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Single gate optimization for plastic injection mold Journal of Zhejiang University - Science A Volume 8, Number 7 (2007), 1077-1083, DOI: 10.1631/jzus.2007.A1077 Ji-quan Li, De-qun Li, Zhi-ying Guo and Hai-yuan LvAbstract: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.INTRODUCTIONPlastic 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 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 designers intuition, because the first step of the method is based on the designers proposition. So the result is to a large ex- tent limited to the designers experience.Lam and Jin (2001) developed a gate location optimization method based on the minimization of the Standard Deviation of Flow Path Length (SDL) and Standard Deviation of Filling Time (SDT) 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 WARPGEDefinition of feature warpageTo 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 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):where 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 separated into two constituents on the reference plane, which are represented on a 2D coordinate system:where x, y are the constituent feature warpages in the X, Y direction, and Lx, Ly are the projected lengths of the feature surface on X, Y component.Evaluation of feature warpageAfter 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 (Wx, Wy, Wz), and the nodal displacement W. W is the vector length of vector sum of Wxi, Wyj, and Wzk, 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 its node is evaluated firstly as follows:where Wi is the deflection in the normal direction of the reference plane of ith node; Wix, Wiy, Wiz are the deflections on X, Y, Z component of ith node; , , are the angles of normal vector of the reference; A and B are the terminal nodes of the feature to projectingdirection (Fig.2); WA and WB are the deflections of nodes A and B:where WAx, WAy, WAz are the deflections on X, Y, Zcomponent of node A; WBx, WBy and WBz are the de- flections on X, Y, Z component of node B; iA and iB are the weighting factors of the terminal node deflections calculated as follows:where LiA is the projector distance between ith node and node A. Ultimately, h is the maximum of the absolute value of Wi: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 surface and the reference platform.GATE LOCATION OPTIMIZATION PROBLEM FORMATIONThe 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 correlation between gate location and warpage distribu- tion is to a large extent independent of the melt and mold temperature, it is assumed that the moldingconditions 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:Subject to: where is the feature warpage; p is the injection pressure at the gate position; p0 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; Xi 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 Np is a function of the total number of nodes N and the total number of gate locations to be optimized n:In this study, only the single-gate location problem is investigated. SIMULATED ANNEALING ALGORITHMThe 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 between 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 combinational (and other) problems.To apply the simulated annealing method to op 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 employs 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 pwhere f is the increase of f, k is Boltzmans 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 Xold with an assigned value Tk of the “temperature” parameter T (the “temperature” counter k is initially set to zero). Proper control parameter c (0c1) in annealing process and Markov chain Ngenerateare given.(2) SA algorithm generates a new gate location Xnew in the neighborhood of Xold and the value of the objective function f(X) is calculated.(3) The new gate location will be accepted with probability determined by the acceptance functionFig.3 The flow chart of the simulated annealing algorithmAPPLICATION AND DISCUSSIONThe application to a complex industrial part is presented in this section to illustrate the proposed quality measure and optimization methodology. The part is provided by a manufacturer, as shown in Fig.4. In this part, the flatness of basal surface is the most important profile precision requirement. Therefore, the feature warpage is discussed on basal surface, in which reference platform is specified as a horizontal plane attached to the basal surface, and the longitudinal direction is specified as projected reference direction. The parameter h is the maximum basal surface deflection on the normal direction, namely the vertical direction, and the parameter L is the projected length of the basal surface to the longitudinal direction.Fig.4 Industrial part provided by the manufacturerThe material of the part is Nylon Zytel 101L (30% EGF, DuPont Engineering Polymer). The molding conditions in the simulation are listed in Table 1. Fig.5 shows the finite element mesh model of the part employed in the numerical simulation. It has1469 nodes and 2492 elements. The objective func- tion, namely feature warpage, is evaluated by Eqs.(1), (3)(6). The h is evaluated from the results of “Flow+Warp” Analysis Sequence in MPI by Eq.(1), and the L is measured on the industrial part immediately, L=20.50 mm.MPI is the most extensive software for the in- jection molding simulation, which can recommend the best gate location based on balanced flow. Gate location analysis is an effective tool for gate location design besides empirical method. For this part, the gate location analysis of MPI recommends that the best gate location is near node N7459, as shown in Fig.5. The part warpage is simulated based on this recommended gate and thus the feature warpage is evaluated: =5.15%, which is a great value. In trial manufacturing, part warpage is visible on the sample work piece. This is unacceptable for the manufacturer.The great warpage on basal surface is caused by the uneven orientation distribution of the glass fiber, as shown in Fig.6a. Fig.6a shows that the glass fiber orientation changes from negative direction to positive direction because of the location of the gate, particularly the greatest change of the fiber orientation appears near the gate. The great diversification of fiber orientation caused by gate location introduces serious differential shrinkage. Accordingly, the feature warpage is notable and the gate location must be optimized to reduce part warpageTo optimize the gate location, the simulated annealing searching discussed in the section “Simulated annealing algorithm” is applied to this part. The maximum number of iterations is chosen as 30 to ensure the precision of the optimization, and the maximum number of random trials allowed for each iteration is chosen as 10 to decrease the probability of null iteration without an iterative solution. Node N7379 (Fig.5) is found to be the optimum gate loca- tion. The feature warpage is evaluated from the war- page simulation results f(X)=0.97%, which is less than that of the recommended gate by MPI. And the part warpage meets the manufacturers requirements in trial manufacturing. Fig.6b shows the fiber orientation in the simulation. It is seen that the optimal gate location results in the even glass fiber orientation, and thus introduces great reduction of shrinkage difference on the vertical direction along the longitudinal direction. Accordingly, the feature warpage is reduced.CONCLUSIONFeature warpage is defined to describe the war- page of injection molded parts and is evaluated based on the numerical simulation software MPI in this investigation. The feature warpage evaluation based on numerical simulation is combined with simulated annealing algorithm to optimize the single gate loca- tion for plastic injection mold. An industrial part is taken as an example to illustrate the proposed method. The method results in an optimal gate location, by which the part is satisfactory for the manufacturer. This method is also suitable to other optimization problems for warpage minimization, such as location optimization for multiple gates, runner system balancing, and option of anisotropic materials.注塑模的单浇口优化 Journal of Zhejiang University - Science A Volume 8, Number 7 (2007), 1077-1083, DOI: 10.1631/jzus.2007.A1077Ji-quan Li, De-qun Li, Zhi-ying Guo and Hai-yuan Lv摘要：本文论述了一种单浇口位置优化注塑模具的方法。客观的浇口优化，尽量减少注塑制品翘曲变形，因为翘曲是一个关键质量问题，对大多数注塑件，这绝大部分受浇口位置影响。专题翘曲的定义是用比例最大位移对特征表面预计长度的表面特征来描述零件翘曲。优化相结合，数值模拟技术，以找到最佳的浇口位置，其中，模拟退火算法就是用来寻找最佳的浇口位置。最后，其中一个例子是讨论有关文件，并可以得出结论认为，所提出的方法是有效的。关键词：注塑模， 浇口位置和结构优化，功能翘曲导言 塑料注塑成型，是一种广泛使用的，复杂的，对大型品种的塑料制品，尤其是那些高产量要求，精密复杂形状的有高效率的技术制作。质量注塑件是一个有功能性，部分几何，模具结构和工艺条件的塑胶材料。最重要的一部分，注塑模，基本上是以下三组组成：腔，浇口和浇道， 和冷却系统。 Lam和Seow（ 2000），Jin和Lam（ 2002）达到平衡腔不同壁厚的一部分。平衡充填过程内部腔给出了一个均匀分布的压力和温度，可大幅度减少该部的翘曲。但腔平衡只是其中一个影响零件质量的重要因素。尤其是零件有其功能要求，其厚度通常不应该多种多样。从这个角度谈了注塑模具设计，浇口是由其尺寸和位置，和浇道系统的规模和布局表征的。浇口尺寸和浇道布局通常定为常量。相对地，浇口位置和浇道的大小是比较有弹性的，能够多样的影响零件质量。因此，他们往往优化设计参数。 Lee和Kim（1996年）为多种注射溶洞优化了浇道和浇口的大小来平衡浇道系统。浇道维持平衡可以理解为有相同腔的多腔模具的不同入口压力，在每一个腔每一个熔体流道底部有不同的情体积和几何形状。该方法已显示压力在整个多腔模具成型周期中的单腔里均匀分布。 Zhai etal.（2005年）发布两个浇口位置优化，它的一个成型腔是由一个在压力梯度的基础上的高效率的搜索方法（ PGSS） ，为由不同尺寸的浇道多浇口零件定位，熔接线向理想的地点（Zhai etal， 2006 ）。作为大容量的一部分，多浇口需要缩短最高流径，与相应减少注射压力。该方法大可成为设计多浇口单型腔的浇口和浇道。 许多注塑件是只制作一个浇口，无论是在单型腔模具或多个腔模具。因此，单浇口的浇口位置是最常见的设计优化参数。形状分析方法是由Courbebaisse和Garcia 2002年提出，是最佳浇口位置的注射成型估计。后来，他们研制的这种理论进一步研究和应用于单一浇口位置优化的一个L形例子（Courbebaisse，2005年）。 它易于使用，而不耗费时间，而且它只不过是提供了简单的有均匀厚度的平面零件。 Pandelidis和Zou（1990年）提出的优化浇口位置，由间接质量相关引起的翘曲和物质降解，这代表着加权温度差，摩擦过热的时间。翘曲是受上述因素的影响， 但它们之间的关系并不明确。 因此，优化效果是受制于测定转归的加权因素。 Lee和Kim（ l996b ）研制出一种自动选择浇口位置的方法，其中一套初步浇口位置，由设计师提出，最优浇口是位于相邻节点。结论在很大程度上取决于设计师的直觉，因为第一步是基于设计师的主张。 所以在相当大的程度上，受限于设计师的经验。 Lam和Jin（2001）开发了浇口位置优化方法，基于最大限度地减少了标准偏差的流径长度（标准差大 ）和在成型充填过程中的标准偏差的灌装时间（标准差 T ）。随后，沈等人（ 2004 年 ） ，优化了浇口位置设计通过最小加权充气压力，灌装时间区别不同的水流路径，温差变化大，以及过度包装的百分比。Zhai等 （ 2005 年）在去年底调查了最佳浇口位置与评价标准的注射压力。这些研究人员介绍目标函数作为注塑成型灌装操作，这对相关产品的品质有益。 但之间的相关性是非常复杂和不清晰在它们之间已经观察到。 人们还很难选择适当的加权因子为每个函数。 一个新的目标函数来评价注塑制品翘曲变形，以优化浇口位置。 直接衡量零件质量，这项调查定义特征翘曲来评价零件翘曲，这是从流加翘曲模拟产出模型流向塑料洞察力（电传等）的软件。目标函数最小化，在浇口位置优化，以达到最低变形。 模拟退火算法是用来寻找最优浇口位置。 给出了一个例子来说明建议优化程序的有效性。质量措施：特征翘曲定义特征翘曲 运用优化理论设计浇口，零件的质量措施必须指定在初审。术语质量可转介许多产品性能，如力学，热学， 电子，光学，工效学或几何性质。有两种零件质量测量：直接和间接。一个有预测性的模型，从数值模拟结果，可作为一个直接的质量测量。 相比之下，间接测量的零件质量是正相关目标质量，但它并不能提供对其质量的直接估计。翘曲，在相关工程的间接质量测量，是一个注塑成型流动行为或加权。 这种行为是作为填充不同流径的时间差，温度差，过度包装的比例问题，等等。这是很明显的，翘曲是受这些因素的影响，但翘曲和这些因素的关系是不明确的，而且决定这些因素所占的比重是相当困难的。因此，用上述目标函数优化大概不会减低零件翘曲，甚至是完美的优化技术。 有时，不恰当加权因素，将导致完全错误的结果。一些统计量计算，节点位移被定性为直接质量测量，以达到最低变形链优化研究。统计数量通常是最多节点位移，平均每年有10%的节点位移，而且整体平均节点位移（Lam和Jin， 1995 ; 1996 ） 。这些节点的位移容易从数值模拟结果获得，统计值，在一定程度上代表着变形。 但统计位移不能有效地描述变形的注塑件。 在工业方面，设计者和制造商通常更加注意，部分上翘曲在某些特点上超过整个变形注射模塑件的程度。在这项研究中，特征翘曲是用来形容变形的注塑件。特征翘曲是表面上的最大位移与表面特征的预计长度之比（图1 ） ： （1） 其中是特征翘曲， h是特征表面偏离该参考平台的最高位移，L是在与参考方向平行的参考平台上的表面特征的预计长度。 对于复杂的特点（这里只讨论平面特征） ，翘曲的特点是通常在参考平面内分为两个区域，它是代表一个二维坐标系统： （2） 其中，是特征翘曲在X ， Y方向， ，是表面特征的预计长度在X ，Y上的投影。特征翘曲的评定 与相应的参考平面和投影方向结合起来测定目标特征后，其L的值可以从图中用解析几何立即计算出来（图2 ） 。在特定的表面特征和预测的方向，L是一个常量。 但H的评定比L复杂得多。 模拟注射成型过程是一种常见的技术，以预测质量来设计零件，设计模具和工艺设置。结果翘曲模拟表达为节点挠度上的X ， Y ， Z分量 ，以及节点位移W。W是向量长度的矢量总和：+ + ，其中 i，j，k是在X，Y，Z方向上的单位矢量。H是在特征表面上的节点的最大位移， 这与通常方向的参考平面相同，并能产生结果的翘曲仿真。 计算h时，节点的挠度提取如下： 其中是挠度在正常方向参考平面内提取节点; ， ， 是对挠度的X ， Y ， Z分量的提取节点;，是角度的向量参考; A和B是终端节点，可以预测方向（图2 ） ; 和是节点A和B的挠度： 其中， ，是对节点A的挠度在X，Y，Z方向上的分量； ，和是对节点B的挠度在X ， Y ， Z方向上的分量; 和是终端节点挠度的加权因子，计算方法如下： 是提取节点和节点A投影间的距离， H是的最大绝对值。 在工业方面，视察该翘曲借助了一个触角衡量，被测工件放在一个
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