Preface ....................................................... xvu
Authors ....................................................... xix
1 Introduction ................................................. 1
1.1 Physical Problems in Engineering ........................ 1
1.2 Numerical Techniques: Practical Solution Tools .......... 2
1.3 Why S-FEM? .............................................. 5
1.4 The Idea of S-FEM ....................................... 8
1.5 Key Techniques Used in S-FEM ............................ 9
1.6 S-FEM Models and Properties ............................. 9
1.7 Some Historical Notes .................................. 11
1.8 Outline of the Book .................................... 12
References .................................................. 15
2 Basic Equations for Solid Mechanics ......................... 21
2.1 Equilibrium Equation: In Stresses ...................... 21
2.2 Constitutive Equation .................................. 23
2.3 Compatibility Equation ................................. 23
2.4 Equilibrium Equation: In Displacements ................. 23
2.5 Equations in Matrix Form ............................... 24
2.6 Boundary Conditions .................................... 26
2.7 Some Standard Default Conventions and Notations ........ 27
2.8 Remarks ................................................ 28
References .................................................. 30
3 The Finite Element Method ................................... 31
3.1 General Procedure of FEM ............................... 31
3.2 Proper Spaces .......................................... 35
3.4 Domain Discretization: Creation of Finite-Dimensional
Space .................................................. 48
3.5 Creation of Shape Functions ............................ 48
3.6 Displacement Function Creation ......................... 53
3.7 Strain Evaluation ...................................... 54
3.8 Formulation of the Discretized System of Equations ..... 55
3.9 FEM Solution: Existence, Uniqueness, Error, and
Convergence ............................................ 57
3.10 Some Other Properties of the FEM Solution .............. 59
3.11 Linear Triangular Element (T3) ......................... 61
3.12 Four-Node Quadrilateral Element (Q4) ................... 64
3.13 Four-Node Tetrahedral Element (T4) ..................... 67
3.14 Eight-Node Hexahedral Element (H8) ..................... 70
3.15 Gauss Integration ...................................... 73
3.16 Remarks ................................................ 80
References .................................................. 81
4 Fundamental Theories for S-FEM .............................. 83
4.1 General Procedure for S-FEM Models ..................... 84
4.2 Domain Discretization with Polygonal Elements .......... 85
4.3 Creating a Displacement Field: Shape Function
Construction ........................................... 87
4.4 Evaluation of the Compatible Strain Field .............. 91
4.5 Modify/Construct the Strain Field ...................... 92
4.6 Minimum Number of Smoothing Domains: Essential to
Stability ............................................. 105
4.7 Smoothed Galerkin Weak Form ........................... 108
4.8 Discretized Linear Algebraic System of Equations ...... 1ll
4.9 Solve the Algebraic System of Equations ............... 113
4.10 Error Assessment in S-FEM and FEM Models .............. 113
4.11 Implementation Procedure for S-FEM Models ............. 123
4.12 General Properties of S-FEM Models .................... 124
4.13 Remarks ............................................... 130
References ................................................. 133
5 Cell-Based Smoothed FEM .................................... 137
5.1 Cell-Based Smoothing Domain ........................... 138
5.2 Discretized System of Equations ....................... 139
5.3 Shape Function Evaluation ............................. 141
5.4 Some Properties of CS-FEM ............................. 146
5.5 Stability of CS-FEM and nCS-FEM ....................... 150
5.6 Standard Patch Test: Accuracy ......................... 153
5.7 Selective CS-FEM: Volumetric Locking Free ............. 156
5.8 Numerical Examples .................................... 157
5.9 Remarks ............................................... 177
References ................................................. 179
6 Node-Based Smoothed FEM .................................... 183
6.1 Introduction .......................................... 183
6.2 Creation of Node-Based Smoothing Domains .............. 184
6.3 Formulation of NS-FEM ................................. 185
6.4 Evaluation of Shape Function Values ................... 187
6.5 Properties of NS-FEM .................................. 190
6.6 An Adaptive NS-FEM Using Triangular Elements .......... 197
6.7 Numerical Examples .................................... 203
6.8 Remarks ............................................... 238
References ................................................. 240
7 Edge-Based Smoothed FEM .................................... 243
7.1 Introduction .......................................... 243
7.2 Creation of Edge-Based Smoothing Domains .............. 244
7.3 Formulation of the ES-FEM ............................. 246
7.4 Evaluation of the Shape Function Values in
the ES-FEM ............................................ 251
7.5 A Smoothing-Domain-Based Selective ES/NS-FEM .......... 254
7.6 Properties of the ES-FEM .............................. 254
7.7 Numerical Examples .................................... 258
7.8 Remarks ............................................... 290
References ................................................. 296
8 Face-Based Smoothed FEM .................................... 299
8.1 Introduction .......................................... 299
8.2 Face-Based Smoothing Domain Creation .................. 300
8.3 Formulation of FS-FEM-T4 .............................. 301
8.4 A Smoothing-Domain-Based Selective FS/NS-FEM-T4
Model ................................................. 305
8.5 Stability, Accuracy, and Mesh Sensitivity ............. 306
8.6 Numerical Examples .................................... 308
8.7 Remarks ............................................... 322
References ................................................. 323
9 TheaFEM .................................................... 325
9.1 Introduction .......................................... 325
9.2 Idea of αFEM-T3 and αFEM-T4 ........................... 327
9.3 αFEM-ТЗ and αFEM-T4 for Nonlinear Problems ............ 333
9.4 Implementation and Patch Tests ........................ 335
9.5 Numerical Examples .................................... 339
9.6 Remarks ............................................... 363
References ................................................. 365
10 S-FEM for Fracture Mechanics ............................... 367
10.1 Introduction .......................................... 367
10.2 Singular Stress Field Creation at the Crack-Tip ....... 368
10.3 Possible sS-FEM Methods ............................... 375
10.4 sNS-FEM Models ........................................ 376
10.5 sES-FEM Models ........................................ 380
10.6 Stiffness Matrix Evaluation ........................... 383
10.7 J-Integral and SIF Evaluation ......................... 384
10.8 Interaction Integral Method for Mixed Mode ............ 386
10.9 Numerical Examples Solved Using sES-FEM-T3 ............ 394
10.10 Numerical Examples Solved Using sNS-FEM-T3 ........... 417
10.11 Remarks .............................................. 434
References ................................................. 436
11 S-FEM for Viscoelastoplasticity ............................ 439
11.1 Introduction .......................................... 439
11.2 Strong Formulation for Viscoelastoplasticity .......... 440
11.3 FEM for Viscoelastoplasticity: A Dual Formulation ..... 443
11.4 S-FEM for Viscoelastoplasticity: A Dual Formulation ... 450
11.5 A Posteriori Error Estimation ......................... 455
11.6 Numerical Examples .................................... 458
11.7 Concluding Remarks .................................... 494
References ................................................. 495
12 ES-FEM for Plates .......................................... 497
12.1 Introduction .......................................... 497
12.2 Weak Form for the Reissner-Mindlin Plate .............. 498
12.3 FEM Formulation for the Reissner-Mindlin Plate ........ 501
12.4 ES-FEM-DSG3 for the Reissner-Mindlin Plate ............ 503
12.5 Numerical Examples: Patch Test ........................ 509
12.6 Numerical Examples: Static Analysis ................... 510
12.7 Numerical Examples: Free Vibration of Plates .......... 517
12.8 Numerical Examples: Buckling of Plates ................ 526
12.9 Remarks ............................................... 536
References ................................................. 536
13 S-FEM for Piezoelectric Structures ......................... 541
13.1 Introduction .......................................... 541
13.2 Galerkin Weak Form for Piezoelectrics ................. 542
13.1 Finite Element Formulation for the Piezoelectric
Problem ............................................... 543
13.4 S-FEM for the Piezoelectric Problem ................... 546
13.5 Numerical Results ..................................... 551
13.6 Remarks ............................................... 565
References ................................................. 565
14 S-FEM for Heat Transfer Problems ........................... 569
14.1 Introduction .......................................... 569
14.2 Strong-Form Equations for Heat Transfer Problems ...... 570
14.3 Boundary Conditions ................................... 571
14.4 Weak Forms for Heat Transfer Problems ................. 572
14.5 FEM Equations ......................................... 577
14.6 S-FEM Equations ....................................... 580
14.7 Evaluation of the Smoothed Gradient Matrix ............ 583
14.8 Numerical Example ..................................... 584
14.9 Bioheat Transfer Problems ............................. 598
14.10 Remarks .............................................. 604
References ................................................. 604
15 S-FEM for Acoustics Problems ............................... 607
15.1 Introduction .......................................... 607
15.2 Mathematical Model of Acoustics Problems .............. 609
15.3 Weak Forms for Acoustics Problems ..................... 611
15.4 FEM Equations ......................................... 614
15.5 S-FEM Equations ....................................... 617
15.6 Error in a Numerical Model ............................ 619
15.7 Numerical Examples .................................... 621
15.8 Remarks ............................................... 642
References ................................................. 643
Index ......................................................... 647
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