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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 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 sepa rated into two constituents on the reference plane which are represented on a 2D coordinate system 100 100 xy hh LL 2 where x 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 W 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 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 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 iA and iB are the weighting factors of the terminal node deflec tions calculated as follows 1 1 iA iA iB iA LL 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 Subject to 0 10 gpp X 1 2 i Xi N X where 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 c 1 in annealing process and Markov chain N generate are given 2 SA algorithm generates a new gate location X new in the neighborhood of X old 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 function accept new old min 1 exp k PkffT XX A uniform random variable P unif is generated in 0 1 If P unif P accept X new is accepted otherwise it is rejected 4 This process is repeated for a large enough number of iterations N generate for T k The sequence of trial gate locations generated in this way is known as Markov chain 5 A new Markov chain is then generated starting from the last accepted gate location in the previous Markov chain for a reduced temperature T k 1 cT k and the same process continues for de creasing values of temperature until the algorithm stops APPLICATION AND DISCUSSION The 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 longitu dinal direction is specified as projected reference Fig 3 The flow chart of the simulated annealing algorithm Li et al J Zhejiang Univ Sci A 2007 8 7 1077 1083 1082 N7459 N7379 Fig 5 Finite element mesh model of the part 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 direc tion The 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 has 1469 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 posi tive direction because of the location of the gate par ticularly 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 fea ture warpage is notable an