the nodal displacements were characterized as direct quality measures to achieve minimum deformation in related optimization studies. The statistical quantities are usually a maximum nodal displacement, an av- erage of top 10 percentile nodal displacements, and an overall average nodal displacement (Lee and Kim, 1995; 1996b). These nodal displacements are easy to obtain from the simulation results, the statistical val- ues, to some extents, representing the deformation. But the statistical displacement cannot effectively describe the deformation of the injection molded parts. In industry, designers and manufacturers usually pay more attention to the degree of part warpage on some specific features than the whole deformation of the injection molded parts. In this study, feature warpage is defined to describe the deformation of the injection parts. The feature warpage is the ratio of the maximum displacement of the feature surface to the projected length of the feature surface (Fig.1): 100%, h L γ =× (1) where γ is the feature warpage, h is the maximum displacement on the feature surface deviating from the reference platform, and L is the projected length of the feature surface on a reference direction paralleling the reference platform. For complicated features (only plane feature discussed here), the feature warpage is usually sepa- rated into two constituents on the reference plane, which are represented on a 2D coordinate system: 100%, 100%, xy hh LL γγ=× =× (2) where γ x , γ y are the constituent feature warpages in the X, Y direction, and L x , L y are the projected lengths of the feature surface on X, Y component. Evaluation of feature warpage After the determination of target feature com- bined with corresponding reference plane and pro- jection direction, the value of L can be calculated immediately from the part with the calculating method of analytic geometry (Fig.2). L is a constant for any part on the specified feature surface and pro- jected direction. But the evaluation of h is more com- plicated than that of L. Simulation of injection molding process is a common technique to forecast the quality of part de- sign, mold design and process settings. The results of warpage simulation are expressed as the nodal de- flections on X, Y, Z component (W x , W y , W z ), and the nodal displacement W. W is the vector length of vector sum of W x ·i, W y ·j, and W z ·k, where i, j, k are the unit vectors on X, Y, Z component. The h is the maximum displacement of the nodes on the feature surface, which is correlated with the normal orientation of the reference plane, and can be derived from the results of warpage simulation. To calculate h, the deflection of ith node is evaluated firstly as follows: cos cos cos ( ), iix iy iz iAAiBB WW W W W Wαβγωω=++?+ (3) where W i is the deflection in the normal direction of the reference plane of ith node; W ix , W iy , W iz are the deflections on X, Y, Z component of ith node; α, β, γ are the angles of normal vector of the reference; A and B are the terminal nodes of the feature to projecting direction (Fig.2); W A and W B are the deflections of nodes A and B: Fig.1 The definition of feature warpage h Reference plane Surface L Fig.2 The projected length evaluation Feature Y X L x B A L y Li et al. / J Zhejiang Univ Sci A 2007 8(7):1077-1083 1080 cos cos cos , cos cos cos , AAx Ay Az BBx By Bz WW W W WW W W α βγ α βγ =++? ? ? =++ ? ? (4) where W Ax , W Ay , W Az are the deflections on X, Y, Z component of node A; W Bx , W By and W Bz are the de- flections on X, Y, Z component of node B; ω iA and ω iB are the weighting factors of the terminal node deflec- tions calculated as follows: 1/, 1, iA iA iB iA LLω ωω=? =? (5) where L iA is the projector distance between ith node and node A. Ultimately, h is the maximum of the absolute value of W i : 12 max{| |,| |, ,| |}. k hWWW=… (6) In industry, the inspection of the warpage is carried out with the help of a feeler gauge, while the measured part should be placed on a reference plat- form. The value of h is the maximum numerical reading of the space between the measured part sur- face and the reference platform. GATE LOCATION OPTIMIZATION PROBLEM FORMATION The quality term “warpage” means the perma- nent deformation of the part, which is not caused by an applied load. It is caused by differential shrinkage throughout the part, due to the imbalance of polymer flow, packing, cooling, and crystallization. The placement of a gate in an injection mold is one of the most important variables of the total mold design. The quality of the molded part is greatly af- fected by the gate location, because it influences the manner that the plastic flows into the mold cavity. Therefore, different gate locations introduce inho- mogeneity in orientation, density, pressure, and temperature distribution, accordingly introducing different value and distribution of warpage. Therefore, gate location is a valuable design variable to minimize the injection molded part warpage. Because the cor- relation between gate location and warpage distribu- tion is to a large extent independent of the melt and mold temperature, it is assumed that the molding conditions are kept constant in this investigation. The injection molded part warpage is quantified by the feature warpage which was discussed in the previous section. The single gate location optimization can thus be formulated as follows: Minimize: min f(X)= γ; Subject to: 0 () / 10,gpp= ?≤X , 1,2,..., , i Xi N∈=X where γ is the feature warpage; p is the injection pressure at the gate position; p 0 is the allowable in- jection pressure of injection molding machine or the allowable injection pressure specified by the designer or manufacturer; X is the coordinate vector of the candidate gate locations; X i is the node on the finite element mesh model of the part for injection molding process simulation; N is the total number of nodes. In the finite element mesh model of the part, every node is a possible candidate for a gate. There- fore, the total number of the possible gate location N p is a function of the total number of nodes N and the total number of gate locations to be optimized n: p (1)( 1) . ! NN N n N n ? ??? ? ? = In this study, only the single-gate location problem is investigated. SIMULATED ANNEALING ALGORITHM The simulated annealing algorithm is one of the most powerful and popular meta-heuristics to solve optimization problems because of the provision of good global solutions to real-world problems. The algorithm is based upon that of Metropolis et al. (1953), which was originally proposed as a means to find an equilibrium configuration of a collection of atoms at a given temperature. The connection be- tween this algorithm and mathematical minimization was first noted by Pincus (1970), but it was Kirkpatrick et al.(1983) who proposed that it formed the basis of an optimization technique for combina- tional (and other) problems. To apply the simulated annealing method to op- Li et al. / J Zhejiang Univ Sci A 2007 8(7):1077-1083 1081 timization problems, the objective function f is used as an energy function E. Instead of finding a low energy configuration, the problem becomes to seek an approximate global optimal solution. The configura- tions of the values of design variables are substituted for the energy configurations of the body, and the control parameter for the process is substituted for temperature. A random number generator is used as a way of generating new values for the design variables. It is obvious that this algorithm just takes the mini- mization problems into account. Hence, while per- forming a maximization problem the objective func- tion is multiplied by (?1) to obtain a capable form. The major advantage of simulated annealing algorithm over other methods is the ability to avoid being trapped at local minima. This algorithm em- ploys a random search, which not only accepts changes that decrease objective function f, but also accepts some changes that increase it. The latter are accepted with a probability p /( ) e, fkT p ?? = where ?f is the increase of f, k is Boltzman’s constant, and T is a control parameter which by analogy with the original application is known as the system “temperature” irrespective of the objective function involved. In the case of gate location optimization, the implementation of this algorithm is illustrated in Fig.3, and this algorithm is detailed as follows: (1) SA algorithm starts from an initial gate loca- tion X old with an assigned value T k of the “tempera- ture” parameter T (the “temperature” counter k is initially set to zero). Proper control parameter c (0

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