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E-raamat: Smooth Bezier Surfaces over Unstructured Quadrilateral Meshes

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Smooth Bezier Surfaces over Unstructured Quadrilateral MeshesSuurem pilt
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Using an elegant mixture of geometry, graph theory and linear analysis, this monograph completely solves a problem lying at the interface of Isogeometric Analysis (IgA) and Finite Element Methods (FEM). The recent explosion of IgA, strongly tying Computer Aided Geometry Design to Analysis, does not easily apply to the rich variety of complex shapes that engineers have to design and analyse. Therefore new developments have studied the extension of IgA to unstructured unions of meshes, similar to those one can find in FEM. The following problem arises: given an unstructured planar quadrilateral mesh, construct a C1-surface, by piecewise Bézier or B-Spline patches defined over this mesh. This problem is solved for C1-surfaces defined over plane bilinear Bézier patches, the corresponding results for B-Splines then being simple consequences. The method can be extended to higher-order quadrilaterals and even to three dimensions, and the most recent developments in this direction are also mentioned here.

 

Arvustused

This well-written monograph provides a way to solve interpolation and partial differential problems for arbitrary structures of quadrilateral meshes. The solution has a linear form, which makes it relatively simple, fast, and stable. The whole theory is illustrated by numerous examples and instructive figures. This book is an important contribution to computer-aided design over unstructured meshes. (Manfred Tasche, Mathematical Reviews, June, 2018)

1 Introduction
1(24)
1.1 Motivation
1(3)
1.2 Problem Definition
4(13)
1.2.1 Bezier Patches
4(2)
1.2.2 Description of the Problem
6(3)
1.2.3 Brief Review of Related Works
9(3)
1.2.4 The Principal Aim of the Present Work
12(4)
1.2.5 Domains of Application
16(1)
1.3 Notations
17(8)
1.3.1 Points and Vectors
17(1)
1.3.2 Polynomials
18(1)
1.3.3 Planar Mesh Data
19(1)
1.3.4 Partial Derivatives of an In-Plane Parameterisation
20(1)
1.3.5 Weight Functions
21(1)
1.3.6 3D Data of the Resulting Surface
21(3)
1.3.7 Definitions of Special Sets, Spaces and Equations
24(1)
2 G1-Smooth Surfaces
25(18)
2.1 Basic Definitions Related to Smoothness of Glued Surfaces
25(2)
2.2 The Vertex Enclosure Problem
27(3)
2.2.1 General Formulation of the Vertex Enclosure Constraint
27(3)
2.3 Linearisation of the Smoothness Constraints and Minimisation
30(9)
2.3.1 Linearisation of the Smoothness Condition
30(4)
2.3.2 Simple Example of Degree 3
34(2)
2.3.3 Linear Form of Additional Constraints
36(2)
2.3.4 Quadratic Form of the Energy Functional
38(1)
2.4 Principles of Construction of an MDS
39(4)
2.4.1 Special Subsets of Control Points and Their Dimensionality
39(1)
2.4.2 Relation Between MDSs and the Additional Constraints
40(1)
2.4.3 The Principle of Locality in the Construction of MDSs
41(1)
2.4.4 Aim of the Classification Process
41(1)
2.4.5 From MDS to the Solution of the Linear Minimisation Problem
42(1)
3 MDS: Quadrilateral Meshes and Polygonal Boundary
43(30)
3.1 Mesh Limitations
43(1)
3.2 In-Plane Parameterisation
44(1)
3.3 Weight Functions and Linear Form of G1 Continuity Conditions
45(1)
3.4 Local MDS
46(27)
3.4.1 Local Classification of E,V-Type Control Points Around a Vertex
47(2)
3.4.2 Local Classification of E,T-Type Control Points for a Single Vertex
49(11)
3.4.3 Local Classification of the Middle Control Points for a Separate Edge
60(1)
3.4.4 Middle Control Points
61(12)
4 Global MDS
73(20)
4.1 MDS of Degree 5 or More
74(1)
4.1.1 Algorithm for the Construction of a Global MDS
74(1)
4.2 MDS of Degree 4
75(13)
4.2.1 Principal Role of Classification of D-Type Control Points
75(1)
4.2.2 Examples of Possible Difficulties
76(1)
4.2.3 Sufficient Conditions and Algorithms for the Global Classification of D,T-Type Control Points
77(8)
4.2.4 Analysis of Different Additional Constraints and the Existence of MDS
85(3)
4.3 Dimensionality of MDS
88(5)
5 MDS for a Smooth Boundary
93(44)
5.1 Definitions, Mesh Limitations and In-Plane Parameterisation
93(4)
5.1.1 Definitions and In-Plane Parameterisation
94(1)
5.1.2 In-Plane Parameterisation
95(1)
5.1.3 Global In-Plane Parameterisation Π(bicubic)
96(1)
5.2 Conventional Weight Functions
97(4)
5.2.1 Weight Functions for an Edge with Two Inner Vertices
97(1)
5.2.2 Weight Functions for an Edge with One Boundary Vertex
97(4)
5.3 Linear Form of G1-Continuity Conditions
101(3)
5.3.1 G1-Continuity Conditions for an Edge with Two Inner Vertices
101(1)
5.3.2 G1-Continuity Conditions for an Edge with One Boundary Vertex
101(3)
5.4 Local MDS
104(29)
5.4.1 Local Templates for a Separate Vertex
104(3)
5.4.2 Local Classification of the Middle Control Points for a Separate Edge
107(26)
5.5 Global MDS
133(4)
5.5.1 Algorithm for the Construction of a Global MDS
133(2)
5.5.2 Existence of a Global MDS of Degree 5 or More
135(1)
5.5.3 Existence of a Global MDS of Degree 4
135(1)
5.5.4 Dimensionality of MDS
136(1)
6 Computational Examples
137(8)
6.1 Examples of MDS
137(2)
6.2 Thin Plate Problem on Irregular Quadrilateral Meshes
139(6)
6.2.1 The Thin Plate Problem
139(1)
6.2.2 Approximate Solution over a Circular Domain
139(2)
6.2.3 Approximate Solution over a Square Domain
141(4)
7 Conclusions and Further Research
145(4)
A Two Patches Geometry and G1 Construction
149(8)
A.1 Geometry
149(1)
A.2 System of Equations for the Degree 4
150(3)
A.3 Application to Degree 5 Control Points in the Plane
153(4)
B Illustrations for the Thin Plate Problem
157(6)
C Mixed MDS of Degrees 4 and 5
163(4)
C.1 Definition of Mixed MDS
163(1)
C.2 Existence of a Suitable Instance of Mixed MDS for Any Type of Additional Constraints
164(3)
D Technical Lemmas
167(8)
D.1 Bicubic In-Plane Parameterisation
167(1)
D.2 Sufficient Conditions for the Parameterisation Regularity
167(8)
E Minimisation Problems
175(8)
E.1 Stiffness Matrix for the Thin Plate Problem
175(2)
E.2 From MDS to the Minimisation Problem
177(2)
E.3 Surface Interpolation
179(4)
E.3.1 Jorg Peters' Algorithm for the Construction of a Smooth Interpolant
179(1)
E.3.2 Comparison with the Current Approach
180(3)
F G1 Is Equivalent to C1
183(4)
References 187(4)
Index 191
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