FINITE ELEMENT METHOD IN COOLING ANALYSIS AND DESIGN OF PLASTIC INJECTION MOULDS SUN YIFENG NATIONAL UNIVERSITY OF SINGAPORE 2003 Founded 1905 FINITE ELEMENT METHOD IN COOLING ANALYSIS AND DESIGN OF PLASTIC INJECTION MOULDS BY SUN YIFENG (B Eng., M Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements ACKNOWLEDGEMENTS First of all, I wish to express my sincere gratitude to my supervisors, Professor Andrew Nee Yeh-Ching and Associate Professor Lee Kim Seng, for their invaluable advice and indispensable guidance throughout the course of this research The breadth and depth of their knowledge in many fields are the key factors in cultivating a conducive environment for me They have been generous with their time and discussions in providing insights and directions that have helped the research and myself in reaching a higher level Their constant enthusiasm and kindness will always be gratefully remembered I would also like to thank Mr Ku Ching Chap, Senior Engineer, Lek Hung moulding Pte Ltd, for advice related to high speed machining and mould-making Thanks also go to Mr Tan Cher Hwee, School of Mechanical & Manufacturing, Singapore Polytechnic, for providing the experimental facilities for this project Special thanks to Dr Jason Wang Huijun, Worley Singapore Pte Ltd, for his help on using ABAQUS Thanks also go to Mr Liew Choan Ann, MSC Software Singapore, for providing useful information on MSC software and CFDesign I would also like to express my gratitude to Associate Professor Wong Yoke San, Associate Professor Jerry Fuh Ying Hsi, and Associate Professor Zhang Yunfeng for their critical suggestions about this project I am also grateful to Computer Centre, NUS, especially the Supercomputing & Visualisation Unit (SVU) for sponsoring useful seminars and providing the I Acknowledgements supercomputers and high-end software that are necessary for the research Thanks go to Mr Yeo Eng Hee, Mr Zhang Xinhuai and Ms Gao Zhihong for their technical supports on using software and hardware in SVU, also to Centre for Applications and IT (CAIT) and the staff, Mr Zhang Lihai and Mr Kong Kian Chay I would also like to express my appreciation to Dr Ye Xiangao, Mr Wang Zheng, Dr Zhang Hua, Dr Liu Xilin, and Mr Wang Ying for their technical expertise Many thanks to my colleagues, Dr Ding Xiaoming, Ms Guo Huaqun, Dr Mohammad Rabiul Alam, Mr Luo Cheng, Mr Wu Shenghui, Mr Xin Yongchun, Mr Gan Pay Yap, Miss Du Xiaojun, Mr Woon Yong Khai, Miss Maria Low Leng Hwa, Ms Cao Jian, Mr Atiqur Rahman, and Mr Saravanakumar Mohanraj, for creating a warm community that made my study in NUS an enjoyable and memorable one I am grateful to the National University of Singapore for providing me a chance to pursue my research work and financing me with a research scholarship to support my studies I wish to thank my parents and in-laws for their moral support Finally, I sincerely thank my wife, Ms Yuan Ping, for her support all the time This thesis is dedicated to her and our son, Sun Ruiqian II Table of contents TABLE OF CONTENTS ACKNOWLEDGEMENTS I TABLE OF CONTENTS III NOMENCLATURE IX LIST OF FIGURES XVI LIST OF TABLES XIX SUMMARY XX CHAPTER INTRODUCTION 1.1 HEAT TRANSFER WITHIN INJECTION MOULDS ············································· 1.2 BACKGROUND OF MOULD COOLING ··························································· 1.2.1 Affecting Factors ··················································································· 1.2.2 Significance of Mould Cooling······························································ 1.2.3 Cooling Methods ··················································································· 1.2.4 Cooling System Design in the Mould Industry ······································ 1.3 CAD/CAM IN MOULD COOLING ANALYSIS AND DESIGN ··························· 1.4 RESEARCH OBJECTIVES ·············································································· 1.5 ORGANIZATION OF THE THESIS ··································································· CHAPTER 2.1 LITERATURE REVIEW 10 THE MATHEMATICAL SOLUTIONS ····························································· 10 2.1.1 Analytical and Numerical Methods ····················································· 11 2.1.2 The Finite Difference and Finite Volume Methods······························ 11 2.1.3 The Finite Element Method ································································· 12 III Table of contents 2.1.4 The Boundary Element Method ··························································· 13 2.1.5 Discussions ·························································································· 15 2.2 REVIEWS ON MOULD COOLING ANALYSIS AND DESIGN ···························· 16 2.2.1 Modelling and Assumptions ································································ 17 2.2.2 Mould Cooling Analysis ······································································ 18 2.2.3 Mould Cooling Design and Optimisation ············································ 20 2.2.4 Discussions ·························································································· 22 2.3 REVIEWS ON THERMAL RESIDUAL STRESS ANALYSIS OF PARTS ··············· 23 2.4 REVIEWS ON MATRIX COMPUTATION ······················································· 25 2.5 REVIEWS ON THE CURVE/SURFACE OFFSET ·············································· 28 CHAPTER 3.1 HEAT TRANSFER MODELLING 31 FACTORS AFFECTING COOLING OF INJECTION MOULDS ···························· 31 3.1.1 Temperature Differences ····································································· 31 3.1.2 Material Thermal Properties ································································ 32 3.1.3 Coolant Flow ······················································································· 33 3.1.4 Cooling Channels Layout ···································································· 34 3.2 HEAT CONDUCTION EQUATION ································································· 35 3.3 INITIAL AND BOUNDARY CONDITIONS ······················································ 37 3.3.1 Initial Conditions ················································································· 38 3.3.2 Boundary Conditions ··········································································· 38 3.4 CONVECTIVE HEAT TRANSFER COEFFICIENT ············································ 40 3.5 CYCLE TIME CALCULATION ······································································ 41 3.5.1 1-D Analytical Formula ······································································· 42 3.5.2 Other Formulas ···················································································· 43 CHAPTER 4.1 FEM IN HEAT TRANSFER ANALYSIS 46 FEM RELATED FUNDAMENTALS ······························································ 46 IV Table of contents 4.1.1 The Method of Weighted Residuals····················································· 46 4.1.2 Bubnov-Galerkin Method ···································································· 47 4.1.3 Integration by Parts·············································································· 48 4.2 INTERPOLATION FUNCTIONS ····································································· 49 4.2.1 Approximations of the Temperature ···················································· 49 4.2.2 Selection of Interpolation Functions ···················································· 50 4.2.3 Interpolation Functions of the Tet-element ·········································· 51 4.3 DERIVING THE ELEMENT EQUATIONS ······················································· 52 4.4 SOLVING THE TIME-DEPENDENT EQUATIONS ············································ 54 4.4.1 Recurrence Method·············································································· 55 4.4.2 Various θ -Methods ············································································· 56 4.4.3 Implicit and Explicit Algorithms ························································· 58 4.4.4 Lumped versus Consistent Mass Methods ··········································· 59 4.5 ASSEMBLING SYSTEM EQUATIONS ···························································· 60 4.6 SOLVING THE MATRIX EQUATION ····························································· 62 CHAPTER 5.1 FEM IN THERMAL STRESS ANALYSIS 64 LINEAR ELASTICITY THEORY ···································································· 64 5.1.1 Stress and Strain ·················································································· 64 5.1.2 Constitutive Equations from Hooke’s Law ·········································· 65 5.1.3 Static Equilibrium Equations ······························································· 66 5.1.4 Thermal Effects ··················································································· 68 5.2 ASSUMPTIONS, INITIAL AND BOUNDARY CONDITIONS ······························ 68 5.3 DERIVING THE ELEMENT EQUATIONS ······················································· 70 5.3.1 Approximating the Displacement ························································ 70 5.3.2 Applying the Galerkin Method ···························································· 71 5.4 ASSEMBLING SYSTEM EQUATIONS ···························································· 72 5.5 SOLVING THE SYSTEM EQUATIONS ··························································· 73 V Table of contents CHAPTER MILLED GROOVE METHODS 74 6.1 POPULAR COOLING METHODS ·································································· 74 6.2 THE UMG AND MGI METHODS ······························································· 76 6.2.1 The UMG Method ··············································································· 77 6.2.2 The MGI Method················································································· 79 6.3 AUTO-DESIGN OF THE UMG AND MGI METHODS ···································· 80 6.4 DISCUSSIONS ON THE UMG AND MGI METHODS ····································· 82 6.4.1 Comparison between the UMG/MGI and the RP Methods ·················· 82 6.4.2 Comparison between the UMG/MGI and the SDCC Methods············· 82 6.4.3 Pros and Cons of the UMG and MGI Methods ···································· 83 CHAPTER 7.1 AUTO-DESIGN AND OPTIMISATION 85 NURBS AND GEOMETRIC FUNDAMENTALS ·············································· 85 7.1.1 Definition and Properties of NURBS··················································· 85 7.1.2 Definitions of Geometric Properties ···················································· 88 7.1.3 Derivatives of Offset Curve/Surface ···················································· 90 7.2 CURVATURE PROPERTIES OF NURBS CURVES AND SURFACES ················· 90 7.2.1 THEOREM on Offsetting NURBS Curve ········································ 91 7.2.2 THEOREM on Offsetting NURBS Surface ······································ 93 7.3 MODIFICATION FOR NON-SELF-INTERSECTING OFFSET ····························· 95 7.3.1 Examining the Curvature ····································································· 96 7.3.2 Modifying Curvature of Knot Points ··················································· 98 7.3.3 Knot Insertion ···················································································· 100 7.3.4 The Modification Algorithm······························································ 102 7.4 EXAMPLE OF OFFSETTING SINGLE CURVE AND SURFACE ························ 103 7.5 MULTIPLE SURFACES OFFSET ································································· 108 7.6 COOLING OPTIMISATION ········································································· 109 VI Table of contents CHAPTER 8.1 CASE STUDIES 113 COOLING ANALYSIS AND COMPARISON ·················································· 114 8.1.1 Moulding Conditions ········································································· 114 8.1.2 Temperature Distributions ································································· 119 8.1.3 Temperature Comparison··································································· 123 8.1.4 Cycle Time ························································································ 126 8.2 COOLING ANALYSIS OF MOULD WITH HOT RUNNER ······························· 126 8.2.1 Moulding Conditions ········································································· 127 8.2.2 Temperature Distribution··································································· 130 8.3 COOLING AND THERMAL STRESS ANALYSES ·········································· 134 8.3.1 Conditions Setting for the Analysis ··················································· 135 8.3.2 Temperature Distributions ································································· 141 8.3.3 Cycle time and flow rate ···································································· 148 8.3.4 Thermal stress and strain ··································································· 149 CHAPTER 9.1 CONCLUSIONS AND RECOMMENDATIONS 152 CONCLUSIONS ························································································ 152 9.1.1 Cooling and Thermal Stress Analysis ················································ 153 9.1.2 The UGM and MGI Methods····························································· 154 9.1.3 Auto-Design and Optimisation of Mould Cooling System················· 155 9.2 RECOMMENDATIONS FOR FUTURE WORK ··············································· 155 PUBLICATIONS RELATED TO THIS THESIS 158 REFERENCES 159 APPENDIX A MATRICES AND VECTORS 169 A.1 CONVENTIONS ························································································ 169 A.2 MATRIX TRANSPOSE ··············································································· 169 A.3 QUADRATIC FORMS ················································································ 169 VII Table of contents A.4 MATRIX INVERSE···················································································· 170 A.5 DIFFERENTIATION OF A MATRIX ····························································· 171 A.6 INTEGRATION OF A MATRIX ···································································· 171 A.7 DIFFERENTIATION OF A QUADRATIC FUNCTION ······································ 171 A.8 DIFFERENTIAL FORMULATION OF VECTOR ·············································· 172 VIII References REFERENCES [Beer 2001] Beer, G., Programming the boundary element method: an introduction for engineers, John Wiley & Sons, 2001 [Bonnet 1995] Bonnet, M., Boundary integral equation methods for solids and fluids, John Wiley & Sons, 1995 [Busch 1988] Busch, J V., Field, F R., and Rosato, D., “Computer-aided part cost analysis for injection molding”, SPE ANTEC, 1988 [Bushko 1995] Bushko, W C and Stokes, V K., “Solidification of Thermoviscoelastic Melts Part II: Effect of Processing Conditions on Shrinkage and Residual Stress”, Polymer Engineering and Science, 36, pp.365-383, 1995 [Chang 1995a] Chang, R Y and Tsaur, B., D., “Experimental and theoretical studies of shrinkage warpage and sink marks of crystalline polymer injection moulded parts”, Polymer Engineering and Science, 35, pp.1222-1230, 1995 [Chang 1995b] Chang, R Y., and Chiou, S Y., “A unified K-BKZ model for residual stress analysis of injection moulded three-dimensional thin shapes”, Polymer Engineering and Science, 35, pp.1733-1745, 1995 [Chen 1990] Chen, S C., Wang, S M., Chang, Y L and Wang, C H., “A study of computer-aided mold cooling simulations based on different methods”, SPE ANTEC, pp359-364, 1990 [Chen 1991a] Chen, S C., Hu, S Y and Davidoff, “Hybrid method for injection mold 159 References cooling process simulation and mold cooling system analysis”, SPE ANTEC, pp 499503, 1991 [Chen 1991b] Chen, S C and Hu, S Y., “Simulations of cycle-averaged mold surface temperature in mold-cooling process by boundary element method”, International Communications in Heat and Mass Transfer, 18, pp.823-832, 1991 [Chen 1992] Chen, S C and Chung, Y C., “Simulations of the cyclic transient mold cavity surface temperature in injection mold-cooling process”, International Communications in Heat and Mass Transfer, 19, pp.559-568, 1992 [Chen 1994] Chen, S C and Chung, Y C., “Numerical simulations of the cyclic, transient mold heat transfer in injection mold cooling process”, International Communications in Heat and Mass Transfer, 21, pp.323-332, 1994 [Chen 2000] Chen, X., Lam, Y C and Li, D Q., “Analysis of thermal residual stress in plastic injection molding”, Journal of Materials Processing Technology, 101, pp.275-280, 2000 [Chiang 1993] Chiang, H H., Himasekhar, K., Santhanam, N and Wang, K K., “Integrated simulation of fluid flow and heat transfer in injection molding for the prediction of shrinkage and warpage”, Journal of Engineering Materials and Technology, 115, pp.37-47, 1993 [Choi 1999] Choi, D S and Im, Y T., “Prediction of shrinkage and warpage in consideration of residual stress in integrated simulation of injection molding”, Composite Structures, 47, pp.655-665, 1999 [Dalgarno 2001] Dalgarno, K W and Stewart, T D., “Manufacture of production 160 References injection mould tooling incorporating conformal cooling channels via indirect selective laser sintering”, Proceedings of the Institution of Mechanical Engineers, Part B (Journal of Engineering Manufacture), 215, pp.1323-1332, 2001 [Elber 1991] Elber, G and Cohen, E., “Error bounded variable distance offset operator for free form curves and surfaces”, International Journal of Computational Geometry and Applications, 1, pp.67-78, 1991 [Farouki, 1985] Farouki, R T., “Exact offset procedures for simple solids”, Computer Aided Geometric Design, 2, pp.257-279, 1985 [Farouki, 1986] Farouki, R T., “The approximation of non-degenerate offset surfaces”, Computer Aided Geometric Design, 3, pp.15-43, 1986 [Farouki, 1990a] Farouki, R T and Neff, C., “Analytical properties of plane offset curves”, Computer Aided Geometric Design, 7, pp.83-100, 1990 [Farouki, 1990b] Farouki, R T and Neff, C., “Algebraic properties of plane offset curves”, Computer Aided Geometric Design, 7, pp.101-127, 1990 [Filip, 1986] Filip, D., Magedson, R and Markot, R “Surface algorithms using bounds on derivatives”, Computer Aided Geometric Design, 3, pp.295-311, 1986 [Hastenberg 1992] Hastenberg, C H V., Wildervanck, P C and Leanen A J H., “The measurement of thermal stress distribution along the flow path in injectionmolded flat plates”, Polymer Engineering and Science, 32, pp.506-515, 1992 [Heinrich, 1999] Heinrich, J C and Pepper, D W., Intermediate finite element method: fluid flow and heat transfer applications, Taylor & Francis, 1999 161 References [Himasekhar 1989a] Himasekhar, K., Hieber, C A and Wang, K K., “Computeraided design software for cooling system in injection molding”, SPE ANTEC, pp 352355, 1989 [Himasekhar 1989b] Himasekhar, K., Wang, K K and Lottey, J., “Mold-cooling simulation in injection molding of three-dimensional thin plastic parts”, in Numerical Heat Transfer with Personal Computers and Supercomputing, Shah, R K (editor), Heat Transfer Division, 110, ASME, pp 129-136, 1989 [Himasekhar 1992] Himasekhar, K., Lottey, J and Wang, K K., “CAE of mold cooling in injection molding using a three-dimensional numerical simulation”, Journal of Engineering for Industry, 114, pp213-221,1992 [HKS 2002] HKS Inc., ABAQUS/Standard User’s manual, HKS Inc, 2002 [Hopkinson 2000] Hopkinson, N and Dickens, P., “Predicting stereolithography injection mould tool behaviour using models to predict ejection force and tool strength” International Journal of Production Research, 38, pp.3747-3757, 2000 [Hu 1995] Hu, S Y., Cheng, N T and Chen, S C., “Effect of cooling system design and process parameters on cyclic variation of mold temperatures – simulation by DRBEM”, Plastics, Rubber and Composites Processing and Applications, 23, pp.221232, 1995 [Huebner 2001] Huebner, K H., Dewhirst, D L., Smith, D E and Byrom, T G., The Finite Element Method for Engineers, 4th ed., John Wiley & Sons, 2001 [Irons 1979] Irons, B M and Ahmad, S., Techniques of Finite Elements, Ellis Horwood, 1979 162 References [Jacques 1982] Jacques, M., “An analysis of warpage in injection molded flat parts due to unbalanced cooling”, Polymer Engineering and Science, 22, pp 241-247, 1982 [Jansen 1996] Jansen, K M B and Titomanlio, G., “Effect of pressure history on shrinkage and residual stresses-injection molding with constrained shrinkage”, Polymer Engineering and Science, 36, pp.2029-2040, 1996 [Jansen 1998a] Jansen, K M B., Pantani, R and Titomanlio, G., “As-molded shrinkage measurements on polystyrene injection molded products”, Polymer Engineering and Science, 38, pp.254-264, 1998 [Jansen 1998b] Jansen, K M B., Dijk, D J V and Husselman, M H., “Effect of processing conditions on shrinkage in injection molding”, Polymer Engineering and Science, 38, pp.838-846, 1998 [Jeppson 1976] Jeppson, R W., Analysis of flow in pipe networks, Butterworth Publishers, 1976 [Kabanemi 1992] Kabanemi, K K and Crochet, M J., “Thermoviscoelastic calculation of residual stresses and residual shape of injection molded parts”, International Polymer Processing,7, pp.60-70, 1992 [Kabanemi 1998] Kabanemi, K K, Vallancourt, H., Wang, H and Salloum, G., “Residual stresses, shrinkage, and warpage of complex injection molded products: numerical simulation and experimental validation”, Polymer Engineering and Science, 38, pp.21-37, 1998 [Kansal 2000] Kansal, G., Atreya, S K and Rao, P N., “Thermal and stress analysis of an injection-molded polystyrene ring”, Polymer-Plastics Technology and 163 References Engineering, 39, pp.61-81, 2000 [Kansal 2001] Kansal, G., Rao, P N and Atreya, S K., “Study: Temperature and residual stress in an injection moulded gear”, Journal of Materials Processing Technology, 108, pp.328-337, 2001 [Li 2001] Li, C L., “A feature-based approach to injection mold cooling system design”, Computer-Aided Design, 33, pp.1073-1090, 2001 [Lin 2002] Lin, J C., “Optimum cooling system design of a free-form injection mold using an abductive network”, Journal of Materials Processing Technology, 120, pp.226-236, 2002 [Liu 1996] Liu S J., “Modeling and simulation of thermally induced stress and warpage in injection molded thermoplastics”, Polymer Engineering and Science, 36, pp.807-818, 1996 [Maekawa 1993] Maekawa, T and Patrikalakis, N M., “Computation of singularities and intersections of offsets of planar curves”, Computer Aided Geometric Design, 10, pp.407-429, 1993 [Maekawa 1997] Maekawa, T., Cho, W., and Patrikalakis, N M., “Computation of self-intersections of offsets of Bezier surface patches”, Journal of Mechanical Design, ASME Transactions, 119, pp.275-283, 1997 [Maekawa 1999] Maekawa, T., “An overview of offset curves and surfaces”, Computer-Aided Design, 31, pp.165-173, 1999 [Maekawa 2001] Maekawa, T and Patrikalakis, N M., Shape interrogation for computer aided design and manufacturing, Springer, 2001 164 References [Martin 1990] Martin, R R and Stephenson, P C., “Sweeping of three dimensional objects”, Computer-Aided Design, 22, pp.223-234, 1990 [Matsumoto 1993] Matsumoto, T., Tanaka, M., Miyagawa, M and Ishii, N., “Optimum design of cooling lines in injection molds by using boundary element design sensitivity analysis”, Finite Elements in Analysis and Design, 14, pp.177-185, 1993 [Matsuoka 1991] Matsuoka, T., Takabatake, J., Koiwai, A., Inoue, Y., Yamamoto, S and Takahashi, H., “Integrated simulation to predict warpage of injection molded parts”, Polymer Engineering and Science, 31, pp.1043-1050, 1991 [Menges 1993] Menges, G and Mohren, P., How to make injection molds, Hanser Publishers, 1993 [Moldex3D 2002] Moldex3D, Coretech Corporation, 2002 [Moldflow 2002] Moldflow Plastic Insight, Moldfolw Corporation, 2002 [Park 1998] Park, S J and Kwon, T H., “Thermal and design sensitivity analyses for cooling system of injection mold, Part 1: Thermal Analysis; Part 2: Design Sensitivity Analysis”, Journal of Manufacturing Science and Engineering, 120, pp.287-305, 1998 [Pham 1992] Pham, B., “Offset curves and surfaces: a brief survey”, Computer-Aided Design, 24, pp.223-229, 1992 [Piegl 1997] Piegl, L A and Tiller, W., The NURBS book, 2nd ed., Springer, 1997 [Piegl 1999] Piegl, L A and Tiller, W., “Computing offsets of NURBS curves and surfaces”, Computer-Aided Design, 31, pp.147-156, 1999 165 References [Pottmann 1995] Pottmann, H., “Rational curves and surfaces with rational offsets”, Computer Aided Geometric Design, 12, pp.175-192, 1995 [Raamachandran 2000] Raamachandran, J., Boundary and finite elements theory and problems, CRC Press, 2000 [Rao 2000] Rao, N S., Schumacher, G., Schott, N R., and O’Brien, K T., “Optimization of cooling systems in injection molds by an easily applicable analytical model”, SPE ANTEC, pp.702-706, 2000 [Ravi Kumar 2002] Ravi Kumar, G V V., Shastry, K G., and Prakash, B G., “Computing non-self-intersecting offsets of NURBS surfaces”, Computer-Aided Design, 34, pp.209-228, 2002 [Ravi Kumar 2003a] Ravi Kumar, G V V., Shastry, K G., and Prakash, B G., “Computing offsets of trimmed NURBS surfaces”, Computer-Aided Design, 35, pp.411-420, 2003 [Ravi Kumar 2003b] Ravi Kumar, G V V., Shastry, K G., and Prakash, B G., “Computing constant offsets of a NURBS B-Rep”, Computer-Aided Design, 35, pp.935-944, 2003 [Rezayat 1990] Rezayat, M and Burton, T., “Boundary-integral formulation for complex three-dimensional geometries”, International Journal for Numerical Methods in Engineering, 29, pp.263-273, 1990 [Rhinoceros 2002] Rhinoceros, Evaluation version 2.0, Robert McNeel & Associates, 2002 [Sachs 2000] Sachs, E., Wylonis, E., Allen, S., Cima, M and Guo, H., “Production of 166 References injection molding tooling with conformal cooling channels using the three dimensional printing processing”, Polymer Engineering and Science, 40, pp.1232-1247, 2000 [Tang 1996a] Tang, L Q., Pochiraju, K., Chassapis, C and Manoochehri, S., “Threedimensional transient mold cooling analysis based on Galerkin finite element formulation with a matrix-free conjugate gradient technique”, International Journal for Numerical Methods in Engineering, 39, pp.3049-3064, 1996 [Tang 1996b] Tang, L Q., Pochiraju, K., Chassapis, C and Manoochehri, S., “Optimal design of cooling system for injection molding”, SPE ANTEC, pp.714-718, 1996 [Tang 1997] Tang, L Q., Pochiraju, K and Manoochehri, S., “Optimal cooling system design for multi-cavity injection molding”, Finite Element in Analysis and Design, 26, pp.229-251, 1997 [Thompson 1984] Thompson, M and White, J R., “The effect of a temperature gradient on residual stresses and distortion in injection molding”, Polymer Engineering and Science, 24, pp.227-241,1984 [Thorne 1979] Thorne, J L., Plastic Process Engineering, Marcel Dekker, 1979 [Titomanlio 1996] Titomanlio, G and Jansen, K M B., “In-mold shrinkage and stress prediction in injection molding”, Polymer Engineering and Science, 36, pp.2041-2049, 1996 [Turng 1990] Turng, L S and Wang, K K., “A computer-aided cooling-line design system for injection molds”, Journal of Engineering for Industry, 112, pp.261-267, 1990 [Turng 1993] Turng, L S., Wang, V W and Wang, K K., “Numerical simulation of 167 References the coinjection molding process”, Journal of Engineering Materials and Technology, 115, pp.46-53, 1993 [Wang 1994] Wang, K K., “Twenty years of CIMP research towards CAE for injection molding”, Advances in Computer-Aided Engineering of Polymer Processing, ASME, pp.1-5, 1994 [Yu 1992] Yu, C J and Sunderland, J E., “Determination of Ejection Temperature and Cooling Time in Injection Molding”, Polymer Engineering and Science, 32, pp.191-197, 1992 [Zhang 2002] Zhang, X., Cheng, X., Stelson, K A., Bhattacharya, M., Sen, A and Voller, V R., “Approximate model of thermal residual stress in an injection molded part”, Journal of Thermal Stresses, 25, pp.523-538, 2002 [Zhou 2001] Zhou, H., Zhang, Y and Li, D., “Injection moulding simulation of filling and post-filling stages based on a three-dimensional surface model”, Proceedings of the Institution of Mechanical Engineers, Part B (Journal of Engineering Manufacture), 215, pp.1459-1463, 2001 [Zoetelief 1996] Zoetelief, W F., Douven, L F A and Ingen, A J., “Residual thermal stresses in injection molded products”, Polymer Engineering and Science, 36, pp.18861896, 1996 [Zou 1992] Zou, Q., “Injection molding cooling optimization”, SPE ANTEC, pp.16141620, 1992 168 Appendix A Matrices and Vectors APPENDIX A MATRICES AND VECTORS A.1 CONVENTIONS Matrix and vector are written in bold, e.g., a matrix A and a vector B The elements of a matrix or vector are written in italic with subscripts, e.g Aij is the ith row and jth column of matrix A, and Bi is the ith element of vector B Sub-matrices are written in bold with subscripts, e.g., A1 and A2 are sub-matrices of matrix A A.2 MATRIX TRANSPOSE The transpose of an m×n matrix A is an n×m matrix AT, whose rows are the columns of A The transpose of a matrix can be formed simply by interchanging its rows and columns so that row i of the matrix become column i of its transpose Applying this definition, it is obvious that a symmetric matrix is its own transpose Also, the transpose of the product of two matrices is the product of their transposes in reverse order, i.e., (AB ) = B T A T T A.3 QUADRATIC FORMS Matrix notation is useful in symbolizing special non-linear expressions called quadratic forms A function of n variables x1, x2,…, xn in quadratic form is defined as n n F ( x1 , x2 , , xn ) = ∑∑ Aij xi x j i =1 j =1 =A x + 11 + A1n x1 xn + A21 x2 x1 + A22 x2 + + A2 n x2 xn + + An1 xn x1 + + Ann xn 169 Appendix A Matrices and Vectors Denoting an n×n matrix A and a vector x = [x1 x2 xn ] , the quadratic form can be express as: F = xAx (A.1) A quadratic form is positive definite if F is non-negative for all possible combinations of real vector x and if F is zero only when vector x is null A useful property of a positive definite quadratic form is that the determinants of the coefficients aij and all of its principle minors are positive, i.e., a11 a12 >0, a21 a22 a12 a22 a1n a2 n an1 a11 > , a11 a21 an ann >0 This property may be used to check whether a quadratic form is positive definite A.4 MATRIX INVERSE An identity matrix I is defined as a square matrix whose elements on the main diagonal are all unity and other elements are all zero Obviously, AI = IA = A If for a given square matrix A, there exists another square matrix A-1 such that A −1A = AA −1 = I , then A-1 is the inverse of A A is non-singular when its inverse A-1 exists In particular, the inverse matrix of a diagonal matrix Λ is given by (Λ ) −1 ii = 1/ Λii (A.2) For a system equations of the form AX = B, the solution can be written as X = A −1B (A.3) 170 Appendix A Matrices and Vectors A.5 DIFFERENTIATION OF A MATRIX If the element of a matrix A are themselves functions of n independent variables, x1, x2,…, xn, the matrix A is called a matrix function of x1, x2,…, xn The nth-order derivative of A exists when the nth-order derivative of each of its elements exists The derivative of A is calculated by differentiating each of its elements Hence the element in the ith row and jth column of ∂A/∂xk is ∂(aij)/∂xk A.6 INTEGRATION OF A MATRIX The integral of a function matrix A exists only when the integral of each element of the matrix exists Integration on each of its elements is necessary to calculate ∫ Adxk , i.e., the element in the ith row and jth column of ∫ Adxk is ∫ aij dxk A.7 DIFFERENTIATION OF A QUADRATIC FUNCTION A quadratic functional I(x1, x2,…, xn) is defined in matrix notation as: I (x1 , x2 , , xn ) = xAx − xB (A.4) where x1, x2,…, xn are n independent variables, A is a positive define matrix, and B is a column vector Both A and B have constant elements To make the functional stationary, it is necessary to compute the n derivatives of I with respect to each independent variable xi and set the result equal to zero Thus, ∂I ∂I = = , i = 1, 2,…,n ∂x ∂xi (A.5) Simple formulas to perform these differentiations are 171 Appendix A Matrices and Vectors ∂xAx = 2Ax , ∂x ∂xB =B ∂x (A.6) Therefore ∂I = Ax − B = ∂x (A.7) Equations (A.7) represent a set of simultaneous equations This result is important in developing element equations in FEM A.8 DIFFERENTIAL FORMULATION OF VECTOR If vectors A and B are r-D, then the Euclidean norm of A and the unit vector of A are defined as: A = r ∑A i =1 a= i A A (A.8) (A.9) The dot product of A and B is given by r A ⋅ B = ∑ Ai Bi (A.10) i =1 In particular, when r = 3, r A ⋅ B = ∑ Ai Bi = A B cos θ (A.11) i =1 where θ is the angle between A and B, ≤ θ ≤ π When A ⊥ B , A ⋅ B = The cross product of two r-D vectors is also an r-D vectors It is most used in 3-D and given by 172 Appendix A Matrices and Vectors A A×B =   B2 A3 A3 , B3 B3 A1 A1 , B1 B1 A2   B2  (A.12) The vector A × B is perpendicular to both A and B, and A × B = A B sinθ (A.13) Therefore, when A = λ B and λ is a real constant, A × B = The derivatives of the norm a vector and a unit vector are obtained by: A′= A ⋅ A′ A  A ′ A A′ − A ′ A A′ − ( a ⋅ A′ ) a a′ =  = =  A   A A   (A.14) (A.15) The derivatives of the dot and cross products of two vectors are obtained by: ( A ⋅ B )′ = A′ ⋅ B + A ⋅ B′ (A.16) ( A × B )′ = A′ × B + A × B′ (A.17) 173 ... cooling designs before constructing the mould Finite Difference Method (FDM), Finite Volume Method (FVM), Boundary Element Method (BEM), and Finite Element Method (FEM) have been applied in cooling. .. FDM Finite Difference Method FEA Finite Element Analysis FEM Finite Element Method FVM Finite Volume Method GA Genetic Algorithm HSM High Speed Machining KP Kont Point MGI Milled Groove Insert... 1905 FINITE ELEMENT METHOD IN COOLING ANALYSIS AND DESIGN OF PLASTIC INJECTION MOULDS BY SUN YIFENG (B Eng., M Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL