Application of real coded genetic algorithm for ship hull surface fitting with a single non uniform b spline surface

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 Depart

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

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TABLE OF CONTENTS

LIST OF FIGURES iv

LIST OF TABLES vii

LIST OF APPENDICES viii

ABSTRACT

Chapter 1 INTRODUCTION 1

1.1 State of the Art 1

1.2 About This Work 7

Chapter 2 CLASSIFICATION OF SURFACE MODELING 9

2.1 Boundary Interpolating Patch Models 9

2.1.1 Ruled Surfaces 9

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

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Chapter 3 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 4 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 5 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

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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 6 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 7 CONCLUSIONS 83

REFERENCES 86

APPENDICES 91

iv

LIST OF FIGURES

Page

Figure 1.1 de Casteljau’s algorithm for cubic curve 2

Figure 1.2 Bézier’s basic curve from intersection of two elliptic cylinders 3

Figure 1.3 Problems in composite surfaces for ship hull form 4

Figure 1.4 The mesh generation process of composite surfaces and single NUB surface 5

Figure 2.1 Linear blending of ruled surface 9

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 = 3 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

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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

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Figure 6.12 The fitness value during generations of complicated shape 78Figure 6.13 The given data points of container ship surface 78Figure 6.14 The single NUB surface and section plan based on the fitted surface at

40000th generation 79Figure 6.15 The Gaussian curvature of the container ship at 1st generation and 40,000thgeneration 80Figure 6.16 The fitness value during generations of container ship 80Figure 6.17 The Gaussian curvature distribution at each population of first and final generation for container ship 81

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Application of Real Coded Genetic Algorithm for Ship Hull Surface Fitting

With a Single Non-Uniform B-Spline Surface

Tat-Hien Le

Department of Naval Architecture and Marine Systems Engineering,

The Graduate School, Pukyong National University

Abstract

In the digital ship design process, surface modeling is required to be as accurate as possible for the effective support of ship production as well as for numerical performance analysis A traditional method for ship hull form reconstruction is skinning operation In this method, a surface model is constructed from a set of cross-sectional data However, the surface quality depends on the cross-sectional spacing and the accuracy of the characteristic curves, such as stern and bow profiles, deck side line, and bottom tangential line In addition, it is impossible to include all of the intersection curves, such as three- dimensional diagonal lines and unconnected curves, into the skinning process This means that valuable shape information is ignored during the ship hull form reconstruction process Therefore, it is difficult to obtain a high quality of hull surface with this approach

The aim of this research is to construct a single non-uniform B-spline (NUB) surface at the initial ship design stage In order to consider many different shapes and features, such as knuckles, discontinuity condition, and bulbous bow with high curvature, various optimization techniques with multiple objectives have been widely used in the surface reconstruction process In recent years, the genetic algorithm (GA) has gained increasing attention as a multimodal optimization solution for efficient surface reconstruction One of the most powerful features of this method is its simplicity

In this research, the GA has been used as a major tool to search for optimal boundary curves and to fit

a surface A preliminary hull surface is assumed to be a gene type The encoded design variables for surface construction are the location of the vertices and knot values Those variables are modified to improve the surface quality until the predefined precision is satisfied

Two algorithms for the fitting method are developed The first algorithm is to determine the boundary curves The simultaneous multi-fitting GA method is developed as an approach to find boundary curves

x

such as stem and stern profiles This method considers both stem and stern curves simultaneously and finds the common knot values for both curves Similarly, the same GA technique is applied for other boundary curves at the bottom and deck The second algorithm is employed to fit the interior data points after fitting the boundary curves The GA technique generates a final surface by manipulating the vertices to fit the interior given data points

In the application examples, four different surfaces are presented The GA technique developed in this research has proven to provide good single NUB surfaces with high efficiency Therefore, the single NUB surface can be translated into other CAD/CAM (Computer Aided Design/Computer Aided Manufacturing) programs easily In the early design stage, the single NUB surface is more convenient for visualization performance and finite element methods

The contributions of this dissertation are a simultaneous fitting of the two different NUB boundary curves and interior NUB surface using the GA and an innovative construction of a single NUB surface for geometrically complicated ship hull forms The simultaneous fitting technique provides optimal knot values and vertices arrangement for the surface reconstruction The single NUB surface can be readily translated into many CAD/CAM packages, which facilitate the smooth data transition across the different design stages These factors provide a powerful tool for the hull form construction at the initial design stage

Key words: Surface Fitting, Hull Form Reconstruction, Genetic Algorithm, Multimodal Optimization,

Simultaneous Multi-Fitting

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Chapter 1 INTRODUCTION

The term CAD/CAM technique is to describe the task of designing and producing throughout a computer process For over a century, a CAD/CAM technique has been developed to visualize the three-dimensional (3D) modeling effectively The 3D surface modeling has been studied by mathematicians, engineers, and scientists A considerable amount of research has been carried out to represent the mathematical of the shape using the CAD system During the 1970s, the earliest applications of CAD were developed so that the geometrical modeling could be used for larger projects such

as aircraft, automobile or ship hull The shape optimization for surface modeling is the ultimate goal of ship hull form design

1.1 State of the Art

Surface modeling is the key to integration of ship design, analysis, manufacturing, and other calculation (Rogers et al 1983) The applications of surface modeling have

to be concerned in manufacturing so that the object can be machined on NC machines From that time, industrial applications for the representation, design, and data exchange of geometric information processed by computer have been in use In the basic ship hull design stage, while a number of difficult representational and integration problems are yet to be solved completely (for example, surface quality in modeling of design and analysis), the ship lines is sufficient to support by intersecting the hull surface with three sets of orthogonal planes The intersecting points are known

as the offset data of ship hull form The shape reconstruction from these offset points is more important than before, because the accurate modeling of hull surface has been considered as important factor for the effective support of the ship production as well

as for numerical ship performance analysis (Lee and Kim 2004) We classify 3D surface reconstruction problems based on shape representations into the following steps:

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(a) Scope A surface representation must be able to describe all classes of shapes in

ship hull form (for example, stern part, chine hull, and bulbous bow)

(b) Accuracy A surface reconstruction must be satisfied the predefined precision (c) Efficiency A modeling must be possible to efficiently compute and visualize by

computer graphics

(d) Support Information of shape must be preserved and, if required, should be able

to exchange to other CAD systems easily

In order to represent the surface models effectively, many research papers have been noticed over the years The US aircraft company Boeing employed the software based

on J Ferguson’s published reports in the late 1950s Ferguson’s bicubic patches were also known as C1 F-patches A cubic Hermite form is defined in terms of two endpoints and two endpoint derivatives Later, Coons used this type to fit a patch between four boundary curves, known as the bilinearly blended Coons patch The extension of a general boundary Coons patch is the Gregory patch which solved the twist compatibility at patch corner of Coons problem (Gregory 1974) The surface which is defined by Gregory patches interpolation can be used for ship hull design (Park and Kim 1994)

Figure 1.1 de Casteljau’s algorithm for cubic curve

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Figure 1.2 Bézier’s basic curve from intersection of two elliptic cylinders

Other researches, in 1959, de Casteljau adopted the use of Bernstein polynomial as

de Casteljau algorithm that was kept a secret by Citroen Automobile for a long time

An iterated affine is combined from polygons or polygonal nets for the cubic case (see

in Fig 1.1) The Casteljau’s technical report was found out by W Boehm in the late seventies Also, during the early 1960, the original formulation of the Bézier was developed by Pierre Bézier at Renault’s design department (see in Fig 1.2) The significance of Bernstein in Bézier’s method was discovered by A R Forest in 1972 Later, Forest’s article helped to popularize the Bézier curves and surface in the Renault CAD/CAM system UNISURF The B-spline formulation followed in 1972 with research by de Boor (1972) In 1974, a first B-spline to Bézier conversion was found

by Gordon and Riesenfeld The combination of Bézier and spline shows that the spline is a generalization of Bézier form Together, Bézier and B-spline techniques become a core technique of almost all CAD systems Especially, in the shipbuilding field, in order to ensure the surface quality of complicated shape at bow, stern and middle part, the ship hull has to be divided into some patches (Westgaard and Nowacki

B-2001, Lee and Kim 2004) However, the disadvantages are caused by the continuing

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patches problem, inconvenient handling in later design (CAE), and difficulties in automatic fairing (for example, the continuity order of Bézier patch and the continuing problem between Gregory patches) Although, the composite surfaces are B-spline surfaces, the boundaries between surfaces should be the same characteristic (knot value of surface, vertices of surface, order of surface, etc) when the continuity condition is required), as shown in Fig 1.3

Figure 1.3 Problems in composite surfaces for ship hull form

Also, the surface approximation from irregular data has been developed In 1976, de Boor first presented a brief description of multivariate simplex splines such as triangular B-spline (DMS spline) In 1993, Fong and Seidel introduced the polynomial surfaces with degree n can be Cn-1 continuity if their knots are in general positions However, the reproduction of Cn-1 piecewise polynomial functions such as C2continuity could not be settled easily In additional, the accuracy of the resulting surface depends on the density of the triangular patch It means that the patch must be very fine to meet accuracy requirements This condition requires too much memory, execution computation for multi-triangular patches, and continuity problem

In an early design stage, geometry performance and mesh generation can be very time consuming Geometry is translated into a CAD system format (IGES, DXF, STEP, etc) and the mesh generation translates it back into the analysis environment for CFD and other finite element methods The accuracy of parametric geometry information

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can be lost if the shape of ship has complex features, especially for the continuities of composite surfaces During the mesh generation process, Fig 1.4 illustrates the grid structure of composite surfaces and single surface at bow part of ship

Figure 1.4 The mesh generation process of composite surfaces and single NUB

surface

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Instead of subdividing a hull form into patches, it is advantageous to work with the single B-spline surface for ship hull The single surface is more convenience for visualization performance, analysis calculation, and easy processing of the fairness of the whole surface, etc The approach of B-spline surface construction has also received lots of attention in recent years B-spline is established as a standard in CAD program B-spline surface approximation has been used widely in the Autoship, Maxsurf ship design software, etc Based on B-spline techniques, the surface reconstruction have been developed quickly The ship hull surface reconstruction from the given data points is also known as a surface fitting technique

There are two principle categories for surface fitting techniques The first one works with reverse engineering in two steps; first the given data points are rearranged into the rectangular mesh, then the surface is constructed by some matrices inversion and B-spline algorithm procedures There are several papers where combinations of above strategies such as Rogers et al (1983) and Choi (1991) The first problem is that the data points are often noisy and unevenly distributed such as the bow part and the stern part of ship The second problem comes from the matrices inversion In fact, the matrices inversion gets the ill conditioned problem, and it must also avoid round off error magnifications in back- substitution calculation and large storage capacity (Mathews and Fink 1992) This approach required mathematical knowledge in order to design with and proved difficult to control (see section 3.4) The second surface fitting technique is to approximate the given data points by B-spline algorithm Many conventional methods have been proposed (Riesenfield and Gordon 1974, Choi 1991) The main problem is the parameterization of B-spline surface It needs to be estimated from an initial unknown surface

The surface fitting with tight tolerances is a highly non-linear problem, since we do not know the ideal number of parameter values (for example, knot vector values and control vertex values) The basic requirements of surface fitting can be summarized as

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follows The surface must be approximated each data point within acceptable tolerance Fitting process is also expected that the surface is obtained in a reasonable way with high quality of shape Many previous researches approached to automated B-spline surface modeling mathematically The approximation of surface fitting is developed by Birmingham and Smith (1998) In this research, the preliminary hull surface is generated in a gene type and genetic algorithm (GA) technique can be applied for the convergent approximation toward a good solution (see section 4.2)

1.2 About This Work

The most frequently used technique in ship design is skinning method (Jensen and Baatrup 1989) The skinning approach to re-creating a set of given data point is to convert the stations into a surface model and take advantage of the capabilities of hull surface modeling To avoid the difficulties of discontinuity problem in skinning section curves to surface, Lu introduced the waterlines skinning method for surface reconstruction (Lu et al 2008) However, the disadvantage of skinning method is that the accuracy of surface is depends on the distance between sections (or waterlines) for complicated shape and the unevenly distributed of given data points at each section (or waterlines), etc If the hull shape is required to recreate as accurately as possible, then the surface construction should be performed as the best solution with high quality The key to approximate a good surface model is to minimize the difference between modeling and data function as small as possible Knot values and location of vertices were considered as variables However, in practice, this multimodal optimization problem cannot be solved easily Some methods have been introduced to determine a good knot by Juup (1978) (good initial knots required) and Dierckx (1993) (error tolerance showed) In fact, these methods are not sufficient, anyway

Yoshimoto described an optimization of knots by using GA and matrices inversion

in which the knot value can be optimized for any type of curves (Yoshimoto et al

8

2003) However, the given data point of ship hull form is irregular at complicated bow and stern part The matrices inversion is not a solution in that case Furthermore, in surface fitting, the knot value in each direction should be the same Although the boundary is good for this part, but the knot optimization for curve fitting may be limited in other sections (or waterlines)

The main contribution of this dissertation are a simultaneous fitting of the two different NUB boundary curves and interior NUB surface using the GA and an innovative construction of a single NUB surface for geometrically complicated ship hull forms The single surface has more advantages than curve representation, surface patches in geometric continuity, boundary conditions, and fairness solution Two algorithms for the fitting were developed The first algorithm was employed to fit through four boundaries of the hull ship (aft profile, fore profile, keel line, and deck side line) by simultaneous multiple curve fitting implementation An alternative adaptive adjustment of multiple vertex point process would be suggested to improve the quality of shape The changes at these vertices will get the stable result The second fitting algorithm was employed to achieve interior given data points Obviously, the difficulties of knot optimization can be solved in this way (see Sec.5.4) In that case, the GA can be robust and efficient for the surface fitting of ship hull form The knot values and location of vertices are modified to improve the surface quality by using the genetic techniques until predefined precision is reached The example applications showed the possibility of the proposed the system model and controlled the shape of NUB as well GA technique, again, improves NUB shape efficiently This procedure is succeeded in the problem of knuckle points and the complicated shape at bow and stern part of ship

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Chapter 2 CLASSIFICATION OF SURFACE MODELING

Wireframe model is one of the earliest geometric modeling techniques The traditional drawing of ship’s lines is a typical wireframe model The buttock lines, waterlines, and sections are created by intersecting the hull surface with three sets of orthogonal planes The main disadvantage of wireframe is the CAD/CAM systems require a full surface description of the hull form The quality of surface modeling is the ultimate goal of ship hull form design Practically, there are three types of surface models available in the literature: boundary interpolating patch models, parametric polynomial patch models, and irregular patches

2.1 Boundary Interpolating Patch Models

The boundary interpolating patch models are constructed by interpolating to a set of boundary curves Popular models in this category include ruled surfaces, loft surfaces, Coons patches, and Gregory patches (Choi 1991)

2.1.1 Ruled Surfaces

A linear blending of the two parametric curves in Eq (2.1), r0(u) and r1(u) with

0u1, defines a ruled surface patch (see Fig 2.1)

 ,  1   0 1  0 , 1

r u w  w r u wr u u w (2.1)

Figure 2.1 Linear blending of ruled surface

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2.1.2 Lofted Surfaces

In Fig 2.2, lofted surface is the case where the boundary curves together with their cross-boundary tangents are given:

Figure 2.2 Lofted surface construction

ri(u) for i = 0,1 : boundary curves and

ti(w) for i = 0,1 : cross-boundary tangents

A lofted surface is constructed by Hermite blending functions 3 

2.1.3 Bilinear Blended Coons Patch

In technical report in 1964, Coons recommended a simple surface to fit a patch between any four arbitrary boundary curves The Coons patch equation contains the sum of the two ruled surfaces r1(u,w), r2(u,w) and the correction surface r3(u,w) (see

Figure 2.3 Bilinear blended Coons patch

The advantages of using bilinear Coons patch are its stability and fast It is also easy

to compute the technique However, bilinear Coons patches are only C0 across their boundaries To overcome the lack of tangent continuity, the bicubic Coons patch is an extension of linear Coons patch using Hermite blending functions

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2.1.4 Bicubic Coons Patches

The bicubic Coons patch has the same form as the bilinear Coons patch except that the ruled surfaces are cubic blended The bicubic Coons patch can be defined if the cross boundary tangent of each boundary is available

b 1 (u)

a 1 (w) w

b 0 (u) u

Figure 2.4 Bicubic Coons patch

In Fig 2.4, s(w) and t(u) are the cross boundary tangents along the lofted curves a(w) and b(u) Then, a cubic (Hermite) blended surface is expressed in Eq (2.4) by two lofted surfaces r1(u,w), r2(u,w) and the correction surface r3(u,w):

a(w), b(u): boundary curves

Pij, xij : position and twist vectors at patch corners

The bicubic Coons patch is better than bilinear blended Coons patch but still has some disadvantages In fact, when some patches are complicated, the bicubic Coons patch tends to include more wiggles In addition, the points close to the boundaries are better than the interior points because of the computation of cross-derivatives (Piegl and Tiller 2001) In piecewise Coons surface, Farin assumed that a surface can be constructed by given network of curves with bicubic blended Coons patches The continuity condition of resulting surface will be C1(first derivative)

Another problem comes from the cross-derivatives (twist) which may influence to local surface oscillation If the two twist xuw and xwu at each corner patch are equal, there is no problem, and the bicubic blended Coons patch is well-defined However, as Fig 2.5 illustrated, two twist values yield the surface that only partially interpolates to the given data points

Figure 2.5 Twist problem at corners in bicubic blended Coons patch

to create a larger surface

C2 continuity could not be settled easily A triangular B-spline surface of degree n over

an arbitrarily triangulated domain is the combination of a set of basis functions with vertices For every triangle I and every triple of indices  with  n, we associate a vertice P,in R3:

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Figure 2.7 Surface reconstruction from rectangular patches and triangular patches

A flexible CAD system should perform a good quality of shape and an interactive manipulation The B-spline vertex point modification actually works with user interface and produces an exactly shape For these reasons, CAD system preferred the rectangular patch techniques

2.3 Parametric Polynomial Patch Model

The parametric polynomial models are widely used in CAD system, especially in the ship design stage because the cubic function is the minimum degree polynomial equation that provides enough flexibility for surface construction (see Fig 2.8) The cubic models are standard polynomial surface patches, Ferguson surface patches, Bézier surface patches and B-spline surface patches The parametric polynomial patch emphasized the geometric intuition, while the boundary interpolating patch required the algebraic concepts such as Boolean sums The distinction is that the boundary of interpolating patch is the arbitrary form, whereas the boundary of polynomial patch must also be polynomial curves

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Figure 2.8 Parametric surface and its parameters

2.3.1 Standard Polynomial Surface Patch

In Fig 2.9, the standard polynomial surface is given by Eq (2.8):

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Figure 2.9 Standard polynomial surface

2.3.2 Ferguson Surface Patch

Around 1960, Ferguson at Boeing announced a method for describing a parametric curve and surface patch with end points and end tangents:

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the end tangents These techniques required mathematical knowledge in order to design with and proved difficult to control

2.3.3 Bézier Surface Patch

In 1970, the problem of interactive control curves and surfaces were overcome when Bézier, an originator of UNISURF, used by Renault car manufactures The concept of shape control was introduced based on vertex points Bézier developed a reformulation

of Ferguson patch in terms of Bernstein polynomials for UNISURF at Renault As shown in Eq (2.10), the cubic Bézier patch is defined by:

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Figure 2.10 Bézier patch

Bézier surfaces in Fig 2.10 exhibit the convex hull properties, end-point interpolation, and Bézier boundaries A major restriction of Bézier model is that:

 The continuity between patches is G1 continuity (at least G2 continuity requirement for ship design stage)

 Global control

However, all of above methods have trouble with continuity conditions between segment joining Therefore, a definition of “spline” was introduced into the field of geometric modeling Gordon and Riesenfeld first introduced B-spline into this area B-splines incorporated the same aspect as Bézier scheme but non-global behavior

2.3.4 Uniform B-Spline Surface Patch

The solution to every problems outlined above is to use piecewise polynomial spline surface The Bézier patch is a special case of B-spline patch The use of B-spline surfaces in CAGD was introduced by Gordon and Riesenfield (1974) B-spline surface consists of more than one patch, each of which has a low degree In Fig 2.11, the bicubic uniform B-spline surface patch (UB) is expressed in Eq (2.12) as a linear combination of the basis functions:

N u  u

Figure 2.11 Uniform B-spline patch

2.3.5 Non-Uniform B-Spline Surface Patch

NUB surfaces possess almost the same properties of Bézier patches They have affine invariance and convex hull properties The bicubic NUB surface patch is defined

N : coefficient matrix with knot spans  j

The discussion of properties of NUB will be continued in the next section with more details and more advantageous techniques

2.3.6 Definition and Properties of Knot Vector

The total number of knots equals the number of vertex points in each u, w degree of freedom plus the surface’s order

1 1

j

i i i

2.3.7 Definition and Properties of Non-Uniform B-Spline Basis Function

The B-spline basis function is generally non-global The non-global behavior is due

to the fact that each vertex point is associated with a unique basis function Thus, the effect of vertices occur only the range of nonzero parameter values For the ithnormalized NUB basis function of order k (degree k-1), the basis function Ni,k(u) and

Mj,l(w) are defined by Cox-de Boor recursive algorithm in Eq (2.16):

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,

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2.3.8 Non-Uniform B-spline Surface from 3D Data Array

In CAD application, the complicated shape cannot be represented by a single Bézier surface Increasing the degree of Bézier adds flexibility to the surface but may cause numerical noise in computation Furthermore, it is difficult to evaluate and manipulate the Bézier surface of high degree For these reasons, Bézier surface construction with piecewise techniques is presented by Bézier (1974) and Choi (1991) Each subdivided Bézier patch can be represented by a lower degree

Although the composite Bézier surface may be used to describe the complex shape

in CAD systems, there are two main disadvantages: order of derivatives continuity and

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global control These problems can be eliminated by using B-spline technique The idea of using NUB surface in shipbuilding is initiated by Rogers (1977)

Figure 2.12 NUB surface fitting from the given data points

This section considers approaches to surface modeling that was fitted from the given data points (shown as Fig 2.12) Of course, the quality of shape depends on the quality

of given data points and the characteristic of surface type Fig 2.13 illustrates the relationship between the vertex points in 3D coordinates and the resulting NUB surface

Figure 2.13 NUB surface and vertices

n and m are the number of vertex points

u and w are the parameter such that: umin uumaxand wmin wwmax

Bi,j is the position vector of the vertex point

Ni,k(u) and Mj,k(w) are the basis functions associated with each vertex point

From Eq (3.1), it can be seen that the NUB surface can be greatly affected by the knot vector and the location of the vertex point In the GA process, the shape of the surface is changed automatically in this way