Methods of Fundamental Solutions in Solid Mechanics

Cover......Page 1
Methods of Fundamental Solutions in Solid Mechanics......Page 2
Copyright......Page 3
Dedication......Page 4
8 -
Meshless analysis of heat transfer in heterogeneous media......Page 240
Preface......Page 7
B.1 Coordinate transformation......Page 284
List of abbreviations......Page 9
Part I: Fundamentals of meshless methods
......Page 10
3 -
Basics of fundamental solutions and radial basis functions......Page 11
1.2 Review of meshless methods......Page 14
1.3.1 Weighted residual method......Page 16
1.3.2 Method of fundamental solutions......Page 18
1.4.1 Problem description......Page 21
1.4.2 MFS formulation......Page 22
1.4.3 Program structure and source code......Page 26
1.4.3.3 Solving the resulting system of linear equations......Page 27
1.4.3.4 Source code......Page 28
1.4.4.1 Circular disk......Page 32
1.4.4.2 Interior region surrounded by a complex curve......Page 37
1.4.4.3 Biased hollow circle......Page 41
1.5 Some limitations for implementing the method of fundamental solutions......Page 42
1.5.2 Location of source points......Page 45
1.5.3.2 Singular value decomposition......Page 47
1.5.5 Multiple domain problems......Page 49
1.6 Extended method of fundamental solutions......Page 51
1.7 Outline of the book......Page 55
References......Page 56
5 -
Meshless analysis for thin plate bending problems......Page 60
2.2.2 Stress components......Page 61
2.3.1 Strain-displacement relation......Page 64
2.3.2 Equilibrium equations......Page 65
2.3.3 Constitutive equations......Page 66
2.3.4 Boundary conditions......Page 67
2.4.1 Plane stress and plane strain......Page 69
2.4.2 Governing equations......Page 71
7.5 Remarks......Page 72
2.5.1 Deformation mode......Page 73
2.5.2 Governing equations......Page 75
2.5.3 Boundary conditions......Page 78
2.5.4 Continuity requirements......Page 79
2.6.1 Deformation mode......Page 80
2.6.2 Governing equations......Page 82
2.6.3 Boundary conditions......Page 84
2.7 Equations for piezoelectricity......Page 87
2.7.1 Governing equations......Page 88
2.8 Remarks......Page 95
References......Page 96
A......Page 98
3.2.2 Fundamental solutions......Page 101
3.3.1 Radial basis functions......Page 104
3.3.2 Radial basis function interpolation......Page 108
References......Page 111
Part II: Applications of the meshless method
......Page 113
4.1 Introduction......Page 114
4.2 Solution procedures......Page 115
4.2.2 Particular solution......Page 116
4.2.3 Approximated full solution......Page 117
4.2.4 Construction of solving equations......Page 118
4.2.5 Treatment of discontinuous loading......Page 119
4.3.1 Statically indeterminate beam under uniformly distributed loading......Page 121
4.3.3 Cantilever beam with end-concentrated load......Page 124
References......Page 129
5.1 Introduction......Page 131
5.2 Fundamental solutions for thin plate bending......Page 132
5.3 Solutions procedure for thin plate bending......Page 134
5.3.1 Particular solution......Page 135
5.3.2 Homogeneous solution......Page 136
5.3.3 Approximated full solution......Page 137
5.3.4 Construction of solving equations......Page 138
5.4 Results and discussion......Page 139
8.2.5 Thermal conductivity matrix......Page 245
5.4.2 Square plate on a winkler elastic foundation......Page 141
References......Page 145
6.1 Introduction......Page 148
6.2 Fundamental solutions for two-dimensional elasticity......Page 151
6.3.1 Solution procedure......Page 154
6.3.2 Program structure and source code......Page 158
6.3.2.1.1 First DOF......Page 159
6.3.2.3 Solving the resulting system of linear equations......Page 160
6.3.2.4 Source code......Page 161
6.3.3.1 Thick-walled cylinder under internal pressure......Page 166
6.3.3.2 Infinite domain with circular hole subjected to a far-field remote tensile......Page 173
6.4 Solution procedure for inhomogeneous elasticity......Page 176
6.4.1 Particular solution......Page 179
6.4.2 Homogeneous solution......Page 184
6.4.4.1 Rotating disk with high speed......Page 186
6.4.4.2 Symmetric thermoelastic problem in a long cylinder......Page 189
6.5.1 Concept of functionally graded material......Page 192
6.5.2 Thermomechanical systems in FGMs......Page 195
6.5.2.1 Strain-displacement relationship......Page 196
6.5.2.2 Constitutive equations......Page 197
6.5.2.3 Static equilibrium equations......Page 198
6.5.3.1 Analog equation method......Page 199
6.5.3.2 Particular solution......Page 200
6.5.3.3 Homogeneous solution......Page 201
6.5.3.4 Approximated full solution......Page 202
6.5.3.5 Construction of solving equations......Page 203
6.5.4.1 Functionally graded hollow circular plate under radial internal pressure......Page 204
6.5.4.2 Functionally graded elastic beam under sinusoidal transverse load......Page 208
6.5.4.3 Symmetrical thermoelastic problem in a long functionally graded cylinder......Page 209
6.6 Remarks......Page 212
References......Page 213
7.1 Introduction......Page 215
7.2 Fundamental solutions for plane piezoelectricity......Page 216
7.3 Solution procedure for plane piezoelectricity......Page 223
7.4.1 Simple tension of a piezoelectric prism......Page 226
7.4.2 An infinite piezoelectric plane with a circular hole under remote tension......Page 229
7.4.3 An infinite piezoelectric plane with a circular hole subject to internal pressure......Page 234
References......Page 238
8.2.1 Energy balance equation......Page 241
8.2.3 Governing equation......Page 244
8.3.1 Solution procedure......Page 247
8.3.1.2 Particular solution......Page 248
8.3.1.4 Approximated full solution......Page 250
8.3.2.1 Isotropic heterogeneous square plate......Page 251
8.3.2.2 Isotropic heterogeneous circular disc......Page 255
8.3.2.3 Anisotropic homogeneous circular disc......Page 257
8.3.2.4 Anisotropic heterogeneous hollow ellipse......Page 258
8.4 Solution procedure of transient heat transfer......Page 263
8.4.1.1 Time marching scheme......Page 264
8.4.1.2 Approximated full solution......Page 265
8.4.2 Results and discussion......Page 266
8.4.2.1 Isotropic homogeneous square plate with sudden temperature jump......Page 267
8.4.2.2 Isotropic homogeneous square plate with nonzero initial condition......Page 270
8.4.2.3.1 Isotropic functionally graded finite strip......Page 273
References......Page 278
Appendix
A - Derivatives of functions in terms of radial variable r......Page 280
B.2 Vector transformation......Page 285
B.3 Stress transformation......Page 286
C.1 Power spline (PS) function......Page 288
C.2 Thin plate spline (TPS) function......Page 289
D......Page 291
H......Page 292
P......Page 293
T......Page 294
Y......Page 295
Back Cover......Page 296