The finite element method(FEM) has been an essential and most important tool for modeling and simulation for practical engineering problems for solids and structures with complex geometry.The first author is one of the FEM users in the past some 30 years,and he wrote his first FEM code to solve a nonlinear mechanics problem for frame structures in 1979,as his university final year research project.In the past two decades he has participated in and directed many engineering projects of very largescale with millions of degrees of freedom.In those practices,we have frequently encountered problems with mesh related issues in using FEM.For accuracy reasons,we want to use quality quadrilateral(for 2D) or hexa he dr on(for 3D) elements,but such ames his quite difficult to generate and require alot of manual operations to cut the domain into proper subdomains.The times penton such a manual operations has been very significant.When using triangular(for 2D) or tetrahedron(for 3D) elements,the mesh generation becomes much easier and it can often be done automatically without manual operations.However,the accuracy of the results from the FEM can often be quite poor.
In searching for alternatives,the authors'group has started to learn and to develop mesh free methods,and quite good progress has been made in the past 15 years.We can now quite safely say that using proper mesh free techniques we can do pretty much what we want using only a background mesh of triangles/tetrahedrons.However,the operations in mesh free methods are generally more complicated,and can be quite costly in terms of computational efforts and resources.We have also found that proper combinations of the mesh free and FEM techniques can be of advantages,and the so-called smoothed finite element method orS-FEM is atypical such a combined approach.It needs only a triangular/tetrahedral mesh,but can deliver excellent solutions inefficient manner.The S-FEM with triangular mesh,However, uses mostly linear interpolations, and the assumed displacement function is still continuous(in aH'space) .When higher order interpolations are used,the assumed displacement function is no longer continuous, and requires a fundamental change in theory,leading to the so-called G space theory.This class of methods is termed as smoothed point interpolation method orS-PIM.It is atypical weakened weak(W²) formulation.It offers additional flexibility in formulation,and can deliver much better solutions.This book is dedicated to the S-PIM.The large part of the research work related to this book was carried out at the Centre for Advanced Computations in Engineering Science(ACES),Department of Mechanical Engineering,National University of Singapore (NUS) .In preparing this book, a number of my colleagues and students atNUS have supported and contributed to the writing.We express sincere thanks to all of them.Special thanks to L.Chen, Y.Jian, Nourbakhsh niaN,Q.Tang, S.C.Wu,A.L.Yang, Z.Q.Zhang and many others.Many of these individuals have contributed examples to this book in addition to their hard work in carrying out a number of projects related to the S-PIM covered in this book.

Preface vii

The Authors xix

Chapter 1 Preliminaries 1

1.1Basic equations for solid mechanics 1

      1.1.1Equilibrium equations in terms of stresses.

      1.1.2Constitutive equations..

      1.1.3Compatibility equations...

      1.1.4Equilibrium equations in terms of displacements..

      1.1.5Boundary conditions..

      1.1.6Strain energy in solids....

      1.1.7Some notations and conventions.

      1.1.8Some basic concepts....11

1.2Numericaltechniques:FEM vs. S-PIM 13

      1.2.1An overview....13

      1.2.2On computational efficiency..16

      1.2.3On shape function creation...19

      1.2.4On integration over the problem domain..19

      1.2.5On the use of weak forms...20

      1.2.6Sampling principle....21

      1.2.7Summarized remarks21

1.3Basic ideas of S-PIM 22

1.4Basic properties of S-PIM 23

1.5Basic steps in S-PIM 24

1.6Basic settings in S-PIM 27

      1.6.1Triangulation.....27

      1.6.2Characteristic length..29

      1.6.3T-schemes for node selection..30

1.7Outline of the book 39

1.8References 42

Chapter 1 G Spaces 47

2.1General issues on function spaces..47

      2.1.1Linear spaces 48

      2.1.2Functional 48

      2.1.3Norm 48

      2.1.4Semi-norm 49

      2.1.5Linear forms 49

      2.1.6Bilinear forms 50

      2.1.7Inner product 50

      2.1.8Cauchy-Schwarz inequality 51

      2.1.9General notation of derivatives 51

2.2Useful spaces in weak formulation 52

      2.2.1Lebesgue spaces 52

      2.2.2Hilbert spaces 58

      2.2.3Sobolev spaces 64

      2.2.4Spaces of continuous functions 65

2.3G spaces:definition 65

      2.3.1Smoothing domain creation 65

      2.3.2Linearly independent smoothing domains 66

      2.3.3Integral representation of function derivatives 66

      2.3.4Derivatives approximation 67

      2.3.5On the treatment of the discontinuity 68

      2.3.6On physical meaning of the gap smoothing 70

      2.3.7Heaviside smoothing function 71

      2.3.8General definition of G space 73

      2.3.9G'space and norms 74

      2.3.10 Minimum number of smoothing domains 77

      2.3.11G’normsfor1D scalar fields 77

      2.3.12G’normsfor2D scalar fields 78

      2.3.13G’normsfor2D vector fields 80

      2.3.14G'norms for 3D vector fields 81

2.4G,space:basic properties 82

      2.4.1Linearity 82

      2.4.2Positivity 83

      2.4.3Scalar modification 83

      2.4.4Completeness 83

      2.4.5Cauchy-Schwarz inequality 84

      2.4.6Triangular inequality 84

2.5G,space:other proper ies 87

      2.5.1Convergence property 87

      2.5.2First inequality 88

      2.5.3Second inequality 88

      2.5.4Third inequality 89

      2.5.5Softening effects 90

2.6Concluding remarks 92

2.7References 93

Chapter 3PIM shape function creation 95

3.1Requirements on shape functions 95

      3.1.1Linear independence 96

      3.1.2Partitions of unity 101

      3.1.3Consistency 101

      3.1.4Delta function property 102

      3.1.5Compatibility 103

      3.1.6Basis:an essential role of shape functions 103

      3.1.7Interpolant 104

3.2PIM shape functions 104

      3.2.1Procedure of creating shape functions 105

      3.2.2Properties of PIM shape functions 108

      3.2.3Methods to avoid singular moment matrix 120

3.3RPIM shape functions 122

      3.3.1Rationale for using RBFs and polynomials 122

      3.3.2Formulation of polynomial augmented RPIM 123

      3.3.3R PIM shape functions with pure RBFs 128

      3.3.4Singularity of the moment matrix 129

      3.3.5On the range of the shape parameters 130

      3.3.6Properties of R PIM shape functions 130

3.4PIM-CT shape functions 133

      3.4.1Coordinate transformation 134

      3.4.2Creation of PIM-CT shape functions 134

      3.4.3Determination of rotation angle 135

3.5Isoparametric PIM shape functions 137

      3.5.1Approach1：using bilinear feature 138

      3.5.2Approach2：coordinate mapping 141

3.6Alpha PIM shape functions 143

      3.6.1Given sets of PIM shape functions 143

      3.6.2Creation of a PIM shape functions 144

      3.6.3Properties of the a PIM shape functions 145

3.7Condensed R PIM shape functions 146

      3.7.1Introduction of virtual nodes 147

      3.7.2Introduction of constraints 147

      3.7.3Properties of the condensed R PIM shape functions 148

      3.7.4 3DcondensedRPIM shape functions 149

3.8Other methods 150

3.9Interpolation error estimation 151

      3.9.1Error in L norm 152

      3.9.2Error in H norm 152

      3.9.3Error in G norm 153

      3.9.4Comparison errors in G' and H' norms 157

3.10 Concluding remarks 159

3.11 References 159

Chapter 4 Strain field construction 165

4.1Why constructing strain field? 166

4.2Discrete models:a base for strain construction 167

4.3General procedure for strain construction 167

4.4Admissible conditions for constructed strain field 169

      4.4.1Condition1:orthogonal condition 169

      4.4.2Condition2a:norm equivalence condition 169

      4.4.3Condition2b:strain convergence condition 173

      4.4.4Condition3:zero-sum condition 173

4.5Strain construction techniques 174

      4.5.1At a glance 174

      4.5.2Strain construction by generalized smoothing 175

      4.5.3Strain construction by point interpolation 179

      4.5.4Strain construction by least square approximation 182

4.6A brief historical note 182

4.7Concluding remarks 183

4.8References 184

Chapter 5 Weak and weakened weak formulations 191

Briefing on weak forms 192

      5.1.1Weak forms 192

      5.1.2Weakened weak forms 192

5.2Galerkin weak form 193

      5.2.1Bilinear form 193

      5.2.2Weak statement 195

5.3Galerkin weak forms:alternative expressions 196

5.4FEM:atypical Galerkin weak formulation 197

      5.4.1Weak statement for FEM models 198

      5.4.2H:A space of FEM displacements 199

      5.4.3Compatible strain field 201

      5.4.4Discrete system equations 202

      5.4.5Imposition of the essential boundary conditions 204

      5.4.6FEM solutions 204

5.5SC-Galerkin:ageneralW'formulation 206

5.6GS-Galerkin:awidelyusedw²form208

      5.6.1Bilinear form sinG,spaces 208

      5.6.2Properties of GS bilinear form 210

      5.6.3GS-Galerkin statement 212

      5.6.4GS-Galerkin:a special case of SC-Galerkin 216

5.7S-PIM：atypicalWformulation 217

      5.7.1PIM displacement field 218

      5.7.2Smoothing domain creation inS-PIM 219

      5.7.3Smoothed strain field 220

      5.7.4Discretized equations for S-PIM 225

      5.7.5Imposition of essential boundary conditions 226

      5.7.6S-PIM solutions 227

5.8Error assessment in S-PIM and FEM models 227

      5.8.1Error in displacement norm 228

      5.8.2Error in energy norm 228

5.9Concluding remarks231

5.10 References 234

Chapter6 Node-based smoothed point interpolation method(NS-PIM) 237

6.1NS-PIMfor2D solids 239

      6.1.1Approximation of displacement field 239

      6.1.2Evaluation of node-based smoothed strains 241

      6.1.3Equally-shared smoothing domains 242

      6.1.4Voronoi smoothing domains243

      6.1.5Stiffness matrix for NS-PIM 244

      6.1.6Comparison of NS-PIM,NS-FEM and FEM 245

      6.1.7Macro flowchart of the NS-PIM 247

      6.1.8Possible NS-PIM models 248

      6.1.9Condition number of NS-PIM models 251

      6.1.10 Estimation of computational cost 251

      6.1.11Issues on the treatments along boundaries 252

      6.1.12Evaluation of nodal strain(stress) 253

      6.1.13Rank test of NS-PIM 253

      6.1.14Numericalexamplesfor2D solids 255

6.2NS-RPIMfor2D solids 279

6.2.1Considerations 279

6.2.2Support node selection 279

6.2.3Possible 2DNS-R PIM models 280

6.2.4Condition number of NS-R PIM models 281

6.2.5Estimation of computational cost for 2DNS-R PIM 282

6.2.6Numerical examples for 2D solids 283

6.3NS-PIM/NS-RPIMfor3D solids 291

      6.3.1Approximation of displacement 291

      6.3.2Computation of node-based smoothed strains.292

      6.3.3Stiffness matrix of 3DNS-PIM 293

      6.3.4Possible 3DNS-PIM/NS-R PIM models 294

      6.3.5Condition number of 3DNS-PIM/NS-R PIM modells 295

      6.3.6Estimation of computational cost 295

      6.3.7Numerical examples for 3D solids 296

6.4Upperbound properties of NS-PIM/NS-R PIM 305

      6.4.1Background 305

      6.4.2Bound properties of NS-PIM models 306

      6.4.3Upperbound solutions:numerical examples 310

6.5Concluding remarks 323

6.6Computer program 325

      6.6.1On the structure of the source codes 325

      6.6.2Source codes in FORTRAN 90 327

      6.6.3Computed results 336

6.7References 337

Chapter7 Edge-based smoothed point interpolation method(ES-PIM) 341

7.1Approximation of displacement field 342

7.2Evaluation of edge-based smoothed strains 344

7.3ES-PIM formulations 345

      7.3.1Dynamic analysis 345

      7.3.2Static analysis 346

      7.3.3Free vibration analysis 346

      7.3.4Forced vibration analysis 347

7.4Numerical implementation 348

      7.4.1Macro flowchart of the ES-PIM 348

      7.4.2Possible ES-PIM models 349

      7.4.3FS-PIMfor3D solids 351

      7.4.4Evaluation of nodal strain(stress) 353

      7.4.5Condition number of ES-PIM models 354

      7.4.6Estimation of computational cost for ES-PIM 355

      7.4.7Rank analysis for ES-PIM 355

      7.4.8Temporal stability analysis 356

7.5Numerical examples 357

7.6Concluding remarks 380

7.7Computer program 381

      7.7.1About the program 381

      7.7.2Source codes in FORTRAN 90 382

      7.7.3Computed results 391

7.8References 392

Chapter8 Cell-based smoothed point interpolation method(CS-PIM) 395

8.1CS-PIMfor2D solids 396

      8.1.1Approximation of displacement field 396

      8.1.2Evaluation of cell-based smoothed strains 397

      8.1.3CS-PIM formulations 398

      8.1.4Numerical implementation 400

      8.1.5Numerical examples for 2D solids 408

8.2CS-PIMfor3D solids 433

      8.2.1Approximation of displacement 433

      8.2.2Evaluation of cell-based smoothed strains 434

      8.2.3Numerical implementation 435

      8.2.4Numerical examples for 3D solids 439

8.3Concluding remarks 447

8.4Computer program 448

      8.4.1About the program 448

      8.4.2Source codes in FORTRAN 90 449

      8.4.3Computed results 456

8.5References 458

Chapter9 The cell-based smoothed alpha radial point interpolation method(CS-aRPIM) 461

CS-aR PIM-Tr4for2D solids 463

      9.1.1Approximation of displacement field 463

      9.1.2Properties of a PIM shape functions 464

      9.1.3Evaluation of cell-based smoothed strains 465

      9.1.4CS-aR PIM formulations 465

9.2CS-aR PIM-Te5 for 3D solids 466

9.3Numerical implementation 466

      9.3.1Macro flowchart of the CS-aR PIM 466

      9.3.2Meshes with the same aspectratio 467

      9.3.3Some possible CS-aR PIM models 468

      9.3.4Condition number of the CS-aR PIM models 469

      9.3.5Estimation of computational cost 470

      9.3.6Evaluation of nodal strain(stress) 471

      9.3.7Rank analysis for CS-aR PIM models 471

      9.3.8Temporal stability analysis 472

9.4Numerical examples for 2D solids 472

9.5Numerical examples for 3D solids 487

9.6Concluding remarks 493

9.7References 494

Chapter10 Strain-constructed point interpolation method(SC-PIM) 497

10.1 Formulation of SC-PIM 498

      10.1.1 Displacement field construction 498

      10.1.2 Strain field construction 498

      10.1.3 Stiffness matrix for SC-PIM 503

10.2 Numerical implementation 504

      10.2.1Macro flowchart of the SC-PIM 504

      10.2.2 Possible SC-PIM models 505

      10.2.3 Condition number of SC-PIM models 506

      10.2.4 Estimation of computational cost for SC-PIM 507

      10.3 Numerical examples 508

      10.4 Concluding remarks 524

      10.5 References 535

Chapter11 S-PIM for heat transfer and thermo elasticity problems 527

11.1Heat transfer problems 528

      11.1.1 Problem statements 528

      11.1.2 Steady-state heat transfer 532

      11.1.3 Temperature field approximation 532

      11.1.4 Generalized gradient smoothing operation 532

      11.1.5 Discrete system equations 535

      11.1.6 Transit-state heat transfer 536

11.2 Thermoelastic problems538

      11.2.1 Modeling of the thermal strain and stress 538

      11.2.2 Discrete system equations 538

11.3 Numerical examples 540

11.4 Concluding remarks 555

11.5 References 556

Chapter12 SingularCS-R PIM for fracture mechanics problems559

12.1 Formulation of the singular CS-R PIM 561

      12.1.1 Approximation of displacement field 561

      12.1.2 Evaluation of cell-based smoothed strains 564

      12.1.3 Discretized system equations 566

      12.1.4 Possible singular CS-R PIM models 566

12.2 Stress intensity factor evaluation 567

      12.2.1J-integral 568

      12.2.2 Domain interaction integral 569

      12.2.3 Determination of area-path 574

12.3 Numerical examples 575

12.4 Concluding remarks 594

12.5 References 595

Chapter 13 Adaptive analysis using S-PIMs 599

13.1 Introduction 599

13.2 Adaptive analysis using S-PIMs 601

      13.2.1 Error indicator based on cell energy error 601

      13.2.2 Local refinement criteria 603

      13.2.3 Refinement strategy 603

      13.2.4 Cell regeneration 605

      13.2.5 General adaptive procedure 606

13.3 Numerical examples 607

13.4 Concluding remarks 617

13.5 References 618

Appendix Program codes library 623

Appendix1:Description of the subroutines 623

Appendix2:A demonstration input file 627

      Inputfile of“DATA INPUT” 627

Appendix3:Source codes of two modules 630

      Module1：MODULE Parameters 630

      Module2：MODULE Variables 630

Appendix4:Source codes of the common subroutines 631

      Program1:source code of“Input” 631

      Program2:source code of“C_materialM” 632

      Program3:source code of“Cell_information” 633

      Program4:source code of“StiffM_Intedomain” 639

      Program5:source code of“Form_GK” 640

      Program6:source code of“Natural_BC” 640

      Program7:source code of“Essential_BC” 641

      Program8:source code of“Solver_LAE” 642

      Program9:source code of“Dispnorm_error” 643

      Program10:source code of“PPIM_SF2D” 644

      Program11:source code of“PPIM_CT_SF2D” 645

      Program12:source code of“Iso_PPIM_SF2D” 647

      Program13:sourcecodeof“RPIM_SF2D” 649

      Program14:source code of“Con_RPIM_SF2D” 650

      Program15:source code of“Poly_Basis2D” 651

      Program16:source code of“Radial_Basis2D” 652

      Program17:source code of“Cell_basedT2L” 652

      Program18:source code of“Edge_basedT2L” 653

      Program19:source code of“FormB_NSPIM” 654

      Program20:source code of“Band_solver” 656

      Program21:source code of“Gausspointcoe_line” 657

      Program22:source code of“Line_gauss” 658

      Program23:source code of“Determinant” 659

      Program24:source code of“Brinv” 660

      Program25:source code of“Inversion" 662

      Program26:source code of“Indexx” 662

      Index 665

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