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
Bibliography
[1] W.E. Boyce and R.C. DiPrima, Elementary differential equations and boundary
value problems. 7th ed. 2001, New York: Wiley.
[2] J.R. Barber, Three-dimensional elasticity solutions for isotropic and generally
anisotropic bodies. Applied Mechanics and Materials 2006. 5-6: 541-550.
[3] C.Y. Lee, Q.H. Qin, and H. Wang, Trefftz functions and application to 3D
elasticity. Computer Assisted Mechanics and Engineering Sciences, 2008. 15(3-
4): 251-263.
[4] R. Piltner, The use of complex valued functions for the solution of three-
dimensional elasticity problems. Journal of Elasticity, 1987. 18: 191-225.
[5] Z.K. Wang and B.L. Zheng, The general solution of three-dimensional problems
in piezoelectric media. International Journal of Solids and Structures, 1995. 32(1):
105-115.
[6] Q.H. Qin, Y.W. Mai, and S.W. Yu, Some problems in plane thermopiezoelectric
materials with holes. International journal of solids and structures, 1999. 36(3):
427-439.
[7] Q.H. Qin and X. Zhang, Crack deflection at an interface between dissimilar
piezoelectric materials. International journal of fracture, 2000. 102(4): 355-370.
[8] Q.-H. Qin, Thermoelectroelastic Green's function for a piezoelectric plate
containing an elliptic hole. Mechanics of Materials, 1998. 30(1): 21-29.
[9] Q.-H. Qin, General solutions for thermopiezoelectrics with various holes under
thermal loading. International Journal of Solids and Structures, 2000. 37(39):
5561-5578.
[10] Q.-H. Qin, Y.-W. Mai, and S.-W. Yu, Effective moduli for thermopiezoelectric
materials with microcracks. International Journal of Fracture, 1998. 91(4): 359-
371.
[11] Q.H. Qin and Y.W. Mai, Crack growth prediction of an inclined crack in a half-
plane thermopiezoelectric solid. Theoretical and Applied Fracture Mechanics,
1997. 26(3): 185-191.
[12] O.C. Zienkiewicz, R.L. Taylor, O.C. Zienkiewicz, and R.L. Taylor, The finite
element method. Vol. 3. 1977: McGraw-hill London.
[13] K.J. Bathe, Finite element procedures. 1996, New Jersey: Prentice-Hall.
[14] Q.H. Qin and C.X. Mao, Coupled torsional-flexural vibration of shaft systems in
mechanical engineering—I. Finite element model. Computers & Structures, 1996.
58(4): 835-843.
[15] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite
Difference Methods. 1985, New York: Oxford University Press.
[16] Q.H. Qin, The Trefftz finite and boundary element method. 2000, Southampton:
WIT Press.
173
[17] Q.H. Qin and H. Wang, Matlab and C programming for Trefftz finite element
methods. 2008, Boca Raton: CRC Press.
[18] T.H.H. Pian and C.C. Wu, Hybrid and incompatible finite element methods. 2006,
Boca Raton: Chapman & Hall/CRC.
[19] H. Wang and Q.H. Qin, FE approach with Green’s function as internal trial
function for simulating bioheat transfer in the human eye. Archives of Mechanics,
2010. 62(6): 493-510.
[20] H. Wang and Q.H. Qin, Special fiber elements for thermal analysis of fiber-
reinforced composites. Engineering Computations, 2011. 28(8): 1079-1097.
[21] H. Wang, Q.H. Qin, and D. Arounsavat, Application of hybrid Trefftz finite
element method to non-linear problems of minimal surface. International journal
for numerical methods in engineering, 2007. 69(6): 1262-1277.
[22] Q.H. Qin, Hybrid-Trefftz finite element method for Reissner plates on an elastic
foundation. Computer Methods in Applied Mechanics and Engineering, 1995.
122(3-4): 379-392.
[23] Q.H. Qin, Hybrid Trefftz finite-element approach for plate bending on an elastic
foundation. Applied Mathematical Modelling, 1994. 18(6): 334-339.
[24] Q.H. Qin, Solving anti-plane problems of piezoelectric materials by the Trefftz
finite element approach. Computational Mechanics, 2003. 31(6): 461-468.
[25] S.N. Atluri and T.L. Zhu, The meshless local Petrov-Galerkin (MLPG) approach
for solving problems in elasto-statics. Computational Mechanics, 2000. 25(2-3):
169-179.
[26] G.R. Liu and M.B. Liu, Smoothed Particle Hydrodynamics: a meshfree particle
method. 2003, Singapore: World Scientific Publishing.
[27] C.A. Brebbia and J. Dominguez, Boundary elements: an introductory course.
1992, Southampton: WIT Press.
[28] X.W. Gao and T.G. Davies, Boundary element programming in mechanics. 2002:
Cambridge University Press.
[29] Q.-H. Qin and Y.-W. Mai, BEM for crack-hole problems in thermopiezoelectric
materials. Engineering Fracture Mechanics, 2002. 69(5): 577-588.
[30] Q.H. Qin, Nonlinear analysis of Reissner plates on an elastic foundation by the
BEM. International Journal of Solids and Structures, 1993. 30(22): 3101-3111.
[31] Y. Weian and H. Wang, Virtual boundary element integral method for 2-D
piezoelectric media. Finite elements in analysis and design, 2005. 41(9): 875-891.
[32] W. Chen and M. Tanaka, A meshless, integration-free, and boundary-only RBF
technique. Computers & Mathematics with Applications, 2002. 43(3): 379-391.
[33] G. Fairweather and A. Karageorghis, The method of fundamental solutions for
elliptic boundary value problems. Advances in Computational Mathematics, 1998.
9(1-2): 69-95.
[34] C.S. Chen, A. Karageorghis, and Y.S. Smyrlis, The method of fundamental
solutions: a meshless method. 2008, Atlanta: Dynamic Publishers.
174
[35] M.K. Chati, S. Mukherjee, and Y.X. Mukherjee, The boundary node method for
three‐dimensional linear elasticity. International Journal for Numerical Methods
in Engineering, 1999. 46(8): 1163-1184.
[36] Y. Gu and G. Liu, A boundary point interpolation method for stress analysis of
solids. Computational Mechanics, 2002. 28(1): 47-54.
[37] Y. Gu and G. Liu, A boundary radial point interpolation method (BRPIM) for 2-
D structural analyses. Structural engineering and mechanics, 2003. 15(5): 535-
550.
[38] G. Liu and Y. Gu, Boundary meshfree methods based on the boundary point
interpolation methods. Engineering analysis with boundary elements, 2004. 28(5):
475-487.
[39] G.B. Kolata, The finite element method: A mathematical revival. Science, 1974.
184(4139): 887-889.
[40] D.L. Logan, A first course in the finite element method. 2011: Cengage Learning.
[41] H. Wang, Y.P. Lei, J.S. Wang, Q.H. Qin, and Y. Xiao, Theoretical and
computational modeling of clustering effect on effective thermal conductivity of
cement composites filled with natural hemp fibers. Journal of Composite
Materials, 2016. 50(11): 1509-1521.
[42] H. Wang, Y. Xiao, and Q.H. Qin, 2D hierarchical heat transfer computational
model of natural fiber bundle reinforced composite. Scientia Iranica. Transaction
B, Mechanical Engineering, 2016. 23(1): 268.
[43] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element
analysis. Computer Methods in Applied Mechanics and Engineering, 1997.
142(1): 1-88.
[44] K.-J. Bathe, Finite element procedures. 1996, New Jersey: Prentice Hall.
[45] C.A. Brebbia and R. Butterfield, Formal equivalence of direct and indirect
boundary element methods. Applied Mathematical Modelling, 1978. 2(2): 132-
134.
[46] Q.-H. Qin, Green's function and boundary elements of multifield materials. 2007,
Elsevier, Oxford.
[47] C. Wang and B.C. Khoo, An indirect boundary element method for three-
dimensional explosion bubbles. Journal of Computational Physics, 2004. 194(2):
451-480.
[48] C.R. Kipp and R.J. Bernhard, Prediction of acoustical behavior in cavities using
an indirect boundary element method. Journal of vibration, acoustics, stress, and
reliability in design, 1987. 109(1): 22-28.
[49] H.C. Sun and W.A. Yao, Virtual boundary element-linear complementary
equations for solving the elastic obstacle problems of thin plate. Finite elements
in analysis and design, 1997. 27(2): 153-161.
[50] W.A. Yao and H. Wang, Virtual boundary element integral method for 2-D
piezoelectric media. Finite elements in analysis and design, 2005. 41(9): 875-891.
[51] A.A. Becker, The boundary element method in engineering: a complete course.
1992, London: McGraw-Hill.
175
[52] G. Beer, Programming the boundary element method. 2001, West Sussex,
England: John Wiley & Sons, Ltd.
[53] P.K. Kythe, Fundamental solutions for differential operators and applications.
1996, Boston: Birkhauser.
[54] N. Phan-Thien, Rubber-like elasticity by boundary element method: Finite
deformation of a circular elastic slice. Rheologica acta, 1988. 27(3): 230-240.
[55] Z. Chen and X. Ji, A new approach to finite deformation problems of
elastoplasticity—Boundary element analysis method. Computer methods in
applied mechanics and engineering, 1990. 78(1): 1-18.
[56] H.J. Al-Gahtani and N.J. Altiero, Application of the boundary element method to
rubber-like elasticity. Applied mathematical modelling, 1996. 20(9): 654-661.
[57] R. Bialecki and A.J. Nowak, Boundary value problems in heat conduction with
nonlinear material and nonlinear boundary conditions. Applied mathematical
modelling, 1981. 5(6): 417-421.
[58] G.S.A. Fam and Y.F. Rashed, The Method of Fundamental Solutions applied to
3D structures with body forces using particular solutions. Computational
Mechanics, 2005. 36(4): 245-254.
[59] D. Nardini and C.A. Brebbia, A new approach to free vibration analysis using
boundary elements. Applied Mathematical Modelling, 1983. 7(3): 157-162.
[60] P.W. Partridge, C.A. Brebbia, and L.C. Wrobel, The Dual Reciprocity Boundary
Element Method. 1992, Southampton: Computational Mechanics Publications.
[61] C. Franke and R. Schaback, Solving partial differential equations by collocation
using radial basis functions. Applied Mathematics and Computation, 1998. 93(1):
73-82.
[62] E. Larsson and B. Fornberg, A numerical study of some radial basis function
based solution methods for elliptic PDEs. Computers & Mathematics with
Applications, 2003. 46(5): 891-902.
[63] L.C. Wrobel and C.A. Brebbia, The dual reciprocity boundary element
formulation for nonlinear diffusion problems. Computer Methods in Applied
Mechanics and Engineering, 1987. 65(2): 147-164.
[64] W.Q. Lu, J. Liu, and Y.T. Zeng, Simulation of the thermal wave propagation in
biological tissues by the dual reciprocity boundary element method. Engineering
Analysis with Boundary Elements, 1998. 22(3): 167-174.
[65] A. Portela, M.H. Aliabadi, and D.P. Rooke, The dual boundary element method:
effective implementation for crack problems. International Journal for Numerical
Methods in Engineering, 1992. 33(6): 1269-1287.
[66] B. Šarler and G. Kuhn, Dual reciprocity boundary element method for convective-
diffusive solid-liquid phase change problems, Part 1. Formulation. Engineering
analysis with boundary elements, 1998. 21(1): 53-63.
[67] B. Šarler and G. Kuhnb, Dual reciprocity boundary element method for
convective-diffusive solid-liquid phase change problems, Part 2. Numerical
examples. Engineering Analysis with Boundary Elements, 1998. 21(1): 65-79.
[68] W.T. Ang, D.L. Clements, and N. Vahdati, A dual-reciprocity boundary element
method for a class of elliptic boundary value problems for non-homogeneous
176
anisotropic media. Engineering Analysis with Boundary Elements, 2003. 27(1):
49-55.
[69] Y.L. Zhang and S.P. Zhu, On the choice of interpolation functions used in the
dual-reciprocity boundary-element method. Engineering Analysis with Boundary
Elements, 1994. 13(4): 387-396.
[70] M. Golberg, C. Chen, H. Bowman, and H. Power, Some comments on the use of
radial basis functions in the dual reciprocity method. Computational Mechanics,
1998. 21(2): 141-148.
[71] M.D. Buhmann, Radial basis functions: theory and implementations. Vol. 5. 2003:
Cambridge university press Cambridge.
[72] H. Wendland, Scattered data approximation. 2005, United Kingdom: Cambridge
University Press.
[73] Y.F. Rashed, BEM for dynamic analysis using compact supported radial basis
functions. Computers & Structures, 2002. 80(16-17): 1351-1367.
[74] H. Wang and Q.H. Qin, Meshless approach for thermo-mechanical analysis of
functionally graded materials. Engineering Analysis with Boundary Elements,
2008. 32(9): 704-712.
[75] M. Golberg, C. Chen, and M. Ganesh, Particular solutions of 3D Helmholtz-type
equations using compactly supported radial basis functions. Engineering
Analysis with Boundary Elements, 2000. 24(7): 539-547.
[76] S.R. Karur and P. Ramachandran, Radial basis function approximation in the dual
reciprocity method. Mathematical and Computer Modelling, 1994. 20(7): 59-70.
[77] C. Chen Y, Y. Rashed, and M. Golberg, A mesh-free method for linear diffusion
equations. Numerical Heat Transfer, Part B, 1998. 33(4): 469-486.
[78] L. Cao, Q.-H. Qin, and N. Zhao, Application of DRM-Trefftz and DRM-MFS to
transient heat conduction analysis. Recent Patents on Space Technology, 2010. 2:
41-50.
[79] L. Cao, Q.-H. Qin, and N. Zhao, An RBF–MFS model for analysing thermal
behaviour of skin tissues. International Journal of Heat and Mass Transfer, 2010.
53(7): 1298-1307.
[80] C.J. Coleman, D.L. Tullock, and N. Phan-Thien, An effective boundary element
method for inhomogeneous partial differential equations. Zeitschrift Fur
Angewandte Mathematik Und Physik, 1991. 42(5): 730-745.
[81] M.R. Dubal, Domain decomposition and local refinement for multiquadric
approximations. I: Second-order equations in one-dimension. J. Appl. Sci. Comp,
1994. 1(1): 146-171.
[82] R.K. Beatson and W.A. Light, Fast evaluation of radial basis functions: methods
for two-dimensional polyharmonic splines. IMA Journal of Numerical Analysis,
1997. 17(3): 343-372.
[83] B. Natalini and V. Popov, Tests of radial basis functions in the 3D DRM-MD.
Communications in Numerical Methods in Engineering, 2006. 22(1): 13-22.
[84] H. Wendland, Piecewise polynomial, positive definite and compactly supported
radial functions of minimal degree. Advances in Computational Mathematics,
1995. 4(1): 389-396.
177
[85] H. Wendland, Error estimates for interpolation by compactly supported radial
basis functions of minimal degree. Journal of Approximation Theory, 1998. 93(2):
258-272.
[86] C.S. Chen, G. Kuhn, J. Li, and G. Mishuris, Radial basis functions for solving
near singular Poisson problems. Communications in Numerical Methods in
Engineering, 2003. 19(5): 333-347.
[87] C.C. Tsai, Meshless BEM for three-dimensional stokes flows. Computer Modeling
in Engineering and Sciences 2001. 3: 117-128.
[88] C. Chen, M. Golberg, and R. Schaback, Recent developments in the dual
reciprocity method using compactly supported radial basis functions.
Transformation of Domain Effects to the Boundary (YF Rashed and CA Brebbia,
eds), WITPress, Southampton, Boston, 2003: 138-225.
[89] C. Chen, C. Brebbia, and H. Power, Dual reciprocity method using compactly
supported radial basis functions. Communications in numerical methods in
engineering, 1999. 15(2): 137-150.
[90] M.P. Wachowiak, X. Wang, A. Fenster, and T.M. Peters. Compact support radial
basis functions for soft tissue deformation. in Biomedical Imaging: Nano to Macro,
2004. IEEE International Symposium on. 2004. IEEE.
[91] R. Schaback, Creating surfaces from scattered data using radial basis functions.
Mathematical methods for curves and surfaces, 1995. 477.
[92] Z. Wu, Compactly supported positive definite radial functions. Advances in
Computational Mathematics, 1995. 4(1): 283-292.
[93] M. Buhmann, A new class of radial basis functions with compact support.
Mathematics of Computation, 2001. 70(233): 307-318.
[94] M.S. Floater and A. Iske, Multistep scattered data interpolation using compactly
supported radial basis functions. Journal of Computational and Applied
Mathematics, 1996. 73(1): 65-78.
[95] G.E. Fasshauer, Solving differential equations with radial basis functions:
multilevel methods and smoothing. Advances in Computational Mathematics,
1999. 11(2-3): 139-159.
[96] C.S. Chen, M. Ganesh, M.A. Golberg, and A.-D. Cheng, Multilevel compact
radial functions based computational schemes for some elliptic problems.
Computers & Mathematics with Applications, 2002. 43(3): 359-378.
[97] V.D. Kupradze, A method for the approximate solution of limiting problems in
mathematical physics. USSR Computational Mathematics and Mathematical
Physics, 1964. 4(6): 199-205.
[98] M. Golberg and C. Chen, The method of fundamental solutions for potential,
Helmholtz and diffusion problems. Boundary integral methods: numerical and
mathematical aspects. Computational engineering, 1998. 1: 103-76.
[99] A. Karageorghis, The method of fundamental solutions for the calculation of the
eigenvalues of the Helmholtz equation. Applied Mathematics Letters, 2001. 14(7):
837-842.
178
[100] A. Poullikkas, A. Karageorghis, and G. Georgiou, The method of fundamental
solutions for three-dimensional elastostatics problems. Computers & structures,
2002. 80(3): 365-370.
[101] A. Bogomolny, Fundamental solutions method for elliptic boundary value
problems. SIAM Journal on Numerical Analysis, 1985. 22(4): 644-669.
[102] R. Mathon and R.L. Johnston, The approximate solution of elliptic boundary-
value problems by fundamental solutions. SIAM Journal on Numerical Analysis,
1977. 14(4): 638-650.
[103] P. Mitic and Y.F. Rashed, Convergence and stability of the method of meshless
fundamental solutions using an array of randomly distributed sources.
Engineering Analysis with Boundary Elements, 2004. 28(2): 143-153.
[104] M. Katsurada and H. Okamoto, A mathematical study of the charge simulation
method I. J. Fac. Sci. Univ. Tokyo Sect. IA Math, 1988. 35(3): 507-518.
[105] M. Katsurada and H. Okamoto, The collocation points of the fundamental solution
method for the potential problem. Computers & Mathematics with Applications,
1996. 31(1): 123-137.
[106] H. Wang and Q.H. Qin, Boundary integral based graded element for elastic
analysis of 2D functionally graded plates. European Journal of Mechanics-
A/Solids, 2012. 33: 12-23.
[107] H. Wang and Q.H. Qin, A new special element for stress concentration analysis
of a plate with elliptical holes. Acta Mechanica, 2012. 223(6): 1323-1340.
[108] H. Wang, Q.H. Qin, and X.P. Liang, Solving the nonlinear Poisson-type problems
with F-Trefftz hybrid finite element model. Engineering analysis with boundary
elements, 2012. 36(1): 39-46.
[109] H. Wang, L.L. Cao, and Q.H. Qin, Hybrid graded element model for nonlinear
functionally graded materials. Mechanics of Advanced Materials and Structures,
2012. 19(8): 590-602.
[110] L.L. Cao, H. Wang, and Q.H. Qin, Fundamental solution based graded element
model for steady-state heat transfer in FGM. Acta Mechanica Solida Sinica, 2012.
25(4): 377-392.
[111] Q.H. Qin and H. Wang, Special circular hole elements for thermal analysis in
cellular solids with multiple circular holes. International Journal of
Computational Methods, 2013. 10(04): 1350008.
[112] J.C. Li, Y.C. Hon, and C.S. Chen, Numerical comparisons of two meshless
methods using radial basis functions. Engineering Analysis with Boundary
Elements, 2002. 26(3): 205-225.
[113] C.C. Tsai, The method of fundamental solutions with dual reciprocity for three-
dimensional thermoelasticity under arbitrary body forces. Engineering
Computations, 2009. 26(3): 229-244.
[114] P.W. Partridge and B. Sensale, The method of fundamental solutions with dual
reciprocity for diffusion and diffusion–convection using subdomains. Engineering
analysis with boundary elements, 2000. 24(9): 633-641.
[115] J.C. Li, Mathematical justification for RBF-MFS. Engineering analysis with
boundary elements, 2001. 25(10): 897-901.
179
[116] R.A. Gingold and J.J. Monaghan, Smoothed particle hydrodynamics: theory and
application to non-spherical stars. Monthly notices of the royal astronomical
society, 1977. 181(3): 375-389.
[117] J.D. Eldredge, A. Leonard, and T. Colonius, A general deterministic treatment of
derivatives in particle methods. Journal of Computational Physics, 2002. 180(2):
686-709.
[118] B. Schrader, S. Reboux, and I.F. Sbalzarini, Discretization correction of general
integral PSE Operators for particle methods. Journal of Computational Physics,
2010. 229(11): 4159-4182.
[119] P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares
methods. Mathematics of computation, 1981. 37(155): 141-158.
[120] G.R. Joldes, H.A. Chowdhury, A. Wittek, B. Doyle, and K. Miller, Modified
moving least squares with polynomial bases for scattered data approximation.
Applied Mathematics and Computation, 2015. 266: 893-902.
[121] W.K. Liu, S. Jun, and Y.F. Zhang, Reproducing kernel particle methods.
International journal for numerical methods in fluids, 1995. 20(8‐9): 1081-1106.
[122] S.N. Atluri and S. Shen, The meshless local Petrov-Galerkin (MLPG) method: A
simple & less-costly alternative to the Finite Element and Boundary Element
methods. Computer Modeling in Engineering & Sciences, 2002. 3: 11-51.
[123] D.P. Bertsekas, Constrained optimization and Lagrange Multiplier methods.
Computer Science and Applied Mathematics, Boston: Academic Press, 1982,
1982. 1.
[124] V.P. Nguyen, T. Rabczuk, S. Bordas, and M. Duflot, Meshless methods: a review
and computer implementation aspects. Mathematics and computers in simulation,
2008. 79(3): 763-813.
[125] Z.W. Zhang, H. Wang, and Q.H. Qin, Meshless method with operator splitting
technique for transient nonlinear bioheat transfer in two-dimensional skin tissues.
International journal of molecular sciences, 2015. 16(1): 2001-2019.
[126] H. Wang and Q.H. Qin, Hybrid FEM with fundamental solutions as trial functions
for heat conduction simulation. Acta Mechanica Solida Sinica, 2009. 22(5): 487-
498.
[127] H. Wang and Q.H. Qin, Fundamental-solution-based finite element model for
plane orthotropic elastic bodies. European Journal of Mechanics-A/Solids, 2010.
29(5): 801-809.
[128] H. Wang and Q.H. Qin, Fundamental-solution-based hybrid FEM for plane
elasticity with special elements. Computational Mechanics, 2011. 48(5): 515-528.
[129] H. Wang, Y.T. Gao, and Q.H. Qin, Green's function based finite element
formulations for isotropic seepage analysis with free surface. Latin American
Journal of Solids and Structures, 2015. 12(10): 1991-2005.
[130] H. Wang and Q.H. Qin, A new special coating/fiber element for analyzing effect
of interface on thermal conductivity of composites. Applied Mathematics and
Computation, 2015. 268: 311-321.
180
[131] G.S. Fam and Y.F. Rashed, An efficient meshless technique for the solution of
transversely isotropic two-dimensional piezoelectricity. Computers &
Mathematics with Applications, 2015. 69(5): 438-454.
[132] G.C. Medeiros, P.W. Partridge, and J.O. Brandão, The method of fundamental
solutions with dual reciprocity for some problems in elasticity. Engineering
analysis with boundary elements, 2004. 28(5): 453-461.
[133] Y.C. Fung, Foundations of Solid Mechanics. 1965, United States of America:
Prentice Hall.
[134] W.A. Strauss, Partial differential equations. 2 ed. 2008, USA: John Wiley & Sons,
Inc.
[135] H. Wang, Q.H. Qin, and Y.L. Kang, A meshless model for transient heat
conduction in functionally graded materials. Computational mechanics, 2006.
38(1): 51-60.
[136] H. Wang, Q.H. Qin, and Y.L. Kang, A new meshless method for steady-state heat
conduction problems in anisotropic and inhomogeneous media. Archive of
Applied Mechanics, 2005. 74(8): 563-579.
[137] W. Chen, Z.-J. Fu, and C.-S. Chen, Recent advances in radial basis function
collocation methods. 2014, New York: Springer.
[138] M.H. Sadd, Elasticity: theory, applications, and numerics. 2009, Boston:
Academic Press.
[139] T.A. Davis, Direct methods for sparse linear systems. 2006, Philadelphia: SIAM.
[140] Z.W. Zhang, H. Wang, and Q.H. Qin, Method of fundamental solutions for
nonlinear skin bioheat model. Journal of Mechanics in Medicine and Biology,
2014. 14(04): 1450060.
[141] H. Wang, M.Y. Han, F. Yuan, and Z.R. Xiao, Fundamental-solution-based hybrid
element model for nonlinear heat conduction problems with temperature-
dependent material properties. Mathematical Problems in Engineering, 2013.
2013.
[142] H. Wang, Q.H. Qin, and W.A. Yao, Improving accuracy of opening-mode stress
intensity factor in two-dimensional media using fundamental solution based finite
element model. Australian Journal of Mechanical Engineering, 2012. 10(1): 41-
51.
[143] H. Wang, Q.H. Qin, and Y. Xiao, Special n-sided Voronoi fiber/matrix elements
for clustering thermal effect in natural-hemp-fiber-filled cement composites.
International Journal of Heat and Mass Transfer, 2016. 92: 228-235.
[144] Q.H. Qin, Variational formulations for TFEM of piezoelectricity. International
Journal of Solids and Structures, 2003. 40(23): 6335-6346.
[145] Q.H. Qin, Trefftz finite element method and its applications. Applied Mechanics
Reviews, 2005. 58(5): 316-337.
[146] M. Dhanasekar, J. Han, and Q.-H. Qin, A hybrid-Trefftz element containing an
elliptic hole. Finite Elements in Analysis and Design, 2006. 42(14): 1314-1323.
[147] Q. Qin, Postbuckling analysis of thin plates by a hybrid Trefftz finite element
method. Computer Methods in Applied Mechanics and Engineering, 1995. 128(1):
123-136.
181
[148] Q.-H. Qin, Formulation of hybrid Trefftz finite element method for elastoplasticity.
Applied Mathematical Modelling, 2005. 29(3): 235-252.
[149] X.W. Gao, The radial integration method for evaluation of domain integrals with
boundary-only discretization. Engineering Analysis with Boundary Elements,
2002. 26(10): 905-916.
[150] H. Wang and Q.H. Qin, Some problems with the method of fundamental solution
using radial basis functions. Acta Mechanica Solida Sinica, 2007. 20(1): 21-29.
[151] C.Y. Lee, H. Wang, and Q.H. Qin, Method of fundamental solutions for 3D
elasticity with body forces by coupling compactly supported radial basis functions.
Engineering Analysis with Boundary Elements, 2015.
[152] C.Y. Lee, H. Wang, and Q.H. Qin, Dual reciprocity boundary element method
using compactly supported radial basis functions for 3D linear elasticity with
body forces. International Journal of Mechanics and Materials in Design, 2015.
DOI:10.1007/s10999-015-9327-9.
[153] Q.-H. Qin and Y. Huang, BEM of postbuckling analysis of thin plates. Applied
Mathematical Modelling, 1990. 14(10): 544-548.
[154] H. Contopanagos, B. Dembart, M. Epton, J.J. Ottusch, V. Rokhlin, J.L. Visher,
and S.M. Wandzura, Well-conditioned boundary integral equations for three-
dimensional electromagnetic scattering. IEEE Transactions on Antennas and
Propagation, 2002. 50(12): 1824-1830.
[155] C. Chien, H. Rajiyah, and S. Atluri, An effective method for solving the hyper‐singular integral equations in 3 ‐D acoustics. The Journal of the Acoustical
Society of America, 1990. 88(2): 918-937.
[156] Y. Liu and F. Rizzo, A weakly singular form of the hypersingular boundary
integral equation applied to 3-D acoustic wave problems. Computer Methods in
Applied Mechanics and Engineering, 1992. 96(2): 271-287.
[157] J.I. Frankel, Regularization of inverse heat conduction by combination of rate
sensor analysis and analytic continuation. Journal of Engineering Mathematics,
2007. 57(2): 181-198.
[158] Q.H. Qin, Fracture mechanics of piezoelectric materials. 2001, WIT Press,
Southampton.
[159] Q.H. Qin, Green's function and boundary elements of multifield materials. 2007,
Elsevier, Oxford.
[160] A.C. Kaya and F. Erdogan, On the solution of integral equations with strongly
singular kernels. Quarterly of Applied Mathematics, 1987: 105-122.
[161] M. Guiggiani, Hypersingular formulation for boundary stress evaluation.
Engineering Analysis with Boundary Elements, 1994. 13(2): 169-179.
[162] L.J. Gray, J. Glaeser, and T. Kaplan, Direct evaluation of hypersingular Galerkin
surface integrals. SIAM Journal on Scientific Computing, 2004. 25(5): 1534-
1556.
[163] A.G. Polimeridis, J.M. Tamayo, J.M. Rius, and J.R. Mosig, Fast and accurate
computation of hypersingular integrals in Galerkin surface integral equation
formulations via the direct evaluation method. IEEE transactions on antennas and
propagation, 2011. 59(6): 2329-2340.
182
[164] P. Martin and F. Rizzo, Hypersingular integrals: how smooth must the density be?
International Journal for Numerical Methods in Engineering, 1996. 39(4): 687-
704.
[165] G. Monegato, Numerical evaluation of hypersingular integrals. Journal of
Computational and Applied Mathematics, 1994. 50(1): 9-31.
[166] G. Monegato, Definitions, properties and applications of finite-part integrals.
Journal of computational and applied mathematics, 2009. 229(2): 425-439.
[167] S. Mikhlin, Multidimensional Singular Integrals and Integral Equations. 1965.
[168] I.K.m. Lifanov, L.N. Poltavskii, and M.M. Vainikko, Hypersingular integral
equations and their applications. Vol. 4. 2003: CRC Press.
[169] B. Ninham, Generalised functions and divergent integrals. Numerische
Mathematik, 1966. 8(5): 444-457.
[170] D. Paget, The numerical evaluation of Hadamard finite-part integrals.
Numerische Mathematik, 1981. 36(4): 447-453.
[171] D. Paget, A quadrature rule for finite-part integrals. BIT Numerical Mathematics,
1981. 21(2): 212-220.
[172] H.R. Kutt, On the numerical evaluation of finite-part integrals involving an
algebraic singularity. 1975, Stellenbosch: Stellenbosch University.
[173] G. Monegato, On the weights of certain quadratures for the numerical evaluation
of Cauchy principal value integrals and their derivatives. Numerische
Mathematik, 1986. 50(3): 273-281.
[174] C.Y. Hui and D. Shia, Evaluations of hypersingular integrals using Gaussian
quadrature. International Journal for Numerical Methods in Engineering, 1999.
44(2): 205-214.
[175] P. Kolm and V. Rokhlin, Numerical quadratures for singular and hypersingular
integrals. Computers & Mathematics with Applications, 2001. 41(3): 327-352.
[176] M. Carley, Numerical quadratures for singular and hypersingular integrals in
boundary element methods. SIAM journal on scientific computing, 2007. 29(3):
1207-1216.
[177] M.P. Brandão, Improper integrals in theoretical aerodynamics-The problem
revisited. AIAA journal, 1987. 25(9): 1258-1260.
[178] M.S. Floater and K. Hormann, Barycentric rational interpolation with no poles
and high rates of approximation. Numerische Mathematik, 2007. 107(2): 315-331.
[179] J.R. Martins, P. Sturdza, and J.J. Alonso, The complex-step derivative
approximation. ACM Transactions on Mathematical Software (TOMS), 2003.
29(3): 245-262.
[180] N.I. Muskhelishvili and J.R.M. Radok, Singular integral equations: boundary
problems of function theory and their application to mathematical physics. 2008:
Courier Corporation.
[181] J. Hadamard, Lectures on Cauchy's problem in linear partial differential
equations. 1923, New Haven: Yale Univ. Press.
[182] J.-P. Berrut and L.N. Trefethen, Barycentric lagrange interpolation. Siam Review,
2004. 46(3): 501-517.
183
[183] J.-P. Berrut, R. Baltensperger, and H.D. Mittelmann, Recent developments in
barycentric rational interpolation, in Trends and applications in constructive
approximation. 2005, Springer. p. 27-51.
[184] J.-P. Berrut and G. Klein, Recent advances in linear barycentric rational
interpolation. Journal of Computational and Applied Mathematics, 2014. 259: 95-
107.
[185] J.-P. Berrut and H.D. Mittelmann, Lebesgue constant minimizing linear rational
interpolation of continuous functions over the interval. Computers & Mathematics
with Applications, 1997. 33(6): 77-86.
[186] C. Schneider and W. Werner, Some new aspects of rational interpolation.
Mathematics of Computation, 1986. 47(175): 285-299.
[187] M. Guiggiani, G. Krishnasamy, T. Rudolphi, and F. Rizzo, A general algorithm
for the numerical solution of hypersingular boundary integral equations. Journal
of applied mechanics, 1992. 59(3): 604-614.
[188] X.-W. Gao, An effective method for numerical evaluation of general 2D and 3D
high order singular boundary integrals. Computer methods in applied mechanics
and engineering, 2010. 199(45): 2856-2864.
[189] W.-Z. Feng, J. Liu, and X.-W. Gao, An improved direct method for evaluating
hypersingular stress boundary integral equations in BEM. Engineering Analysis
with Boundary Elements, 2015. 61: 274-281.
[190] M. Guiggiani, Hypersingular boundary integral equations have an additional free
term. Computational Mechanics, 1995. 16(4): 245-248.
[191] M. Guiggiani and A. Gigante, A general algorithm for multidimensional Cauchy
principal value integrals in the boundary element method. Journal of Applied
Mechanics, 1990. 57(4): 906-915.
[192] A. Sutradhar, J. Reeder, and L.J. Gray, Symmetric Galerkin boundary element
method. 2008, Heidelberg, Berlin: Springer Science & Business Media.
[193] L. Ricci-Vitiani, D.G. Lombardi, E. Pilozzi, M. Biffoni, M. Todaro, C. Peschle,
and R. De Maria, Identification and expansion of human colon-cancer-initiating
cells. Nature, 2007. 445(7123): 111-115.
[194] N.A. Peppas, Y. Huang, M. Torres-Lugo, J.H. Ward, and J. Zhang,
Physicochemical foundations and structural design of hydrogels in medicine and
biology. Annual review of biomedical engineering, 2000. 2(1): 9-29.
[195] B. Mehta, Intravascular hydrogel implant. 1993, Google Patents.
[196] W.M. Lai, J.S. Hou, and V.C. Mow, A triphasic theory for the swelling and
deformation behaviors of articular cartilage. Journal of biomechanical
engineering, 1991. 113(3): 245-258.
[197] J.M. Huyghe and J. Janssen, Quadriphasic mechanics of swelling incompressible
porous media. International Journal of Engineering Science, 1997. 35(8): 793-802.
[198] A.J.H. Frijns, J. Huyghe, and J.D. Janssen, A validation of the quadriphasic
mixture theory for intervertebral disc tissue. International Journal of Engineering
Science, 1997. 35(15): 1419-1429.
[199] H. Li, Smart hydrogel modelling. 2010, Berlin Heidelberg, Germany: Springer
Science & Business Media.
184
[200] Q.-S. Yang, Q.-H. Qin, L.-H. Ma, X.-Z. Lu, and C.-Q. Cui, A theoretical model
and finite element formulation for coupled thermo-electro-chemo-mechanical
media. Mechanics of Materials, 2010. 42(2): 148-156.
[201] X. Zhao, W. Hong, and Z. Suo, Stretching and polarizing a dielectric gel
immersed in a solvent. International Journal of Solids and Structures, 2008. 45(14):
4021-4031.
[202] P.J. Flory and J. Rehner Jr, Statistical Mechanics of Cross ‐Linked Polymer
Networks I. Rubberlike Elasticity. The Journal of Chemical Physics, 1943. 11(11):
512-520.
[203] D.N. Sun, W.Y. Gu, X.E. Guo, W.M. Lai, and V.C. Mow, A mixed finite element
formulation of triphasic mechano-electrochemical theory for charged, hydrated
biological soft tissues. International Journal for Numerical Methods in
Engineering, 1999. 45(10): 1375-1402.
[204] X. Zhou, Y.C. Hon, S. Sun, and A.F.T. Mak, Numerical simulation of the steady-
state deformation of a smart hydrogel under an external electric field. Smart
materials and structures, 2002. 11(3): 459.
[205] T. Dutta-Roy, A. Wittek, and K. Miller, Biomechanical modelling of normal
pressure hydrocephalus. Journal of biomechanics, 2008. 41(10): 2263-2271.
[206] J. Kim, Sequential methods for coupled geomechanics and multiphase flow. 2010,
Citeseer.
[207] A. Verruijt, An introduction to soil dynamics. 2010, Dordrecht: Springer.
[208] M.A. Biot, General theory of three‐dimensional consolidation. Journal of applied
physics, 1941. 12(2): 155-164.
[209] T. BELYTSCHKO, W.K. LIU, and B. MORAN, Nonlinear Finite Elements for
Continua and Structures. 2004, West Sussex: John Wiley & Sons Ltd.
[210] T.T. Wong, D.G. Fredlund, and J. Krahn, A numerical study of coupled
consolidation in unsaturated soils. Canadian Geotechnical Journal, 1998. 35(6):
926-937.
[211] J. Chen, Time domain fundamental solution to Biot's complete equations of
dynamic poroelasticity. Part I: two-dimensional solution. International Journal of
Solids and Structures, 1994. 31(10): 1447-1490.
[212] J. Chen, Time domain fundamental solution to Biot's complete equations of
dynamic poroelasticity part II: Three-dimensional solution. International Journal
of Solids and Structures, 1994. 31(2): 169-202.
[213] P. Li and M. Schanz, Time domain boundary element formulation for partially
saturated poroelasticity. Engineering Analysis with Boundary Elements, 2013.
37(11): 1483-1498.
[214] M. Schanz, Application of 3D time domain boundary element formulation to wave
propagation in poroelastic solids. Engineering Analysis with Boundary Elements,
2001. 25(4): 363-376.
[215] M. Cavalcanti and J. Telles, Biot's consolidation theory—application of BEM with
time independent fundamental solutions for poro-elastic saturated media.
Engineering Analysis with Boundary Elements, 2003. 27(2): 145-157.
185
[216] D. Soares, J. Telles, and W. Mansur, A time-domain boundary element
formulation for the dynamic analysis of non-linear porous media. Engineering
analysis with boundary elements, 2006. 30(5): 363-370.
[217] A. Papoulis, A new method of inversion of the Laplace transform. Quarterly of
Applied Mathematics, 1957. 14(4): 405-414.
[218] H. Stehfest, Algorithm 368: Numerical inversion of Laplace transforms [D5].
Communications of the ACM, 1970. 13(1): 47-49.
[219] F. Durbin, Numerical inversion of Laplace transforms: an efficient improvement
to Dubner and Abate's method. The Computer Journal, 1974. 17(4): 371-376.
[220] K.S. Crump, Numerical inversion of Laplace transforms using a Fourier series
approximation. Journal of the ACM (JACM), 1976. 23(1): 89-96.
[221] A.H. Cheng, P. Sidauruk, and Y. Abousleiman, Approximate inversion of the
Laplace transform. Mathematica Journal, 1994. 4(2): 76-82.
[222] C. Cryer, A comparison of the three-dimensional consolidation theories of Biot
and Terzaghi. The Quarterly Journal of Mechanics and Applied Mathematics,
1963. 16(4): 401-412.
[223] D.P. Mason, A. Solomon, and L.O. Nicolaysen, Evolution of stress and strain
during the consolidation of a fluid‐saturated porous elastic sphere. Journal of
applied physics, 1991. 70(9): 4724-4740.
186
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.