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Optimum topology design for the stationary platen of a plastic injection machine
Computers in Industry 55 (2004) 147–158 www.elsevier.com/locate/compind
Optimum topology design for the stationary platen of a plastic injection machine Shu Huang Sun* Assistant Professor, Department of Mechanical Engineering, Kun Shan University of Technology, Yung Kang City, Tainan County 710, Taiwan, ROC Received 8 October 2002; accepted 5 July 2004 Available online 15 September 2004
Abstract Tie bars are key components of a plastic injection machine. They very easily fatigue in periodically long term operations due to the bending moment transferred to them by the bending of the stationary platen. This problem can be easily overcome by reducing the deflection of the stationary platen through topology optimization of the platen structure by applying a cost or weight constraint. In this paper, the self-organization method was introduced to optimize the topology of the stationary platen. Topology design optimizes material allocation, i.e. strengthening locations of high loads and minimizing material usage at other locations. By applying this method to the stationary platen design, the deflection of the platen could be reduced, which correspondingly reduces the bending load of the tie bars and thus extending their operating life. # 2004 Elsevier B.V. All rights reserved. Keywords: Optimum topology; Plastic injection machine; Tie bars; Stationary platen
1. Introduction 1.1. Fatigue of tie bars The clamping unit of an injection molding machine provides the motion needed for mold closing and opening, and produces the forces that are necessary to clamp the mold. It has principal components including * Tel.: + 886 6 2050496; fax: +886 6 2050509. E-mail address: [email protected].
tie bars, stationary and movable platens, and a mechanism for mold opening, closing and clamping. The stationary platen is fixed to the frame, whereas the movable platen is pushed and pulled by a driving mechanism, such as direct hydraulic, direct electric or a toggle system. Due to the high cavity pressure when molten plastic is injected into a mold, a clamping force is applied to the movable and stationary platens of the mold to prevent it from opening. The mold also gives a reaction force to these two platens at their center
0166-3615/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.compind.2004.07.001
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Fig. 1. Bending deflection of tie bars.
regions. At the same time, the stationary platen carries a force from the four nuts at its four corner regions to balance the reaction force as shown in Fig. 1. Since these two forces that are applied to the stationary platen are not co-linear, the stationary platen experiences a bending moment, which is transferred to the tie bars as a counter-reacting bending moment. Therefore, in long term operations, fatigue is inevitable. This obvious problem has been an on-going concern for injection machine manufacturers and is still an important issue for consideration when designing plastic injection machine. The traditional but inefficient method is to strengthen the tie bars by increasing their diameters. More efficiently, however, is to strengthen the stationary platen instead as fatigue experienced by the tie bars is a direct effect from the bending deflection of the stationary platen. Increasing the thickness would, of course, strengthen the stationary platen, but this translates to an increase in raw material cost, a key consideration in the design of any machines. Therefore, strength improvement must be balanced by increased in cost. In this paper, an explicit and forward idea of using the topology optimization method is proposed to determine the best topology for the stationary platen in order to reduce its deflection but not increase its weight.
1.2. Topology optimization The three main fields in engineering optimization are: optimum topology design, optimum shape design and optimum size design. This is also the step sequence in the optimization process. The first step specifies a desirable material distribution while the second step shapes the subject to avoid stress concentration regions. Detail dimensions are then given in the third step. Optimum topology design methods have been well developed in recent years and can be divided into two categories, viz., the layout theory method and the design domain method. A few references are available for the layout theory method. Haftka and Grandhi [1] introduced the fully stressed design concept. Prager [2] proposed an optimality criterion to solve layout problems. Rozvany and Zhou [3] extended this criterion and proposed a continuum-based optimality criterion to solve the general layout problem. The design domain method can be sub-divided again into three categories, the homogenization method, the density function method, and the simulating biological growth method. Bendsfe and Kikuchi [4] applied the homogenization theory to find the optimum topology. Mlejnek and Schirrmacher [5]
S.H. Sun / Computers in Industry 55 (2004) 147–158
proposed an energy approach and treated the density of each element as the design variable. Young and Chuang [6] modified this approach by sequential linear programming and sensitivity analysis. Both of the above methods, however, involved complex theory and calculations. A means of avoiding these complications is to simulate the biological growth for an optimum topology. Burkhardt and Mattheck [7] proposed a soft kill option to simulate bone mineralization. Weinans et al. [8] and Huiskes and co-workers [9] proposed the simulated bone remodeling method to find the real density distribution in human leg bone. Inou et al. [10,11] proposed a selforganization method simulating human cell growth behavior to find an optimum topology design. In this algorithm, the Young’s modulus of each element was treated as a design variable and adjusted in each finite element method (FEM) analysis. Kita et al. [12] extended the self-organization method and improved it by adding local rules and penalty factors. Although many references discussed the optimum topology design method, few papers [13–16], however, dealt with 3-D applications.
149
In this study, the self-organization method was modified and applied to determine the optimum topology design of the stationary platen of a plastic injection machine.
2. Algorithm of the self-organization method The concept of the self-organization method [10,11] is to modify the Young’s modulus of each element according to the ratio of its stress and the average stress of the entire model after each FEM analysis. In the first iteration, all elements are given the same initial Young’s modulus value. After the first iteration, all elements are reassigned Young’s modulus values. Elements with high stress values mean that they were subjected to high loads and thus must retain high Young’s modulus values while low stressed elements are assigned low Young’s modulus values. As the iteration continues, the differences in the Young’s modulii of the elements gradually increase. At the condition when the Young’s modulii of certain elements falls below a pre-set value, the elements are
Fig. 2. Flowchart of self-organization method.
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deleted and the region would become a void. This process continues until a stop criterion is satisfied and the optimum topology design is obtained. Fig. 2 shows the process flowchart and the detailed procedure of the algorithm is discussed below. (1) The design domain is assigned and an initial FEM model is generated, filling up the domain with elements. (2) All elements are then assigned an arbitrary initial Young’s modulus value E0. (3) Using FEM analysis, the von Mises stress of each element is obtained. (4) The desired von Mises stress sC of each element is then determined by the following formulae: s C ¼ s C0 f1 þ b½eaðEE0 =E0 Þ 1g 2 a ¼ a1 and b ¼ ð2a Þ 1 e 1 when E E0
(1) (2)
1 (3) 1 ea2 when E < E0.Here sC0 is the average von Mises stress value of the entire model. The assignment of the parameters a and b is described in detail as follows. In the self-organization algorithm, the decision as to which an element would be retained or deleted is entirely based on its Young’s modulus a ¼ a2
and
b¼
value. The condition is set such that the elements with high Young’s modulus would be retained while those with low Young’s modulus would be deleted. The in-between elements are known as ‘gray’ elements, which should be minimized as they often lead to indecisions. The two parameters a1 and a2 in formulae (2) and (3) must be assigned properly for this reason. As shown in Fig. 3, Inou and Uesugi [10] proved that when a1 = 7.5 and a2 = 10 in the formulae (1)– (3), the obtained value of sC has the optimum effects in avoiding the generation of ‘gray’ elements. (5) The Young’s modulus of each element is subsequently updated after each iteration by the following formula: s ðtþ1Þ ðtÞ E ¼E 1þa 1 (4) sC where E(t+1) and E(t) are the Young’s modulus of each element in the (t+1)th and (t) th FEM analysis and is a constant used to control the step size in each Young’s modulus update. In this study, it is set to 0.5. In the iteration process, extreme Young’s modulus values may cause instability in the FEM analysis. To counter this, two rules are introduced to restrict the ‘growth’ or ‘decline’ of the Young’s modulus and they are [17] If E > 3E0 ;
then
If E > 0:3E0 ;
then
E ¼ Emax ¼ 3E0 E ¼ Emin ¼ 107 E0
(5) (6)
(6) Elements are grouped into four sets according to their Young’s modulus values as shown in Table 1. In the table, N1, N2, N3, and N4 are respectively, the number of elements whose Young’s modulus are larger than 200 GPa, between 100 and 200 GPa, between 30 and 100 GPa and under
Table 1 Elements are grouped into four sets according to their Young’s modulus values
Fig. 3. Mapping curve of sC and E.
Young’s modulus value
Number of elements
200 E 100 E < 200 30 E < 100 E < 30
N1 N2 N3 N4
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(7) The elements in N2 and N3 are the ‘gray’ elements described in the previous section. One of the main objectives of this algorithm is to minimize the number of such elements. The method is to force the elements to ‘jump’ from N2 and N3 to either N1 or N4 such that the optimum topology can be obtained by finally retaining the elements in N1 and deleting the elements in N4. Therefore, the number of residual elements in N2 and N3 can be used as a stop criteria. A parameter b called gray volume percentage and defined as the ratio of the sum of N2, N3 and Nall is used as the stop criteria in this study. It is b¼
Fig. 4. A rough structure of the stationary platen.
30 GPa. The decision is then to delete the elements in N4, resulting in a residual volume of a new topology that is different from the initial one.
ðN2 þ N3 Þ Nall
(7)
If the gray volume percentage is smaller than the desired value, the algorithm stops and the final optimum topology is obtained. Otherwise, steps (3)–(6) are repeated with the new FEM model. (8) In the whole iteration history, the gray volume percentage curve will decrease rapidly at the beginning and then saturates. A desired value can be specified as the stop criteria. As described in step (7), once the curve converges to a value under the desired value, the process ceases. However, if the chosen value falls below the saturation value, a maximum iteration number can be assigned to
Fig. 5. The initial FEM model.
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Fig. 6. FEM model including boundary conditions and loading situations in this study.
Fig. 7. The curves of gray volume percentage.
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Fig. 8. The curves of number of elements N1–N4 vs. iteration number.
suspend the process while the iteration curve is examined. From the trend of the curve, a decision can easily be made as to whether to continue or terminate the process. When the curve has reached saturation, the eventual value is of no significance as the number of gray elements has been minimized and their consequent effect on the final topology design is minimal.
3. Application of the self-organization method to the stationary platen The stationary platen of a plastic injection machine is usually a large cast-iron-made block. The mold is fixed on its left side plane and the injection barrel approaches from the right and moves to the center to make contact with the mold as illustrated in Fig. 1. Four tie bars secure the four corners of the platen with screws so as to build up the clamping force. The initial design domain in this optimization study is defined according to these descriptions and shown in Fig. 4. It is a solid block with one large circular hole in its
center, which functions as an entry for the injection barrel, and four small holes at its four corners for securing the tie bars. The platen is attached to the machine frame by means of two small ‘legs’ at the left and right bottom corners. A commercial machine FT90 produced by Fu Chun Shin, a famous plastic injection machine manufacture in Taiwan, has a typical platen size of 520 mm wide, 520 mm high and 180 mm thick. This machine is a 90 tons injection machine, i.e. the loading in the center region by the mold and the four corners by the nuts are both 90 tons. For simplification, the two securing ‘legs’ are considered as small features and therefore neglected in the FEM analysis. As the platen is symmetrical in the X and Y directions, one quarter of the volume would suffice for the FEM analysis. The volume that is bounded by the bold line in Fig. 4 is modeled in the FEM analysis. The elements are first generated to fill the design space as shown in Fig. 5 for the initial FEM analysis. Owing to symmetry, translation in the X direction of the left surface and the Y direction of the bottom surface are both restricted. These constraints
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Table 2 The final values of N1–N4 N1 N2 N3 N4
1810 16 28 2706
are represented by triangle pyramids at the left and bottom surface of the block as shown in Fig. 6. In the injection process, the molten plastic is injected into the mold, thereby producing high cavity pressure that has a tendency to push open the mold. This force is transferred from the mold to the center of the stationary platen where it spreads the loading on a projection area of the mold. In actual operating environment, the area is a 286 mm square not including the center hole. Four nuts lock the four tie bars to the four corners of the platen to provide a circular region of close contact between the platen and the nuts. The nuts also provide the platen with reaction force, which is equal to the loading from mold but opposite in direction to prevent the platen from moving. The loading situation of this model therefore results in a total of 90 tons distribution load in the center square region of the back surface of this platen and restricts the Z direction translation in the circular
region around the four corner holes at the front surface of the platen. The loading is represented by arrows and the translation constraint in Z direction is shown by triangle pyramids around the circular region as shown in Fig. 6. ANSYS 5.5 was used to perform the FEM analysis and computation was carried out with a Pentium III800 PC computer. There are a total of 4560 elements and 5824 nodes in the initial FEM model as shown in Fig. 5. The desired gray volume percentage is set to 1%. The initial Young’s modulus is set to 100 GPa for all elements and the initial value of N2 equals to the number of total elements and N1, N3, N4 are all zeros. An automatic iteration process was performed by subroutines written in C language programs. Upon completing one FEM analysis, the stress of each element is output and formulae (1)–(6) are applied to update the new Young’s modulus for each element, from which the new values of N2–N4 are obtained. These data, including the new Young’s modulus, are then automatically written into a new ANSYS input file for the next FEM analysis. As the iteration progresses, N2 and N3 will decrease gradually while N1 and N4 increase. The iteration cycle continues until the desired gray volume percentage is reached where the process stops and the optimum topology design is obtained.
Fig. 9. Result of topology optimum design (skin elements are included).
S.H. Sun / Computers in Industry 55 (2004) 147–158
Fig. 10. Result of topology optimum design (skin elements are not included).
Fig. 11. A brief design according to the result of topology optimum method.
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4. Results and discussion The proposed process resulted in 52 iterations and the gray volume percentage decreased from 92.92% in the 1st iteration to 0.96% in the 52nd iteration. The total computational time was 3462.16 s. Fig. 7 shows the gray volume percentage curve and Fig. 8 shows the number of elements N1–N4 versus the iteration number. The final values of N1–N4 are summarized in Table 2. Evident from the iteration history, N2 and N3 are reduced from 2626 and 1780, respectively, after the first iteration to 16 and 28 at the end of the 52nd iteration. The final number of gray elements is only 44, which should have no significant effect on the optimum topology design. At the end of the last iteration, only 1854 elements remained. In other words, only about 40% of the initial volume is retained. The result of topology optimization of the stationary platen is shown in Fig. 9. It is evident that most elements that are retained were from the skin while most of the deleted elements were from within
the inner side of the skin. If the skin element mesh is hidden, the geometrical model of the platen can be clearly seen in Fig. 10. As the objective of this research is to design the structure of the stationary platen with optimized stiffness at minimal raw material cost, the limited material shown in Fig. 10 must be arranged in the ‘right’ location. Excepting the elements around the five holes and skin, all other elements are placed regularly in ‘V’ shape spreading from the center hole to the corner holes as shown by the bold lines in Fig. 10. ‘V’ shape structures are chosen as they function as strengthening ribs, which increase the platen stiffness at minimal cost. Fig. 11 shows the proposed design of the stationary platen obtained from this study. To illustrate the effectiveness of the method, a photo of a commercial plastic injection machine is shown in Fig. 12. It is clear that the stationary platen in the commercial machine is almost identical to the proposed design in this design. The V-shape strengthened ribs in the commercial machines were developed
Fig. 12. A commercial plastic injection machine.
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over the years as the structure of the stationary platen evolved through trial and error to satisfy the key requirements of high stiffness and low cost. The proposed method herein, proved the effectiveness of the existing platen design through a simple and systematic scientific way.
5. Conclusions In this study, the self-organization method was successfully applied to optimize the structure of the stationary platen of a plastic injection machine. In summary, the following conclusions can be drawn. (1) The self-organization algorithm was shown to be a reliable method in obtaining an optimum topology design of a stationary platen that was strong and yet economical. It would be a useful tool in industrial applications and could be employed in future works to optimize the topology of any key components of industrial machines within the constraints of strength and cost. (2) V-shape was shown to be the best topology structure for the requirement of the stationary platen as it provided higher stiffness and lower cost. (3) The proposed stationary platen design was almost identical to that used in a commercial machine, which reaffirmed the effectiveness of the approach. Also, the proposed method herein, proved the effectiveness of the existing platen design through a simple and systematic scientific way. Acknowledgements The author gratefully acknowledges the support from NSC 89-2218-E-168-003. Thanks are also due to Mr. Hsiang-Tang Cheng for kindly help in C language coding.
References [1] R.T. Haftka, R.V. Grandhi, Structural shape optimization—a survey, Computer Methods in Applied Mechanics and Engineering 57 (1986) 91–106. [2] W. Prager, A note on discretized michell structure, Computer Methods in Applied Mechanics and Engineering 3 (3) (1974) 349–355.
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[3] G.I.N. Rozvany, M. Zhou, Continuum-based optimality criteria (COC) method: an introduction, in: Optimization of Large Structural System, NATO/DFG Advanced Study Institute, Kluwer Academic Publishers, The Netherlands, 1991. [4] M.P. Bendsfe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering 71 (2) (1988) 197–224. [5] H.P. Mlejnek, R. Schirrmacher, An engineer’s approach to optimal material distribution and shape finding, Computer Methods in Applied Mechanics and Engineering 106 (1–2) (1993) 26. [6] R.J. Young, C.H. Chuang, Optimal topology design using linear programming, Computers and Structures 52 (2) (1994) 265–275. [7] S. Burkhardt, C. Mattheck, A new method of structural shape optimization based on biological growth, International Journal of Fatigue 12 (3) (1990) 185–190. [8] H. Weinans, R. Huiskes, H.J. Grootenboer, The behavior of adaptive bone-remodeling simulation models, Journal of Biomechanics 25 (12) (1992) 1425–1441. [9] M.G. Mullender, R. Huiskes, H. Weinans, A physiological approach to the simulation of bone remodeling as a selforganizational control process, Journal of Biomechanics 27 (11) (1994) 1389–1394. [10] N. Inou, T. Uesugi, Cellular automation self-organizing the mechanical structure (generation of various types of topological structures and a comparison), Transactions of JSME (A) 61 (585) (1995) 241–246 (in Japanese). [11] N. Inou, N. Shimotai, H. Kobayashi, Cellular automation selforganizing a mechanical structure (behavior of system generated by local rules), Transactions of JSME (A) 61 (586) (1995) 272–278 (in Japanese). [12] E. Kita, T. Toyoda, T. Nomura, Structural design using cellular automation (a consideration on derivation of local rule), Transactions of JSME (A) 64 (622) (1998) 283–289 (in Japanese). [13] J. Rodriguez-Velazquez, A.A. Seireg, Optimizing the shapes of structures via a rule-based computer program, Computers in Mechanical Engineering 4 (1) (1985) 20–28. [14] M. Chirehdast, S. Sankaranarayanan, S.D. Ambo, R.P. Johanson, Validation of topology optimization for component design, AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 5th, Panama City Beach, FL, 7–9 September 1994, pp. 132–137 (AIAA Paper 94-4265). [15] G.C.A. DeRose Jr., A.R. Diaz, Hierarchical solution of largescale three-dimensional topology optimization problems, in: Proceedings of the 1996 ASME Design Engineering Technical Conference and Computers in Engineering Conference, Irvine, CA, 96-DETC/DAC-1486, August 1996. [16] A. Diaz, R. Lipton, Optimal material layout for 3D elastic structures, Structural Optimization l3 (1997) 60–64. [17] H.T. Cheng, MS dissertation, The study of topology optimization method on the frame design of vehicles, Department of Mechanical Engineering, National Cheng Kung University, Taiwan, ROC, 2000.
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S.H. Sun / Computers in Industry 55 (2004) 147–158 Shu-Huang Sun is currently an assistant professor in the Department of Mechanical Engineering at the Kun Shan University of Technology. Before being an assistant professor, he worked in plastic machines producing companies for 4 years and had been the R&D manager of Fu Chun Shin machinery manufacture
cooperation, a well known plastic injection machine producer in Taiwan. In that period, he joins many research projects, such as mechanism optimization, FEM analysis, design automation, multi-functions plastic machines developments, etc. He obtained his MS and PhD degrees from the National Cheng Kung University at 1991 and 1996. His research interests include optimization, topology design, mechanism synthesis, FEM and precision injection.
Optimum topology design for the stationary platen of a plastic injection machine Shu Huang Sun* Assistant Professor, Department of Mechanical Engineering, Kun Shan University of Technology, Yung Kang City, Tainan County 710, Taiwan, ROC Received 8 October 2002; accepted 5 July 2004 Available online 15 September 2004
Abstract Tie bars are key components of a plastic injection machine. They very easily fatigue in periodically long term operations due to the bending moment transferred to them by the bending of the stationary platen. This problem can be easily overcome by reducing the deflection of the stationary platen through topology optimization of the platen structure by applying a cost or weight constraint. In this paper, the self-organization method was introduced to optimize the topology of the stationary platen. Topology design optimizes material allocation, i.e. strengthening locations of high loads and minimizing material usage at other locations. By applying this method to the stationary platen design, the deflection of the platen could be reduced, which correspondingly reduces the bending load of the tie bars and thus extending their operating life. # 2004 Elsevier B.V. All rights reserved. Keywords: Optimum topology; Plastic injection machine; Tie bars; Stationary platen
1. Introduction 1.1. Fatigue of tie bars The clamping unit of an injection molding machine provides the motion needed for mold closing and opening, and produces the forces that are necessary to clamp the mold. It has principal components including * Tel.: + 886 6 2050496; fax: +886 6 2050509. E-mail address: [email protected].
tie bars, stationary and movable platens, and a mechanism for mold opening, closing and clamping. The stationary platen is fixed to the frame, whereas the movable platen is pushed and pulled by a driving mechanism, such as direct hydraulic, direct electric or a toggle system. Due to the high cavity pressure when molten plastic is injected into a mold, a clamping force is applied to the movable and stationary platens of the mold to prevent it from opening. The mold also gives a reaction force to these two platens at their center
0166-3615/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.compind.2004.07.001
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Fig. 1. Bending deflection of tie bars.
regions. At the same time, the stationary platen carries a force from the four nuts at its four corner regions to balance the reaction force as shown in Fig. 1. Since these two forces that are applied to the stationary platen are not co-linear, the stationary platen experiences a bending moment, which is transferred to the tie bars as a counter-reacting bending moment. Therefore, in long term operations, fatigue is inevitable. This obvious problem has been an on-going concern for injection machine manufacturers and is still an important issue for consideration when designing plastic injection machine. The traditional but inefficient method is to strengthen the tie bars by increasing their diameters. More efficiently, however, is to strengthen the stationary platen instead as fatigue experienced by the tie bars is a direct effect from the bending deflection of the stationary platen. Increasing the thickness would, of course, strengthen the stationary platen, but this translates to an increase in raw material cost, a key consideration in the design of any machines. Therefore, strength improvement must be balanced by increased in cost. In this paper, an explicit and forward idea of using the topology optimization method is proposed to determine the best topology for the stationary platen in order to reduce its deflection but not increase its weight.
1.2. Topology optimization The three main fields in engineering optimization are: optimum topology design, optimum shape design and optimum size design. This is also the step sequence in the optimization process. The first step specifies a desirable material distribution while the second step shapes the subject to avoid stress concentration regions. Detail dimensions are then given in the third step. Optimum topology design methods have been well developed in recent years and can be divided into two categories, viz., the layout theory method and the design domain method. A few references are available for the layout theory method. Haftka and Grandhi [1] introduced the fully stressed design concept. Prager [2] proposed an optimality criterion to solve layout problems. Rozvany and Zhou [3] extended this criterion and proposed a continuum-based optimality criterion to solve the general layout problem. The design domain method can be sub-divided again into three categories, the homogenization method, the density function method, and the simulating biological growth method. Bendsfe and Kikuchi [4] applied the homogenization theory to find the optimum topology. Mlejnek and Schirrmacher [5]
S.H. Sun / Computers in Industry 55 (2004) 147–158
proposed an energy approach and treated the density of each element as the design variable. Young and Chuang [6] modified this approach by sequential linear programming and sensitivity analysis. Both of the above methods, however, involved complex theory and calculations. A means of avoiding these complications is to simulate the biological growth for an optimum topology. Burkhardt and Mattheck [7] proposed a soft kill option to simulate bone mineralization. Weinans et al. [8] and Huiskes and co-workers [9] proposed the simulated bone remodeling method to find the real density distribution in human leg bone. Inou et al. [10,11] proposed a selforganization method simulating human cell growth behavior to find an optimum topology design. In this algorithm, the Young’s modulus of each element was treated as a design variable and adjusted in each finite element method (FEM) analysis. Kita et al. [12] extended the self-organization method and improved it by adding local rules and penalty factors. Although many references discussed the optimum topology design method, few papers [13–16], however, dealt with 3-D applications.
149
In this study, the self-organization method was modified and applied to determine the optimum topology design of the stationary platen of a plastic injection machine.
2. Algorithm of the self-organization method The concept of the self-organization method [10,11] is to modify the Young’s modulus of each element according to the ratio of its stress and the average stress of the entire model after each FEM analysis. In the first iteration, all elements are given the same initial Young’s modulus value. After the first iteration, all elements are reassigned Young’s modulus values. Elements with high stress values mean that they were subjected to high loads and thus must retain high Young’s modulus values while low stressed elements are assigned low Young’s modulus values. As the iteration continues, the differences in the Young’s modulii of the elements gradually increase. At the condition when the Young’s modulii of certain elements falls below a pre-set value, the elements are
Fig. 2. Flowchart of self-organization method.
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S.H. Sun / Computers in Industry 55 (2004) 147–158
deleted and the region would become a void. This process continues until a stop criterion is satisfied and the optimum topology design is obtained. Fig. 2 shows the process flowchart and the detailed procedure of the algorithm is discussed below. (1) The design domain is assigned and an initial FEM model is generated, filling up the domain with elements. (2) All elements are then assigned an arbitrary initial Young’s modulus value E0. (3) Using FEM analysis, the von Mises stress of each element is obtained. (4) The desired von Mises stress sC of each element is then determined by the following formulae: s C ¼ s C0 f1 þ b½eaðEE0 =E0 Þ 1g 2 a ¼ a1 and b ¼ ð2a Þ 1 e 1 when E E0
(1) (2)
1 (3) 1 ea2 when E < E0.Here sC0 is the average von Mises stress value of the entire model. The assignment of the parameters a and b is described in detail as follows. In the self-organization algorithm, the decision as to which an element would be retained or deleted is entirely based on its Young’s modulus a ¼ a2
and
b¼
value. The condition is set such that the elements with high Young’s modulus would be retained while those with low Young’s modulus would be deleted. The in-between elements are known as ‘gray’ elements, which should be minimized as they often lead to indecisions. The two parameters a1 and a2 in formulae (2) and (3) must be assigned properly for this reason. As shown in Fig. 3, Inou and Uesugi [10] proved that when a1 = 7.5 and a2 = 10 in the formulae (1)– (3), the obtained value of sC has the optimum effects in avoiding the generation of ‘gray’ elements. (5) The Young’s modulus of each element is subsequently updated after each iteration by the following formula: s ðtþ1Þ ðtÞ E ¼E 1þa 1 (4) sC where E(t+1) and E(t) are the Young’s modulus of each element in the (t+1)th and (t) th FEM analysis and is a constant used to control the step size in each Young’s modulus update. In this study, it is set to 0.5. In the iteration process, extreme Young’s modulus values may cause instability in the FEM analysis. To counter this, two rules are introduced to restrict the ‘growth’ or ‘decline’ of the Young’s modulus and they are [17] If E > 3E0 ;
then
If E > 0:3E0 ;
then
E ¼ Emax ¼ 3E0 E ¼ Emin ¼ 107 E0
(5) (6)
(6) Elements are grouped into four sets according to their Young’s modulus values as shown in Table 1. In the table, N1, N2, N3, and N4 are respectively, the number of elements whose Young’s modulus are larger than 200 GPa, between 100 and 200 GPa, between 30 and 100 GPa and under
Table 1 Elements are grouped into four sets according to their Young’s modulus values
Fig. 3. Mapping curve of sC and E.
Young’s modulus value
Number of elements
200 E 100 E < 200 30 E < 100 E < 30
N1 N2 N3 N4
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151
(7) The elements in N2 and N3 are the ‘gray’ elements described in the previous section. One of the main objectives of this algorithm is to minimize the number of such elements. The method is to force the elements to ‘jump’ from N2 and N3 to either N1 or N4 such that the optimum topology can be obtained by finally retaining the elements in N1 and deleting the elements in N4. Therefore, the number of residual elements in N2 and N3 can be used as a stop criteria. A parameter b called gray volume percentage and defined as the ratio of the sum of N2, N3 and Nall is used as the stop criteria in this study. It is b¼
Fig. 4. A rough structure of the stationary platen.
30 GPa. The decision is then to delete the elements in N4, resulting in a residual volume of a new topology that is different from the initial one.
ðN2 þ N3 Þ Nall
(7)
If the gray volume percentage is smaller than the desired value, the algorithm stops and the final optimum topology is obtained. Otherwise, steps (3)–(6) are repeated with the new FEM model. (8) In the whole iteration history, the gray volume percentage curve will decrease rapidly at the beginning and then saturates. A desired value can be specified as the stop criteria. As described in step (7), once the curve converges to a value under the desired value, the process ceases. However, if the chosen value falls below the saturation value, a maximum iteration number can be assigned to
Fig. 5. The initial FEM model.
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Fig. 6. FEM model including boundary conditions and loading situations in this study.
Fig. 7. The curves of gray volume percentage.
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Fig. 8. The curves of number of elements N1–N4 vs. iteration number.
suspend the process while the iteration curve is examined. From the trend of the curve, a decision can easily be made as to whether to continue or terminate the process. When the curve has reached saturation, the eventual value is of no significance as the number of gray elements has been minimized and their consequent effect on the final topology design is minimal.
3. Application of the self-organization method to the stationary platen The stationary platen of a plastic injection machine is usually a large cast-iron-made block. The mold is fixed on its left side plane and the injection barrel approaches from the right and moves to the center to make contact with the mold as illustrated in Fig. 1. Four tie bars secure the four corners of the platen with screws so as to build up the clamping force. The initial design domain in this optimization study is defined according to these descriptions and shown in Fig. 4. It is a solid block with one large circular hole in its
center, which functions as an entry for the injection barrel, and four small holes at its four corners for securing the tie bars. The platen is attached to the machine frame by means of two small ‘legs’ at the left and right bottom corners. A commercial machine FT90 produced by Fu Chun Shin, a famous plastic injection machine manufacture in Taiwan, has a typical platen size of 520 mm wide, 520 mm high and 180 mm thick. This machine is a 90 tons injection machine, i.e. the loading in the center region by the mold and the four corners by the nuts are both 90 tons. For simplification, the two securing ‘legs’ are considered as small features and therefore neglected in the FEM analysis. As the platen is symmetrical in the X and Y directions, one quarter of the volume would suffice for the FEM analysis. The volume that is bounded by the bold line in Fig. 4 is modeled in the FEM analysis. The elements are first generated to fill the design space as shown in Fig. 5 for the initial FEM analysis. Owing to symmetry, translation in the X direction of the left surface and the Y direction of the bottom surface are both restricted. These constraints
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Table 2 The final values of N1–N4 N1 N2 N3 N4
1810 16 28 2706
are represented by triangle pyramids at the left and bottom surface of the block as shown in Fig. 6. In the injection process, the molten plastic is injected into the mold, thereby producing high cavity pressure that has a tendency to push open the mold. This force is transferred from the mold to the center of the stationary platen where it spreads the loading on a projection area of the mold. In actual operating environment, the area is a 286 mm square not including the center hole. Four nuts lock the four tie bars to the four corners of the platen to provide a circular region of close contact between the platen and the nuts. The nuts also provide the platen with reaction force, which is equal to the loading from mold but opposite in direction to prevent the platen from moving. The loading situation of this model therefore results in a total of 90 tons distribution load in the center square region of the back surface of this platen and restricts the Z direction translation in the circular
region around the four corner holes at the front surface of the platen. The loading is represented by arrows and the translation constraint in Z direction is shown by triangle pyramids around the circular region as shown in Fig. 6. ANSYS 5.5 was used to perform the FEM analysis and computation was carried out with a Pentium III800 PC computer. There are a total of 4560 elements and 5824 nodes in the initial FEM model as shown in Fig. 5. The desired gray volume percentage is set to 1%. The initial Young’s modulus is set to 100 GPa for all elements and the initial value of N2 equals to the number of total elements and N1, N3, N4 are all zeros. An automatic iteration process was performed by subroutines written in C language programs. Upon completing one FEM analysis, the stress of each element is output and formulae (1)–(6) are applied to update the new Young’s modulus for each element, from which the new values of N2–N4 are obtained. These data, including the new Young’s modulus, are then automatically written into a new ANSYS input file for the next FEM analysis. As the iteration progresses, N2 and N3 will decrease gradually while N1 and N4 increase. The iteration cycle continues until the desired gray volume percentage is reached where the process stops and the optimum topology design is obtained.
Fig. 9. Result of topology optimum design (skin elements are included).
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Fig. 10. Result of topology optimum design (skin elements are not included).
Fig. 11. A brief design according to the result of topology optimum method.
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4. Results and discussion The proposed process resulted in 52 iterations and the gray volume percentage decreased from 92.92% in the 1st iteration to 0.96% in the 52nd iteration. The total computational time was 3462.16 s. Fig. 7 shows the gray volume percentage curve and Fig. 8 shows the number of elements N1–N4 versus the iteration number. The final values of N1–N4 are summarized in Table 2. Evident from the iteration history, N2 and N3 are reduced from 2626 and 1780, respectively, after the first iteration to 16 and 28 at the end of the 52nd iteration. The final number of gray elements is only 44, which should have no significant effect on the optimum topology design. At the end of the last iteration, only 1854 elements remained. In other words, only about 40% of the initial volume is retained. The result of topology optimization of the stationary platen is shown in Fig. 9. It is evident that most elements that are retained were from the skin while most of the deleted elements were from within
the inner side of the skin. If the skin element mesh is hidden, the geometrical model of the platen can be clearly seen in Fig. 10. As the objective of this research is to design the structure of the stationary platen with optimized stiffness at minimal raw material cost, the limited material shown in Fig. 10 must be arranged in the ‘right’ location. Excepting the elements around the five holes and skin, all other elements are placed regularly in ‘V’ shape spreading from the center hole to the corner holes as shown by the bold lines in Fig. 10. ‘V’ shape structures are chosen as they function as strengthening ribs, which increase the platen stiffness at minimal cost. Fig. 11 shows the proposed design of the stationary platen obtained from this study. To illustrate the effectiveness of the method, a photo of a commercial plastic injection machine is shown in Fig. 12. It is clear that the stationary platen in the commercial machine is almost identical to the proposed design in this design. The V-shape strengthened ribs in the commercial machines were developed
Fig. 12. A commercial plastic injection machine.
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over the years as the structure of the stationary platen evolved through trial and error to satisfy the key requirements of high stiffness and low cost. The proposed method herein, proved the effectiveness of the existing platen design through a simple and systematic scientific way.
5. Conclusions In this study, the self-organization method was successfully applied to optimize the structure of the stationary platen of a plastic injection machine. In summary, the following conclusions can be drawn. (1) The self-organization algorithm was shown to be a reliable method in obtaining an optimum topology design of a stationary platen that was strong and yet economical. It would be a useful tool in industrial applications and could be employed in future works to optimize the topology of any key components of industrial machines within the constraints of strength and cost. (2) V-shape was shown to be the best topology structure for the requirement of the stationary platen as it provided higher stiffness and lower cost. (3) The proposed stationary platen design was almost identical to that used in a commercial machine, which reaffirmed the effectiveness of the approach. Also, the proposed method herein, proved the effectiveness of the existing platen design through a simple and systematic scientific way. Acknowledgements The author gratefully acknowledges the support from NSC 89-2218-E-168-003. Thanks are also due to Mr. Hsiang-Tang Cheng for kindly help in C language coding.
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S.H. Sun / Computers in Industry 55 (2004) 147–158 Shu-Huang Sun is currently an assistant professor in the Department of Mechanical Engineering at the Kun Shan University of Technology. Before being an assistant professor, he worked in plastic machines producing companies for 4 years and had been the R&D manager of Fu Chun Shin machinery manufacture
cooperation, a well known plastic injection machine producer in Taiwan. In that period, he joins many research projects, such as mechanism optimization, FEM analysis, design automation, multi-functions plastic machines developments, etc. He obtained his MS and PhD degrees from the National Cheng Kung University at 1991 and 1996. His research interests include optimization, topology design, mechanism synthesis, FEM and precision injection.