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E-raamat: Scaled Boundary Finite Element Method: Introduction to Theory and Implementation

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  • Ilmumisaeg: 19-Jun-2018
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119388463
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Scaled Boundary Finite Element Method: Introduction to Theory and ImplementationSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 19-Jun-2018
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119388463
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An informative look at the theory, computer implementation, and application of the scaled boundary finite element method 

This reliable resource, complete with MATLAB, is an easy-to-understand introduction to the fundamental principles of the scaled boundary finite element method. It establishes the theory of the scaled boundary finite element method systematically as a general numerical procedure, providing the reader with a sound knowledge to expand the applications of this method to a broader scope. The book also presents the applications of the scaled boundary finite element to illustrate its salient features and potentials. 

The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation covers the static and dynamic stress analysis of solids in two and three dimensions. The relevant concepts, theory and modelling issues of the scaled boundary finite element method are discussed and the unique features of the method are highlighted. The applications in computational fracture mechanics are detailed with numerical examples. A unified mesh generation procedure based on quadtree/octree algorithm is described. It also presents examples of fully automatic stress analysis of geometric models in NURBS, STL and digital images.

  • Written in lucid and easy to understand language by the co-inventor of the scaled boundary element method
  • Provides MATLAB as an integral part of the book with the code cross-referenced in the text and the use of the code illustrated by examples
  • Presents new developments in the scaled boundary finite element method with illustrative examples so that readers can appreciate the significant features and potentials of this novel method—especially in emerging technologies such as 3D printing, virtual reality, and digital image-based analysis

The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation is an ideal book for researchers, software developers, numerical analysts, and postgraduate students in many fields of engineering and science.

Preface xv
Acknowledgements xix
1 Introduction 1(26)
1.1 Numerical Modelling
1(5)
1.2 Overview of the Scaled Boundary Finite Element Method
6(4)
1.3 Features and Example Applications of the Scaled Boundary Finite Element Method
10(16)
1.3.1 Linear Elastic Fracture Mechanics: Crack Terminating at Material Interface
11(2)
1.3.2 Automatic Mesh Generation Based on Quadtree/Octree
13(1)
1.3.3 Treatment of Non-matching Meshes
14(3)
1.3.4 Crack Propagation
17(1)
1.3.5 Adaptive Analysis
17(2)
1.3.6 Transient Wave Scattering in an Alluvial Basin
19(1)
1.3.7 Automatic Image-based Analysis
19(5)
1.3.7.1 Two-dimensional Elastoplastic Analysis of Cast Iron
20(2)
1.3.7.2 Three-dimensional Concrete Specimen
22(2)
1.3.8 Automatic Analysis of STL Models
24(2)
1.4 Summary
26(1)
Part I: Basic Concepts and MATLAB Implementation of the Scaled Boundary Finite Element Method in Two Dimensions 27(210)
2 Basic Formulations of the Scaled Boundary Finite Element Method
31(42)
2.1 Introduction
31(1)
2.2 Modelling of Geometry in Scaled Boundary Coordinates
31(19)
2.2.1 S-domains: Scaling Requirement on Geometry, Scaling Centre and Scaling of Boundary
31(6)
2.2.2 S-elements: Boundary Discretization of S-domains
37(3)
2.2.3 Scaled Boundary Transformation
40(13)
2.2.3.1 Scaled Boundary Coordinates
40(2)
2.2.3.2 Coordinate Transformation of Partial Derivatives
42(2)
2.2.3.3 Geometrical Properties in Scaled Boundary Coordinates
44(6)
2.3 Governing Equations of Linear Elasticity in Scaled Boundary Coordinates
50(1)
2.4 Semi-analytical Representation of Displacement and Strain Fields
51(2)
2.5 Derivation of the Scaled Boundary Finite Element Equation by the Virtual Work Principle
53(10)
2.5.1 Virtual Displacement and Strain Fields in Scaled Boundary Coordinates
54(1)
2.5.2 Nodal Force Functions
54(1)
2.5.3 The Scaled Boundary Finite Element Equation
55(8)
2.6 Computer Program Platypus: Coefficient Matrices of an S-element
63(10)
2.6.1 Element Coefficient Matrices of a 2-node Line Element
63(4)
2.6.2 Assembly of Coefficient Matrices of an S-element
67(6)
3 Solution of the Scaled Boundary Finite Element Equation by Eigenvalue Decomposition
73(76)
3.1 Solution Procedure for the Scaled Boundary Finite Element Equations in Displacement
73(4)
3.2 Pre-conditioning of Eigenvalue Problems
77(1)
3.3 Computer Program Platypus: Solution of the Scaled Boundary Finite Element Equation of a Bounded S-element by the Eigenvalue Method
78(6)
3.4 Assembly of S-elements and Solution of Global System of Equations
84(3)
3.4.1 Assembly of S-elements
84(1)
3.4.2 Surface Tractions
85(2)
3.4.3 Enforcing Displacement Boundary Conditions
87(1)
3.5 Computer Program Platypus: Assembly and Solution
87(15)
3.5.1 Assembly of Global Stiffness Matrix
87(8)
3.5.2 Assembly of Load Vector
95(1)
3.5.3 Solution of Global System of Equations
96(1)
3.5.4 Utility Functions
97(5)
3.6 Examples of Static Analysis Using Platypus
102(9)
3.7 Evaluation of Internal Displacements and Stresses of an S-element
111(3)
3.7.1 Integration Constants and Internal Displacements
111(1)
3.7.2 Strain/Stress Modes and Strain/Stress Fields
112(2)
3.7.3 Shape Functions of Polygon Elements Modelled as S-elements
114(1)
3.8 Computer Program Platypus: Internal Displacements and Strains
114(18)
3.9 Body Loads
132(3)
3.10 Dynamics and Vibration Analysis
135(14)
3.10.1 Mass Matrix and Equation of Motion
135(5)
3.10.2 Natural Frequencies and Mode Shapes
140(3)
3.10.3 Response History Analysis Using the Newmark Method
143(6)
4 Automatic Polygon Mesh Generation for Scaled Boundary Finite Element Analysis
149(60)
4.1 Introduction
149(1)
4.2 Basics of Geometrical Representation by Signed Distance Functions
150(4)
4.3 Computer Program Platypus: Generation of Polygon S-element Mesh
154(21)
4.3.1 Mesh Data Structure
157(8)
4.3.2 Centroid of a Polygon
165(1)
4.3.3 Converting a Triangular Mesh to an S-element Mesh
166(5)
4.3.4 Use of Polygon Meshes Generated by PolyMesher in a Scaled Boundary Finite Element Analysis
171(1)
4.3.5 Dividing Edges of Polygons into Multiple Elements
172(3)
4.4 Examples of Scaled Boundary Finite Element Analysis Using Platypus
175(34)
4.4.1 A Deep Beam
178(15)
4.4.1.1 Static Analysis
186(3)
4.4.1.2 Modal Analysis
189(1)
4.4.1.3 Response History Analysis
190(1)
4.4.1.4 Pure Bending of a Beam: 2 Line Elements on an Edge of Polygons
190(3)
4.4.2 A Circular Hole in an Infinite Plane Under Remote Uniaxial Tension
193(4)
4.4.3 An L-shaped Panel
197(15)
4.4.3.1 Static Analysis
203(1)
4.4.3.2 Modal Analysis
204(3)
4.4.3.3 Response History Analysis
207(2)
5 Modelling Considerations in the Scaled Boundary Finite Element Analysis
209(28)
5.1 Effect of Location of Scaling Centre on Accuracy
209(3)
5.2 Mesh Transition
212(6)
5.2.1 Local Mesh Refinement
212(2)
5.2.2 Rapid Mesh Transition
214(2)
5.2.3 Effect of Nonuniformity of Line Element Length on the Boundary of S-elements
216(2)
5.3 Connecting Non-matching Meshes of Multiple Domains
218(16)
5.3.1 Computer Program Platypus: Combining Two Non-matching Meshes
220(3)
5.3.2 Computer Program Platypus: Modelling of a Problem by Multiple Domains with Non-matching Meshes
223(2)
5.3.3 Examples
225(9)
5.4 Modelling of Stress Singularities
234(3)
Part II: Theory and Applications of the Scaled Boundary Finite Element Method 237(212)
6 Derivation of the Scaled Boundary Finite Element Equation in Three Dimensions
239(42)
6.1 Introduction
239(1)
6.2 Scaling of Boundary
239(3)
6.3 Boundary Discretization of an S-domain
242(7)
6.3.1 Isoparametric Quadrilateral Elements
243(3)
6.3.1.1 Four-node Quadrilateral Element
243(2)
6.3.1.2 Quadrilateral Element of Variable Number of Nodes
245(1)
6.3.2 Isoparametric Triangular Elements
246(13)
6.3.2.1 Three-node Triangular Elements
247(1)
6.3.2.2 Six-node Triangular Elements
248(1)
6.4 Scaled Boundary Transformation of Geometry
249(4)
6.5 Geometrical Properties in Scaled Boundary Coordinates
253(4)
6.6 Governing Equations of Elastodynamics with Geometry in Scaled Boundary Coordinates
257(2)
6.7 Derivation of the Scaled Boundary Finite Element Equation by the Galerkin's Weighted Residual Technique
259(8)
6.7.1 Displacement, Strain Fields and Nodal Force Functions in Scaled Boundary Coordinates
259(3)
6.7.2 The Scaled Boundary Finite Element Equation
262(5)
6.8 Unified Formulations in Two and Three Dimensions
267(1)
6.9 Formulation of the Scaled Boundary Finite Element Equation as a System of First-order Differential Equations
268(1)
6.10 Properties of Coefficient Matrices
269(3)
6.10.1 Coefficient Matrices [ E0] and [ M0]
270(1)
6.10.2 Coefficient Matrix [ E2]
270(1)
6.10.3 Matrix [ Zp]
271(1)
6.11 Linear Completeness of the Scaled Boundary Finite Element Solution
272(6)
6.11.1 Constant Displacement Field
272(1)
6.11.2 Linear Displacement Field
273(5)
6.12 Scaled Boundary Finite Element Equation in Stiffness
278(3)
7 Solution of the Scaled Boundary Finite Element Equation in Statics by Schur Decomposition
281(48)
7.1 Introduction
281(2)
7.2 Basics of Matrix Exponential Function
283(4)
7.3 Schur Decomposition
287(4)
7.3.1 Introduction
287(1)
7.3.2 Treatment of the Diagonal Block of Eigenvalues of 0
288(3)
7.4 Solution Procedure for a Bounded S-element by Schur Decomposition
291(4)
7.4.1 Transformation of the Scaled Boundary Finite Element Equation
291(1)
7.4.2 Enforcing the Boundary Condition at the Scaling Centre
292(2)
7.4.3 Determining the Solution for Displacement and Nodal Force Functions
294(1)
7.4.4 Determining the Static Stiffness Matrix
295(1)
7.5 Solution of Displacement and Stress Fields of an S-element
295(2)
7.5.1 Integration Constants
295(1)
7.5.2 Stress Modes and Stresses on the Boundary
296(1)
7.6 Block-diagonal Schur Decomposition
297(6)
7.7 Solution Procedure by Block-diagonal Schur Decomposition
303(7)
7.7.1 General Solution of the Scaled Boundary Finite Element Equation
303(2)
7.7.1.1 [ Zp]Having No Eigenvalues of Zero
304(1)
7.7.1.2 [ Zp]Having Eigenvalues of Zero
304(1)
7.7.2 Solution for Bounded S-elements
305(2)
7.7.3 Solution for Unbounded S-elements
307(3)
7.7.3.1 [ Zp] Having No Eigenvalues of Zero
307(1)
7.7.3.2 [ Zp] Having Eigenvalues of Zero
308(2)
7.8 Displacements and Stresses of an S-element by Block-diagonal Schur Decomposition
310(3)
7.8.1 Integration Constants and Displacement Fields
310(1)
7.8.2 Stress Modes and Stress Fields
311(1)
7.8.3 Shape Functions of Polytope Elements
312(1)
7.9 Body Loads
313(2)
7.10 Mass Matrix
315(2)
7.11 Remarks
317(2)
7.12 Examples
319(8)
7.12.1 Circular Cavity in Full-plane
319(3)
7.12.2 Bi-material Wedge
322(3)
7.12.3 Interface Crack in Anisotropic Bi-material Full-plane
325(2)
7.13 Summary
327(2)
8 High-order Elements
329(26)
8.1 Lagrange Interpolation
330(3)
8.2 One-dimensional Spectral Elements
333(8)
8.2.1 Shape Functions
334(3)
8.2.2 Numerical Integration of Element Coefficient Matrices
337(4)
8.2.2.1 Gauss-Legendre Quadrature
337(1)
8.2.2.2 Gauss-Lobatto-Legendre Quadrature
338(3)
8.3 Two-dimensional Quadrilateral Spectral Elements
341(3)
8.3.1 Shape Functions
341(1)
8.3.2 Integration of Element Coefficient Matrices by Gauss-Lobatto-Legendre Quadrature
342(2)
8.4 Examples
344(11)
8.4.1 A Cantilever Beam Subject to End Loading
345(2)
8.4.2 A Circular Hole in an Infinite Plate
347(2)
8.4.3 An L-shaped Panel
349(2)
8.4.4 A 3D Cantilever Beam Subject to End-shear Loading
351(1)
8.4.5 A Pressurized Hollow Sphere
352(3)
9 Quadtree/Octree Algorithm of Mesh Generation for Scaled Boundary Finite Element Analysis
355(40)
9.1 Introduction
355(5)
9.1.1 Mesh Generation
355(2)
9.1.2 The Quadtree/Octree Algorithm
357(3)
9.2 Data Structure of S-element Meshes
360(1)
9.3 Quadtree/Octree Mesh Generation of Digital Images
361(9)
9.3.1 Illustration of Quadtree Decomposition of Two-dimensional Images by an Example
361(5)
9.3.2 Octree Decomposition
366(4)
9.4 Solutions of S-elements with the Same Pattern of Node Configuration
370(4)
9.4.1 Two-dimensional S-elements
370(2)
9.4.2 Three-dimensional S-elements
372(2)
9.5 Examples of Image-based Analysis
374(4)
9.5.1 A 2D Concrete Specimen
374(2)
9.5.2 A 3D Concrete Specimen
376(2)
9.6 Quadtree/Octree Mesh Generation for CAD Models
378(5)
9.6.1 Quadtree/Octree Grid
380(1)
9.6.2 Trimming of Boundary Cells
381(2)
9.7 Examples Using Quadtree/Octree Meshes of CAD Models
383(11)
9.7.1 Square Body with Multiple Holes
384(1)
9.7.2 An Evolving Void in a Square Body
385(1)
9.7.3 Adaptive Analysis of an L-shaped Panel
386(1)
9.7.4 A Mechanical Part
387(2)
9.7.5 STL Models
389(5)
9.8 Remarks
394(1)
10 Linear Elastic Fracture Mechanics
395(54)
10.1 Introduction
395(2)
10.2 Basics of Fracture Analysis: Asymptotic Solutions, Stress Intensity Factors, and the T-stress
397(9)
10.2.1 Crack in Homogeneous Isotropic Material
397(4)
10.2.2 Interfacial Cracks between Two Isotropic Materials
401(1)
10.2.3 Interfacial Cracks between Two Anisotropic Materials
402(3)
10.2.4 Multi-material Wedges
405(1)
10.3 Modelling of Singular Stress Fields by the Scaled Boundary Finite Element Method
406(1)
10.4 Stress Intensity Factors and the T-stress of a Cracked Homogeneous Body
407(9)
10.5 Definition and Evaluation of Generalized Stress Intensity Factors
416(16)
10.6 Examples of Highly Accurate Stress Intensity Factors and T-stress
432(8)
10.6.1 A Single Edge-cracked Rectangular Body Under Tension
433(2)
10.6.2 A Single Edge-cracked Rectangular Body Under Bending
435(2)
10.6.3 A Centre-cracked Rectangular Body Under Tension
437(1)
10.6.4 A Double Edge-cracked Rectangular Body Under Tension
438(1)
10.6.5 A Single Edge-cracked Rectangular Body Under End Shearing
439(1)
10.7 Modelling of Crack Propagation
440(9)
10.7.1 Modelling of Crack Paths by Polygon Meshes
442(1)
10.7.2 Modelling of Crack Paths by Quadtree Meshes
443(1)
10.7.3 Examples of Crack Propagation Modelling
444(5)
10.7.3.1 Fatigue Crack Propagation Using Polygon Mesh
444(3)
10.7.3.2 Crack Propagation in a Beam with Three Holes
447(2)
Appendix A: Governing Equations of Linear Elasticity 449(10)
A.1 Three-dimensional Problems
449(5)
A.1.1 Strain
449(1)
A.1.2 Stress and Equilibrium Equation
450(1)
A.1.3 Stress-strain Relationship and Material Elasticity Matrix
451(2)
A.1.4 Boundary Conditions
453(1)
A.2 Two-dimensional Problems
454(3)
A.2.1 Elasticity Matrix in Plane Stress
455(1)
A.2.2 Elasticity Matrix in Plane Strain
456(1)
A.3 Unified Expressions of Governing Equations
457(2)
Appendix B: Matrix Power Function 459(4)
B.1 Definition of Matrix Power Function
459(1)
B.2 Application to Solution of System of Ordinary Differential Equations
460(1)
B.3 Computation of Matrix Power Function by Eigenvalue Method
461(2)
Bibliography 463(12)
Index 475
Chongmin Song, PhD, is a Professor of Civil Engineering and the Acting Director of the Centre for Infrastructure Engineering and Safety at the University of New South Wales, Australia. He is a Member of the General Council of the Asia-Pacific Association of Computational Mechanics (APCOM) and an Executive Member of the Australian Association for Computational Mechanics (AACM). Professor Song is also the co-author of Finite-Element Modelling of Unbounded Media.
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