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Extended Finite Element Method
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Table of Contents

Series Preface xv Preface xvii 1 Introduction 1 1.1 Introduction 1 1.2 An Enriched Finite Element Method 3 1.3 A Review on X-FEM: Development and Applications 5 1.3.1 Coupling X-FEM with the Level-Set Method 6 1.3.2 Linear Elastic Fracture Mechanics (LEFM) 7 1.3.3 Cohesive Fracture Mechanics 11 1.3.4 Composite Materials and Material Inhomogeneities 14 1.3.5 Plasticity, Damage, and Fatigue Problems 16 1.3.6 Shear Band Localization 19 1.3.7 Fluid-Structure Interaction 19 1.3.8 Fluid Flow in Fractured Porous Media 20 1.3.9 Fluid Flow and Fluid Mechanics Problems 22 1.3.10 Phase Transition and Solidification 23 1.3.11 Thermal and Thermo-Mechanical Problems 24 1.3.12 Plates and Shells 24 1.3.13 Contact Problems 26 1.3.14 Topology Optimization 28 1.3.15 Piezoelectric and Magneto-Electroelastic Problems 28 1.3.16 Multi-Scale Modeling 29 2 Extended Finite Element Formulation 31 2.1 Introduction 31 2.2 The Partition of Unity Finite Element Method 33 2.3 The Enrichment of Approximation Space 35 2.3.1 Intrinsic Enrichment 35 2.3.2 Extrinsic Enrichment 36 2.4 The Basis of X-FEM Approximation 37 2.4.1 The Signed Distance Function 39 2.4.2 The Heaviside Function 43 2.5 Blending Elements 46 2.6 Governing Equation of a Body with Discontinuity 49 2.6.1 The Divergence Theorem for Discontinuous Problems 50 2.6.2 The Weak form of Governing Equation 51 2.7 The X-FEM Discretization of Governing Equation 53 2.7.1 Numerical Implementation of X-FEM Formulation 55 2.7.2 Numerical Integration Algorithm 57 2.8 Application of X-FEM in Weak and Strong Discontinuities 60 2.8.1 Modeling an Elastic Bar with a Strong Discontinuity 61 2.8.2 Modeling an Elastic Bar with a Weak Discontinuity 63 2.8.3 Modeling an Elastic Plate with a Crack Interface at its Center 66 2.8.4 Modeling an Elastic Plate with a Material Interface at its Center 68 2.9 Higher Order X-FEM 70 2.10 Implementation of X-FEM with Higher Order Elements 73 2.10.1 Higher Order X-FEM Modeling of a Plate with a Material Interface 73 2.10.2 Higher Order X-FEM Modeling of a Plate with a Curved Crack Interface 75 3 Enrichment Elements 77 3.1 Introduction 77 3.2 Tracking Moving Boundaries 78 3.3 Level Set Method 81 3.3.1 Numerical Implementation of LSM 82 3.3.2 Coupling the LSM with X-FEM 83 3.4 Fast Marching Method 85 3.4.1 Coupling the FMM with X-FEM 87 3.5 X-FEM Enrichment Functions 88 3.5.1 Bimaterials, Voids, and Inclusions 88 3.5.2 Strong Discontinuities and Crack Interfaces 91 3.5.3 Brittle Cracks 93 3.5.4 Cohesive Cracks 97 3.5.5 Plastic Fracture Mechanics 99 3.5.6 Multiple Cracks 101 3.5.7 Fracture in Bimaterial Problems 102 3.5.8 Polycrystalline Microstructure 106 3.5.9 Dislocations 111 3.5.10 Shear Band Localization 113 4 Blending Elements 119 4.1 Introduction 119 4.2 Convergence Analysis in the X-FEM 120 4.3 Ill-Conditioning in the X-FEM Method 124 4.3.1 One-Dimensional Problem with Material Interface 126 4.4 Blending Strategies in X-FEM 128 4.5 Enhanced Strain Method 130 4.5.1 An Enhanced Strain Blending Element for the Ramp Enrichment Function 132 4.5.2 An Enhanced Strain Blending Element for Asymptotic Enrichment Functions 134 4.6 The Hierarchical Method 135 4.6.1 A Hierarchical Blending Element for Discontinuous Gradient Enrichment 135 4.6.2 A Hierarchical Blending Element for Crack Tip Asymptotic Enrichments 137 4.7 The Cutoff Function Method 138 4.7.1 The Weighted Function Blending Method 140 4.7.2 A Variant of the Cutoff Function Method 142 4.8 A DG X-FEM Method 143 4.9 Implementation of Some Optimal X-FEM Type Methods 147 4.9.1 A Plate with a Circular Hole at Its Centre 148 4.9.2 A Plate with a Horizontal Material Interface 149 4.9.3 The Fiber Reinforced Concrete in Uniaxial Tension 151 4.10 Pre-Conditioning Strategies in X-FEM 154 4.10.1 Bechet's Pre-Conditioning Scheme 155 4.10.2 Menk-Bordas Pre-Conditioning Scheme 156 5 Large X-FEM Deformation 161 5.1 Introduction 161 5.2 Large FE Deformation 163 5.3 The Lagrangian Large X-FEM Deformation Method 167 5.3.1 The Enrichment of Displacement Field 167 5.3.2 The Large X-FEM Deformation Formulation 170 5.3.3 Numerical Integration Scheme 172 5.4 Numerical Modeling of Large X-FEM Deformations 173 5.4.1 Modeling an Axial Bar with a Weak Discontinuity 173 5.4.2 Modeling a Plate with the Material Interface 177 5.5 Application of X-FEM in Large Deformation Problems 181 5.5.1 Die-Pressing with a Horizontal Material Interface 182 5.5.2 Die-Pressing with a Rigid Central Core 186 5.5.3 Closed-Die Pressing of a Shaped-Tablet Component 188 5.6 The Extended Arbitrary Lagrangian-Eulerian FEM 192 5.6.1 ALE Formulation 192 5.6.1.1 Kinematics 193 5.6.1.2 ALE Governing Equations 194 5.6.2 The Weak Form of ALE Formulation 195 5.6.3 The ALE FE Discretization 196 5.6.4 The Uncoupled ALE Solution 198 5.6.4.1 Material (Lagrangian) Phase 199 5.6.4.2 Smoothing Phase 199 5.6.4.3 Convection (Eulerian) Phase 200 5.6.5 The X-ALE-FEM Computational Algorithm 202 5.6.5.1 Level Set Update 203 5.6.5.2 Stress Update with Sub-Triangular Numerical Integration 204 5.6.5.3 Stress Update with Sub-Quadrilateral Numerical Integration 205 5.7 Application of the X-ALE-FEM Model 208 5.7.1 The Coining Test 208 5.7.2 A Plate in Tension 209 6 Contact Friction Modeling with X-FEM 215 6.1 Introduction 215 6.2 Continuum Model of Contact Friction 216 6.2.1 Contact Conditions: The Kuhn-Tucker Rule 217 6.2.2 Plasticity Theory of Friction 218 6.2.3 Continuum Tangent Matrix of Contact Problem 221 6.3 X-FEM Modeling of the Contact Problem 223 6.3.1 The Gauss-Green Theorem for Discontinuous Problems 223 6.3.2 The Weak Form of Governing Equation for a Contact Problem 224 6.3.3 The Enrichment of Displacement Field 226 6.4 Modeling of Contact Constraints via the Penalty Method 227 6.4.1 Modeling of an Elastic Bar with a Discontinuity at Its Center 231 6.4.2 Modeling of an Elastic Plate with a Discontinuity at Its Center 233 6.5 Modeling of Contact Constraints via the Lagrange Multipliers Method 235 6.5.1 Modeling the Discontinuity in an Elastic Bar 239 6.5.2 Modeling the Discontinuity in an Elastic Plate 240 6.6 Modeling of Contact Constraints via the Augmented-Lagrange Multipliers Method 241 6.6.1 Modeling an Elastic Bar with a Discontinuity 244 6.6.2 Modeling an Elastic Plate with a Discontinuity 245 6.7 X-FEM Modeling of Large Sliding Contact Problems 246 6.7.1 Large Sliding with Horizontal Material Interfaces 249 6.8 Application of X-FEM Method in Frictional Contact Problems 251 6.8.1 An Elastic Square Plate with Horizontal Interface 251 6.8.1.1 Imposing the Unilateral Contact Constraint 252 6.8.1.2 Modeling the Frictional Stick-Slip Behavior 255 6.8.2 A Square Plate with an Inclined Crack 256 6.8.3 A Double-Clamped Beam with a Central Crack 259 6.8.4 A Rectangular Block with an S-Shaped Frictional Contact Interface 261 7 Linear Fracture Mechanics with the X-FEM Technique 267 7.1 Introduction 267 7.2 The Basis of LEFM 269 7.2.1 Energy Balance in Crack Propagation 270 7.2.2 Displacement and Stress Fields at the Crack Tip Area 271 7.2.3 The SIFs 273 7.3 Governing Equations of a Cracked Body 276 7.3.1 The Enrichment of Displacement Field 277 7.3.2 Discretization of Governing Equations 280 7.4 Mixed-Mode Crack Propagation Criteria 283 7.4.1 The Maximum Circumferential Tensile Stress Criterion 283 7.4.2 The Minimum Strain Energy Density Criterion 284 7.4.3 The Maximum Energy Release Rate 284 7.5 Crack Growth Simulation with X-FEM 285 7.5.1 Numerical Integration Scheme 287 7.5.2 Numerical Integration of Contour J-Integral 289 7.6 Application of X-FEM in Linear Fracture Mechanics 290 7.6.1 X-FEM Modeling of a DCB 290 7.6.2 An Infinite Plate with a Finite Crack in Tension 294 7.6.3 An Infinite Plate with an Inclined Crack 298 7.6.4 A Plate with Two Holes and Multiple Cracks 300 7.7 Curved Crack Modeling with X-FEM 304 7.7.1 Modeling a Curved Center Crack in an Infinite Plate 307 7.8 X-FEM Modeling of a Bimaterial Interface Crack 309 7.8.1 The Interfacial Fracture Mechanics 310 7.8.2 The Enrichment of the Displacement Field 311 7.8.3 Modeling of a Center Crack in an Infinite Bimaterial Plate 314 8 Cohesive Crack Growth with the X-FEM Technique 317 8.1 Introduction 317 8.2 Governing Equations of a Cracked Body 320 8.2.1 The Enrichment of Displacement Field 322 8.2.2 Discretization of Governing Equations 323 8.3 Cohesive Crack Growth Based on the Stress Criterion 325 8.3.1 Cohesive Constitutive Law 325 8.3.2 Crack Growth Criterion and Crack Growth Direction 326 8.3.3 Numerical Integration Scheme 328 8.4 Cohesive Crack Growth Based on the SIF Criterion 328 8.4.1 The Enrichment of Displacement Field 329 8.4.2 The Condition for Smooth Crack Closing 332 8.4.3 Crack Growth Criterion and Crack Growth Direction 332 8.5 Cohesive Crack Growth Based on the Cohesive Segments Method 334 8.5.1 The Enrichment of Displacement Field 334 8.5.2 Cohesive Constitutive Law 335 8.5.3 Crack Growth Criterion and Its Direction for Continuous Crack Propagation 336 8.5.4 Crack Growth Criterion and Its Direction for Discontinuous Crack Propagation 339 8.5.5 Numerical Integration Scheme 341 8.6 Application of X-FEM Method in Cohesive Crack Growth 341 8.6.1 A Three-Point Bending Beam with Symmetric Edge Crack 341 8.6.2 A Plate with an Edge Crack under Impact Velocity 343 8.6.3 A Three-Point Bending Beam with an Eccentric Crack 346 9 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM 351 9.1 Introduction 351 9.2 Large FE Deformation Formulation 353 9.3 Modified X-FEM Formulation 356 9.4 Large X-FEM Deformation Formulation 359 9.5 The Damage-Plasticity Model 364 9.6 The Nonlocal Gradient Damage Plasticity 368 9.7 Ductile Fracture with X-FEM Plasticity Model 369 9.8 Ductile Fracture with X-FEM Non-Local Damage-Plasticity Model 372 9.8.1 Crack Initiation and Crack Growth Direction 372 9.8.2 Crack Growth with a Null Step Analysis 375 9.8.3 Crack Growth with a Relaxation Phase Analysis 377 9.8.4 Locking Issues in Crack Growth Modeling 379 9.9 Application of X-FEM Damage-Plasticity Model 380 9.9.1 The Necking Problem 380 9.9.2 The CT Test 383 9.9.3 The Double-Notched Specimen 385 9.10 Dynamic Large X-FEM Deformation Formulation 387 9.10.1 The Dynamic X-FEM Discretization 388 9.10.2 The Large Strain Model 390 9.10.3 The Contact Friction Model 391 9.11 The Time Domain Discretization: The Dynamic Explicit Central Difference Method 393 9.12 Implementation of Dynamic X-FEM Damage-Plasticity Model 396 9.12.1 A Plate with an Inclined Crack 398 9.12.2 The Low Cycle Fatigue Test 400 9.12.3 The Cyclic CT Test 401 9.12.4 The Double Notched Specimen in Cyclic Loading 405 10 X-FEM Modeling of Saturated/Semi-Saturated Porous Media 409 10.1 Introduction 409 10.1.1 Governing Equations of Deformable Porous Media 411 10.2 The X-FEM Formulation of Deformable Porous Media with Weak Discontinuities 414 10.2.1 Approximation of Displacement and Pressure Fields 415 10.2.2 The X-FEM Spatial Discretization 418 10.2.3 The Time Domain Discretization and Solution Procedure 419 10.2.4 Numerical Integration Scheme 421 10.3 Application of the X-FEM Method in Deformable Porous Media with Arbitrary Interfaces 422 10.3.1 An Elastic Soil Column 422 10.3.2 An Elastic Foundation 424 10.4 Modeling Hydraulic Fracture Propagation in Deformable Porous Media 427 10.4.1 Governing Equations of a Fractured Porous Medium 428 10.4.2 The Weak Formulation of a Fractured Porous Medium 430 10.5 The X-FEM Formulation of Deformable Porous Media with Strong Discontinuities 434 10.5.1 Approximation of the Displacement and Pressure Fields 434 10.5.2 The X-FEM Spatial Discretization 437 10.5.3 The Time Domain Discretization and Solution Procedure 438 10.6 Alternative Approaches to Fluid Flow Simulation within the Fracture 442 10.6.1 A Partitioned Solution Algorithm for Interfacial Pressure 442 10.6.2 A Time-Dependent Constant Pressure Algorithm 444 10.7 Application of the X-FEM Method in Hydraulic Fracture Propagation of Saturated Porous Media 445 10.7.1 An Infinite Saturated Porous Medium with an Inclined Crack 446 10.7.2 Hydraulic Fracture Propagation in an Infinite Poroelastic Medium 449 10.7.3 Hydraulic Fracturing in a Concrete Gravity Dam 452 10.8 X-FEM Modeling of Contact Behavior in Fractured Porous Media 455 10.8.1 Contact Behavior in a Fractured Medium 455 10.8.2 X-FEM Formulation of Contact along the Fracture 456 10.8.3 Consolidation of a Porous Block with a Vertical Discontinuity 457 11 Hydraulic Fracturing in Multi-Phase Porous Media with X-FEM 461 11.1 Introduction 461 11.2 The Physical Model of Multi-Phase Porous Media 463 11.3 Governing Equations of Multi-Phase Porous Medium 465 11.4 The X-FEM Formulation of Multi-Phase Porous Media with Weak Discontinuities 467 11.4.1 Approximation of the Primary Variables 469 11.4.2 Discretization of Equilibrium and Flow Continuity Equations 473 11.4.3 Solution Procedure of Discretized Equilibrium Equations 476 11.5 Application of X-FEM Method in Multi-Phase Porous Media with Arbitrary Interfaces 477 11.6 The X-FEM Formulation for Hydraulic Fracturing in Multi-Phase Porous Media 482 11.7 Discretization of Multi-Phase Governing Equations with Strong Discontinuities 487 11.8 Solution Procedure for Fully Coupled Nonlinear Equations 493 11.9 Computational Notes in Hydraulic Fracture Modeling 497 11.10 Application of the X-FEM Method to Hydraulic Fracture Propagation of Multi-Phase Porous Media 499 12 Thermo-Hydro-Mechanical Modeling of Porous Media with X-FEM 509 12.1 Introduction 509 12.2 THM Governing Equations of Saturated Porous Media 511 12.3 Discontinuities in a THM Medium 513 12.4 The X-FEM Formulation of THM Governing Equations 514 12.4.1 Approximation of Displacement, Pressure, and Temperature Fields 515 12.4.2 The X-FEM Spatial Discretization 517 12.4.3 The Time Domain Discretization 520 12.5 Application of the X-FEM Method to THM Behavior of Porous Media 521 12.5.1 A Plate with an Inclined Crack in Thermal Loading 521 12.5.2 A Plate with an Edge Crack in Thermal Loading 522 12.5.3 An Impermeable Discontinuity in Saturated Porous Media 524 12.5.4 An Inclined Fault in Porous Media 527 References 533 Index 557

About the Author

Amir R. Khoei, Sharif University of Technology, Iran

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