Cover......Page 1
EXTENDED FINITEELEMENT ANDMESHFREE METHODS......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 13
Latin symbols......Page 18
1.1 Partition of unity methods......Page 20
1.2 Moving boundary problems......Page 25
1.3 Fracture mechanics......Page 27
1.4 Level set methods......Page 29
1.4.1 Implicit interface and signed distance functions......Page 30
1.4.3 Capturing motion interface......Page 31
1.4.4 Level sets for 3D fracture modeling......Page 33
References......Page 34
2.1.1 One dimensional model problem......Page 38
2.1.2 Model problem in higher dimensions......Page 39
2.1.3 Total Lagrangian formulation......Page 40
2.1.4 Updated Lagrangian formulation......Page 41
2.2.1 Weak form for the one-dimensional model problem......Page 43
2.2.2 Weak form for the total Lagrangian formulation......Page 45
2.3 Variational formulation......Page 46
3.1.1 Standard XFEM......Page 48
3.1.1.1 Application to strong discontinuities......Page 49
3.1.1.2 Application to weak discontinuities......Page 51
3.1.2 Hansbo-Hansbo XFEM......Page 53
3.2.1 Blending......Page 55
3.2.2 Isoparametric 2D quadrilateral XFEM element for linear elasticity......Page 59
3.2.3 Shape functions......Page 60
3.2.4 The B-operator......Page 61
3.2.5 The element stiffness matrix......Page 63
3.2.6 Integration......Page 65
3.3.1 XFEM approximation for cracks......Page 69
3.3.2 Discrete equations......Page 73
3.3.3 Crack branching and crack junction......Page 76
3.3.4 Crack opening and crack closure......Page 78
3.4.1 Diagonalized mass matrix......Page 79
3.4.2 Limitations......Page 83
3.5 Smoothed extended finite element method......Page 84
3.5.1 Introduction to SFEM......Page 86
3.5.2 Enrichment in SXFEM and selection of enriched nodes......Page 89
3.5.3 Displacement-, strain field approximation and discrete equations......Page 91
3.5.4 Numerical integration......Page 94
3.6.1 Hydro-mechanical problems......Page 96
3.6.1.1 Strong and weak form of the coupled hydro-mechanical problem......Page 97
3.6.1.2 Constitutive relation......Page 100
3.6.1.3 Discretization and discrete system of equations......Page 105
3.6.2 Thermo-mechanical problems......Page 108
3.6.3 Piezoelectric materials......Page 111
3.6.3.1 Strong form and weak form......Page 114
3.6.3.2 XFEM formulation for piezoelectric materials......Page 116
3.6.4.1 Governing equations and weak form......Page 119
3.6.4.2 XIGA discretization......Page 123
3.7.1 Inverse problem......Page 124
3.7.1.1 Void and inclusion detection in piezoelectric materials......Page 128
3.7.1.3 Selection of the regularization parameter β......Page 130
3.7.1.4 The forward and adjoint problem......Page 131
3.7.2 Optimization problems......Page 134
3.7.3 Mathematical form of a structural optimization problem......Page 135
3.7.4 Solid isotropic material with penalization (SIMP)......Page 136
3.7.6 Nanoelasticity......Page 137
3.7.6.1 Discretization using XFEM......Page 141
3.7.6.2 Material derivative......Page 145
3.7.6.3 Sensitivity analysis......Page 146
Velocity extension......Page 147
3.7.6.4 Numerical example......Page 148
3.7.7 Nanopiezoelectricity......Page 149
3.7.7.1 Strong and weak form of surface piezoelectricity......Page 151
3.7.7.2 XFEM formulation for surface piezoelectricity......Page 153
3.7.7.3 Topology optimization of nanoscale piezoelectric energy harvesters......Page 157
3.7.7.4 Numerical examples......Page 159
3.8 Conditioning and solution of ill-conditioned systems......Page 165
References......Page 166
4.1 Formulation and concepts......Page 171
4.2.1 Three-node triangular element......Page 172
4.2.2 Four-node quadrilateral element......Page 175
4.3 Multiple crack modeling......Page 176
References......Page 177
5.1.1 Basic approximation......Page 179
5.1.2 Completeness and conservation......Page 180
5.1.3 Consistency, stability and convergence......Page 182
5.1.5 Partition of unity......Page 183
5.1.6.1 Construction of the kernel function......Page 185
5.1.6.2 Eulerian and Lagrangian kernels......Page 186
5.2.1.1 The SPH-method......Page 189
5.2.1.2 Krongauz-Belytschko correction......Page 190
5.2.1.3 Randles-Libersky correction......Page 192
5.2.1.4 The MLS-approximation......Page 193
5.2.2 Spatial integration......Page 195
5.2.2.1 Nodal integration......Page 196
5.2.2.2 Stabilized conforming nodal integration......Page 199
5.2.2.3 Stress-point integration......Page 200
5.2.2.4 Cell integration......Page 202
5.2.3 Essential boundary conditions......Page 203
5.2.4 Comparison of different methods......Page 204
5.2.4.1 Cantilever beam......Page 206
5.3 Numerical instabilities......Page 208
5.3.1 Instability due to rank deficiency......Page 210
5.3.3 Attempts to remove instabilities......Page 211
5.3.4.1 Material stability for continua......Page 212
5.3.4.2 Material stability analyses of meshfree methods......Page 214
Hyperelastic material model with strain softening......Page 215
Example of an instability for hyperelastic material......Page 224
5.4.1 The visibility method......Page 227
5.4.2 The diffraction method......Page 230
5.4.3 The transparency method......Page 233
5.5 The concept of enrichment......Page 235
5.5.1 Intrinsic enrichment......Page 237
5.5.2.1 Extrinsic MLS enrichment......Page 240
5.6 (Extrinsically) enriched local PU meshfree methods......Page 243
5.6.1 Enriched methods with crack tip enrichment......Page 244
5.6.2 Enriched methods without crack tip enrichment......Page 248
5.6.2.1 Domain decrease method......Page 250
5.6.2.2 Lagrange multiplier method......Page 253
5.6.3 Crack branching and crack junction......Page 254
5.7 Extended local maximum entropy (XLME)......Page 256
5.7.1 Local Maximum Entropy (LME) approximants......Page 257
5.7.2.2 Numerical integration for enriched LME......Page 261
5.8 Cracking particle methods......Page 263
5.8.1 The enriched cracking particles method......Page 264
5.8.3 The cracking particles method without enrichment......Page 268
5.8.4 Cracking rules for cracking particle methods......Page 269
5.9.1 The mode I crack problem......Page 271
5.9.1.2 PU methods with intrinsic and extrinsic enrichments......Page 273
5.9.1.3 Cracking particle method......Page 274
5.9.1.4 XLME method......Page 275
5.9.2 The mixed mode problem......Page 278
5.9.2.2 PU methods with intrinsic and extrinsic enrichments......Page 279
5.9.2.3 XLME method......Page 280
5.10.1 Enriching in the shear band plane......Page 281
5.11.1 Methods without enrichment......Page 283
5.11.2.2 The extrinsic MLS-method......Page 285
5.11.2.3 The extrinsic PU-method......Page 286
5.11.3.1 The PU-method with crack tip enrichment......Page 288
5.11.3.2 The PU-method without crack tip enrichment......Page 292
5.11.3.3 The cracking particles method......Page 294
5.11.3.4 The cracking particles method for shear bands......Page 298
5.12 Spatial integration......Page 301
5.13.1 Explicit-implicit time integration......Page 304
5.13.2 Explicit time integration, critical time step and mass lumping......Page 305
5.13.2.1 Critical time step and consistent mass matrix......Page 306
5.13.2.2 Mass lumping strategy 1 (MLS1)......Page 307
5.13.2.3 Mass lumping strategy 2 (MLS2)......Page 309
5.13.2.4 Critical time step analysis......Page 312
5.13.2.5 Analytical critical time step estimates......Page 319
5.13.3 Crack propagation in time......Page 322
References......Page 324
6.1.1 B-splines and NURBS......Page 332
6.1.2 Bézier extraction......Page 334
6.2 Hierarchical refinement with PHT-splines......Page 337
6.2.1 PHT-spline space......Page 338
6.2.2 Computing the control points......Page 340
6.3 Analysis using splines......Page 341
6.3.1 Galerkin method......Page 342
6.3.2 Linear elasticity......Page 344
6.4.1 Infinite plate with circular hole......Page 346
6.4.2 Open spanner......Page 347
6.4.3 Pinched cylinder......Page 348
6.4.4 Hollow sphere......Page 349
6.5.1 Determining the superconvergent point locations......Page 350
6.5.2 Superconvergent patch recovery......Page 354
6.5.3 Marking algorithm......Page 357
6.7 XIGA for interface problems......Page 358
6.7.1 Governing and weak form equations......Page 359
6.7.2 Enriched basis functions selection......Page 362
6.7.3.1 Ramp enrichment function......Page 364
6.7.4 Greville Abscissae......Page 365
6.7.5 Repeating middle neighbor knots......Page 366
6.7.6 Inverse mapping......Page 367
6.7.7 Curve fitting......Page 368
6.7.8 Intersection points......Page 370
6.7.9 Triangular integration......Page 371
References......Page 372
7.1.1 Weak form......Page 376
7.1.2 Implementation based on the Q4 element......Page 378
7.1.3 Shear locking......Page 379
7.1.4 Curvature strain smoothing......Page 380
7.1.5 Extended finite element method for shear deformable plates......Page 382
7.1.6 Smoothed extended finite element method......Page 384
7.1.7 Integration......Page 385
7.2.1.1 Shell formulation with fracture......Page 387
7.2.1.2 Element formulation......Page 389
7.2.1.3 Representation of the discontinuity......Page 391
7.2.1.4 Representation of multiple discontinuities: crack branching......Page 393
7.2.1.5 Discretization......Page 394
7.2.2.1 The uncracked element......Page 395
7.2.2.2 Overlapping paired elements for cracked elements......Page 398
7.2.2.3 Constrained overlapping paired elements for tip elements......Page 399
7.2.2.4 Equilibrium equations and numerical integration......Page 402
Domain form for J-integral calculation......Page 404
Extraction of mixed-mode stress intensity factors......Page 407
7.3 Extended meshfree methods for fracture in shells......Page 409
7.3.1.1 Kinematics......Page 410
7.3.1.2 Virtual work......Page 411
7.3.1.3 Discretization......Page 412
7.3.2 Continuum constitutive models......Page 413
7.3.3.1 Method 1: cracked particles......Page 414
7.3.3.2 Method 2: local partition of unity approach......Page 415
7.4 An immersed particle method for fluid-structure interaction......Page 419
7.5.1 Kinematics of the shell......Page 425
7.5.2 Weak form......Page 427
7.5.3.1 In-plane enrichment......Page 429
7.5.3.2 Out-of-plane enrichments......Page 430
7.5.3.3 Discretization of the displacement field......Page 431
7.5.3.4 Essential boundary conditions, numerical integration and compatibility enforcement......Page 432
7.5.4 Discrete system of equations......Page 436
7.5.4.1 Membrane B-matrix for XIGA......Page 437
7.5.4.2 Bending B-matrix for XIGA......Page 438
7.5.5.1 Results of the phantom node MITC3 elements......Page 439
7.5.5.2 Results obtained by XIGA......Page 443
7.5.6 Pressurized cylinder with an axial crack......Page 445
7.5.6.1 Results of the phantom node MITC3 element......Page 446
7.5.6.2 Results obtained by XIGA......Page 448
References......Page 449
8.2.1 Criteria in LEFM......Page 453
8.2.3 Rankine criterion......Page 456
8.2.4 Loss of material stability condition......Page 457
8.2.5 Rank-one-stability criterion......Page 459
8.2.7 Computation of the crack length......Page 460
8.3 Crack surface representation and tracking the crack path......Page 461
8.3.1 The level set method to trace the crack path......Page 463
8.3.2 Tracking the crack path in 3D......Page 467
8.3.2.1 Tracking the crack path in 3D with plane segments......Page 469
8.3.2.2 Smooth crack path representation with level sets......Page 476
8.3.3 Adaptive crack propagation technique......Page 478
8.3.4 Comments......Page 480
References......Page 482
9 Multiscale methods for fracture......Page 487
9.1 Extended Bridging Domain Method......Page 488
9.1.1.1 Variational formulation......Page 490
9.1.1.2 Coupling method......Page 491
9.1.1.3 Adaptive coarsening and refinement......Page 493
9.1.1.4 Size of the fine-scale domain......Page 494
9.2 Extended bridging scale method......Page 495
9.2.1 Consistency of material properties......Page 497
9.2.2.1 Detection of `fine-scale' fractures......Page 499
Centro symmetry parameter (CSP)......Page 500
9.2.2.2 Upscaling, downscaling and adaptivity......Page 501
Downscaling-adaptive refinement......Page 502
Upscaling-adaptive coarse graining......Page 503
Crack surface orientation......Page 505
9.3.1 Overview of the method......Page 507
9.3.2 Coarse graining method......Page 510
9.3.3 Micro-macro linkage......Page 516
9.4 Crack opening in unit cells with the hourglass mode......Page 519
9.5 Stability of the macromaterial......Page 520
9.6 Implementation......Page 523
9.7.2 Hierarchical multiscale example......Page 524
9.7.3 Semi-concurrent FE-FE coupling example......Page 526
9.7.4 Concurrent FE-XFEM coupling example......Page 528
9.7.5 MD-XFEM coupling example......Page 529
References......Page 532
10.1 Numerical manifold method (finite cover method)......Page 536
10.1.1 The cover approximation......Page 537
10.1.2 The least square-based physical cover functions......Page 538
10.1.4 Fracture modeling......Page 539
10.1.5 Geometric and material nonlinear analysis......Page 542
10.2 Peridynamics and dual-horizon peridynamics......Page 543
10.2.1.1 The shortcomings of constant horizons......Page 546
10.2.1.2 Horizon and dual-horizon......Page 547
10.2.1.3 Equation of motion for dual-horizon peridynamics......Page 549
10.2.1.4 Numerical implementation of DH-PD......Page 550
10.2.1.5 The bond force density fxx'......Page 551
10.2.1.6 Volume correction......Page 552
10.2.2 The dual property of dual-horizon......Page 554
10.2.2.1 Proof of basic physical principles......Page 557
10.2.3 Wave propagation in 1D homogeneous bar......Page 558
10.2.4.1 Two-dimensional wave reflection in a rectangular plate......Page 559
10.2.4.2 Multiple materials......Page 565
10.2.4.3 Simulation of the Kalthoff-Winkler experiment......Page 571
10.2.4.4 Plate with pre-crack subjected to traction......Page 575
10.3 Phase field models......Page 577
10.3.1 Concepts......Page 578
10.3.2 Governing equations......Page 583
10.3.3 Discretization......Page 584
10.3.4.1 Monolithic scheme......Page 587
10.3.5.2 COMSOL implementation......Page 588
References......Page 590
11.1 Computer implementation of enriched methods......Page 595
11.1.1 Pre-processing......Page 596
11.1.2 Processing......Page 599
11.1.3 Post-processing......Page 603
11.2 Numerical examples......Page 604
11.2.2 Hydro-mechanical model with center cracks......Page 605
11.2.3 Hydro-mechanical model with edge crack......Page 606
11.2.3.1 Hydro-mechanical model with edge crack under fluid flux loading......Page 608
References......Page 611
APPENDIX A. Derivation of shape derivative for the nanoelasticity problem......Page 613
APPENDIX B. Derivation of the adjoint problem for the nanopiezoelectricity problem......Page 615
Index......Page 618
Back Cover......Page 629