A novel meshfree smoothed least squares(SLS) method with applications to dielectrophoresis simulations

Summary Mesh-based numerical methods, such as finite element method FEM, and finite difference method FDM, have been the primary numerical techniques in engineering computations.. a The

Acknowledgements

I would like to express my deepest gratitude and sincerest appreciation to my

supervisor, Professor Liu Gui-Rong, for his invaluable guidance, dedicated support

and continuous encouragement throughout my four years Ph.D study His passion

and enthusiasm in research has inspired me enormously and will continue to influence

me for a life time I would also like to extend my gratitude to my co-supervisor,

Assistant Professor Li Hua for his great help and valuable guidance in my research

work

Many thanks are conveyed to my fellow colleagues and friends in Center for

ACES, Dr Gu Yuan Tong, Dr Dai Keyang, Dr Zhang Guiyong, Dr Zhao Xin, Dr

Deng Bin, Mr Li Zirui, Mr Bernard Kee Buck Tong, Mr Zhang Jian, Mr Khin Zaw, Ms

Chen Yuan, Mr Trung, and Mr George Xu I would like to thank them all for their

helpful discussions, constructive suggestions, as well as their inspirations and

encouragement throughout the course of my Ph.D study I sincerely appreciate their

friendship and support

I am grateful to every one of my family members, my parents, my younger sister

and younger brother, for their continuous support and encouragement which made my

Ph.D years meaningful and happy Last but not least, I must thank the National

University of Singapore for granting me research scholarship Many thanks are due to

Mechanical department and Center for ACES for their material support to every aspect

of this work

Table of contents

Acknowledgements i

Table of contents i

Summary vi

Nomenclature ix

List of Figures xi

List of Tables xviii

Chapter 1 Introduction 1

1.1 Background 1

1.1.1 Meshfree methods 1

1.1.2 Classification of meshfree method 6

1.1.3 Dielectrophoresis background 8

1.2 Literature review 10

1.2.1 A review of meshfree methods 10

1.2.1.1 SPH and RKPM method 10

1.2.1.2 The EFG method 11

1.2.1.3 The MLPG method 12

1.2.1.4 Point interplolation method (PIM) 12

1.2.2 Studies of dielectrophoresis 13

1.3 A review of meshfree shape functions 17

1.3.1 Moving least-squares (MLS) approximation 18

1.3.1.1 Formulation procedure of MLS 18

1.3.1.2 Weight functions 22

1.3.1.3 Properties of MLS shape functions 24

1.3.2 Polynomial point interpolation method (Polynomial PIM) 25

1.3.2.1 Formulation procedure of Polynomial PIM 25

1.3.2.2 Properties of polynomial PIM Shape Functions 28

1.3.2.3 Techniques for overcoming singularity in moment matrix 31

1.3.3 Radial point interpolation method (RPIM) 33

1.3.3.1 Formulation procedure of RPIM 33

1.3.3.2 Property of RPIM shape function 36

1.3.3.3 Implementation Issues 38

1.4 Objectives of the thesis 39

1.5 Organization of the thesis 41

Chapter 2 Development of a novel meshfree smoothed least-squares (SLS) method 45

2.1 Introduction 45

2.2 Meshfree smoothed least-squares (SLS) formulation 47

2.2.1 General least-squares formulations 48

2.2.2 Gradient smoothing 50

2.3 Numerical Examples 52

2.3.1 One-dimensional problems 52

2.3.1.1 Convection-diffusion problem 52

2.3.1.2 Pure convection problem 54

2.3.2 Two-dimensional problems 56

2.4 Remarks 61

Chapter 3 Validation of the developed meshfree smoothed least-squares (SLS) method for linear elasticity 72

3.1 Introduction 72

3.2 The SLS formulation for linear elasticity problem 73

3.3 Elasticity problems 77

3.3.1 2-D Standard patch test 77

3.3.2 Cantilever beam subjected to a parabolic shear traction at the end 78

3.3.3 An infinite plate subjected to uniaxial traction along horizontal direction ……… 80

3.4 Remarks 82

Chapter 4 Validation of the developed meshfree smoothed least-squares (SLS) method for steady incompressible flow 92

4.1 Introduction 92

4.2 The Navier-Stokes equations in the velocity-pressure-vorticity formulation ……… 94

4.3 The SLS formulation for Navier-Stokes equations 97

4.4 Steady incompressible flow problems 99

4.4.1 A model problem for Stokes equations 99

4.4.2 Driven cavity flow problem for Stokes equations 101

4.4.3 Driven cavity flow problem for Navier -Stokes equations 102

4.4.4 Backward-facing step flow problem 103

4.5 Remarks 103

Chapter 5 Application of the meshfree smoothed least-squares (SLS) method for dielectrophoresis 117

5.1 Introduction 117

5.2 Dielectrophoresis theory 118

5.3 Meshfree smoothed least-squares formulation for dielectrophoresis 119

5.4 Dielectrophoresis simulation 122

5.5 Remarks 126

Chapter 6 Simulation of an extruded quadrupolar dielectrophoretic trap 133

6.1 Introduction 133

6.2 Radial point collocation method (RPCM) 134

6.3 Meshless finite difference method 138

6.4 Simulation of extruded quadruple trap 143

6.4.1 Governing equations and boundary conditions 143

6.4.2 Determination of dielectrophoretic forces 145

6.4.3 Determination of hydrodynamic forces 146

6.4.4 Determination of the total resultant force 147

6.4.5 Validation with experimental results 147

6.4.5.1 Comparison between RPCM and MFD 148

6.4.5.2 Comparison between numerical prediction and experimental results ………148

6.4.6 Results and discussion 150

6.4.6.1 Results for resultant force field 150

6.4.6.2 Variation of holding characteristic with trap geometry 152

6.4.6.3 Variation of holding characteristic with particle radius 154

6.4.6.4 Variation of holding characteristic with Clausius-Mossotti factor 155 6.5 Remarks 155

Chapter 7 Simulation of an interdigitated dielectrophoretic array 166

7.1 Introduction 166

7.2 Additional dielectrophoresis theories 167

7.3 Linearly conforming point interpolation method (LC-PIM) 169

7.3.1 Node selection 169

7.3.2 Gradient smoothing 171

7.3.3 Variational form 173

7.4 Results and discussion 175

7.4.1 Simulation of the DEP array 176

7.4.1.1 Linear potential change in the gap 177

7.4.1.2 Exact boundary condition in the gap 178

7.4.2 Simulation of the traveling wave DEP array 179

7.4.2.1 Study of the traveling wave DEP array 179

7.4.3 Simulation results using RPIM shape function 180

7.5 Remarks 182

Chapter 8 Conclusion and future work 201

8.1 Conclusion remarks 201

8.2 Recommendations for future research 205

References 207

Publications arising from thesis 216

Summary

Mesh-based numerical methods, such as finite element method (FEM), and finite

difference method (FDM), have been the primary numerical techniques in engineering

computations Due to mesh related problems of these methods, a new group of

numerical techniques called meshfree methods have been proposed and developed in

recent years Many different methods and techniques have been developed for

applications in different engineering fields It has been a standard practice to employ

different numerical schemes for different types of differential equations in engineering

problems

This thesis focuses on the development and application of a unified meshfree

method applicable for all types of differential equations that govern practical

engineering problems The objectives of the present study are two-fold: One is to

develop new meshfree method with a unified formulation so that it can be potentially

applied to all engineering problems; the other is to apply the developed and existing

meshfree methods to simulations of dielectrophoresis (DEP) based devices, which have

attracted great attention in recent Micro-Electro-Mechanical Systems (MEMS)

researches

The first contribution of this thesis is development of the meshfree smoothed

least-squares (SLS) method based on first-order least-squares formulation The

meshfree SLS method uses a unified formulation for all types of partial differential

equations: elliptic, parabolic, hyperbolic or mixed As long as the equations are well

posed and have a unique solution, the SLS method can always produce a good

approximate solution The properties of the SLS method have been studied in details

The SLS method is found particularly effective for solving non-self-adjoint system

such as the convection dominated problem, which is difficult to solve by conventional

Galerkin methods The SLS method always leads to symmetric positive-definite

matrices which can be efficiently solved by iterative methods Using the SLS method,

no special treatments, such as upwinding, artificial dissipation, staggered grid or

non-equal-order elements, operator-preconditioning, etc are needed

In the second part, the SLS method is devoted to numerical analysis of various

engineering problems, including linear elastic problems, incompressible steady flow

problem, and dielectrophoresis problem, etc It is found that the SLS method achieves

better accuracy and convergence rate, comparing with other methods based on Galerkin

formulation The SLS method is based on first-order least-squares method, so that the

primary variables and the derivatives can be solved simultaneously and with the same

order of accuracy This unique feature is of very importance for many practical

problems where it is essential to obtain accurate solutions in the derivatives, such as

strain and stress in elasticity problems, flux in fluid problems

The last part of the thesis deals with simulations of DEP based systems using

meshfree techniques A strong-form meshfree method termed radial point collocation

method (RPCM) is used to simulate the extruded quadrupolar DEP trap Compared to

weak-form methods, strong-form methods are easy to implement and have lower

computational cost The model developed is able to approximate the strength of the

trap, and it can also be used for design optimization purpose The model is validated

with good accuracy by comparing with experimental data Another meshfree

technique, linear conforming point interpolation method (LC-PIM) is used for

simulation of the dielectrophoretic array as well as the traveling wave

dielectrophoretic array LC-PIM has been found to be very effective to capture the

high gradient feature of the electric field, and can produce accurate results for

derivatives of the shape functions, which are important for computing the DEP forces

in DEP related simulations The results have been compared with the analytical

solution obtained using Fourier series analysis, good accuracy has been demonstrated

Nomenclature

N(x) Vector of shape functions

( )

m

List of Figures

Figure 1.1 Pascal triangle of monomials for two dimensional spaces .44

Figure 2.1.Illustration of background triangular cells and formation of nodal representative domain 65

Figure 2.2 Results of ufor convection-diffusion problem with different Pelect numbers using SLS meshfree method A total of 21 regularly distributed nodes are used 65

Figure 2.3 Results of du / dxfor convection-diffusion problem with different Pelect numbers using SLS meshfree method A total of 21 regularly distributed nodes are used 66

Figure 2.4 Results of ufor convection-diffusion problem with Pelect =0.25 66

Figure 2.5 Results of ufor convection-diffusion problem with Pelect =1.25 67

Figure 2.6 Comparing solutions of pure convective problem withε=0.05 for difference numerical methods 67

Figure 2.7. Comparing solutions of pure convective problem withε =0.05 for difference numerical methods 68

Figure 2.8 Convergence of primary variables for Laplace example 1 68

Figure 2.9 Convergence of dual variables for Laplace example 1 69

Figure 2.10 Irregular nodal distribution in Laplace example 2 70

Figure 2.11 Convergence of primary variables for Laplace example 2 70

Figure 2.12 Convergence of dual variables for Laplace example 2 71

Figure 3.1 (a) Patch a with 16 irregular distributed nodes and (b) Patch b with 25 irregularly distributed nodes (c) Patch c with 36 irregularly distributed nodes 84

Figure 3.2 A two-dimensional cantilever solid subjected to a parabolic traction on the right edge 84

Figure 3.3 Boundary conditions for cantilever beam problem 85

Figure 3.4 Deflection distribution along the neutral line .85

Figure 3.5 Shear stress distribution along the line x=L/2. 86

Figure 3.6 Relative errors in displacement norm for beam problem 86

Figure 3.7 Relative errors in energy norm for beam problem 87

Figure 3.8 Infinite two-dimensional solid with a circular hole subjected to a tensile force and its quarter model 87

Figure 3.9 Boundary conditions for Infinite plate with a circular hole problem 88 Figure 3.10 Distribution of U along the bottom edge for infinite plate with a x circular hole problem 88

Figure 3.11 Distribution of U along the left edge for infinite plate with a y circular hole problem 89

Figure 3.12 Stress distribution along the left edge for infinite plate with a circular hole problem 89

Figure 3.13 Relative errors in displacement norm for plate with a circular hole problem 90

Figure 3.14 Relative errors in energy norm for plate with a circular hole problem .90

Figure 3.15 Change of relative errors in energy norm with Poisson ratio for plate with a circular hole problem 91

Figure 4.1. Boundary conditions for Stokes model problem 105

Figure 4.2. Node distributions for Stokes model problem 106

Figure 4.3 Computed convergence rate for Stokes model problem (a) 2nd-order polynomial basis with boundary condition 1 from Figure 4.1 (b) 2nd-order polynomial basis with boundary condition 2 from Figure 4.1 (c) 2nd-order polynomial basis with boundary condition 3 from Figure 4.1 107

Figure 4.4 The boundary conditions for driven cavity flow 108

Figure 4.5.Computed results for lid-driven cavity flow problem (a) Pressure (b) Vorticity (c) Streamline (d) velocity .108

Figure 4.6.Node distribution for driven cavity flow problem 109

Figure 4.7 Computed results for lid-driven cavity flow problem at Re = 1000 (a) Pressure (b) Vorticity (c) Streamline (d) velocity 109

Figure 4.8. Computed results for lid-driven cavity flow problem at Re = 5000 (a) Pressure (b) Vorticity (c) Streamline (d) velocity 110

Figure 4.9 (a)Comparison of v-velocity along horizontal line through geometric center; (b) (a)Comparison of u-velocity along vertical line through geometric center; 111

Figure 4.10.Boundary conditions for backward facing step flow problem 112

Figure 4.11 Computed results for backward facing step flow problem at Re =200 .113

Figure 4.12 Computed results for backward facing step flow problem at Re =400 .114

Figure 4.13. Computed results for backward facing step flow problem at Re =600 .115

Figure 4.14 Reattachement length versus Reynolds number for backward-facing flow .116

Figure 5.1. Ilustration of dielectrophoresis in a point-plane electrode system If particle is more polarizable than surrounding medium, it moves towards highest electric field region due to positive dielectrophoresis; If particle is less polarizable than surrounding medium, it is repelled from highest electric field region due to negative dielectrophoresis 127

Figure 5.2 A typical interdigitated electrode array 127

Figure 5.3 Schematic of the boundary condition on the bottom 128

Figure 5.4 Boundary condition of one unit cell .128

Figure 5.5.Contour plot of electric potential in one unit cell 129

Figure 5.6. Contour plot of electric field in one unit cell 129

Figure 5.7 Computed solution for distribution of F'DEP x, 130

Figure 5.8 Computed solution for distribution of F'DEP y, 130

Figure 5.9 Computed solution for distribution of F'DEP 131

Figure 5.10 Computed solution for distribution of F DEP y, 131

Figure 5.11 Computed solution on middle vertical line x’=1 132

Figure 6.1 A problem governed by PDEs in domainΩ 158

Figure 6.2 (a) Extruded quadrupolar DEP trap (Voldman, website); (b) Simulation domain for the simplified 2-D model (in mμ ) 159

Figure 6.3 llustration of boundary conditions 159

Figure 6.4 Comparison of electric potential between RPCM and LSFD on line y =-30 160

Figure 6.5 Comparison of convergence between RPCM and LSFD .160

Figure 6.6 Non-dimensional Electric potential field obtained using RPCM 161

Figure 6.7 Numerical and experimental results comparison for 13.2 mμ diameter beads .161

Figure 6.8 Numerical and experimental results comparison for 10 mμ diameter beads 162

Figure 6.9 Determination of release flow velocity 162

Figure 6.10 Variation of release flow velocity with voltage .163

Figure 6.11 Effect of trap length on strength of the trap 163

Figure 6.12 Effect of entrance posts separation on strength of the trap 164

Figure 6.13 Effect of exit posts separation on strength of the trap .164

Figure 6.14 Change of CM factor with frequency in different fluid 165

Figure 7.1 Illustration of background cells of triangles and the selection of

supporting nodes 183

Figure 7.2 Illustration of background triangular cells and formation of nodal

representative domain 183

Figure 7.3 Interdigitated electrode array used for dielectrophoretic separation and

traveling wave dielectrophoresis 184

Figure 7.4 Boundary conditions of a unit cell in dielectrophoresis array 184

Figure 7.5 Solution of the problem near the electrode (a) The electric potential

'

φ (b) The magnitude of electric field | ∇φ' |R (c) The magnitude of

vector ∇ ∇ | φ' |R 2 (d) The direction of vector ∇ ∇ | φ' |R 2 186

Figure 7.6 Comparison of numerical and analytical solution on middle vertical

line x’=1 (a) comparison of electric field magnitude | ∇φR | (b)

comparison of magnitude of vector ∇|∇φR |2 .187

Figure 7.7 Comparison of numerical and analytical solution on horizontal line

y’=0.1 (a) comparison of electric field magnitude | ∇ φR| (b) comparison of magnitude of vector ∇ | ∇ φR| 2 188

Figure 7.8 Comparing magnitude of ∇ | ∇ φR| 2 189

Figure 7.9 Boundary conditions for a unit cell of traveling wave array 189

Figure 7.10 Solution of electric potential for traveling wave array (a) Real part

of potential phasor φ (b) Imaginary part of potential phasor 'R φ'I 190

Figure 7.11 Solution of the cDEP force component (a)Magnitude of the vector

)

| ' '

|

| ' ' (|

∇ (b) Direction of the vector ∇ ' × ( ∇ 'φ'R×∇ 'φ'I) 192

Figure 7.13 Solution of the problem near the electrode (RPIM shape function is

used) (a) The electric potential 'φ (b) The magnitude of electric field |∇′φR | (c) The magnitude of vector ∇′|∇′φR |2 (d) The

direction of vector ∇′|∇′φR |2 194

Figure 7.14 Comparison of numerical and analytical solution on middle vertical

line x’=1(RPIM shape function is used) (a) comparison of electric field magnitude |∇φR | (b) comparison of magnitude of vector

2

|

|∇φR

∇ 195

Figure 7.15 Comparison of numerical and analytical solution on horizontal line

y’=0.1(RPIM shape function is used) (a) comparison of electric field magnitude |∇φR | (b) comparison of magnitude of vector

Figure 7.17 Solution of electric potential for traveling wave array (RPIM shape

function is used) (a) Real part of potential phasor φ'R (b) Imaginary

part of potential phasor φ'I 198

Figure 7.18 Solution of the cDEP force component (RPIM shape function is

used) (a)Magnitude of the vector '(| ' ' |2 | ' ' |2)

Direction of the vector '(| ' ' |2 | ' ' |2)

Figure 7.19. Solution of the cDEP force component (RPIM shape function is

used) (a)Magnitude of the vector ' ( ' '∇ × ∇ φ R×∇' ' )φ I (b) Direction of the vector ∇ × ∇' ( ' 'φ R×∇' ' )φ I 200

List of Tables

Table 1.1 Typical conventional form of radial basis functions .44

Table 3.1 Error norm of SLS method for linear patch test .83

Table 6.1 Schematic of geometrical parameters (R =10μm is used as reference

parameter) .157

Table 6.2 Influence of particle radius on the release velocity 157

Chapter 1

Introduction

1.1 Background

1.1.1 Meshfree methods

In order to analyze an engineering system, a mathematical model is first developed

with some possible simplifications and assumptions to describe the physical

phenomenon of the system These mathematical models are usually expressed in form

of governing equations with proper boundary conditions (BCs) and/or initial conditions

(ICs) The governing equations are generally differential equations, which are usually

difficult to solve analytically With rapid development of computer technology, various

numerical techniques have been developed and applied to solve numerous complex

practical problems in engineering and applied science The most popular numerical

methods include the finite difference method (FDM), the finite volume method (FVM),

the finite element method (FEM) and boundary element method (BEM), etc In

particular, the FEM has become one of the major numerical solution techniques, and

widely used in engineering fields including solid mechanics, fluid flow, heat transfer,

electric fields, etc, due to its versatility for complex geometry and flexibility for various

linear and non-linear problems

The FEM does not operate on the differential equations directly, instead, the

governing differential equations, whether being ordinary differential equations (ODEs)

or partial differential equations (PDEs), are transformed into equivalent variational

forms by means of certain principles, such as variational method, minimum potential

energy principle or principle of virtual work The solution appears in an integral of a

quality over the problem domain The integral of a function over a domain can be

divided into the sum of integrals over a collection of subdomains called finite element

These elements are connected together by a topological map termed as mesh As long

as the mesh is fine enough or the elements are sufficiently small, polynomial functions

can approximately represent the local behavior of the solution One of the advantages

of the FEM is that it is essentially independent of geometry, and many domains of

complex shapes can be handled by the FEM with ease The clear structure of the FEM

makes it possible to construct general purpose software, many commercial software

packages are made available nowadays e.g ABAQUS, ANSYS, etc The FEM has a

solid mathematical basis due to the extensive work done in the past decades, and this

adds the reliability and in many cases makes it possible to analyze mathematically

Despite of its robustness and effectiveness in numerical analysis, the FEM has the

inherent shortcomings of numerical methods that rely on meshes which are connected

together by nodes in a predefined manner A number of mesh related problems have

become increasingly evident:

1) High cost in FEM mesh generation

Mesh generation is the first part of FEM analysis, and a prerequisite in using any

FEM code or software The quality of the mesh plays an important role in the accuracy

of the final solution Computer auto-generated meshes are oftentimes of poor quality

and non-desirable, human intervention is needed in most cases especially for problems

of complex three-dimensional domains such increases the labour cost of the computer

aided design (CAD) projects

2) Low accuracy in derivatives of primary variables

Due to the assumption of piecewise continuous displacement (primary variables)

made in the FEM formulation, the derivatives or secondary variables such as strain and

stress obtained from the FEM are usually discontinuous at the interface of the elements,

and much less accurate Special post-processing techniques are required to restore the

accuracy of the derivatives

3) Difficulty in adaptive analysis

Adaptive analysis is an important step in numerical analysis to improve the

accuracy of the solution In using the FEM, re-meshing is necessary at each adaptive

process to ensure the proper connectivity, and add additional expensive computational

cost The mapping of field variables between meshes of successive steps also adds

additional cost of computation time and reduce the accuracy in the solution Adaptive

analysis for three-dimensional problems is an extremely burdensome and

time-consuming task

4) Difficulty in dealing with certain special classes of problems

z Large deformation that leads to extremely skewed meshes

z Crack growth with arbitrary and complex paths which do not coincide

with the original element interfaces

z Breakage of material with large number of fragments, since the FEM is based on continuum mechanics and the predefined connectivity between

elements can not be broken

To overcome the mesh related problems, a group of new numerical methods called

meshfree, meshless or element-free method are emerging and achieved remarkable

progress in recent years In these methods, no predefined mesh structure is required,

and the problem domain is represented by a set of scattered nodes

Meshfree methods and techniques developed so far include the general finite

difference method (Liszka and Orkisz, 1979), smooth particle hydrodynamics (SPH)

(Gingold and Monaghan, 1977), the diffuse element method (DEM) (Nayroles et al.,

1992), the element-free Galerkin (EFG) method (Belytschko at el., 1994), the

reproducing kernel particle method (RKPM) (Liu et al, 1995), the point interpolation

method (Liu and Gu , 2001a,b,c, Liu and Zhang, 2005), the hp clouds method (Liszka at

el., 1996), meshless local Petrov-Galerkin (MLPG) (Atluri and Zhu 1998), etc These

meshfree methods do not need meshes to discretize the problem domain, but a set of

irregularly scattered nodes The shape functions are constructed entirely based on

nodes, no meshes or connectivity are needed Great flexibility is provided in the nodal

selection for constructing shape functions The adaptive analysis can be handled with

ease using meshfree methods even for problems that pose great challenges for the FEM

such as crack growth and large deformation problems In addition, some meshfree

methods such as the LC-PIM (Liu and Zhang, 2005; Liu and Zhang, 2007) and

LC-RPIM (Li et al., 2007) can provide upper bound solutions This is a very important

and attractive feature A procedure proposed by Liu et al (2006b) is able to determine

both upper bound and lower bound for the exact solution in energy norm to elasticity

problems without knowing it in advance In this procedure, LC-RPIM is used to

compute the upper bound, and the standard FEM is used to compute the lower bound

based on the same mesh for the problem domain

The development of meshfree methods is still in its infant stage, there are still

some technical difficulties that need to be solved Some of the most frequently

addressed concerns for the existing meshfree methods are listed as follows

1) Generally, the computational cost of the meshfree methods is higher than the

FEM due to the complexity in constructing the shape functions The resulting

system matrix has a wider bandwidth that adds more computational cost

2) Some meshfree methods still require background meshes for the integration of

the weak-form formulation over the problem domain, and therefore are not truly

meshfree, e.g., element-free Galerkin (EFG) method etc

3) Some meshfree shape functions do not possess Kronecker Delta property so that

additional techniques such as penalty method are required to impose the essential

boundary conditions

1.1.2 Classification of meshfree method

According to its formulation procedure, meshfree method can be classified into

one of the three categories, namely meshfree weak-form method, meshfree strong-form

method and meshfree weak-strong form method The Diffuse Element Method (DEM)

(Nayroles et al., 1992), Element Free Galerkin(EFG) method (Belytschko et al., 1994),

meshless local Petrov-Galerkin (MLPG) method (Atluri and Zhu, 1998), Local radial

point interpolation method (LRPIM) (Liu and Gu, 2001 b,c), the point interpolation

method (PIM) (Liu and Gu, 2001a), etc, are based on Galerkin weak formulation and

under the category of meshfree weak-form method Meshfree strong-form methods,

which are formulated, based on strong form, include finite point method (Onate et al,

1996), radial point collocation method (Liu et al., 2005; Liu et al., 2006a; Kee et al.,

2007a,b), smooth Particle Hydrodynamics (SPH) method(Liu and Liu, 2003;

Monaghan, 1988), etc The third category of meshfree methods are based on a

combined formulation using both strong-form and weak-form, such as the meshfree

weak-strong form method (Liu and Gu, 2003b; Liu et al., 2004)

Comparing to strong-form methods, meshfree weak-from methods are more stable

and accurate, and have been applied successfully to problems in many engineering

fields such as solid and structure mechanics In meshfree weak-form methods, the

Neumann boundary conditions can be imposed naturally However, most of the

above-mentioned weak-form methods still have to use a background mesh for

integration, they are only meshfree in the sense of not requiring mesh for function

approximation, hence not truly meshfree Weak-form methods are more computational

expensive due to the integrations, and very inefficient for large scale problems Due to

this reason, some methods based on local Petrov-Galerkin weak formulation have been

proposed to avoid the use of background mesh, such as the above mentioned MLPG,

LRPIM, etc In these methods, weak-form integration is only performed on local

subdomains which are easy to form

Meshfree strong-form methods do not involve integration, therefore no

background mesh are needed, and are considered as truly meshfree methods The most

attractive advantage of meshfree strong-form methods is that they are very easy to

implement and computationally efficient They have been widely used in fluid

mechanics However, strong-form methods suffer from stability problem, especially

when Neumann boundary conditions are involved

In order to combine the advantages of strong-form and weak-form methods, the

meshfree weak-strong form method (MWS) was proposed by Liu and Gu (2003b) The

stability problem of meshfree strong-form methods is raised primarily by the

imposition of Neumann boundary conditions Therefore, in the MWS method, the local

weak-form is used to enforce Neumann boundary conditions for nodes on or close to

the natural boundaries, and strong-form formulation is used for the rest of the nodes

Since the number of nodes near the natural boundaries is relatively small compared to

the total nodes, the computational cost from the weak-form integration is nearly

negligible

Another type of classification categorizes the meshfree methods according to the

way to construct shape functions This type of classification leads to three main

categories:

1) Finite integral representation methods, such as smooth particle

hydrodynamics (SPH) method and reproducing kernel particle method

(RKPM);

2) Finite series representation methods, such as moving least squares (MLS)

method, partition of unity (PU) method, and point interpolation method (PIM)

3) Finite differential representation methods, such as finite difference method

and finite point method

1.1.3 Dielectrophoresis background

Recent research has shown growing interest in biological particle investigation,

such as cells, DNA Cells sized from less than a micron up to several hundred microns

make up all living organisms Manipulation, separation and handling of individual

bio-particles have become a hot topic in recent scientific research

Many methods have been used for the purpose of manipulating, concentrating and

separating biological particles These methods employ some kinds of physical forces

such as mechanical, hydrodynamic, ultrasonic, and optical, etc Among these methods,

dielectrophoresis (Pohl, 1978；Jones, 1995) is becoming increasingly popular because

of its ease of micro-scale generation and structuring an electric field on microchips It

also has the advantages of speed, flexibility and controllability Fabrication of the DEP

devices is also inexpensive

Particles suspended in fluid exhibit motion when subjected to AC electric fields

The applied field results in force on both the particles and the fluid, the study of which

is referred to as AC electrokinetics The AC electrokinetics techniques have been used

for the controlled manipulation and characterization of particles, and the separation of

mixtures A number of phenomena could arise from the interaction of the field with a

suspension of particles When exposed to an electric field, a charged particle will

experience a Coulomb force and the resulting movement is termed electrophoresis

When a neutral particle is subjected to a non-uniform AC electric field, a dipole

moment is induced in the particle The polarized particle experiences a force that can

cause them to move to region of high or low electric field, depending on the particle

polarizability compared with the suspending medium This force was termed

dielectrophoresis (DEP) （ Pohl, 1978 ； Jones, 1995 ） As one of the attractive

technologies for manipulating particles in micrometer scale, DEP has a wide variety of

applications in micro electromechanical system (MEMS), especially in biomedical

field It has been used for trapping, focusing, translation, fractionation of chemical and

biological particles in fluid medium It is particularly suitable for applications at

microscale fluidic device that can be fabricated by inexpensive fabrication methods

DEP methods are applicable to purification and characterization of a wide variety of

biological and clinical components

1.2 Literature review

1.2.1 A review of meshfree methods

1.2.1.1 SPH and RKPM method

The smooth particle hydrodynamics (SPH) method (Lucy, 1977; Gingold and

Managhan, 1977) is one of the earliest developed meshfree methods, which was

originally used for modeling astrophysical phenomena without boundaries such as

exploding stars and dust clouds In contrast to many conventional meshfree methods,

the SPH uses an integral representation of a function In the formulation of SPH

approximation, the field function u at an interest point x can be expressed in the

the kernel approximation to be valid and converge Four types of weight functions have

been proposed, including cubic spline, quartic spline, exponential spline and new

quartic smoothing function Details on weight functions will be discussed in the

following section when we introduce the Moving Least-Squares (MLS) shape

functions

Reproducing kernel particle method (RKPM) is another well-known meshfree

method proposed by Liu et al (1993, 1995) The field function u is represented in an

integral form by adding a correction function into the SPH approximation given in the

is the correction function Example of correction function for one

dimensional case is:

RKPM has been successfully applied to solve many practical problems in area of solids,

structures, and acoustics, etc (Liu et al, 1993, 1995, 1997)

1.2.1.2 The EFG method

The element free Galerkin (EFG) method was proposed by Belytschko et al

(1994), in which the MLS approximation was used for the first time in the Galerkin

procedure to establish the weak form of PDEs The EFG method is accurate and stable

(Belytschko et al 1994; 1996), and the convergence rate is even higher than that of

FEM (Belytschko et al 1994) Furthermore, the EFG method does not seem to exhibit

volumetric locking even when the linear basis functions are used, and the irregularity of

the node distribution does not affect the performance of the EFG method (Belytschko et

al 1994)

1.2.1.3 The MLPG method

In the EFG method, the global background integration mesh is needed This makes

the EFG method a non-truly meshfree method Atluri and his co-workers proposed a

so-called “truly” meshfree method termed as meshfree local Petrov-Galerkin (MLPG)

method (Atluri and Zhu, 1998), in which the concept of local weak-form is first

introduced The MLPG method does not need a background integration mesh over the

entire problem domain; instead, a local sub-domain is defined around each node for the

integration of the local weak form The continuity between neighboring local

sub-domains is not required

The MLPG method has been applied to elastic-static problems (Atluri and Zhu,

2000), 4th order thin beams (Atluri et al., 1999a) and thick beams (Cho and Atluri,

2001), linear fracture problems (Ching and Batra, 2001), fluid mechanics problems

(Lin and Atluri, 2000; 2001), and so on

1.2.1.4 Point interplolation method (PIM)

The point interpolation method (PIM) in weak-form formulation was originated

by Liu and his co-workers (Liu and Gu, 2001; Wang and Liu, 2001 a, b; Liu, 2002)

Either polynomial PIM shape functions or radial point interpolation method (RPIM)

shape functions can be used These shape functions possess the Delta function property,

so that the essential boundary conditions can be imposed easily More details about

PIM and RPIM shape function construction will be discussed in the following sections

when we review the commonly used shape functions

1.2.2 Studies of dielectrophoresis

DEP has been widely used in manipulating particles on the micrometer scale and

for the separation of particles from a heterogeneous mass Detailed theoretical

background of DEP will be discussed in the related chapters in this thesis A brief

review of some common DEP researches will be presented in this section

a) Flow separation

Flow separation is the simplest method of practical dielectrophoretic separation

The separation is carried out in a chamber which has an electrode array on the bottom,

and is enclosed by sides and a lid There is a single inlet and outlet The mixture that is

to be separated is pumped into the chamber by using a syringe pump Then the

electrodes are energized and the mixture will be separated due to the different

properties of the two populations One experiencing positive dielectrophoresis will be

attracted to the electrode edges, and the other experiencing negative dielectrophoresis

will be repelled to local minima at the center or between electrodes The later

population will be pushed through the outlet by the flow and the particles will be

collected Then the electric field will be removed and former population is released to

the chamber The flow will then push the former population to the outlet where they are

collected in a separate container

This method can be very effective if the dielectric properties of the particle types

are greatly dissimilar For instance, DEP has been applied for separation of bacteria

from mammalian cells (Wang et al., 1993), blood cells from cancer cells (Becker et al.,

1995; Cheng et al., 1998), normal from malaria-infected blood cells (Gascoyne et al.,

1998), CD34 stem-cells from blood (Stephens et al., 1996), live cells from dead cells

(Markx et al., 1994), and cells from debris If the difference between different cell

population is small, separation efficiency may be improved by passing the elute

thourgh the separator several times (Stephens et al., 1996) This type of separation is

sufficient for separating such distinctly different particles as bacteria from blood cells,

but inadequate for many mammalian cell applications

b) Field-flow fractionation

Field-flow fractionation (FFF) (Wang et al., 1998; Huang et al., 1999; Yang et al.,

2000; Wang et al., 2000; Muller et al., 2000; Markx and Tethig, 1995) is a family of

methods in which force fields are applied to particles to position them characteristically

within the velocity profile of a fluid flow stream The applied force field will place

different types of particles at different heights above the surface in accordance with

their characteristics Due to the effect of viscous force, the particles will be travelling at

different speeds according to their distance from the surface Hence, when particles are

introduced to the force field at the same point and time, they will exit at different time

according to their heights above the surface such that the particles will be separated

Typical fields used for dielectrophoretic FFF include gravity (sedimentation FFF),

temperature gradient (thermal FFF) and viscous properties of the particle in a crossflow

(flow FFF) Giddings elucidated three primary modes of FFF: normal, steric, and

hyperlayer Normal FFF involves thermal diffusion profiles of sub- mμ -sized particles

As a rough guide, Brownian motion and thermal diffusion are negligible for particles

above 1 mμ in diameter at room temperature In steric FFF, the applied force causes the particles to impact one side of the separation chamber causing them to experience steric

hindrances that diminish their velocity in the flow stream In hyperlayer FFF, the

particles are positioned away from the chamber walls at an equilibrium height in the

flow stream and are carried at the velocity of the fluid at the specific height

c) Stepped flow seperation

Another effective separation method which could possibly achieve 100%

efficiency is called stepped flow separation The interdigitated castellated electrode

array configuration was first devised by Markx et al (1995) It has two-port fluid entry

and exit, and has been shown to be effective for the separation of bacteria, yeast and

plant cells Two types of particles are brought to center of the array by flow from one of

the port, then the flow is removed and electric field is applied, the particles will be

trapped by positive and negative DEP Those particles experiencing negative DEP are

confined weakly, so that when a flow is re-introduced, the particles are displaced

towards the outlet The field is then removed, and both populations are released, a brief

flow is introduced in the opposite direction and it will move those particles trapped by

positive DEP towards the original inlet, whilst those trapped by negative DEP have still

moved a net distance towards the original outlet

The process is repeated over and over again until the two populations are moved to

the opposite ends of the array The disadvantage of this method is that it has very low

speed However, the repeated action will bring the efficiency to nearly 100%

d) Travelling wave dielectrophoresis

The idea of using a traveling electric field to induce controlled translational

motion of bioparticles was first found in Masuda’s work (Masuda et al., 1987; Masuda

et al., 1988) The traveling fields were generated by applying three-phase voltages of

frequency 0.1-100 Hz to a series of bar-shaped electrodes Masuda et al proposed that

such traveling fields could eventually find application in the separation of particles

according to their size or electrical charge It has been shown in Huang’s study (Huang

et al., 1993) that traveling fields of frequency between 1 kHz and 10 MHz can be used

to manipulate yeast cells and to separate them selectively when they are mixed with

bacteria It was shown by Fuhr (Fuhr et al., 1991; Hagedorn et al., 1992) traveling fields

of frequency between 10 kHz and 30 MHz are capable of imparting linear motion to

pollen and cellulose particles It has been shown in later work (Talary et al., 1996; Hugh

et al., 1996) that by changing the frequency of the traveling field, it is possible to switch

between conventional and travelling wave DEP to enhance separation

The discovery and utilization of traveling wave dielectrophoresis have received a

great deal of attention in laboratory-on-a-chip systems application, since the force

exerted can be made to act in a direction parallel to the plane of the electrodes Four

signal phased 0o, 90 o, 180 o, and 270 o is the most commonly used in traveling wave

DEP electrode arrays

f) Dielectrophoresis simulations

Many simulation works have been done for various types of DEP devices, such as

the castellated electrode array (Green and Morgan, 1997), interdigitated electrode array

(Markx et al., 1997), planer quadrupole trap (Hartley et al., 1999), and extruded

quadrupole trap (Voldman et al., 2003) etc All these researchers have used finite

element commercial software to simulate the electric field The biggest disadvantage of

using FEM method for DEP simulation is the difficulty in design optimization The

ultimate purpose of doing DEP simulation is to optimize the design parameters so that

we can reduce the trial and errors in real fabrications While using finite element

method, every time we change the parameters, a re-meshing process is needed This

will greatly increase the expanses of computational time

1.3 A review of meshfree shape functions

Shape function construction is an important part of meshfree methods The

challenge lies on how to construct shape functions using scattered nodes without

predefined connectivity The quality of the numerical solution highly depends on the

meshfree shape functions

In the following sections, three of the most commonly used methods for

constructing meshfree shape functions are introduced, namely, moving least-squares

(MLS) approximation, polynomial point interpolation method (PIM) approximation

and radial point interpolation method (RPIM) approximation The MLS shape function

is used in many of the meshfree methods developed so far, and gain the most popularity

However, there are two problems remains unsolved for MLS shape function: first is the

difficulty of implementing the essential boundary conditions due to its lacking of delta

function property (Belytschko et al., 1994), second is the complexity in numerical

algorithm for computing the shape functions and the derivatives (Liu, 2003) The

Polynomial PIM (Liu and Gu, 2001a, b) and RPIM (Wang and Liu, 2002 a, b) are

constructed by letting the approximate functions pass through all field nodes in the

local support domain, therefore they possess the Kronecker delta function property,

which allows easy imposition of essential boundary condition The Polynomial PIM

and RPIM shape functions can be constructed in a much simpler way and more

efficiently than MLS shape functions The derivatives are also straightforward to obtain

Comprehensive details are provided for Polynomial PIM and RPIM shape functions

construction and their properties in the following sections, and they are used in all the

meshfree methods in this thesis As a comparison, MLS shape function construction

procedure is also introduced

1.3.1 Moving least-squares (MLS) approximation

Moving least-squares (MLS) is one of the most widely used approximation

scheme used in meshfree methods It is originated by mathematicians for data fitting

and surface construction MLS has been adopted for constructing shape functions in

many meshfree methods, such as diffuse element method (DEM) (Nayroles et al.,

1992), element free Galerkin (EFG) method (Belytschko et al., 1994), etc The detailed

formulation of MLS shape function is presented in this section

1.3.1.1 Formulation procedure of MLS

In the formulation of MLS approximation, a field function u at any interest point

x can be approximated in the following form:

where m is the number of basis, p x i( ) is a complete basis of monomials of the lowest

order of m, a x i( ) is the corresponding coefficient

The coefficients vector in the Eq.(1.4) are chosen so that h( )

u x approximates the

given function in a least-squares sense This yields the following quadratic form as a

function of weighted residual:

( ) ( ) ( ) ( ) ( )

2 1

2 1

Note that the number of supporting nodes is equal to or greater than the number of basis

in the approximation, i.e, n m≥ The approximated function h( )

u x does not pass

through the nodal values

The coefficient vector a x( )can then be obtained by minimizing the functional of the weighted residual:

where U is a vector that collects nodal parameters of the filed variables for all the s

nodes in the support domain, A(x) is called the weighted moment matrix given by

Note that matrix A(x) is symmetric, while matrix B(x) is non-symmetric The

coefficient vector a x( ) can then be obtained by

( )h

where N(x) is the vector of MLS shape functions corresponding to n support nodes in

the support domain:

where i ,j denotes x, y coordinates A comma designates a partial derivative with respect

to the indicated spatial variable The partial derivatives of the shape functions can be

We should note that the MLS shape functions do not satisfy the Kronecker delta

criterionN1(x )j ≠δij, which leads to the fact that the nodal parameters u are not the i