The Scaled Boundary Finite Element Method E-Book

Chongmin Song

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 647

Veröffentlichungsjahr: 2018

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Preface

Structure of the Book

Accompanying Computer Program Platypus

1 Introduction

1.1 Numerical Modelling

1.2 Overview of the Scaled Boundary Finite Element Method

1.3 Features and Example Applications of the Scaled Boundary Finite Element Method

1.4 Summary

Part I: Basic Concepts and MATLAB Implementation of the Scaled Boundary Finite Element Method in Two Dimensions

2 Basic Formulations of the Scaled Boundary Finite Element Method

2.1 Introduction

2.2 Modelling of Geometry in Scaled Boundary Coordinates

2.3 Governing Equations of Linear Elasticity in Scaled Boundary Coordinates

2.4 Semi‐analytical Representation of Displacement and Strain Fields

2.5 Derivation of the Scaled Boundary Finite Element Equation by the Virtual Work Principle

2.6 Computer Program Platypus: Coefficient Matrices of an S‐element

3 Solution of the Scaled Boundary Finite Element Equation by Eigenvalue Decomposition

3.1 Solution Procedure for the Scaled Boundary Finite Element Equations in Displacement

3.2 Pre‐conditioning of Eigenvalue Problems

3.3 Computer Program Platypus: Solution of the Scaled Boundary Finite Element Equation of a Bounded S‐element by the Eigenvalue Method

3.4 Assembly of S‐elements and Solution of Global System of Equations

3.5 Computer Program Platypus: Assembly and Solution

3.6 Examples of Static Analysis Using Platypus

3.7 Evaluation of Internal Displacements and Stresses of an S‐element

3.8 Computer Program Platypus: Internal Displacements and Strains

3.9 Body Loads

3.10 Dynamics and Vibration Analysis

4 Automatic Polygon Mesh Generation for Scaled Boundary Finite Element Analysis

4.1 Introduction

4.2 Basics of Geometrical Representation by Signed Distance Functions

4.3 Computer Program Platypus: Generation of Polygon S‐element Mesh

4.4 Examples of Scaled Boundary Finite Element Analysis Using Platypus

5 Modelling Considerations in the Scaled Boundary Finite Element Analysis

5.1 Effect of Location of Scaling Centre on Accuracy

5.2 Mesh Transition

5.3 Connecting Non‐matching Meshes of Multiple Domains

5.4 Modelling of Stress Singularities

Part II: Theory and Applications of the Scaled Boundary Finite Element Method

6 Derivation of the Scaled Boundary Finite Element Equation in Three Dimensions

6.1 Introduction

6.2 Scaling of Boundary

6.3 Boundary Discretization of an S‐domain

6.4 Scaled Boundary Transformation of Geometry

6.5 Geometrical Properties in Scaled Boundary Coordinates

6.6 Governing Equations of Elastodynamics with Geometry in Scaled Boundary Coordinates

6.7 Derivation of the Scaled Boundary Finite Element Equation by the Galerkin’s Weighted Residual Technique

6.8 Unified Formulations in Two and Three Dimensions

6.9 Formulation of the Scaled Boundary Finite Element Equation as a System of First‐order Differential Equations

6.10 Properties of Coefficient Matrices

6.11 Linear Completeness of the Scaled Boundary Finite Element Solution

6.12 Scaled Boundary Finite Element Equation in Stiffness

7 Solution of the Scaled Boundary Finite Element Equation in Statics by Schur Decomposition

7.1 Introduction

7.2 Basics of Matrix Exponential Function

7.3 Schur Decomposition

7.4 Solution Procedure for a Bounded S‐element by Schur Decomposition

7.5 Solution of Displacement and Stress Fields of an S‐element

7.6 Block‐diagonal Schur Decomposition

7.7 Solution Procedure by Block‐diagonal Schur Decomposition

7.8 Displacements and Stresses of an S‐element by Block‐diagonal Schur Decomposition

7.9 Body Loads

7.10 Mass Matrix

7.11 Remarks

7.12 Examples

7.13 Summary

8 High‐order Elements

8.1 Lagrange Interpolation

8.2 One‐dimensional Spectral Elements

8.3 Two‐dimensional Quadrilateral Spectral Elements

8.4 Examples

9 Quadtree/Octree Algorithm of Mesh Generation for Scaled Boundary Finite Element Analysis

9.1 Introduction

9.2 Data Structure of S‐element Meshes

9.3 Quadtree/Octree Mesh Generation of Digital Images

9.4 Solutions of S‐elements with the Same Pattern of Node Configuration

9.5 Examples of Image‐based Analysis

9.6 Quadtree/Octree Mesh Generation for CAD Models

9.7 Examples Using Quadtree/Octree Meshes of CAD Models

9.8 Remarks

10 Linear Elastic Fracture Mechanics

10.1 Introduction

10.2 Basics of Fracture Analysis: Asymptotic Solutions, Stress Intensity Factors, and the

T

‐stress

10.3 Modelling of Singular Stress Fields by the Scaled Boundary Finite Element Method

10.4 Stress Intensity Factors and the

T

‐stress of a Cracked Homogeneous Body

10.5 Definition and Evaluation of Generalized Stress Intensity Factors

10.6 Examples of Highly Accurate Stress Intensity Factors and

T

‐stress

10.7 Modelling of Crack Propagation

Appendix A: Governing Equations of Linear Elasticity

A.1 Three‐dimensional Problems

A.2 Two‐dimensional Problems

A.3 Unified Expressions of Governing Equations

Appendix B: Matrix Power Function

B.1 Definition of Matrix Power Function

B.2 Application to Solution of System of Ordinary Differential Equations

B.3 Computation of Matrix Power Function by Eigenvalue Method

Bibliography

Index

End User License Agreement

List of Tables

Chapter 03

Table 3.1 Example 3.3 Nodal connectivity of S‐elements.

Table 3.2 Strains modes of S‐element 1 of a rectangular body under uniaxial tension.

Chapter 04

Table 4.1 Relative error norm in displacement of a beam subject to pure bending. The polygon S‐element meshes are converted from triangular meshes generated by DistMesh.

Table 4.2 Relative error norm in displacement of a beam subject to pure bending. The polygon S‐element meshes are generated by PolyMesher.

Table 4.3 Convergence of the first four natural frequencies (rad/s) of a deep beam.

Table 4.4 Displacement error norm (Eq. (4.12)) of an infinite plate under remote uniaxial tension. (a) The polygon S‐element meshes (see Fig 4.20a for an example) are converted from triangular meshes generated via DistMesh. (b) The polygon S‐element meshes (see Fig 4.21a for an example) are converted from polygon meshes generated via PolyMesher.

Table 4.5 Vertical displacement at upper‐right corner of L‐shaped panel obtained by the scaled boundary finite element method.

Table 4.6 Vertical displacement at upper‐right corner of L‐shaped panel obtained by the finite element method.

Table 4.7 Convergence of the first four natural frequencies (rad/s) of L‐shaped panel.

Chapter 05

Table 5.1 Relative error norm in displacement of a beam subject to pure bending. The meshes are grids of square S‐elements with 2 line elements on each edge of a square (see Figure 5.1).

Chapter 07

Table 7.1 Matrix [

s

] with eigenvalues leading to stress singularities in bi‐material wedge.

Chapter 08

Table 8.1 Legendre polynomials

P

p

(

η

) from order 0 to order 6 and their first‐order derivatives d

P

p

(

η

)/d

η

.

Table 8.2 Natural nodal coordinates of one‐dimensional spectral elements of orders 1–6 and weight factors for Gauss‐Lobatto‐Legendre quadrature.

Table 8.3 Sampling points and weight factors of Gauss‐Legendre quadrature of orders 1 to 5.

Table 8.4 Convergence of vertical displacement

v

A

at point

A

of an L‐shaped panel with increasing order of elements:

p

‐refinement.

Table 8.5 Convergence of vertical displacement

v

A

at point

A

of an L‐shaped panel with increasing order of elements:

h

s

‐refinement with 2nd‐ and 4th‐order elements.

Table 8.6 Relative error norm of displacement of 3D cantilever beam subject to end‐shear loading with mesh refinement.

Chapter 10

Table 10.1 Edge‐cracked square body: Eigenvalues (

λ

i

).

Table 10.2 An edge‐cracked square body: Normalized stress intensity factors

K

I

,

K

II

and the

T

‐stress.

Table 10.3 An edge‐cracked square body modelled by spectral elements: Normalized stress intensity factors

K

I

,

K

II

and the

T

‐stress.

Table 10.4 An angled crack in a rectangular orthotropic body with

: Convergence of matrix of orders of singularity with increasing order of elements.

Table 10.5 An angled crack in a rectangular orthotropic body with

: Convergence of stress intensity factors with increasing order of elements.

Table 10.6 Stress intensity factors of an angled crack in a rectangular orthotropic body.

Table 10.7 An interfacial central crack between two anisotropic materials: Matrix of order of singularity.

Table 10.8 An interfacial central crack between two anisotropic materials: Generalized stress intensity factors.

Table 10.9 A single edge‐cracked rectangular body under tension: Convergence of stress intensity factor

.

Table 10.10 A single edge‐cracked rectangular body under tension: Convergence of

T

‐stress

.

List of Illustrations

Chapter 01

Figure 1.1 Discretization process of a rectangular plate with a circular hole in the scaled boundary finite element method. (a) Problem domain. (b) Discretization of the problem domain into S‐domains. (c) Discretization of the boundaries of S‐domains with 3‐node line elements. The red nodes indicate the end nodes of line elements. (d) Modification of the boundary discretization of S‐domains: Replacing the 3‐node element at the common edge of S‐domains 1 and 6 with a 5‐node line element, and dividing the common edge between S‐domains 1 and 7 and between S‐domains 6 and 8 into two 3‐node line elements.

Figure 1.2 Examples of S‐elements. (a) in two dimensions. (b) in three dimensions.

Figure 1.3 Examples of octagons that satisfy the scaling requirement.

Figure 1.4 Example of shape functions of a cracked S‐element. (a) Polygon S‐element with edges discretized using quadratic elements. (b) Shape function for the crack mouth node. (c) Shape function near the crack tip, showing the

distribution.

Figure 1.5 Crack terminating at material interface. (a) Geometry. (b) Mesh.

Figure 1.6 Crack terminating at material interface with varying crack angle

α

. (a) Generalized stress intensity factors. (b) Order of singularity.

Figure 1.7 Crack terminating at material interface. Angular variation of generalized stress intensity factors at crack inclination angle

.

Figure 1.8 A square body with multiple holes under tension.

Figure 1.9 A cube with a cutout corner under tension. (a) Geometry and boundary conditions. (b) Scaled boundary finite element mesh and

z

‐displacement contour. (c) Finite element mesh and

z

‐displacement contour. (d) Convergence of displacements.

Figure 1.10 Modelling of two square domains meshed independently. The meshes of the two domains are linked at the interface by inserting the nodes of one mesh into the other.

Figure 1.11 Connecting 3D non‐matching mesh. (a) Mesh of domain 1. (b) Surface mesh (enlarged) of the interface, where the surface elements represent the boundary of the 3D S‐elements connected to the interface. (c) Mesh of domain 2. (d) Combined mesh. (e) Interior view of the combined mesh. (f) A modal shape where the right end is fixed.

Figure 1.12 Hertz contact problem between a cuboid and a parallel semi‐cylinder.

Figure 1.13 Crack path deflection by a circular hole.

Figure 1.14 Adaptive analysis of a mechanical part.

Figure 1.15 Transient response of an alluvial basin subject to a plane

wave propagating with incident angle of

.

Figure 1.16 Two‐dimensional image‐based elastoplastic analysis of cast iron.

Figure 1.17 Three‐dimensional image‐based analysis of concrete specimen. (a) Digital image (256

3

voxel). (b) Octree mesh of S‐elements. (c) Contour of z‐displacement of aggregates. (d) Contour of z‐displacement of mortar.

Figure 1.18 Stress analysis of STL model of Lucy. (a) STL model. (b) von Mises stress. (c) First principal stress.

Chapter 02

Figure 2.1 Illustration of the scaling requirement. (a) A bounded domain

V

that satisfies the scaling requirement. The scaling centre is indicated by the marker

from where every point on the boundary

S

(bold solid line) is directly visible. The dashed lines show examples of direct lines of sight. Such a domain will be referred to as an S‐domain. (b) A bounded domain

V

that does not satisfy the scaling requirement.

Figure 2.2 Representation of an S‐domain

V

by scaling its boundary

S

with respect to the scaling centre

O

selected inside the domain. The dash lines indicate the lines of sight from the scaling centre. The thin lines indicate two typical internal curves resulting from scaling the boundary.

Figure 2.3 (a) Scaling of the boundary of an S‐domain using the radial coordinate

ξ

(

at the boundary and

at the scaling centre) as the scaling factor. (b) An S‐domain is transformed to a unit circular domain in the system of radial coordinate

ξ

and angular coordinate

θ

.

Figure 2.4 Star‐shaped polygonal domain. (a) The dashed lines are the straight line segments that connect a point with the vortices of the polygonal domain. (b) The shaded region shows the kernel of the star‐shaped polygonal domain.

Figure 2.5 An unbounded S‐domain. The unbounded domain

V

is covered by scaling its boundary

S

with respect to the scaling centre

O

selected outside of the domain. The thin lines indicate internal curves resulting from scaling the boundary with a factor larger than 1. The dashed lines indicate the lines of sight from the scaling centre.

Figure 2.6 (a) Modelling of an edge‐cracked square by scaling the defining curve. The two side‐faces (crack faces) are formed by scaling the two ends of the defining curve. (b) Modelling of a three‐material corner by scaling the defining curves. The two side‐faces and the material interfaces are formed by scaling the ends of the defining curves.

Figure 2.7 Modelling of an excavation in a half‐plane as an S‐domain by scaling the defining curve. The two side‐faces (free surface) are formed by scaling the two ends of the defining curve.

Figure 2.8 (a) An S‐domain with small angles between the boundary and the lines of sight from the scaling centre, as indicated by the shaded areas, leading to low visibility of the boundary. (b) Subdivision of the S‐domain to increase angles between the boundary and the lines of sight from the scaling centre and, thus, the visibility of boundary.

Figure 2.9 An S‐element obtained by discretizing the boundary

S

of S‐domain

V

. Displacement‐based elements of different orders are used. The direction of the elements has to follow the counter‐clockwise direction around the scaling centre.

Figure 2.10 Two‐node line element on boundary. The element direction must follow the counter‐clockwise direction around the scaling centre. (a) Physical element. A point (

x

b

, 

y

b

) on the element is obtained by interpolating nodal coordinates, see Eq. (2.17). (b) Parent element in natural coordinate

η

. (c) Shape functions in natural coordinate

η

.

Figure 2.11 Three‐node line element on boundary. The element direction must follow the counter‐clockwise direction around the scaling centre. (a) Physical element. A point (

x

b

, 

y

b

) on the element is obtained by interpolating nodal coordinates, see Eq. (2.17). (b) Parent element in natural coordinate

η

. (c) Shape functions in natural coordinate

η

.

Figure 2.12 A quadtree mesh illustrating the simplicity of mesh generation and remeshing using S‐elements. This mesh of S‐elements satisfies displacement compatibility requirement.

Figure 2.13 Representation of an S‐element by scaling the line elements on boundary.

Figure 2.14 Scaled boundary coordinates defined by scaling a line element at the boundary.

Figure 2.15 Representation of the polygonal domain shown in Figure 2.13 in the scaled boundary coordinates.

Figure 2.16 A sector of an S‐element covered by scaling an element at the boundary.

Figure 2.17 Scaled boundary coordinate transformation of a 2‐node line element.

Figure 2.18 Semi‐analytical representation of displacement field in an S‐element by interpolating nodal displacement functions element‐by‐element independently. (a) Displacement functions are introduced along radial lines connecting the scaling centre and nodes at the boundary. (b) Displacement field in a sector covered by scaling one line element on boundary. The dashed lines show the deformed shapes. Displacements along the circumferential direction

η

are obtained by interpolating the nodal displacement functions.

Figure 2.19 Nodal displacement and nodal forces of a pentagon S‐element. (a) Nodal displacements. (b) Nodal forces.

Figure 2.20 A square S‐element (Unit: meter).

Chapter 03

Figure 3.1 A regular pentagon S‐element (Unit: meter).

Figure 3.2 (a) Surface traction applied to a 2‐node line element. (b) Components of nodal values of surface traction. (c) Equivalent nodal forces.

Figure 3.3 Modelling of a rectangular body by 3 S‐elements. (a) Mesh. (b) Isolated S‐elements with line elements defining the boundary.

Figure 3.4 Example 3.3 Connectivity of line element in S‐element 1. (a) Global nodal numbers. (b) Local nodal numbers.

Figure 3.5 Deformed shape of a rectangular body under uniaxial tension.

Figure 3.6 A deep cantilever beam subject to a force couple. (a) Geometry. (b) Mesh.

Figure 3.7 Deformed shape of deep cantilever beam subject to a force couple.

Figure 3.8 An edge‐cracked rectangular body subject to tension. (a) Geometry. (b) Mesh.

Figure 3.9 Deformation of edge‐cracked rectangular body subject to tension.

Figure 3.10 Patch of two irregular polygon S‐elements. (a) Geometry. (b) Mesh of patch with

. (c) Mesh of patch with

and

.

Figure 3.11 Edge‐cracked circular body under uniform radial tension. (a) Geometry and boundary conditions. (b) Mesh. The crack faces are modelled by scaling the two nodes at the crack mouth and not discretized.

Figure 3.12 Deformed mesh of edge‐cracked circular plate under uniform radial tension.

Figure 3.13 (a) Contour of stress

σ

yy

of edge‐cracked circular plate under uniform radial tension. (b) Enlarged view around the crack tip.

Figure 3.14 Shape function of a regular pentagon element. (a) Surface plot. The locations of the nodal numbers are changed from the MATLAB output for clearer view. (b) Contour plot.

Figure 3.15 Five‐node square S‐element.

Figure 3.16 Mode shapes of deep beam.

Figure 3.17 Time variation factor

f

(

t

) of force couple applied on deep beam.

Figure 3.18 Response history of deep beam: vertical displacement at Node 16.

Chapter 04

Figure 4.1 Signed distance function. The distance from a given point

P

to a point

B

at the boundary

S

of the domain

V

is denoted as

d

(

P

, 

B

). When the point

P

is outside of the domain

V

, the signed distance function

d

V

is the distance to the closest boundary point. When the point

P

is inside the domain

V

, the signed distance function

d

V

(

P

) is the minus distance to the closest boundary point. When the point

P

is at the boundary, the signed distance function is equal to zero.

Figure 4.2 Signed distance function of a straight line passing through two points.

Figure 4.3 Signed distance function of a circle.

Figure 4.4 Signed distance function of a rectangle.

Figure 4.5 Set operations. (a) Union. (b) Difference of

A

from

B

. (c) Intersection.

Figure 4.6 Geometry of a circle (domain A) and a rectangle (domain B).

Figure 4.7 Signed distance functions of the circle and rectangle.

Figure 4.8 Signed distance functions obtained from set operations on the circle (A) and rectangle (B).

Figure 4.9 Example of triangular and polygon meshes. The dimension of the square domain is

.

Figure 4.10 Generation of a polygon S‐element mesh from a triangular mesh. (a) Triangular mesh. (b) The centroids “

” of triangular elements, boundary nodes “

” of the triangular mesh and the midpoint “

” of edges at the boundary are identified. (c) Initial numbering of nodes of polygon S‐elements. (d) Construction of three types of polygon S‐element: Interior (S‐element 9), Boundary (S‐element 3) and Concave (S‐element 13). The scaling centres are indicated by the symbol

. (e) Final S‐element mesh after removing unconnected nodes.

Figure 4.11 Example of polygon mesh (see Figure 4.12 on page 179) with each edge subdivided into 2 line elements.

Figure 4.12 Geometry and boundary conditions of a deep beam. (a) Static analysis with surface tractions caused by pure bending. (b) Modal analysis of the beam as a cantilever. (c) Response history analysis of the beam as a cantilever subject to a uniform distributed load. (d) Time variation factor

f

(t) of the surface traction.

Figure 4.13 A beam subject to pure bending. (a) Mesh generated from the pre‐stored triangular mesh in the problem definition function

ProbDefDeepBeam.m

on page 181 (see Line 155). (b) Deformed mesh.

Figure 4.14 Convergence of displacements of a beam subject to pure bending. (a) The S‐element meshes are generated via triangular meshes of DistMesh. (b) The S‐element meshes are generated via PolyMesher.

Figure 4.15 Mode shapes of a deep beam.

Figure 4.16 Response history of a deep beam subject to a triangular impulse of uniform surface traction.

Figure 4.17 Beam subject to pure bending. (a) Mesh. Every edge of the polygons is discretized with 2 line elements. (b) Deformed mesh.

Figure 4.18 Convergence of displacement of a beam subject to pure bending. The results are obtained by discretizing every edge of the polygon mesh with 2 line elements (see Figure 4.17).

Figure 4.19 A circular hole in an infinite plane under remote uniaxial tension.

Figure 4.20 (a) Polygon S‐element mesh of a square with a circular hole converted from a triangular mesh generated by DistMesh with the element size

. (b) Deformed mesh simulating a circular hole in an infinite plane under uniform remote tension.

Figure 4.21 (a) Polygon S‐element mesh of a square with a circular hole generated by PolyMesher with 128 polygons. (b) Deformed mesh simulating a circular hole in an infinite plane under uniform remote tension.

Figure 4.22 Convergence of displacements of an infinite plate under remote uniaxial tension. (a) The polygon S‐element meshes are converted from the triangular meshes generated by DistMesh. (b) The polygon S‐element meshes are generated by PolyMesher.

Figure 4.23 An L‐shaped panel. (a) Geometry. (b) Time variation factor

f

(

t

) of surface traction in response history analysis.

Figure 4.24 Mesh of L‐shaped panel.

Figure 4.25 Convergence of vertical displacement at point

A

of L‐shaped panel.

Figure 4.26 Mode shapes of L‐shaped panel.

Figure 4.27 Response history of L‐shaped panel subject to a triangular impulse of uniform surface traction.

Chapter 05

Figure 5.1 Variation of location of scaling centre of square S‐elements for the modelling of beam under pure bending. (a) A square S‐element is discretized with eight 2‐node elements on boundary, where the smallest angle between the radial line and the boundary is denoted as

α

. (b) Mesh of square S‐elements with length

. The scaling centres are placed in locations such that

(

). (c)

(

).

Figure 5.2 Effect of locations of scaling centres of square S‐elements on the convergence of displacement of a beam under pure bending.

Figure 5.3 S‐element mesh of L‐shaped panel. The line elements at the edges of S‐element 13 is refined locally. (1) Two line elements per edge for S‐element 13 and one line element per edge elsewhere. (2) Four line elements per edge for S‐element 13 and two line elements per edge elsewhere.

Figure 5.4 Convergence of vertical displacement at point

A

of L‐shaped panel as the edges of polygon mesh (Figure 5.3) are divided into increasing number (

nDiv

) of line elements.

Figure 5.5 Illustration of mesh transition using a beam discretized with square S‐elements. (a) Every edge of the squares is discretized with 1 line element. (b) Edges (except for the common edge with S‐element 8) connected to S‐element 7 are discretized with 5 line elements and edges connected to S‐element 8 with 9 line elements.

Figure 5.6 Effect of nonuniform subdivision of edges on accuracy of vertical displacement at point

A

of L‐shaped panel. The S‐element mesh on the left is obtained by subdividing the elements in the S‐element mesh shown in Figure 5.3a into 2 elements. The length ratio between the shorter element to the original element is 0.1.

Figure 5.7 Connecting of non‐matching meshes by S‐elements. (a) Two non‐matching meshes on two rectangular domains to be connected by shifting them vertically. (b) Close‐up view of the S‐elements adjacent to the left part of the interface. At the interface, an edge of an S‐element of a mesh is subdivided at the locations marked by squares to match the nodes of the other mesh at the opposite side (indicated by arrows). (c) The combined mesh satisfies compatibility at the interface. (d) Close‐up view of the S‐elements adjacent to the left part of the interface after connecting the two meshes. The upper edge of S‐element A is subdivided into 4 line elements and the lower edge of S‐element B into 2 line elements.

Figure 5.8 Find points on a line defined by points 1 and 2.

Figure 5.9 Mesh of a rectangular plate with a circular hole obtained by combining a mesh of the upper square domain with the hole and a mesh of the lower rectangular domain. The meshes of the two domains are generated independently.

Figure 5.10 Deformed mesh of a circular hole in an infinite plate under remote uniaxial tension.

Figure 5.11 Mode shapes of the first two modes of a rectangular plate with a circular hole.

Figure 5.12 A mosaic composed of five domains meshed independently.

Figure 5.13 Modelling of L‐shaped panel by one scaled boundary finite element S‐element.

Chapter 06

Figure 6.1 A three‐dimensional S‐domain.

Figure 6.2 Surface discretization of an S‐element. Faces with more than four vertices are divided into triangular elements. A node is inserted at the centroid of such a face (optional).

Figure 6.3 Four‐node parent element in natural coordinates

η

,

ζ

.

Figure 6.4 Nine‐node parent element in natural coordinates

η

,

ζ

.

Figure 6.5 Area coordinates

L

1

,

L

2

,

L

3

in a triangle.

Figure 6.6 Three‐node parent element in natural coordinates

η

,

ζ

.

Figure 6.7 Six‐node parent element in natural coordinates

η

,

ζ

.

Figure 6.8 Scaled boundary transformation of geometry defined on one surface element.

Figure 6.9 Geometrical properties in the scaled boundary coordinates of a part of an S‐domain obtained by scaling a surface element.

Chapter 07

Figure 7.1 Circular cavity in full‐plane subject to vertical loading.

Figure 7.2 Displacement on wall of circular cavity subject to vertical loading.

Figure 7.3 Vertical displacement along the radial line at

of circular cavity subject to vertical loading.

Figure 7.4 Stresses on wall of circular cavity subject to vertical loading.

Figure 7.5 Variation of stress error norm in radial direction for circular cavity subject to vertical loading.

Figure 7.6 Stresses along radial line

for circular cavity subject to vertical loading.

Figure 7.7 Bi‐material wedge with traction‐free wedge surfaces.

Figure 7.8 Angular distribution of singular stress modes in bi‐material wedge at opening angle

.

Figure 7.9 Interface crack in anisotropic bi‐material full‐plane.

Figure 7.10 Angular distribution of stresses at

for interface crack in anisotropic bi‐material full‐plane.

Chapter 08

Figure 8.1 Refinement schemes of S‐element mesh. The solid dots (•) indicate the end nodes and the circles (

) the internal nodes of line elements (a) Initial mesh. (b)

h

‐refinement by subdividing S‐elements into smaller ones. (c)

p

‐refinement by increasing the order of elements on boundary of S‐elements. (d)

h

s

‐refinement by subdividing the elements on the boundary of S‐elements into smaller ones. (e) and (f) examples of mixed types of refinement.

Figure 8.2 Lagrange interpolation.

Figure 8.3 A cubic element with uniformly spaced nodes.

Figure 8.4 Shape functions of cubic elements with uniformly spaced nodes (indicated by circles).

Figure 8.5 A 4th order spectral element.

Figure 8.6 Shape functions of a 4th order spectral element. The nodes are indicated by circles.

Figure 8.7 Cubic spectral element (

) in two dimensions.

Figure 8.8 Cantilever beam subject to a vertical force at the free end.

Figure 8.9 Mesh of a cantilever beam with quadratic line elements (

) on edges of S‐elements.

Figure 8.10 Convergence of displacement of a cantilever beam subject to vertical force at free end.

Figure 8.11 Mesh of a cantilever beam with cubic line elements (

) on edges of S‐elements.

Figure 8.12 Mesh of a quarter of a square body with a circular hole using one S‐domain with one spectral elements on each edge.

Figure 8.13 Convergence of displacements of a square body with a circular hole with increasing order of spectral elements.

Figure 8.14 Mesh of an L‐shaped panel by one S‐element with spectral elements on the edges.

Figure 8.15 Convergence of vertical displacement at point

A

of a L‐shaped panel.

Figure 8.16 A 3D cantilever beam subject to end‐shear loading.

Figure 8.17 Convergence of displacement of 3D cantilever beam subject to end‐shear loading.

Figure 8.18 A pressurized hollow sphere under uniform inner pressure.

Figure 8.19 Convergence of displacement of a pressurized hollow sphere.

Chapter 09

Figure 9.1 Generation of quadtree mesh on a circular domain. (a) Problem domain. (b) Quadtree mesh covering the problem domain. The cells are refined to fit the bonudary (c) Quadtree mesh after removing external cells.

Figure 9.2 An octree with three levels.

Figure 9.3 Generation of an octree mesh of a cylinder: (a) Problem domain and initial division of bounding box. (b) Cells are refined to fit the boundary. (c) Balanced octree mesh (2:1 rule). (d) Octree mesh after removing external cells.

Figure 9.4 Quadtree mesh of the top‐right quadrant of a circular domain. Demonstration of subdivision (dashed lines) is given in two quadtree cells with hanging nodes on their sides.

Figure 9.5 A quadtree mesh of S‐elements.

Figure 9.6 Types of objects in a three‐dimensional polyhedral mesh. (a) Node. (b) Edge. (c) Face. (d) S‐element.

Figure 9.7 A digital image of

pixels for use as an example to demonstrate quadtree decomposition. The colour intensity is given in a range between 0 (darkest colour) and 15 (lightest colour).

Figure 9.8 The square image matrix [

I

2

] of dimension of an integer power of 2 obtained by padding matrix [

I

1

] in Figure 9.7b with a background colour (100) that is outside the colour range of the image.

Figure 9.9 Quadtree decomposition of the image matrix in Figure 9.8. The parameters are: actual threshold of colour intensity is equal to 2,

mindim=1

 pixel and

maxdim=4

 pixels.

Figure 9.10 Quadtree decomposition of the image matrix in Figure 9.8 into cells of uniform colour intensity (

threshold=0

). The maximum dimension of the edge is

maxdim=4

 pixels and the minimum dimension is

mindim=1

 pixel.

Figure 9.11 The balanced quadtree decomposition of the image matrix in Figure 9.8 obtained by balancing the decomposition in Figure 9.9c.

Figure 9.12 Balanced quadtree mesh of the image in Figure 9.7a obtained with the actual threshold of colour intensity being equal to 2,

mindim=1

 pixel and

maxdim=4

 pixels.

Figure 9.13 A balanced quadtree cell. The solid dots (•) indicate the four corners and the circles (

) the locations of possible midpoints connected to adjacent cells.

Figure 9.14 Unique patterns of S‐elements in a balanced quadtree mesh. The numbers in circles indicate the element numbers on the boundary of the S‐elements.

Figure 9.15 A balanced quadtree decomposition of a circular inclusion in a square domain.

Figure 9.16 Example of octree decomposition of a 3D digital image into cells of uniform colour intensity (

threshold=0

). The maximum dimension of the edge is

maxdim=8

 voxels and the minimum dimension is

mindim=1

 voxel. (a) A cube of

voxels with a grey spot of 1 voxel and a grey spot of

voxels. (b)–(e) The first, second, third and final decomposition. (f) Balanced octree decomposition.

Figure 9.17 Balanced octree decomposition of one octant of a spherical inclusion in a cube.

Figure 9.18 A balanced octree cell. The solid dots (•) indicate the corners. The circles (

) and the crosses (

) show, respectively, the locations of possible centre points of square faces and midpoints of edges connected to adjacent cells.

Figure 9.19 Unique patterns of configuration of nodes, indicated by solid dots (•), on a face of an octree cell. The faces are discretized by (isosceles) triangular and rectangular (square) elements as depicted by the additional thin solid lines. A centre node, shown as a square (■), is introduced in the configurations shown in Figures b, c and f.

Figure 9.20 The surface discretization of S‐elements in a balanced octree mesh with hanging nodes.

Figure 9.21 Scaling of S‐elements generated from the same quadtree cell pattern. The solid dots (•) indicate the corner nodes. The circles (

) indicate the possible midside nodes at the midpoints of the edges.

Figure 9.22 Modelling of 2D concrete specimen.

Figure 9.23 Vertical displacement (µm) of 2D concrete specimen under uniaxial extension.

Figure 9.24 Maximum principal stress (MPa) for the 2D concrete specimen under uniaxial extension.

Figure 9.25 Model of 3D concrete specimen.

Figure 9.26 Vertical displacement (in

μ

m) 3D concrete specimen under uniaxial extension.

Figure 9.27 Vertical displacement (µm) on 3

slices of 3D concrete specimen under uniaxial extension.

Figure 9.28 Maximum principal stress (MPa) of 3D concrete specimen.

Figure 9.29 Maximum principal stress (in MPa) on 3

slices of 3D concrete specimen under uniaxial extension.

Figure 9.30 Quadtree grid of a square domain. The density of cells are controlled by the mesh seeds given on a inclined straight line and two circles.

Figure 9.31 Sequence of operation in trimming an octree.

Figure 9.32 Splitting an edge. The signed distance function of a point is positive, zero or negative when the point is located outside the domain, on the boundary and inside the domain.

Figure 9.33 Trimming a face. (a) a face with two edges cut by boundary. (b) Lines inside the domain. (c) Trimmed face.

Figure 9.34 Trimming a cell. (a) Cell with faces trimmed by boundary. (b) Construction of boundary face. (c) Trimmed cell formed by trimmed faces and boundary face.

Figure 9.35 Square body with multiple holes under uniaxial tension. (a) Geometry and boundary conditions. (b) Quadtree mesh.

Figure 9.36 Convergence of displacements at Point A of a square body with multiple holes under uniaxial tension with

h

‐ and

p

‐refinements.

p

is the element order and

m

is the slope of the trendline representing the rate of convergence.

Figure 9.37 Contour plots of stress

σ

y

in of a square body with multiple holes under uniaxial tension.

Figure 9.38 An evolving void in a squared body.

Figure 9.39 Meshes around the evolving void in Figure 9.38.

Figure 9.40 Von Mises stress in a square body with an evolving void.

Figure 9.41 L‐shaped panel under a tensile load.

Figure 9.42 Adaptive mesh refinement of L‐shaped panel using quadratic elements.

Figure 9.43 Relative error in the energy norm for the L‐shaped domain for both uniform and adaptive refinement.

Figure 9.44 A mechanical part subject to tension. (a) Geometry and boundary conditions. (b) Mesh and contour plot of vertical displacement.

Figure 9.45 Vertical displacement of a mechanical part along the dotted line shown in Figure 9.44a.

Figure 9.46 A triangular facet of STL and its ASCII representation.

Figure 9.47 Mesh generation of the sphinx. (a) STL model. The close‐up view shows some elongated triangles. (b) Mesh of S‐elements generated by octree algorithm. The close‐up view of mesh is at the same location as the close‐up view in the STL model.

Figure 9.48 Vertical displacement of a sphinx under self‐weight. (a) Contour on the whole model. (b) Contour on half of the model showing interior of mesh.

Figure 9.49 An STL model of Lucy. A flaw is found in the rectangle in the close‐up view.

Figure 9.50 Stress analysis of Lucy under self‐weight. Left is von Mises stress and right is the maximum principal stress. The locations of maximum values are shown in the close‐up views.

Chapter 10

Figure 10.1 A crack in a homogeneous isotropic material.

Figure 10.2 An interfacial crack in a bimaterial material.

Figure 10.3 A multi‐material wedge.

Figure 10.4 Modelling of stress field near singularity points. (a) Interfacial crack. (b) Multi‐material wedge.

Figure 10.5 An edge‐cracked square body: Geometry and boundary discretization with cubic elements.

Figure 10.6 An edge‐cracked square body: Stress modes.

Figure 10.7 An edge‐cracked square body: Convergence of stress intensity factors and the

T

‐stress with increasing numbers of cubic elements.

Figure 10.8 An edge‐cracked square body modelled by spectral elements: Convergence of stress intensity factors and the

T

‐stress.

Figure 10.9 An angled crack a in rectangular orthotropic body. (a) Geometry. (b) Mesh of 6th order elements. Total number of nodes is 60. (c) Mesh of 8th order elements. Total number of nodes is 80. (d) Mesh of 10th order elements. Total number of nodes is 100. (e) Mesh of 14th order elements. Total number of nodes is 140.

Figure 10.10 Interfacial central crack between two anisotropic materials. (a) Geometry. (b) Mesh.

Figure 10.11 A V‐notched bimaterial body. (a) Geometry. (b) Mesh.

Figure 10.12 A V‐notched bimaterial body of varying opening angle

α

: Generalized stress intensity factors.

Figure 10.13 An V‐notched bimaterial body of varying opening angle

α

. (a) Matrix of orders of singularity. (b) Eigenvalues of matrix of orders of singularity.

Figure 10.14 A crack terminating at a material interface. (a) Geometry. (b) Mesh.

Figure 10.15 A crack terminating at a material interface for varying crack angle

α

: Generalized stress intensity factors.

Figure 10.16 A crack terminating at a material interface for varying crack angle

α

. (a) Matrix of orders of singularity. (b) Eigenvalues of matrix of orders of singularity.

Figure 10.17 Angular variation of generalized stress intensity factors at crack inclination angle

Figure 10.18 A single edge‐cracked rectangular body under tension. (a) Geometry. (b) Mesh consisting of 3 S‐elements with scaling centres and the S‐element numbers. Only the end nodes of line elements are shown. (c) Mesh of 5th order line elements of S‐element 1.

Figure 10.19 A single edge‐cracked rectangular body under tension: QR‐codes of MATLAB functions of Padé approximants of the normalized stress intensity factor and

T

‐stress.

Figure 10.20 A single edge‐cracked rectangular body under bending: Geometry (left), and the normalized stress intensity factor

) and

T

‐stress

T

/

σ

0

(right).

Figure 10.21 A single edge‐cracked rectangular body under bending: QR‐codes of MATLAB functions of Padé approximants of stress intensity factor and

T

‐stress.

Figure 10.22 Centre‐cracked rectangular body under tension.

Figure 10.23 A centre‐cracked rectangular body under tension: QR‐codes of MATLAB functions of Padé approximants of stress intensity factor and

T

‐stress.

Figure 10.24 A double edge‐cracked rectangular body under tension.

Figure 10.25 A double edge‐cracked rectangular body under tension: QR‐codes of MATLAB functions of Padé approximants of stress intensity factor and

stress.

Figure 10.26 A single edge‐cracked rectangular body under end shearing.

Figure 10.27 A single edge‐cracked rectangular body under end shearing: QR‐codes of MATLAB functions of Padé approximants of stress intensity factors and

T

‐stress.

Figure 10.28 Illustration of modelling of crack propagation by S‐elements.

Figure 10.29 Model crack propagation by local remeshing of a polygon S‐element mesh (medium solid line) generated from a triangular mesh (thin dashed line). (a) Mesh around crack tip (Point O) at current step. (b) Identification of a group of triangles surrounding the crack increment for remeshing. (c) Construction of a patch incorporating the crack increment. (d) Generate a triangular mesh on the patch conforming to the new crack path. (e) The polygon mesh of S‐elements of the patch is generated from the triangular mesh incorporating the crack increment.

Figure 10.30 Modelling crack propagation by local remeshing of a quadtree mesh.

Figure 10.31 Arcan specimen.

Figure 10.32 Predicted crack paths for Arcan specimen.

Figure 10.33 Final polygon meshes for Arcan specimen.

Figure 10.34 Predicted fatigue life for Arcan specimen.

Figure 10.35 Cracked beam with three holes. The two cases of crack location and depth are indicated in the table.

Figure 10.36 Initial meshes of the cracked beam with three holes.

Figure 10.37 Final crack paths for cracked beam with three holes.

Figure 10.38 Comparison of predicted crack paths for the cracked beam with three holes with published results in the literature.

Guide

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The Scaled Boundary Finite Element Method

The Scaled Boundary Finite Element Method

Introduction to Theory and Implementation

Chongmin Song

University of New SouthWalesSydney, Australia

This edition first published 2018© 2018 John Wiley & Sons Ltd

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Chongmin Song to be identified as the author has been asserted in accordance with law.

Registered Office(s)John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK

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Limit of Liability/Disclaimer of WarrantyMATLAB® is a trademark ofThe MathWorks, Inc. and is used with permission.The MathWorks does not warrant the accuracy of the text or exercises in this book. This work’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging‐in‐Publication DataNames: Song, Chongmin, author.Title: The scaled boundary finite element method: introduction to theory and implementation / by Chongmin Song.Description: Hoboken, New Jersey: John Wiley & Sons, 2018. | Includes bibliographical references and index. |Identifiers: LCCN 2018003949 (print) | LCCN 2018010627 (ebook) | ISBN 9781119388463 (pdf) | ISBN 9781119388456 (epub) | ISBN 9781119388159 (cloth)Subjects: LCSH: Finite element method. | Boundary element methods.Classification: LCC QC20.7.F56 (ebook) | LCC QC20.7.F56 S66 2018 (print) | DDC 518/.25–dc23LC record available at https://lccn.loc.gov/2018003949

Cover design: WileyCover image: Courtesy of Chongmin Song

To Feng, Helena and Sophiefor their support, patienceand many sacrifices

Preface

The finite element method is nowadays the most widely applied numerical method in computational stress analysis. Many finite element software packages are available commercially or in the public domain. This book presents the scaled boundary finite elements, a type of finite elements based on the scaled boundary technique. These elements are practically arbitrary polygons/polyhedra in shape, leading to a substantial reduction of the mesh generation burden. The semi‐analytical representation of the displacement field in an element is advantageous when modelling singularity problems. Furthermore, the scaled boundary finite elements are seamlessly compatible with standard displacement‐based finite elements. Incorporating these kinds of elements in a finite element software package has great potential in directly integrating geometric models and finite element analysis, in modelling singularity problems and crack propagation and in adaptive analysis.

A basic knowledge of the finite element method, linear algebra and ordinary differential equations at the level of typical undergraduate courses would be beneficial to the understanding of this book. The theoretical derivations are presented step‐by‐step in great detail and are thus self‐contained.

This book should appeal to numerical analysts and software developers in various fields of engineering and science, such as civil engineering, mechanical engineering, computational mathematics and material science. This book can be used for an advanced course on finite elements and numerical methods.

Structure of the Book

This book consists of ten chapters covering the theory, computer implementation and application of the scaled boundary finite element method. This book commences with an overview of the scaled boundary finite element method in Chapter 1. The salient features of this method are illustrated with examples. The remaining chapters are organized into two Parts.

Part I, including Chapters 2–5, provides an introduction to the fundamental principles and the MATLAB programming of the scaled boundary finite element method. When selecting the contents, the primary consideration is simplicity, allowing the reader to follow and to learn. The theory is limited to the two‐dimensional case. In developing the scaled boundary finite element formulations, only the simplest 2‐node line element is addressed. All theoretical developments are explained step‐by‐step and with numerical examples. A computer program written in MATLAB is presented along with the theory and forms an integral part of the text. All the MATLAB functions and scripts necessary for the reader to reproduce all the numerical examples in this Part are provided.

In

Chapter 2

, the key concepts of the scaled boundary finite element method are presented. The scaling requirement, scaled boundary coordinates and transformation are introduced. A detailed derivation of the scaled boundary finite element equation by applying the virtual work principle is provided.

In

Chapter 3

, a solution procedure of the scaled boundary finite element equation is addressed. The quantities commonly employed in the finite element method, such as the stiffness matrix, the mass matrix and equivalent nodal force vector, are obtained. The shape functions of polygon elements are constructed, which allows the scaled boundary finite element method to be applied as a general finite element method. The static and dynamic analyses are demonstrated by examples.

In

Chapter 4

, the automatic generation of polygon mesh is addressed. Examples are presented to demonstrate the salient features and to evaluate the accuracy and convergence of the scaled boundary finite element method.

In

Chapter 5

, modelling considerations in applying scaled boundary finite element method are discussed. The guidelines that are not conventional or intuitive from the experience of the finite element analysis are discussed. Some of the applications for which the scaled boundary finite element method is advantageous are illustrated by examples. They provide a basis for establishing good practice in carrying out scaled boundary finite element analyses.

Part II, including Chapters 6– 10 , expands on the basics introduced in Part I to systematically develop the theory of the scaled boundary finite element method. The significant advantages that the scaled boundary finite element method offers in several types of challenging applications are demonstrated.

In

Chapter 6

, the scaled boundary finite element equation in three dimensions is derived for elastodynamics by applying the Galerkin’s weighted residual technique. The properties of the equation and the linear completeness of the solution are examined theoretically.

In

Chapter 7

, a solution procedure of the scaled boundary finite element equation based on the matrix exponential function and block‐diagonal Schur decomposition is presented. It provides a robust representation of singularities, to which the polynomial interpolations in standard finite elements are unsuitable, and contributes to the advantages of the scaled boundary finite element method in fracture analysis.

In

Chapter 8

, the use of high‐order elements is presented with the focus on the spectral elements. The scaled boundary finite element method is applied as high‐order polygon/polyhedral elements of arbitrary edges, which eases the difficulties of spectral elements in modelling complex geometries. Numerical examples demonstrate that the spectral elements lead to exponential convergence.

In

Chapter 9

, automatic mesh generation for use with the scaled boundary finite element method in two and three dimensions is addressed. Based on a quadtree/octree algorithm, mesh techniques are developed to handle CAD models and digital images. Fully automatic analyses of complex digital images of meso‐structures of materials and STL models of statues are demonstrated with examples.

In

Chapter 10

, the application of the scaled boundary finite element method to linear elastic fracture mechanics is covered. The stress singularities at crack tips and vertices of multi‐material wedges are treated. Generalized stress intensity factors and the

T

‐stress are evaluated. This approach does not require local mesh refinements, enrichments with analytical functions or special post‐processing techniques. Crack propagation problems are conveniently handled owing to the simplicity in computing the stress intensity factors and the flexibility in meshing.

Two appendices are provided. Appendix A presents the governing equations of linear elasticity, and Appendix B introduces the matrix power function.

Accompanying Computer Program Platypus

A computer program Platypus implementing the scaled boundary finite element method for stress analysis in MATLAB is integrated into the text of the book. Platypus includes a mesh generator for use to discretize simple problem domains into meshes of polygon elements of arbitrary number of edges. The polygon elements are constructed by modelling the boundary of the polygon elements with 2‐node line elements. After obtaining the element solutions using the scaled boundary finite element theory contained in this book, standard finite element procedures are followed to perform linear static and dynamic analyses. Basic post‐processing functions are included. All the examples in Part I of this book are produced with Platypus.

The MATLAB functions are documented in detail. In particular, cross‐references to the equations implemented in a function are provided within the function for the purpose of clear documentation. The MATLAB code will be helpful in understanding the theory presented in the text and provides an unambiguous interpretation of the mathematical theory and equations. The code can also be incorporated into a general finite element programme and forms the basis for developing more advanced versions.

The computer program Platypus is available at:

https://www.dropbox.com/sh/0nrc7tb9rjhfya3/AABHNwY5oo6SuLI2N4bnNh19a?dl=0

The above internet address can be acquired by scanning the QR‐code below:

Electronic files on the installation and use of Platypus are found in the same place.

Acknowledgements

The scaled boundary finite element method originated in research led by my mentor Dr. John P. Wolf at the Swiss Federal Institute of Technology, in Lausanne. I gratefully acknowledge his significant scientific contributions. Without his leadership, scientific insight, strong support and trust, this research would not have been brought to fruition.

I would like to thank my collaborators and students at the University of New South Wales (UNSW Sydney) for their contributions to the work contained in this book. In particular, the efforts of Ean Tat Ooi, Albert Suputra, Carolin Birk, Hauke Gravenkamp, Yan Liu, Hossein Talebi, Sundararajan Natarajan, Hou Man, Mohammad Bazyar, Suriyon Prempramote, Irene Chiong, Morsaleen Chowdhury, Chao Li, Xiaojun Chen, Ke He, Tingsong Xiang, Junchao Wang, Lei Liu, Weiwei Xing, Junqi Zhang and Duc Tran are greatly appreciated.

I express my deepest gratitude to Professor Gao Lin at the Dalian University of Technology, China, Fellow of the Chinese Academy of Sciences, for his very significant contributions, encouragement and support throughout the development of the scaled boundary finite element method.

Some of the research leading to this book was financially supported by the Australia Research Council. This support is gratefully acknowledged.

Sydney, AustraliaOctober, 2017

Chongmin Song

About the Companion Website

This book is accompanied by a companion website:

The URL is:

www.wiley.com/go/author/scaledboundaryfiniteelementmethod

The website includes Source codes

Platypus

1Introduction

1.1 Numerical Modelling

The advances in numerical modelling techniques and computer technology during the past few decades have transformed the analysis and design of many types of engineering structures. Today, numerical simulations as an integral part of Computer‐Aided Engineering (CAE) are routinely performed in civil engineering, mechanical engineering, aerospace and other industries.

Many problems in engineering and science are formulated as field problems by mathematical models in terms of field variables, such as displacements, potentials, etc. The mathematical models consist of governing differential equations to describe physical laws (such as equilibrium and compatibility in a stress analysis), material constitutive models and boundary conditions enforced on the problem domains. Classical methods of engineering analysis are often based on analytical solutions to the mathematical model. The analytical solutions are expressed as mathematical functions and can be evaluated at any locations of interest. However, analytical solutions are available only for very simple problems. For many engineering problems, considerable simplifications have to be made in order to apply the classical analysis methods. This often leads to over‐conservative designs.