Chongmin Song

The Scaled Boundary Finite Element Method (eBook, ePUB)

Introduction to Theory and Implementation

• Format: ePub

Produktbeschreibung

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.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 504
• Erscheinungstermin: 19. Juni 2018
• Englisch
• ISBN-13: 9781119388456
• Artikelnr.: 53058980

Autorenporträt

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.

Inhaltsangabe

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