01 Prefix & Chapt 01

DEVELOPMENT OF MESHFREE TECHINQUES

FOR ANALYZING SMART/FGM STRUCTURES

AND NONLINEAR PROBLEMS

DAI KEYANG

NATIONAL UNIVERSITY OF SINGAPORE

2004

DEVELOPMENT OF MESHFREE TECHNIQUES

FOR ANALYZING SMART/FGM STRUCTURES

AND NONLINEAR PROBLEMS

DAI KEYANG (B.Eng., M. Eng., Harbin IT, CHINA)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2004

ACKNOWLEDGEMENTS

i

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my supervisors, Associate Professor,

Dr. Liu Gui-Rong and Assistant Professor, Dr. Lim Kian-Meng for their guidance and

support throughout this study. Their extensive knowledge, serious research attitude,

constructive suggestions and encouragement were extremely valuable to me. Their

influence on me is far beyond this thesis and will benefit me in my whole life.

I would also like to thank Dr. Han Xu and Dr. Gu Yuan-Tong for their encouragement,

continuous discussions and helpful assistance when problems were encountered. I am

also thankful to the colleagues and friends my research center, especially Mr. Li Wei,

Dr. Chen Xi-Liang, Dr. Liu Mou-Bin, Mr. Deng Bin, Dr. Liu Xin, Dr. Quek Siu Sin

Jerry, Mr. Chong Chiet Sing for their active discussions, valuable comments and

useful suggestions they provided.

To my family, I highly appreciate their eternal love and strong support. Special thanks

are conveyed to my wife, Chu Yue, who sacrificed all her time and favorite job to

support my study in Singapore. I am really indebted to her a lot. Without her endless

encouragement, understanding and full support, it is impossible to finish this thesis.

I am also grateful to the National University of Singapore for providing the financial

support during my study. I would like to thank the Center for Advance Computations

in Engineering Science (ACES), Department of Mechanical Engineering, for the use of

its office facilities including computers, printers and a nice study environment which

are all of great assistance to my research work.

TABLE OF CONTENTS

ii

TABLE OF CONTENTS

Acknowledgements i

Table of contents ii

Summary viii

Nomenclature x

List of Figures xiii

List of Tables xxi

Chapter 1 Introduction 1

1.1 Background ............... 1

1.1.1 The necessity for meshfree method .. 1

1.1.2 Classification of meshfree method ... 3

1.1.3 Procedure of meshfree method . 6

1.1.4 Main features of meshfree method... 8

1.1.5 Properties of meshfree shape functions 9

1.2 Literature review ....... 10

1.2.1 Some meshfree methods . 10

1.2.2 Research methods in smart structures 13

1.2.3 Functionally graded materials (FGMs) .. 16

1.2.4 Meshfree methods for nonlinear problems .... 18

1.3 Objectives of the research........ 20

1.4 Organization of the thesis ...... 21

1.5 Original contributions..... 24

Chapter 2 Moving least squares approximation 25

2.1 Introduction 25

2.2 Moving least squares (MLS) approximation . 26

2.2.1 MLS procedure . 26

2.2.2 Weight function 28

2.3 Weak form formulations based on MLS approximation .... 30

2.3.1 Lagrange multipliers method . 31

TABLE OF CONTENTS

iii

2.3.2 Penalty method . 34

2.3.3 Orthogonal transform method .. 35

2.3.4 Numerical examples . 37

2.4 Elimination of shear locking in the EFG method .. 38

2.5 Concluding remarks 42

Chapter 3 Point interpolation method 46

3.1 Introduction 46

3.2 Polynomial PIM .... 48

3.2.1 Polynomial PIM formulation 48

3.2.2 Consistency 50

3.2.3 Techniques to overcome singularity in moment matrix 51

3.3 Radial PIM . 53

3.3.1 Radial PIM formulation 53

3.3.2 Some useful radial basis functions 56

3.3.3 Discussions ... 57

3.4 Properties for the PIM shape function ... 58

3.5 Discrete form of PIM . 60

3.6 Compatibility of PIM . 62

3.6.1 Patch test 62

3.6.2 Requirements for compatibility . 62

3.7 Numerical examples 64

3.7.1 Closed-form solution.. 64

3.7.2 Effect of shape parameters...... 65

3.7.3 Convergence rate ... 67

3.7.4 Effect of polynomial terms..... 67

3.7.5 Effect of irregular/regular node distribution....... 68

3.8 Concluding remarks .... 68

Chapter 4 Radial point interpolation method based on Kriging formulation 82

4.1 Introduction . 82

4.2 Kriging-based formulation . 83

4.2.1 Universal Kriging .. 83

TABLE OF CONTENTS

iv

4.2.2 Relation between covariance and semivariogram ......... 88

4.2.3 Properties of Kriging shape function 89

4.2.4 Comparison between Kriging formulation and RPIM ...... 90

4.3 Spherical basis for radial PIM 91


4.4.1 Parameter study for spherical basis 93

4.4.2 Comparison study . 95

4.4.3 Shape parameter study for modified Gauss basis . 97

4.4.4 Infinite plate with a hole 99

4.4.5 Pressured thick-walled cylinder 100

4.4.6 Dam subjected to hydrostatic pressure . 100

4.5 Concluding remarks .. 101

Chapter 5 Analysis of two-dimensional piezoelectric structures 115

5.1 Introduction 115

5.2 Discrete model for piezoelectric structures 116

5.2.1 Variational weak form formulation . 116

5.2.2 Discrete model based on PIM . 118

5.3 Numerical examples .. ...... 121

5.3.1 Numerical procedure ....... 121

5.3.2 Static analysis .. 122

5.3.3 Eigenvalue analysis . 124

5.4 Concluding remarks . 129

Chapter 6 Analysis of shear deformable laminated plates 140

6.1 Introduction .. 140

6.2 Plate theories 143

6.2.1 Classical laminated plate theory (CLPT) 143

6.2.2 First-order shear deformation theory (FSDT) . 144

6.2.3 Third-order shear deformation theory (TSDT) 145

6.2.4 Higher-order shear and normal deformable plate theory (HOSNDPT) .. 147

6.3 Constitutive relations for orthotropic lamina 148

6.4 Variational form and discrete system equations .. 150

6.4.1 Variational form of system equations . 150

TABLE OF CONTENTS

v

6.4.2 Approximation of field variables . 152

6.4.3 Discrete system equations 153

6.5 Numerical examples ... . 155

6.5.1 Static deflection analysis . 156

6.5.2 Natural frequency analysis .. 158


Chapter 7 Control of laminated plates using integrated sensors and actuators 172


7.2 Theory and formulation 173

7.2.1 Variational form .. 173

7.2.2 Constitutive equations . 174

7.2.3 Meshfree model .. 176

7.2.4 Vibration control mechanism .. 179

7.3 Numerical results . 180

7.3.1 Verification case . 181

7.3.2 Static analysis .. 182

7.3.3 Dynamic analysis . 185


Chapter 8 Thermomechanical analysis of functionally graded plates 197


8.2 Functionally graded material and thermal analysis . 198

8.2.1 Effective moduli estimate ....... 198

8.2.2 Thermal analysis . 201

8.3 Theory and formulation 202

8.3.1 Weak form of the governing equations ...... 202

8.3.2 Discrete meshfree model 204

8.3.3 Control Mechanism 208

8.4 Numerical implementation ...... 210

8.4.1 Treatment of boundary condition 210

8.4.2 Numerical procedure .. 211


8.5.1 Static analysis using RPIM . 213

TABLE OF CONTENTS

vi

8.5.2 Dynamic analysis using RPIM. 215

8.5.3 Static analysis under thermal loadings using EFG method.. 216

8.5.4 Response repression using EFG method.. 218


Chapter 9 Inelastic analysis of 2-D solids based on deformation theory 235

9.1 Introduction . 235

9.2 Stress-strain relation based on the deformation theory 237

9.3 Computer implementation issues 238

9.3.1 Parameter selection in the meshfree method .. 238

9.3.2 Determination of effective material parameters . 239

9.3.3 Numerical procedure .. 241

9.4 Case study 242

9.4.1 Long thick-walled cylinder . 242

9.4.2 V-notched tension specimen 246

9.4.3 Simply supported circular plate .. 247


Chapter 10 Geometrically nonlinear analysis 261


10.2 Solution of nonlinear algebraic equations . 262

10.2.1 Direct iteration method 263

10.2.2 Newton-Raphson method ......... 263

10.2.3 Modified Newton-Raphson method . 266

10.3 Theory and formulations ... 266

10.3.1 Stress and strain measures 266

10.3.2 Variational form equations .......... 269

10.3.3 Discretization of linearized equations .. 271

10.4 Constitutive model . 275

10.5 Numerical implementation . 277

10.5.1 Newton-Raphson iteration procedure ....... 277

10.5.2 Procedure to establish the stiffness matrix ....... 278

10.5.3 Comparison between the RPIM and FEM in algorithm ....... 279

10.6 Numerical examples .. 280

TABLE OF CONTENTS

vii

10.6.1 Uniaxial tension of a solid 280

10.6.2 Upsetting of a billet .. 281

10.6.3 A cantilever beam and a clamped-clamped beam 281

10.7 Concluding remarks ... 282

Chapter 11 Explicit transient nonlinear analysis 293


11.2 Discrete governing equations . 294

11.3 Determination of internal nodal forces .. 295

11.3.1 Elasto-plastic analysis .. 295

11.3.2 Geometrically nonlinear analysis . 298

11.4 Numerical implementation issues .. 300

11.4.1 Numerical procedures for computing internal nodal forces .. 300

11.4.2 Special lumped mass matrix . 301

11.4.3 Stability limit . 302


11.5.1 Simply supported beam . 303

11.5.2 Spherical shells .. 304

11.6 Concluding remarks 306

Chapter 12 Conclusions and recommendations 313

12.1 Conclusions 313

12.2 Recommendations for future research ....... 318

References 320

Publications arising from thesis 334

SUMMARY

viii

SUMMARY

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

boundary element method (BEM), have been the primary numerical techniques in

engineering computations. There are still some limitations arising from a prearranged

topological environment which sets an artificial constraint to ensure a physical

compatibility that a continuum possesses. To overcome these difficulties, meshfree

methods have been proposed. The origin of meshfree methods could be traced back to

a few decades, but it was not a hot topic until after the beginning of 1990s that

substantial and significant advances were made in this field. As meshfree methods are

relatively new, there are still many technical problems to be resolved before they can

become a powerful new-generation numerical tool in computational mechanics for

solids and structures.

This thesis focuses on the development and application of meshfree methods in

computational mechanics. The objectives of the present study are two-fold: one is to

develop new meshfree interpolation schemes that can relieve the burden of meshing

and successive mesh generation; the other is to apply the existing meshfree methods to

solve relatively complicated engineering structures and systems.

The main work of the present research can be described as three parts. The

contribution of the first part is that a meshfree interpolation scheme is proposed and

developed based on moving Kriging method, which is then formulated in Galerkin

weak form for solving elastic problems. It is found that the Kriging interpolation

function is identical with the existing radial point interpolation method (RPIM). Its

SUMMARY

ix

property of compatibility is studied in detail. In addition, the constrained element free

Galerkin (EFG) method based on moving least squares (MLS) method is also

developed and applied to static and free vibration analysis of structures.

In the second part, meshfree methods are devoted to the numerical analysis and

active control of complex structures and systems, which cover piezoelectric structures,

shear deformable laminated composite plates, functionally graded material (FGM)

plates integrated with smart materials. In these meshfree models, both the mechanical

and electrical properties as well as thermal loadings are coupled and acting in

combination. The static shape and dynamic responses can be actively controlled

through bonded piezoelectric sensors and actuators.

The last part of the thesis deals with the nonlinear static and dynamic structural

analysis. The material inelastic analysis is performed based on Henckys deformation

theory. Material nonlinearity is considered by regularly updating the effective material

constants that can be obtained from one-dimensional uniaxial material curve in an

iterative procedure. Geometrically nonlinear analyses are carried out by using Newton-

Raphson iteration procedure for problems with large deformation. Variational

equations are established for total and updated Lagrange formulations. Nonlinear

transient dynamic problems are analyzed through an explicit integration scheme.

Numerical implementation issues and parameter studies are detailed and described

throughout the chapters. A large number of numerical experiments are conducted to

demonstrate the validity and efficiency of the proposed meshfree models. Many

important results are compared with analytical solutions, experimental results or those

obtained by other numerical methods; some valuable findings are reported accordingly.

To sum up, the developed meshfree techniques in this study are successful and

versatile for solving a variety of engineering problems.

NOMENCLATURE

x

NOMENCLATURE

a coefficient vector

b body force ib , left Cauchy-Green deformation tensor ijb

C right Cauchy-Green deformation tensor ijC , elastic constants

ijklC

ad CC , Rayleigh / active damping matrix

d general displacement vector

D rate-of-deformation (velocity strain)

de , ee error in displacement / energy norm

e piezoelectric constant

E, TE Youngs modulus, tangent Youngs modulus

E electric field iE , Green strain tensor ijE

f force vector

F deformation gradient matrix

G shear modulus

dG , vG displacement/velocity feedback control gain

I moment inertia of section

210 ,, III mass moments

321 ,, III first, second and third invariants of Cauchy-Green

deformation

J (= |F|) Jacobian determinant

NOMENCLATURE

xi

k thermal conductivity

K, TK stiffness matrix, tangent stiffness matrix

cC KK , material and spatial stiffness

KK ,S geometric stiffness in material and spatial space

L velocity gradient matrix

M mass matrix

n volume fraction exponent

n, 0n unit normal vector in current and reference configuration

N total number of nodes in the problem domain

LN number of layers for a laminated plate

p pyroelectric constants

p(x) polynomial basis functions

P nominal stress

QP moment matrix of polynomial PIM

q parameter of MQ radial basis function

q dielectric displacement, electrical charge vector

r distance

r residual force vector

QR moment matrix of radial basis

s 1-D curvilinear coordinate

S second Piola-Kirchhoff stress (PK2)

t field temperature

0, tt traction vector on current and reference surface

Twvu ],,[=u displacements in x, y, and z direction

NOMENCLATURE

xii

hu approximation of function u

w deflection of beams or plates

w nondimensionalized deflection

w weight function

x, X current and initial coordinates

penalty constant , 0 problem boundary in current and reference configuration

u , / q , mechanical/electrical boundary

Kronecker delta function pe , elastic strain, plastic strain

strain tensor

dielectric matrix

, Lame constants v Poisson ratio

mass density

0 initial yield stress stress tensor, Cauchy stress

electric potential vector

yx , rotations about (y, -x) axis

strain energy potential shape function matrix

natural frequency nondimensionalized frequency

0, problem domain in current and reference configuration

LIST OF FIGURES

xiii

LIST OF FIGURES

Fig. 2.1 Nodal distributions for patch test: (a) Evenly distributed nodes; (b) Irregularly scattered inner nodes 44

Fig. 2.2 Elimination of shear locking by the increase the order of MLS

shape functions 45 Fig. 2.3 Elimination of shear locking, S1: Matching field scheme; S2:

Original scheme 45 Fig. 3.1 Two-dimensional Pascal triangle 72 Fig. 3.2 Moving nodes in random directions 72 Fig. 3.3 A local coordinate system (, ) defined in a global coordinate

(x, y) 73 Fig. 3.4 A patch with 6 6 uniformly distributed nodes 73 Fig. 3.5 Variation of displacement along interface between element

#12 and #13 (r = 1.5): (a) xu ; (b) yu 74 Fig. 3.6 Variation of displacement along interface between element #8 and #13 (r = 2.0): (a) xu ; (b) yu 75 Fig. 3.7 A cantilever beam 76 Fig. 3.8 (a) Regular nodal distribution; and (b) irregular nodal

distribution for the cantilever beam 76 Fig. 3.9 Effect of shape parameter EXPc on error of energy for EXP

radial basis function 77 Fig. 3.10 Effect of shape parameter q on relative error of deflection and

energy for MQ radial basis function without linear reproduction 77

Fig. 3.11 Effect of shape parameter MQc on relative error of deflection

and energy for MQ radial basis function 78

LIST OF FIGURES

xiv

Fig. 3.12 Convergence rate of radial PIM using MQ and EXP radial basis function 78

Fig. 3.13 Convergence rate of polynomial PIM 79 Fig. 3.14 Relative error of displacement using the PIM with different

bases 79 Fig. 3.15 Effect of shape parameter q on relative error of deflection and

energy for MQ radial basis function with linear reproduction 80 Fig. 3.16 Shear stress distribution at section 2/Lx = using regular and

irregular node patterns for EXP basis 80 Fig. 3.17 Shear stress distribution at section 2/Lx = using regular and

irregular node patterns for MQ basis 81 Fig. 4.1 Relation between the semivariogram and covariance 104 Fig. 4.2 Gaussian semivariogram model )2,1( 00 == ca 104 Fig. 4.3 Three basis functions ( maxmin dd = = 1.0): (a) Original forms;

(b) modified forms 105 Fig. 4.4 (a) Regular nodal distribution ( 921 nodes); and (b) irregular

nodal distribution for the cantilever beam (189 nodes) 106 Fig. 4.5 Deflections of the cantilever beam using different values of c

in the spherical basis function for 921 regular nodes 106 Fig. 4.6 Deflections of the cantilever beam for different regular node

distributions (c=2.0) 107 Fig. 4.7 Shear stress on the section x=L/2 for 1131 regular nodes 107 Fig. 4.8 Shear stress on the section x=L/2 for 1141 regular nodes 108 Fig. 4.9 Errors in energy norm for different node densities 108 Fig. 4.10 Comparison of relative error of deflection between regular and

irregular nodes 109 Fig. 4.11 Comparison of errors in energy norm between MQ and

spherical bases 109 Fig. 4.12 Comparisons of errors in energy norm between the EXP and

spherical bases 110

LIST OF FIGURES

xv

Fig. 4.13 Tip deflections of the cantilever beam using different regular node density 110

Fig. 4.14 Shear stress xy on the section x=L/2 obtained using different

regular nodes densities 111 Fig. 4.15 Nodal arrangement in a plate with a central hole 111 Fig. 4.16 Stress distribution in an infinite plate with a central hole

subjected to a unidirectional tensile load ( 0=xatx ) 112 Fig. 4.17 Node arrangement for the pressured cylinder problem 112 Fig. 4.18 Displacements in the pressured cylinder problem 113 Fig. 4.19 Stresses in the pressured cylinder problem ( 5.10 =a ) 113 Fig. 4.20 A dam subjected to hydrostatic pressure 114 Fig. 4.21 (a) Regular nodes; and (b) irregular nodes arrangement for the

dam 114 Fig. 5.1 Parallel connection of a bending motor 135 Fig. 5.2 Nodal arrangements of the bending motor: (a) regular 326

nodes; and (b) regular 351 nodes 135 Fig. 5.3 Nodal deflection of the piezoelectric bimorph beam 136 Fig. 5.4 A one-layer pizeoelectric disk element 136 Fig. 5.5 Nodal distribution of the disk: (a) Coarse nodal distribution;

(b) Irregular coarse nodal distribution; (c) Regular fine nodal distribution; and (d) Irregular fine nodal distribution 137

Fig. 5.6 Eigenmodes for the piezoelectric disk by polynomial PIM

method 137 Fig. 5.7 Schematic Representation of a transducer 138 Fig. 5.8 Nodal arrangements of the transducer: (a) Regular coarse

nodal distribution and (b) Regular fine nodal distribution 139 Fig. 5.9 Eigenmodes for the piezoelectric transducer by polynomial

PIM method 139 Fig. 6.1 A typical laminated plate and its coordinate system 169 Fig. 6.2 Simple supported boundary conditions for (a) cross-ply

laminates (SS1); and (b) angle-ply laminates (SS2) 170

LIST OF FIGURES

xvi

Fig. 6.3 Nodal distribution for the square laminate: (a) 2121 regular

nodes; (b) 441 irregular nodes 171 Fig. 7.1 Schematic diagram of a cantilever laminate with integrated

piezoelectric sensors and actuators 190 Fig. 7.2 A piezoelectric polymeric bimorph beam 190 Fig. 7.3 (a) 26 5 regularly distributed nodes; (b) 130 irregularly

distributed nodes 191 Fig. 7.4 Static deflections of a piezoelectric bimorph beam 191 Fig. 7.5 Centerline deflections of the cantilever laminate [p/-45/45]as

(Actuator input voltage =10V) 192 Fig. 7.6 Centerline deflection of the cantilever laminate [p/-45/45]as

under uniform load and different actuator input voltages 192 Fig. 7.7 Deflections of the laminate with different boundary conditions

under different input actuator voltages 193 Fig. 7.8 Centerline deflections of the simply supported laminates with

different locations of piezo layers under the uniform load and different actuator input voltages 193

Fig. 7.9 Centerline deflections of the simply supported laminate with

different ply angles under the uniform load and different actuator input voltages 195

Fig. 7.10 First four vibration modes of the laminate [p/-45/45]as 196 Fig. 7.11 Effect of the feedback control gain G on the response of point

A on the cantilever laminate [p/-45/45]as 196 Fig. 8.1 Variation of the volume fraction function versus the non-

dimensional thickness 224 Fig. 8.2 Two-phase material with (a) particulate microstructure and (b)

skeletal microstructures Fig. 8.3 Through-the-thickness distribution of the (a) Youngs modulus

and (b) the density and the comparison of (c) Youngs modulus and (d) Poissons ratio using MTm and MTc 225

Fig. 8.4 Schematic representation of an FGM plate integrated with

piezoelectric sensor and actuator layers 225

LIST OF FIGURES

xvii

Fig. 8.5 Nodal distribution of the plate: (a) 1111 regular nodes; (b) 121 irregular nodes 226

Fig. 8.6 Nondimensionalized deflection versus volume fraction

exponent for a simply supported square plate (a/h=10) under sinusoidal load 227

Fig. 8.7 Nondimensionalized deflection versus aspect ratio for a simply

supported square plate (a/h=10) 227 Fig. 8.8 Nondimensionalized natural frequency versus volume fraction

exponent for a simply supported square plate (a/h=10) 228 Fig. 8.9 Nondimensionalized natural frequency versus aspect ratio for a

simply supported square plate (a/h=10) 228 Fig. 8.10 The first six vibration modes of the cantilever FGM plate with

n = 229 Fig. 8.11 Relative deflection responses for an FGM plate (n = 1.0): (a)

Model I; (b) Model II 229 Fig. 8.12 Decay envelopes of the transient response for the centerline tip

point of the cantilever FGM plate: (a) Model I; (b) Model II 230 Fig. 8.13 Tip deflection w versus the volume fraction exponent n for a

cantilever FGM plate under uniformly distributed load 100 2/ mN 231

Fig. 8.14 Temperature profile through the thickness of the FGM plate 232 Fig. 8.15 Centerline deflection of the cantilever FGM plate under

thermal gradient 232 Fig. 8.16 Tip deflection w versus the volume fraction exponent n for a

cantilever FGM 233 Fig. 8.17 Shape control of the FGM plate under thermal gradient using

various displacement feedback control gains 233 Fig. 8.18 Dynamic response repression for the FGM plate (n = 1.0)

using various velocity feedback control gains 234 Fig. 9.1 Projection method for determination of effective elastic

modulus: (a) Elastic-perfectly plastic material; (b) Linearly work-hardening material 249

Fig. 9.2 Arc length method for effE determination 250

LIST OF FIGURES

xviii

Fig. 9.3 Neubers method for effE determination 250 Fig. 9.4 A thick-walled cylindrical pressure vessel subjected to internal

surface loading: (a) Geometric model; (b) One typical segment of unit length; (c) Computational model 251

Fig. 9.5 Nodal arrangement for domain discretization: (a) Evenly

distributed nodes; (b) Irregularly distributed nodes 252 Fig. 9.6 Normalized stress distributions for elastic-perfectly plastic

material. Lines: ANSYS solutions; Nodes: present solutions 252 Fig. 9.7 Normalized stress distributions under different pressure ratios

for elastic-perfectly plastic material. Lines: ANSYS solutions; Nodes: present solutions 253

Fig. 9.8 State of stress for a set of points in radial direction after

convergence 253 Fig. 9.9 Convergence path for a particular point 254 Fig. 9.10 Normalized stress distributions for linearly work-hardening

material. Lines: ANSYS solutions; Nodes: present solutions. (a) 0.1=pr using evenly distributed nodes; (b) 0.2=pr using irregularly distributed nodes 254

Fig. 9.11 Normalized stress distributions under different pressure ratios

for linearly work-hardening material. Lines: ANSYS solutions; Nodes: present solutions 255

Fig. 9.12 State of stress for a set of material points in radial direction

after convergence 256 Fig. 9.13 Convergence path for a particular point 256 Fig. 9.14 Normalized stress distributions under different pressure ratios

using the Ramberg-Osgood material model. Lines: ANSYS solutions; Nodes: present solutions 257

Fig. 9.15 State of stress for a set of points in radial direction 257 Fig. 9.16 Convergence path for a particular point 258 Fig. 9.17 V-notched tension specimen under plane stress 258 Fig. 9.18 Distributions of normal stress across the minimum section 259 Fig. 9.19 Load-extension response of V-notched tension specimen 259

LIST OF FIGURES

xix

Fig. 9.20 Geometry of a uniformly loaded simply supported circular plate ( 0',16000,24.0,10 0

7 ==== HvE ) 260 Fig. 9.21 Load/Central deflection response for a uniformly loaded

simply supported circular plate. (Collapse load Pc = 260) 260 Fig. 10.1 Possibility of multiple solutions and diverged solution case 284 Fig. 10.2 The Newton-Raphson method 284 Fig. 10.3 The Modified Newton-Raphson method 285 Fig. 10.4 Initial and current configurations of a Lagrangian element and

their relationships to the parent element 285 Fig. 10.5 Deformation of a solid under uniaxial tension 286 Fig. 10.6 Deformation of a solid under uniaxial tension with left side

clamped 287 Fig. 10.7 Tension/load relationships for a solid 288 Fig. 10.8 The initial and final configurations for a billet 288 Fig. 10.9 Bending of a cantilever beam 290 Fig. 10.10 Tip-deflection/load relationships for a cantilever beam 291 Fig. 10.11 Initial and final configurations for a clamped-clamped beam

subjected to uniform loading 292 Fig. 10.12 Mid-deflection/load relationships for a clamped-clamped beam

under uniform loading 292 Fig. 11.1 Factoring process for just yielding points 308 Fig. 11.2 Model of a simply support beam 308 Fig. 11.3 Transient response of the simply supported beam 309 Fig. 11.4 Transient response with damping effect 309 Fig. 11.5 Model of a spherical shell 310 Fig. 11.6 Transient dynamic response of a spherical shell using

meshfree method 310 Fig. 11.7 Linear and nonlinear elastic response of a spherical shell under

concentrated load 311

LIST OF FIGURES

xx

Fig. 11.8 Prediction of static solutions from transient response with

damping 311 Fig. 11.9 Relationship between concentrated load versus vertical apex

deflection of a spherical cap 312 Fig. 11.10 The deformed configurations of the spherical shell 312

LIST OF TABLES

xxi

LIST OF TABLES

Table 2.1 Numerical solutions of standard patch test using evenly distributed nodes 43

Table 2.2 Numerical solutions of standard patch test using irregular inner

nodes 43 Table 3.1 Relative errors of displacements at 4 inner nodes for a standard

patch test 71 Table 3.2 Comparison of error with/without linear reproduction (MQ

basis, 297 nodes) 71 Table 3.3 Tip deflections of the cantilever beam using two nodal patterns

( 310 m) 71 Table 4.1 Tip deflections of a cantilever beam using different regular

nodes (Analytical solution: 3109.8 m) 103 Table 4.2 Tip deflections of a cantilever beam using 189 irregular nodes

(Analytical solution: 3109.8 m) 103

Table 4.3 Displacements at point A and B of the dam ( 310 m) 103 Table 5.1 Piezoelectric material properties 131 Table 5.2 Static deflections of the piezoelectric bending motor by

ABAQUS ( 810 m) 131 Table 5.3 Static deflections of the piezoelectric bending motor by

polynomial PIM ( 810 m) 131 Table 5.4 Eigenvalue estimates for a piezodisk element by ABAQUS 132 Table 5.5 Eigenvalue estimates for a piezodisk element by polynomial

PIM 132 Table 5.6 Piezoelectric transducer eigenvalue estimates by ABAQUS 133 Table 5.7 Effect of the shape parameter q on the cylinder transducer

eigenvalues ( 0 =2.0, 0c =1, m=3, 175 nodes) (kHz) 133

LIST OF TABLES

xxii

Table 5.8 Effect of the size of influence domain on the cylinder

transducer eigenvalues (q=1.03, 0c =1, m=3, 175 nodes) (kHz) 133 Table 5.9 Effect of the polynomial terms on the cylinder transducer

eigenvalues (q=1.03, 0c =1, 69 nodes) (kHz) 134 Table 5.10 Effect of node density for discretization on the cylinder

transducer eigenvalues when q=0.98 and 0c =1 (kHz) 134 Table 5.11 Comparison of CPU time and memory between the FEM and

RPIM 134 Table 6.1 Nondimensionalized maximum deflections w in simply

supported symmetric cross-ply (0/90/0) square laminates under sinusoidally distributed transverse load using different nodal densities (EFG: a/h=10) 163

Table 6.2 Nondimensionalized maximum deflections w in simply

supported symmetric cross-ply (0/90/0) square laminates under sinusoidally distributed transverse load using different nodal densities (RPIM: a/h=10, q=1.03, C=2) 163


supported symmetric cross-ply (0/90/90/0) square laminates under sinusoidally distributed transverse load (Regularly distributed nodes) 163


supported symmetric cross-ply (0/90/0) square laminates under sinusoidally distributed transverse load 164


supported symmetric cross-ply (0/90/0) square laminates under sinusoidally distributed transverse load 164

Table 6.6 Nondimensionalized center deflections w of antisymmetric

cross-ply (0/90) square plates with various boundary conditions 164

Table 6.7 Nondimensionalized center deflection w of a thick square

plate 165 Table 6.8 Natural frequency coefficients of a lateral free vibration of

a free square plate 165 Table 6.9 Nondimensionalized frequencies in simply supported

(0/90/90/0) cross-ply laminates as functions of modulus ratio 165

LIST OF TABLES

xxiii

Table 6.10 Effect of side-to-thickness ratio on the dimensionless frequencies of antisymmetric cross-ply (0/90) square plates with various boundary conditions (2 layers) 166

Table 6.11 Natural frequencies parameters of laminated square plates (BC: SSSS) (h = 0.06, h/a = 0.006) 166 Table 6.12 Natural frequencies parameters of laminated square plates (BC: CCCC) (h=0.06, h/a = 0.006) 167 Table 6.13 Natural frequencies parameters of laminated square plates (BC: SSSS) (h = 3, h/a = 0.3) 167 Table 6.14 Nondimensionalized fundamental frequencies, , of simply

supported (SS-2), antisymmetric angle-ply )///( L square plates by TSDT 168

Table 6.15 Nondimensionalized fundamental frequencies, Eh / = ,

of a thick isotropic square plate with v=0.3 168 Table 7.1 Material properties of the Piezoelectric PVDF 188 Table 7.2 Static deflection of the piezoelectric bimorph beam ( m710 ) 188 Table 7.3 Tip deflection of the piezoelectric bimorph beam using

different methods and nodal densities ( m710 ) 188 Table 7.4 Material properties of PZT G1195N piezoceramics and

T300/976 graphite-epoxy composites 189 Table 7.5 Central node deflections of the simply supported laminate

under the uniform load and different actuator input load ( m510 ) 189

Table 7.6 Natural frequencies of the laminated plate (Hz) 189 Table 8.1 Mechanical and electrical properties of applied materials 221 Table 8.2 Natural fundamental frequencies of a square Al/ZrO2

functionally graded thick plate ( 1,0 == + cc VV ) 221 Table 8.3 Non-dimensionalized center deflections of the square plate

(Model I, a/h=10, n = 2.0) 221 Table 8.4 Non-dimensionalized natural fundamental frequencies of the

square plate (Model I, a/h=10, n = 2.0) 222

LIST OF TABLES

xxiv

Table 8.5 Variation of the natural frequencies (Hz) with the volume fraction exponent n for a cantilever FGM plate using FEM 222

Table 8.6 Variation of the natural frequencies (Hz) with the volume

fraction exponent n for a simple FGM plate using FEM 222 Table 8.7 Variation of the natural frequencies (Hz) with the volume

fraction exponent n for a cantilever FGM plate using the RPIM 223 Table 8.8 Variation of the natural frequencies (Hz) with the volume

fraction exponent n for a simply supported FGM plate using the RPIM 223


fraction exponent n for a cantilever FGM plate (CFFF) using the EFG method (Model III) 223


fraction exponent n for a simply supported FGM plate (SSSS) using the EFG method (Model III) 223

Table 9.1 Number of iterations for convergence using FEM and the

present method 248

CHAPTER ONE INTRODUCTION

1

CHAPTER ONE

INTRODUCTION

1.1 Background

1.1.1 The necessity for meshfree method

When designing an advanced engineering system, engineers must undertake a very

important process of modeling, simulation, analysis and visualization. Before

analyzing an engineering system, they need to develop a mathematical model that

describes the system. In the process of developing the mathematical model, some

assumptions are made for simplification. Then the governing mathematical expressions

are established to describe the behavior of the system. The mathematical expressions

usually consist of differential equations as well as some prescribed conditions. Except

for some simple models, these differential equations are usually very difficult to solve

directly in order to obtain their analytical solutions that describe the behavior of the

given engineering system. With the advent and rapid development of high-

performance computers, it has become possible to solve such differential equations

using numerical methods. Various numerical techniques have been developed such as

finite difference method (FDM), finite element method (FEM), boundary element

method (BEM), etc. In particular, the FEM has become one of the major numerical

solution techniques. One of the advantages of the FEM is that a general-purpose

computer program can be easily developed to analyze various kinds of problems. In

particular many complex shape of problem domain with prescribed conditions can be


2

handled with ease using the finite element method. To date, numerous engineering

problems have been solved using the FEM including mechanics of solids and

structures, fluid flows, heat transfer, ion diffusion and electric fields.

The FEM requires division of the problem domain into many subdomains and each

one is called an element. Consequently the problem domain consists of many finite

element patches. In FDM, the problem domain is discretized into grids. All these

elements or grids can be generally termed as meshes. The common characteristic of the

meshes is that each of them has several connecting nodes and there is some

information concerning the relation of the nodes. The continuity of field variables

within the domain spreads through the adjacent meshes and related nodes. The

governing differential equations, whether they are ordinary differential equations

(ODEs) or partial differential equations (PDEs), can be transformed into weak-form

formulations on the discretized sub-domains by means of certain principles, such as

variational method, minimum potential energy principle or principle of virtual work.

Using a set of properly predefined meshes and the field discretization method, a set of

algebraic equations are generated. After assembling the equations of all the meshes and

imposition of proper boundary conditions, the system equations governing the entire

problem domain can be founded and thereafter solved for desired variables.

Despite of its robustness in numerical simulations, there still exist some limitations

or inconveniences in the FEM. For instance the data preparation in course of mesh

generation and model conversion from physical model to finite element data is an

extremely burdensome and time-consuming task. Due to the rapid development of

computational facilities, it becomes expedient and viable to explore a method which

may be somewhat more expensive from the viewpoint of computer time but cost less

time in the preparation of data for engineers. As the problems of computational


3

mechanics grow ever more challenging, the limitations of the FEM are also becoming

evident gradually. For example, in the simulation of manufacturing process such as

molding and extrusion, it is necessary to deal with extremely large deformations. The

meshes are often so seriously distorted that accurate solutions cannot be retrieved.

Although adaptive algorithm to re-mesh the domain is an effective solution, it is still

an expensive task and sometimes causes mesh-dependent results. In the simulation of

failure process, it is difficult to model the propagation of the crack growth with

arbitrary and complex paths. Due to the inherited property of FEM that the predefined

continuity between elements cannot be broken, it is also difficult to simulate the

breakage of material with large number of fragments or problems with sliding

boundaries. In addition the secondary variables such as strain and stress by FEM

methods are much less accurate than the primary variables such as displacement,

temperature, etc. A post-processing procedure is required to fit their distributions,

which is also an annoying problem in the stress analysis.

To overcome these problems, meshfree, meshless or element-free method has been

proposed and achieved remarkable progress in recent years. In this method, the

problem domain and the boundaries are represented by a set of scattered nodes and

predefined mesh structure is not required.

1.1.2 Classification of meshfree method

There are a number of versions of meshfree methods developed so far and some

new ones will continue to appear in the future. According to the approaches to arrive at

the discrete governing equations, they largely fall into three categories. The first one is

the meshfree methods based on strong-form formulation (or in short, strong-form

meshfree methods), such as the meshfree collocation method (Zhang et al., 2001), the


4

smooth particle hydrodynamics (SPH) method (Gingold and Monaghan, 1977), the

general finite difference method (Liszka and Orkisz, 1980), the finite point method

(FPM) (Oate et al., 1996). The second category includes meshfree methods based on

weak-form formulation (or briefly, weak-form meshfree methods), such as the

element-free Galerkin (EFG) method (Belytschko et al., 1994), the meshless local

Petrov-Galerkin (MLPG) method (Atluri and Zhu, 1998; Atluri et al., 1999), point

interpolation method (Liu and Gu, 1999), and the boundary-type meshfree methods

(Mukherjee and Mukherjee, 1997). The third category comprises the meshfree

methods based on a combined formulation using both weak and strong forms, such as

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

The attractive advantages of the strong-form meshfree methods are that they are

simple to implement and computationally efficient. As they require no mesh for both

field variable approximation and integration, they are truly meshfree methods. Due to

these virtues, strong-form methods have been widely used in fluid mechanics.

However, the defects of strong-form meshfree methods are also evident. They are

often unstable and inaccurate in dealing PDEs with Neumann (derivative) boundary

conditions.

On the contrary, the weak-form meshfree methods are capable of imposing

Neumann boundary conditions naturally and easily. They exhibit very good stability

and satisfactory accuracy. Therefore weak-form methods have been successfully

applied in problems in solid and structural mechanics. However the efficiency is a big

problem for weak-form methods due to their requirement for weak-form integration.

Hence background cells are needed over the global problem domain. In order to avoid

the global integration background mesh, some methods based on local Petrov-Galerkin

weak formulation have been proposed, such as the above-mentioned MLPG, method


5

of finite spheres (De and Bathe, 2000), local point interpolation method (LPIM) (Gu

and Liu, 2001b). In these methods, weak-form integration is only performed in local

subdomains and very simple integral domain can be used such as circles, triangles and

rectangles.

From the brief review of the two types of meshfree methods, one naturally thinks if

they could be combined together? Through close examination one may find that the

two methods both construct and assemble discrete equations node by node. The local

weak form can be used to enforce Neumann boundary conditions for nodes on or close

to the natural boundaries, while strong-form formulation can be employed for the rest

nodes, i.e., the inner nodes or those on essential boundaries. As the number of nodes

on natural boundaries is much smaller by comparison, the computational work for

local integration is nearly negligible. This is the main idea of the so-called meshfree

weak-strong (MWS) form method. The advantages of the MWS are two-fold. One is

the Neumann boundary conditions can be imposed accurately and straightforwardly for

arbitrarily distributed nodes; the other is the method is of high efficiency as strong-

form formulation is applied for most nodes in the domain. The MWS method have

been successfully used for both solid and fluid analysis recently (Liu and Gu, 2003b;

Liu et al., 2004). As the thesis mainly deals with solid problems, attention is

concentrated on the weak-form methods.

Another way to classify the meshfree methods is based on the approaches used to

construct shape functions. Hence it is feasible to review them by the methods of

creating the shape functions. According to Liu (2002), these methods can be classified

into three main categories: (1) Finite integral representation methods, which include

smooth particle hydrodynamics (SPH) method and reproducing kernel particle method

(RKPM); (2) Finite series representation methods, which include moving least squares


6

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

(3) Finite differential representation methods, which include finite difference method

(regular grids) and finite point method (irregular grids).

Based on the global and local weak-form formulations Category (2) can be further

classified into two groups. If MLS method is formulated in global weak form in the

entire problem domain, the famous EFG method can be recovered. If MLS method is

formulated in the local weak form, the meshless local Petrov-Galerkin (MLPG)

method can be obtained. Similarly, the PIM can also be formulated in both global and

local weak forms.

1.1.3 Procedure of meshfree method

The basic step of weak-form meshless method is addressed below, which follows

nearly the same procedure as FEM except the formation of shape functions and

imposition of boundary conditions. It can be outlined briefly as such:

Step I: Domain representation. The problem domain and its boundaries are

represented by a set of properly distributed nodes. The node density depends on the

stress gradient for solid problems. As these nodes carry information of the field

variables, they are also called field nodes.

Step II: Field interpolation or approximation. In meshfree method, the field

variable at any point say x=(x, y, z) within the problem domain is constructed using the

field values of nodes in the local support domain of x, which has the form of

uxxx )()()(1

== =

i

n

ii uu (1.1)

where n is the number of nodes included in the support domain. The support domain is

defined as a local subdomain around node of interest x that the nodes within the


7

subdomain have influence on the field variable x; on the contrary the nodes outside of

the subdomain have no effect on it. The domain may or may not be weighted using a

weight function. The shape of the support domain can be defined as any reasonable

shape. A circle and a rectangular are two commonly used shapes. Detailed description

of support domain is given in the literature (Liu, 2002).

Step III: Transformation of strong-form governing differential equations into weak

form. After discretizing the entire problem domain into a set of scattered nodes, some

principle will be used to transform the strong-form differential equations into weak-

form formulation. The principles used in meshfree method are much the same as those

in finite element method. The commonly used principles include variational method,

principle of virtual work, minimum potential energy principle and residual method,

etc. The global weak-form formulation is commonly used in solids and structural

mechanics, which often generate very stable and accurate results. The Petrov-Galerkin

method is applied in some meshfree methods in which integration is only carried out in

local subdomain and the formulation is therefore called local weak form.

Step IV: Formation of system algebraic equations. Using the approximate field

variable functions constructed in Step II, the weak-form governing equations can be

changed into a set of algebraic equations. According to idea of the weak-form

formulation, the algebraic equations at any quadrature point can be assembled into

global ones, hence system algebraic equations are established.

Step V: Solving equations. The system equations in static problems can be solved

using a linear standard algebraic equation solver, such as Gauss elimination, LU

decomposition and iterative method. For eigenvalue and buckling problems, the

equations can be solved by eigenvalue equation solvers, such as Jacobi method, QR

decomposition, sub-space iteration, etc. In dynamic problems, the time history of the


8

variables needs to be considered. The following methods are selected depending on the

complexity of problem: direct integration, modal superposition, Newmark method and

Wilson method, etc.

Step VI: Post processing of desired results. When primary variables are obtained

from system equations, the secondary variables can be subsequently solved using

constitutive equations or other relationships.

1.1.4 Main features of meshfree method

In order to understand what meshfree method is, one needs to know its

characteristics. From the outlined procedure, some prerequisites need to be maintained

for a meshfree method. Firstly, predefined meshes are not required at least in the stage

of field variable interpolation. Some versions even do not need global background cells

for matrix integration. Next the problem domain can be represented by a group of

relatively irregular nodes, or the nodes used to discretize the domain can be freely

located to some extent. Furthermore the number of nodes covered in local support

domain should not affect the accuracy greatly. Finally the method should be stable,

accurate and have desirable convergence rate. Apart from the abovementioned

attributes, compared with the FEM, the currently developed meshfree method has

some other main features listed below.

(1) In the FEM, its shape function is identical for element of the same type in an

intrinsic coordinate system. They can be predetermined before the simulation starts.

However, in meshfree method, shape function generally changes from point to point

and it is constructed during the process of analysis.


9

(2) Meshfree shape function may or may not satisfy Kronecker delta conditions,

depending on the method adopted for creating it. If not, like the MLS-based meshfree

methods, special techniques are needed to impose the essential boundary conditions.

(3) Some meshfree methods require background cells for the integration of the

weak-form formulations over the problem domain (such as the global weak-form

methods). These methods are still practical because any simply shaped cell can be

employed as long as sufficient accuracy of integration is ensured. On the contrary, for

other meshfree methods only local cells are required (such as the local weak-form

methods) and or no integration is conducted (like the strong-form methods). They are

the so-called truly mesh free methods.

(4) Mesh automation and adaptive analysis can be easily realized. Hence it is

suitable for problems related to large deformation, crack propagation or elastodynamic

fracture.

(5) The results are generally more accurate than the FEM.

(6) The method is computationally more expensive than the FEM due to the

complexity in construction of shape functions and imposition of boundary conditions.

1.1.5 Properties of meshfree shape functions

As the FEM, the shape function plays an important role in meshfress method. It

determines the compatibility, accuracy and even the validity of the method. A

necessary requirement that a shape function must satisfy is the partition of unity, as

given by

1)(1

==

xn

ii (1.2)


10

This is a necessary requirement to reproduce the rigid body motion of the problem.

Similarly, the shape function may preferably have the ability to reproduce the linear

field, that is

ii

n

ii xxx =

=)(

1 (1.3)

We know that most shape functions in the FEM satisfy the Kronecker delta

function property, that is,

===

jiji

ijji ,0,1

)( x (1.4)

It is also preferable if the shape functions in meshfree methods satisfy this

condition because the property ensures the easy and convenient imposition of essential

boundary conditions. Unfortunately, only a few meshfree shape functions have this

property, which will be detailed in following chapters.

1.2 Literature review

1.2.1 Some meshfree methods

1.2.1.1 SPH and RKPM method

SPH method seems to be the oldest method (Lucy, 1977; Gingold and Monaghan,

1977), which was originally used for modeling astrophysical phenomena without

boundaries such as exploding stars and dust clouds. It uses the integral representation

of a function in a finite sub-domain in the following form

= xx dhWuu h ),()()( (1.5)

where )(xhu represents the approximate function )(xu , W is a kernel or weight

function, and h is the smoothing length in SPH. The finite integral representation,


11

which is also termed as kernel approximation, is valid and converges on condition that

its weight functions satisfy some conditions (Monaghan, 1982). Four kinds of weight

functions have been proposed, i.e., cubic spline, quartic spline, exponential spline and

new quartic smoothing function (Liu, 2002). Detailed formulations are given in

Chapter 2.

Recently, Liu et al. (1993, 1995) proposed a correction function for kernels both in

the discrete and continuous cases and named it the reproducing kernel particle method

(RKPM). The correction function is particularly useful to improve the SPH

approximation near the boundaries as well as to make it linearly or C1 consistent near

the boundary. The finite integral form of a function with a correction function can be

expressed as

= xxx dhWCuu h ),(),()()( (1.6)

where ),( xC is the correction function. The RKPM has been successfully applied to

solve many problems such as solids, structures, and acoustics, etc. (Liu et al., 1993,

1995, 1997).

1.2.1.2 MLS and EFG method

Moving least squares (MLS) method was originated by mathematicians in curve or

surface fitting (Mclain, 1974; Lancaster, 1981). Nayroles et al. (1992) were the first to

use a moving least square procedure but developed it from the notion of diffuse

elements. Belytschko et al. (1994) refined and modified the method and called it the

element-free Galerkin (EFG) method. Their major contributions in EFG are as follows:

(a) the EFG method does not seem to exhibit any volumetric locking even when the

basis functions are linear; (b) the convergence rate is faster than that of the FEM; and

(c) a high resolution of localized steep gradients can be achieved. This part will be

covered in Chapter 2.


12

1.2.1.3 Local Weak form and MLPG method

As the EFG is based on a global weak form, a background cell structure over the

entire domain is required to perform the integration. It is actually not a meshfree

method. In order to avoid the background cells in the whole domain, the Meshless

Local Petrov-Galerkin (MLPG) method was proposed by Atluri et al. (Atluri and Zhu,

1998; Atluri et al., 1999), in which a local weak form is used over a local subdomain.

Detailed description of this method was given in the monograph by Atluri and Shen

(2002). Compared with the global weak form, the local weak form has two advantages.

One is that it does not need any background integration cell over the entire problem

domain. Therefore it leads to a natural way to construct the system stiffness matrix: not

through the assembly of the stiffness matrices, but through the integration over local

sub-domain. Another advantage is that the continuity between neighboring local

subdomains is not required; but they may overlap each other.

Remarkable success of the MLPG method has been reported in solving shear-

deformable beams (Cho and Atluri, 2001) and plate bending problems (Gu and Liu,

2001; Long and Atluri, 2002; Qian et al., 2003a, b); fracture mechanics problems (Kim

and Atluri, 2000; Ching and Batra, 2001; Batra and Ching, 2002); and Navier-Stokes

flows (Lin and Atluri, 2001).

1.2.1.4 Point interpolation method (PIM)

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

proposed by Liu and his coworkers (Liu and Gu, 2001; Wang and Liu, 2001a, b; Liu,

2002). It uses the nodal field values in the local support domain of considered point to

exactly interpolate the field variables at its location. The shape functions so

constructed possess the Kronecker delta function properties and hence the imposition

of boundary condition is as easy as in conventional FEM method. Compared to MLS


13

method, as weighted functions are not necessary, it is more efficient than MLS

algorithm. Chapter 3 will establish the PIM using direct interpolation concepts while

Chapter 4 will give an alternative formulation based on the moving Kriging method.

1.2.2 Research methods in smart structures

In the past two decades, the area of smart materials and structures has experienced

tremendous growth in research and engineering. Scientists and engineers from

aerospace, civil, mechanical, material and electrical engineering fields are involved in

some parts of the development of smart materials and systems. The reasons may be

two-fold (Reddy, 1999). One reason for these activities is that it may be possible to

create certain types of structures and systems capable of adapting to or correcting for

changing operation condition. The advantage of incorporating some special types of

materials into the structures is that the sensing and actuating mechanisms become part

of the functions of the structure. Another reason may be that, with the advent of the

new century, the next generation of smart materials and systems will come out quickly,

which will feature thermo-electro-mechanical coupling, intelligence and

miniaturization. These systems operate under varying conditions and some of the

environments pose serious problems to design and maintenance of the smart structures.

Experimental investigation of the smart structures and systems, though possible, may

be prohibitively expensive, and therefore, theoretical analyses are essentially required.

Many definitions appear in literature on what are smart and intelligent structures.

We prefer to follow Newnhams definition (1993), that is, the structures with surface

mounted or embedded sensors and actuators that have the capability to sense and to

take corrective actions are referred to as smart structures. Typically, a smart structural

system is composed of a load-bearing part that is usually passive, an active material


14

part that performs the operations of sensing and actuating and a feedback and control

system made of feedback circuitry linking sensors and actuators. Different materials

have been used in smart structures such as piezoelectric materials, electrostrictive

materials, magnetostrictive materials, shape memory alloys and electro-rheological

fluids, etc. Among the currently available sensors and actuators, the piezoelectric

material is a popular one. The thesis will only focus on this material. Piezoelectricity

is a phenomenon in which some materials develop polarization upon application of

strains (Mason, 1950; Cady, 1964; Maugin, 1988), or vice versa.

An experimental method is a basic and essential way to study the piezoelectric

structures, but its models are limited by size, cost, noise, and many other laboratory

uncertainties. Different generic theories on piezoelectric structures of arbitrary shape

have been proposed for many applications. Dokmeci (1978) established a theory on

coated thermopiezoelectric lamina. Senik and Ludriavtsev (1980) formulated the

equations of motion for piezoelectric shells polarized in the normal of the shell middle

surface. Chau (1986) proposed a variational formulation to solve the equilibrium

problem of anisotropic piezoelectric shells.

Distributed active vibration controls of flexible beams were studied with embedded

or surface-distributed piezoelectric sensors and actuators. Analytical methods were

initially used for the analysis of smart beams (Bailey and Hubbard, 1985; Crawly and

Luis, 1987; Tzou, 1987; Im and Atluri, 1989; Shen, 1995). Based on the piezoelectric

plate theory, an analytical model was proposed by Lee (1992) and applied to the

vibration control of laminated structures with distributed sensors and actuators. A set

of theoretical formulations were also established by Lam and Ng (1999) for the

dynamic and active control of piezoelectric laminated composite plates subject to

different mechanical and electrical loadings. An integrated distributed sensing and


15

control theory for thin shells was also proposed (Tzou, 1988, 1991). Generic theories

on structural identification and vibration control of continua using electroded

piezoelectric layers can be found in literature by Tzou (1992). In the literature (Tzou,

1993), generic distributed structural identification and vibration control theories of a

generic deep shell continuum were presented. Open and closed-loop dynamic system

equations and state equations of the continuum were formulated. Simple reduction

procedures were proposed and applications to other common geometries were

demonstrated in case studies.

Though theoretical models can be more general, analytical solutions are only

restricted to relatively simple geometries and boundary conditions. Finite element

method (FEM) becomes a powerful technique for the numerical analysis of smart

structures and systems of complicated shape. Distributed vibration control of a beam

modeled by piezoelectric beam finite elements was investigated by Obal (1987). A

piezoelectric thin hexahedron solid element was developed by Tzou and Tseng (1990)

and applied to the modeling and analysis of flexible continua plates and shells with

distributed piezoelectric sensors and actuators. Hwang and Park (1993) proposed a

model for the vibration control of a laminated plate using a four-node quadrilateral

plate element with one electrical degree of freedom. Based on classical plate theory

(CPT), a finite-element model was also established (Lam et al. 1997; Peng et al, 1998;

Liu et al, 1999) for the active vibration control of beams and plates containing

distributed sensors and actuators. A simple negative velocity feedback control

algorithm was used to actively control the dynamic response of an integrated structure

through a closed control loop. Detailed theoretical formulations, the Navier solutions

and finite element models based on classical and shear deformation theories were

presented by Reddy (1999) for the analysis of laminated composite plates with


16

integrated sensors and actuators. Analytical solutions for piezoelectric laminates were

also presented by Vel and Batra (2000, 2001, 2004).

1.2.3 Functionally graded materials (FGMs)

Functionally graded materials, as a special class of composite materials, were first

proposed by the Japanese in the late 1980s (Yamanouchi et al., 1990). Many

techniques were developed for fabricating FGMs since then (Sasaki et al. 1989;

Watanabe and Kawasaki 1990). FGMs are generally a mixture made of two or more

materials whose volume fractions are changing continuously along certain dimension

of structures to achieve a required function (Reddy, 2000). The main applications have

been found in high temperature environments (Koizumi, 1993; Sata, 1993). For

example, thermal barrier plate structures are fabricated from a mixture of ceramic and

metal in a high-temperature environment. The composition is varied from a ceramic-

rich surface to a metal-rich surface with a desirable variation of the volume fractions of

the two materials between the two surfaces. The ceramic constituent of the material

provides the high-temperature resistance due to its low-thermal conductivity while the

metal acts as the load-bearing elements. The gradual change of material properties can

be tailored to meet different requirements in applications and environments.

The continuous variation in the microstructure of FGMs shows advantages over the

laminated composite materials in several aspects (Reddy, 2000). Firstly, due to the

mismatch of mechanical properties across the interface of two adjacent bonded layers,

the fiber-matrix composites are prone to debonding at extremely high thermal loading.

Furthermore, cracks are likely to initiate at the interfaces and grow into weaker

material sections. Additional problems include the presence of residual stresses due to

the different coefficients of thermal expansion of the fiber and matrix in the composite


17

materials. FGMs overcome these problems by gradually varying the volume fraction of

the constituents rather than abruptly changing them at the interfaces.

Noda presented an extensive review that covers a wide range of topics from

thermoelastic to thermo-inelastic problems (1991). Some analytical models were

developed in the early 1990s on the thermal behavior of FGMs and collected in

literature (Tanigawa, 1995). Fukui and Yamanaka (1992, 1993) carried out elastic

analysis of thick-walled tubes made of FGM subjected to internal high pressure and

later extended their analysis under uniform thermal loading. Fuchiyama et al. (1993)

studied the transient stresses and particularly the stress intensity factors of FGMs with

cracks using eight-node quadrilateral axisymmetric finite elements. Jin and Batra

(1996, 1998) have given general expressions for stress concentration under thermal

loadings. Obata and Noda (1992, 1994) studied the in-plane thermal stress

distributions on an assumed temperature distribution in the thickness direction of a

plate and examined the relationship between the volume fraction of its components and

the distributions of temperature and thermal stresses. A set of finite element

formulations (He et al. 2001; 2002) were also established for the active control of

functionally graded material (FGM) plates and shells with integrated piezoelectric

sensor/actuator layers subjected to a thermal gradient.

Optimization design regarding the composition and variation of components was

also investigated to achieve desired stress distributions. Tanigawa (1992) formulated

the optimization problem of the material composition to reduce the thermal stress

distribution. Tanaka et al. (1993a, b) designed FGM property profiles using sensitivity

and optimization methods to reduce thermal stresses. Jin and Noda (1993, 1994)

studied the steady state and transient heat conduction problems, and particularly the

thermal stress intensity factors for a crack in a strip and semi-infinite plate and


18

optimized the material property variation based on the minimization of these factors.

The optimum material design for FGMs was also covered in the work by Obata and

Noda (1996).

The wave propagation in FGM plates was investigated by Liu et al. (1990, 1991).

They presented the concept of functionally graded piezoelectric materials (FGPMs) for

the first time and analyzed them using a hybrid numerical method (Liu and Tani, 1991,

1994). Transient waves in FGM plates and cylinders were studied by Han et al. (2001

a, b).

Recently, the nonlinear transient analysis was carried out for functionally graded

ceramic-metal plates under thermal loading using the finite element method by

Praveen and Reddy (1998). They found that the response of the plates with material

properties between those of the ceramic and metal is not intermediate to the response

of the pure ceramic and metal plates. Theoretical formulations and FEM models for

FGM plates based on the third-order shear deformation theory were also presented by

Reddy (2000), which account for thermomechanical coupling, time dependency and

the von Karman-type geometric nonlinearity. Vel and Batra (2002, 2004) have given

exact solutions for static and vibration problems of functionally graded rectangular

plates.

1.2.4 Meshfree methods for nonlinear problems

As mentioned above, in meshfree methods, the domain of interest is discretized by

a set of scattered points. One of the successes of meshfree methods is due to the

development of new shape functions that allow the approximation of field variables to

be accomplished at a local subdomain and therefore a mesh structure is avoided.


19

Meshfree methods are ideal for model refinement and adaptive analysis, fracture

problems and large deformation problems.

The EFG method has been widely used in fracture problems such as dynamic

fracture (Belytschko et al. 1995; Belytschko and Tabarra, 1996), crack growth

(Belytschko et al. 1995; Xu and Saigal, 1998; Krysl and Belytschko, 1999). Later on,

Barry and Saigal (1999) applied the EFG method to the analysis of 3-D elastic and

elastoplastic problems. As the modified MLS approximation functions in their work

are true interpolants satisfying the delta function property, the direct imposition of

essential boundary conditions is allowed. Kargarnovin et al. (2003) extended the EFG

method to solve field equations for the incremental plastic behavior of materials. In

particular they showed that the method is applicable and useful for the stress analysis

around cracks.

The RKPM has also been used for nonlinear problems. Chen et al. (1996)

implemented this method for the large deformation analysis of non-linear structures.

Their applications included both path-independent and path-dependent materials with

emphasis on hyperelasticity and elasto-plasticity. In their work, a material kernel

function was introduced which covers the same set of particles throughout the course

of deformation and hence instability is avoided in large deformation analysis. Chen et

al. (1997) also applied the RKPM for the large deformation of rubber materials which

are considered to be hyperelastic and nearly incompressible. A modified RKPM shape

function that possesses Kronecker delta property was developed by a transformation

method to impose essential boundary conditions. Higher-order rubber strain energy

density functions were used to better represent the nonlinear behavior of rubber. They

found volume locking is avoided when applying RKPM to large deformation analysis


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of nearly incompressible hyperelastic problems. For the same level of accuracy RKPM

requires fewer degrees of freedom compared to FEM.

1.3 Objectives of the research

From above sections it is seen that the meshfree methods are still in the infancy.

They still offer many strategies as well as opportunities for researchers to develop

them into a powerful, robust and versatile numerical method.

As discussed earlier, the element-free Galerkin method has two key advantages:

one is it does not require any element connectivity information for field approximation

and the other is it does not suffer much degradation in accuracy when nodal

distribution is irregular. As the MLS approximation used in EFG is a local regression

approach, the Kronecker delta property does not hold for EFG shape functions. Hence

the application of essential boundary conditions has been a major limitation of EFG as

well as other meshfree methods employing this approximation.

Kriging method is already a very well-known geostatistical technique for special

interpolation in geology and mining. Recently it has been discovered to have the

capability in computational mechanics. If moving Kriging method can be employed in

meshfree method, field variables may be interpolated, rather than approximated by

MLS, from nodal values of the neighborhood. Therefore, the first objective is to

develop meshfree method based on Kriging interpolation and to study its numerical

properties.

The effectiveness and capability of a new numerical method can only be

demonstrated and verified by a set of numerical simulations in different engineering

areas. The testing is not confined to 2-D elastic problems. It should also include some


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other complicated structures and systems as well. The second objective is to apply the

developed meshfree method to complex structures and systems which cover the

piezoelectric structures, composite laminates, functionally graded material (FGM)

plates and the realization of static shape control as well as active vibration repression

using integrated piezoelectric sensors and actuators. The material should include not

only mechanical properties, but also the coupling with thermal and electrical behavior

as well.

The third objective is to apply the developed meshfree method to the elasto-plastic

analysis (including material nonlinearity) and geometrically nonlinear analysis with

large deformation. For the former material nonlinear problems could be formulated

based on the deformation theory and numerical implementation can be very easily

realized, while for the latter part the classical incremental theory will be applied

together with the Newton-Raphson method. Both total Lagrangian (TL) and updated

Lagrangian (UL) formulations may be established within the framework of meshfree

method. Both static and dynamic nonlinear analysis may be conducted.

1.4 Organization of the thesis

This thesis is divided into three parts. The first part, consisting of chapters 2-4,

deals with the field approximation methods and in particular, the construction of the

meshfree shape functions. The related meshfree methods are also described in detail.

The second part, ranging from chapter 5 to chapter 8, concerns the analysis of

complicated structures and systems made of composite materials, smart materials,

FGMs as well as static and dynamic active controlling mechanism. The last part,


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covering chapters 9, 10 and 11, treats elasto-plastic and geometrically nonlinear static

as well as transient analysis using meshfree method.

Chapter 2 describes the moving least squares method and its derivative, the

element-free Galerkin method. Three techniques are introduced to impose the essential

boundary conditions in process of weak-form formulation. Shear locking mechanism is

investigated in solving beam or plate problems using higher order theory and a

practical and effective technique is recommended. Chapter 3 deals with the point

interpolation method and the construction of its shape functions. Both polynomial

terms and radial basis functions are employed as basis for interpolation. Particular

emphasis is put on the study of the compatibility of PIM. Chapter 4 gives an

alternative formulation for radial PIM based on the moving Kriging method.

Comparison between Kriging formulation and RPIM is made and some important

conclusions are drawn there. Spherical function is introduced into RPIM and its

properties are examined through numerical examples.

Chapter 5 treats 2-D piezoelectric structures. Piezoelectric structures with arbitrary

shape are formulated based on Galerkin weak-form formulation and linear constitutive

piezoelectric equations. Both the mechanical and electrostatic properties are

considered. Numerical examples are presented to investigate the deflection of a

piezoelectric bending motor, and the natural frequencies and vibration modes of

transducers. Chapter 6 is devoted to the analysis of shear deformable laminated plates

using both PIM and EFG method. Three kinds of laminated plate theories are

discussed and compared in numerical ways. Many numerical examples are performed

in this chapter for different laminates with different side-to-thickness ratios, materials,

boundary conditions, or ply angles. Chapter 7 describes the static shape control as well

as dynamic response repression of laminated plates containing distributed piezoelectric


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sensors and actuators. Both mechanical and electric variables are interpolated using the

PIM shape functions. A constant feedback control algorithm is applied for the dynamic

response control through a closed loop. Chapter 8 deals with functionally graded

material (FGM) plates subject to both mechanical and thermal loadings. A meshfree

model is presented for the thermomechanical analysis and active control of FGM

plates through piezoelectric sensors and actuators. The influence of the power law

exponents on the static deflection and natural frequency analysis are studied in detail.

Chapter 9 concerns the inelastic analysis of 2-D solids based on the deformation

theory. Material nonlinearity is considered by suitable updating of material properties

in terms of effective material parameters that can be easily obtained from the one-

dimensional uniaxial material curve in an iterative procedure. Compared with the

conventional inelastic analysis using classical incremental method and flow theory, the

present scheme can be easily implemented numerically and the external loadings can

be applied totally in one step. Chapter 10 deals with geometrically nonlinear analysis

of solids with large deformation. Meshfree models are established based on

incremental theory. Variational equations are derived for both total Lagrange (TL) and

update Lagrange (UL) formulations. A hyperelastic constitutive model for

compressible neo-Hookean material is introduced. Several benchmark problems are

analyzed and compared with FEM solutions. Chapter 11 studies the nonlinear transient

dynamic analysis. Both the material and geometric nonlinearities are considered in the

formulations and explicit central difference method is used for time integration.

Geometrically nonlinear static solutions are predicted through transient dynamic

responses.


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1.5 Original contributions

The original contributions of the present research mainly lie on the following

aspects. The first one is moving Kriging method is formulated in meshfree method and

applied for analyzing 2-D elastic problems in solid mechanics for the first time. The

relationship between RPIM and Kriging shape functions is unveiled. A new basis

function is found effective for meshfree Kriging method. Many techniques have been

proposed for numerical implementation, such as the penalized EFG method, the

conforming PIM, etc.

Some complicated structures involving thermo-electro-mechanical coupling have

been successfully analyzed using developed meshfree methods, which include smart

structures, functionally-graded-material (FGM) structures, shear deformable laminated

composite plates, piezoelectric structures. Static shape control as well as dynamic

response repression has been realized successfully through introduced feedback control

algorithms.

Both static and dynamic nonlinear problems have been well resolved considering

material or geometrical nonlinearities. Elasto-plastic problems can be easily and

accurately analyzed based on the Henckys deformation theory. Total Lagrange

formulations are established for geometrically nonlinear solids with large

deformations. Static nonlinear solution can be exactly predicted from nonlinear

transient dynamic responses by introducing suitable damping effects.

Documents

01 Prefix & Chapt 01