1. Introduction ................................................. 1
1.1. Defining Mesh Free Methods .............................. 1
1.2. Need for MFree Methods .................................. 3
1.3. The Idea of MFree Methods ............................... 4
1.4. Outline of the Book ..................................... 4
2. Mesh Free Methods for Engineering Problems ................... 9
2.1. Physical Phenomena in Engineering ....................... 9
2.2. Solution Procedure ...................................... 9
2.3. Modeling the Geometry .................................. 10
2.4. Node Generation ........................................ 13
2.5. Shape Function Creation ................................ 15
2.6. Property of Material or Media .......................... 15
2.7. Boundary, Initial, and Loading Conditions .............. 15
2.8. Simulation ............................................. 16
2.8.1. Discrete System Equations ....................... 16
2.8.2. Equation Solvers ................................ 17
2.9. Visualization .......................................... 18
2.10.MFree Method Procedure ................................. 18
2.10.1.Basic Steps ..................................... 19
2.10.2.Determination of the Dimension of a Support
Domain .......................................... 22
2.10.3.Determination of the Average Nodal Spacing ...... 22
2.10.4.Concept of the Influence Domain ................. 23
2.10.5.Property of MFree Shape Functions ............... 24
2.11.Remarks ................................................ 25
3. Mechanics of Solids and Structures .......................... 27
3.1. Basics ................................................. 27
3.2. Equations for Three-Dimensional Solids ................. 28
3.2.1. Stress and Strain ............................... 28
3.2.2. Constitutive Equations .......................... 32
3.2.3. Dynamic Equilibrium Equations ................... 33
3.3. Equations for Two-Dimensional Solids ................... 34
3.3.1. Stress and Strain ............................... 34
3.3.2. Constitutive Equations .......................... 36
3.3.3. Dynamic Equilibrium Equations ................... 37
3.4. Equations for Truss Members ............................ 37
3.4.1. Stress and Strain ............................... 38
3.4.2. Constitutive Equations .......................... 38
3.4.3. Dynamic Equilibrium Equations ................... 38
3.5. Equations for Beams .................................... 38
3.5.1. Stress and Strain ............................... 39
3.5.2. Constitutive Equations .......................... 40
3.5.3. Moments and Shear Forces ........................ 40
3.5.4. Dynamic Equilibrium Equations ................... 42
3.5.5. Equations for Thick Beams ....................... 42
3.6. Equations for Plates ................................... 43
3.6.1. Thin Plates ..................................... 43
3.6.2. Mindlin Plates .................................. 48
3.6.3. Third-Order Theory of Plates .................... 50
3.7. Remarks ................................................ 51
4. Principles for Weak Forms ................................... 53
4.1. Strong Forms vs. Weak Forms ............................ 53
4.2. Hamilton's Principle ................................... 54
4.3. Constrained Hamilton's Principle ....................... 55
4.3.1. Method of Lagrange Multipliers .................. 56
4.3.2. Penalty Method .................................. 56
4.3.3. Determination of Penalty Factor ................. 58
4.4. Galerkin Weak Form ..................................... 58
4.5. Constrained Galerkin Weak Form ......................... 61
4.5.1. Galerkin Weak Form with Lagrange Multipliers .... 62
4.5.2. Galerkin Weak Form with Penalty Factors ......... 62
4.6. Minimum Total Potential Energy Principle ............... 62
4.7. Weighted Residual Method ............................... 63
4.8. Weighted Residual Method with Constraints .............. 64
4.9. Points to Note ......................................... 65
4.10.Remarks ................................................ 65
5. MFree Shape Function Construction ........................... 67
5.1. Overview ............................................... 67
5.2. Smoothed Particle Hydrodynamics Approach ............... 70
5.2.1. Choice of Weight Function ....................... 72
5.2.2. Consistency ..................................... 74
5.3. Reproducing Kernel Particle Method ..................... 77
5.4. Moving Least Squares Approximation ..................... 79
5.4.1. MLS Procedure ................................... 79
5.4.2. Consistency ..................................... 84
5.4.3. Continuous Moving Least Square Approximation .... 86
5.5. Point Interpolation Method ............................. 87
5.5.1. Polynomial PIM .................................. 87
5.5.2. Consistency ..................................... 90
5.5.3. Properties of PIM Shape Functions ............... 91
5.5.4. Difference between PIM Interpolation
and MLS Approximation ........................... 94
5.5.5. Methods to Avoid Singular Moment Matrix ......... 94
5.6. Radial PIM ............................................. 96
5.6.1. Rationale for Radial Basis Functions ............ 96
5.6.2. PIM Formation Using Radial Basis Functions ...... 96
5.6.3. Nonsingular Moment Matrix ....................... 98
5.6.4. Consistency ..................................... 99
5.6.5. Radial Functions with Dimensionless Shape
Parameters ...................................... 99
5.7. Radial PIM with Polynomial Reproduction ............... 101
5.7.1. Rationale for Polynomials ...................... 101
5.7.2. Formulation Using Radial-Polynomial Basis ...... 101
5.7.3. Singularity Issue of the Transformed Moment
Matrix ......................................... 104
Example 5.1. Sample RPIM Shape Functions .............. 104
Example 5.2. Effects of Shape Parameters of RBFs
on Shape Function ........................ 107
5.8. Polynomial PIM with Coordinate Transformation ......... 107
5.8.1. Coordinate Transformation ...................... 108
5.8.2. Choice of Rotation Angle ....................... 109
5.9. Matrix Triangularization Algorithm .................... 110
5.9.1. MTA Procedure .................................. 1ll
5.9.2. Normalization of the Support Domain ............ 112
5.9.3. MTA Flowchart .................................. 113
5.9.4. Test Examples .................................. 113
Example 5.3. Interpolation Using 6 Nodes in
Parallel Lines .................... 113
Example 5.4. Interpolation Using 12 Nodes in
Parallel Lines .................... 115
5.10.Comparison Study via Examples ......................... 116
Example 5.5. Comparison of Shape Functions
Obtained Using Different Methods
(ID Case) ......................... 116
Example 5.6. Comparison of Shape Functions
Obtained Using Different Methods
(2D Case) ......................... 118
Example 5.7. Curve Fitting Using MFree Shape
Functions ......................... 118
Example 5.8. Effects of Shape Parameters on
the Condition Number of Moment
Matrices and Curve Fitting ........ 125
Example 5.9. Surface Fitting Using MFree
Shape Functions (Effects of
Parameters) ....................... 130
Example 5.10.Surface Fitting Using MFree
Shape Functions (Accuracy in
Derivatives of the Fitted
Surface) .......................... 137
Example 5.11.Surface Fitting Using MFree
Shape Functions (Effects of the
Support Domain) ................... 138
5.11.Compatibility of MFree Function Approximation ......... 138
5.12.On the Concept of Reproduction ........................ 143
5.13.Other Methods ......................................... 144
5.14.Remarks ............................................... 144
6. Element Free Galerkin Method ............................... 147
6.1. EFG Formulation with Lagrange Multipliers ............. 147
6.1.1. Formulation .................................... 147
6.1.2. EFG Procedure .................................. 156
6.1.3. Background Integration ......................... 156
6.1.4. Numerical Examples ............................. 158
Example 6.1. Patch Test ........................ 158
Example 6.2. Cantilever Beam (Numerical
Integration) ...................... 161
6.1.5. Remarks ........................................ 168
6.2. EFG with Penalty Method ............................... 169
6.2.1. Formulation .................................... 169
6.2.2. Penalty Method for Essential Boundary
Conditions ..................................... 170
6.2.3. Penalty Method for Continuity Conditions ....... 171
6.2.4. Numerical Examples ............................. 174
Example 6.3. Patch Test ........................ 174
Example 6.4. Timoshenko Beam ................... 174
Example 6.5. Cantilever Beam of Bi-Material .... 178
Example 6.6. Sandwich Composite Beam ........... 179
6.2.5. Remarks ........................................ 181
6.3. Constrained Moving Least Square Method for EFG ........ 181
6.3.1. Formulation .................................... 182
6.3.2. Constrained Surfaces Generated by CMLS ......... 185
Example 6.7. Linear Constraint ................. 185
Example 6.8. Parabolic Constraint .............. 185
6.3.3. Weak Form and Discrete Equations ............... 188
6.3.4. Examples for Mechanics Problems ................ 189
Example 6.9. Patch Test ........................ 189
Example 6.10.Cantilever Beam ................... 190
Example 6.11.Hole in an Infinite Plate ......... 191
6.3.5. Computational Time ............................. 195
6.3.6. Remarks ........................................ 197
6.4. EFG for Nonlinear Elastic Problems .................... 198
6.4.1. Basic Equations ................................ 198
6.4.2. Weak Form for Nonlinear Elastic Problems ....... 200
6.4.3. Discretization and Numerical Strategy .......... 200
6.4.4. Numerical Procedure ............................ 201
6.4.5. Numerical Example .............................. 202
Example 6.12. Soil Foundation .................. 202
6.4.6. Remarks ........................................ 208
6.5. Summary ............................................... 210
7. Meshless Local Petrov-Galerkin Method ...................... 211
7.1. MLPG Formulation ...................................... 212
7.1.1. The Idea of MLPG ............................... 212
7.1.2. Formulation of MLPG ............................ 213
7.1.3. Types of Domains ............................... 217
7.1.4. Procedures for Essential Boundary Conditions ... 218
7.1.5. Numerical Investigation ........................ 219
7.1.6. Examples ....................................... 220
Example 7.1. Patch Test ........................ 220
Example 7.2.High-Order Patch Test .............. 221
Example 7.3.Cantilever Beam .................... 223
Example 7.4.Infinite Plate with a Circular
Hole ............................... 226
Example 7.5.Half-Plane Problem ................. 226
7.2. MLPG for Dynamic Problems ............................. 229
7.2.1. Statement of the Problem ....................... 229
7.2.2. Free-Vibration Analysis ........................ 230
7.2.3. Imposition of Essential Boundary Conditions
for Free Vibration ............................. 232
7.2.4. Numerical Examples ............................. 233
Example 7.6. Cantilever Beam ................... 233
Example 7.7. Cantilever Beam with Variable
Cross Section ..................... 236
Example 7.8. Shear Wall ........................ 236
7.2.5. Forced Vibration Analysis ...................... 237
7.2.6. Direct Analysis of Forced Vibration ............ 239
7.2.7. Numerical Examples ............................. 240
Example 7.9. Cantilever Beam ................... 240
Example 7.9a.Simple Harmonic Loading ........... 241
Example 7.9b.Transient Loading ................. 244
7.3. Remarks ............................................... 246
8. Point Interpolation Methods ................................ 249
8.1. Polynomial Point Interpolation Method ................. 250
8.1.1. Domain Discretization .......................... 250
8.1.2. Enclosure of Nodes ............................. 251
8.1.3. Variational Form of Galerkin PIM ............... 253
8.1.4. Comparison of PIM, EFG, and FEM ................ 255
8.1.5. Numerical Examples ............................. 256
Example 8.1. Patch Test ........................ 256
Example 8.2. Cantilever Beam ................... 258
Example 8.3. Hole in an Infinite Plate ......... 263
Example 8.4. Bridge Pier ....................... 264
8.1.6. Remarks ........................................ 265
8.2. Application of PIM to Foundation Consolidation
Problem ............................................... 266
8.2.1. Biot's Consolidation Theory and Its Weak
Form ........................................... 266
8.2.2. Discretization of Weak Form .................... 268
8.2.3. Numerical Examples ............................. 270
Example 8.5. One-Dimensional Consolidation
Problem ........................... 270
Example 8.6. Two-Dimensional Consolidation
Problem ........................... 274
8.3. Radial Point Interpolation Method ..................... 276
8.3.1. Key Considerations ............................. 276
8.3.2. Numerical Examples ............................. 281
Example 8.7. Patch Test ........................ 281
Example 8.8. Cantilever Beam ................... 282
Example 8.9. Infinite Plate with a Hole ........ 292
Example 8.10.Parallel Tunnel ................... 297
8.3.3. Remarks ........................................ 299
8.4. Local Point Interpolation Method (LPIM) ............... 300
8.4.1. LPIM Formulation ............................... 301
8.4.2. Weight Function ................................ 302
8.4.3. Numerical Examples ............................. 303
Example 8.11.Standard Patch Test (LPIM +
MTA) .............................. 303
Example 8.12.Higher-Order Patch Test ........... 305
Example 8.13.Cantilever Beam ................... 306
Example 8.14.Infinite Plate with a Hole ........ 311
Example 8.15.Stress Distribution in a Dam ...... 312
8.4.4. Remarks ........................................ 313
8.5. Local Radial Point Interpolation Method ............... 314
8.5.1. Examples of Static Problems .................... 314
Example 8.16.Patch Test ........................ 315
Example 8.17.High-Order Patch Test ............. 315
Example 8.18.Cantilever Beam ................... 316
Example 8.19.Infinite Plate with a Circular
Hole .............................. 322
Example 8.20.Half-Plane Problem ................ 323
8.5.2. Examples of Dynamic Problems ................... 323
Example 8.21.Cantilever Beam ................... 324
Example 8.22.Free Vibration Analysis of
a Shear Wall ...................... 334
8.5.3. Remarks ........................................ 334
8.6. Application of LRPIM to Diffusion Equations ........... 335
8.6.1. Terzaghi's Consolidation Theory ................ 335
8.6.2. Discretized System Equation in the Time
Domain ......................................... 337
8.6.3. Numerical Example .............................. 338
Example 8.23.Two-Dimensional Foundation ........ 338
8.7. Comparison Study ...................................... 339
8.7.1. Convergence Comparison ......................... 339
Example 8.24.Cantilever Beam (Convergence
of LPIM-MTA, MQ-LRPIM, and
MLPG) ............................. 339
8.7.2. Efficiency Comparison. 341
Example 8.25.Cantilever Beam (Efficiency of
LPIM-MTA, MQ-LRPIM, and MLPG) ..... 341
8.8. Summary ............................................... 341
9. Mesh Free Methods for Fluid Dynamics Problems .............. 345
9.1. Introduction .......................................... 345
9.2. Smoothed Particle Hydrodynamics Method ................ 346
9.2.1. SPH Basics ..................................... 347
9.2.2. SPH Formulations for Navier-Stokes Equation .... 348
9.2.3. Major Numerical Implementation Issues .......... 352
9.2.4. SPH Code Structure ............................. 359
9.2.5. Applications ................................... 359
Example 9.1. Poiseuille Flow ................... 361
Example 9.2. Couette Flow ...................... 361
Example 9.3. Shear-Driven Cavity Problem ....... 362
Example 9.4. Free Surface Flows ................ 362
Example 9.5. Explosion in Vacuum ............... 365
Example 9.6. Simulation of Explosion
Mitigated by Water ................ 366
9.2.6. Remarks ........................................ 368
9.3. Local Petrov-Galerkin Method .......................... 369
9.3.1. MLPG Formulation ............................... 369
9.3.2. Numerical Integration in MLPG .................. 370
9.3.3. Governing Equations and Their Discretized
Form ........................................... 376
9.3.4. Boundary Condition for Vorticity ............... 378
9.3.5. Numerical Results and Discussion ............... 379
Example 9.7. Natural Convection in a Square
Cavity Problem .................... 379
9.3.6. Remarks ........................................ 381
9.4. Local Radial Point Interpolation Method ............... 382
9.4.1. LRPIM Formulation .............................. 383
9.4.2. Implementation Issue in LRPIM for CFD
Problems ....................................... 383
9.4.3. Numerical Results and Discussion ............... 384
Example 9.8. Natural Convection in a Square
Cavity ............................ 384
Example 9.9. Natural Convection in
a Concentric Annulus .............. 386
9.4.4. Remarks ........................................ 388
10.Mesh Free Methods for Beams ................................ 391
10.1. PIM Shape Function for Thin Beams .................... 392
10.1.1.Formulation ................................... 392
10.1.2.Example ....................................... 394
Example 10.1.PIM Shape Functions for Thin
Beams ......................................... 394
10.2. Elastostatic Analysis of Thin Beams .................. 396
10.2.1.Local Weighted Residual Weak Form ............. 396
10.2.2.Discretized System Equations .................. 398
10.2.3.Numerical Example for Static Problems ......... 399
Example 10.2.Simply-Simply Supported Beams
under Various Loads ............... 399
Example 10.3.Beams under Uniformly
Distributed Load with Different
Boundary Conditions ............... 401
10.3.Buckling Analysis of Thin Beams (Eigenvalue
Problem) .............................................. 403
10.3.1.Local Weak Form ................................ 403
10.3.2.Discretized System Equations ................... 403
10.3.3.Numerical Example .............................. 404
Example 10.4.Bulking Analysis of Thin Beams .... 404
10.4.Free-Vibration Analysis of Thin Beams (Eigenvalue
Problem) .............................................. 405
10.4.1.Local Weak Form ................................ 405
10.4.2.Discretized System Equations ................... 405
10.4.3.Numerical Results .............................. 406
Example 10.5.Free-Vibration Analysis of
Thin Beams ........................ 406
10.5.Forced Vibration Analysis of Thin Beams (Time-
Dependent Problem) .................................... 408
10.5.1.Local Weak Form ................................ 408
10.5.2.Discretized System Equations ................... 409
10.5.3.Numerical Results .............................. 410
Example 10.6.Vibration of a Pinned-Pinned
Thin Uniform Beam Subject to
Harmonic Loading .................. 410
Example 10.7.Vibration of a Pinned-Pinned
Thin Uniform Beam Subject to
Transient Loading ................. 411
10.6.Timoshenko Beams ...................................... 413
10.6.1.Local Weak Form ................................ 414
10.6.2.Discretized System Equations ................... 415
10.6.3.Numerical Example .............................. 416
Example 10.8.Static Deflection of Timoshenko
Beams ............................. 416
10.7.Remarks ............................................... 419
11.Mesh Free Methods for Plates ............................... 421
11.1.EFG Method for Thin Plates ............................ 421
11.1.1.Approximation of Deflection .................... 422
11.1.2.Variational Forms .............................. 423
11.1.3.Discrete Equations ............................. 425
11.1.4.Eigenvalue Problem ............................. 429
11.1.5.Numerical Examples ............................. 431
Example 11.1.Static Deflection of
Rectangular Thin Plates ........... 431
Example 11.2.Natural Frequency Analysis of
Thin Square Plates ................ 433
Example 11.3.Natural Frequency Analysis of
Elliptical Plates ................. 435
Example 11.4.Natural Frequency Analysis of
Polygonal Plates .................. 435
Example 11.5.Natural Frequency Analysis of
a Plate of Complex Shape .......... 437
11.2.EFG Method for Thin Composite Laminates ............... 439
11.2.1.Governing Equation for Buckling ................ 440
11.2.2.Discretized Equation for Buckling Analysis ..... 442
11.2.3.Discretized Equation for Free-Vibration
Analysis ....................................... 444
11.2.4.Numerical Examples for Buckling Analysis ....... 444
Example 11.6.Static Buckling of Rectangular
Plates (Validation) ............... 444
Example 11.7.Static Buckling of a Square
Plate (Efficiency) ................ 446
Example 11.8.Static Buckling of a Plate with
Complicated Shape (Application) ... 447
Example 11.9.Static Buckling of a Laminated
Plate (Application) ............... 448
11.2.5.Numerical Examples for Free-Vibration
Analysis ....................................... 449
Example 11.10.Frequency Analysis of Free
Vibration of Orthotropic
Square Plates ..................... 449
Example 11.11.Natural Frequency Analysis of
Composite Laminated Plates ........ 451
11.3.EFG Method for Thick Plates ........................... 457
11.3.1.Field Variables for Thick Plates ............... 458
11.3.2.Approximation of Field Variables ............... 459
11.3.3.Variational Forms of System Equations .......... 460
11.3.4.Discrete System Equations ...................... 461
11.3.5.Discrete Form of Essential Boundary
Conditions ..................................... 462
11.3.6.Equations for Static Deformation Analysis ...... 464
11.3.7.Numerical Examples of Static Deflection
Analyses ....................................... 465
Example 11.12.Comparison of Deflection of
Thin and Thick Square Plates
with Different Types of
Boundary Conditions ............... 465
Example 11.13.Convergence of Deflection of
a Thin Square Plate ............... 466
Example 11.14.Convergence of Deflection of a
Thick Square Plate ................ 466
Example 11.15.Maximum Deflections of Thick
Plates under Several Kinds of
Boundaries ........................ 467
Example 11.16.Elimination of Shear Locking ..... 467
11.3.8.Numerical Examples of Vibration Analyses ....... 470
Example 11.17.Frequency Analysis of Thick
Plates (FSDT) ..................... 471
Example 11.18.Frequency Analysis of Thick
Plates (FSDT and TSDT) ............ 471
11.3.9.Numerical Examples of Vibration Analyses ....... 472
Example 11.19.Buckling Analysis of Thick
Plates (FSDT and TSDT) ............ 472
Example 11.20.Buckling Loads of a Square
Plate Based on FSDT and TSDT
with Different Loads and
Boundaries ........................ 472
Example 11.21.Buckling Loads of a Square
Plate with a Circular Hole
Based on FSDT and TSDT ............ 473
11.4.RPIM for Thick Plates ................................. 475
11.4.1.Formulation .................................... 475
11.4.2.Numerical Examples ............................. 475
Example 11.22.Deflection of a Thick Square
Plate (Effects of the EXP
Shape Parameters) ................. 476
Example 11.23.Deflection of a Thick Square
Plate (Effects of the MQ Shape
Parameters) ....................... 478
Example 11.24.Deflection of a Thick Square
Plate (Effects of Polynomial
Terms) ............................ 481
Example 11.25.Deflection of a Thick Square
Plate (Convergence of Maximum
Deflections) ...................... 483
Example 11.26.Deflection of a Thick Square
Plate (Effects of Irregularly
Distributed Nodes) ................ 484
Example 11.27.Deflection of a Thick Square
Plate (Effects of Shear
Locking) .......................... 484
11.5.MLPG for Thin Plates .................................. 486
11.5.1.Governing Equations ............................ 486
11.5.2.Local Weak Form of MLPG ........................ 487
11.5.3.Discretized System Equations ................... 488
11.5.4.Weight Function ................................ 490
11.5.5.Numerical Integration .......................... 490
11.5.6.Numerical Examples ............................. 491
Example 11.28.Static Analysis of Thin Square
Plates ............................ 491
Example 11.29.Square Plate under Different
Load with Different Support ....... 494
Example 11.30.Static Analysis of Thin
Rectangular Plates ................ 497
Example 11.31.Static Deflection Analysis of
a Circular Plate .................. 497
Example 11.32.Free-Vibration Analysis of
Thin Plates ....................... 498
11.6.Remarks ............................................... 499
12.Mesh Free Methods for Shells ............................... 501
12.1.EFG Method for Spatial Thin Shells .................... 502
12.1.1.Moving Least Squares Approximation ............. 502
12.1.2.Governing Equation for Thin Shell .............. 503
12.1.3.Strain-Displacement Relations .................. 506
12.1.4.Principle of Virtual Work ...................... 507
12.1.5.Surface Approximation .......................... 508
12.1.6.Discretized Equations .......................... 508
12.1.7.Static Analysis ................................ 509
12.1.8.Free Vibration ................................. 509
12.1.9.Forced (Transient) Vibration ................... 510
12.1.10.Numerical Example for Static Problems ......... 511
Example 12.1.Static Deflection of a Barrel
Vault Roof under Gravity Force .... 511
12.1.11.Numerical Examples for Free Vibration of
Thin Shells .................................... 514
Example 12.2.Free Vibration of a Clamped
Cylindrical Shell Panel ........... 514
Example 12.3.Free Vibration of a
Hyperbolical Shell ................ 517
Example 12.4.Free Vibration of a Cylindrical
Shell ............................. 517
12.1.12.Numerical Examples for Forced Vibration of
Thin Shells .................................... 519
Example 12.5.Clamped Circular Plate Subject
to an Impulsive Load .............. 519
Example 12.6.Clamped Cylindrical Shell
Subject to a Sine Load ............ 519
Example 12.7.Clamped Spherical Shell Subject
to a Sine Curve Load .............. 521
12.1.13.Remarks ....................................... 523
12.2.EFG Method for Thick Shells ........................... 523
12.2.1.Fundamental Relations .......................... 523
12.2.2.Principle of Virtual Work ...................... 524
12.2.3.Numerical Examples ............................. 525
Example 12.8.Static Deflection of a Barrel
Vault Roof under Gravity Force .... 525
Example 12.9.Pinched Cylindrical Shell ......... 526
Example 12.10.Pinched Hemispherical Shell ...... 530
12.2.4.Remarks ........................................ 532
12.3.RPIM for Thick Shells ................................. 534
12.3.1.Formulation Procedure .......................... 534
12.3.2.Numerical Examples ............................. 534
Example 12.11.Barrel Vault Roof ................ 534
Example 12.12.Pinched Cylindrical Shell ........ 541
Example 12.13.Pinched Hemispherical Shell ...... 541
12.3.3.Remarks ........................................ 543
12.4.Summary ............................................... 544
13.Boundary Mesh Free Methods ................................. 545
13.1.BPIM Using Polynomial Basis ........................... 546
13.1.1.Point Interpolation on Curves .................. 546
13.1.2.Discrete Equations of BPIM ..................... 549
13.1.3.Implementation Issues in BPIM .................. 550
13.1.4.Numerical Examples ............................. 551
Example 13.1.Cantilever Beam ................... 551
Example 13.2.Plate with a Hole ................. 553
Example 13.3.A Rigid Flat Punch on a Semi-
Infinite Foundation ............... 554
13.2.BPIM Using Radial Function Basis ...................... 557
13.2.1.Radial Basis Point Interpolation ............... 557
13.2.2.BRPIM Formulation .............................. 558
13.2.3.Comparison of BPIM, BNM, and BEM ............... 559
13.2.4.Numerical Examples.............................. 560
Example 13.4.Cantilever Beam ................... 560
Example 13.5.Plate with a Hole ................. 562
Example 13.6.Internally Pressurized Hollow
Cylinder .......................... 563
13.3.Remarks ............................................... 565
14.Mesh Free Methods Coupled with Other Methods ............... 567
14.1.Coupled EFG/BEM ....................................... 567
14.1.1.Basic Equations of Elastostatics ............... 568
14.1.2.Discrete Equations of EFG ...................... 568
14.1.3.BE Formulation ................................. 569
14.1.4.Coupling of EFG and BE System Equations ........ 570
14.1.5.Numerical Results .............................. 574
Example 14.1.Cantilever Beam ................... 574
Example 14.2.Hole in an Infinite Plate ......... 576
Example 14.3.A Structure on a Semi-Infinite
Soil Foundation ................... 579
14.2.Coupled EFG and Hybrid BEM ............................ 580
14.2.1.EFG Formulation ................................ 582
14.2.2.Hybrid Displacement BE Formulation ............. 583
14.2.3.Coupling of EFG and HBE ........................ 584
14.2.4.Numerical Results .............................. 586
Example 14.4.Cantilever Beam ................... 587
Example 14.5.Hole in an Infinite Plate ......... 587
Example 14.6.Structure on a Semi-Infinite
Foundation ........................ 588
14.3.Coupled MLPG/FE/BE Methods ............................ 589
14.3.1.MLPG Formulation ............................... 590
14.3.2.FE Formulation ................................. 590
14.3.3.Coupling of MLPG and FE or BE .................. 591
14.3.4.Numerical Results .............................. 593
Example 14.7.Cantilever Beam ................... 593
Example 14.8.Hole in an Infinite Plate ......... 594
Example 14.9.Internal Pressurized Hollow
Cylinder .......................... 595
Example 14.10.A Structure on a Semi-Infinite
Foundation ........................ 596
14.4.Remarks ............................................... 599
15.Implementation Issues ...................................... 601
15.1.Definition of the Support Domain or Influence
Domain ................................................ 601
15.2.Triangular Mesh and Size of the Influence Domain ...... 602
15.3.Node Numbering and Bandwidth of the Stiffness
Matrix ................................................ 603
15.4.Bucket Algorithm for Node Searching ................... 604
15.5.Relay Model for Domains with Irregular Boundaries ..... 605
15.5.1.Problem Statement .............................. 605
15.5.2.Visibility Method .............................. 607
15.5.3.Diffraction Method ............................. 607
15.5.4.Transparency Method ............................ 609
15.5.5.The Relay Model ................................ 610
15.6.Adaptive Procedure Based on Background Cells .......... 625
15.6.1.Issues of Adaptive Analysis .................... 625
15.6.2.Existing Error Estimates ....................... 628
15.6.3.Cell Energy Error Estimate ..................... 628
15.6.4.Numerical Examples ............................. 631
Example 15.1.Cantilever Beam (Error
Estimation) ....................... 631
Example 15.2.Infinite Plate with a Circular
Hole (Error Estimation) ........... 631
Example 15.3.A Square Plate Containing
a Crack ........................... 634
15.7.Strategy for Local Adaptive Refinement ................ 634
15.7.1.Update of the Density Factor ................... 636
15.7.2.Local Delaunay Triangulation Algorithm ......... 636
Example 15.4.Infinite Plate with a Circular
Hole (Adaptive Analysis) .......... 638
Example 15.5.Square Plate with a Square Hole
(Adaptive Analysis) ............... 638
Example 15.6.Square Plate with a Crack
(Adaptive Analysis) ............... 638
Example 15.7.Square Plate with Two
Parallel Cracks (Adaptive
Analysis) ......................... 638
Example 15.8.Arbitrary Complex Domain
(Adaptive Analysis) ............... 638
15.8.Remarks ............................................... 644
16.MFree2D© ................................................... 645
16.1.Overview .............................................. 645
16.2.Techniques Used in MFree2D ............................ 646
16.3.Preprocessing in MFree2D .............................. 646
16.3.1.Main Windows ................................... 647
16.3.2.Geometry Creation .............................. 648
16.3.3.Boundary Conditions and Loads .................. 650
16.3.4.Modify and Delete Boundary Conditions and
Loads .......................................... 654
16.3.5.Node Generation ................................ 655
16.3.6.Materials Property Input ....................... 656
16.3.7.Miscellaneous .................................. 661
16.4.Postprocessing in MFree2D ............................. 661
16.4.1.Start of MFreePost ............................. 661
16.4.2.Window of MFreePost ............................ 661
References ................................................. 675
Index ...................................................... 685
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