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Finite Element Mesh Generation [Kõva köide]

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  • Formaat: Hardback, 672 pages, kõrgus x laius: 254x178 mm, kaal: 1460 g, 64 Tables, black and white; 607 Illustrations, black and white
  • Ilmumisaeg: 15-Jan-2015
  • Kirjastus: CRC Press
  • ISBN-10: 041569048X
  • ISBN-13: 9780415690485
Teised raamatud teemal:
Finite Element Mesh GenerationSuurem pilt
  • Formaat: Hardback, 672 pages, kõrgus x laius: 254x178 mm, kaal: 1460 g, 64 Tables, black and white; 607 Illustrations, black and white
  • Ilmumisaeg: 15-Jan-2015
  • Kirjastus: CRC Press
  • ISBN-10: 041569048X
  • ISBN-13: 9780415690485
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780415690485.html
Märksõnad:
Preface Nowadays, the finite element method has diverse applications to problems in science and engineering ranging from simple two-dimensional static elasticity, non-linear large deformation analysis to three-dimensional fluid dynamic problems with shock waves. The prerequisite for a finite element analysis is a sound and valid finite element mesh, which can only be constructed efficiently by means of some well-devised and thoroughly tested computer algorithms. In contrast to the numerous textbooks, monographs, journal papers, etc., on the finite element method, comprehensive and concise accounts on mesh generation technologies seem to have been missing, except perhaps the book Mesh Generation: Application to Finite Elements written by P.J. Frey and P.L. George some 15 years ago. Anyway, finite element mesh generation has not been taken as a formal subject of teaching in universities, as it encompasses several disciplines including classical geometry, computational geometry and topology, finite elementmethod, data structures and algorithms, computer programming and, to a certain extent, even computer graphics. With the ever-improving performance of PCs, large-scale challenging engineering simulations and scientific computations by means of the finite element method are more accessible to daily design operations and even to research students. In line with this development, the mesh generation methodology is becoming increasingly recognised as a subject in its own right. As meshing technologies and their applications in new areas have developed pretty rapidly over the recent years, it is imperative to review and consolidate the progress in meshing technologies achieved thus far into a concise yet comprehensive text with a logical-- Highlights the Progression of Meshing Technologies and Their ApplicationsFinite Element Mesh Generation provides a concise and comprehensive guide to the application of finite element mesh generation over 2D domains, curved surfaces, and 3D space. Organised according to the geometry and dimension of the problem domains, it develops from the basic meshing algorithms to the most advanced schemes to deal with problems with specific requirements such as boundary conformity, adaptive and anisotropic elements, shape qualities, and mesh optimization.It sets out the fundamentals of popular techniques, including:Delaunay triangulationAdvancing-front (ADF) approachQuadtree/Octree techniquesRefinement and optimization-based strategiesFrom the geometrical and the topological aspects and their associated operations and inter-relationships, each approach is vividly described and illustrated with examples. Beyond the algorithms, the book also explores the practice of using metric tensor and surface curvatures for generating anisotropic meshes on parametric space. It presents results from research including 3D anisotropic meshing, mesh generation over unbounded domains, meshing by means of intersection, re-meshing by Delaunay-ADF approach, mesh refinement and optimization, generation of hexahedral meshes, and large scale and parallel meshing, along with innovative unpublished meshing methods. The author provides illustrations of major meshing algorithms, pseudo codes, and programming codes in C++ or FORTRAN.Geared toward research centers, universities, and engineering companies, Finite Element Mesh Generation describes mesh generation methods and fundamental techniques, and also serves as a valuable reference for laymen and experts alike.

Arvustused

" the present monograph fills a gaping hole in the literature on scientific computing it could be subtitled by 'All you want to know about mesh generation'." Zentralblatt MATH, 2015

"This book provides a well-structured and thorough treatment of very recent research on mesh generation, in a single well organized document. The descriptions of methods and algorithms are complete and provide readers with all the necessary information needed to implement on their own the algorithms and methods discussed in the book. Numerical examples provide concrete measures of the performance of the algorithms and can serve as a reference for those interested in validating their own implementations."

Francois Guibault, Polytechnique Montreal, Canada

"there are very few books on this subject. Daniel Los book (examining the contents) offers a practical point of view and gives details on some topics which, I think, are not cover[ ed] by the other books on meshing techniques." Houman Borouchaki, Université de Technologie de Troyes, France

"This book brings together the major propulsion system components with control oriented models and actuators to enable software and hardware-in-the-loop simulations. This book will provide students with a detailed set of component models and simulation tools to learn Rapid Control Prototyping methods." Douglas J. Nelson, Professor of Mechanical Engineering, Virginia Tech

"This book is aimed at those who want a comprehensive overview of the techniques of finite-element mesh generation. The techniques and algorithms are clearly explained and there are good references to follow up where greater detail is required. However, there is probably a broader readership among practising engineers, who use the finite-element method on a daily basis, and who want a better understanding of the tools they rely on as a basis for their calculations."

Stephen Hendry, Engineering and Computational Mechanics " the present monograph fills a gaping hole in the literature on scientific computing it could be subtitled by 'All you want to know about mesh generation'." Zentralblatt MATH, 2015

"This book provides a well-structured and thorough treatment of very recent research on mesh generation, in a single well organized document. The descriptions of methods and algorithms are complete and provide readers with all the necessary information needed to implement on their own the algorithms and methods discussed in the book. Numerical examples provide concrete measures of the performance of the algorithms and can serve as a reference for those interested in validating their own implementations."

Francois Guibault, Polytechnique Montreal, Canada

"there are very few books on this subject. Daniel Los book (examining the contents) offers a practical point of view and gives details on some topics which, I think, are not cover[ ed] by the other books on meshing techniques." Houman Borouchaki, Université de Technologie de Troyes, France

"This book brings together the major propulsion system components with control oriented models and actuators to enable software and hardware-in-the-loop simulations. This book will provide students with a detailed set of component models and simulation tools to learn Rapid Control Prototyping methods." Douglas J. Nelson, Professor of Mechanical Engineering, Virginia Tech

"This book is aimed at those who want a comprehensive overview of the techniques of finite-element mesh generation. The techniques and algorithms are clearly explained and there are good references to follow up where greater detail is required. However, there is probably a broader readership among practising engineers, who use the finite-element method on a daily basis, and who want a better understanding of the tools they rely on as a basis for their calculations."

Stephen Hendry, Engineering and Computational Mechanics

Preface xvii
Acknowledgements xix
1 Introduction 1(10)
1.1 Finite element method
1(1)
1.2 What is finite element mesh generation?
1(1)
1.3 Why finite element mesh generation?
2(1)
1.4 Problem definition, scope and philosophy: Science or art?
3(1)
1.5 General strategies, robustness, difficulties and methodologies
4(1)
1.6 Mathematics
4(1)
1.7 Historical development
5(2)
1.8 So far achieved and what lies ahead
7(1)
1.9 Topics discussed in the chapters
8(3)
2 Fundamentals 11(66)
2.1 Introduction
11(1)
2.2 Notations, symbols and abbreviations
12(2)
2.2.1 Notations
12(1)
2.2.2 Symbols
12(1)
2.2.3 Abbreviations
13(1)
2.3 Terminologies and data structures
14(5)
2.3.1 Triangulation
14(1)
2.3.2 Delaunay triangulation
14(1)
2.3.3 Constrained triangulation
14(1)
2.3.4 Mesh and FE mesh
14(1)
2.3.5 Structured and unstructured meshes
15(1)
2.3.6 Mixed and hybrid meshes
15(1)
2.3.7 Discretised manifold
16(1)
2.3.8 Control space
16(1)
2.3.9 Adaptive mesh
16(1)
2.3.10 Data structure
16(3)
2.3.10.1 Nodal points
17(1)
2.3.10.2 Boundary of a planar domain
17(1)
2.3.10.3 Boundary of a 3D domain
17(1)
2.3.10.4 Node labelling of FEs
18(1)
2.4 Geometrical operations and formulas
19(14)
2.4.1 Distance from a point P to a line segment AB, d(P, AB)
19(1)
2.4.2 Distance from a point P to a triangular facet ABC, d(P, ABC)
20(1)
2.4.3 Distance between line segments in space, d(AB, CD)
20(1)
2.4.4 Intersection between two line segments on a plane
21(2)
2.4.4.1 Analytical method
21(1)
2.4.4.2 Vectorial method
22(1)
2.4.4.3 Parametric method
22(1)
2.4.4.4 The max—min method
23(1)
2.4.5 Solid angle
23(2)
2.4.6 Normal at a node
25(1)
2.4.7 Intersection between a line segment and a triangular facet
25(1)
2.4.8 Distance between a line segment and a triangular facet in space, d(PQ, ABC)
26(1)
2.4.9 Dividing an edge into segments
27(3)
2.4.9.1 Element size is specified at nodal points
27(1)
2.4.9.2 Element size is specified along the edge
28(2)
2.4.10 y Value of a tetrahedron cannot exceed the a value of its face
30(2)
2.4.11 Determine whether a point is inside or outside of the problem domain
32(1)
2.4.11.1 Two-dimensional domain
32(1)
2.4.11.2 Three-dimensional domain
32(1)
2.5 Topological operations and algorithms
33(10)
2.5.1 Find the neighbouring elements of a triangular mesh
33(1)
2.5.2 Find the neighbouring elements of a tetrahedral mesh
34(1)
2.5.3 Find the elements connected to each node in a mesh
35(2)
2.5.4 Find the edges (unique line segments) of a triangular mesh
37(1)
2.5.5 Find the faces (unique triangular facets) of a tetrahedral mesh
38(1)
2.5.6 Find the edges (unique line segments) of a tetrahedral mesh
38(1)
2.5.7 Retrieve the boundary (loop of line segments) of a triangular mesh
39(1)
2.5.8 Retrieve the boundary (triangular facets) of a tetrahedral mesh
40(1)
2.5.9 Find the tetrahedral elements connected to an edge
41(1)
2.5.10 Delete flagged elements from a tetrahedral mesh
41(1)
2.5.11 Find the tetrahedral elements within the boundary surface
42(1)
2.6 Sorting
43(10)
2.6.1 Bubble sort
43(1)
2.6.2 Insertion sort
44(1)
2.6.3 Quick sort
45(2)
2.6.4 Bin sort
47(3)
2.6.5 Comparison of the sorting methods
50(3)
2.7 Background grid
53(24)
2.7.1 Regular (uniform) grid (2D)
53(1)
2.7.2 Regular (uniform) grid (3D)
54(2)
2.7.3 Searching for general objects by means of a background grid
56(2)
2.7.3.1 Method 1: Search by neighbourhood
56(1)
2.7.3.2 Method 2: By checking the distance
57(1)
2.7.3.3 Method 3: By elimination
57(1)
2.7.4 Determine the cells intersected by a triangular facet
58(1)
2.7.5 Irregular grid
59(2)
2.7.6 Quadtree
61(5)
2.7.7 Octree
66(3)
2.7.8 Kd-tree
69(9)
2.7.8.1 Construction of 2-d tree
69(4)
2.7.8.2 Construction of 3-d tree
73(4)
3 Mesh generation on planar domain 77(88)
3.1 Introduction
77(1)
3.2 Structured mesh on planar domain
78(5)
3.2.1 FE interpolation
78(3)
3.2.2 Trans finite mapping
81(2)
3.2.3 Drag method and sweeping method
83(1)
3.3 Unstructured mesh on planar domain
83(3)
3.3.1 MG using contour
84(1)
3.3.2 Coring method
84(1)
3.3.3 Mesh refinement by subdivision
85(1)
3.4 Meshing by quadtree decomposition
86(5)
3.4.1 Boundary specification
86(1)
3.4.2 Spatial partition of the bounding box
86(4)
3.4.3 Creation of internal points and elements
90(1)
3.4.4 Connection of the interior elements with the boundary segments
90(1)
3.5 Delaunay triangulation
91(30)
3.5.1 Introduction
91(1)
3.5.1.1 The convex hull of a given point set
92(1)
3.5.2 Properties of DT
92(1)
3.5.3 Time complexity in the construction of DT
93(1)
3.5.4 FE meshing by DT
93(13)
3.5.4.1 Fundamentals and strategy
95(2)
3.5.4.2 Point insertion algorithm
97(1)
3.5.4.3 Determination of the CORE
98(1)
3.5.4.4 Searching for the BASE
98(2)
3.5.4.5 Steps in locating the BASE
100(1)
3.5.4.6 Circumcentre and circumcircle
101(1)
3.5.4.7 Procedure for the creation of the CORE
102(3)
3.5.4.8 Correction of the CORE
105(1)
3.5.4.9 Construction of triangles in the CORE and establishment of the adjacency relationship
106(1)
3.5.5 Details in computer programming
106(2)
3.5.6 Generation of interior points
108(9)
3.5.6.1 Specification of nodal spacing
108(1)
3.5.6.2 Control space
108(1)
3.5.6.3 Element size based on domain boundary
109(1)
3.5.6.4 Element size based on a previous analysis
110(1)
3.5.6.5 Creation of interior points
111(6)
3.5.7 Boundary recovery in two dimensions
117(3)
3.5.7.1 Determination of the pipe
118(1)
3.5.7.2 Divide-and-conquer
118(1)
3.5.7.3 Swapping of diagonals
119(1)
3.5.8 Closure
120(1)
3.6 Advancing front approach
121(19)
3.6.1 Introduction
121(2)
3.6.2 Adaptive meshing by the AFT
123(5)
3.6.3 Use of background grid
128(6)
3.6.3.1 Construction of the background grid
129(1)
3.6.3.2 Setting the size of each cell in the grid
129(2)
3.6.3.3 Marking and unmarking cells intersected by a line segment
131(1)
3.6.3.4 Marking cells intersected by a line segment L
132(1)
3.6.3.5 Unmarking cells intersected by a line segment L
133(1)
3.6.3.6 Search for nearby line segments with the help of the background grid
133(1)
3.6.3.7 Updating boundary segments
133(1)
3.6.4 Test examples
134(5)
3.6.5 Closure
139(1)
3.7 Meshing by a combined scheme of DT and ADF approach
140(7)
3.7.1 Introduction
140(1)
3.7.2 Advancing-front—Delaunay scheme
140(4)
3.7.2.1 DT of non-convex planar domains
140(1)
3.7.2.2 Delaunay and non-Delaunay triangles
140(1)
3.7.2.3 Delaunay and non-Delaunay segments
141(1)
3.7.2.4 Triangulation process
141(1)
3.7.2.5 Updating Γ1 and Γ2
142(1)
3.7.2.6 Existence and Delaunay property of the triangulation
142(1)
3.7.2.7 Delaunay property of triangulation
143(1)
3.7.3 Delaunay—advancing-front scheme
144(3)
3.8 Enhanced quadtree meshing
147(4)
3.8.1 Quadtree partition of the bounding box
148(1)
3.8.2 Removal of quadrilaterals near domain boundary
148(1)
3.8.3 Boundary recovery for triangulation
149(1)
3.8.4 Advancing-front MG
149(2)
3.9 Quadrilateral mesh
151(14)
3.9.1 Direct method
151(1)
3.9.2 Indirect method
152(1)
3.9.3 Quadrilateral-dominated mesh
153(4)
3.9.3.1 Distortion coefficient β of a quadrilateral
154(1)
3.9.3.2 Merging of triangles to form quadrilaterals
155(2)
3.9.4 All-quad unstructured mesh
157(2)
3.9.4.1 Initialisation of the merging front
158(1)
3.9.4.2 Merging of triangles
158(1)
3.9.4.3 Updating merging front
159(1)
3.9.4.4 Complete conversion to quadrilateral mesh
159(1)
3.9.5 Mesh quality enhancement
159(3)
3.9.5.1 Elimination of node
161(1)
3.9.5.2 Elimination of element
161(1)
3.9.5.3 Swapping of diagonals
161(1)
3.9.5.4 Elimination of segment
162(1)
3.9.6 Examples of quadrilateral meshes
162(3)
4 Mesh generation over curved surfaces 165(74)
4.1 Introduction
165(4)
4.1.1 Parametric meshing for curved surfaces
165(2)
4.1.2 Direct mesh generation on surfaces
167(2)
4.1.3 Surface meshing by means of intersection
169(1)
4.2 Parametric mapping method
169(32)
4.2.1 Introduction
169(1)
4.2.1.1 The mapping φ from planar domain Ω to the surface S
169(1)
4.2.1.2 Gap between a triangular facet and the curved surface
170(1)
4.2.1.3 Metric for curved surface geometry
170(1)
4.2.2 Fundamental forms and the related metric
170(2)
4.2.2.1 Tangent and normal vectors
170(2)
4.2.3 Principal curvatures
172(2)
4.2.3.1 Gaussian curvature and mean curvature
174(1)
4.2.4 Metric and principal curvatures
174(2)
4.2.5 Geometrical control
176(2)
4.2.6 Metric on parametric planar domain
178(1)
4.2.7 Metric tensor and Green—Cauchy deformation tensor
179(3)
4.2.7.1 Change in length by metric M
180(1)
4.2.7.2 Change in area by metric M
180(2)
4.2.8 Interpolation of metric
182(2)
4.2.8.1 Metric interpolation over a line segment
182(1)
4.2.8.2 Metric interpolation within a triangular element
183(1)
4.2.9 Lengths controlled by multiple metrics
184(2)
4.2.10 Element shape measure with respect to anisotropic metric
186(2)
4.2.11 Metric tensor field of parametric curved surfaces and its characteristics
188(3)
4.2.12 Generation of anisotropic mesh by the Delaunay—ADF method
191(7)
4.2.12.1 Steps for anisotropic meshing
191(5)
4.2.12.2 The completed mesh
196(2)
4.2.13 Optimisation of anisotropic meshes
198(3)
4.2.13.1 Node smoothing
198(1)
4.2.13.2 Diagonal swapping
199(1)
4.2.13.3 Optimisation of the wavy surface
200(1)
4.3 Mesh generation by packing ellipses
201(11)
4.3.1 Ellipse-packing algorithm
202(4)
4.3.1.1 Data structure
202(1)
4.3.1.2 Three criteria for ellipse packing
202(1)
4.3.1.3 Unit metric
202(1)
4.3.1.4 Initial pack and coefficient β
203(1)
4.3.1.5 Fitting an ellipse to the existing pack
204(1)
4.3.1.6 Checking intersection and mesh generation
204(2)
4.3.2 Efficiency and complexity
206(2)
4.3.3 Examples of surface meshing by ellipse packing
208(4)
4.4 Direct mesh generation on surface
212(10)
4.4.1 Initial generation front
213(1)
4.4.2 Forming triangular elements on a surface
214(5)
4.4.2.1 Find the best node on the generation front
215(1)
4.4.2.2 Locate interior node
216(1)
4.4.2.3 Space-to-surface projection
217(2)
4.4.3 Examples of direct construction
219(3)
4.5 Mesh generation by surface intersection
222(15)
4.5.1 Introduction
222(1)
4.5.1.1 The determination of neighbours
223(1)
4.5.2 Background grid
223(2)
4.5.2.1 Determination of cells intersected by a triangular facet
224(1)
4.5.3 Find all the candidate triangles
225(2)
4.5.3.1 Calculating the intersection between a pair of triangular facets
226(1)
4.5.4 Tracing neighbours of intersecting triangles
227(2)
4.5.5 Time complexity and memory management
229(1)
4.5.6 Mesh generation along intersection lines
230(1)
4.5.7 Work examples
231(5)
4.5.8 Intersection of surfaces of quadrilateral elements
236(4)
4.5.8.1 Closure
236(1)
4.6 Quadrilateral surface mesh
237(2)
5 Mesh generation in three dimensions 239(88)
5.1 Introduction
239(1)
5.2 Delaunay triangulation (3D)
240(9)
5.2.1 Introduction
240(1)
5.2.2 The insertion algorithm
240(8)
5.2.2.1 Determination of the CORE
241(1)
5.2.2.2 Search for the BASE
241(1)
5.2.2.3 Determination of the CORE
242(1)
5.2.2.4 Triangulation of the CORE
243(1)
5.2.2.5 Adjacency relationship
244(2)
5.2.2.6 Heredity of geometrical quantities
246(2)
5.2.2.7 Memory management
248(1)
5.2.3 Examples
248(1)
5.3 Boundary recovery for 3D DT
249(21)
5.3.1 Introduction
249(1)
5.3.2 Boundary recovery by local mesh reconnection
250(1)
5.3.3 Boundary recovery by introducing Steiner points
251(13)
5.3.3.1 Introduction
251(1)
5.3.3.2 Insertion algorithm and boundary recovery
251(13)
5.3.4 Worked examples and industrial applications
264(6)
5.3.4.1 Worked examples
264(3)
5.3.4.2 Industrial applications
267(3)
5.4 Boundary protection in DT
270(12)
5.4.1 Introduction
270(1)
5.4.2 2D conforming DT
271(3)
5.4.2.1 Insert Steiner points at the mid-points of missing edges
272(1)
5.4.2.2 Insert Steiner points at the intersections of missing edges
273(1)
5.4.3 Algorithm RBR: Retrieving bounded region
274(2)
5.4.4 3D conforming DT
276(3)
5.4.4.1 Recovery of boundary edges
276(1)
5.4.4.2 Recovery of boundary faces
277(2)
5.4.5 Practical examples
279(3)
5.5 Generation of tetrahedral mesh by ADF approach
282(9)
5.5.1 Introduction
282(2)
5.5.1.1 γ-quality of tetrahedral element
283(1)
5.5.2 ADF meshing procedures
284(5)
5.5.2.1 The generation front
284(1)
5.5.2.2 Generation of interior node
284(2)
5.5.2.3 Construction of tetrahedral elements
286(2)
5.5.2.4 No tetrahedron found on triangle J1J2J3
288(1)
5.5.2.5 Check for intersections
288(1)
5.5.3 Efficiency consideration and mesh quality
289(1)
5.5.4 ADF meshing of 3D objects
289(2)
5.6 Delaunay—ADF meshing
291(8)
5.6.1 Delaunay—ADF mesh procedure
292(5)
5.6.1.1 Initial generation front
292(1)
5.6.1.2 Boundary triangulation
292(1)
5.6.1.3 Zonal division and MG front
293(1)
5.6.1.4 Generation of tetrahedral elements on a frontal triangle
293(1)
5.6.1.5 Updating the generation front
294(1)
5.6.1.6 Termination of the meshing process
295(1)
5.6.1.7 Strategy in placing interior nodes
295(2)
5.6.2 Example
297(2)
5.7 Generation of tetrahedral mesh by sphere packing
299(13)
5.7.1 Introduction
299(2)
5.7.2 Sphere packing and MG algorithm
301(5)
5.7.2.1 Data structure
301(1)
5.7.2.2 Criteria for sphere packing
301(1)
5.7.2.3 The controlled space
302(1)
5.7.2.4 Generation of the initial pack
302(1)
5.7.2.5 Packing spheres
302(4)
5.7.2.6 MG by Delaunay point insertion
306(1)
5.7.2.7 Termination of the meshing process
306(1)
5.7.3 Efficiency and time complexity
306(1)
5.7.4 Examples of sphere packing
307(5)
5.8 Generation of hexahedral mesh
312(15)
5.8.1 Introduction
312(2)
5.8.2 Direct methods
314(1)
5.8.3 Indirect methods
314(1)
5.8.4 Subdivision, mapping and transformation
315(1)
5.8.5 Block decomposition
316(1)
5.8.6 Drag method and extrusion
317(1)
5.8.7 Meshing by revolution
317(2)
5.8.8 Grid-based or voxel-based method
319(1)
5.8.9 Medial surface method
319(1)
5.8.10 Plastering method
320(1)
5.8.11 Whisker weaving method
321(1)
5.8.12 H-morph approach
322(1)
5.8.13 Generation of transition elements
322(1)
5.8.14 Generation of transition quadrilateral mesh
323(2)
5.8.15 Generation of transition hexahedral mesh
325(2)
6 Mesh optimisation 327(58)
6.1 Introduction
327(1)
6.2 Shape measure and quality coefficient
328(12)
6.2.1 Common simplex shape measures
329(6)
6.2.1.1 Minimum solid angle θ
329(2)
6.2.1.2 Radius ratio ρ
331(1)
6.2.1.3 Mean ratio η
331(3)
6.2.1.4 Shape measures based on condition number κ
334(1)
6.2.1.5 Minimum dihedral angle is not a valid shape measure
335(1)
6.2.1.6 Edge ratio is not a valid shape measure
335(1)
6.2.2 Relationship between shape measures
335(1)
6.2.3 Extension to Riemann space
336(1)
6.2.4 Shape measure for polyhedron
337(3)
6.3 Optimisation by shifting of nodes
340(34)
6.3.1 Optimisation of triangular meshes
342(9)
6.3.1.1 QL smoothing
342(1)
6.3.1.2 LO of triangular mesh
343(1)
6.3.1.3 GETMe (2D)
344(4)
6.3.1.4 Examples: Node smoothing for triangular meshes
348(3)
6.3.2 Optimisation of quadrilateral and mixed meshes
351(5)
6.3.2.1 Shape quality of a mixed mesh of triangles and quadrilaterals
351(1)
6.3.2.2 GETMe transformation for quadrilaterals
352(1)
6.3.2.3 Examples: Node smoothing for mixed meshes
353(3)
6.3.3 Node smoothing for 3D meshes
356(18)
6.3.3.1 QL smoothing (3D)
357(1)
6.3.3.2 LO of polyhedral mesh
357(2)
6.3.3.3 GETMe (3D)
359(5)
6.3.3.4 Examples: Node smoothing for tetrahedral meshes
364(4)
6.3.3.5 Examples: Node smoothing for hexahedral meshes
368(6)
6.4 Optimisation by topological operations
374(11)
6.4.1 Triangular meshes
375(1)
6.4.2 Quadrilateral meshes
375(1)
6.4.3 Tetrahedral meshes
376(3)
6.4.4 Examples of optimisation by face/edge swap
379(2)
6.4.5 Optimisation by both geometrical and topological operations
381(4)
7 Mesh generation by parallel processing 385(44)
7.1 Introduction
385(2)
7.2 Fundamentals and strategies
387(3)
7.2.1 Partition of points and insertion algorithm
388(1)
7.2.2 The zonal insertion scheme
389(1)
7.3 Parallel Delaunay triangulation in 2D
390(12)
7.3.1 Points partitioned into cells
390(1)
7.3.2 Grouping cells into zones
390(2)
7.3.3 Simultaneous insertion within zones
392(3)
7.3.4 Elimination of redundant triangles
395(1)
7.3.5 Minimum vertex-allocation scheme
396(1)
7.3.6 Efficiency analysis
397(2)
7.3.7 Memory requirement
399(1)
7.3.8 Test on OpenMP shared memory systems
399(3)
7.4 Parallel Delaunay triangulation in 3D
402(14)
7.4.1 Points partitioned into cells
402(1)
7.4.2 Grouping cells into zones
403(1)
7.4.3 Simultaneous insertion in 3D
403(2)
7.4.4 Elimination of redundant tetrahedra
405(1)
7.4.5 Zonal insertion is Delaunay and complete
406(1)
7.4.6 Efficiency analysis
407(3)
7.4.7 Memory requirement
410(1)
7.4.8 Treatment of degeneracy
410(1)
7.4.9 Test on OpenMP shared memory systems
411(5)
7.5 Partition of discretised surface for parallel processing
416(13)
7.5.1 Introduction
416(2)
7.5.2 Problem definition and preliminaries
418(1)
7.5.2.1 Triangulated surfaces
418(1)
7.5.3 How the surface is cut into n pieces
418(2)
7.5.4 Euler—Poincare characteristics
420(1)
7.5.5 Procedure for surface decomposition
421(4)
7.5.5.1 Read in the surface S and carry out some basic topological computation
421(1)
7.5.5.2 Determination of the cutting zone
421(1)
7.5.5.3 Subdivide S by the cutting zone
422(1)
7.5.5.4 Balancing the two resulting surface parts
422(1)
7.5.5.5 Distance from the cut line
422(1)
7.5.5.6 Marching on the surface
423(1)
7.5.5.7 Optimisation to improve the quality of the cut
424(1)
7.5.6 Examples
425(2)
7.5.7 Conclusion
427(2)
8 Auxiliary meshing techniques 429(140)
8.1 Surface verification and preparation
430(13)
8.1.1 Introduction
430(1)
8.1.1.1 Boundary surface of solid objects
431(1)
8.1.2 Preliminary checks and preparations
431(1)
8.1.2.1 Limits of points
431(1)
8.1.2.2 Normalisation of co-ordinates
431(1)
8.1.2.3 Check if any node is outside the range [ 1,Np]
432(1)
8.1.2.4 Find out all the connected node points
432(1)
8.1.2.5 Check the spacing between nodes
432(1)
8.1.2.6 Verification of individual elements
432(1)
8.1.3 Analysis of topology
432(3)
8.1.3.1 Search for all the edges on boundary surface B
433(1)
8.1.3.2 Elements connected to each edge
433(1)
8.1.3.3 Elimination of redundant triangles
434(1)
8.1.3.4 Surface construction
434(1)
8.1.3.5 Flagging unused surface parts
435(1)
8.1.4 Region identification
435(3)
8.1.4.1 Boundary edges
435(1)
8.1.4.2 Formation of regions
435(2)
8.1.4.3 Validity check of the formation of regions
437(1)
8.1.4.4 Convergence
438(1)
8.1.5 Geometrical aspects
438(2)
8.1.5.1 Intersection
438(1)
8.1.5.2 Touch
439(1)
8.1.5.3 Sharp angle
440(1)
8.1.5.4 Use of background grid
440(1)
8.1.6 Examples
440(3)
8.2 Multi-grid insertion of non-uniform point distributions (2D)
443(28)
8.2.1 Introduction
443(1)
8.2.2 Review on insertion schemes
444(3)
8.2.2.1 Random order
444(1)
8.2.2.2 Biased randomised insertion order
445(1)
8.2.2.3 Hilbert curve
445(1)
8.2.2.4 Space partition (background grid)
446(1)
8.2.3 Kd-tree insertion scheme
447(4)
8.2.3.1 Kd-tree construction
447(2)
8.2.3.2 Kd-tree partition of points
449(1)
8.2.3.3 Sequence of cell insertion
450(1)
8.2.3.4 Kd-tree grid insertion
451(1)
8.2.4 Multi-grid insertion
451(4)
8.2.4.1 Regular grid insertion
452(1)
8.2.4.2 Multi-grid as a repeated application of the regular grid
453(1)
8.2.4.3 Pseudo-code for the recursive insertion algorithm by multi-grid
454(1)
8.2.5 Tests on non-uniform point distributions
455(16)
8.2.6 Closure
471(1)
8.3 Multi-grid insertion of non-uniform point distributions (3D)
471(23)
8.3.1 Introduction
471(1)
8.3.2 Kd-tree insertion (3D)
472(2)
8.3.2.1 3d-tree partition of space and points
472(2)
8.3.2.2 Insertion by a sandwich sequence
474(1)
8.3.2.3 Enhanced kd-tree insertion
474(1)
8.3.3 Multi-grid insertion
474(3)
8.3.3.1 Regular grid (3D)
475(1)
8.3.3.2 Point insertion by regular grid
476(1)
8.3.3.3 Multi-grid as a repeated application of the regular grid
476(1)
8.3.4 Tests on non-uniform point distributions
477(15)
8.3.5 Possibility for parallelisation
492(2)
8.3.6 Closure
494(1)
8.4 Mesh generation and adaptation by edge refinement
494(17)
8.4.1 Introduction
494(3)
8.4.2 Refinement of discretised surfaces
497(2)
8.4.2.1 Statement of the problem
497(1)
8.4.2.2 Algorithm: Refinement of triangular mesh
497(2)
8.4.3 3D refinement in compliance with a specified node-spacing function
499(11)
8.4.3.1 The algorithm
499(3)
8.4.3.2 Optimisation of element shape
502(1)
8.4.3.3 Examples
503(4)
8.4.3.4 Refinement according to an anisotropic metric field
507(3)
8.4.4 Refinement of non-simplicial elements
510(1)
8.5 Meshing volume bounded by analytical curved surfaces
511(6)
8.5.1 Introduction
511(1)
8.5.2 MG algorithm by refinement and boundary fitting
512(3)
8.5.2.1 Initial embedding mesh
513(1)
8.5.2.2 Mesh refinement over object boundary
513(1)
8.5.2.3 Projection of nodes close to boundary surface
513(1)
8.5.2.4 Cutting of intersecting edges
514(1)
8.5.2.5 Elimination of elements not belonging to the object
514(1)
8.5.2.6 Boundary point projection
514(1)
8.5.3 Example of mesh adaptation by refinement
515(2)
8.6 Merging of tetrahedral meshes
517(22)
8.6.1 Introduction
517(2)
8.6.2 Algorithm: Merging tetrahedral mesh
519(12)
8.6.2.1 Intersection of boundary surfaces
520(2)
8.6.2.2 Incorporating intersection loops into meshes Ω and Ω
522(1)
8.6.2.3 Volume (region) of intersection
523(2)
8.6.2.4 Identification of intersection volumes (regions)
525(2)
8.6.2.5 Mesh compatibility
527(2)
8.6.2.6 Merging of tetrahedral meshes
529(2)
8.6.3 Examples
531(6)
8.6.4 Closure
537(2)
8.7 Merging of hexahedral meshes
539(13)
8.7.1 Introduction
539(1)
8.7.2 Algorithm: Merging hexahedral mesh
539(8)
8.7.2.1 Hexahedron decomposed into tetrahedra
540(3)
8.7.2.2 Merging of hexahedral meshes
543(1)
8.7.2.3 Recovery of hexahedral elements from tetrahedral elements
543(1)
8.7.2.4 Compatibility between hexahedral and tetrahedral elements
544(3)
8.7.3 Examples
547(1)
8.7.4 Closing remarks
547(5)
8.8 Curvilinear finite element mesh
552(5)
8.8.1 Introduction
552(1)
8.8.2 Generation of curvilinear meshes
553(2)
8.8.2.1 Generation of a linear element mesh
553(1)
8.8.2.2 Snap of boundary node and mesh subdivision
553(1)
8.8.2.3 Quality improving by mesh optimisation
554(1)
8.8.3 Examples in 2D
555(2)
8.9 Adaptive refinement analysis
557(12)
8.9.1 Fundamentals in solid mechanics and error in FE solution
557(1)
8.9.2 A priori and a posteriori error estimates
558(1)
8.9.3 Super-convergence and optimal sampling points
559(4)
8.9.3.1 One-dimensional example
559(1)
8.9.3.2 Super-convergent patch recovery
560(2)
8.9.3.3 The Herrmann theorem and optimal sampling points
562(1)
8.9.4 Adaptive refinement strategy
563(2)
8.9.5 Examples
565(4)
References 569(30)
Appendix 599(20)
Index 619
Daniel S.H. Lo received his Doc-Ing from LEcole Nationale des Ponts et Chaussees in France. He is currently a professor at the Department of Civil Engineering of the University of Hong Kong, and has been working on mesh generation for more than 30 years. Apart from numerous journal papers on mesh generation and finite element technology, Lo has also been the guest editor for two special issues on finite element mesh adaptation, co-author of a book on the finite element method, and author of book chapters on mesh generation.