Thesis for the Degree of Doctor of Philosophy Application of Real Coded Genetic Algorithm for Ship Hull Surface Fitting With a Single Non-Uniform B-spline Surface by Tat-Hien Le Department of Naval Architecture and Marine Systems Engineering, The Graduate School Pukyong National University August 2009 Application of Real Coded Genetic Algorithm for Ship Hull Surface Fitting With a Single Non-Uniform B-spline Surface (유전알고리즘을 이용한 단일 B-spline 선체 곡면 표현) Advisor: Prof Dong-Joon Kim by Tat-Hien Le A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Naval Architecture and Marine Systems Engineering, The Graduate School, Pukyong National University August 2009 Application of Real Coded Genetic Algorithm for Ship Hull Surface Fitting With a Single Non-Uniform B-spline Surface A dissertation by Tat-Hien Le Approved by: Prof In Chul Kim, Ph.D.(Chairman) Prof Yong Jig Kim, Ph.D.(Member) Prof Dong-Joon Kim, Ph.D.(Member) Prof Jong-Ho Nam, Ph.D.(Member) Prof Won Don Kim, Ph.D.(Member) August 2009 TABLE OF CONTENTS LIST OF FIGURES iv LIST OF TABLES vii LIST OF APPENDICES viii ABSTRACT Chapter INTRODUCTION 1.1 State of the Art 1.2 About This Work Chapter CLASSIFICATION OF SURFACE MODELING 2.1 Boundary Interpolating Patch Models 2.1.1 Ruled Surfaces 2.1.2 Lofted Surfaces 10 2.1.3 Bilinear Blended Coons Patch 10 2.1.4 Bicubic Coons Patches 12 2.2 Irregular Patch 14 2.3 Parametric Polynomial Patch Model 16 2.3.1 Standard Polynomial Surface Patch 17 2.3.2 Ferguson Surface Patch 18 2.3.3 Bézier Surface Patch 19 2.3.4 Uniform B-Spline Surface Patch 20 2.3.5 Non-Uniform B-Spline Surface Patch 21 2.3.6 Definition and Properties of Knot Vector 22 2.3.7 Definition and Properties of Non-Uniform B-Spline Basis Function 23 2.3.8 Non-Uniform B-spline Surface from 3D Data Array 24 i Chapter OVERVIEW OF THE NON-UNIFORM B-SPLINE FITTING ALGORITHM 27 3.1 Non-Uniform B-Spline Surface Fitting Application in Ship Hull Design 27 3.2 Hull Form Modeling Requirements 29 3.2.1 Shape Requirements 30 3.2.2 Continuity between Patches 31 3.2.3 End Condition 33 3.2.4 Irregular Patch Constraints 34 3.2.5 Effect of Multiple Knot Vector and Multiple Vertex Point 35 3.3 Matrices Inversion Problems in Non-Uniform B-spline Surface Fitting 37 Chapter APPLICATION OF REAL CODED GENETIC ALGORITHM FOR SURFACE FITTING 40 4.1 The Goals of Optimization 40 4.1.1 What Is Optimization? 40 4.1.2 Local and Global Optimization 41 4.2 Overview of Real Coded Genetic Algorithm 42 4.3 Fitness Function For Non-Uniform B-spline Surface Fitting 43 4.4 Encoding for Initial Population 43 4.5 Reproduction Process 44 4.6 Crossover Process 46 4.7 Mutation Process 47 4.8 Crossover and Mutation Probability 48 Chapter SINGLE NON-UNIFORM B-SPLINE SURFACE FITTING 50 5.1 Non-Uniform B-spline Curve Fitting for Boundary Curves 50 5.1.1 Yoshimoto’s Method for Boundary Curves 50 5.1.2 Different Sets of Knot Value at Stern and Bow Boundaries 51 5.1.3 The Other Problems of Boundary Curves Fitting 53 ii 5.2 New Approach to Boundary Curves Fitting 54 5.2.1 Simultaneous GA Fitting for Multiple Curves 54 5.2.2 Handling Weakly Knuckle Point and Twist Problem 56 5.3 New Approach to Surface Fitting for the Given Interior Data Point by Using GA 58 5.3.1 Vertices Encoding for Initial Population 58 5.3.2 Reproduction Process 60 5.3.3 Crossover Process 61 5.3.4 Mutation Process 62 5.3.5 Nearest Point Finding for Fitness Function 63 5.4 Summary 65 Chapter APPLICATION EXAMPLES 68 6.1 Simple Surface 68 6.2 Yacht Hull Surface 72 6.3 Complicated Surface 75 6.4 Container Ship Hull Form 78 Chapter CONCLUSIONS 83 REFERENCES 86 APPENDICES 91 iii LIST OF FIGURES Page Figure 1.1 de Casteljau’s algorithm for cubic curve Figure 1.2 Bézier’s basic curve from intersection of two elliptic cylinders Figure 1.3 Problems in composite surfaces for ship hull form Figure 1.4 The mesh generation process of composite surfaces and single NUB surface Figure 2.1 Linear blending of ruled surface Figure 2.2 Lofted surface construction 10 Figure 2.3 Bilinear blended Coons patch 11 Figure 2.4 Bicubic Coons patch 12 Figure 2.5 Twist problem at corners in bicubic blended Coons patch 13 Figure 2.6 Bézier Triangular surface 14 Figure 2.7 Surface reconstruction from rectangular patches and triangular patches 16 Figure 2.8 Parametric surface and its parameters 17 Figure 2.9 Standard polynomial surface 18 Figure 2.10 Bézier patch 20 Figure 2.11 Uniform B-spline patch 21 Figure 2.12 NUB surface fitting from the given data points 25 Figure 2.13 NUB surface and vertices 25 Figure 3.1 Surface fitting process 28 Figure 3.2 Surface curvature analysis 28 Figure 3.3 The requirements of shape at complicated parts 30 Figure 3.4 The discontinuity requirement between surfaces 31 Figure 3.5 The continuity condition for surface 32 Figure 3.6 Single NUB surface visualization 32 Figure 3.7 End condition at corners of surface 33 Figure 3.8 Irregular data points in each section of ship hull form 34 Figure 3.9 Effect of multiple knot on NUB curve, k = 35 Figure 3.10 Effect of multiple vertices at vertex point B2 on NUB curve 36 Figure 3.11 No multiple vertices at knuckle 36 Figure 3.12 Multiple vertices at knuckle 36 Figure 3.13 The effect of multiple knots and multiple vertices at stern part 37 Figure 3.14 The difficulties of fitted surface in matrices inversion method 39 Figure 4.1 Diagram of a function or process that is to be optimized 40 Figure 4.2 Overview the genetic algorithm procedure 43 Figure 4.3 Real coded individual 44 Figure 4.4 The convergence of fitness value with and without reproduction 46 Figure 4.5 Crossover mechanism 47 iv Figure 4.6 Mutation mechanism 47 Figure 5.1 GA application for boundary curve fitting 50 Figure 5.2 The different knot value sets at stern and bow curve 51 Figure 5.3 The surface construction from one of the knot value sets at boundaries 52 Figure 5.4 Twist problem in Stern boundary 53 Figure 5.5 Unwanted knuckle problem 53 Figure 5.6 The simultaneous GA fitting for multiple curves at stern and bow boundaries 55 Figure 5.7 Double vertices for knuckle point 57 Figure 5.8 Multiple curves fitting implementation 57 Figure 5.9 The generation of vertices from the given data points at initial population 58 Figure 5.10 The small deviation estimation in y direction in the initial population 59 Figure 5.11 Elite mechanism 60 Figure 5.12 Reproduction technique 60 Figure 5.13 Real coded value in individuals 61 Figure 5.14 Crossover procedure 61 Figure 5.15 The effect of moving one of the vertex points of the NUB surface 62 Figure 5.16 The given data point and the closest point on NUB surface 64 Figure 5.17 The data range of surface modeling 64 Figure 5.18 Simultaneous multiple curves fitting implementation for boundaries 65 Figure 5.19 Regenerate the rectangular vertices for NUB surface 65 Figure 5.20 GA for NUB surface fitting process 66 Figure 5.21 NUB surface after GA application 67 Figure 6.1 The mesh of given data points of simple surface 68 Figure 6.2 The Gaussian curvature of the simple surface at 1st generation and 20,000th generation 69 Figure 6.3 The fitness value during generations of simple surface 70 Figure 6.4 The Gaussian curvature distribution at each population of first and final generation for simple surface 71 Figure 6.5 The given data points of yacht surface 72 Figure 6.6 The Gaussian curvature of yacht surface at 1st generation and 20,000th generation 73 Figure 6.7 The Gaussian curvature distribution at each population of first and final generation for hull surface of yacht without keel 74 Figure 6.8 The fitness value during generations of complicated shape 75 Figure 6.9 The given data points of complicated surface 75 Figure 6.10 The Gaussian curvature of the complicated surface at 1st generation and 20,000th generation 76 Figure 6.11 The Gaussian curvature distribution at each population of first and final generation for complicated shape 77 v Figure 6.12 The fitness value during generations of complicated shape 78 Figure 6.13 The given data points of container ship surface 78 Figure 6.14 The single NUB surface and section plan based on the fitted surface at 40000th generation 79 Figure 6.15 The Gaussian curvature of the container ship at 1st generation and 40,000th generation 80 Figure 6.16 The fitness value during generations of container ship 80 Figure 6.17 The Gaussian curvature distribution at each population of first and final generation for container ship 81 vi points and in the knuckle requirement In fact, the knuckle points should be defined from the given data points for the adaptive adjustment of double vertices in the multiple curve fitting process After that, the fully automatic process is applied for a single NUB surface by using GA effectively For future research in this study, emphasis should be on finding a faster speed of optimization for ship hull surface and the required fairness criterion for other analysis applications after achieving the fitted NUB surface 85 REFERENCES Bézier, P., 1974 “Mathematical and practical possibilities of UNISURF.” Computer Aided Geometric Design (Edited by Barnhill and Riesenfeld), Academic Press, pp 127-152 Birmingham, R W and Smith, T A G, 1998, “Automatic Hull Form Generation: a practical tool for design and research.” Proceedings of the Seventh International Symposium on Practical Design of Ships and Mobile Units, pp 281-287 Choi, B.K., 1991, Surface Modeling for CAD/CAM Elsevier Amsterdam – Oxford – New York – Tokyo Chiyokura H and Kimura, G., 1984, “A New Surface Interpolation Method for Irregular Curve Models.” Computer Graphic Forum Chiyokura H., Takamura T., Konno K., and Harada T., 1991,” Surface Interpolation over Irregular Meshes with Rational Curves.” In: Farin G., ed., NURBS for Curves and Surfaces Design, SIAM, Philadelphia PA, pp.15-34 Cho, D.Y., Lee, K.Y, and Kim, T.W, 2006, “Interpolating Bézier Surfaces over Irregular Curve Networks for Ship Hull Design.” Computer Aided Design 38, pp.641 - 660 Dahmen, W., Micchelli, C A., and Seidel, H P., 1992, “Blossoming begets B-spline bases built better by B-patches.” Mathematics of Computation, Vol 59, No 199, pp 97-115 de Boor C., 1972, “On calculating with B-spline.” Journal of approximation theory, Vol 6, pp 50-52 86 deJong, K A., 1975, An Analysis of the Behavior of a class of Genetic Adaptive Systems PhD thesis, University of Michigan, Ann Arbour Department of Computer and Communication Sciences Dierckx P.,1993, Curve and Surface Fitting with Splines, Oxford: Oxford University Press Farin, G., 1990, Curves and Surfaces for Computer Aided Geometric Design – A practical guide Academic Press, INC Fong, P and Seidel, H P., 1993, “An Implementation of Triangular B-spline Surfaces over Arbitrary Triangulation.” Computer Aided Design, Vol 10, pp 267-275 Goldberg, D.E., 1989, Genetic algorithm in search, optimization, and machine learning, Addison-Wesley Gordon, W and Riesenfeld, R., 1974, “B-spline Curves and Surfaces.” Computer Aided Geometric Design (Edited by Barnhill and Riesenfeld), Academic Press, pp 95-126 Gregory J.A., 1974, “Smooth Interpolation without Twist Constrains.” Computer Aided Geometric Design, Academic-Press, pp.27-33 Hazen, G.S., 2002, “Fastship & NURBS modeling: a historical note.” Computer Aided Design, Vol 34, No 7, pp 541-543 Hoschek, J., 1988, “Intrinsic Parameterization for Approximation.” Computer Aided Geometric Design 5, 27–31 Holland, J., 1975, Adaptation In Natural and Artificial Systems The University of Michigan Press, Ann Arbour Jensen J.J and Baatrup J, 1989, Transformation of body planes to a B-spline surface, Elvier Science Publishers B V 87 Jupp DLB., 1978, “Approximation to Data by Splines with Free Knots.” SIAM J Number Anal, Vol 15, pp 328-43 Le, T.H., Kim, D.J., Min, K.C., and Pyo, S.W., 2009, “B-spline Surface Fitting using Genetic Algorithm.” Journal of the Society of Naval Architectures of Korea, Vol 46, No 1, pp 87-95 Lee, J.H and Kim, D.J., 2004, “A Study on the Surface Modeling of Hull Form for General purpose CAD program.” Journal of the Society of Naval Architectures of Korea, Vol 41, No 1, pp 75-81 Lee, S.Y., Wolberg, G., and Shin, S.Y., 1997, “Scattered data interpolation with multilevel B-splines.” Transactions of visualization and computer graphics, Vol 3, No 3, pp 228-244 Lu, C., Lin, Y., Ji, Z., and Chen, M., 2008, “NURBS based Ship Form Design Using Adaptive Genetic Algorithm.” Proceedings of the Eighteenth (2008) International Offshore and Polar Engineering Conference Marson, A., 2002, “MacSurf an early NURBS ship hull design system: a historical note.” Computer Aided Design, Vol 34, No.7, pp 545-546 Mathews, J.H and Fink, K.D., 1992, Numerical Methods Using Matlab, Pearson Prentice Hall Olfe, D B., 1995, Computer graphics for Design from algorithm to Autocad Prentice Hall, INC Park, J S and Kim, D J., 1994, “Definition of Ship Hull using GC1 Surface.” Transactions of the Society of Naval Architecture of Korea, Vol 31, No.4, pp 3240 Piegl, L., 1991, “On NURBS: A survey.” IEEE Journal of computer graphics & Applications, pp 55-71 88 Piegl, L and Tiller, W., 1997, The NURBS Book – Monographs in Visual Communications Springer Rao, S.S., 1996, Engineering optimization: Theory and practice New York, WileyInterscience Riesenfield, R and Gordon, W., 1974, “B-spline curves and surfaces.” Computer Aided Geometric Design, pp 95-126 Rogers, D F and Adams, J A., 1989, Mathematical elements for computer graphic McGraw Hill International editions Rogers, D F, Satterfield, S G and Rodriguez F A., 1983, “Ship Hulls, B-spline Surfaces and CAD/CAM.” IEEE 0272-1716 Weiss, V., Andor, L., Renner, G., and Varady, T., 2002, “Advanced Surface Fitting Techniques.” Computer Aided Design, Vol.19, pp.19-42 Westgaard, G and Nowacki, H., 2001, "Construction of Fair Surfaces over Irregular Meshes." Journal of Computing and Information Science in Engineering, Vol 1, No 4, pp 376-384 Whitley, D and Kauth, J., 1988, “GENITOR: A different Genetic Algorithm.” Proc of the Rocky Mountain Conf on Artificial Intelligence, pp 118-130 Yoon, B.H., Sur, S.W., Kim, W.D., and Kim, K.W., 1985, “Representation of Ship Hull Surface Using B-Spline.” Journal of the Society of Naval Architectures of Korea, Vol 22, No 3, pp 19-26 Yoshimoto, F., Harada, T., Moriyama, M and Yoshimoto, Y., 2003, “Automatic Knot Placement by a Genetic algorithm for data fitting with a spline.” Shape modeling International ’99, IEEE Computer Society Press, p.162 - 89 Yoshimoto, F., Harada, T., Moriyama, M and Yoshimoto, Y., 2003, “Data Fitting with a Spline using a Real Coded Genetic Algorithm.” Computer Aided Design, Vol.35, pp.751-760 90 APPENDIX A THE GIVEN DATA POINTS OF CONTAINER SHIP IN 3-D COORDINATE Number of given data points: 251 points 12.053 12.053 12.053 12.053 12.053 x,y,z coordinate in each section of ship hull form 12.053 14.338 16.817 0.947 0.706 0.4362 0.172 14.59 15.915 16.103 0.507 0.7612 0.848 0.736 0.088 0 11.151 3.13 3.3 3.6 4.4 4.82 6.2 8.3 10.8 13.275 15.931 18.586 20.313 18.532 18.532 18.532 18.532 18.532 18.532 18.532 18.532 18.532 18.532 18.532 18.532 18.532 1.602 1.568 1.502 1.416 1.305 0.836 0.145 10.239 14.78 16.048 16.104 16.104 0.022 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.32 23 23 23 23 23 23 23 23 23 23 23 23 23 0.001 1.684 1.654 1.598 1.526 1.438 1.117 1.8805 11.852 15.387 16.103 16.104 16.104 0.001 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.328 29 29 29 29 29 29 29 29 29 29 29 29 29 0.001 1.878 1.87 1.852 1.829 1.806 1.856 7.206 13.705 15.912 16.104 16.104 16.104 0.01 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.343 37.064 37.064 37.064 37.064 37.064 37.064 37.064 37.064 37.064 37.064 37.064 37.064 37.064 0.001 2.538 2.609 2.715 2.84 3.016 4.848 11.451 15.388 16.103 16.104 16.104 16.104 0.035 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.376 52 52 52 52 52 52 52 52 52 52 52 52 52 0.001 6.09 6.69 7.6 8.531 9.507 12.576 15.612 16.104 16.104 16.104 16.104 16.104 0.103 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.465 63 63 63 63 63 63 63 63 63 63 63 63 63 0.001 10.862 11.503 12.314 13.064 13.771 15.413 16.094 16.104 16.104 16.104 16.104 16.104 0.133 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.516 74.128 74.128 74.128 74.128 74.128 74.128 74.128 74.128 74.128 74.128 74.128 74.128 74.128 0.001 14.285 14.646 15.049 15.369 15.624 16.035 16.104 16.104 16.104 16.104 16.104 16.104 0.122 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.525 92.66 92.66 92.66 92.66 92.66 92.66 92.66 92.66 92.66 92.66 92.66 92.66 92.66 0.001 15.401 15.546 15.706 15.832 15.933 16.087 16.104 16.104 16.104 16.104 16.104 16.104 0.108 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.535 110 110 110 110 110 110 110 110 110 110 110 110 110 0.001 13.944 14.235 14.576 14.868 15.126 15.709 16.081 16.104 16.104 16.104 16.104 16.104 91 0.108 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.558 129.72 129.72 129.72 129.72 129.72 129.72 129.72 129.72 129.72 129.72 129.72 129.72 129.72 0.001 9.355 9.832 10.425 10.972 11.501 12.96 14.673 15.767 16.104 16.104 16.104 16.104 0.108 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.583 145 145 145 145 145 145 145 145 145 145 145 145 145 0.001 5.957 6.33 6.813 7.284 7.7637 9.236 11.221 13.056 14.743 15.961 16.104 16.104 0.105 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.603 158.16 158.16 158.16 158.16 158.16 158.16 158.16 158.16 158.16 158.16 158.16 158.16 158.16 0.001 3.868 4.108 4.419 4.725 5.042 6.067 7.635 9.365 11.19 13.167 15.18 16.104 0.084 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 20.654 169 169 169 169 169 169 169 169 169 169 169 169 169 2.627 2.775 2.959 3.132 3.307 3.863 4.766 5.874 7.402 9.511 11.809 16.103 0.049 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 23.523 175.90 175.90 175.90 175.90 175.90 175.90 175.90 175.90 175.90 175.90 175.90 175.90 175.90 2.113 2.218 2.341 2.445 2.539 2.797 3.232 3.92 5.025 6.842 9.274 16.103 0.034 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 25.246 184 184 184 184 184 184 184 184 184 184 184 184 184 0.001 1.707 1.821 1.953 2.059 2.141 2.201 1.915 1.883 2.539 3.968 6.158 13.998 0.047 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 25.681 193 193 193 193 193 193 193 193 193 193 193 193 193 1.375 1.481 1.614 1.735 1.847 2.076 1.744 0.437 0.535 1.31 2.8 10.388 0.262 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 26.05 195 195 195 195 195 195 195 195 195 195 195 195 195 0.001 1.307 1.413 1.545 1.668 1.782 2.027 1.78 0.108 0.213 0.794 2.044 9.375 0.524 3.3 3.6 4.4 4.82 6.2 8.4 10.8 13.276 15.931 18.586 26.127 197 199 200.85 202.09 202.73 202.78 201.49 197 195.18 195.62 196.96 198.62 203.85 0.0124 0 0 0 0 0 0 1.0683 1.9651 3.3 4.82 6.2 8.4 9.6 9.987 10.8 13.276 15.931 18.586 26.05 BLANK LINE 12.053 0.947 0.088 4.82 10.8 x,y,z coordinate of cusp points or irregular data points 92 APPENDIX B THE OUTPUT DATA OF CONTAINER SHIP (INCLUDED VERTICES AND KNOT VALUES) Data Output (Container ship hull form) *Can be output to IGES file Order of surface: w = 4, u = Number of vertex points: 17 (number of vertices on w direction) 21 (number of vertices on u direction) 12.0530 12.0530 12.0530 12.0530 12.0530 12.0530 12.0630 12.0030 16.0180 18.6470 1.1370 0.3170 0.8760 0.4162 0.1720 0.0625 0.0010 0.0010 0.1992 0.5530 0.7023 0.9114 0.7349 0.0298 0.0108 -0.0090 0.0310 -0.3890 -0.3390 11.6132 14.8076 15.9101 16.4057 16.1036 3.1300 3.1350 3.3200 3.4100 3.9700 4.5800 4.7300 5.1000 6.9500 9.1000 10.7800 10.7600 12.7258 15.7710 18.6560 19.6920 20.3130 14.7178 15.3902 14.5629 14.5390 14.5970 14.5584 14.5307 15.2327 16.6607 16.0595 10.9818 8.8528 8.8650 8.8356 8.7042 10.3738 8.4918 0.0010 1.5795 0.7084 2.2984 1.2334 3.4123 1.2070 3.6959 1.1556 4.1648 0.9145 4.5132 0.6850 4.8468 0.5725 5.7654 -0.0467 7.3198 -0.0477 8.9188 2.5686 10.1320 6.4037 11.1157 11.1388 12.7638 14.3074 14.9495 15.4963 17.1248 16.1005 19.0287 16.1083 20.3152 16.2519 18.5420 18.5214 18.5221 18.5067 18.5736 18.5377 18.4388 18.5392 18.5729 18.5204 18.5124 18.4950 18.5175 18.5455 18.5711 18.5320 0.0010 1.2820 1.7555 1.6384 1.5865 1.5191 1.4403 1.2864 0.1373 0.5262 5.9198 12.4570 15.5154 16.2330 16.9450 15.7348 16.1049 0.0220 2.4592 3.3075 3.5975 3.9984 4.4777 4.8294 5.8136 7.3088 8.7886 9.7758 11.1365 13.4271 15.7007 17.6028 19.2867 20.3200 20.8900 23.0008 23.0220 22.9931 23.0091 23.0095 23.0060 22.9980 23.0002 22.9865 22.9600 22.9750 22.9450 23.0100 22.9900 22.9800 23.0200 0.0010 1.3251 1.7982 1.5832 1.6006 1.5167 1.4634 1.0820 0.9102 2.4979 8.4209 14.0078 15.7152 15.9984 16.3615 15.8653 16.1048 0.0010 2.4048 3.3269 3.6272 3.9360 4.3145 4.7079 5.6953 7.2267 8.8425 9.7537 11.2751 13.5627 15.7303 17.6855 19.3066 20.3280 28.9900 29.0450 28.9807 29.0049 29.0053 29.0584 28.9892 29.0599 29.0166 28.9681 28.9634 28.9731 29.0117 29.0005 28.9600 28.9739 28.9500 0.0010 1.5704 1.8309 1.7430 1.7770 1.6241 1.6320 1.2849 2.5807 6.9543 12.1546 15.6814 16.4548 16.0033 16.2777 16.2447 16.0987 37.0340 37.0699 37.0795 37.1551 37.0937 37.1090 37.0190 37.1040 37.1190 37.0490 37.1140 37.0640 37.1040 37.0840 37.0940 37.0340 37.1040 0.0010 1.8448 2.6466 2.4270 2.5787 2.5640 3.1187 3.4214 5.9170 11.4244 14.9145 15.8847 16.0550 16.2817 15.6334 16.2097 16.1078 0.0250 2.5195 3.3950 3.7020 4.0296 4.6898 4.8604 5.8022 6.6913 8.0871 10.0153 11.7792 13.7623 15.6764 17.6073 19.1540 20.3660 93 0.0100 2.4443 3.4097 3.6378 3.9058 4.4350 4.7337 5.7353 6.8582 8.0967 9.7712 11.6072 13.7879 15.7446 17.6274 19.0989 20.3430 51.8400 52.0100 52.0400 51.9900 51.9850 52.0050 51.9900 52.0150 51.9000 52.0650 52.0600 52.0250 52.0000 52.0000 51.9800 51.9350 52.0400 0.0010 4.5885 6.5642 6.6146 7.2383 7.8790 9.1767 10.4367 13.7932 15.0683 15.3357 15.5618 16.6092 16.0919 15.8019 16.3201 16.1064 0.1130 2.6761 3.5823 3.8502 4.1025 4.2670 4.7077 5.4128 6.4212 8.1525 9.9102 11.7709 13.7679 15.7472 17.6324 19.2804 20.4650 63.0600 62.9903 62.9783 62.9851 63.0350 63.0150 62.9950 63.0000 62.9900 63.0700 63.0661 62.9751 62.9634 62.9794 62.9765 62.9559 62.9900 0.0010 8.0514 11.4454 11.3212 12.8064 12.8536 13.8920 15.1762 16.2792 16.5122 15.5903 15.9081 16.1855 16.3180 15.6819 16.3129 16.1058 0.1330 2.4045 3.2233 3.4116 3.8111 4.0928 4.4680 5.2720 6.5961 8.1918 10.0047 11.8416 13.7839 15.6908 17.6101 19.1820 20.5160 74.1780 74.1072 74.0959 74.0689 74.0677 74.0930 74.0827 74.1330 74.1430 74.0512 74.0835 74.1650 74.0948 74.0820 74.1495 74.1250 74.1180 0.0010 10.6011 14.4452 15.1132 15.6603 15.1994 15.9034 16.4292 16.1750 15.9466 15.7772 16.0186 15.8560 16.0107 16.2605 16.0489 16.1052 0.1220 2.4767 3.1957 3.4281 3.8826 4.2181 4.4239 5.2032 6.3042 8.1837 10.0225 11.8161 13.7714 15.7916 17.6332 19.2093 20.5250 92.8400 92.6600 92.6300 92.6304 92.6448 92.6153 92.6554 92.6700 92.7050 92.6250 92.6850 92.6400 92.6700 92.7050 92.6500 92.6750 92.6100 0.0010 11.4649 15.0423 16.3081 15.7331 15.8231 15.5905 16.4763 16.1316 16.5243 16.2864 16.4739 15.9764 16.3741 15.8868 16.1488 16.1086 0.1080 2.4431 3.1627 3.3939 3.7961 4.2301 4.5523 5.2654 6.6030 8.2604 10.0092 11.8131 13.7842 15.6957 17.5968 19.1479 20.5350 110.1400 110.0308 110.0271 110.0391 109.9651 110.0008 109.9188 110.0055 110.0211 109.9500 109.9600 109.9850 109.9450 110.0100 109.9800 109.9950 110.0800 0.0010 10.9125 14.5772 13.8313 15.0822 15.2934 15.2198 15.8314 16.1372 16.4280 15.9250 15.7234 16.3171 15.5501 15.9639 16.4375 16.1000 0.1080 2.4848 3.2857 3.3259 3.7253 4.1965 4.6152 5.4046 6.4094 8.1364 10.1358 11.7982 13.7219 15.7312 17.6297 19.2251 20.5580 129.7040 129.6797 129.6861 129.6931 129.7493 129.6339 129.5793 129.7891 129.7440 129.7340 129.7190 129.6690 129.7991 129.7140 129.7491 129.7240 129.6940 0.0010 7.1666 9.7579 10.2565 10.3967 10.9445 11.2519 12.2572 13.0601 14.4252 15.1351 15.4736 16.4514 16.2777 16.4978 15.9296 16.0996 0.1080 2.4848 3.3325 3.6274 3.9614 4.2315 4.3874 5.2159 6.2954 7.9836 10.0775 11.8890 13.7900 15.9219 17.6663 19.2141 20.5730 144.9100 145.0042 144.9729 144.9158 144.9749 144.9642 145.0815 145.2081 145.0375 145.0016 145.2336 145.0782 145.0868 144.9756 144.9585 145.0391 146.0094 0.0010 4.3218 6.0446 5.9748 6.4731 6.8464 7.4414 8.4766 9.6584 10.9397 12.1037 13.8250 15.4100 16.1728 15.9507 15.9135 16.0984 0.1050 2.4746 3.3065 3.5068 3.8670 4.1966 4.3696 5.5683 6.5997 7.9981 9.6258 11.8829 13.5790 15.7068 17.6352 19.1552 20.5630 158.3169 158.1670 158.0919 158.1520 158.1670 158.1370 158.2621 158.3222 158.3071 158.2471 158.1920 158.1770 158.1670 158.1270 158.1720 158.1970 159.9361 0.0010 2.9639 3.8316 3.9378 4.2218 4.5295 4.8848 5.7108 6.5838 7.4216 9.0878 10.2233 11.8651 12.6491 14.0194 15.6083 16.1536 0.0840 2.4883 3.2921 3.4785 3.7715 4.2158 4.6774 5.7211 6.8617 8.3116 9.9425 11.9328 13.3292 15.4455 17.4573 19.1279 20.5640 169.0100 169.0100 168.9550 169.0250 168.9750 169.0000 169.0050 169.0200 169.0451 168.9800 168.9750 169.0401 168.9800 169.0300 168.9900 168.9950 167.9406 0.0010 1.9845 2.7323 2.7973 2.8134 3.0842 3.0606 3.7002 4.0626 4.4989 5.2722 6.3979 7.6610 9.4943 10.9247 13.3970 15.8699 0.0490 2.4705 3.3272 3.4942 3.7461 4.2168 4.5710 5.3734 6.5228 7.9727 9.9552 11.8071 13.8360 15.6816 17.7261 20.0182 23.0530 175.9590 0.0010 0.0340 175.8889 1.5567 2.4913 184.0800 0.0010 0.0370 184.0300 1.3676 2.4802 94 192.8901 0.0010 0.1420 193.0017 0.9800 2.5630 175.8540 175.9090 175.9240 175.9090 175.9040 175.8540 175.9190 175.9390 175.9301 175.8991 175.8524 175.8183 175.9005 175.9550 175.1194 2.1388 2.2354 2.2026 2.3046 2.4165 2.5081 2.9007 3.0866 3.6590 4.1007 5.3087 6.7771 8.2574 10.9197 16.7097 3.3740 3.6208 3.7877 4.1698 4.5932 5.2834 6.8964 8.2947 10.0547 11.8599 13.8775 15.8946 17.6506 20.2840 25.6360 183.9800 183.9900 183.9700 184.0350 183.9800 184.0100 183.9200 183.9350 183.9889 183.9799 183.9615 184.0207 183.9735 184.0141 184.0700 1.6628 1.7285 1.8571 2.0895 2.0471 2.1606 2.0434 1.7639 1.9272 2.1575 2.8211 3.7396 5.3412 7.7757 14.2381 3.4107 3.4543 3.8011 4.2464 4.6670 5.2024 6.5541 8.2528 10.0375 11.7909 13.8903 15.8868 17.6955 20.5774 25.6210 193.0183 192.9851 192.9950 193.0350 193.0000 192.9250 192.8098 192.9149 192.9650 192.9850 192.9500 192.9400 192.8849 192.9700 193.1000 1.4038 1.4355 1.5032 1.6490 1.7665 2.0418 1.9899 1.8067 0.8290 0.2720 0.7151 1.2593 2.3337 4.8896 10.1689 3.3564 3.4884 3.7974 4.1733 4.6520 5.3266 6.6878 8.2262 9.9619 11.7185 13.7756 15.8061 17.9501 20.7794 26.0800 195.1200 194.9892 195.0380 195.0359 195.0199 195.0343 195.0464 195.0196 195.0290 195.0182 195.0094 194.9469 195.0230 194.9950 194.8892 194.9099 195.1300 0.0010 0.9868 1.3656 1.4876 1.6576 1.8262 1.9231 1.9535 2.0610 1.8513 0.5134 -0.1776 0.3532 0.7355 1.3784 3.8187 9.8793 0.5340 2.5706 3.2316 3.3968 3.7919 4.1918 4.5419 5.3229 6.5750 8.3004 10.0158 11.7944 13.6540 15.7550 17.9093 20.7285 26.1270 196.3032 195.6074 197.7047 198.3726 198.8274 199.1454 199.2984 199.0259 198.0729 196.4850 195.5772 195.4158 195.7583 196.3230 196.9492 197.3046 199.6588 0.0066 0.3820 0.7223 0.7551 0.7687 0.8923 0.9107 0.8662 0.8655 0.9131 0.1771 -0.1763 0.1213 0.2868 0.4220 3.0219 4.5716 0.7998 1.7858 2.8237 3.5363 4.3033 5.1262 6.0502 7.2125 8.2230 9.3163 10.2864 11.7759 13.7312 15.8222 17.7387 21.4367 26.1013 197.0000 197.6841 198.5566 200.2557 201.5315 202.4782 202.9164 202.8927 201.6759 197.4598 194.8998 195.1575 196.1366 197.6153 199.0782 202.0025 203.8510 0.0124 0.0081 0.0014 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 1.0683 1.2166 1.8316 2.4506 4.1772 5.1888 6.7403 8.8656 9.8300 9.8148 10.3666 11.9619 14.8250 16.7902 19.5106 23.3913 26.0500 Output W Knot value after fitting & fairing: 0.0000 0.0000 0.0000 0.0000 0.0725 0.1439 0.2182 0.2882 0.3481 0.4145 0.4849 0.5658 0.6280 0.7067 0.7805 0.8404 0.9378 1.0000 1.0000 1.0000 1.0000 Output U Knot value after fitting & fairing: 0.0000 0.0000 0.0000 0.0000 0.0255 0.0719 0.1480 0.3359 0.4678 0.4780 0.6099 0.6659 0.7419 0.7723 0.8739 0.8840 0.8989 0.9043 0.9292 0.9548 0.9803 1.0000 1.0000 1.0000 1.0000 95 Total errors value: 38.3210 Output Stern data points after fitting & fairing: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6.0000 0.4000 0.0000 0.2000 0.0000 0.0000 2.6000 7.0000 7.0000 Output Bow data points after fitting & fairing: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Output Bottom data points after fitting & fairing: 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1000 0.0000 0.0000 0.0000 0.0000 Output Deck data points after fitting & fairing: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1000 0.0000 0.0000 0.0000 0.2000 1.5000 1.6000 4.8000 1.2000 2.1000 0.2000 0.4000 0.1000 0.0000 Output Interior errors after 0.0316 0.0023 0.0006 0.0007 0.0027 0.0003 0.0015 0.2035 0.0009 0.1598 0.4002 0.0031 0.0994 0.0017 0.0033 0.0004 0.1420 0.0942 0.1108 0.0764 0.2160 0.1321 0.1349 0.0588 0.2198 0.5953 0.8963 0.3721 0.3235 0.6727 0.1971 0.2587 0.2380 0.0023 0.1191 0.0044 0.0073 0.0031 0.7888 1.2475 1.2434 0.9255 0.3213 0.1121 0.0001 fitting & fairing: 0.0014 0.0026 0.0004 0.4181 0.0364 0.0397 0.0005 0.0006 0.0034 0.0005 0.2907 0.1484 0.0430 0.1469 0.0457 0.2448 0.2561 0.0410 0.3507 0.1709 0.1030 0.0298 0.0024 0.0006 0.0074 0.7028 0.9487 2.5731 0.0077 0.0274 0.0005 0.0051 0.0004 0.0946 0.0027 0.0010 0.2249 0.0042 0.0029 0.0042 0.0004 0.0530 1.0039 1.1336 0.7806 0.0065 0.0556 0.0676 0.0005 0.1135 0.1077 0.1966 0.4263 0.0021 0.1175 0.0016 3.4799 0.0901 96 0.0547 0.0001 0.2446 0.1466 0.0398 1.8892 0.0022 0.0893 0.2918 0.0035 0.1221 0.0007 0.0730 0.0023 0.0787 0.0029 0.3329 0.0471 0.0011 0.1716 0.0041 0.5504 0.1313 0.3184 0.0228 0.0030 0.0137 0.0009 0.0007 0.0082 0.3063 0.3658 1.0384 0.0969 0.0001 0.3675 0.1051 0.0684 0.0008 0.0248 0.1518 0.0015 0.0006 0.0009 0.2939 0.1037 0.9315 0.0011 0.1200 0.1350 0.2610 0.0023 0.0022 0.0310 0.0851 0.0001 0.0031 0.0036 0.3544 0.9647 0.0024 0.0006 0.0473 0.1171 0.0015 0.0943 0.0006 0.1837 0.0432 0.0348 0.0004 0.0309 0.0999 1.2008 0.0205 0.0463 0.0473 0.0476 0.1855 0.2157 0.0008 0.0825 0.0492 0.2510 유전알고리즘을 이용한 단일 B-spline 선체 곡면 표현 Tat-Hien Le 부경대학교 대학원 조선해양시스템공학과 요 약 디지털 방식으로 배를 설계하는 과정에서 선형의 곡면 모델링은 배의 생산과 성능수치해석에 있어서 매우 정밀함을 요구한다 곡면 모델링의 전통적인 방법은 skinning 법 이다 이 방법에서 곡면 모델은 단면 곡선의 데이터 집합으로부터 만들어진다 그러나 외판의 품질은 stern, bow profile, deck side lines, bottom tangential line 과 같은 특성곡선의 정확도와 단면곡선의 간격에 달려있으며 차원 diagonal lines, 불연속곡선과 같은 차원 교차곡선을 모두 포함하는 것이 skinning 법에서는 불가능하다 이것은 배의 외판을 만드는 과정에서 유효한 형태의 정보가 무시된다는 것을 의미한다 그러므로 고품질의 외판모델링은 쉽지 않은 설계과정이다 이 연구의 목표는 선체곡면을 단일 B-spline 표현하는 것이다 너클(Knuckle), 불연속조건, 곡률변화가 심한 구상선수(bulbous bow)와 같이 다양하고 독특한 형상을 고려하기 위해서 많은 최적화 기술이 사용되었다 최근 몇 년사이에 GA 를 이용하여 multimodal 최적화 해결책을 얻을 수 있었으며 이 방법의 가장 큰 장점은 이전보다 간단하다는 것이다 이 연구에서는, 최적화된 경계곡선을 찾아 곡면을 피팅(fitting)하기 위하여 GA 를 이용하였다 선체의 곡면을 유전자 타입으로 가정하였으며 설계 변수의 입력 값은 조정점(vertex)과 노트(knot) 값이다 이러한 변수들은 미리 정의한 정도(precision)값을 만족할 때까지 곡면의 붐질을 향상 시키기 위하여 수정되었다 두 가지 알고리즘이 개발되었다 첫 번째는 경계곡선을 고려한 알고리즘이다 Simultaneous multi-fitting GA 방법은 선수와 선미 경계곡선을 찾아내기 위해서 개발되었다 이 방법은 선수, 선미곡선부의 공통적인 노트 값을 찾는데 사용되었으며 최적화된 노트 값과 조정점을 제공한다 유사하게, bottom tangential line 이나 deck side line 같은 다른 경계곡선에도 같은 GA 기법이 적용된다 두 번째 알고리즘은 경계곡선이 최적화된 후 내부 곡면 데이터를 피팅하기 97 위해서 개발되었으며 주어진 데이터 값을 만족시키기 위해 조정점을 움직이는 GA 기술로 곡면이 생성된다 네 가지 곡면에 대해 본 연구에서 개발된 기법을 적용하였으며 적용 결과 GA 기법은 좋은 결과를 보여주고 있다 이를 이용하면 초기 설계 단계에서, 단일 NUB 곡면은 다른 CAD/CAM 프로그램으로 쉽게 변환할 수 있으며, 시각적 표현과 유한요소 방법 등에서 쉽게 이용될 수 있게 된다 즉 다음 설계단계로 쉽게 전환이 용이하다 본 연구의 결과는 초기설계 단계에서 곡면을 모델링 함에 있어서 강력한 도구를 제공한다 98 ACKNOWLEDGMENTS Korea, especially in Busan, the dynamic city with active people, kind, working hard and high technology performance, those things impress me much First and foremost, I would like to especially thank Professor Dong-Joon Kim of the Department of Naval Architecture and Marine System Engineering, my supervisor with a spirit of enterprise, who helped me to find my path and has been providing continual support, advices, and wonderful research opportunities during my studying I promise that I will my best most in the future for a better work to go in my country and in here as well In addition, I would like to thank to the committee members for my dissertation and your lectures, professor In Chul Kim, professor Yong Jig Kim, professor Ja Sam Goo, professor Sung Yong Bae, professor Dong Myung Bae, professor Sang-Mook Shin, professor Wan Don Kim and professor Jong-Ho Nam have provided wonderful feedback on my study and great suggestions for the better contribution of my dissertation I would like to thank all members of Ship Design Lab for giving me a comfortable and active environment to achieve my work: Dr 윤태경, Dr 민경철, Mr 신승우 and all of members in Lab You all, by this way or another one, have helped me a lot from the time of my first step here to the time of my graduation Also, thanks to Korean teachers in Pukyong National University and all my Vietnamese friends for their friendship and encouragement Finally, I express my gratitude and appreciation to my parents and my wife Quyen, for their love and support Pukyong National University, Busan, Korea Le Tat Hien TO MY FAMILY 99 ... Pukyong National University August 2009 Application of Real Coded Genetic Algorithm for Ship Hull Surface Fitting With a Single Non-Uniform B-spline Surface A dissertation by Tat-Hien Le Approved... data of container ship (included vertices and knot values) 93 viii Application of Real Coded Genetic Algorithm for Ship Hull Surface Fitting With a Single Non-Uniform B-Spline Surface Tat-Hien... the advantages of single rectangular B-spline surface in ship hull form design 3.1 Non-Uniform B-Spline Surface Fitting Application in Ship Hull Design The ship hull can be generated in one of