FUNDAMENTAL SOLUTION BASED NUMERICAL

METHODS FOR THREE DIMENSIONAL

PROBLEMS: EFFICIENT TREATMENTS OF

INHOMOGENEOUS TERMS AND

HYPERSINGULAR INTEGRALS

By

Cheuk Yu Lee

A thesis submitted for the degree of

Doctor of Philosophy

of The Australian National University

4th of November 2016

© Copyright by Cheuk Yu Lee 2016

All Rights Reserved

I

Declaration

I certify that this thesis is my own research work at the Research School of Engineering

of The Australian National University, Acton, Australia. This submission contains no

material which has been accepted for the award of any other degree or diploma in my

name, in any university or other tertiary institution and, to the best of my knowledge and

belief, contains no material previously published or written by another person, except

where due reference is made in the text of the thesis.

Cheuk Yu Lee

November 2016

II

Dedication

To my grandmother, my family, my dog Mimi and to the loving memory of my

grandfather

For their unconditional love and support

III

Acknowledgements

I would like to express my sincere gratitude to my supervisor Prof. Qinghua Qin for his

continuous support of my PhD research direction and academic freedom. Under Prof.

Qin’s guidance, I can always obtain very insightful academic advice, especially at the

time of publishing research papers. I am thankful for Prof. Qin’s time devoted to my PhD

project and his 24 hours a day, 7 days a week accessibility. It is always motivational when

you have a supervisor working harder than you. Research takes time and in my case it has

taken me longer than anticipated. Without Prof. Qin’s continuous support, including

research funding, I certainly wouldn’t be able to finish my PhD project. I am greatly

indebted to Prof. Qin for all he has done for me and words are not enough to repay him.

I would like to express my regards and gratitude to my PhD advisors: Prof. Hui Wang of

Henan University of Technology, China and Dr. Zbigniew Henryk Stachurski of the

Australian National University. Prof. Wang has given me a lot of valuable and insightful

advice on computational mechanics research and I enjoyed every moment of our

collaboration. My sincere thanks also go to Dr. Stachurski, who was always there to

motivate me and give me courage throughout my PhD study.

I would also like to thank my former university classmate and good friend Albert Dessi,

who has given me a lot of advice on how to write English efficiently in a concise manner.

He is always here to help me out whenever I seek for his advice and I am very thankful

for that.

I would also like to thank all my research group colleagues, in particular Song Chen,

Shuang Zhang, Haiyang Zhou and Bobin Xing for all their independent and helpful advice

whenever I encounter an obstacle in research.

IV

I greatly acknowledge the generous University Research Scholarships, the ANU

Supplementary Scholarship and ANU Miscellaneous Scholarship offered from the

Australian National University. I would also like to acknowledge the university’s

invaluable support through the ANU Tuition Fee Exemption Sponsorship.

Last but not the least, I would like to thank my family: my parents, my aunt Marisa and

my brother for their unconditional, endless and spiritual support throughout my research

studies and my life in general.

Supervisory Panel

Prof. Qinghua Qin, The Australian National University, Chair

Dr. Zbigniew Henryk Stachurski, The Australian National University, Advisor

V

Abstract

In recent years, fundamental solution based numerical methods including the meshless

method of fundamental solutions (MFS), the boundary element method (BEM) and the

hybrid fundamental solution based finite element method (HFS-FEM) have become

popular for solving complex engineering problems. The application of such fundamental

solutions is capable of reducing computation requirements by simplifying the domain

integral to the boundary integral for the homogeneous partial differential equations. The

resulting weak formulations, which are of lower dimensions, are often more

computationally competitive than conventional domain-type numerical methods such as

the finite element method (FEM) and the finite difference method (FDM).

In the case of inhomogeneous partial differential equations arising from transient

problems or problems involving body forces, the domain integral related to the

inhomogeneous solutions term will need to be integrated over the interior domain, which

risks losing the competitive edge over the FEM or FDM. To overcome this, a particular

treatment to the inhomogeneous term is needed in the solution procedure so that the

integral equation can be defined for the boundary. In practice, particular solutions in

approximated form are usually applied rather than the closed form solutions, due to their

robustness and readiness. Moreover, special numerical treatment may be required when

evaluating stress directly on the domain surface which may give rise to hypersingular

integral formulation. This thesis will discuss how the MFS and the BEM can be applied

to the three-dimensional elastic problems subjected to body forces by introducing the

compactly supported radial basis functions in addition to the efficient treatment of

hypersingular surface integrals. The present meshless approach with the MFS and the

VI

compactly supported radial basis functions is later extended to solve transient and coupled

problems for three-dimensional porous media simulation.

VII

CONTENTS

Declaration ......................................................................................................................... I

Dedication ........................................................................................................................ II

Acknowledgements ......................................................................................................... III

Abstract ............................................................................................................................ V

List of Tables................................................................................................................. XII

List of Figures .............................................................................................................. XIV

Acronyms ...................................................................................................................... XX

Chapter 1 Introduction ..................................................................................................... 1

1.1 Classification of numerical methods in continuum mechanics .............................. 1

1.2 Finite element method ............................................................................................ 3

1.3 Boundary element method ..................................................................................... 6

1.4 Method of fundamental solutions......................................................................... 18

1.5 Other meshless methods ....................................................................................... 21

1.6 Summarisation of literature review and Research Gaps ...................................... 22

1.7 Our work .............................................................................................................. 23

1.8 Structure of the thesis ........................................................................................... 24

Chapter 2 Method of fundamental solutions for three-dimensional elasticity with body

forces by coupling compactly supported radial basis functions...................................... 25

VIII

2.1 Introduction .......................................................................................................... 25

2.2 Problem description ............................................................................................. 26

2.3 Method of particular solutions ............................................................................. 28

2.3.1 Dual reciprocity method ................................................................................. 29

2.3.2 Particular solution kernels with CSRBF ........................................................ 31

2.4 Method of fundamental solutions for homogeneous solutions ............................ 38

2.5 Numerical examples and discussions ................................................................... 40

2.5.1 Prismatic bar subjected to gravitational load ................................................. 41

2.5.2 Cantilever beam under gravitational load ...................................................... 46

2.5.3 Thick cylinder under centrifugal load ............................................................ 50

2.6 Further discussions on the sparseness of CSRBF ................................................ 56

2.7 Conclusions .......................................................................................................... 63

Chapter 3 Dual reciprocity boundary element method using compactly supported radial

basis functions for 3D linear elasticity with body forces ................................................ 65

3.1 Introduction .......................................................................................................... 65

3.2 Problem description ............................................................................................. 67

3.3 Dual reciprocity method....................................................................................... 68

3.4 Formulation of dual reciprocity boundary element method ................................. 70

3.5 Numerical Examples and discussions .................................................................. 73

3.5.1 Prismatic bar subjected to gravitational load ................................................. 73

3.5.2 Thick cylinder under centrifugal load ............................................................ 78

3.5.3 Axle bearing under internal pressure and gravitational load ......................... 82

IX

3.6 Conclusions .......................................................................................................... 84

Chapter 4 Evaluation of hypersingular line integral by complex-step derivative

approximation ................................................................................................................. 86

4.1 Introduction .......................................................................................................... 86

4.2 Definitions and properties of hypersingular line integrals ................................... 89

4.2.1 Cauchy principal value integral .................................................................... 90

4.2.2 Hypersingular integral ................................................................................... 90

4.2.3 Simpler hypersingular integrals values ......................................................... 92

4.3 Barycentric rational interpolation ........................................................................ 93

4.3.1 Interpolatory quadrature formulation ............................................................ 94

4.3.2 Barycentric rational polynomial .................................................................... 95

4.3.3 Numerical example ....................................................................................... 97

4.4 Complex-step derivative approximation ............................................................ 102

4.5 Conclusions ........................................................................................................ 106

Chapter 5 Evaluation of hypersingular surface integral by complex-step derivative

approximation ............................................................................................................... 107

5.1 Introduction ........................................................................................................ 107

5.2 Definition of hypersingular surface integral ...................................................... 108

5.3 Regularisation and numerical procedures of hypersingular surface integrals ... 111

5.3.1 Surface discretisation .................................................................................. 111

5.3.2 Polar coordinate transformation .................................................................. 115

5.3.3 Asymptotic expansions of r ......................................................................... 119

X

5.3.4 Asymptotic expansions of density function ψ ............................................ 120

5.3.5 Proposed formulation of hypersingular surface integral ............................. 121

5.4 Numerical examples ........................................................................................... 125

5.4.1 Example 1A: Trapezium meshed with a distorted rectangular element ..... 126

5.4.2 Example 1B: Improved meshing of trapezium using rectangular elements 130

5.4.3 Example 2A: Rectangle meshed with regular rectangular elements ........... 133

5.4.4 Example 2B: Rectangle meshed with distorted triangular elements of various

degrees .................................................................................................................. 137

5.4.5 Example 3: Quarter cylindrical panel with curved rectangular element ..... 141

5.5 Conclusions ......................................................................................................... 146

Chapter 6 Simulation of three-dimensional porous media using MFS-CSRBF .......... 148

6.1 Introduction ........................................................................................................ 148

6.2 Model description .............................................................................................. 151

6.3 Iteratively coupled method................................................................................. 154

6.3.1 Uncoupled porous media equations ............................................................ 155

6.3.2 Time discretisation with Laplace transform ................................................ 157

6.3.3 MFS-CSRBF kernels for flow equation...................................................... 158

6.3.4 Laplace inversion algorithm........................................................................ 159

6.4 Numerical simulation ......................................................................................... 161

6.5 Conclusions ........................................................................................................ 164

Chapter 7 Summary and outlook ................................................................................. 166

7.1 Summary of present research ............................................................................. 166

XI

7.2 Research limitations ........................................................................................... 169

7.3 Future research ................................................................................................... 170

Bibliography .................................................................................................................. 172

List of publications ........................................................................................................ 186

XII

List of Tables

Table 1. 1: Globally supported RBFs .............................................................................. 17

Table 2. 1: Displacement results for the prismatic bar ................................................... 44

Table 2. 2: Stress results for the prismatic bar numerical simulation ............................. 44

Table 2. 3: Displacement results for the cantilever numerical simulation ...................... 48

Table 2. 4: 𝜎𝑟 results for the thick cylinder numerical simulation ................................. 53

Table 2. 5: 𝜎𝑡 results for the thick cylinder numerical simulation ................................. 53

Table 2. 6: 𝑢𝑟 results for the thick cylinder numerical simulation ................................. 54

Table 3. 1: Displacement results for the prismatic bar ................................................... 76

Table 3. 2: Stress results for the prismatic bar numerical simulation ............................. 76

Table 3. 3: 𝜎𝑟 results for the thick cylinder numerical simulation ................................. 80

Table 3. 4: 𝜎𝑡 results for the thick cylinder numerical simulation ................................. 81

Table 3. 5: 𝑢𝑟 results for the thick cylinder numerical simulation ................................. 81

Table 3. 6: Displacement results for the axle bearing numerical simulation .................. 83

Table 3. 7: Stress results at various test points ............................................................... 84

Table 4. 1: Numerical results of barycentric rational interpolation scheme and the Kolm

and Rokhlins’ method in comparison to analytical results ........................................... 100

XIII

Table 4. 2: 14-node quadratures for t=-0.9862838086968120 ..................................... 101

Table 5. 1: Variables and their spatial quantities for evaluating regular integrand

, , , P Pξ ξ ξ ....................................................................................................... 125

Table 5. 2: Numerical evaluation of hypersingular surface integrals on trapezium meshed

with a distorted 8-node rectangular element at various test locations with respect to the

different orders of Gaussian quadrature ........................................................................ 129

Table 5. 3: Numerical evaluation of hypersingular surface integral on trapezium at test

point a with improved meshing using 4 rectangular elements ...................................... 132

Table 5. 4: Numerical evaluation of hypersingular surface integrals on rectangle meshed

with two 8-node rectangular elements at test points a, b and c ..................................... 136

Table 5. 5: Numerical evaluation of hypersingular surface integrals using rectangular

elements and triangular elements with various aspect ratios ........................................ 140

Table 5. 6: Convergence rate of the proposed method and the literature [170] at various

singularity points for increasing number of Gaussian quadrature points ...................... 144

Table 5. 7: Hypersingular integral results for the quarter cylindrical panel at 3 different

singularity points (Note that 0’s are added in the reference results as only 6 significant

figures are found in their publications) ......................................................................... 145

XIV

List of Figures

Fig. 1. 1: Classification of numerical methods for boundary value problems .................. 2

Fig. 1. 2: Fundamental solution for two-dimensional Laplace equation ........................ 10

Fig. 1. 3: Displacement fundamental solution for three-dimensional Navier equations . 10

Fig. 1. 4: Thin plate splines with different orders ........................................................... 17

Fig. 1. 5: Profile of normalised Wendland CSRBFs 2

1 r .......................................... 18

Fig. 2. 1: Schematic representation of a 2D section of possible computational domain 28

Fig. 2. 2: Cut off parameter 𝛼 for various support radii ................................................. 33

Fig. 2. 3: Schematic figure illustrating the MFS-CSRBF numerical procedures ........... 40

Fig. 2. 4: Prismatic bar under gravitational load ............................................................. 42

Fig. 2. 5: Field points, source points and interpolation points distributing for the prismatic

bar.................................................................................................................................... 42

Fig. 2. 6: Displacements and normal stresses along the centreline of the prismatic bar

subjected to gravitational load ........................................................................................ 45

Fig. 2. 7: Contour plots of the vertical displacement component by ABAQUS (left) and

the present method (right) ............................................................................................... 45

Fig. 2. 8: Bending of cantilever beam under gravitational load ...................................... 46

Fig. 2. 9: Field points, source points and interpolation points distributing for the cantilever

......................................................................................................................................... 47

XV

Fig. 2. 10: Displacements along the centre axis of the cantilever subjected to gravitational

load .................................................................................................................................. 49

Fig. 2. 11: Contour plots of the displacement results from ABAQUS (left) and the present

method (right).................................................................................................................. 49

Fig. 2. 12: Thick cylinder under centrifugal load............................................................ 51

Fig. 2. 13: Field points, source points and interpolation points distributing for the thick

cylinder ............................................................................................................................ 51

Fig. 2. 14: Plot of radial displacements and radial stresses along the radial direction of the

cylinder subjected to centrifugal load (left) and plot of hoop stresses along the same

reference locations (right) ............................................................................................... 54

Fig. 2. 15: Contour plots of the 𝜎𝑟 results simulated in ABAQUS (left) and the present

method (right).................................................................................................................. 55

Fig. 2. 16: Contour plot of the 𝜎𝑡 results simulated in ABAQUS (left) and the present

method (right).................................................................................................................. 55

Fig. 2. 17: Contour plot of the 𝑢𝑟 results simulated in ABAQUS (left) and the present

method (right).................................................................................................................. 56

Fig. 2. 18: Cumulative frequency plot of radial distance for prismatic bar, cantilever and

quarter cylinder. std is the standard deviation of the radial distance .............................. 57

Fig. 2. 19: Quasi-random distribution (left) and pseudo-random distribution (right) of 790

interpolation points (49 points per unit area) .................................................................. 58

Fig. 2. 20: Convergence study of the pseudo-random and quasi-random distribution

schemes: MAPE% of the vertical displacements versus number of interpolation points

......................................................................................................................................... 59

XVI

Fig. 2. 21: Structures of the interpolation matrix at 10% sparseness for prismatic bar after

reordering using Reverse Cuthill-McKee algorithm (left column), approximate minimum

degree (centre column) and nested dissection (right column). First row presents the

interpolation matrices after reordering. Second row presents the Cholesky’s upper

triangular matrices after reordering. bw is the bandwidth and nz is the number of nonzero

entries of the matrices ..................................................................................................... 61

Fig. 2. 22: Varied sparseness of interpolation matrix versus sparseness of Cholesky’s

upper triangular matrix for prismatic bar after reordering using Reverse Cuthill-McKee

algorithm, approximate minimum degree and nested dissection .................................... 63

Fig. 3. 1: Prismatic bar under gravitational load ............................................................. 74

Fig. 3. 2: Boundary element meshing of prismatic bar ................................................... 75

Fig. 3. 3: Boundary element meshing and interior points distribution of a prismatic bar: 9

elements (left) and 36 elements (right) per unit area. ..................................................... 77

Fig. 3. 4: Convergence study of the present method with various degrees of sparseness:

MAPE% of the vertical displacements versus meshing density ..................................... 78

Fig. 3. 5: Thick cylinder under centrifugal load ............................................................. 79

Fig. 3. 6: Boundary element meshing of a quarter thick cylinder ................................... 79

Fig. 3. 7: Numerical model of the axle bearing .............................................................. 82

Fig. 4. 1: Regularisation process for hypersingular line integral using Brandao’s

formulation ...................................................................................................................... 92

XVII

Fig. 4. 2: Mean absolute percentage error of hypersingular integral evaluation using

numerical quadrature of barycentric rational interpolation scheme and the Kolm and

Rokhlins’ method ............................................................................................................ 99

Fig. 4. 3: Mean absolute percentage error of hypersingular integral evaluation using

complex-step derivative approximation of various complex-step sizes and Richardson

extrapolation orders ....................................................................................................... 105

Fig. 5. 1: Exclusion surface around a singularity point at corner ................................. 109

Fig. 5. 2: Local orthogonal curvilinear coordinates of eight-node plate element and its

nodal points’ distribution .............................................................................................. 112

Fig. 5. 3: Local orthogonal curvilinear coordinates of six-node triangular element and its

nodal points’ distribution .............................................................................................. 114

Fig. 5. 4: Regions of integration for the eight-node plate element in polar coordinate 116

Fig. 5. 5: Regions of integration for the 6-node triangular element in polar coordinate

....................................................................................................................................... 116

Fig. 5. 6: Radial path ρL0,0, θ for singularity point located at the centre of the eight-node

plate element ................................................................................................................. 118

Fig. 5. 7: Schematic of trapezium meshed with an 8-node rectangular element and with

various singularity points a, b and c .............................................................................. 128

Fig. 5. 8: Contour plot of the hypersingular singular integrals with varying singularity

points inside the enclosed region by test points a, b and c ............................................ 130

Fig. 5. 9: Trapezium meshed with four 8-node rectangular element ............................ 131

XVIII

Fig. 5. 10: Relative errors of hypersingular surface integral with singularity point a in

Examples 1A and 1B..................................................................................................... 133

Fig. 5. 11: On the left: Schematic of the rectangle example with test points a, b and c. On

the right: Distribution of Gaussian quadrature points inside an 8-node rectangular element

....................................................................................................................................... 135

Fig. 5. 12: Upper left: Case II rectangle subdivision using 6-node triangular element with

aspect ratio of 1.4142; Upper right: Case III rectangular subdivision using triangular

element with aspect ratio of 1.7889; Lower left: Case IV rectangular subdivision using

triangular element with aspect ratio of 2.3851; Lower right: Case V rectangular

subdivision using triangular element with aspect ratio of 3.5777 ................................. 139

Fig. 5. 13: Relative errors of hypersingular surface integrals using rectangular elements

and triangular elements with various aspect ratios ....................................................... 141

Fig. 5. 14: Schematic of a curved boundary element representing the quarter cylindrical

panel .............................................................................................................................. 143

Fig. 5. 15: Convergence rate of the proposed method and the literature [187] at singularity

point c for increasing number of Gaussian quadrature points ...................................... 145

Fig. 5. 16: Contour plot of the hypersingular integral values with respect to various

singularity points ........................................................................................................... 146

Fig. 6. 1: Schematic of smart gels subjected to environmental stimuli ........................ 150

Fig. 6. 2: Flow chart of iteratively coupled solutions finding process for poroelasticity

simulation ...................................................................................................................... 160

Fig. 6. 3: Geometrical model of the computational domain ......................................... 162

XIX

Fig. 6. 4: Configuration of collocation on one-eighth of sphere ................................... 163

Fig. 6. 5: Simulated pore fluid pressure against the time at the origin point ................ 163

Fig. 6. 6: Analytical result [223] of pore fluid pressure against the time at the origin point

....................................................................................................................................... 164

XX

Acronyms

BEM Boundary element method

BIE Boundary integral equation

BKM Boundary knot method

BNM Boundary node method

BPIM Boundary point interpolation method

BVP Boundary value problem

CSRBF Compactly supported radial basis function

DRBEM Dual reciprocity boundary element method

DRM Dual reciprocity method

FDM Finite difference method

FEM Finite element method

GSRBF Globally supported radial basis function

HFEM Hybrid finite element method

HFS-FEM Hybrid fundamental solution based finite element method

MAPE Mean absolute percentage error

MFS Method of fundamental solutions

MLPG Meshless local Petrov-Galerkin method

MLS Moving Least Squares

MMLS Modified Moving Least Squares

XXI

MQ Multiquadrics

PDE Partial differential equation

PSE Particle Strength Exchange

RBF Radial basis function

RIM Radial integration method

RKPM Reproducing Kernel Particle Methods

SPH Smoothed particle hydrodynamics

TPS Thin plate splines

VBEM Virtual boundary element method

1

Chapter 1 Introduction

1.1 Classification of numerical methods in continuum

mechanics

It is well known that engineering problems are usually described by mathematically

defined boundary value problems (BVPs) consisting of differential equations together

with boundary conditions [1]. Very often, analytical solutions are only available for a few

boundary value problems with simple geometries and boundary conditions [2-11].

Numerical methods, i.e. the finite element method (FEM) [12-14], the finite difference

method (FDM) [15], the hybrid finite element method (HFEM) [16-24], the meshless

local Petrov-Galerkin (MLPG) method [25], the smoothed particle hydrodynamics (SPH)

meshless method [26], the boundary element/integral equation method (BEM/BIE) [27-

30], the virtual boundary element method (VBEM) [31], the boundary knot method

(BKM) [32], the method of fundamental solutions (MFS) [33, 34], boundary node method

(BNM) [35] and boundary point interpolation method (BPIM) [36-38] provide alternative

approaches to approximate solutions for boundary value problems. Generally, these

numerical methods can be classified into two types: domain-type methods and boundary-

type methods, as shown in Fig. 1.1. The domain-type methods such as the FEM, the FDM,

the HFEM, the MLPG and the SPH require domain element or collocation discretisation.

By contrast, the boundary-type methods like the BEM/BIE and the MFS require only

boundary element or collocation discretisation for the homogeneous partial differential

equations (PDE).

Unlike the domain-type methods, the boundary-type methods are generally dependent on

the application of fundamental solutions of problems, which are capable of reducing the

2

computation requirements by simplifying the domain integrals to the boundary integrals.

The resulting weak formulations, which are of lower dimensions, are often more

computationally competitive than the conventional domain-type methods such as the

FEM.

As is typical of novel research involving the domain-type and boundary-type numerical

methods, the theoretical basis of the FEM, the BEM and the MFS are reviewed to

demonstrate the advantages and disadvantages of boundary-type methods.

Fig. 1. 1: Classification of numerical methods for boundary value problems

3

1.2 Finite element method

The FEM, devised by Courant in 1943 [39], adopts Ritz’s approach and has become a

powerful numerical tool these years for solving various static engineering problems such

as structural, fluid flow and heat transfer problems as illustrated in some popular

textbooks by Zienkiewicz [12] and Logan [40]. Its trial functions, known as interpolation

functions or shape functions satisfy a priori the boundary conditions but violate the

governing differential equations. The variational functional is employed to minimise the

nodal potential so as to enforce the governing differential equations. To illustrate the finite

element approach, a linear elastic problem defined in an elastic solid domain Ω is taken

into consideration. The conventional strain energy functional or variational principle Π is

a one-field (displacement field) principle and is stated as follows [12, 13]

T T1d dS Minimum

2 t ε Cε t u (1.1)

in which the strain field 𝛆 is in terms of the displacement field 𝐮 through the strain-

displacement relation, 𝐂 is the elasticity tensor and 𝐭 is the specified tractions along the

boundary Γ𝑡.

ε Du (1.2)

where 𝐃 is the matrix consisting of the shape functions’ derivatives.

When the domain is represented by a finite number of connecting elements, the principle

of minimum potential energy (1.1) can be restated as

T T1( ) ( )d d Minimum

2n tnn

S

Du C Du t u (1.3)

4

where Ω𝑛 is the nth element domain, and Γ𝑡𝑛 the traction boundary of the nth element

domain.

In the finite element formulation, the element displacement field 𝐮 is interpolated in terms

of element nodal displacements 𝐝

u Nd (1.4)

Substituting into Eq. (1.2) one obtains

ε Du Bd (1.5)

where B DN

Further, the substitution of Eqs. (1.4) and (1.5) into Eq. (1.3) gives

T1

2n n

n

d K d f d (1.6)

where

T( )dn

n

K B CB (1.7)

T( )dtn

n S

f t N (1.8)

After assembling the elementary stiffness matrix 𝐊𝑛 and the elementary nodal force

vector 𝐟𝑛 into the global stiffness matrix 𝐊 and the global nodal force vector 𝐟 , the

variational principle Π becomes

T1

2 d Kd fd (1.9)

Minimizing the functional (1.9) in terms of the nodal displacement 𝐝

0

d (1.10)

yields the basic stiffness equation for finite element analysis

5

Kd f (1.11)

As seen from the above procedure, the core of the FEM’s methodology is to discretise the

domain into a finite number of elements and to construct the corresponding energy

functional, thus the FEM can be effectively used for complicated engineering problems

involving complicated boundary conditions, material nonlinear, large deformation,

multiple domains and multi-field problems [41, 42]. Because the potential energy

variational principle is only in terms of displacement variables, the derivation of finite

element equations is straightforward and simple. Moreover, comprehensive error

estimation for this popular method has been established [43]. Also, the stiffness matrix in

the FEM is banded, sparse and symmetric. This allows storage of only the non-zero

elements in the matrix, saving storage space and memory. However, this method has

several shortcomings [44]:

(1) Standard FEM requires volumetric meshing and evaluation of the domain

integrals, which are time-consuming and complicated procedures for multi-

dimensional problems. Inside the domain, the solutions are not exact due to

the use of shape functions for their approximations.

(2) The FEM requires more computational effort than the BEM, due to volumetric

domain discretisation. Some modelling approaches, such as large deformation

and fracture mechanics, require iterative re-meshing of the computational

domain.

(3) Excessive element distortion will cause elemental interpolation to fail.

(4) It is difficult for the conventional FEM to maintain compatibility of the normal

derivative of displacements along the inter-element boundaries.

6

(5) Shear locking occurs when using lower-order linear elements for bending

problems. One effective solution is to employ shape functions with higher

degrees of order at the cost of more computational resources.

(6) Volumetric locking also occurs for incompressible materials.

(7) FEM is not convenient for modelling unbounded regions as they would require

specially built infinite elements.

1.3 Boundary element method

In contrast to the FEM, the BEM is a meshed boundary-type method based on the

boundary integral equation technique and just requires boundary element discretisation.

Boundary elements can be formulated using two different approaches called the direct

and indirect BEMs [45, 46]. The indirect BEM uses fictitious density functions or sources

that have no physical meaning in the boundary integral equation [47-50]. It requires the

placement of a second set of points representing the fictitious density distribution over

the boundary. The choice of location of the density functions can result in weak continuity

of the physical solutions and the possibility of an ill conditioned global interpolation

matrix. By contrast, the direct BEM employs physical parameters such as boundary

displacement and traction values in its integral equation applicable over the boundary [27,

51]. As the result, the solutions in this direct method retain their physical meanings and

the convergence of solutions can always be guaranteed [52]. Based on these merits, the

direct boundary element procedure is adopted in this thesis and thus is reviewed and

compared to the FEM. The development of the direct BEM can trace back to Somigliana’s

identity, which was established in 1886 and forms the backbone of the direct boundary

element formulation. Such characteristics are achieved by the use of fundamental

7

solutions of problems. In the BEM, the application of such fundamental solutions is

capable of reducing the domain integrals to the boundary integrals. The resulting weak

formulations, which are of lower dimensions, are often more computationally competitive

than the conventional FEM. For example, for the elastic solid domain Ω considered above,

the boundary integral equation can be derived by the reciprocal work theorem (also called

Betti’s theorem). Following [51], the reciprocal work theorem states that the work done

by the stresses of system (a) on the displacements of system (b) is equal to the work done

by the stresses of system (b) on the displacements of system (a). Now let’s consider two

different sets of stresses and strains as interacting pairs producing work done in the

equilibrium states of an elastic body Ω [27, 28, 51]:

Set (a): stresses 𝜎𝑖𝑗(𝑎)

that gives rise to strains 휀𝑖𝑗(𝑎)

Set (b): stresses 𝜎𝑖𝑗(𝑏)

that gives rise to strains 휀𝑖𝑗(𝑏)

Thus, we have the following integral relationship

( ) ( ) ( ) ( )d da b b a

ij ij ij ij

(1.12)

Substituting the strains expressed in terms of displacements

1

2

jiij

j i

uu

x x

(1.13)

into the integral equation (1.12) results in

( ) ( )( ) ( )( ) ( )1 1

d d2 2

b ab aj ja bi i

ij ij

j i j i

u uu u

x x x x

(1.14)

Considering the symmetry of stresses, i.e. 𝜎𝑖𝑗 = 𝜎𝑗𝑖, Eq. (1.14) can be rewritten as

( ) ( )( ) ( )d d

b aa bi i

ij ij

j j

u u

x x

(1.15)

8

In Eq. (1.15), expanding the left integral term in the following way gives

( )( )

( ) ( ) ( ) ( )d d d

abija a b bi

ij ij i i

j j j

uu u

x x x

(1.16)

Applying the divergence theorem and the stress-traction relation 𝜎𝑖𝑗𝑛𝑗 = 𝑡𝑖 to the first

term in the right of Eq. (1.16) yields

( ) ( ) ( ) ( ) ( ) ( )d d da b a b a b

ij i ij i j i iS S

j

u u n S t u Sx

(1.17)

Simultaneously, introducing the equilibrium relation 𝜕𝜎𝑖𝑗 𝜕𝑥𝑗⁄ + 𝑓𝑖 = 0 in the second

term in the right of Eq. (1.16) leads to

( )

( ) ( ) ( )d d

a

ij b a b

i i i

j

u f ux

(1.18)

Finally, we have

( )( ) ( ) ( ) ( ) ( )d d d

ba a b a bi

ij i i i iS

j

ut u S f u

x

(1.19)

Similarly, the right term of Eq. (1.15) can be rewritten as

( )( ) ( ) ( ) ( ) ( )d d d

ab b a b ai

ij i i i iS

j

ut u S f u

x

(1.20)

Hence, the Betti’s theorem can be finally expressed as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )d d d da b a b b a b a

i i i i i i i iS St u S f u t u S f u

(1.21)

Eq. (1.21) is the so-called Betti’s equation for elastic bodies, from which the boundary

integral equation can be derived by replacing the set (a) with actual quantities and

replacing the set (b) with virtual quantities expressed by fundamental solutions of

problems, that is

9

New Set (a): 𝑢𝑖(𝑎)

= 𝑢𝑖( Qx ), 𝑡𝑖(𝑎)

= 𝑡𝑖( Qx ), 𝑓𝑖(𝑎)

= 𝑓𝑖( Qx )

New Set (b): 𝑢𝑖(𝑏)

= 𝑈𝑖𝑗∗ ( ,P Qx x )𝑒𝑗, 𝑡𝑖

(𝑏)= 𝑇𝑖𝑗

∗ ( ,P Qx x )𝑒𝑗, 𝑓𝑖(𝑏)

= 0

Substituting into Betti’s equation yields the classic boundary integral equation (BIE)

* * *( ) ( , ) ( )d ( , ) ( )d ( , ) ( )di ij j ij j ij jS S

u T u S U t S U f

P P Q Q P Q Q P Q Qx x x x x x x x x x

(1.22)

where 𝑈𝑖𝑗∗ ( ,P Qx x ) and 𝑇𝑖𝑗

∗ ( ,P Qx x ) represent the displacement and traction fundamental

solutions at the boundary point Qx (field point) caused by the interior point P

x (source

point), respectively. This equation is known as the Somigliana’s identity for

displacements.

During the derivation of BIE, the basic physical features of fundamental solutions of

problems are employed for converting the domain integrals into boundary integrals. The

fundamental solutions are singular ones dependent on the reciprocal distance of two

points, i.e. Px and

Qx . As these two points approach, the fundamental solutions tend to

infinity. Such distinctive characteristics can be shown from Figs. 1.2 and 1.3. In Fig. 1.2,

the fundamental solution for the Laplace equation [53] with the source point Px at the

origin is plotted

1ln ( , )

2r

P Qx x (1.23)

and Fig. 1.3 illustrates the displacement fundamental solutions (Kelvin’s solutions) for

three-dimensional Navier equations [53] with the source point Px at the origin

( , ) ( , )1 1(3 4 )

16 (1 ) ( , )ij

i j

r r

G r x x

P Q P Q

P Q

x x x x

x x (1.24)

In Eqs. (1.23) and (1.24), 𝑟( ,P Qx x ) denotes the distance of point Px and

Qx .

10

Fig. 1. 2: Fundamental solution for two-dimensional Laplace equation

Fig. 1. 3: Displacement fundamental solution for three-dimensional Navier equations

11

Typically, in the absence of body forces 𝑓𝑖, the boundary integral equation (1.22) can be

simplified as

* *( ) ( , ) ( )d ( , ) ( )di ij j ij jS S

u T u S U t S P P Q Q P Q Qx x x x x x x (1.25)

which is the most popular form used in the standard BEM, because all integrals are

performed just along the domain boundary.

For stress at the source point 𝒙𝑷, a constitutive relationship between the stress and the

strain can be applied to the differentiation form of (1.25). Consequently, after substituting

the now known nodal displacements and the nodal tractions, we have

* *( ) ( , ) ( )d ( , ) ( )dij kij k kij kS SS u S D t S P P Q Q Q P Q Q Qx x x x x x x x x (1.26)

where 𝑆𝑘𝑖𝑗∗ and 𝐷𝑘𝑖𝑗

∗ are third order tensors derived using Hooke’s law which makes use

of traction and displacement fundamental solutions *

ijT and *

ijU .

For boundary stress problems, the first integral on the right hand side of (1.26) will give

rise to hypersingular surface integrals. Since evaluating such integrals is not a

straightforward task, we will defer the discussions to Chapter 4 and 5. Alternatively, the

traction recovery method [51] which transforms the local stress to global stress indirectly

instead of the hypersingular integral evaluation can be employed for the boundary stresses.

It is obvious from the simplified BIE (1.25) that the standard BEM has some advantages

over the conventional FEM depending on domain elements [51]:

(1) The boundary modelling in the BEM reduces dimensionality by one. This

means less data preparation time and a simpler meshing approach.

(2) Stresses at interior points have high accuracy, because no further

approximation is imposed on the solution inside the domain. Thus, the BEM

is very suitable for simulating stress concentrations.

12

(3) The BEM requires less computation time and storage, because only boundary

information is required.

(4) The BEM is easily applicable to unbounded regions.

(5) The BEM is easily applicable to incompressible materials. The integral kernels

of the BEM will not become singular when volumetric strain approaches to

zero. However, the same cannot be said for the FEM.

However, there are some disadvantages for the standard BEM:

(1) The numerical procedure of the BEM requires a strong mathematical

background.

(2) Inconvenient procedures for finite deformation applications. The domain

integral representing the finite deformation term is treated as a fictitious body

force and solved via iterative methods [54-56].

(3) Inconvenient procedures for non-linear materials problems. The governing

equations need to be first linearised by employing Kirchhoff transformations

[57] or to be solved iteratively.

(4) Treating multi-material problems using the BEM is difficult, because each

material domain needs to form its BIE and additional equations related to

interfacial conditions are required to connect these BIEs.

(5) The BEM has difficulty treating dynamic problems with time-dependent

variables.

(6) Due to the need for the domain integral to be used for inhomogeneous terms,

the BEM is inconvenient for problems with body forces.

13

(7) The solution matrix in the BEM is full and non-symmetrical, requiring more

computational resources, whereas the FEM has a sparse and symmetric

stiffness matrix.

(8) The fulfilment of the BEM requires the fundamental solutions of problems,

which are difficult to derive for some problems such as those involving large

deformations.

(9) The computation of hypersingular or near-singular integrals is complicated

and affects the solution accuracy of the BEM.

As one of key issues of the BEM, the efficient treatment of domain integrals, caused by

generalised inhomogeneous terms which may be related to actual body forces, time

domain discretisation, or thermoelasticity, is always challenging. Only for some special

cases, particular solutions related to inhomogeneous terms can be found analytically [58].

In many other cases, finding such analytical solutions is not a trivial task. In recent years,

in order to simplify the effort of domain discretisation in the BEM for inhomogeneous

cases, the dual reciprocity method (DRM) [59] has been commonly employed to couple

with the BEM to avoid the domain integration by applying collocation discretisation in

the domain [60]. Apart from the direct radial basis function (RBF) method [61, 62], the

dual reciprocity method aims to efficiently approximate the particular solution by finding

its solution kernels while prescribing the inhomogeneous terms such as body forces with

a series of linearly independent basis functions. This allows any known or unknown body

forces terms to be reconstructed using a finite set of discrete data. The use of the dual

reciprocity method to solve partial differential equations by boundary integrals was first

proposed by Nardini and Brebbia [59] for vibration problems, in which the domain

integrals are transferred to the boundary by introducing basis function approximation of

inhomogeneous terms. This approach was then extended for nonlinear diffusion problems

14

[63], thermal wave propagation in biological tissues [64], crack problems [65], phase

change problems [66, 67] and thermal transfer in non-homogeneous anisotropic media

[68]. The choice of the approximation function for the inhomogeneous terms will have a

significant impact on computation complexity, stability and accuracy [69]. When a

problem consists of irregular boundaries or inhomogeneous terms with localised

irregularities, approximation functions such as trigonometric or Chebyshev polynomials

are usually problematic since they would involve a large number of terms for the

approximation. Hence, the employment of the DRM method requires an accurate

approximation of the inhomogeneous term, which is usually constructed by a finite series

of basis functions.

Clearly, the choice of the basis functions is critical in the DRM-BEM to provide accurate

numerical solutions [70]. In most of the literature, the most common choices are the radial

basis functions (RBFs) [71]. RBFs are real-valued functions whose value depends only

on the Euclidean distance variable so that it is suitable to approximate given functions in

arbitrary dimensional space and does not increase computational cost. Many attractive

properties of RBF such as good convergence power, positive definiteness and ease of

smoothness control are widely reported [71, 72]. Due to such features, RBF has been

employed to approximate the inhomogeneous terms in many elliptic partial differential

equations, as done in some studies [73-75]. In the early development of DRM [60], the

simple radial basis function 1 + 𝑟 was employed. Other RBF choices were later studied,

including thin plate splines (TPS) and multiquadrics (MQ). Golberg et al. [70] provided

a convergence proof of the DRM on Poisson’s problem. Karur and Ramachandran [76]

numerically examined the convergence of different RBFs, namely the distance function,

thin plate spline and scaled linear function in non-linear poisson type problems, showing

that the smoothness of RBF will have an impact on the rate of convergence. With the help

15

of DRM associated with RBF, time dependent problems also can be reduced to solving

Helmholtz equations via Laplace transform [77] or finite difference schemes [78]. For

example, this method was applied to simulate the transient thermal behaviour of skin

tissues by Cao et al. [79]. Successful BIE applications coupling with RBFs for 3D linear

elasticity problems include the globally supported Gaussian [80], power splines and thin

plate splines [17].

In the literature above, the globally supported RBFs (GSRBFs), i.e. Polyharmonic splines,

TPS and MQ as shown in Table 1.1, are widely used for the approximation of

inhomogeneous terms. They are defined over the whole domain with global support.

Although the global support functions are infinitely smooth, they are, however, only

conditionally positive definite. This means that additional polynomials are always

required to supplement the globally supported basis functions to ensure invertibility of

the interpolation matrix. In addition, when RBFs are globally supported, the resulting

matrix for interpolation is dense and may be highly ill-conditioned for large scale

problems, or problems with higher dimensions, or large number of interpolation points,

or high-order RBFs. To resolve this issue, domain decomposition [81] and localisation

methods [82] were proposed to work around the computation difficulty.

By contrast to the globally supported RBFs, the compactly supported radial basis

functions (CSRBFs) defined with local support domains are unconditionally positive

definite, making them truly multivariate without the need of supplementation by the

dimensionally dependent polynomials as is the case for the globally supported RBF.

Moreover, RBFs with local support such as the Wendland’s CSRBF are capable of

producing sparse interpolation matrices and improving matrix conditioning while

maintaining competitive accuracy due to the fact that they feature positive definite and

banded interpolation matrices [83-85]. As the result, CSRBF has become a natural choice

16

for solving higher dimensional problems [86, 87]. For example, Chen and Golberg [88]

gave a general review of the recent developments of the DRM-CSRBF method. Chen and

Brebbia [89] provided the detailed procedure for the CSRBF solution kernel derivation

for Laplacian operators. Golberg and others [75] employed MFS-CSRBF to solve

Helmholtz type equations in three dimensions, showing that the CSRBF with a high

degree of sparseness is competitive in providing accurate results with less computation

time and resources. The efficiency of using CSRBF approximation for simulating soft

tissue deformation is demonstrated by Wachowiak and others [90]. Besides, CSRBF was

also employed to handle multivariate surface reconstruction, demonstrating that the

method is capable of solving large scale interpolation problems [91]. Apart from

Wendlend’s CSRBF, Wu [92] had also proposed different types of CSRBF, starting with

very smooth, positive definite functions in low dimensions and gaining less smooth

functions in higher dimensional spaces, while Wendlend’s CSRBF takes the opposite

approach. Both methods were later generalised by Buhmann [93]. In order to optimise

the degree of support, multilevel schemes for CSRBFs was suggested by Schaback [91]

and illustrated by Floater and Iske [94] and later by Fasshauer [95] and Chen et al. [96]

on elliptic problems. Overall, the CSRBFs have significant merits over the GSRBFs,

especially in terms of stability. For illustration, Figs. 1.4 and 1.5 respectively display the

TPS GSRBF and the scaled Wendland CSRBFs for comparison.

17

Table 1. 1: Globally supported RBFs

Linear 𝑟

Cubic 𝑟3

Polyharmonic splines 𝑟𝑛

Thin-plate splines 𝑟𝑛𝑙𝑛(𝑟)

Multiquadrics (𝑟2 + 𝑐2)𝑛2

(a) 𝑟2ln(𝑟)

(b) 𝑟4ln(𝑟)

Fig. 1. 4: Thin plate splines with different orders

18

Fig. 1. 5: Profile of normalised Wendland CSRBFs 2

1 r

1.4 Method of fundamental solutions

In addition to the fundamental solution-dependent BEM, the meshless method of

fundamental solutions (MFS) is another popular fundamental solution-dependent

approach. Unlike the BEM which is based on boundary integral equations, the MFS uses

boundary collocation approaches. Its implementation is very straightforward and simple,

without excessive mathematical derivations [34]. The MFS was originally formulated by

Kupradze and Aleksidze [97], and subsequently was used for engineering problems with

simple geometrical domains. For example, Fairweather and Karageorghis applied the

MFS for general two-dimensional elliptic boundary value problems including potential

19

problems, elastic problems and acoustics problems [33]. Golberg and Chen discussed the

application of the MFS for potential, Helmholtz and diffusion problems [98].

Karageorghis solved the eigenvalues of the Helmholtz equation using the MFS [99]. Most

of applications of the MFS are limited to two-dimensional problems with simple shapes,

because of the relatively simple distribution of collocations. The use of the MFS for three-

dimensional elasticity problems without body forces was first demonstrated by Brebbia

and Dominguez [27], and then was further illustrated by Poullikkas et al. [100].

In the MFS procedure, the linear combination of Green’s functions, also known as

fundamental solutions, is used to approximate the field variable of interest, e.g.

temperature, displacements and stresses, to ensure the approximate field to analytically

satisfy the governing differential equations of problems. For example, for elastic

problems without the body forces, the approximate displacement and traction solutions

are expressed in terms of fundamental solutions

*

1

( ) ( , )N

i j ij j

j

u c U

P P Qx x x (1.27)

*

1

( ) ( , ), N

i j ij j

j

t c T

P P Q Px x x x (1.28)

with the singularities 1 N

jQx placed outside the domain Ω of the problem. The locations

of the singularities can be preassigned along with the coefficients 1 N

j jc of the

fundamental solutions so that the approximate solutions satisfy the boundary conditions

by a least squares fit of the boundary data [101]. If the locations of the singularities are to

be determined, the resulting minimisation problem is nonlinear and should be solved

carefully [102].

20

In the MFS, the most significant feature is that singularities are avoided by the

employment of a fictitious boundary outside the problem domain. The main merits of the

MFS over the domain based numerical methods include:

(1) No domain nor boundary element discretisation.

(2) More rapid convergence of the MFS than the BEM for uncomplicated

geometry where stability issue is not a concern [103-105].

(3) Simple formulation and fast implementation.

(4) No singular or hypersingular integrals.

However, like the BEM and other fundamental solution based numerical methods [106-

111], the MFS is only applicable when a fundamental solution of the differential equation

in question is known. Moreover, the stability of the MFS is severely affected by the

number and the location of singularities. Besides, the MFS formulation is just valid for

homogeneous problems without inhomogeneous terms. If there are inhomogeneous terms

in the governing equations, the MFS should couple with other techniques, i.e. RBF

approximation or the dual reciprocal method (DRM). In this mixed approach, the whole

solution of problem is divided into the homogeneous part and the particular part. The

MFS is used to construct the homogeneous solution part by boundary collocations while

the RBF is used to construct the particular solution part by domain collocations. Due to

this feature, this mixed approach is called MFS-DRM [112-114] or MFS-RBF [74, 79,

115] and has become popular in recent years. However, due to the complexity of three-

dimensional elasticity, few studies are available, e.g. Tsai applied the MFS-DRM for

three-dimensional thermoelasticity [113].

21

1.5 Other meshless methods

Apart from the RBF and MFS, there exists other meshless schemes such as Smooth

Particle Hydrodynamics (SPH), Particle Strength Exchange (PSE), Moving Least Squares

(MLS), Modified Moving Least Squares (MMLS) and Reproducing Kernel Particle

Methods (RKPM). The SPH [116] is based on the interpolations of field quantities by

finite sets of smoothed particles. The PSE [117] approximates the differential operators

of the governing equations with carefully chosen integral kernels [118]. Furthermore, the

MLS [119] employs weighted least squares approximation using polynomials while the

MMLS [120] augments the error functional in MLS with additional terms to allow for

usage of higher order polynomial bases. The RKPM [121] is a generalisation of the MLS

for the same weight functions and the linear basis functions. Its approximation spans a

linear space and is similar to the concept of partition of unity. While the RBF and MFS

are truly meshless, some of the above meshless methods such as MLS, MMLS and RKPM

require integration of the Galerkin weak formulation and background meshing. These

meshless methods mainly differ in the choices of the shape functions, with closed support

for interpolating the trial solutions, but computation of their shape functions is generally

a time-consuming operation requiring matrix inversion at each interpolating node. When

comparing MLS type performance to RBF, one study found that the former scheme could

be more accurate [122]. However, it is not without drawbacks. Since the MLS type shape

functions do not possess the Kronecker delta property and their interpolated solutions do

not pass through the nodal points, imposing the essential boundary conditions is not a

straight forward process. In the literature, the Lagrange multiplier or penalty method [123]

can be employed to resolve the issue at the cost of more computational resources. For

example, the Lagrange multiplier technique could yield a non-banded and non-positive

definite matrix, while the latter method could become ill-conditioned and sensitive to a

22

large penalty parameter as a trade-off for accuracy. Several of these meshless methods

are described in a review paper by Nguyen et al [124], who also present a detailed

computer implementation of the popular meshless methods.

1.6 Summarisation of literature review and Research Gaps

To summarise, standard FEM requires volumetric meshing and evaluation of the domain

integrals, which are time-consuming and complicated for multi-dimensional problems.

By contrast, the BEM can reduce modelling dimensionality by one, while the MFS

requires no domain nor boundary element discretisation. Since no further approximation

is imposed on the solutions inside the domain, the solutions of the BEM and the MFS

would have high accuracy at interior points. Although the fundamental solution based

numerical methods have merits, several research gaps would need to be bridged. First,

the BEM is inconvenient for problems with body forces due to the need for the domain

integral. For the case of MFS, only homogeneous problems can be solved. A more

efficient way of handling the inhomogeneous terms using the DRM is preferable to

simplify the underlying problems. Furthermore, when employing the DRM to

approximate the particular solutions, choosing sparse interpolation matrices with

unconditional invertibilities for the approximations would be more computationally

efficient. Finally, the computation of hypersingular or near-singular integrals in the BEM

is complicated and would affect the solution accuracy. A direct evaluation of these

integrals would enhance the feasibility of this fundamental solution based method.

23

1.7 Our work

This research project aims at acquiring better three-dimensional numerical modelling for

the fundamental solution based numerical methods, including the meshless method of

fundamental solutions (MFS) and the boundary element method (BEM). This thesis will

revolve around two significant issues in these boundary-based methods, namely the

efficient treatments of inhomogeneous terms and the direct evaluation of hypersingular

integrals. The new developments include:

Coupling CSRBFs and the meshless MFS for three-dimensional elasticity

problems involving inhomogeneous generalised body force terms. This meshless

scheme is computationally efficient as it can produce sparse interpolation matrices

with unconditional invertibilities.

Coupling CSRBFs and the BEM for three-dimensional elasticity problems

involving inhomogeneous terms. This boundary-type meshed method simplifies

domain discretisation in the BEM with sparse interpolation matrices, resulting in

a more robust and efficient numerical scheme.

Accurately and efficiently computing hypersingular linear integrals for two-

dimensional problems. The proposed general algorithm can directly evaluate

improper integrals arising from the fundamental solution based numerical

methods.

Accurately and efficiently computing hypersingular surface integrals for three-

dimensional problems. The developed algorithm allows direct integration of the

hypersingular third order kernels for the boundary stresses in the BEM as

presented in Eq. (1.26).

Developing an iteratively coupled boundary based numerical method for solving

poroelasticity problems. The proposed method makes use of the readily available

24

solutions of the uncoupled phases. This has the advantages of reducing the

computational cost of developing and implementing a fully coupled model.

1.8 Structure of the thesis

The thesis is organised as follows. After an introductory chapter, Chapter 2 puts emphasis

on the MFS solutions of three-dimensional linearly elastic problems with body forces. It

presents the coupling of the classic boundary-type meshless MFS algorithm with the

Wendland’s CSRBFs for calculating the displacement and stress distributions in the 3D

solids with body forces. Next, Chapter 3 establishes an algorithm combining the

boundary-type integral BEM approach and the Wendland’s CSRBFs for solving the

three-dimensional linearly elastic problems with body forces. Chapter 4 and 5 address the

key issues of implementing the two- or three-dimensional boundary integral formulations,

i.e. the calculation of hypersingular linear and surface integrals. Finally, Chapter 6 carries

out the numerical simulation of porous media using MFS-CSRBF, and develops an

iterative strategy for such coupling problems.

25

Chapter 2 Method of fundamental solutions for three-

dimensional elasticity with body forces by coupling

compactly supported radial basis functions

2.1 Introduction

As a boundary-type meshless numerical method, the method of fundamental solutions

(MFS) represents the desired solution as a series of fundamental solutions, with sources

located outside the computational domain. This approach can reduce the computing

dimensionalities compared to the domain-type meshless numerical methods, but its use

is unfortunately limited to homogeneous solutions of partial differential equations [33,

125]. This disadvantage also holds for other fundamental solution based methods [125-

130]. Specifically, when exact particular solutions can be derived for known

inhomogeneous terms, the MFS can be directly applied for inhomogeneous problems. For

example, Fam and Rashed applied the MFS with analytical particular solutions for three-

dimensional structures with body force [58]. Similarly, the same authors applied the MFS

with RBF interpolated particular solutions for solving two dimensional piezoelectricity

problem [131]. However, most cases require approximated treatment of inhomogenous

terms by the dual reciprocity method (DRM) with suitable radial basis functions (RBFs).

Currently, the MFS-DRM with globally supported RBFs has been utilised to solve

thermoelasticity problems with general body forces [113, 132]. However, the globally

supported RBFs may cause ill-conditioning of the resulting matrix, especially for large-

scale three-dimensional problems.

26

In this chapter, a mixed MFS-DRM with locally supported radial basis functions, i.e.

CSRBF, is developed for three-dimensional (3D) linear elasticity in the presence of

general body forces. In our approach, we consider using the MFS for the approximation

of homogeneous terms and the dual reciprocity method, which is also named as the

method of particular solutions, fulfilled with CSRBF instead of the conventional globally

supported basis functions for the approximation of inhomogeneous terms. During the

computation, we can freely control the sparseness of the interpolation matrix by varying

the support radius without trading off too much of the accuracy. Using Galerkin vectors

in the linear elastic theory, the particular solution kernels with respect to the CSRBF

approximation are firstly derived and then the displacement and stress particular solutions

are obtained to modify the boundary conditions. Subsequently, the homogeneous

solutions are evaluated by the MFS using the modified boundary conditions. Finally,

several examples are presented to demonstrate the accuracy and efficiency of the present

method.

This chapter is organised as follows. Section 2.2 describes the basics of three-dimensional

elasticity. Section 2.3 presents the derivation of the particular solution kernels associated

with the Wendland’s CSRBF, and in Section 2.4, the method of fundamental solutions is

presented for the homogeneous terms. Several examples are considered in Section 2.5

and a further discussion on the sparseness of the CSRBF is given in Section 2.6. Finally,

some concluding remarks on the present method are presented in Section 2.7.

2.2 Problem description

Consider a 3D isotropic linear elastic body (see Fig. 2.1) with inhomogeneous terms in

the domain Ω. The governing equations at point x are [133]

27

, 0ij j ib x x (2.1)

2ij kk ij ijG x x x (2.2)

, ,

1

2ij i j j iu u x x x (2.3)

where 𝜎𝑖𝑗 is the stress tensor, 휀𝑖𝑗 the strain tensor, 𝑢𝑖 the displacement vector, 𝑏𝑖 the

known body force vector, 𝜆 and 𝐺 the Lame constants, and 𝛿 the Kronecker delta.

Combining the above equations yields the following Navier’s equations in terms of

displacement components

, , 01 2

i jj j ij i

GGu u b

v

x x x (2.4)

where 𝑣 is the Poisson’s ratio. Later in the numerical examples, Young modulus 𝐸 and 𝑣

will be employed, and the shear modulus 𝐺 can be computed using the conversion

formula

2(1 )

EG

(2.5)

For a well-posed boundary value problem, the suitable boundary conditions should be

applied on the boundary of the computing domain. Here, the related boundary conditions

are given as

Displacement boundary condition

i iu ux x , Γux (2.6)

Traction boundary condition

i ij j it n t x x x x , Γtx (2.7)

where 𝑡𝑖 is the traction field,𝑖 and 𝑡 the prescribed displacement and traction, and Γ =

Γ𝑢 ∪ Γ𝑡. From the elastic theory, the traction component can be expressed in terms of

stress components

28

i ij jt nx x x (2.8)

where 𝑛𝑖 the unit vector outward normal to the boundary .

Fig. 2. 1: Schematic representation of a 2D section of possible computational domain

2.3 Method of particular solutions

Using the method of particular solutions, the full solution variables 𝑢𝑖 can be expressed

as the summation of particular solutions 𝑢𝑖𝑝 and homogeneous solutions 𝑢𝑖

ℎ [134-136],

that is

p h

i i iu u u x x x , Ωx (2.9)

where 𝑢𝑖𝑝

should satisfy the inhomogeneous equations (2.4) and 𝑢𝑖ℎ satisfies the

homogeneous equations with modified boundary conditions:

, , 01 2

h h

i jj j ij

GGu u

v

x x , Ωx (2.10)

h p

i i iu u u x x x , Γux (2.11)

29

h p

i i it t t x x x , Γtx (2.12)

In order to determine the particular solution, it is convenient to express the particular

solutions of displacement 𝑢𝑖𝑝 in terms of Galerkin vector 𝑔𝑖 as [133]

, ,

1

2 1

p

i i kk k iku g gv

x x x (2.13)

Upon substituting Eq. (2.13) into Eq. (2.4) yields the following bi-harmonic equation

,

i

i jjkk

bg

G

xx (2.14)

By means of derivation of displacement variables, the corresponding stress particular

solutions in terms of Galerkin vector is

, , , ,11

ij k mmk ij k ijk i jkk j ikk

Gvg g v g g

v

x x x x x (2.15)

2.3.1 Dual reciprocity method

Sometimes, inhomogeneous terms of Eq. (2.14) could be a well described function such

as gravitational load, for which special particular solution can be found analytically. In

many other cases, finding such analytical solution is not a trivial task. The dual reciprocity

method [60] aims to efficiently approximate the particular solution by finding its solution

kernels while prescribing the inhomogeneous terms such as body forces with a series of

linearly independent basis functions so that any known or unknown body forces terms

can be reconstructed using finite set of discrete data

1

Nn

i l li n

n

b

x x , x (2.16)

where 𝜑𝑛 is the chosen series of functions to approximate body forces from the

inhomogeneity terms of Eq. (2.14), 𝑁 the number of interpolation points in the domain,

𝛼𝑙𝑛 the interpolation coefficients to be determined. The use of the Kronecker delta δ is to

30

separate the basis functions for approximating the body forces in each direction

independently, i.e.

1

1

1

2

1

3

2

1

1 12

2

1 22

3

1 3

1

2

3

0 0 0 0

0 0 0 0

0 0 0 0

N

N

N

N

N

N

b

b

b

x x x

x x x

x x x

(2.17)

Similarly, the Galerkin vector 𝑔𝑖 and the particular solution 𝑢𝑖𝑝 and 𝜎𝑖𝑗

𝑝 can be expressed

as

1

Nn

i l li n

n

g

x x (2.18)

1

Np n n

i l li

n

u

x x (2.19)

1

Np n n

ij l lij

n

S

x x (2.20)

where 𝜙𝑛 is the respected Galerkin vector solution kernels, 𝜓𝑙𝑖𝑛 the displacement

particular solution kernels,𝑆𝑙𝑖𝑗𝑛 the stress particular solution kernels.

By linearity, it suffices to analytically determine 𝜙𝑛 by substituting Eqs. (2.17) and (2.18)

into Eq. (2.14)

,

n

n jjkkG

xx (2.21)

It’s clear that by enforcing Eq. (2.14) to satisfy the known inhomogeneity terms in Ω, we

can obtain 𝑁 linear equations to uniquely solve for the interpolation coefficients 𝛼𝑙𝑛.

31

2.3.2 Particular solution kernels with CSRBF

Generally [137], the function 𝜑 in Eq. (2.21) can be chosen as radial basis function such

that

j i i j ijr x x x (2.22)

where 𝐱𝑖 represents the collocation points and 𝐱𝑗 represents the reference points.

Herein, RBF is employed to approximate the inhomogeneous terms of Eq. (2.14). Since

RBF is expressed in terms of Euclidian distance, it usually works well in arbitrary

dimensional space and doesn’t increase computational cost. Furthermore, many attractive

properties of RBF such as good convergence power, positive definiteness and ease of

smoothness control are widely reported [71].

Then, from Eqs. (2.13), (2.18), (2.19) the displacement particular solution kernels can be

expressed as

1 2 , , 3

1

2 1li li li i lr r r

v

(2.23)

Similarly, the stress particular solution kernels can be expressed as

, , , 4 , , , 5 , , , 611

lij ij l li j lj i lj i li j ij l i j l

GS r r v v r r r r r r r r

v

(2.24)

where

32

1 , ,

2 ,

3 , ,

4 , , ,2

5 , ,2

6 , , ,2

2

1

1

2 2

1 1

3 3

rr r

r

rr r

rrr rr r

rr r

rrr rr r

r

r

r

r r

r r

r r

(2.25)

To analytically determine 𝜙, 𝜓𝑙𝑖 and 𝑆𝑙𝑖𝑗, an explicit function needs to be chosen first for

𝜑. For the Wendland’s CSRBF in three-dimensional cases [84], 𝜑 is defined as

22

0

4

2

6 2

1 01 for

, smoothness

0 ,

1 4 1 for smoothness

1 35 18 3

rrr

r C

r

r rr C

r r rr

4

8 3 2

6

for smoothness

1 35 25 8 1 for smoothn s

es

C

r r r rr C

(2.26)

where the subscript + denotes that the bracket function will be forced to be zero when the

bracketed value is less than zero. 𝛼 is a cut off parameter for varying the support radius

of interpolation matrix 𝜑(𝑟) as illustrated in Fig. 2.2.

The sparseness of the CSRBF interpolation matrix can be interpreted as the cumulative

frequency of 𝑟, which is defined as

N N

ij ij

i j

sparseness f r r (2.27)

with

33

2

N Nij

i j

r rf r

N

(2.28)

where 𝑓 is the frequency function of 𝑟𝑖𝑗 and [ ] denotes the use of Iverson Bracket. For

case 𝛼 = 𝑚𝑎𝑥(𝑟), sparseness of the interpolation matrix is equal to 100%. For case 𝛼 =

0, the sparseness is equal to 0%. In practice α can be chosen according to the sparseness

requirement.

Fig. 2. 2: Cut off parameter 𝛼 for various support radii

Since the radial part of the bi-harmonic operator in Eq. (2.21) can be written as

4 34

4 3

4

r r r

(2.29)

the Galerkin vector solution kernels 𝜙 can be analytically determined for points located

within the compact support radius by solving the ordinary differential equation

4 3

4 3

4 for 0

r r rr

r r r G

(2.30)

For points located outside the compact support radius, 𝜙 satisfies the following

homogeneous equation

34

4 3

4 3

40 for

r rr

r r r

(2.31)

with solution

2 41 2 3r

Cr C r C r C

r (2.32)

The four constants 𝐶1, 𝐶2, 𝐶3, 𝐶4 in Eq. (2.32) are to be chosen so that 𝜙 satisfies the

continuity conditions at the compact support radius, that is

0

0

0

0

' '

'' ''

''' '''

r r

r r

r r

r r

(2.33)

Particularly, the corresponding 𝜙 for the first three Wendland’s CSRBF defined in Eqs.

(2.26) are expressed in Eqs., (2.34)-(2.39) for points located within the compact support

radius (0 ≪ 𝑟 ≪ 𝛼) and for points located outside the compact support radius (𝑟 > 𝛼).

For 𝐶0 smoothness

4 5 6

0 2120 180 840r

r r r

G G G

(2.34)

5 4 3 2 2

630 120 60 72r

r r

Gr G G G

(2.35)

For 𝐶2 smoothness

4 6 7 8 9

0 2 3 4 5

5

120 84 84 1008 1260r

r r r r r

G G G G G

(2.36)

5 4 3 2 25

1260 1008 84 84r

r r

Gr G G G

(2.37)

35

For 𝐶4 smoothness

4 6 8 9 10 11 12

0 2 4 5 6 7 8

5 4 7 8 7

40 30 72 45 132 495 3432r

r r r r r r r

G G G G G G G

(2.38)

5 4 3 2 28 7 4

6435 792 165 36r

r r

Gr G G G

(2.39)

It should be noted that the 𝜙 across the support radius are at least thrice differentiable for

the minimal smoothness of the CSRBF as evidenced by Eq. (2.32) and Eq. (2.34). By

substituting Eqs. (2.34)-(2.39) into Eqs. (2.23)-(2.25), the displacement and stress

particular solution kernels can be found:

For 𝐶0 smoothness

2 2 2 22

, ,, ,

,0 2

2

, ,

2

126 117 3684 378 420

2520 1 2520 1

105 385 420

2520 1

li li i li l li li

li r

i l li li

r r v r r r rr r r vr

G v G v

r r r r r rv

G v

(2.40)

2 3 32, ,

, 3

2 2

, ,

3

2 6175 210

2520 1 2520 1

63 21 84

2520 1

li i lli li

li r

li i l li

r rvr

G v Gr v

r r r v

Gr v

(2.41)

2 2

, , ,

,0 2

2 2 2 2 2 2

, , , , , ,

2

2

, , , ,

10 15 6

30

28 112 28 72 12 12

420 1

24 175 35

li j ij l lj i

lij r

li j ij l lj i ij l li j lj i

i j l ij l li

r r r r r rS r

r r r r r r r r r r

v

r r r r r r r r

, , , , ,

2

35 35

420 1

j lj i i j lr r r r r r r

v

(2.42)

36

3

, , , , , , , , ,

, 2

5

, , , , , ,

4

3 2 2 2

60 1

5

210 1

li j ij l lj i i j l ij l li j lj i

lij r

ij l li j lj i i j l

r r r r r r r v r v r vS r

r v

r r r r r r

r v

(2.43)

For 𝐶2 smoothness

2 5 5 52

, ,, ,

,0 5

2 4 4 4 3 2 2 2 2

, , , ,

5 5

180 171 6384 378 420

2520 1 2520 1

900 850 300 1260 1170 360 +

2520 1 2520 1

li li i li l li li

li r

li li i l li li i l

r r v r r r rr r r vr

G v G v

r r v r r r r r r v r r r r

G v G v

2 2 3 3 3

, ,

5

1680 1575 525

2520 1

li li i lr r v r r r r

G v

(2.44)

2 3 3 2 22, , , ,

, 3 3

3 45 15 60150 180

2520 1 2520 1 2520 1

li i l li i l lili li

li r

r r r r r vvr

G v Gr v Gr v

(2.45)

5 3 2 2 3 4 5

, , ,

,0 5

5 5 5 5 5 5 4

, , , , , , ,

5

4 4

, , , , ,

14 84 140 90 21

42

28 112 28 189 21 21 800

420 1

100 100 105

li j ij l lj i

lij r

li j ij l lj i ij l li j lj i ij l

li j lj i i j l

r r r r r r r rS r

r r r r r r r r r r r r

v

r r r r r r r r

5 2 3 2 3 2 3

, , ,

5

3 2 3 2 3 2 2 3 3 2 4

, , , , , , , , , , , ,

5

1225 175 175

420 1

720 120 120 525 240 400

420 1

ij l li j lj i

ij l li j lj i i j l i j l i j l

r r r r r r r

v

r r r r r r r r r r r r r r r r r r r

v

(2.46)

3

, , , , , , , , ,

, 2

5

, , , , , ,

4

3 2 2 2

84 1

5

420 1

li j ij l lj i i j l ij l li j lj i

lij r

ij l li j lj i i j l

r r r r r r r v r v r vS r

r v

r r r r r r

r v

(2.47)

37

For 𝐶4 smoothness

88 82 2, , , ,

,0 8

7 27 7 2 22 6 2, , , ,

8 8

3

359 7 175

1 10 20 2 1 22 572 286

4 232 92 7 13

1 15 45 5 1 5 10 5

i l i lli li li lili r

i l i lli li li li

r r r r rv r v rr rr

G v G v

r r r r r rr v r r v rr r

G v G v

5 45 42 4 2, , , ,5 4

8 8

66 62 2, ,

8

14 538 858 5

1 5 5 1 18 3

7035 245

1 6 44 33

i l i lli lili li

i lli li

r r r r r rr rr rr v r v

G v G v

r r rr v rr

G v

(2.48)

2 3 32, ,

, 3

2 2

, ,

3

16 483575 4290

25740 1 25740 1

936 312 1248

25740 1

li i lli li

li r

li i l li

r rvr

G v Gr v

r r r v

Gr v

(2.49)

3

, , , , , , , , ,

,0 2

, , , , , ,

5

, , ,

4 6 6 2 7 7 7

5 1

4 4 5 5 5

5 1

10 8 8

ij l li j lj i i j l ij l li j lj i

lij r

ij l li j lj i ij l li j lj i

ij l li j lj

r r r r r r r r v r v r vS r

v

r r r r r v r v r v

v

r r r r

, , , , , ,

4

6

, , , , , , , , ,

5

7

, , , , , ,

4 9 9 9

3 1

28 9 9 5 10 10 10

5 1

140 10 10 6 1

i i j l ij l li j lj i

ij l li j lj i i j l ij l li j lj i

ij l li j lj i i j l

r r r r v r v r v

v

r r r r r r r r v r v r v

v

r r r r r r r

, , ,

6

8

, , , , , , , , ,

7

9

, , , , , , ,

1 11 11

33 1

8 11 11 7 12 12 12

5 1

35 12 12 8 13 1

ij l li j lj i

ij l li j lj i i j l ij l li j lj i

ij l li j lj i i j l ij l

r v r v r v

v

r r r r r r r r v r v r v

v

r r r r r r r r v

, ,

8

3 13

143 1

li j lj ir v r v

v

(2.50)

3

, , , , , , , , ,

, 2

5

, , , , , ,

4

4 3 2 2 2

165 1

8 5

2145 1

li j ij l lj i i j l ij l li j lj i

lij r

ij l li j lj i i j l

r r r r r r r v r v r vS r

r v

r r r r r r

r v

(2.51)

38

2.4 Method of fundamental solutions for homogeneous

solutions

In the MFS, the homogeneous displacement and stress solutions satisfying the

homogeneous system consisting of Eqs. (2.10)-(2.12) can be approximated by a series of

fundamental solutions 𝐺𝑙𝑖𝑚, 𝐻𝑙𝑖𝑗

𝑚 with coefficients 𝛽𝑙𝑚

1

Mh m m

i l li

m

u G

x x (2.52)

1

Mh m m

ij l lij

m

H

x x (2.53)

where 𝑀 is number of source points placed outside the domain.

Similar to that in Eq. (2.22), the fundamental solution 𝐺𝑙𝑖 makes use of Euclidian distance

between two points

m

li n li n m li nmG G G r x x x (2.54)

where nx represents the collocation points on Γ and mx represents the source points

placed outside Ω.

As is well known in the literature, there is a trade-off between numerical accuracy and

stability that the MFS equations could become highly ill-conditioned with increased

radial distances in the fundamental solutions. Usually, the source points can be put on a

virtual boundary, which is geometrically similar to the physical boundary of the solution

domain. In particular, the source points location can be systematically generated by the

following equation

m n n cc x x x x (2.55)

39

where 𝐱𝑐 is the centre of Ω and 𝑐 is a dimensionless parameter to be specified for placing

the source points outside Ω . The magnitude of 𝑐 is a dominant factor of numerical

accuracy due to its impact on the radial distances in the fundamental solutions. In our

practical computation, the parameter 𝑐 can be feasibly chosen according to the number of

source points to avoid the ill-conditioning of the computing matrix.

For 3D isotropic linear elastic problems, the fundamental solutions in Eqs. (2.52) and

(2.53) is given as

, ,3 4

16 1

li l im

li n

v r rG

G v r

x (2.56)

, , , , , ,2

11 2 3

8 1

m

lij n ij l li j lj i l i jH v r r r r r rv r

x (2.57)

From the basic definition of fundamental solutions, the homogeneous solutions in Eqs.

(2.52) and (2.53) analytically satisfy the homogeneous governing equation (2.10). Thus,

only the modified boundary conditions (2.11)-(2.12) need to be considered to determine

the unknown coefficients 𝛽𝑙𝑚. For example, by making number of collocation points on

the physical boundary Γ equal to the number of source points, we can obtain 𝑀 linear

equations to uniquely solve for the coefficients 𝛽𝑙𝑚, i.e.

1

1

, 1 ΓM

m m p

l li k i k i k k u

m

G u u k N

x x x x (2.58)

2

1

, 1 ΓM

m m p

l lij q j q i q i q q t

m

H n t t q N

x x x x x (2.59)

where 𝑁1 and 𝑁2 are the numbers of nodes on the displacement boundary Γu and the

traction boundary Γt, respectively. Meanwhile, 𝑀 = 𝑁1 + 𝑁2 . Finally 𝛽𝑙𝑚 can be

determined by solving this square system of linear equations.

A schematic figure illustrating the MFS-CSRBF numerical procedures is shown in Fig.

2.3.

40

Fig. 2. 3: Schematic figure illustrating the MFS-CSRBF numerical procedures

2.5 Numerical examples and discussions

To demonstrate the accuracy and efficiency of the derived formulation, three benchmark

examples, which are solved by the proposed meshless collocation method, are considered

in this section. The examples include: 1) a prismatic bar subjected to gravitational load,

2) a cantilever beam under gravitational load, and 3) a thick cylinder under centrifugal

load. For simplification, only Wendland’s CSRBF with smoothness 𝐶0 is considered

here. Simulation results obtained from the proposed method and the conventional FEM

method are compared against the analytical solutions. We also compute the mean absolute

percentage error (MAPE) as an effective description for quantifying the average

performance accuracy of the present method

1

1MAPE 1 100%

tnsimulation i

it analytical i

f

n f

(2.60)

41

where 𝑓𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 and 𝑓𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 are the analytical and simulation values evaluated at test

point 𝑖. 𝑛𝑡 is the total number of the test points.

2.5.1 Prismatic bar subjected to gravitational load

In the first example, we consider a straight prismatic bar subjected to gravitational load,

as shown in Fig. 2.4. The dimensions of the bar are 1m×1m×2m and it is fixed at the

top. Assuming the bar being loaded along the z-direction by its gravitational load, the

corresponding body forces can be expressed as

0, 0, x y zb b b g (2.61)

where 𝜌 is density and 𝑔 is gravity.

The material parameters used in the simulation are: 𝐸 = 4x107Pa, 𝑣 = 0.25, 𝜌 =

2000kg ∙ m−3, 𝑔 = 10m ∙ s−2 . A total number of 490 collocation points are equally

spaced on Γ and an additional of 300 points are arranged inside Ω for interpolation using

the CSRBF. The number of source points is equal to the number of collocation points on

Γ, in which the geometric parameter 𝑐 of Eq. (2.55) is chosen as 3.0 for the generation of

source points, as shown in Fig. 2.5.

42

Fig. 2. 4: Prismatic bar under gravitational load

Fig. 2. 5: Field points, source points and interpolation points distributing for the

prismatic bar

43

Test points are chosen along the centreline of the prismatic bar. The corresponding

displacement and stress results are compared to the analytical solutions [133] and the

FEM solutions, which are evaluated by the commercial software ABAQUS. Numerical

results in Tables 2.1 and 2.2 show the variations of displacement and stress in terms of

the sparseness of CSRBF. It is found that by increasing the sparseness from 20% to 100%,

the MAPE of the present method reduces from 2.87% to 1.55% for the displacement

component 𝑢𝑧 and from 2.23% to 0.67% for the stress component 𝜎𝑧𝑧. Meanwhile, the

MAPE of the Abaqus records at 3.73% and 1.98% respectively. Fig. 2.6 displays the

distribution of displacement and stress components for the sake of clearness. Overall, the

present method gives good accuracy and good stability of numerical results for different

sparseness values as demonstrated in Table 2.1, Table 2.2 and Fig. 2.6. It is noted that in

the ABAQUS, a total number of 9537 elements of type 20-node quadratic brick are

employed for the prismatic bar. Besides, the isoline plot of 𝑢𝑧 is provided in Fig. 2.7,

from which a similar colour distribution of the displacement is observed for both the

ABAQUS and the present method.

44

Table 2. 1: Displacement results for the prismatic bar

z (m)

−𝑢𝑧(10−3m)

Present method

ABAQUS Analytical

solutions Sparseness

=20%

Sparseness

=60%

Sparseness

=100%

-0.25 0.2424 0.2404 0.2397 0.2055 0.2345

-0.50 0.4526 0.4487 0.4471 0.4141 0.4375

-0.75 0.6291 0.6237 0.6211 0.5944 0.6100

-1.00 0.7729 0.7663 0.7627 0.7392 0.7500

-1.25 0.8845 0.8770 0.8726 0.8500 0.8600

-1.50 0.9642 0.9561 0.9511 0.9285 0.9450

-1.75 1.0119 1.0036 0.9982 0.9754 0.9900

MAPE (%) 2.871 2.003 1.552 3.728

Table 2. 2: Stress results for the prismatic bar numerical simulation

z (m)

𝜎𝑧𝑧(kPa)

Present method

ABAQUS Analytical

solutions Sparseness

=20%

Sparseness

=60%

Sparseness

=100%

-0.25 36.34 36.02 35.88 36.30 35.6

-0.50 30.87 30.60 30.44 31.81 30.0

-0.75 25.57 25.35 25.19 25.91 25.0

-1.00 20.42 20.24 20.10 20.31 20.0

-1.25 15.32 15.18 15.06 15.08 15.0

-1.50 10.20 10.13 10.04 10.01 10.0

-1.75 5.11 5.07 5.02 5.00 5.0

MAPE (%) 2.232 1.390 0.673 1.975

45

Fig. 2. 6: Displacements and normal stresses along the centreline of the prismatic bar

subjected to gravitational load

Fig. 2. 7: Contour plots of the vertical displacement component by ABAQUS (left) and

the present method (right)

46

2.5.2 Cantilever beam under gravitational load

Next we consider the bending problem of a cantilever beam under gravitational load. The

cantilever beam fixed at 𝑦 = 0 is assumed to have dimensions 1m×2m×1m as shown in

Fig. 2.8. If the gravitational force is along the z-axial direction, the corresponding body

forces are the same as those described in Eq. (2.61).

For the sake of convenience, the material parameters used in the simulation are taken to

be the same as those in the first example. A total number of 250 collocation points are

equally spaced on the physical boundary and additional 72 interior points are uniformly

distributed in the domain for the CSRBF interpolation. However, due to the shear locking,

the solving matrix in the MFS could be highly ill-conditioned. To counter this, a number

of 1000 source points are generated with geometric parameter 𝑐 = 1.0. Fig. 2.9 displays

the geometrical configuration of the source points, collocations and interior interpolation

points used in the computation.

Fig. 2. 8: Bending of cantilever beam under gravitational load

47

Fig. 2. 9: Field points, source points and interpolation points distributing for the

cantilever

To investigate bending shape of the beam, the numerical results of deflection along the

y-axis from the present method are compared to those from ABAQUS in Table 2.3. It is

found that the numerical results obtained from the various degrees of sparseness do not

deviate more than 1.7% from the full sparseness. This implies that the computational

accuracy of the present method is not sensitive for low sparseness. The numerical

solutions seem to converge when the degree of sparseness increases and the discrepancy

between the present method at full sparseness and ABAQUS is only 0.32%. For better

illustration, the deflection results from the present method are plotted in Fig. 2.10 and

well agreement between the present method and ABAQUS is demonstrated. In ABAQUS,

a total number of 9537 elements of type 20-node quadratic brick elements are employed.

Fig. 2.11 illustrates the isoline maps of the beam deflection from the conventional FEM

48

implemented by ABAQUS and the present method, and similar distribution can be

observed.

Table 2. 3: Displacement results for the cantilever numerical simulation

y (m)

𝑢𝑧(10−3m)

Present method

ABAQUS Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

0.25 -0.953 -0.949 -0.940 -0.939

0.50 -2.469 -2.459 -2.436 -2.429

0.75 -4.314 -4.294 -4.253 -4.249

1.00 -6.347 -6.317 -6.254 -6.263

1.25 -8.468 -8.425 -8.340 -8.373

1.50 -10.605 -10.548 -10.440 -10.500

1.75 -12.704 -12.633 -12.501 -12.586

49

Fig. 2. 10: Displacements along the centre axis of the cantilever subjected to

gravitational load

Fig. 2. 11: Contour plots of the displacement results from ABAQUS (left) and the

present method (right)

50

2.5.3 Thick cylinder under centrifugal load

In the third example, a cylinder with 10m internal radius, 10m thickness and 20m height

is assumed to be subjected to centrifugal load. Due to the rotation, this cylinder is

subjected to apparent generalised body force. If the cylinder is assumed to rotate about its

z-axis as shown in Fig. 2.12, the generalised body forces in terms of spatial coordinates

can be written as

2 2, , 0x y zb w x b w y b (2.62)

where𝑤 is the angular velocity. In this example, 𝑤 = 10 is chosen.

The problem is solved with dimensionless material parameters 𝐸 = 2.1x105, 𝑣 = 0.3,

𝜌 = 1.According to the symmetry of the model, only one quarter of the cylinder domain

needs to be considered for establishing the computing model. Proper symmetric

displacement constraints are then applied on the symmetric planes (see Fig. 2.12). In the

quarter cylinder model, a total number of 430 collocation points are equally spaced on Γ

and an additional of 216 points are arranged in Ω for the CSRBF interpolation. For

convenience, the number of source points is chosen to be equal to the number of

collocation points on Γ, and their configuration is shown in Fig. 2.13 by setting the

geometric parameter 𝑐 = 1.0.

51

Fig. 2. 12: Thick cylinder under centrifugal load

Fig. 2. 13: Field points, source points and interpolation points distributing for the thick

cylinder

52

For the rotating cylinder, the displacement and the stress fields are more complicated than

those in the straight prismatic bar and the cantilever beam as discussed above. The results

of radial and hoop stresses and radial displacement at specified locations are tabulated

respectively from the present method (see Table 2.4-Table 2.6). These results are then

compared to ABAQUS with 10881 elements of type 20-node quadratic brick elements

and the analytical solutions [138]. As similar to the former two examples, the MAPE of

the present method reduces with increased sparseness. That is, from 17.69% to 7.07% for

the radial stress, 5.54% to 1.2% for the hoop stress and 7.69% to 0.09% for the radial

displacement while the MAPE of ABAQUS is recorded at 2.39% for the stress fields and

2.23% for the radial displacement. The present method seems to provide better accuracy

for the radial displacement as well as the hoop stress. For the radial stress, an optimal

MAPE of 1.59% is found at 80% sparseness of the present method comparing to the

2.39% obtained from the ABAQUS. Since the MFS is sensitive to the location of the

source points, this method tends to have stability issue for irregular or complex geometry.

As the result, a small change of the modified boundary conditions due to the particular

solutions interpolations of various sparseness would impact the numerical results slightly.

Overall, it is found that reasonable agreement with the analytical solutions is obtained as

shown in Fig. 2.14. The present method with small number of collocations can produce

better results than the conventional FEM solutions, which use more elements and nodes,

particularly for the case of large sparseness. To investigate the distribution of stresses and

displacement in the entire computing domain, some contour plots are given in Fig. 2.15-

Fig. 2.17, from which similar variation is observed between ABAQUS and the present

method.

53

Table 2. 4: 𝜎𝑟 results for the thick cylinder numerical simulation

r

𝜎𝑟(kPa)

Present method

ABAQUS Analytical

solutions Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 80%

Sparseness

= 100%

11.25 2.723 2.221 2.424 2.340 2.434 2.367

12.50 4.068 3.603 3.673 3.691 3.740 3.620

13.75 4.562 4.247 4.132 4.271 4.233 4.099

15.00 4.482 4.269 4.027 4.245 4.108 4.010

16.25 4.005 3.789 3.489 3.751 3.568 3.484

17.50 3.194 2.902 2.582 2.888 2.647 2.604

18.75 1.942 1.652 1.357 1.684 1.442 1.430

MAPE (%) 17.694 7.492 1.594 7.073 2.394

Table 2. 5: 𝜎𝑡 results for the thick cylinder numerical simulation

r

𝜎𝑡 (kPa)

Present method

ABAQUS Analytical

solutions Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

11.25 33.88 28.69 29.92 31.48 30.66

12.50 29.71 26.29 27.00 28.24 27.47

13.75 26.27 24.24 24.53 25.55 24.86

15.00 23.27 22.45 22.38 23.21 22.61

16.25 20.51 20.84 20.43 21.10 20.60

17.50 17.96 19.36 18.63 19.12 18.74

18.75 15.78 18.03 17.05 17.21 16.97

MAPE (%) 5.542 3.520 1.200 2.392

54

Table 2. 6: 𝑢𝑟 results for the thick cylinder numerical simulation

𝑢𝑟(m)

r

Present method

ABAQUS Analytical

solutions Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

11.25 1.721 1.562 1.601 1.641 1.604

12.50 1.684 1.531 1.568 1.606 1.571

13.75 1.659 1.509 1.546 1.582 1.547

15.00 1.643 1.491 1.529 1.563 1.529

16.25 1.630 1.474 1.514 1.546 1.513

17.50 1.619 1.455 1.498 1.529 1.497

18.75 1.607 1.432 1.478 1.510 1.477

MAPE (%) 7.692 2.649 0.092 2.225

Fig. 2. 14: Plot of radial displacements and radial stresses along the radial direction of

the cylinder subjected to centrifugal load (left) and plot of hoop stresses along the same

reference locations (right)

55

Fig. 2. 15: Contour plots of the 𝜎𝑟 results simulated in ABAQUS (left) and the present

method (right)

Fig. 2. 16: Contour plot of the 𝜎𝑡 results simulated in ABAQUS (left) and the present

method (right)

56

Fig. 2. 17: Contour plot of the 𝑢𝑟 results simulated in ABAQUS (left) and the present

method (right)

2.6 Further discussions on the sparseness of CSRBF

As we know, the merits of using the CSRBF are that the resulting interpolation matrix is

sparse for saving computation time and storage. The sparseness of the CSRBF matrix is

defined as the cumulative frequency of radial pairs inside the support radius α. As

described in Eq. (2.27), the cumulative frequency plot of radial distance against the

normalised radial distance can be computed as shown in Fig. 2.18. For instance, a 50%

sparseness requirement on the quarter cylinder simulation would imply a 0.4 cut off value

for α. Since the interpolation points are evenly distributed inside the simulation domains,

we anticipate that the standard deviation of the radial distances as well as the sensitivity

of α with respect to the sparseness requirement will not differ much for different model

geometries.

57

Fig. 2. 18: Cumulative frequency plot of radial distance for prismatic bar, cantilever and

quarter cylinder. std is the standard deviation of the radial distance

At this point, one may wonder about the performance of the proposed method with

unevenly distributed interpolation points. Herein, the interpolation points are generated

by two random processes: quasi-random distribution representing an evenly spaced

interpolation scheme; and pseudo-random distribution representing an unevenly spaced

interpolation scheme as shown in Fig. 2.19. Then, a convergence study is performed on

each distribution scheme by varying the number of interpolation points per unit area from

9 to 49. Each of the simulations is run 100 times with different random seeds so that an

average error can be obtained. The MAPE% of their vertical displacements is plotted

against the number of interpolation points as shown in Fig. 2.20. It is evident that both

distribution schemes of the present method converge efficiently with increased number

of interpolation points. That is, convergence is obtained with a satisfactory combination

of speed and computational efficiency. However, the evenly distributed interpolation

58

scheme is capable of producing more accurate results with much fewer interpolation

points. For example, around 5 MAPE% can be obtained with only 9 evenly distributed

interpolation points per unit area. By comparison, the unevenly distributed interpolation

scheme would require 25 interpolation points per unit area for the same result. Thus, the

evenly distributed interpolation scheme provides additional computational advantages.

Fig. 2. 19: Quasi-random distribution (left) and pseudo-random distribution (right) of 790

interpolation points (49 points per unit area)

59

Fig. 2. 20: Convergence study of the pseudo-random and quasi-random distribution

schemes: MAPE% of the vertical displacements versus number of interpolation points

Besides, sparse matrix could potentially save up computational resources when

performing matrix inversion due to the many zero entries enforced on the CSRBF

interpolation matrix. Since the Wendland’s CSRBF is derived to be positive definite,

symmetric and sparse, Cholesky decomposition can always be employed as an effective

mean to factorise the CSRBF matrix 𝜑 into upper triangular matrix: 𝜑 = 𝑈𝑇𝑈 for

solving the system of linear equations. During the decomposition, however, fill-in may

be created resulting in a less sparse system. Reducing the CSRBF matrix fill-in can be

achieved by reordering 𝜑 of sparse structure before computing the Cholesky

decomposition.

Three popular reordering methods, namely the reverse Cuthill-McKee, approximate

minimum degree and nested dissection algorithms are chosen to study the effectiveness

60

of minimising the fill-in for the CSRBF matrix. The reverse Cuthill-McKee algorithm

seeks to reorder the interpolation matrix with narrow bandwidth. The approximate

minimum degree algorithm seeks to reorder the matrix with large blocks of continuous

zeroes. While the minimum degree algorithm prioritises the matrix permutation based on

the sparsest pivot row and column, the nested dissection algorithm searches for a node

separator, which in turn recursively splits a matrix graph into sub graphs from a top down

perspective. This paper employs the Matlab build-in functions: symrcm and symamd to

perform the Cuthill-Mckee reordering and the approximate minimum degree reordering

respectively. The nested dissection implementation follows the algorithm described by

Davis [139]. These three reordering algorithms are employed to illustrate the sparse

structures of the CSRBF interpolation matrix and the corresponding Cholesky’s upper

triangular matrix for the prismatic bar simulation. For the reverse Cuthill-Mckee

reordering as shown in the left column of Fig. 2.21, one can clearly see that the usage of

the CSRBF can produce banded interpolation matrix which also results in a banded

Cholesky’s upper triangular matrix. It is noted that the change of the matrix bandwidth is

inversely proportional to the value of the sparseness. In the top centre of Fig. 2.21, the

approximate minimum degree algorithm forms a vastly different pattern of matrix graph

as comparing to the former algorithm. There we can see that a lower degree of sparseness

(less non zero entries) tends to produce larger and more blocks of zero entries scattering

inside the interpolation matrix. Similarly the nested dissection algorithm produces graph

with large blocks of zero entries, in which the non-zero entries are ordered in leaf up

shape as shown in the upper right corner of Fig. 2.21. Neither of the approximate

minimum degree nor the nested dissection algorithm produces Cholesky’s upper

triangular matrix with pattern consistent with the CSRBF matrix before the

decomposition (lower mid and lower right of Fig. 2.21).

61

Fig. 2. 21: Structures of the interpolation matrix at 10% sparseness for prismatic bar

after reordering using Reverse Cuthill-McKee algorithm (left column), approximate

minimum degree (centre column) and nested dissection (right column). First row

presents the interpolation matrices after reordering. Second row presents the Cholesky’s

upper triangular matrices after reordering. bw is the bandwidth and nz is the number of

nonzero entries of the matrices

To illustrate the effectiveness of each of the reordering algorithms, the sparseness of the

Cholesky’s upper triangular matrix with respect to the varied sparseness of the CSRBF

matrix for the prismatic bar simulation is plotted in Fig. 2.22. Ideally, the sparseness of

the Cholesky’s upper triangular matrix is at best equal to the sparseness of the original

interpolation matrix provided that there is no fill-in during the Cholesky’s decomposition

process. Conversely, a full interpolation matrix will result in a full Cholesky’s upper

triangular matrix. As seen in Fig. 2.22, CSRBF matrix with sparseness higher than 90%

seems to create the same amount of fill-in with or without the reordering. For sparseness

62

lower than 90%, Cuthill-Mckee algorithm is capable of reducing the fill-in consistently

and noticeably. The approximate minimum degree algorithm seems to be effective for

sparseness lower than 40% and could only outperform the Cuthill-Mckee algorithm

slightly for very sparse CSRBF matrix, i.e. roughly below 20%. The performance of the

nested dissection algorithm is the mix of the former two algorithms. At sparseness 50%

or above, its performance follows that of the Cuthill-Mckee algorithm with occasional

outliners; for sparseness below 50%, its performance follows that of the approximate

minimum degree algorithm with moderate and continuous fluctuations. Sparse matrices

with reordering algorithms are generally better than the ones without reordering. This can

be observed by comparing the curve without reordering (green scattered crosses) to the

very few other points that have worse sparseness in the Cholesky’s upper triangular

matrix after the reordering. The curves of the Cuthill-Mckee and the approximate

minimum degree reordering algorithms can be adequately described by the use of

exponential functions:

2 4

1 3

a x a xy a e a e (2.63)

where 𝑎1, 𝑎2, 𝑎3, 𝑎4 are the coefficients for the curve fitting.

63

Fig. 2. 22: Varied sparseness of interpolation matrix versus sparseness of Cholesky’s

upper triangular matrix for prismatic bar after reordering using Reverse Cuthill-McKee

algorithm, approximate minimum degree and nested dissection

2.7 Conclusions

In this chapter, the mixed meshless collocation method is developed through the MFS

framework to more efficiently analyse 3D isotropic linear elasticity with the presence of

body forces. The particular solution kernels using the CSRBF interpolation for

inhomogeneous body forces are derived using the Galerkin vectors and then coupled with

the method of fundamental solutions, based on a linear combination of fundamental

solutions for the full displacement and stress solutions in the solution domain. Numerical

results presented in this paper demonstrate that the proposed meshless collocation method

is capable of solving three-dimensional solid mechanics problems with inhomogeneous

terms efficiently, in addition to obtaining high accuracy with varied degrees of sparseness.

64

In contrast to the GSRBF, the CSRBF interpolation can provide unconditionally stable

and efficient computational treatment of various body forces. The computational speed

of interpolating the GSRBF is the same as the CSRBF with 100% sparseness where the

interpolating matrix becomes dense. In this case, the CSRBF will still maintain the

unconditionally positive definite property in contrast to the conventional GSRBF. Similar

application with GSRBF had been studied before and can be found in [80]. Last but not

least, the particular solution kernels derived in this paper are directly applicable to the

boundary element formulation and other boundary-type methods for determining

particular solutions related to inhomogeneous terms in the solution domain.

65

Chapter 3 Dual reciprocity boundary element method

using compactly supported radial basis functions for

3D linear elasticity with body forces

3.1 Introduction

It is well known that the BEM/BIM requires only boundary element discretisation for the

homogeneous partial differential equations (PDE) and thus has advantage of dimension

reduction over the FEM/FDM, as with other fundamental solution based numerical

methods like MFS [140] and HFEM [141-148]. However, domain discretisation is

generally unavoidable in the BEM/BIE for inhomogeneous PDE problems like 3D linear

elasticity problems with arbitrary body forces, as discussed in Chapter 2. To make the

BEM a true boundary discretisation method for the inhomogeneous cases, a variety of

domain transformation methods such as the radial integration method (RIM) [149] and

the dual reciprocity boundary element method (DRBEM) [60] have been proposed to

obtain equivalent boundary terms and bypass the need for domain integration caused by

the inhomogeneous terms. The essence of the former method is to transform the domain

integrals into surface integrals consisting of radial integral functions, while the latter

method aims to directly interpolate the inhomogeneous terms by a series of linearly

independent basis functions and then analytically determines the respective particular

solution kernels. In both methods, the choice of the basis functions is critical to provide

accurate numerical solutions [70, 150]. In most of the literature, globally supported radial

basis functions (RBF) are common choices. However, the choice of globally supported

66

RBFs has been questioned in relation to their accuracy and the number and position of

internal nodes required to obtain satisfactory results, especially for irregular domains. The

severe drawback of using the above globally supported RBFs is their dense interpolation

matrices, which often become highly ill-conditioned as the number of interpolation points

or the order of basis functions increases. Conversely, RBFs with locally supported

features such as the Wendland’s CSRBF are capable of producing sparse interpolation

matrices and improving matrix conditioning while maintaining competitive accuracy [84,

85]. As the result, CSRBF has become a natural choice for solving higher dimensional

problems [75, 89]. To our knowledge, the application of the CSRBF has only been applied

to two-dimensional linear elasticity problems [73].

In this chapter, the dual reciprocity boundary element formulation with CSRBF

approximation is developed for 3D linear elasticity with arbitrary body forces. In our

approach, we consider using the dual reciprocity technique to convert the domain

integrals into equivalent boundary integrals due to the presence of body forces in the

boundary element formulation, and using the CSRBF instead of the conventional globally

supported basis functions for the dual reciprocity approximation. During the computation,

we can freely control the sparseness of the interpolation matrix by varying the support

radius without trading off too much of the accuracy. Subsequently, the coefficients

associated with the particular solution kernels are determined by the addition of internal

nodes and the full solutions are evaluated by the DRBEM formulation. Finally, several

examples are presented to demonstrate the accuracy and efficiency of the present method.

A brief outline of the chapter is as follows. Section 3.2 describes the basics of three-

dimensional elasticity. Section 3.3 presents the concept of the particular solution kernels

associated with the Wendland’s CSRBF, and in Section 3.4, the formulation of the dual

reciprocity boundary element method with CSRBF is presented. Several examples are

67

considered in Section 3.5. Finally, some concluding remarks on the present method are

presented in Section 3.6.

3.2 Problem description

In this chapter, same 3D isotropic linear elastic body with inhomogeneous body force

terms in the domain Ω is taken into consideration. The related governing equations

including equilibrium equations, stress-strain equations and strain-displacement relations

are described as [138]

,

, ,

0

2

2

ij j i

ij kk ij ij

ij i j j i

b

G

u u

x x

x x x

x x x

, x (3.1)

where 𝐱 is a point in the domain Ω, 𝜎𝑖𝑗 the stress tensor, 휀𝑖𝑗 the strain tensor, 𝑢𝑖 the

displacement vector, 𝑏𝑖 the known body force vector, 𝜆 and 𝐺 the Lame constants and 𝛿

the Kronecker delta. Herein and after, an index after a comma denotes a differentiation

with respect to the coordinate component corresponding to the index.

Combining the above equations yields the following Navier’s equations in terms of

displacement components

0i iu b x (3.2)

with the differential operator ℒ

, ,1 2

i i jj j ij

Gu Gu u

v

x x (3.3)

where 𝑣 is the Poisson’s ratio which can be expressed as 𝑣 =𝜆

2(𝜆+𝐺). Later in the

numerical examples, Young modulus 𝐸 and Poisson’s ratio 𝑣 will be employed, in which

𝐺 can be computed using the conversion formula 𝐺 = 𝐸 2(1 + 𝜈)⁄ .

68

Besides, for a direct problem with the linear elastic governing equations, the

corresponding boundary conditions should be supplemented for the determination of the

unknown displacement and stress fields. The boundary conditions considered in this work

include the displacement boundary conditions defined on Γ𝑢 and the traction boundary

conditions defined on Γ𝑡:

,

,

i i u

i ij j i t

u u

t n t

x x x

x x x x x (3.4)

where 𝑡𝑖 is the traction field,𝑖 and 𝑡 the prescribed displacement and traction, 𝑛𝑖 the

unit vector outward normal to the boundary Γ𝑡. It is assumed as usual that the boundaries

Γ𝑢 and Γ𝑡 are not overlapped so that Γ = Γ𝑢 ∪ Γ𝑡 and ∅ = Γ𝑢 ∩ Γ𝑡.

3.3 Dual reciprocity method

As similar to the procedure in Section 2.3, the particular solutions of displacement 𝑢𝑖𝑝

and

stress 𝜎𝑖𝑗𝑝

can be expressed in terms of Galerkin vector 𝑔𝑖 as [138]

, ,

1,

2 1

p

i i kk k iku g gv

x x x x (3.5)

, , , ,1 , 1

p

ij k mmk ij k ijk i jkk j ikk

Gvg g v g g

v

x x x x x x (3.6)

Upon substituting Eq. (3.5) into the governing equation (3.2) yields the following bi-

harmonic equation

,

i

i jjkk

bg

G

xx (3.7)

The dual reciprocity method (DRM) aims to efficiently approximate the particular

solution by finding its solution kernels while prescribing the inhomogeneous terms such

69

as body forces with a series of linearly independent basis functions, so that the unknown

body forces terms in Eq. (3.7) can be reconstructed using finite set of discrete data

1

Nn

i l li n

n

b

x x (3.8)

where 𝜑𝑛 is the chosen series of basis functions, 𝑁 the number of interpolation points

including the boundary points and interior points, 𝛼𝑙𝑛 the interpolation coefficients to be

determined. The use of the Kronecker delta δ is to separate the basis functions for

approximating the body forces in each direction independently.

Similarly, substituting Eq. (3.8) into Eq. (3.7) produces the following approximated

Galerkin vector 𝑔𝑖

1

Nn

i l li n

n

g

x x (3.9)

with

,

n

n jjkkG

xx (3.10)

Further, from Eqs. (3.5) and (3.6) the particular solution 𝑢𝑖𝑝, 𝜎𝑖𝑗

𝑝 can be expressed as

1

1

Np n n

i l li

n

Np n n

ij l lij

n

u

S

x x

x x

(3.11)

where 𝜙𝑛 is the respected Galerkin vector solution kernels, 𝜓𝑙𝑖𝑛 the displacement

particular solution kernels,𝑆𝑙𝑖𝑗𝑛 the stress particular solution kernels.

From Eq. (3.11), the traction particular solution 𝑡𝑖𝑝

can be written as

1

Np n n

i l j lij

n

t Sn

x x x (3.12)

70

To analytically determine 𝜙, 𝜓𝑙𝑖 and 𝑆𝑙𝑖𝑗, an explicit function needs to be chosen first for

𝜑. For Wendland’s CSRBF in 3D [84, 85], 𝜑 is defined as

22

0

4

2

6 2

1 01 for smoothn

ess

0 ,

1 4 1 for smoothness

1 35 18

,

3

rrr

r C

r

r rr C

r r rr

4

8 3 2

6

for smoot

hness

1 35 25 8 1 for smoothness

C

r r r rr C

(3.13)

where the subscript + denotes that the bracket function will be forced to be zero when the

bracketed value is less than zero. 𝛼 is a cut off parameter for varying the support radius

of interpolation matrix 𝜑(𝑟) as illustrated in Fig. 2.2.

Meanwhile, the related particular solution using compactly supported functions can be

derived, as given in Section 2.3.

3.4 Formulation of dual reciprocity boundary element method

In the classical BEM, the domain integral arises due to the presence of body forces is

illustrated as [28]

, ,

,

ij j ij j ij j

S S

ij j

V

c u T u dS U t dS

U b dV

∮P P Q Q Q P Q Q Q

P Q Q Q

x x x x x x x x x

x x x x

(3.14)

where 𝑈𝑖𝑗 and 𝑇𝑖𝑗 are the fundamental solutions for displacements and surface tractions,

respectively, 𝒙𝑷 is a source point which can be any point within the domain or on the

71

boundary, 𝒙𝑸 is an arbitrary integration point and 𝑐𝑖𝑗 are the boundary geometry

coefficient [28]

lim ,ij ij ij

S

c T dS

Q P

P Q Qx x

x x x (3.15)

The dual reciprocity method makes use of the particular solution kernels from Eq. (3.11),

in which the domain integral containing the body forces term becomes [60]

1

, ,

,

p

ij j j ij

V V

Nn n

l li ij

n V

U b dV u U dV

U dV

P Q Q Q Q P Q Q

Q P Q Q

x x x x x x x x

x x x x

(3.16)

Integrating by parts the differential operator term yields

1

,

, ,

ij j

V

Nn n n n

l ij lj ij lj ij j lij

n S S

U b dV

c T dS U n S dS

∮

P Q Q Q

P P Q Q Q P Q Q Q Q

x x x x

x x x x x x x x x x

(3.17)

Substituting Eq. (3.17) in Eq. (3.14), we have

1

, ,

, ,

ij j ij j ij j

S S

Nn n n n

l ij lj ij lj ij j lij

n S S

c u T u dS U t dS

c T dS U n S dS

∮

∮

P P Q Q Q P Q Q Q

P P Q Q Q P Q Q Q

x x x x x x x x x

x x x x x x x x x

(3.18)

Next, to write Eq. (3.18) in discretised form, the whole boundary is modelled with 𝐸

surface elements so that we can use summations over the boundary elements to replace

the integrals in Eq. (3.18). For example, the first two terms in Eq. (3.18) can be written

in discretised form as

72

1

, ,

e

E

ij j ij j ij j ij j

eS S

c u T u dS c u T u dS

∮ ∮P P Q Q P P Q Q Qx x x x x x x x x (3.19)

where 𝑆𝑒 is the surface of 𝑒𝑡ℎ boundary element.

Further, introducing the interpolation functions and numerically integrating over each

boundary surface element, one gets for the surface integral in Eq. (3.19)

1 2 1 2 1 2 1 2

1 21 1 1 1

, , , ,ˆE G G N

n en

ij g g g g j g g g g

e g g n

T P P N P P u J P P W W

P ex x (3.20)

where

1 2 1 2

1

ˆ, ,N

n en

g g g g m

n

P P N P P x

e mx e (3.21)

The 𝒆𝒎 denotes the standard basis for each of the directions 𝑚 in the Cartesian coordinate system,

𝑁𝑛 is the element shape functions, 𝑥𝑗𝑒𝑛, 𝑗

𝑒𝑛are element nodal coordinates and

displacements, respectively, 𝐽𝜉 is the determinant of the local Jacobian matrix related to

the global coordinate to local coordinate derivatives, 𝑃𝑔1 , 𝑃𝑔2 are element natural

coordinates of integration points, 𝐺 is the number of integration points, and 𝑊𝑔1 ,𝑊𝑔2 are

the associated weight factors.

After substituting Eq. (3.20) into Eq. (3.19) and replacing the local nodal indices 𝑥𝑗𝑒𝑛

and 𝑗𝑒𝑛 with global indices 𝑗

𝑘 and 𝑗𝑘, we have, for 𝑝 collocation points

1 2 1 2 1 2

1 21 1

, ˆ,G G

pk pk k

pk ij ij g g g g g g j

g g

c H P P J P P W W u

(3.22)

where

1 2 1 2 1 2 1 2, , , , , ,pk en n

ij g g ij g g ij g g g gH P P H P P T P P N P P p p ex x x (3.23)

Similar procedure can be applied for the third integral term in Eq. (3.18). Finally, one

obtains the following expression in matrix form

73

ˆ' ˆ H u Gt H U GT α (3.24)

Equation (3.24) is the basis for the application of the DRBEM for solving 3D linear

elasticity with body forces and involves discretisation of the boundary only. Moreover,

the displacement and traction fundamental solutions used in the above derivation are

written as [28]

, ,

, , , , , ,2

3 4,

16 1

1, 1 2 3 1 2

8 1

li l i

li

li li l i l i i l

v r rU

G v r

rT v r r v r n r n

v r n

P Q

P Q

x x

x x

(3.25)

3.5 Numerical Examples and discussions

To demonstrate the accuracy and efficiency of the derived formulation, three benchmark

examples, which are solved by the proposed method, are considered in this section. These

examples include: 1) a prismatic bar subjected to gravitational load, 2) a thick cylinder

under centrifugal load, and 3) axle bearing under internal pressure and gravitational load.

For simplification, only Wendland’s CSRBF with smoothness 𝐶0 is considered here.

Simulation results obtained from the proposed method and the conventional FEM method

are compared against the analytical solutions. We also compute the mean absolute

percentage error (MAPE) as an effective description for quantifying the average

performance accuracy of the present method, as done in Section 2.5.

3.5.1 Prismatic bar subjected to gravitational load

In the first example, we consider a straight prismatic bar subjected to gravitational load,

as shown in Fig. 3.1. The dimensions of the bar are 1m×1m×2m and it is fixed at the

74

top. Assuming the bar being loaded along the z-direction by its gravitational load, the

corresponding body forces can be expressed as

0, 0, x y zb b b g (3.26)

where 𝜌 is density and 𝑔 is gravity. The material parameters used in the simulation are:

𝐸 = 4x107Pa, 𝑣 = 0.25, 𝜌 = 2000kg ∙ m−3, 𝑔 = 10m ∙ s−2.

Fig. 3. 1: Prismatic bar under gravitational load

75

Fig. 3. 2: Boundary element meshing of prismatic bar

The numerical model is composed of 160 boundary elements as shown in Fig. 3.2. Test

points are chosen along the centreline of the prismatic bar. The corresponding

displacement and stress results are compared to the analytical solutions [138] and the

FEM solutions, which are evaluated by the commercial software ABAQUS. Numerical

results in Tables 3.1 and 3.2 show the variations of displacement and stress in terms of

the sparseness of CSRBF. It is found that there is good agreement between the numerical

results from the present method and the FEM results and the available analytical solutions.

76

Table 3. 1: Displacement results for the prismatic bar

z (m)

−𝑢𝑧(10−3m)

Present method

ABAQUS Analytical

solutions Sparseness

=20%

Sparseness

=60%

Sparseness

=100%

-0.25 0.2354 0.2314 0.2249 0.2055 0.2345

-0.50 0.4781 0.4760 0.4476 0.4141 0.4375

-0.75 0.6733 0.6422 0.6206 0.5944 0.6100

-1.00 0.8218 0.8141 0.7672 0.7392 0.7500

-1.25 0.9371 0.9238 0.8754 0.8500 0.8600

-1.50 1.0008 0.9871 0.9403 0.9285 0.9450

-1.75 1.0611 1.0431 0.9919 0.9754 0.9900

MAPE (%) 7.3808 5.5058 0.5330 3.728

Table 3. 2: Stress results for the prismatic bar numerical simulation

z (m)

𝜎𝑧𝑧(kPa)

Present method

ABAQUS Analytical

solutions Sparseness

=20%

Sparseness

=60%

Sparseness

=100%

-0.25 39.07 36.46 35.61 36.30 35.6

-0.50 31.64 31.84 30.21 31.81 30.0

-0.75 27.51 27.07 25.05 25.91 25.0

-1.00 21.73 21.38 20.04 20.31 20.0

-1.25 15.80 15.13 15.00 15.08 15.0

-1.50 10.16 10.31 10.07 10.01 10.0

-1.75 5.66 5.61 5.42 5.00 5.0

MAPE (%) 7.72 5.70 1.46 1.975

77

To better compare the performance of the present method to FEM, a convergence study

is carried out for various degrees of sparseness. The surface elements and interior nodes

are discretised and aligned with the same nodal points of the FEM as shown in Fig. 3.3.

The MAPE% of their vertical displacements are plotted against the meshing density for

the various sparseness levels: 40%, 60% and 80% as shown in Fig. 3.4. As expected,

interpolation with the higher degree of sparseness can consistently produce better

accuracy. As we increase the number of surface elements in the BEM, the results

converge exponentially across all the sparseness schemes. In particular, the performance

of the 40% sparseness scheme surpasses the FEM at meshing density of 25 elements per

unit area. All sparseness schemes of the present method have errors less than 0.1% which

is twice as accurate as the FEM at meshing density of 36 elements per unit area.

Fig. 3. 3: Boundary element meshing and interior points distribution of a prismatic bar:

9 elements (left) and 36 elements (right) per unit area.

78

Fig. 3. 4: Convergence study of the present method with various degrees of sparseness:

MAPE% of the vertical displacements versus meshing density

3.5.2 Thick cylinder under centrifugal load

In the second example, a cylinder with 10m internal radius, 10m thickness and 20m height

is assumed to be subjected to centrifugal load. Due to the rotation, this cylinder is

subjected to apparent generalised body force. If the cylinder is assumed to rotate about its

z-axis as shown in Fig. 3.5, the generalised body forces in terms of spatial coordinates

can be written as

2 2, , 0x y zb w x b w y b (3.27)

where𝑤 is the angular velocity. In this example, 𝑤 = 10 is chosen.

The problem is solved with dimensionless material parameters 𝐸 = 2.1x105, 𝑣 = 0.3,

𝜌 = 1.According to the symmetry of the model, only one quarter of the cylinder domain

79

needs to be considered for establishing the computing model. Proper symmetric

displacement constraints are then applied on the symmetric planes (see Fig. 3.5).

Fig. 3. 5: Thick cylinder under centrifugal load

Fig. 3. 6: Boundary element meshing of a quarter thick cylinder

80

The numerical model is composed of 460 boundary elements as shown in Fig. 3.6. For

the rotating cylinder, the displacement and the stress fields are more complicated than

those in the straight prismatic bar and the cantilever beam as discussed above. The results

of radial and hoop stresses and radial displacement at specified locations are tabulated

respectively from the present method (see Table 3.3-3.5). These results are then

compared to ABAQUS with 10881 elements of type 20-node quadratic brick elements

and the analytical solutions [138]. The result shows that the accuracy of the DRM-BEM

method improves with an increase of the sparseness for the body force terms.

Table 3. 3: 𝜎𝑟 results for the thick cylinder numerical simulation

𝑟

𝜎𝑟(kPa)

Present method

ABAQUS Analytical

solutions Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

11.25 2.57 2.52 2.40 2.434 2.367

12.50 3.93 3.87 3.68 3.740 3.620

13.75 4.47 4.37 4.17 4.233 4.099

15.00 4.33 4.25 4.04 4.108 4.010

16.25 3.77 3.71 3.52 3.568 3.484

17.50 2.87 2.83 2.69 2.647 2.604

18.75 1.54 1.51 1.44 1.442 1.430

MAPE (%) 8.6125 6.6752 1.5096 2.394

81

Table 3. 4: 𝜎𝑡 results for the thick cylinder numerical simulation

𝑟

𝜎𝑡 (kPa)

Present method

ABAQUS Analytical

solutions Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

11.25 33.26 33.19 31.43 31.48 30.66

12.50 30.07 29.39 28.05 28.24 27.47

13.75 27.19 26.77 25.13 25.55 24.86

15.00 24.32 24.45 22.91 23.21 22.61

16.25 22.88 22.10 20.79 21.10 20.60

17.50 20.47 18.73 18.73 19.12 18.74

18.75 19.17 17.02 17.04 17.21 16.97

MAPE (%) 9.735 5.512 1.188 2.392

Table 3. 5: 𝑢𝑟 results for the thick cylinder numerical simulation

𝑢𝑟(m)

𝑟

Present method

ABAQUS Analytical

solutions Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

11.25 1.7382 1.7027 1.6213 1.641 1.604

12.50 1.7107 1.6837 1.5951 1.606 1.571

13.75 1.6666 1.6466 1.5656 1.582 1.547

15.00 1.6440 1.6272 1.5410 1.563 1.529

16.25 1.6410 1.6106 1.5388 1.546 1.513

17.50 1.6228 1.5927 1.5135 1.529 1.497

18.75 1.6020 1.5611 1.4908 1.510 1.477

MAPE (%) 8.2626 6.3893 1.1916 2.225

82

3.5.3 Axle bearing under internal pressure and gravitational load

To show the ability of the present method for complicated geometrical domain, a

numerical model of axle bearing, as taken from [28], is used for the third example. In this

model, the displacements are fixed along the boundary surface at the lower right of the

bearing as shown in Fig. 3.7 while normal stress of 1GPa is uniformly applied to the

surface of the top inner circle. Gravitational loading is applied as similar to Eq. (3.26).

The material parameters used in the simulation are: 𝐸 = 4x107Pa, 𝑣 = 0.25, 𝜌 =

2000kg ∙ m−3, 𝑔 = 10m ∙ s−2 . This numerical model is meshed by 392 boundary

elements as shown in Fig. 3.7.

Fig. 3. 7: Numerical model of the axle bearing

𝑢𝑥 = 𝑢𝑦 = 𝑢𝑧 = 0

𝑃

83

The results are compared to ABAQUS and listed Table 3.6 and Table 3.7. Test points

are chosen along the vertical centre line from the tip of the body to the tip of the upper

inner circle. Again, the result shows that the present method is as accurate as ABAQUS

while requiring less computational resources.

Table 3. 6: Displacement results for the axle bearing numerical simulation

y (m)

𝑢𝑦(10−3m)

Present method

ABAQUS Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

0.04 -1.7544 -1.7341 -1.6974 -1.7214

0.05 -6.4197 -6.3919 -6.3255 -6.1702

0.06 -18.1341 -17.6372 -17.4276 -17.2109

0.10 70.3576 69.1086 67.7661 67.0053

0.11 58.9133 57.7927 56.7390 56.0975

0.12 53.9164 52.9418 51.8259 51.3768

84

Table 3. 7: Stress results at various test points

y (m)

𝜎𝑥𝑥 (MPa)

Present method

ABAQUS Sparseness

= 20%

Sparseness

= 60%

Sparseness

= 100%

0.04 13.1720 12.7956 12.6637 12.4969

0.05 40.5788 39.8536 39.1310 38.6931

0.06 62.8807 61.7737 60.4393 59.8852

0.10 66.8648 65.5202 64.3152 63.6428

0.11 49.1604 48.1644 47.2435 46.7450

0.12 38.4089 37.5319 36.7898 36.4924

3.6 Conclusions

In this study, the dual reciprocity boundary element method using compactly supported

radial basis functions is developed to more effectively analyse 3D isotropic linear

elasticity problems in the presence of body forces. The particular solution kernels using

the CSRBF interpolation for inhomogeneous body forces are derived using the Galerkin

vectors and then coupled with equivalent boundary element integrals based on the dual

reciprocity method in the solution domain. Numerical results presented in this paper

demonstrate that the proposed method is capable of solving three-dimensional solid

mechanics problems with inhomogeneous terms efficiently, in addition to obtaining high

accuracy with varied degrees of sparseness. In contrast to the globally supported RBF,

the CSRBF interpolation can provide stable and efficient computational treatment of

various body forces and complicated geometrical domains. Moreover, the particular

solution kernels derived in this paper are directly applicable to other boundary-type

85

methods for determining particular solutions related to inhomogeneous terms in the

solution domain.

86

Chapter 4 Evaluation of hypersingular line integral by

complex-step derivative approximation

4.1 Introduction

Hypersingular integrals are important for many problems in engineering, physics and

mathematics. Such integrals arise when the finite domain of integration contains

singularity points. This is often due to the employment of fundamental solution based

numerical methods such as method of fundamental solutions (MFS) [33, 151], the

boundary element method (BEM) [28, 152, 153] and hybrid fundamental solution based

finite element method (HFS-FEM) [143] for solving boundary value problems. Popular

applications of the hypersingular integrals include problems in electromagnetic scattering

[154], acoustics [155, 156], heat transfer [157], piezoelectric materials[158, 159], fracture

mechanics [160], boundary stress problems in elasticity [161], and it is vital in developing

the symmetric Galerkin boundary integral equations [162, 163]. Like the Cauchy

principal value integral, an accurate evaluation of the hypersingular integrals often

requires the knowledge of their conditions for existence [164] along with their

geometrical definitions and properties [165, 166], or is defined as pseudo-differential

operators [167, 168]. Consequently, the so called Hadamard finite part of the

hypersingular integral can be taken for evaluation.

Unlike the Cauchy principal value integral, which is well known and is implemented in

popular and high level numerical software such as Mathematica and Matlab, the

numerical evaluation of hypersingular integrals is still not mainstreamed and remains of

87

research interest. In the literature, the first numerical attempt to evaluate the hypersingular

integrals dates back to 1966, with Ninham [169] using the asymptotic expansion of the

Euler Maclaurin formula in conjunction with the midpoint trapezoidal rule. Other

approaches by means of quadrature rules can be found in the works by Paget [170, 171].

If one chooses to split a hypersingular integral into regular and singular parts, the regular

part of the integrand can be approximated by interpolation such as the generalised

Lagrange polynomial [172, 173]. The merit of this interpolatory quadrature is that the

regular function’s derivative, which arises naturally for singular integrals of higher order,

need not be found explicitly. Instead, the derivative is simply approximated by

differentiating the corresponding interpolants. Alternatively, Hui and Shia [174] derived

a formulation using the explicit derivative of the regular function with Gaussian

quadrature to evaluate the hypersingular integrals. Apart from the usual interpolatory

quadrature, Kolm and Rokhlin [175] employed the Fourier-Legendre series, which is a

type of spectral method, to approximate the regular function and its derivative for various

orders of principal value integrals simultaneously. This approach was further simplified

and generalised for endpoint singularities by Carley [176] using the semi-analytical

formulation for improper integrals proposed by Brandao [177]. It is worth noting that in

Brandao’s formulation, the simpler finite part of the singular integral is extracted and

determined analytically. The numerical error is mainly due to the integration of the

regular integral in addition to the error from the regular integrand and its derivatives’

approximations.

Although the method proposed by Kolm and Rokhlin [175] demonstrates convergence

results with Legendre polynomials of higher degree, it is not without drawbacks for

practical applications. As with the interpolatory quadrature, the modified quadrature

weights in Kolm and Rokhlin’s method need to be re-evaluated when there is a change of

88

bounds in the hypersingular integral. As a result, such a method is not suitable for higher

dimensional applications when the bounds of the inner integral are a function of the outer

integral. Without a simple change of interval bypassing the weight re-evaluation, the

computational cost will be so high that it would defeat the purpose of constructing a

generalised numerical quadrature scheme.

Herein, our method of evaluating hypersingular integral should serve three purposes.

Firstly, the solution’s accuracy is comparable to machine precision. Secondly, the method

is straight forward and generalised enough for fast numerical implementation with

minimal computational effort and is comparable to the interpolatory quadrature method.

Lastly, the method is robust and flexible enough to handle hypersingular surface integrals

which have applications in many fundamental solution-based numerical methods.

In this chapter, we pay attention to the effective evaluation of the high-order

hypersingular line integral, as given in Eq. (4.1). The hypersingular integral is expanded

into regular and singular integrals by employing Brandao’s formulation. The regular

function is first interpolated by barycentric rational polynomial [178] for comparison to

the results reproduced by the Kolm and Rokhlin’s method. It was suggested in literature

that such an interpolation scheme could give better approximation than polynomial

interpolation due to its rational form with adjustable degree of order. Since the numerical

quadrature is not reusable for hypersingular integrals with different bounds, further

improvement can be made by dropping the interpolation entirely and instead, resorting to

finding the derivative of the regular function numerically. In order to achieve machine

precision like accuracy for the derivative approximation, complex-step derivative

approximation is employed [179]. Unlike the derivative approximation by the finite

difference method, the complex-step approach makes use of a finite step on the complex

axis while avoiding difference operation and the subsequent catastrophic cancellation on

89

the real axis for small step size. As a result of this, the complex-step derivative

approximation is capable of producing machine precision like accuracy.

The remainder of this chapter is organised as follows. Section 4.2 describes the definitions

and semi-analytical values of the hypersingular line integrals. Section 4.3 presents the

implementation of the barycentric rational interpolation and its results for the

hypersingular integral evaluation. Section 4.4 presents the formulations and results by

means of complex-step derivative approximation. Finally, some concluding remarks on

the present method are presented in Section 4.5.

4.2 Definitions and properties of hypersingular line integrals

In this chapter, we are interested in the efficient evaluation of hypersingular line integral

as given in Eq. (4.1)

2

b

a

xI dx

x t

(4.1)

where 𝜑 is the density function in terms of variable 𝑥, 𝑡 the singularity point, 𝑎 and 𝑏 the

lower and upper bounds of the hypersingular line integral.

As is well known in the literature [180, 181], there holds a close relationship between the

Cauchy principal value integral and the hypersingular integral. To ensure the existence of

the singular integrals, the corresponding density function 𝜑(𝑥) must satisfy some

smoothness and continuity conditions [164], which will be briefly introduced in the

following sections.

90

4.2.1 Cauchy principal value integral

For a Holder continuous function 𝜑(𝑥) ∈ 𝐶0,𝛼 at 𝑥 = 𝑡 , the Cauchy principal value

integral in a symmetric neighbourhood of 𝑡 ∈ (𝑎, 𝑏) is defined as [166]

0

1 1lim

b b t b

a a a t

x x tdx dx t dx dx

x t x t x t x t

for a t b

(4.2)

where 𝒞 indicates that the singular integral is defined as Cauchy principal value and 𝜖 the

symmetric neighbourhood of 𝑡.

After taking the limit, we have

b b

a a

x x t b tdx dx t ln

x t x t t a

for a t b (4.3)

4.2.2 Hypersingular integral

The Hadamard finite-part integral in a symmetric neighbourhood of 𝑡 ∈ (𝑎, 𝑏) is defined

as [166]

2 2 2 20

1 1lim

b b t b

a a a t

x x tdx dx t dx dx

x t x t x t x t

for

a t b

(4.4)

where ℋ indicates the integral is defined as Hadamard finite-part integral

After taking the limit, we have

2 2

1 1 2b b

a a

x x tdx dx t

a t b tx t x t

for a t b (4.5)

91

Suppose the above limit exists and the term 2

𝜖 is ignored, then (4.5) represents the finite-

part of the otherwise divergent integral.

By repeating the above procedures, we have, for a Holder continuous function 𝜑(𝑥) ∈

𝐶𝑝,𝛼 at 𝑥 = 𝑡, the finite-part integral in a neighbourhood of 𝑡 ∈ (𝑎, 𝑏),

1

1b b

papa

x xddx dx

p dtx t x t

for a t b (4.6)

where 𝑝 ≥ 1 is the order of singularity.

To simplify the above procedures of taking numerical limits, Brandao proposed a

formulation which expands the Taylor series of 𝜑(𝑥) about 𝑡 and subsequently

transforming the finite-part integrals into regular integrals and simpler finite-part integrals

for semi-analytical evaluation [177] as illustrated in Fig. 4.1,

1 1

0

1

0

1

!

1

!

kkpb b

p pa ak

kpb

p kak

x t x tdx x dx

kx t x t

tdx

k x t

for a t b (4.7)

Using (4.7), the finite-part value of the hypersingular integral (4.1) can be expressed as

2

1

2

1

2

1 1

b b b

a a a

b

a

xdx t dx t dx

x tx t x t

x t t x tdx

x t

for a t b (4.8)

92

Fig. 4. 1: Regularisation process for hypersingular line integral using Brandao’s

formulation

4.2.3 Simpler hypersingular integrals values

Supposing (4.8) exists, its simpler finite-part integrals can be analytically determined by

once again ignoring the divergent terms [160].

For one sided Cauchy principal value integrals

0

1 1lim ln ln

b b

t tdx dx b t

x t x t

for t a (4.9)

1

lnt

adx t a

x t

for t b (4.10)

For two sided Cauchy principal value integral, we have from (4.3),

1ln

b

a

b tdx

x t t a

for a t b (4.11)

For one sided finite-part integral

2 20

1 1 1 1lim

b b

t tdx dx

b tx t x t

for t a (4.12)

93

2

1 1t

adx

a tx t

for t b (4.13)

For two sided finite-part integral, we have from (4.5),

2

1 1 1b

adx

a t b tx t

for a t b (4.14)

Similarly, for higher order of finite-part integral, we have

1

1 1b

p ptdx

x t p b t

for t a (4.15)

1

1 1t

p padx

x t p a t

for t b (4.16)

1

1 1 1 1b

p p padx

px t a t b t

for a t b (4.17)

4.3 Barycentric rational interpolation

Provided that the density function 𝜑(𝑥) is known, (4.8) can be numerically evaluated

with the analytical results for the simpler finite-part integrals from (4.9)-(4.14). For

example, one may choose to analytically derive the derivative of 𝜑(𝑥) and employ

quadrature rule to evaluate the regular integral in (4.8). However, such approach is not

convenient for implementation when there is a change of function 𝜑(𝑥) due to different

applications or when it is of very complicated forms. Instead, 𝜑(𝑥) and its derivative are

often approximated. In the work by Kolm and Rokhlin [175], they employed Fourier-

Legendre expansion to approximate 𝜑(𝑥) and achieve convergence result. In this section,

we investigate the performance of interpolatory scheme using barycentric rational

polynomial proposed by Floater and Hormann [178]. As suggested in the literature, the

rational interpolation scheme could give better approximations than polynomial

interpolation such as the generalised Lagrange interpolation due to the adjustable degree

94

of order on the interpolant’s denominator. Furthermore, the barycentric form’s property

also ensures that the computation time for the interpolants will be much less and stable

[182]. In the followings, we first derive an explicit formulation for the interpolatory

quadrature scheme, followed by making use of the barycentric rational polynomial for

the interpolation.

4.3.1 Interpolatory quadrature formulation

Let ℓ𝑖(𝑥) be the interpolants of function 𝜑(𝑥), then

1

0

n

i i

i

x x x

(4.18)

where 𝜑𝑖 is the nodal value at interpolation point 𝑥𝑖 and 𝑛 is the number of interpolatory

points

Substitution of (4.18) into (4.8) yields,

1*

2

0

n

i i

i

b

a

xdx x W

x t

(4.19)

where 𝑊𝑖∗ is the modified quadrature weight expressed as

1

1*

2 2

1 1b b bi i i

i i ia a a

x t t x tW t dx t dx dx

x tx t x t

(4.20)

Since ℓ𝑖(1)

can be found directly by differentiating the interpolant ℓ𝑖(𝑥), the numerical

task remains to integrate the regular integral in (4.20). Using the popular Gaussian

quadrature scheme with weights 𝑊𝑗 and points 𝑃𝑗 and making use of the analytical

solutions from (4.9)-(4.14), we obtain

95

For case 𝑎 < 𝑡 < 𝑏,

1* *

1*

2*

1

1 1ln

2

mi j i i j

i i i j

jj

P t t P tb t b aW t t W

a t b t t a P t

(4.21)

For case 𝑡 = 𝑎,

1* *

1*

2*

1

ln2

mi j i i ji

i i j

jj

P t t P tt b aW t b t W

b t P t

(4.22)

For case 𝑡 = 𝑏,

1* *

1*

2*

1

ln2

mi j i i ji

i i j

jj

P t t P tt b aW t t a W

a t P t

(4.23)

where 𝑃𝑗∗ is the normalisation over the interval [−1,1] for arbitrary limits [𝑎, 𝑏],

*

2 2j j

b a a bP P

(4.24)

As was seen from (4.21)-(4.23), the modified quadrature weight for the generalised

numerical quadrature scheme depends on the bounds [𝑎, 𝑏] of the regular integral. The

spectral method by Kolm and Rokhlin [175] also suffers from this setback.

4.3.2 Barycentric rational polynomial

So far, we have yet specified in what form the interpolant ℓ𝑖(𝑥) should possess. The most

popular interpolant is the generalised Lagrange polynomial [172, 173]. In recent research

[182-184], it turns out that it is often more advantageous to have a barycentric

representation for better computation speed and stability. In fact, there exists a generalised

barycentric formula for every interpolation scheme [182, 185]. The interpolant ℓ𝑖(𝑥) in

barycentric form can be expressed as

96

1

0

i

ii n

k

k k

w

x xx

w

x x

(4.25)

where 𝑤𝑘 is called the barycentric weights.

For instance, the weights for the barycentric Lagrange interpolation is

1

0,

1k n

k l

l l k

w

x x

(4.26)

As was mentioned earlier, the density function could be in complicated forms depending

on the different practical applications. In the case of 𝜑(𝑥) possessing a rational form such

as having polynomials on both of its numerator and denominator, one may expect a

rational interpolation to perform better than the more traditional Lagrange interpolation.

The barycentric rational polynomial as proposed by Floater and Hormann [178] aims to

generalise the family of rational interpolants. Essentially, this method avoid the formation

of poles by blending the local interpolants to form a global ones. The barycentric weights

for the rational interpolation can be expressed as

,0 1 ,

11

k s ds

k

s k d s n d l s l k k l

wx x

(4.27)

The parameter 𝑑 ∈ [0, 𝑛 − 1] controls the degree of blending and has implication for the

rational function’s 𝜑(𝑥) degree of order.

The derivative of order 𝛽 of the barycentric rational interpolants share some simple

differentiation formulas [186] by iterative process,

1

0

n

i i

i

x x x

(4.28)

97

For 𝛽 = 1,

1 ii j

j j i

wx

w x x

for jix x (4.29)

1

1 1

0,

n

j j k j

k k j

x x

for jix x (4.30)

For 𝛽 > 1,

1

1 1 i j

i j i j j j

j i

xx x x

x x

for jix x (4.31)

1

0,

n

j j k j

k k j

x x

for jix x (4.32)

By substituting (4.25), (4.27), (4.29) and (4.30) into equations (4.21)-(4.23) for the

modified weights 𝑊𝑖∗ , the finite-part of the hypersingular integral with interpolatory

quadrature scheme, i.e. (4.19) can then be evaluated.

4.3.3 Numerical example

To demonstrate the efficiency of the interpolatory quadrature scheme using the

barycentric rational polynomial, a benchmark example for a specified density function

𝜑(𝑥) over the interval [−1, 1] is considered.

sin 2 cos 3x x x (4.33)

The analytical solution 𝐼(𝑡) of the hypersingular integral for the above density functions

is,

1 2

1 xI t dx

x t

(4.34)

98

For varying 𝑡, Eq. (4.19) can be modified to represent the numerical quadrature 𝐼(𝑡),

1

*

0

ˆ ,n

i i

i

I t tx W

(4.35)

To illustrate the competitiveness of the scheme using (4.35), results computed using

Matlab are compared to those reproduced by the Kolm and Rokhlin’s method [175] and

analytical solutions by Mathematica. The approximated integrals are evaluated at 20

different singularity point 𝑡 coinciding with the normalised Legendre points 𝑃𝑘∗ . The

number of quadrature points 𝑚 for the regular part integration is fixed at 40 and the

parameter 𝑑 for the barycentric rational interpolation is set as 𝑛 − 1 to maximise the

interpolation’s effectiveness. To measure the precision of this method, the mean absolute

percentage error for incremental number of interpolation points 𝑛 from 6 to 30 is

computed as

*

*1

11 100%

ˆtnk

kt k

I PE

n I P

(4.36)

where tn is the number of test points

The results for the mean absolute percentage error of hypersingular integral evaluation

using numerical quadrature of barycentric rational interpolation scheme and the Kolm

and Rokhlins’ method are plotted in Fig. 4.2. The numerical results for the 20 different

singularity point 𝑡 of both schemes evaluated by 20 interpolation points are compared to

the analytical results as tabulated in Table 4.1 while the numerical quadrature weights

𝑊𝑖∗ for 14 Legendre nodes are tabulated in Table 4.2.

99

Fig. 4. 2: Mean absolute percentage error of hypersingular integral evaluation using

numerical quadrature of barycentric rational interpolation scheme and the Kolm and

Rokhlins’ method

100

Table 4. 1: Numerical results of barycentric rational interpolation scheme and the Kolm

and Rokhlins’ method in comparison to analytical results

t Barycentric rational

interpolation scheme

Kolm and Rokhlins’

method Analytical results

-0.9931 2.82658253003066E+02 2.82658253003065E+02 2.82658253003064E+02

-0.9640 6.04993211321839E+01 6.04993211321839E+01 6.04993211321840E+01

-0.9122 3.04414367754299E+01 3.04414367754293E+01 3.04414367754293E+01

-0.8391 2.08303094565043E+01 2.08303094565043E+01 2.08303094565044E+01

-0.7463 1.55919962833773E+01 1.55919962833773E+01 1.55919962833772E+01

-0.6361 1.10381851055952E+01 1.10381851055951E+01 1.10381851055951E+01

-0.5109 6.12588689503621E+00 6.12588689503620E+00 6.12588689503617E+00

-0.3737 8.82850065480757E-01 8.82850065480740E-01 8.82850065480749E-01

-0.2278 -4.06487665677184E+00 -4.06487665677179E+00 -4.06487665677183E+00

-0.0765 -7.86743441239314E+00 -7.86743441239313E+00 -7.86743441239311E+00

0.0765 -9.85009422568689E+00 -9.85009422568686E+00 -9.85009422568686E+00

0.2278 -9.81486163370438E+00 -9.81486163370438E+00 -9.81486163370438E+00

0.3737 -8.10535878905558E+00 -8.10535878905560E+00 -8.10535878905559E+00

0.5109 -5.42044736384153E+00 -5.42044736384153E+00 -5.42044736384152E+00

0.6361 -2.50470104424188E+00 -2.50470104424189E+00 -2.50470104424189E+00

0.7463 1.14594810576479E-01 1.14594810576473E-01 1.14594810576490E-01

0.8391 2.25489829273077E+00 2.25489829273076E+00 2.25489829273079E+00

0.9122 4.13007085595155E+00 4.13007085595157E+00 4.13007085595156E+00

0.9640 6.73228110033772E+00 6.73228110033772E+00 6.73228110033771E+00

0.9931 1.81807044870749E+01 1.81807044870751E+01 1.81807044870748E+01

101

Table 4. 2: 14-node quadratures for t=-0.9862838086968120

𝑃𝑖

Barycentric rational

interpolation scheme Kolm and Rokhlins’ method

𝑊𝑖∗ 𝑊𝑖

∗

-0.9862838086968120 -113.0556007318071700 -113.0556007318075900

-0.9284348836635736 23.4363574230437730 23.4363574230444160

-0.8272013150697651 20.5297025668623970 20.5297025668613640

-0.6872929048116855 -13.7624046215441350 -13.7624046215422880

-0.5152486363581543 14.5515561694639360 14.5515561694615630

-0.3191123689278899 -11.2557672047745850 -11.2557672047714930

-0.1080549487073440 10.0571741702014350 10.0571741701980550

0.1080549487073438 -7.7749595174036301 -7.7749595174002240

0.3191123689278899 6.3823314060363003 6.3823314060329679

0.5152486363581543 -4.6269487646271443 -4.6269487646242613

0.6872929048116854 3.3657256472462649 3.3657256472440924

0.8272013150697650 -2.0459924078164597 -2.0459924078150067

0.9284348836635736 1.0830056717666725 1.0830056717659142

0.9862838086968125 -0.2941688960408579 -0.2941688960406454

As can be seen from Fig. 4.2, both the barycentric rational interpolation scheme and the

method by Kolm and Rokhlins have similar numerical accuracy when the number of

quadrature points used is concerned. This is evidenced by the fact that the modified

quadrature weights 𝑊𝑖∗ do not vary much between these two methods. For instance, the

𝑊𝑖∗ of the 14-node quadrature of both methods only differ by less than 11 decimal places

as shown in Table 4.2. At 𝑛 = 20, the precision of both methods starts to reach the

maximum and is unable to improve further beyond 10−15. This is in agreement with the

Matlab’s default setting of double precision floating point format. Results show that

numerical quadrature using rational interpolation scheme is an alternative to the spectral

method. When approximating integrand consisting of different order of singularities, we

anticipate that the barycentric rational interpolation scheme would have the similar

accuracy as to the method by Kolm and Rokhlins. To this end, it is noted that the above

numerical schemes are attractive only if the hypersingular integrals are to be evaluated

102

within the same interval [𝑎, 𝑏] and with the same singularity point 𝑡 so that the modified

quadrature weights can be recycled.

4.4 Complex-step derivative approximation

Although the above methods yield convergence result when polynomials of higher degree

are employed, the corresponding modified quadrature weights are not reusable and

therefore need to be re-evaluated when there is a change of bounds in the hypersingular

integral. In order to reduce the computational cost, further improvement can be made by

dropping the interpolation entirely and instead, resorted to finding the derivative of the

regular function numerically. In order to achieve machine precision like accuracy with

minimal computational effort for the derivative approximation, complex-step derivative

approximation [179] is employed in this section. Unlike the conventional finite difference

approach, the complex-step approach makes use of finite step on the complex axis while

avoiding difference operation and the subsequent catastrophic cancellation on the real

axis for small step size.

Let i and ℎ represent the imaginary unit and the complex step of a real and analytic

function 𝜑(𝑥 + iℎ). The complex-step derivative approximation is formulated by first

expanding the Taylor series of 𝜑(𝑥 + iℎ) around 𝑥,

0

ii

!

kk

k

xx h

k

h

(4.37)

The first derivative is then extracted and rearranged from the imaginary part of the above

expansion,

21Im ix h

x hh

(4.38)

where Im represents the operation of taking imaginary part.

103

Based on the above strategy, one can also construct a recursive Richardson extrapolation

to further increase the convergence rate for the complex-step derivative approximation.

For instance, the 𝑛th order of the first derivative approximation 𝜑(1)(𝑥) = 𝐷𝑘=𝑛,𝑙=𝑛(𝑥)

can be formulated as

,0

1

Im i Im i2 2

2

k k

k

k

h hx x

D xh

(4.39)

, 1 1, 1

, 1

4

4 1

l

k l k l

k l l

D x D xD x

(4.40)

where 𝑘, 𝑙 = 0,1,2, … , 𝑛

The hypersingular integral in the finite-part form of (4.8) can be efficiently evaluated

using equations (4.38) or (4.40). It is noted that the convergence rate for the 0th order

Richardson extrapolation in (4.39) is the same as the one in (4.38). For simplicity, the

numerical formulations of hypersingular integral using one complex-step derivative

approximation i.e. (4.38) are listed as follows,

For case 𝑎 < 𝑡 < 𝑏,

* *

2*

1

2

1 1

Im i

Im i ln

2

b

a

m j j

j

jj

xdx t

a t b tx t

t hP t P tt h b t b a hW

h t a P t

(4.41)

104

For the case 𝑡 = 𝑎,

2

* *

2*

1

Im iln

Im i

2

b

t

m j j

j

jj

t hx tdx b t

b t hx t

t hP t P t

b a hWP t

(4.42)

For the case 𝑡 = 𝑏,

2

* *

2*

1

Im iln

Im i

2

t

a

m j j

j

jj

t hx tdx t a

a t hx t

t hP t P t

b a hWP t

(4.43)

Again, the same benchmark example as in (4.33) is considered. The results for the mean

absolute percentage error (4.36) of hypersingular integral evaluation using complex-step

derivative approximation of various complex-step sizes and Richardson extrapolation

orders are plotted in Fig. 4.3.

105

Fig. 4. 3: Mean absolute percentage error of hypersingular integral evaluation using

complex-step derivative approximation of various complex-step sizes and Richardson

extrapolation orders

As can be seen from Fig. 4.3, the hypersingular integral evaluation using complex-step

derivative approximation has a very high convergence rate. For instance, the maximum

accuracy can be easily achieved by choosing the size ℎ be smaller than 10−7 in a one-

step approximation according to the 0th order Richardson extrapolation results. The

convergence rate also improves significantly with the use of higher order Richardson

extrapolations. In practice, the one-step approximation will suffice provided that the

decimal places of precision gained are higher than the respected complex-step size, i.e.

−log10ℎ. As this would imply there always exists a sufficiently small size of ℎ in a one-

step approximation capable of producing machine precision accuracy. Lastly, it should

be emphasised that by transforming the derivative finding to complex-step operation, the

hypersingular integral can be evaluated directly bypassing the interpolatory quadrature

106

which would otherwise require around 20 interpolatory points to achieve similar accuracy

as comparing to the complex-step derivative scheme of using just one complex-step.

4.5 Conclusions

In this chapter, the hypersingular line integral is accurately evaluated by the present

numerical scheme. The hypersingular integrals are first separated into regular and

singular parts, in which the singular integrals are defined as limits around the singularity

and their values determined analytically by taking the finite part values. The remaining

regular integrals can be evaluated using the barycentric rational interpolatory quadrature,

or the complex-step derivative approximation for the regular function when machine

precision like accuracy is required. It should be emphasised that the core novelty of the

line integrals treatment is the employment of the complex-step derivative approach to

approximate the derivative of the density function so that a near machine precision

accuracy is obtained. Numerical results show that the present method is accurate and

efficient for the evaluation of the hypersingular line integral, compared to the existing

numerical schemes.

107

Chapter 5 Evaluation of hypersingular surface integral

by complex-step derivative approximation

5.1 Introduction

In Chapter 4, the hypersingular line integral, as commonly used in two-dimensional

boundary integral formulation, is evaluated by the complex-step derivation

approximation. However, for three-dimensional problems, the implementation of the

BEM would often involve the computation of hypersingular surface integrals, which are

more complex than hypersingular line integrals. To the author’s best knowledge, there

are few numerical studies on the computation of hypersingular surface integrals. For

instance, in a seemingly less generalised approach, the hypersingular integrals can also

be analytically examined so as to have the singularity terms regularised per application.

This direct approach often leads to the more accurate result as demonstrated by Guiggiani

et al. [187] for evaluating the hypersingular surface integral for three-dimensional

potential problems, in which the Laurent series are employed to identify and regularise

the singularity terms. However, Guiggiani’s approach may not be easily implemented

when the underlying hypersingular kernels are of complicated forms. More recently, Gao

[188] and his co-workers [189] proposed the use of power series for regular function

approximation and for the regularisation of the singularity terms. Apart from the error

due to the numerical integration, even though the approximation using the power series

may converge, Gao’s method suffers some truncation error due to the power series,

limiting its application in high precision applications.

108

Herein, our method of evaluating hypersingular integrals should serve three purposes.

Firstly, the solution’s accuracy should be comparable to machine precision. Secondly, the

method should be straight forward and generalised enough for fast numerical

implementation with minimal computational effort and is comparable to the interpolatory

quadrature method. Lastly, the method should be robust and flexible enough to handle

hypersingular surface integrals which have applications in many fundamental solution-

based numerical methods.

In this chapter, the method proposed in Chapter 4 is generalised for evaluating

hypersingular surface integrals, in which coordinate transformations are performed such

that the singularity and the finite part can be methodically identified and regularised. The

resulting inner integral containing the singularity will have a form similar to the

hypersingular line integral which can now be evaluated efficiently. The remaining

integration task on the outer integral can be simply solved. Numerical results of the

proposed method using 8-node rectangular boundary elements and 6-node triangular

elements are then compared to the reference results produced by Guiggiani et al [187]

and Gao [188].

5.2 Definition of hypersingular surface integral

Let us define hypersingular surface integral

,S

I f dS P Q Qx x x (5.1)

with the hypersingular integrand

3

,,

,f

r

P Q

P Q

P Q

x xx x

x x (5.2)

109

where 𝜓 is the density function of the hypersingular surface integral and 𝑟 is the

Euclidean distance between the source point 𝒙𝑷 and the arbitrary integration point 𝒙𝑸.

As we know, hypersingular surface integrals, usually arisen from the use of fundamental

solution as trial function in boundary integral equations, is first formulated by subtracting

a vanishing exclusion zone around the singularity point 𝒙𝑷,

3 3 0 30

, , ,lim lim

, , ,S S S CdS dS dS

r r r

P Q P Q P Q

Q Q Q

P Q P Q P Q

x x x x x xx x x

x x x x x x (5.3)

In Eq. (5.3), 𝑆𝜖 is the exclusion surface around the singularity point 𝒙𝑷 with vanishing

neighbourhood ϵ and 𝐶𝜖 is the region of spherical surface replacing 𝑆𝜖 around the

vanishing neighbourhood of 𝒙𝑷 as shown in Fig. 5.1.

Fig. 5. 1: Exclusion surface around a singularity point at corner

If the singularity point 𝒙𝑷 is a corner point with discontinuous outward normal, the last

integral of (5.3) will give rise to additional free terms and the coefficients of which can

110

be evaluated analytically or numerically by integrating the vanishing region 𝐶𝜖 as was

demonstrated in [190, 191]. The task is then simplified to evaluating the finite part of the

hypersingular surface integral 𝐼,

3

,

,SI dS

r

P Q

Q

P Q

x xx

x x (5.4)

In analogy to the regularisation process for hypersingular line integral (4.7), the

hypersingular surface integral can also be regularised as follows,

23

3

,1 1,

, ,

, , , ,

,

S S

S

I dS dSrr r

r rdS

r

P P

P P Q Q

P Q P Q

P Q P P P Q P P

Q

P Q

x xx x x x

x x x x

x x x x x x x xx

x x

(5.5)

The above formulation is difficult to materialise for two reasons: firstly, the

differentiation of the density function 𝜓 with respect to 𝑟 needs to be analytically derived

or numerically approximated which may not be a straightforward process, given that 𝑟

needs not be an explicit variable of 𝜓. Secondly, the principal value integrals in (5.5)

would require proper coordinate transformation such that it is aligned with 𝑟 before

taking the finite part value. We will resolve these issues by employing polar coordinate

transformation onto the local orthogonal curvilinear coordinate of the discretised surface

in boundary element setting. Then, the complex-step derivative scheme of approximating

hypersingular line integral as was discussed previously in (4.42) is employed to evaluate

the hypersingular surface integral now equipped with compatible integration variable and

the respected finite part value.

111

5.3 Regularisation and numerical procedures of hypersingular

surface integrals

5.3.1 Surface discretisation

If we discretise the surface of integration in (5.4) by boundary elements, the global

Cartesian coordinate 𝑥𝑖=1,2,3 can then be interpolated by element’s nodal points 𝑖𝑘 using

shape functions 𝑁𝑘 of local orthogonal curvilinear coordinate 𝜉𝑗=1,2 in two dimensions,

1 2

1

ˆ, i

nk

i k

k

x N x

(5.6)

where 𝑘 and 𝑛 are the nodal index and the total number of nodal points in an element

respectively.

For surface area represented by an eight-node rectangular element as illustrated in Fig.

5.2, the shape functions 𝑁𝑘 and their derivatives 𝜕𝑁𝑘/𝜕𝜉𝑗 can be expressed as

1 2 1 1 2 2 1 1 2 2ˆ ˆ ˆ ˆ 1

1, 1

41k k k k

kN for 1,4k (5.7)

2 2

1 2 1 1 2 2 1 2 2 1ˆ ˆ ˆ ˆ1

1,

21k k k k

kN

for 5,8k (5.8)

1 2

1 1 1 2 2 2 2

1

ˆ ˆ, 12 ˆ 1

4ˆk k k k k

N

for 1,4k (5.9)

1 2

2 1 1 2

2

12 1ˆ ˆ ˆ ˆ2 1

, 1

4

k k k k kN

for 1,4k (5.10)

2 2 21 2

1 1 2 2 1 2 1 1 1 2 2

1

ˆ ˆ ˆ ˆ ˆ ˆ1, 1

21

k k k k k k kN

for 5,8k (5.11)

112

2 2 21 2

2 1 2 2 1 1 2 1 1 2 2

2

ˆ ˆ ˆ ˆ ˆ ˆ1 1, 1

2

k k k k k k kN

for 5,8k (5.12)

where 𝜉𝑙=1,2𝑘 are the nodal points expressed in the local coordinate 𝜉𝑗 of the element.

Fig. 5. 2: Local orthogonal curvilinear coordinates of eight-node plate element and its

nodal points’ distribution

In the case of a six-node triangular element as shown in Fig. 5.3, the orthogonal

curvilinear coordinate system plays a central role in simplifying the Jacobian for the later

polar coordinate transformation. The corresponding shape functions and their derivatives

can be expressed in the followings [192],

1 2 1 11 2 2

1,

63 3 3N (5.13)

1 2 1 2 212

1,

63 3 3N (5.14)

23 21 2

1, 2 3

3N (5.15)

𝜉2

𝜉1 (0,0)

2 1 5

6

3 7 4

8 𝜉1 = −1 𝜉1 = 1

𝜉2 = 1

𝜉2 = −1

113

14 21 22 131

33 3, 3N (5.16)

15 21 2 2 3 32

,3

N (5.17)

16 1 2 2 23 32

,3

N (5.18)

1

1

1 2

1 2

3 1

3 2

,N

(5.19)

1

1

2 2

1 2

3 1

3 2

,N

(5.20)

1

1

3 2,0

N

(5.21)

1 2

1

4

1

,2

N

(5.22)

1 2

2

5

1

, 2 3

3

N

(5.23)

2

1

6 1

2

, 2 3

3

N

(5.24)

1 21

2

1 2

1 33

3 2

,N

(5.25)

1 2

2

1

2

2

1 33

3 2

,N

(5.26)

1 23

2

2

14 3

3

,N

(5.27)

24

2

1

2

, 23

3

N

(5.28)

15

1

2

2

2

23 2 3

,

3

N

(5.29)

114

2

2

6

1

1

2

23 3

3

,2

N

(5.30)

Fig. 5. 3: Local orthogonal curvilinear coordinates of six-node triangular element and its

nodal points’ distribution

The surface integral of (5.4) using the above local coordinate transformation becomes

1 1

1 21 1S

dS J d d

Qx ξ (5.31)

1 2 3

1 2 1 2

1 2 1 1 2 1 2 1 2 1 3 1 2 1

1 2

1 1 2 2 2 1 2 2 3 1 2 2

, ,, , / , / , /

, / , / , /

e e e

J x x x

x x x

Q Qx x

(5.32)

115

where 𝐽𝜉(𝜉1, 𝜉2) is the Jacobian for the transformation from the global coordinate to the

local coordinate and 𝑒1, 𝑒2, 𝑒3 is a set of basis vectors in the Cartesian coordinate. The

term 𝜕𝑥𝑖(𝝃)/𝜕𝜉𝑗 are computed by differentiating Eq. (5.6) and substitution from Eq. (5.9)

-(5.12) for the 8-noded rectangular element or substitution from Eq. (5.19)-(5.30) for the

6-noded triangular element.

5.3.2 Polar coordinate transformation

To evaluate the surface integral consisting of radial singularity 𝑟3(𝝃𝑷, 𝝃) , polar

coordinate transformation around the singularity point 𝝃𝑷

1 1

2 2

, , cos

, , sin

P

P

P

P

ξ

ξ (5.33)

is often performed. The surface integration can then be expressed as

2π ,

0 0, ,

L

SdS J J d d

Pξ

P

Qx ξ ξ (5.34)

where 𝜌 and 𝜃 are the radial distance and polar angle, 𝐽𝜌(𝜌) = 𝜌 is the corresponding

Jacobian for the transformation from the local coordinate to the polar coordinate.

𝜌𝐿(𝝃𝑷, 𝜃) is the path of radial integration depending on the singularity point and the value

of the polar angle. 𝐽𝜉(𝝃𝑷, 𝝃(𝝃𝑷, 𝜌, 𝜃)) is numerically computed by directly substituting

(5.33) into (5.9)-(5.12) then into (5.32) using the differential form of (5.6).

For the eight-node rectangular element, 𝜃 and 𝜌𝐿(𝝃𝑷, 𝜃) are partitioned into four

subregions as illustrated in Fig. 5.4 while for the six-node triangular element, they are

partitioned into three subregions as illustrated in Fig. 5.5.

116

Fig. 5. 4: Regions of integration for the eight-node plate element in polar coordinate

Fig. 5. 5: Regions of integration for the 6-node triangular element in polar coordinate

The surface integration then becomes

,

01

, ,R

l L

a

b

l

m

Sl

dS J J d d

P P

P

ξ ξP

Qξ

x ξ ξ (5.35)

117

where 𝑚𝑅 is the total number of regions with respect to the choice of the boundary

element. 𝜃𝑙𝑎(𝝃𝑷) and 𝜃𝑙

𝑏(𝝃𝑷) are respectively the lower and upper bounds of the outer

integration for each partitioned region 𝑙. For the eight-node rectangular element as shown

in Fig. 5.4, they can be expressed as

1 1 1 2

2 2 2 3

3 3 3 4

4 4 4 1

,

2π

a b

a b

a b

a b

P P P P

P P P P

P P P P

P P P P

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

(5.36)

with

1 21

1

1tan

1

P

P

Pξ , 1 22

1

1π tan

1

P

P

Pξ

1 23

1

1π tan

1

P

P

Pξ , 1 24

1

12π tan

1

P

P

Pξ (5.37)

21 2

12 3

23 4

14 1

1 for

sin

1 for

cos

1 for

sin

1 for 2π

c

,

os

P

P

L P

P

Pξ (5.38)

For example, the radial path 𝜌𝐿(𝝃𝑷, 𝜃) when the singularity point is located at 𝝃𝑷 = (0,0)

is shown in Fig. 5.6.

118

Fig. 5. 6: Radial path ρL((0,0), θ) for singularity point located at the centre of the eight-

node plate element

For the six-node triangular element, 𝜃 and 𝜌𝐿(𝝃𝑷, 𝜃) are partitioned into three subregions

as shown in Fig. 5.5. The lower and upper bounds of 𝜃 in the partitioned regions are

defined as

1 1 1 2

2 2 2 3

3 3 3 1

,

2π

a b

a b

a b

P P P P

P P P P

P P P P

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

(5.39)

where

2

2

2

1 11

2

1

1

2

1

1

32

1

cos

3

π tan1

2π tan1

P

P P

P

P

P

P

P

P

P

ξ

ξ

ξ

(5.40)

The upper bounds of the inner integration 𝜌𝐿(𝝃𝑷, 𝜃) are derived using the point-line

perpendicular distance formula and simple trigonometric functions,

119

21

1 2

1

3

22 3

2

1

3 3for

5π2cos

6

, for sin

3 3for

π2 os

6

2π

c

P P

P

L

P P

P P

P P P

P P

ξ ξ

ξ ξ ξ

ξ ξ

(5.41)

5.3.3 Asymptotic expansions of r

Apart from the transformation for the surface integral and its regular integrand, the

singular part, i.e. the Euclidean distance 𝑟 of the integrand also needs to be transformed

accordingly. The Taylor expansion of the Euclidean distance projected on the Cartesian

axes 𝑥𝑖(𝝃𝑷, 𝝃(𝝃𝑷, 𝜌, 𝜃)) about the singularity point 𝑥𝑖

𝑃 is,

2, , , , ,i i i ir A B C

P P P Pξ ξ ξ ξ (5.42)

where

, , , , , P

i i ir x x P P Pξ ξ ξ ξ (5.43)

1 2

, cos sini ii

x xA

P P

P

ξ ξ

ξ (5.44)

2 2 2

2 2

2 2

1 1 2 2

1 1, cos cos sin sin

2 2

i i ii

x x xB

P P P

P

ξ ξ ξ

ξ (5.45)

3 3

2 2

2 2

1 2 1 2

1 1, cos sin cos sin

2 2

i ii

x xC

P P

P

ξ ξ

ξ (5.46)

After extracting the radial distance 𝜌 term which represents the singularity order, we have

the following simplified form of 𝑟(𝝃𝑷, 𝜌, 𝜃) comprising the singular part 𝜌 and the

regular part 𝜌𝑅(𝝃𝑷, 𝜌, 𝜃),

120

, , , ,Rr P Pξ ξ (5.47)

where

3 2

2

1

, , , , ,R i i i

i

A B C

P P P Pξ ξ ξ ξ (5.48)

5.3.4 Asymptotic expansions of density function ψ

Owing to the fact that the density function 𝜓 also depends on 𝜌 and 𝜃 after the coordinate

transformations, we seek to derive the asymptotic expansion of 𝜓(𝝃𝑷, 𝝃(𝝃𝑷, 𝜌, 𝜃))

subjected to the same transformation process of 𝑟(𝝃𝑷, 𝜌, 𝜃). For fundamental solution

based hypersingular integrand, it will suffice to deal with spatial variables of 𝜓. The most

commonly seen of such variables include the already discussed components of 𝑟 with

respect to the Cartesian direction, that is 𝑟𝑖 from (5.42). The other useful variables include

the unit outward normal 𝒏, the spatial derivative of the Euclidean distance 𝜕𝑟

𝜕𝑥𝑖 and its

normal derivative 𝜕𝑟

𝜕𝑛. They can be easily evaluated as follows,

1 2, , / , , /

, , ,, , ,J

P P

Q QP P

P P

x ξ ξ x ξ ξn ξ ξ ξ

ξ ξ ξ (5.49)

2

3 22

1

, , , , , , ,

, ,, , ,

i i i i

i

i i i

i

r r A B C

x rA B C

P P P P P

P

P P P

ξ ξ ξ ξ ξ

ξξ ξ ξ

(5.50)

3

1

, , , , ,, , ,i

i i

r rn

n x

P P P

P Pξ ξ ξ ξ

ξ ξ ξ (5.51)

121

5.3.5 Proposed formulation of hypersingular surface integral

We now have acquired all necessary tools to further progress the hypersingular surface

integral from (5.4). Applying firstly the local coordinate transformation to the

hypersingular integral (5.4) yields,

1 2

1

1 3

1

1

,

,J d d

rI

P

P

ξ ξξ

ξ ξ (5.52)

Then, after applying the polar coordinate transformation, we have

2π ,

0 0 2

, , ,L

I d d

P

P Pξ ξ ξ ξ

(5.53)

where

3

, , , , , ,, ,

, ,R

J

P P P P

P

P

ξ ξ ξ ξ ξ ξξ

ξ (5.54)

The equation (5.54) is significant in the sense that all the regular parts of the hypersingular

integrand (5.4) have been grouped together and expressed in a form depending on 𝜌

where the finite part value is based upon. As the result, the inner integral of (5.53) can be

perceived and treated as the hypersingular line integral as in (4.1). Finally, by substituting

(4.8) into (5.53), we obtain the following proposed formulation for the generalised

evaluation of hypersingular surface integral,

, ,

0 2

2

02π

0

,

0

, ,0,1 1, ,0,

, , , , ,0, , ,0,

L L

L

d d

I d

d

P P

P

P Pξ ξ

P P

P P P P P Pξ

ξ ξ ξξ ξ ξ

ξ ξ ξ ξ ξ ξ ξ ξ ξ

(5.55)

122

Note that this equation is essentially the equivalent representation form of (5.5). The

remaining task is to obtain an explicit formulation using the previously proposed

complex-step derivative scheme (4.41)-(4.43) for efficient numerical implementation.

To obtain the numerical formula for the hypersingular surface integral, we first apply

(5.36) to partition the area of integration in (5.53),

,

01

21

, , ,R Rl L

al

bm m

l

l l

I d d I

P P

P

P Pξ ξ

ξ

ξ ξ ξ (5.56)

where 𝐼𝑙 are the hypersingular surface integrals of each partitioned regions.

To perform outer integration in (5.56) using Gaussian quadrature, a change of interval for

the normalisation of 𝜃 is required,

,

20

, , ,l L

l

b

alI d d

P P

P

P Pξ ξ

ξ

ξ ξ ξ

2

*

1 ,

1 0

, , , ,

2

L lb

l l

a

d d

PP P PP P

ξ ξ ξ ξ ξξ ξ (5.57)

where

* ,2 2

l l

b a a b

l l

l

P P P P

Pξ ξ ξ ξ

ξ (5.58)

Further, let 𝑃𝑘𝑜𝑢𝑡 and 𝑊𝑘

𝑜𝑢𝑡 be the points and weights of the Gaussian quadrature with

𝑚𝑜𝑢𝑡 be the total number of points for the outer integral, then

*

,

0 21

, , , ,

2

out

L

outm

l kl l out

a

l

k

b

k

PI W d

P

P P PP Pξ ξ ξ ξ ξξ ξ

(5.59)

After employing the formerly proposed complex-step derivative scheme from (4.42) for

the inner hypersingular line integral evaluation, the numerical formulation for the

hypersingular surface integral 𝐼𝑙 is,

123

* *

1

* *

*

* * *

1

, ,0, , / , ,2

Im , , , , ln , , /

, ,, , , , , ,0, ,

2

Im , ,

ˆ

,

outml l out out out

l k l k L l k

k

out out

l k L l k

outm

L l k out out

j j l l k

j

b

k

a

I W P P

ih P P h

PW P P P

ih

P P

P P P P P

P P P P P

P P

P P P P P P

P P

ξ ξξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

ξ ξξ ξ ξ ξ ξ ξ ξ ξ

ξ ξ ξ * * *2, / /out

l k j jP P h P

Pξ

(5.60)

where

* *

* *, , , ,

, ,2 2

out out

L l k L l kout

j l k j

P PP P P

P P P P

P Pξ ξ ξ ξ

ξ ξ (5.61)

This completes the numerical implementation of hypersingular surface integral using the

proposed method.

To elucidate thoroughly on how the algorithms of the proposed method works, we provide

the following pseudocode for numerically evaluating the hypersingular surface integral 𝐼.

124

Choose Pξ

Compute 1 2( ), ( ), , ( )Rm P P P

ξ ξ ξ from (5.37) or (5.40)

Loop for 𝑙 = 1,2, … ,𝑚𝑅

Loop for 1,2,..., outk m

Compute * ,l Pξ for

out

kP from (5.58)

Compute ,L Pξ according to the range of for

* , out

l kP Pξ from (5.38) or (5.41)

Compute * ,, ,0, out

l kP P P Pξ ξ ξ ξ and

*, , , , out

l kih P P P Pξ ξ ξ ξ by using Table 5.1.

Loop for 𝑗 = 1,2, … ,𝑚

Compute * ,jP Pξ for * , out

l kP Pξ from (5.61)

Compute * *, , , , out

j l kP P P P Pξ ξ ξ ξ by using Table

5.1 with changed substitutions: * *, , out

j l kP P Pξ

End loop 𝑗

End loop 𝑘

Compute 𝐼𝑙

End loop 𝑙

Compute 𝐼 from (5.56)

125

Table 5. 1: Variables and their spatial quantities for evaluating regular integrand

, , , P Pξ ξ ξ

Substitutions *0, , out

l kP Pξ *, , out

l kih P Pξ

1,2,3 , ,ir

Pξ 0 From (5.42)

2 , ,p

R Pξ From (5.48)

, ,r Pξ 0 From (5.47) and with R

, , / ir x Pξ From (5.50)

, , Pξ ξ P

= ξ From (5.33)

, , ,J P Pξ ξ ξ From (5.6), (5.32), (5.9)-(5.12) and with ξ

, , , P Pn ξ ξ ξ From (5.6), (5.49), (5.9)-(5.12) and with ξ

, , , /r n P Pξ ξ ξ From (5.51) and with / ir x and n

, , , P Pξ ξ ξ Use all variables from above and from (5.54)

5.4 Numerical examples

The hypersingular surface integrals in boundary element type methods arise when

accurate information is needed at the surface such as boundary stress. As mentioned in

Eq. (1.26), the boundary stress can be directly evaluated through the integration of the

third order tensors 𝑆𝑘𝑖𝑗∗ and 𝐷𝑘𝑖𝑗

∗ which are hypersingular and strongly singular

respectively [51]. Herein, numerical experiments concerning the integrations of a single

126

boundary element with benchmark singularity points are carried out to demonstrate the

proposed integration scheme in the BEM setting. The proposed method of hypersingular

surface integral evaluation is applied to three benchmark examples so as to demonstrate

its accuracy, efficiency and robustness for solving distinct complex problems. The

examples include: (1) a trapezium meshed with distorted 8-node rectangular elements, (2)

a rectangle meshed with the regular 8-node rectangular elements comparing to the

distorted 6-node triangular elements of various degrees, and (3) a quarter cylindrical panel

represented by curved 8-node rectangular element. Similar to the numerical example for

hypersingular line integral, the computations are performed in Matlab with the default

double precision floating point format. Since our previous study on line integral showed

that further reduction on the size of the complex-step won’t affect the numerical accuracy

once the result has reached the machine precision, a complex-step size of 10−12 will be

employed throughout the simulations.

5.4.1 Example 1A: Trapezium meshed with a distorted rectangular element

In this first example, distortion effects on the 8-node rectangular element is studied for

the hypersingular surface integral of the following form

,3 33

13

S

rr n

nI dS

r

(5.62)

where the surface of integration is over a trapezium region with corner points located at

−1,−1,0, 1.5, −1,0, 0.5,1,0, −1,1,0 as shown in Fig. 5.7. Three singularity points

namely, 𝑎(0,0) , 𝑏(0.66,0) and 𝑐(0.66,0.66) are placed with respect to the local

orthogonal curvilinear coordinate 𝝃. In order to cast the hypersingular surface integral

into our proposed formulation of (5.53), we have the following modified density function

𝜑 from (5.54),

127

,3 3 3, 3,

R

rr

Jn

n

Pξ (5.63)

We employ a maximum of 30-point Gaussian quadrature for the hypersingular surface

integral evaluation at three of the previously discussed test points. The corresponding

integral values are compared to the reference results by Guiggiani et al [187]. Our

numerical results in Table 5.2 show the convergences of the hypersingular integrals

values at the test locations with respect to the increasing number of Gaussian quadrature

orders of up to 20 units. As comparing to the results by Guiggiani for up to 10-point

Gaussian quadrature, both methods show almost identical results in this particular

example. But beyond the Gaussian quadrature order of 10, our results exemplify further

that if the location of the singularity is well away from the element’s boundary such as

the test point 𝑎, the proposed method is highly efficient in obtaining convergence result

of 7 significant figures with the use of only 10 Gaussian integration points on a distorted

8-node rectangular element. The convergence will be slightly degraded when the

singularity point is close to the element’s boundary. For instance, the test points 𝑏 and 𝑐

will require 4 and 6 higher orders of Gaussian quadrature respectively to achieve results

in 7 significant figures. Overall, the present method is able to provide highly accurate and

stable results with exponential convergence for economically viable number of Gaussian

integration points. To further demonstrate the capability of our algorithm and to illustrate

the continuity of the hypersingular surface integral, we also provide a contour plot of the

hypersingular surface integral with varying singularities inside the enclosed region by the

test points, that is, the grey area as indicated in Fig. 5.7. As shown in Fig. 5.8, this contour

plot displays an expected smooth transition of the integral values subjecting to the

distorted geometric boundaries.

128

Fig. 5. 7: Schematic of trapezium meshed with an 8-node rectangular element and with

various singularity points a, b and c

129

Table 5. 2: Numerical evaluation of hypersingular surface integrals on trapezium

meshed with a distorted 8-node rectangular element at various test locations with

respect to the different orders of Gaussian quadrature

𝑚 Point a Point b Point c

Reference

[187]

Proposed Reference Proposed Reference Proposed

4 -5.749091 -5.749091 -9.222214 -9.222141 -15.72221 -15.72221

6 -5.749244 -5.749244 -9.157439 -9.157439 -15.30541 -15.30541

8 -5.749236 -5.749236 -9.154546 -9.154546 -15.31768 -15.31768

10 -5.749237 -5.749237 -9.154525 -9.154525 -15.32806 -15.32806

12 N/A -5.749237 N/A -9.154587 N/A -15.32887

14 N/A -5.749237 N/A -9.154586 N/A -15.32850

16 N/A -5.749237 N/A -9.154585 N/A -15.32849

18 N/A -5.749237 N/A -9.154585 N/A -15.32850

20 N/A -5.749237 N/A -9.154585 N/A -15.32850

exact -5.749237 -9.154585 -15.32850

130

Fig. 5. 8: Contour plot of the hypersingular singular integrals with varying singularity

points inside the enclosed region by test points a, b and c

5.4.2 Example 1B: Improved meshing of trapezium using rectangular elements

To further improve the convergence of the hypersingular surface integral, the trapezium

is divided at the test point 𝑎(0,0) into 4 regions consisting of 2 non-distorted elements

and 2 distorted elements as shown in Fig. 5.9. As discussed previously in (5.3), there is

no difficulty in evaluating the hypersingular surface integral at geometric corner with the

addition of free terms. It is further noted that since the corner point 𝑎 has a continuous

outward normal across all 4 divided elements, the free terms can subsequently be dropped

when evaluating the integral value. The numerical results employing the improved

meshing are presented in Table 5.3. Comparing to the results in example 1a, the results

with improved meshing show better convergence with reduced order of Gaussian

quadrature. To achieve the same numerical accuracy of 7 significant figures, only 6

Gaussian integration points are required in contrast to the 10 Gaussian points in the

131

previous example. To put the numerical performance into perspective, we compute the

relative errors with respect to the order of Gaussian quadrature in both examples. The

results in Fig. 5.10 show that the improved meshing scheme is able to provide

significantly faster convergence rate until it reaches and settles on the machine precision.

It is in line with Table 5.3 that the precision limit of 13 significant figures can be

efficiently achieved by using a mere of 12 Gaussian integration points.

Fig. 5. 9: Trapezium meshed with four 8-node rectangular element

132

Table 5. 3: Numerical evaluation of hypersingular surface integral on trapezium at test

point a with improved meshing using 4 rectangular elements

𝑚 Point a

4 -5.749136555405

6 -5.749236707179

8 -5.749236751240

10 -5.749236751229

12 -5.749236751228

14 -5.749236751228

16 -5.749236751228

18 -5.749236751228

20 -5.749236751228

133

Fig. 5. 10: Relative errors of hypersingular surface integral with singularity point a in

Examples 1A and 1B

5.4.3 Example 2A: Rectangle meshed with regular rectangular elements

In this example, the effect of the singularity point on the accuracy of the numerical results,

when close to the boundary, is studied. The hypersingular surface integral has the

following form,

,1

3SI dS

r

r (5.64)

where the surface of integration is over a rectangular region with corner points at

−1,0,0, 1,0,0, 1,4,0, −1,4,0 as shown in Fig. 5.11. Again, three test points of

𝑎(0.5, −0.5) , 𝑏(0.75,−0.5) and 𝑐(0.99,−0.5) are chosen for hypersingular surface

integral evaluation using the following modified density function 𝜑,

,1

3, ,

R

r J

P

ξ (5.65)

134

For this numerical computation, the integration surface is divided into an element with

hypersingularity and a regular element which can be evaluated using normal integration

procedure without any special treatment. Gaussian quadrature order of up to 52 points is

employed in this study. For instance, the distribution of 4-point Gaussian quadrature in

the triangular integration subregions (cf. (5.36)) is illustrated in Fig. 5.11. The

hypersingular surface integral values at the three test points are compared to the reference

results by Gao [188]. The numerical results in Table 5.4 show that the proximity of the

singularity point to the element’s boundary doesn’t seem to have a noticeable impact on

the convergence of the evaluated results as comparing to the distorted element in the

previous example. The computational cost of attaining 12 significant figures accuracy are

shown to be at 16-point Gaussian quadrature. Indeed, the present method demonstrates

an exponential convergence as shown in Fig. 5.13, in which the current example is

labelled as case I. These results also outperform the recent power series method by Gao

[188] which indicates an accuracy of 5 significant figures at best.

135

Fig. 5. 11: On the left: Schematic of the rectangle example with test points a, b and c.

On the right: Distribution of Gaussian quadrature points inside an 8-node rectangular

element

136

Table 5. 4: Numerical evaluation of hypersingular surface integrals on rectangle meshed

with two 8-node rectangular elements at test points a, b and c

𝑚 Point a Point b Point c

4 -1.94777472320 -5.17913149206 -155.889540666

6 -1.94776345248 -5.18070780168 -156.048370304

8 -1.94774656956 -5.18069814188 -156.048470319

10 -1.94774594840 -5.18069754252 -156.048470573

12 -1.94774592732 -5.18069752027 -156.048470580

14 -1.94774592674 -5.18069751929 -156.048470581

16 -1.94774592673 -5.18069751926 -156.048470581

18 -1.94774592673 -5.18069751926 -156.048470581

20 -1.94774592673 -5.18069751926 -156.048470581

22 -1.94774592673 -5.18069751926 -156.048470581

24 -1.94774592673 -5.18069751926 -156.048470581

26 -1.94774592673 -5.18069751926 -156.048470581

28 -1.94774592673 -5.18069751926 -156.048470581

30 -1.94774592673 -5.18069751926 -156.048470581

Exact -1.94774592673 -5.18069751926 -156.048470581

Gao

[188]

-1.94782 -5.18065 -156.048

137

5.4.4 Example 2B: Rectangle meshed with distorted triangular elements of various

degrees

Based on the former Example 2A, the viability of employing 6-node triangular elements

to evaluate hypersingular surface integral and their distortion effects are studied. It is

noted that the special treatment using the proposed method is only applied to the

triangular elements containing the singularity point. The distribution of Gaussian point

inside these elements will have an implication on the result accuracy. To examine the

progressive impacts of the degree of distortion, we first mesh the rectangle with regular

6-node triangular elements and compute its integral value at test point 𝑎 for later

comparison. The aspect ratio (AR) of this meshing is 1.4142 and is labelled as case II in

our numerical simulation. In case III, the same integral is subjected to distorted meshing

where the elements division is at the centre of the rectangle, of which the aspect ratio is

26.5% higher than the case II meshing. It is noted that the singularity point is now located

at edges between the two triangular elements on the lower right where the hypersingular

integral is applied to. In case IV, a higher order of distortion with an aspect ratio of 2.3851

is examined. The division point is placed at the halfway between the rectangle’s centre

and the test point 𝑎. In case V, the elements division is at the test point 𝑎 such that the

proposed hypersingular integral treatment will need to be applied to all 4 triangular

elements, of which the highest aspect ratio is 3.5777. The schematics of case II to case V

along with their Gaussian quadrature distributions corresponding to the proposed

hypersingular integral treatment are illustrated in Fig. 5.12.

The numerical results in Table 5.5 and the respected relative errors in Fig. 5.13 show that

elements with lower degree of distortion, which can be measured by the magnitude of the

aspect ratio, are consistently able to provide better convergences. For instance, to obtain

7 significant figures of accuracy, a 10-point Gaussian quadrature would be required for

138

the regular rectangular element in case I, a 16-point Gaussian quadrature for the highly

regular triangular element in case II, or a 34-point Gaussian quadrature for the highly

distorted triangular element in case V. Generally, the 8-node rectangular element enjoys

a better convergence than the 6-node triangular element due to the higher degree of

freedom its polynomial interpolation imposes on. Furthermore, the number of integration

subregions resulting from the polar coordinate transformation is 4 for the rectangular

element as comparing to 3 for the triangular element. The implication of this is the

increased number of integration points which further contributes to the stronger result by

the rectangular element. Overall, both the rectangular element, the highly regular and the

highly distorted triangular elements using the present method are able to produce accurate

results that converge to near machine precision as illustrated in Fig. 5.13.

139

Fig. 5. 12: Upper left: Case II rectangle subdivision using 6-node triangular element

with aspect ratio of 1.4142; Upper right: Case III rectangular subdivision using

triangular element with aspect ratio of 1.7889; Lower left: Case IV rectangular

subdivision using triangular element with aspect ratio of 2.3851; Lower right: Case V

rectangular subdivision using triangular element with aspect ratio of 3.5777

140

Table 5. 5: Numerical evaluation of hypersingular surface integrals using rectangular

elements and triangular elements with various aspect ratios

𝑚 Case I

AR=1

Case II

AR=1.4142

Case III

AR=1.7889

Case IV

AR=2.3851

Case V

AR=3.5777

4 -1.947775 -1.842528 -2.813580 -2.533568 -1.284457

7 -1.947749 -1.946756 -1.962373 -2.140702 -1.752741

10 -1.947746 -1.947638 -1.932850 -1.987446 -1.892835

13 -1.947746 -1.947748 -1.946350 -1.953851 -1.933926

16 -1.947746 -1.947746 -1.947857 -1.948453 -1.944561

19 -1.947746 -1.947746 -1.947773 -1.947794 -1.947059

22 -1.947746 -1.947746 -1.947746 -1.947743 -1.947605

25 -1.947746 -1.947746 -1.947746 -1.947744 -1.947718

28 -1.947746 -1.947746 -1.947746 -1.947745 -1.947741

31 -1.947746 -1.947746 -1.947746 -1.947746 -1.947745

34 -1.947746 -1.947746 -1.947746 -1.947746 -1.947746

37 -1.947746 -1.947746 -1.947746 -1.947746 -1.947746

40 -1.947746 -1.947746 -1.947746 -1.947746 -1.947746

Exact -1.947746

141

Fig. 5. 13: Relative errors of hypersingular surface integrals using rectangular elements

and triangular elements with various aspect ratios

5.4.5 Example 3: Quarter cylindrical panel with curved rectangular element

In this final example, the performance of using the curved rectangular element is studied.

The hypersingular surface integral has form similar to Example 1,

,3 33

1

43

SI d

rr n

nS

r

(5.66)

where the surface of integration is over a quarter cylindrical panel of radius 1 unit by

length 2 units with corner points located at 1,0,0, 1,2,0, 0,2,1, 0,0,1 as shown in

Fig. 5.14. Three test points namely, 𝑎(0,0) , 𝑏(0.66,0) and 𝑐(0.66,0.66) of local

orthogonal curvilinear coordinate 𝝃 are chosen for evaluations. The modified density

function 𝜑 of the present method has the following form,

,3 3 3

1, ,

43

R

rr n

n

J

Pξ (5.67)

142

Gaussian quadrature of up to 30 points are employed for evaluating the hypersingular

surface integrals. The corresponding integral values of each test point are compared to

the reference results by Guiggiani et al [187] and Feng et al [189]. As similar to the

previous examples, our numerical results in Table 5.6 show that the present method has

convergence results of up to 12 significant figures. While the convergence rate will

slightly deteriorate when the singularity point approaches to the element’s boundary, e.g.

at test point c, it is nevertheless still very economical to obtain 8 significant figures

accuracy with only 14-point Gaussian quadrature, or the near machine precision of 12

significant figures for 22-point Gaussian quadrature in our worst case scenario. To give a

more substantial picture on the competitiveness of our proposed method, we list our

convergence results of each test point in 12 significant figures and directly compare them

to the reference results [187, 189] in Table 5.7. If we take our convergence results of 12

significant figures as reference value, then the results of the Guiggiani’s method are

deviated from that by 10−6 unit while the results of the Feng’s method are deviated from

10−4 to 10−6 units. Moreover, Fig. 5.15 shows that the proposed method features

exponential decay behaviour for the relative error and a faster convergence as comparing

to the method by Guiggiani. Finally, we also provide a contour plot of the hypersingular

surface integral with continuing singularities inside the enclosed region by the test points

as indicated by the grey area in Fig. 5.14. As similar to our first example, the contour plot

in Fig. 5.16 provides the expected smooth transition of the integral values subjecting to

the curved geometric boundaries.

143

Fig. 5. 14: Schematic of a curved boundary element representing the quarter cylindrical

panel

144

Table 5. 6: Convergence rate of the proposed method and the literature [170] at various

singularity points for increasing number of Gaussian quadrature points

𝑚

Point a Point b Point c

Reference

[187] Proposed

Reference

[187] Proposed

Reference

[187] Proposed

4 -0.343645 -0.343645913994 -0.496925 -0.496925913068 -0.876300 -0.876523782050

6 -0.343804 -0.343805515720 -0.497091 -0.497091926609 -0.877106 -0.877141548455

8 -0.343807 -0.343808345446 -0.497099 -0.497099921348 -0.877203 -0.877206361150

10 -0.343807 -0.343808387316 -0.497099 -0.497100266325 -0.877214 -0.877214733871

12 N/A -0.343808387967 N/A -0.497100302627 N/A -0.877215684952

14 N/A -0.343808387977 N/A -0.497100303642 N/A -0.877215775171

16 N/A -0.343808387977 N/A -0.497100303642 N/A -0.877215778657

18 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777981

20 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777810

22 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777788

24 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777788

26 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777788

28 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777788

30 N/A -0.343808387977 N/A -0.497100303638 N/A -0.877215777788

145

Table 5. 7: Hypersingular integral results for the quarter cylindrical panel at 3 different

singularity points (Note that 0’s are added in the reference results as only 6 significant

figures are found in their publications)

Method Point a Point b Point c

By Guiggiani et al. [187] -0.343807000000 -0.497099000000 -0.877214000000

By Feng et al. [189] -0.343808000000 -0.497095000000 -0.877370000000

Proposed -0.343808387977 -0.497100303638 -0.877215777788

Fig. 5. 15: Convergence rate of the proposed method and the literature [187] at

singularity point c for increasing number of Gaussian quadrature points

146

Fig. 5. 16: Contour plot of the hypersingular integral values with respect to various

singularity points

5.5 Conclusions

In this chapter, for the evaluation of hypersingular surface integral, coordinate

transformations are first carried out such that the integration variables are oriented to the

singularity. The resulting inner integral can be readily cast into the previously proposed

numerical form for hypersingular line integrals so as to have the singularity and the finite

part automatically regularised. Despite more mathematical manipulations being involved

in the numerical procedures, the underlying principle is that the corresponding density

functions must be transformed into modified ones compatible with the regularisation

process. This transformation enables the previously developed hypersingular line integral

formulation to be directly invoked to solve the surface integral problems.

There are some other notable features associated with the present work:

147

The singularity regularisation process for the hypersingular surface integral is

generalised by using Brandao’s formulation which bypasses the need for the

limiting processes that are often required in other reference methods.

When it comes to the density function approximation, the one complex-step

derivative approximation will suffice to provide near machine precision accuracy.

Equipped with this powerful tool, the present method is free of stability issues

while capable of providing highly accurate results. In contrast, the approximation

process in other reference methods often relies on series expansion or

interpolation schemes which inevitably would cost more computational resources

but not necessarily improve the accuracy of results.

The outcome of the present work is a highly accurate, fast and generalised method for

numerically evaluating surface integrals with principal values. The developed

regularisation process and the numerical treatment are directly applicable to integrals with

higher order singularities.

148

Chapter 6 Simulation of three-dimensional porous

media using MFS-CSRBF

6.1 Introduction

Smart gel materials, a type of porous media, are an important class of biomaterials. Owing

to their highly hydrated state, the gel materials are capable of providing excellent

biocompatibility with multi-functional properties [193]. Smart gel materials are networks

of molecules or monomers with cross-linked long chains. The networks are insoluble

because of the presence of tie-points and junctions in the chemical cross-links; or

entanglements and crystallites in the physical cross-links [194]. The free spaces within

the monomer networks, i.e. the interstitial spaces, of the gels can contain as much as 99%

fluid by weight [195]. These water-swollen gels can swell or shrink dramatically with

small changes in environmental conditions such as temperature, pH, ionic strength, salt

type, solvent, electric field, magnetic field, light or pressure, as shown in Fig. 6.1. Once

the external stimuli are released, the gels can recover to their original shapes subjected to

chain dynamics inside the three-dimensional cross-linked networks. These unique

features including the high water content of the materials have given smart gels many

successful applications for biological systems such as drug delivery devices,

bioseparation, biosensors, artificial muscles, linings for artificial hearts, and actuators for

adaptive structures.

Regardless of the wide applications of smart gel materials, there are only a limited number

of theoretical modelling studies due to the complexity of the problem. Usually, numerical

149

simulation is employed to predict the solid-fluid coupling behaviours of the smart gels.

For example, Lai et al proposed a model for the swelling and deformation behaviour of

articular cartilage using a triphasic theory framework [196]. Huyghe and Janssen [197]

developed a quadriphasic, i.e. solid, fluid, cation and anion, mixture theory and showed

that the corresponding numerical model can provide a realistic estimation for many

phenomena observed in biological tissues and gel systems [198]. More recently, Li [199]

employed the Poisson-Nernst-Planck equation to take into account the ion concentration

and electric potential in his hydrogel simulation. Yang et al. [200] investigated the linearly

coupled thermo-electro-chemo-elastic behaviours and proposed the corresponding finite

element formulation. Zhao et al. [201] adopted the free-energy function studied by Flory

and Rehner [202] to describe the mechanics of dielectric gels due to stretching, mixing

and polarizing. However, these mathematical models are often restricted to limited

numerical applications. For instance, the mixture model proposed by Lai [196] forbids

the gels to deform quickly under transient loads. The mixed finite element formulation

employed by Sun et al. [203] is unable to incorporate the effect of an external electric

field. In a meshless model developed by Zhou et al. [204], its computational domain only

covers the material’s interior but excluding the surrounding buffer solution. As the result,

development of a more robust numerical scheme is essential to better simulate the

coupling behaviours of the smart gel systems.

150

Fig. 6. 1: Schematic of smart gels subjected to environmental stimuli

In the literature, numerical schemes for solving a coupled system can be classified into

fully coupled methods and iteratively coupled methods. The former method seeks

simultaneous solutions [205] while the latter one seeks sequential solutions [206]. Since

the iteratively coupled method makes use of the readily available solutions of the

uncoupled phases, this has the advantage of reducing the computational cost of

developing and implementing a fully coupled model.

In this chapter, the meshless method of MFS-CSRBF is applied to a coupled solid-fluid

problem in porous media. To reuse the solution kernels derived in the previous chapters,

the governing equations of the solid and fluid parts are first uncoupled. Then, the Laplace

transform method is applied to the flow equation for time discretisation. To recouple the

differential governing equations, solutions in the Laplace domain are iteratively solved

for the solid and fluid parts. Finally, the coupled solutions are obtained by applying the

inverse Laplace transform to the converged results.

151

6.2 Model description

The basic mathematical relationship for the coupled displacement field 𝑢𝑖 of the solid

skeleton and the pressure field 𝑃 of the pore fluid in the homogeneous and isotropic

porous media are described by two sets of governing equations which represent the

equilibrium equations for the mechanical part and a continuity equation for the fluid flow

part [207, 208].

(1) Mechanical equilibrium equations

The governing equations describing the mechanical equilibrium of porous media follows

the classical elasticity equations from (2.1). For simplicity, body forces such as

gravitational force are assumed to be zeros. In addition, the solid skeleton in porous media

is assumed to deflate at a sufficiently slow rate and consequently, the inertial force is also

neglected. Thus, the total stress tensor ij of the bulk material satisfies the equilibrium

equations

0ij

jx

(6.1)

In porous media, the total stress is composed of pore pressure 𝑃 and effective stress eff

ij

sustained by the solid skeleton. The relationship is obtained by the generalised Terzaghi’s

principle

eff

ij ij ijP (6.2)

with

2eff

ij ij ij (6.3)

for homogeneous isotropic material.

152

In Eq. (6.3), (1 )(1 2 )

E

and

2(1 )

E

are the Lame constants of the elastic

skeleton with Young’s modulus E and Poisson’s ratio , ij is the elastic strain of the

solid skeleton

1

2

jiij

j i

uu

x x

(6.4)

and is the dilation strain of the skeleton defined by

11 22 33 u (6.5)

Thus substituting Eq. (6.2) into Eq. (6.1) yields

0

eff

ij

j i

P

x x

(6.6)

It is noted that infinitesimal displacements are employed in Eq. (6.3) and Eq. (6.4) for

model simplification. This approach is built into the assumption that the gel materials do

not undergo large rotation, which is a prerequisite for large deformation [209]. This

assumption holds for the porous sphere example demonstrated later in this chapter, in

which the porous body does not undergo large rotation due to the imposed fixed boundary

conditions on its symmetrical interior planes. Future research could evaluate the effects

of relaxing this assumption.

(2) Continuity equation for fluid flow

For an immiscible and fully saturated porous material, let 𝑠 and 𝑓 represent the solid and

fluid parts of the media. There holds a close relationships between the volume fractions

𝜙𝑠, 𝜙𝑓 and the apparent density 𝜌𝑠 and 𝜌𝑓,

t

a a a (6.7)

153

where 𝑎 = 𝑠, 𝑓 and 𝜌𝑎𝑡 is the true density of each constituent which is assumed to be

constant based on the incompressibility assumption on both the solid skeleton and the

pore fluid parts.

Substitute (6.7) into the mass balance equation for each constituent and divided by the

constant 𝜌𝑎𝑡 , we have the flow balance equation

0aa av

t

(6.8)

where 𝑣𝑎 is the mean velocity of each constituent. The physical interpretation of (6.8) is

that the volumetric rate of a constituent is equal to the rate of its volumetric inflow.

Owing to the immiscible and the fully saturated properties, the total apparent volume

fractions, i.e. 𝜙𝑠 + 𝜙𝑓 is always equal to a unity. The sum of the flow balance equations

therefore yields

0s f f sv v v (6.9)

The diffusion velocity 𝑣𝑓 − 𝑣𝑠 and the volume fraction of fluid 𝜙𝑓 can be understood as

the Darcy’s velocity and the porosity. The product of them represents the rate of discharge.

Through the employment of Darcy’s law, such term can then be related to the pressure

term

f f s K Pv v (6.10)

where K is a symmetric tensor of second rank related to the permeability ijk of the

porous medium and the coefficient of shear viscosity f of the pore fluid. For isotropic

permeability, K is expressed as

154

11

11

11

0 0

0 0

0 0

f

f

f

k

kK

k

(6.11)

Substitute equations (6.5) and (6.10) into (6.9) for the fluid flow, the final equation

consisting of only variables 𝒖 and 𝑃 is

K Pt

(6.12)

where t denotes the time variable for transient problem.

6.3 Iteratively coupled method

As seen from the previous section, equations (6.6) and (6.12) form the mathematical

model for porous media. Evidently, this is a coupled system since the displacement field

𝑢𝑖 of the elastic skeleton and the fluid pressure field 𝑃 of the pore fluid contain a total of

4 unknowns which cannot be solved by just considering any one set of the above

equations. In the literature, numerical scheme of solving the above coupled system can

be classified into fully coupled method and iteratively coupled method. The former

method seeks for the simultaneous solutions while the latter one seeks for the sequential

solutions [206]. For instance, the fully coupled method has seen implementation into

FEM by Wong et al. for simulating coupled consolidation in unsaturated soils [210]. For

the boundary-type implementation, Chen derived the fundamental solution of the coupled

dynamic poroelasticity for 2D and 3D applications [211, 212]. Later, Schanz and his co-

worker made use of Chen’s fundamental solution for BEM implementation [213, 214].

Meanwhile for the iteratively coupled method, Cavalcanti and Telles proposed the use of

time independent fundamental solution to solve for the coupled system iteratively in 2D

155

[215]. Soares et al. later extended the iteratively coupled approach for solving dynamic

problems [216]. More recently, Kim implemented the iteratively coupled method in FEM

and examined the stability issues associated with the different sequential schemes namely,

the drained split, undrained split, fixed-strain split and the fixed-stress split [206].

Herein, we propose a meshless iteratively coupled scheme using the previously developed

MFS-CSRBF method for our porous media simulation. The essence of this scheme is to

uncouple the governing equations so that the mechanical equation of (6.6) and the flow

equation of (6.12) can be reduced to simpler form and solved sequentially. To initiate the

process and to uncouple the system, the total stress ij is assumed to be a constant in

analogy to the fixed-stress split. The equations (6.6) and (6.12) are then solved

sequentially with the updated total stress value. The final coupled solutions are obtained

only when the total stress value converges.

6.3.1 Uncoupled porous media equations

To uncouple the porous media equations, we first introduce the linear relationship

between the dilation strain and the isotropic effective stress eff for homogeneous

isotropic porous material

1 eff

(6.13)

where the isotropic effective stress

11 22 33

3

eff eff effeff

(6.14)

and is the bulk modulus of the elastic skeleton

3(1 2 )

E

(6.15)

156

From the generalised Terzaghi principle (6.2), we have

eff P (6.16)

with

11 22 33

3

(6.17)

Thus Eq. (6.13) can be rewritten as

1

P

(6.18)

Differentiate the dilatation strain with time while assuming the total stress is a constant,

i.e. a fixed-stress split process, the flow equation (6.12) can now be simplified in terms

of just one single pressure term

21 PK P

t

(6.19)

The rest of the unknown can be easily computed by substituting the computed pressure

back into the mechanical governing equation (6.6), which can be solved by treating the

gradient of pressure as generalised body force, that is

eff

ij

i

j i

Pb

x x

(6.20)

After solving (6.20), the rate of the isotropic total stress is approximated.

Consequently, the left hand side of (6.19) can then be updated accordingly

21 PK P

t t

(6.21)

Thereafter, (6.21) and (6.20) are solved sequentially until the isotropic total stress rate

converges within a pre-set tolerance.

157

6.3.2 Time discretisation with Laplace transform

In this section, the solving procedure of the transient system consisting of Eq. (6.19)-

(6.21) and related boundary conditions is described. While (6.20) can be easily solved by

employing the MFS-CSRBF numerical scheme from Chapter 2, the flow equations (6.19)

and (6.21) would require their time domain be first discretised. One popular approach is

the use of backward finite difference scheme which is unconditionally stable for the time-

stepping [79]. When evaluating solutions at the current time, solutions at the previous

time step are required. A more efficient scheme is to instead relate the solutions at any

time to a reference state. In this regard, we apply Laplace transform onto the decoupled

flow equations (6.19) and (6.21) with constant boundary conditions. The coupled

solutions are iteratively solved between (6.20) and (6.21) in the Laplace domain until the

isotropic total stress rate , also operated in Laplace domain, converges within a pre-set

tolerance. To this end, the time domain solutions are obtained from the converged

solutions in the Laplace domain by approximating the inverse of the Laplace transform.

The Laplace transform of equations (6.19) and (6.21) are respectively,

20Ps

P sK K

(6.22)

2 10 0

sP s P s s

K K

(6.23)

where s is the Laplace space and the 0s s term is approximated from the

mechanical equations (6.20).

158

6.3.3 MFS-CSRBF kernels for flow equation

As can be seen, equations (6.22) and (6.23) are the modified Helmholtz equations, also

known as the Screened Poisson equation, which can be solved using the meshless MFS-

CSRBF scheme. The respected fundamental solutions 𝑃𝑝 for the homogeneous part and

the particular solutions kernels 𝑃ℎ of CSRBF type for the inhomogeneous part [75] are:

Particular solutions kernels within the support 𝛼

e er r

p

r

A B C rP r

r

(6.24)

where

s

K

(6.25)

5 2

e 3 4A

(6.26)

5 2

e 3B

(6.27)

2 3

4 2 4 2 2 2 2

4 1 6 2 1C r r r r r

(6.28)

Particular solutions kernels outside the support 𝛼

e r

p

r

DP r

r

(6.29)

where

5 2

3e e 3 4 eD

(6.30)

Fundamental solution of the modified Helmholtz equation

e

4

rhP r

r

(6.31)

159

6.3.4 Laplace inversion algorithm

As was discussed, the last step of the present iteratively coupled method is to convert the

convergence solutions in the Laplace domain back to the time domain by approximating

the inverse of the Laplace transform. In the literature, there exists many Laplace inversion

algorithms. For instance, the popular ones include the Papoulis method [217], the Stehfest

method [218], and the Durbin-Crump method [219, 220]. The Papoulis method makes

use of exponential series function for approximating the Laplace inversion function.

Meanwhile, the Stehfest method employs a delta-convergent series and the Durbin-

Crump method uses Fourier series for the approximation. A comprehensive study on their

performances can be found in the work by Cheng [221].

In consolidation process, the solutions profile will follow a diffusion like behaviour the

longer the time progresses. It is therefore desirable to acquire the coupled poroelasticity

solutions in the early stage of the simulation. For this reason, the Laplace inversion

algorithm is ideally be insensitive to time due to the fact that a smaller value of time will

incur a large value of the Laplace parameter s which in turn causes instability issue when

solving for the homogeneous part of the flow equation using MFS in (6.31). Based on the

numerical experience, it is found that of the three mentioned algorithms, the Durbin-

Crump’s method is capable of producing results with high stability and its inversion

algorithm is the least sensitive to time. Therefore, Durbin-Crump’s method will be

employed for discretising the time. The inversion formulation [220] for our

implementation is

1

e1 Re

0.8 2

at Nn

n

n

F af t F s

t

(6.32)

160

πi

0.8n

ns a

t (6.33)

ln

2 0.8

tolEa

t (6.34)

where f is the function to be inverted back to the time domain, F is the function in

Laplace domain and 610tolE is the pre-set error tolerance.

With the help of the present MFS-CSRBF, the inhomogeneous modified Helmholtz

equation can be solved by combining the specific boundary conditions expressed in terms

of the pressure variable to determine the distribution of pressure in the pore. This

completes the iteratively coupled method for poroelasticity simulation. A flow chart

illustrating the solutions finding process is shown in Fig. 6.2.

Fig. 6. 2: Flow chart of iteratively coupled solutions finding process for poroelasticity

simulation

161

6.4 Numerical simulation

As a numerical example, a three-dimensional porous sphere is taken into consideration

and subjected to the following boundary conditions of Cryer’s ball problem [222]

Boundary conditions at the centre of the porous sphere

0P

r

(6.35)

0u (6.36)

Boundary conditions on the surface of the porous sphere

0P (6.37)

0 , for 0

100000Pa , for 0r

t

t

(6.38)

where r is the radial stress

Due to the symmetry of the sphere and its symmetric boundary conditions, only one-

eighth of it is chosen as the computation domain, as displayed in Fig. 6.3. To implement

the numerical simulation by the present method, the computational domain surface is

modelled by 114 collocation points (Fig. 6.4). Additionally, 62 interpolation points are

placed for the inhomogeneous terms evaluation. Fig. 6.5 shows the time evolution of the

pore fluid pressure at 𝑟 = 0 within the sphere for 710 PaE , 0.25 , 54.91x10K

and 5

0 10 PaP . It is seen from Fig. 6.5 that at the onset of consolidation, the pore fluid

pressure in the origin of the sphere is equal to the applied surface traction 0P , and then it

begins to increase for some time and reaches a maximum value before following the

diffusion like behaviour and dissipating to zero starts. This Mandel-Cryer effect [222]

162

can be interpreted physically as a stress transfer effect. Similar trend can be found from

the dimensionless analytical result by Mason et al. [223] as regenerated in Fig. 6.6. As

can be seen, apart from the matching trend between the simulated and the analytical

results, the peak value of the simulated pressure as indicated in Fig. 6.5 has only less than

1% of deviation from the analytical results.

Fig. 6. 3: Geometrical model of the computational domain

163

Fig. 6. 4: Configuration of collocation on one-eighth of sphere

Fig. 6. 5: Simulated pore fluid pressure against the time at the origin point

164

Fig. 6. 6: Analytical result [223] of pore fluid pressure against the time at the origin

point

6.5 Conclusions

In this chapter, the meshless method of MFS-CSRBF is applied to the coupled solid-fluid

problem of porous media. The governing equations of the coupled problem are first

uncoupled by a fixed stress split sequential process. Then, Laplace transform is applied

to the flow equation for time discretisation. To recouple the differential governing

equations, solutions in the Laplace domain are iteratively solved for the solid and fluid

parts. Finally, the coupled solutions are obtained by performing inverse Laplace

transform onto the converged results. As a numerical example, the fluid pressure profile

of Cryer’s ball is simulated. This simulation demonstrates that the present method is

capable of solving three-dimensional poroelasticity problems efficiently. The numerical

results from the present method show a similar trend and is comparable to the available

165

analytical solutions. To conclude, this chapter provides insights into possible applications

of fundamental solution based numerical methods. Initial results are provided to support

the direction of future research and are subject to further developments. For instance, the

application of meshed methods may have potential to further improve the computational

efficiency and accuracy of the results.

166

Chapter 7 Summary and outlook

7.1 Summary of present research

This thesis research project aims at acquiring better three-dimensional numerical

modelling for the fundamental solution based numerical methods including the meshless

method of fundamental solutions (MFS) and the boundary element method (BEM) by

coupling compactly supported radial basis functions (CSRBFs). CSRBFs are used for

dealing with inhomogeneous generalised body force terms in three-dimensional elastic

formulations, due to their efficient sparse structure and unconditional invertibility.

Simultaneously, two important issues, hypersingular linear and surface integrals, which

exist in the two-dimensional and three-dimensional boundary-based methods are to be

resolved. They are summarised as follows:

(1) Method of fundamental solutions for three-dimensional elasticity with body forces

The standard method of fundamental solutions cannot be applied for three-dimensional

linear elastic problems in homogeneous isotropic solids. To keep the advantage of the

MFS such as boundary collocation only, the mixed meshless strategy was developed by

combining the MFS and the locally supported RBFs, which can provide stable and

accurate approximation of inhomogeneous terms in the governing equations, and several

numerical examples were solved to verify the present mixed meshless approach.

(2) Dual reciprocity boundary element method using compactly supported radial basis

functions for 3D linear elasticity with body forces

167

Unlike the meshless boundary-type method, i.e. MFS, the boundary integral method can

provide better stability than the MFS. However, the BEM cannot directly solve three-

dimensional elasticity problems with body forces. To expand the application of the BEM

for efficient body forces treatment, the locally supported RBFs, mainly the Wendland’s

CSRBF, are coupled with the BEM so as to numerically determine the particular solutions

of the problem, which are related to the specified body forces. Numerical experiments

show that the present method is capable of solving three-dimensional elasticity problems

with body forces.

(3) Evaluation of hypersingular line integral by complex-step derivative approximation

The highly efficient computation of hypersingular line integrals has confused researchers

for a long time. In contrast to the existing work, this study establishes a novel numerical

scheme to evaluate the hypersingular line integral efficiently. The hypersingular linear

integrals are first separated into regular and singular parts, in which the singular integrals

are defined as limits around the singularity and their values determined analytically by

taking the finite-part values. The remaining regular integrals can be evaluated by the

barycentric rational interpolatory quadrature or the complex-step derivative

approximation for the regular function when machine precision like accuracy is required.

Numerical results show that the present method is accurate and efficient.

(4) Evaluation of hypersingular surface integral by complex-step derivative

approximation

For three-dimensional problems, the implementation of the BEM would often involve the

computation of hypersingular surface integrals, which are more complex than the

hypersingular line integrals. To the author’s best knowledge, there are only few numerical

studies on the computation of hypersingular surface integrals. The research work

introduces two different coordinate transformations such that the integration variables are

168

oriented to the singularity. The resulting inner integral can be readily cast into the

previously proposed numerical form for hypersingular line integrals so as to have the

singularity and the finite part automatically regularised. Despite more mathematical

manipulations being involved in the numerical procedures, the underlining principle is

that the corresponding density functions must be transformed into modified ones

compatible with the regularisation process. This transformation enables the previously

developed hypersingular line integral formulation to be directly invoked to solve surface

integral problems. Numerical results have demonstrated the correctness and the greater

efficiency of the present method.

(5) Simulation of three-dimensional porous media using MFS-CSRBF

As an initial application of the methods set out in Chapter 2, the three-dimensional porous

media is simulated by the MFS-CSRBF to determine the pore fluid pressure. The

complicated three-dimensional porous model is firstly simplified by introducing

sequential schemes to obtain a decoupled fluid flow equation in terms of pore fluid

pressure. Then the simplified model is solved by the present MFS-CSRBF, in which the

time discretisation is carried out by Laplace transform technique and then the iterative

procedure is designed for obtaining the approximated relation between pressure and time.

Numerical simulation of a three-dimensional porous sphere shows that the numerical

results from the present meshless method are comparable to the available analytical

solutions. Meshed methods may have potential to further improve the computational

efficiency and accuracy of the results.

169

7.2 Research limitations

This thesis develops and applies new numerical schemes, making use of fundamental

solution based methods to solve elliptic boundary value problems. Therefore, results of

this study are applicable within the scope of the modelling assumptions applied, and

within the capabilities of the numerical methods selected, as follows:

(1) The present numerical methods simplify boundary value problems by assuming

infinitesimal displacements, isotropic elasticity and isotropic permeability.

(2) The present numerical methods are limited to a single region. The ability to handle

multi-regions and contact problems would require the posing of the interface conditions

across the material boundaries.

(3) The meshless method by MFS-CSRBF tends to have stability issues for irregular or

complex geometries as well as being sensitive to the locations of source points.

(4) In the present work on hypersingular integrals, the order of singularity is limited to

two for line integrals and three for surface integrals.

(5) The proposed iteratively coupled meshless method is limited to biphasic model

simulation. More complex and coupled models would require different sets of iteration

schemes.

170

7.3 Future research

Although some topics related to the MFS and the BEM were studied in the thesis, due to

time limitations, some interesting problems can be considered in future research:

(1) Numerical evaluation of integrals with higher order singularities

In the present work on hypersingular integrals, the order of singularity is limited to two

for line integrals and three for surface integrals. The developed regularisation process and

the numerical treatment can be generalised to integrals with higher order singularities,

provided that the complex-step derivative approach is able to approximate the derivatives

of the density function for higher orders.

(2) Three-dimensional thermoelasticity caused by temperature changes in elastic media

The present MFS-CSRBF numerical scheme has seen applications to potential and

elasticity problems. Multi-field problems such as thermoelasticity can also be solved by

the MFS-CSRBF scheme, provided that the corresponding particular solution kernels of

the multi-field problem are derived.

(3) Full implementation of meshed methods for porous media simulation

The present iteratively coupled solution scheme for simulating poroelasticity problems is

implemented in the meshless method using MFS-CSRBF. However, this method tends to

have stability issues for irregular or complex geometries as well as being sensitive to the

locations of source points. Hence, potential improvements in computational efficiency

and accuracy can be made by a full implementation of meshed method.

(4) Computational domain extends to surrounding bathing solution

The present computational domain for porous media simulation is limited to a single

region, i.e. the material itself. In practice, a more comprehensive simulation would need

to take into account the surrounding bathing solution, in which the porous media is

171

immersed. This would require an understanding of the interface conditions across the

material boundaries which could be examined in further research.

(5) Extend isotropy assumption to anisotropy

Throughout this thesis, isotropic elasticity and isotropic permeability assumptions are

taken so as to simplify the solution finding processes, e.g. the derivation of CSRBF

particular solution kernels, the respected fundamental solutions for the MFS, and the

iteratively coupled solutions scheme for porous media simulation. Hence, extending the

studies to anisotropy applications is highly desirable for future research projects.

172

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List of publications

Lee, Cheuk-Yu, Wang, H., and Qin, Q. H. (2016). “Accurate, fast and generalised

hypersingular line and surface integrals direct evaluation by complex-step

derivative approximation”. Applied Mathematics and Computation. Submitted on

15th November 2016.

Lee, Cheuk-Yu, Wang, H., and Qin, Q. H. (2016). “Dual reciprocity boundary

element method using compactly supported radial basis functions for 3D linear

elasticity with body forces”. Journal of Mechanics and Materials in Design 12:

463-476.

Lee, Cheuk-Yu, Wang, H., and Qin, Q. H. (2015). “Method of fundamental

solutions for 3D elasticity with body forces by coupling compactly supported

radial basis functions”. Engineering Analysis with Boundary Elements 60: 123-

136.

Lee, Cheuk-Yu, Qin, Q. H., Walpole, G. (2013) “Numerical modelling on

electric response of fibre-orientation of composites with piezoelectricity”,

International Journal of Research and Reviews in Applied Science, 16 (3): 377-

386.

Publications not related to this thesis

Chen, S., Lee, C. Y., Li, R. W., Smith, P. N. & Qin, Q. H. (2017). “Modelling

osteoblast adhesion on surface-engineered biomaterials: optimisation of

nanophase grain size”. Computer Methods in Biomechanics and Biomedical

Engineering 20: 905-914.

187

See, T. L., Feng, R. X., Lee, C. Y., & Stachurski, Z. H. (2012). “Phonon thermal

conductivity of a nanowire with amorphous structure”. Computational Materials

Science 59: 152-157.

Lee, Cheuk-Yu, Stachurski, Z. H. and Welberry, R. T. (2010). “The geometry,

topology and structure of amorphous solids”. Acta Materialia 58 (2): 615-625.