Extended Finite Element Method

Amir R. Khoei, Sharif University of Technology, Iran

• Series Preface xv Preface xvii1 Introduction 11.1 Introduction 11.2 An Enriched Finite Element Method 31.3 A Review on X-FEM: Development and Applications 51.3.1 Coupling X-FEM with the Level-Set Method 61.3.2 Linear Elastic Fracture Mechanics (LEFM) 71.3.3 Cohesive Fracture Mechanics 111.3.4 Composite Materials and Material Inhomogeneities 141.3.5 Plasticity, Damage, and Fatigue Problems 161.3.6 Shear Band Localization 191.3.7 Fluid–Structure Interaction 191.3.8 Fluid Flow in Fractured Porous Media 201.3.9 Fluid Flow and Fluid Mechanics Problems 221.3.10 Phase Transition and Solidification 231.3.11 Thermal and Thermo-Mechanical Problems 241.3.12 Plates and Shells 241.3.13 Contact Problems 261.3.14 Topology Optimization 281.3.15 Piezoelectric and Magneto-Electroelastic Problems 281.3.16 Multi-Scale Modeling 292 Extended Finite Element Formulation 312.1 Introduction 312.2 The Partition of Unity Finite Element Method 332.3 The Enrichment of Approximation Space 352.3.1 Intrinsic Enrichment 352.3.2 Extrinsic Enrichment 362.4 The Basis of X-FEM Approximation 372.4.1 The Signed Distance Function 392.4.2 The Heaviside Function 432.5 Blending Elements 462.6 Governing Equation of a Body with Discontinuity 492.6.1 The Divergence Theorem for Discontinuous Problems 502.6.2 The Weak form of Governing Equation 512.7 The X-FEM Discretization of Governing Equation 532.7.1 Numerical Implementation of X-FEM Formulation 552.7.2 Numerical Integration Algorithm 572.8 Application of X-FEM in Weak and Strong Discontinuities 602.8.1 Modeling an Elastic Bar with a Strong Discontinuity 612.8.2 Modeling an Elastic Bar with a Weak Discontinuity 632.8.3 Modeling an Elastic Plate with a Crack Interface at its Center 662.8.4 Modeling an Elastic Plate with a Material Interface at its Center 682.9 Higher Order X-FEM 702.10 Implementation of X-FEM with Higher Order Elements 732.10.1 Higher Order X-FEM Modeling of a Plate with a Material Interface 732.10.2 Higher Order X-FEM Modeling of a Plate with a Curved Crack Interface 753 Enrichment Elements 773.1 Introduction 773.2 Tracking Moving Boundaries 783.3 Level Set Method 813.3.1 Numerical Implementation of LSM 823.3.2 Coupling the LSM with X-FEM 833.4 Fast Marching Method 853.4.1 Coupling the FMM with X-FEM 873.5 X-FEM Enrichment Functions 883.5.1 Bimaterials, Voids, and Inclusions 883.5.2 Strong Discontinuities and Crack Interfaces 913.5.3 Brittle Cracks 933.5.4 Cohesive Cracks 973.5.5 Plastic Fracture Mechanics 993.5.6 Multiple Cracks 1013.5.7 Fracture in Bimaterial Problems 1023.5.8 Polycrystalline Microstructure 1063.5.9 Dislocations 1113.5.10 Shear Band Localization 1134 Blending Elements 1194.1 Introduction 1194.2 Convergence Analysis in the X-FEM 1204.3 Ill-Conditioning in the X-FEM Method 1244.3.1 One-Dimensional Problem with Material Interface 1264.4 Blending Strategies in X-FEM 1284.5 Enhanced Strain Method 1304.5.1 An Enhanced Strain Blending Element for the Ramp Enrichment Function 1324.5.2 An Enhanced Strain Blending Element for Asymptotic Enrichment Functions 1344.6 The Hierarchical Method 1354.6.1 A Hierarchical Blending Element for Discontinuous Gradient Enrichment 1354.6.2 A Hierarchical Blending Element for Crack Tip Asymptotic Enrichments 1374.7 The Cutoff Function Method 1384.7.1 The Weighted Function Blending Method 1404.7.2 A Variant of the Cutoff Function Method 1424.8 A DG X-FEM Method 1434.9 Implementation of Some Optimal X-FEM Type Methods 1474.9.1 A Plate with a Circular Hole at Its Centre 1484.9.2 A Plate with a Horizontal Material Interface 1494.9.3 The Fiber Reinforced Concrete in Uniaxial Tension 1514.10 Pre-Conditioning Strategies in X-FEM 1544.10.1 Béchet’s Pre-Conditioning Scheme 1554.10.2 Menk–Bordas Pre-Conditioning Scheme 1565 Large X-FEM Deformation 1615.1 Introduction 1615.2 Large FE Deformation 1635.3 The Lagrangian Large X-FEM Deformation Method 1675.3.1 The Enrichment of Displacement Field 1675.3.2 The Large X-FEM Deformation Formulation 1705.3.3 Numerical Integration Scheme 1725.4 Numerical Modeling of Large X-FEM Deformations 1735.4.1 Modeling an Axial Bar with a Weak Discontinuity 1735.4.2 Modeling a Plate with the Material Interface 1775.5 Application of X-FEM in Large Deformation Problems 1815.5.1 Die-Pressing with a Horizontal Material Interface 1825.5.2 Die-Pressing with a Rigid Central Core 1865.5.3 Closed-Die Pressing of a Shaped-Tablet Component 1885.6 The Extended Arbitrary Lagrangian–Eulerian FEM 1925.6.1 ALE Formulation 1925.6.1.1 Kinematics 1935.6.1.2 ALE Governing Equations 1945.6.2 The Weak Form of ALE Formulation 1955.6.3 The ALE FE Discretization 1965.6.4 The Uncoupled ALE Solution 1985.6.4.1 Material (Lagrangian) Phase 1995.6.4.2 Smoothing Phase 1995.6.4.3 Convection (Eulerian) Phase 2005.6.5 The X-ALE-FEM Computational Algorithm 2025.6.5.1 Level Set Update 2035.6.5.2 Stress Update with Sub-Triangular Numerical Integration 2045.6.5.3 Stress Update with Sub-Quadrilateral Numerical Integration 2055.7 Application of the X-ALE-FEM Model 2085.7.1 The Coining Test 2085.7.2 A Plate in Tension 2096 Contact Friction Modeling with X-FEM 2156.1 Introduction 2156.2 Continuum Model of Contact Friction 2166.2.1 Contact Conditions: The Kuhn–Tucker Rule 2176.2.2 Plasticity Theory of Friction 2186.2.3 Continuum Tangent Matrix of Contact Problem 2216.3 X-FEM Modeling of the Contact Problem 2236.3.1 The Gauss–Green Theorem for Discontinuous Problems 2236.3.2 The Weak Form of Governing Equation for a Contact Problem 2246.3.3 The Enrichment of Displacement Field 2266.4 Modeling of Contact Constraints via the Penalty Method 2276.4.1 Modeling of an Elastic Bar with a Discontinuity at Its Center 2316.4.2 Modeling of an Elastic Plate with a Discontinuity at Its Center 2336.5 Modeling of Contact Constraints via the Lagrange Multipliers Method 2356.5.1 Modeling the Discontinuity in an Elastic Bar 2396.5.2 Modeling the Discontinuity in an Elastic Plate 2406.6 Modeling of Contact Constraints via the Augmented-Lagrange Multipliers Method 2416.6.1 Modeling an Elastic Bar with a Discontinuity 2446.6.2 Modeling an Elastic Plate with a Discontinuity 2456.7 X-FEM Modeling of Large Sliding Contact Problems 2466.7.1 Large Sliding with Horizontal Material Interfaces 2496.8 Application of X-FEM Method in Frictional Contact Problems 2516.8.1 An Elastic Square Plate with Horizontal Interface 2516.8.1.1 Imposing the Unilateral Contact Constraint 2526.8.1.2 Modeling the Frictional Stick–Slip Behavior 2556.8.2 A Square Plate with an Inclined Crack 2566.8.3 A Double-Clamped Beam with a Central Crack 2596.8.4 A Rectangular Block with an S–Shaped Frictional Contact Interface 2617 Linear Fracture Mechanics with the X-FEM Technique 2677.1 Introduction 2677.2 The Basis of LEFM 2697.2.1 Energy Balance in Crack Propagation 2707.2.2 Displacement and Stress Fields at the Crack Tip Area 2717.2.3 The SIFs 2737.3 Governing Equations of a Cracked Body 2767.3.1 The Enrichment of Displacement Field 2777.3.2 Discretization of Governing Equations 2807.4 Mixed-Mode Crack Propagation Criteria 2837.4.1 The Maximum Circumferential Tensile Stress Criterion 2837.4.2 The Minimum Strain Energy Density Criterion 2847.4.3 The Maximum Energy Release Rate 2847.5 Crack Growth Simulation with X-FEM 2857.5.1 Numerical Integration Scheme 2877.5.2 Numerical Integration of Contour J–Integral 2897.6 Application of X-FEM in Linear Fracture Mechanics 2907.6.1 X-FEM Modeling of a DCB 2907.6.2 An Infinite Plate with a Finite Crack in Tension 2947.6.3 An Infinite Plate with an Inclined Crack 2987.6.4 A Plate with Two Holes and Multiple Cracks 3007.7 Curved Crack Modeling with X-FEM 3047.7.1 Modeling a Curved Center Crack in an Infinite Plate 3077.8 X-FEM Modeling of a Bimaterial Interface Crack 3097.8.1 The Interfacial Fracture Mechanics 3107.8.2 The Enrichment of the Displacement Field 3117.8.3 Modeling of a Center Crack in an Infinite Bimaterial Plate 3148 Cohesive Crack Growth with the X-FEM Technique 3178.1 Introduction 3178.2 Governing Equations of a Cracked Body 3208.2.1 The Enrichment of Displacement Field 3228.2.2 Discretization of Governing Equations 3238.3 Cohesive Crack Growth Based on the Stress Criterion 3258.3.1 Cohesive Constitutive Law 3258.3.2 Crack Growth Criterion and Crack Growth Direction 3268.3.3 Numerical Integration Scheme 3288.4 Cohesive Crack Growth Based on the SIF Criterion 3288.4.1 The Enrichment of Displacement Field 3298.4.2 The Condition for Smooth Crack Closing 3328.4.3 Crack Growth Criterion and Crack Growth Direction 3328.5 Cohesive Crack Growth Based on the Cohesive Segments Method 3348.5.1 The Enrichment of Displacement Field 3348.5.2 Cohesive Constitutive Law 3358.5.3 Crack Growth Criterion and Its Direction for Continuous Crack Propagation 3368.5.4 Crack Growth Criterion and Its Direction for Discontinuous Crack Propagation 3398.5.5 Numerical Integration Scheme 3418.6 Application of X-FEM Method in Cohesive Crack Growth 3418.6.1 A Three-Point Bending Beam with Symmetric Edge Crack 3418.6.2 A Plate with an Edge Crack under Impact Velocity 3438.6.3 A Three-Point Bending Beam with an Eccentric Crack 3469 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM 3519.1 Introduction 3519.2 Large FE Deformation Formulation 3539.3 Modified X-FEM Formulation 3569.4 Large X-FEM Deformation Formulation 3599.5 The Damage–Plasticity Model 3649.6 The Nonlocal Gradient Damage Plasticity 3689.7 Ductile Fracture with X-FEM Plasticity Model 3699.8 Ductile Fracture with X-FEM Non-Local Damage-Plasticity Model 3729.8.1 Crack Initiation and Crack Growth Direction 3729.8.2 Crack Growth with a Null Step Analysis 3759.8.3 Crack Growth with a Relaxation Phase Analysis 3779.8.4 Locking Issues in Crack Growth Modeling 3799.9 Application of X-FEM Damage-Plasticity Model 3809.9.1 The Necking Problem 3809.9.2 The CT Test 3839.9.3 The Double-Notched Specimen 3859.10 Dynamic Large X-FEM Deformation Formulation 3879.10.1 The Dynamic X-FEM Discretization 3889.10.2 The Large Strain Model 3909.10.3 The Contact Friction Model 3919.11 The Time Domain Discretization: The Dynamic Explicit Central Difference Method 3939.12 Implementation of Dynamic X-FEM Damage-Plasticity Model 3969.12.1 A Plate with an Inclined Crack 3989.12.2 The Low Cycle Fatigue Test 4009.12.3 The Cyclic CT Test 4019.12.4 The Double Notched Specimen in Cyclic Loading 40510 X-FEM Modeling of Saturated/Semi-Saturated Porous Media 40910.1 Introduction 40910.1.1 Governing Equations of Deformable Porous Media 41110.2 The X-FEM Formulation of Deformable Porous Media with Weak Discontinuities 41410.2.1 Approximation of Displacement and Pressure Fields 41510.2.2 The X-FEM Spatial Discretization 41810.2.3 The Time Domain Discretization and Solution Procedure 41910.2.4 Numerical Integration Scheme 42110.3 Application of the X-FEM Method in Deformable Porous Media with Arbitrary Interfaces 42210.3.1 An Elastic Soil Column 42210.3.2 An Elastic Foundation 42410.4 Modeling Hydraulic Fracture Propagation in Deformable Porous Media 42710.4.1 Governing Equations of a Fractured Porous Medium 42810.4.2 The Weak Formulation of a Fractured Porous Medium 43010.5 The X-FEM Formulation of Deformable Porous Media with Strong Discontinuities 43410.5.1 Approximation of the Displacement and Pressure Fields 43410.5.2 The X-FEM Spatial Discretization 43710.5.3 The Time Domain Discretization and Solution Procedure 43810.6 Alternative Approaches to Fluid Flow Simulation within the Fracture 44210.6.1 A Partitioned Solution Algorithm for Interfacial Pressure 44210.6.2 A Time-Dependent Constant Pressure Algorithm 44410.7 Application of the X-FEM Method in Hydraulic Fracture Propagation of Saturated Porous Media 44510.7.1 An Infinite Saturated Porous Medium with an Inclined Crack 44610.7.2 Hydraulic Fracture Propagation in an Infinite Poroelastic Medium 44910.7.3 Hydraulic Fracturing in a Concrete Gravity Dam 45210.8 X-FEM Modeling of Contact Behavior in Fractured Porous Media 45510.8.1 Contact Behavior in a Fractured Medium 45510.8.2 X-FEM Formulation of Contact along the Fracture 45610.8.3 Consolidation of a Porous Block with a Vertical Discontinuity 45711 Hydraulic Fracturing in Multi-Phase Porous Media with X-FEM 46111.1 Introduction 46111.2 The Physical Model of Multi-Phase Porous Media 46311.3 Governing Equations of Multi-Phase Porous Medium 46511.4 The X-FEM Formulation of Multi-Phase Porous Media with Weak Discontinuities 46711.4.1 Approximation of the Primary Variables 46911.4.2 Discretization of Equilibrium and Flow Continuity Equations 47311.4.3 Solution Procedure of Discretized Equilibrium Equations 47611.5 Application of X-FEM Method in Multi-Phase Porous Media with Arbitrary Interfaces 47711.6 The X-FEM Formulation for Hydraulic Fracturing in Multi-Phase Porous Media 48211.7 Discretization of Multi-Phase Governing Equations with Strong Discontinuities 48711.8 Solution Procedure for Fully Coupled Nonlinear Equations 49311.9 Computational Notes in Hydraulic Fracture Modeling 49711.10 Application of the X-FEM Method to Hydraulic Fracture Propagation of Multi-Phase Porous Media 49912 Thermo-Hydro-Mechanical Modeling of Porous Media with X-FEM 50912.1 Introduction 50912.2 THM Governing Equations of Saturated Porous Media 51112.3 Discontinuities in a THM Medium 51312.4 The X-FEM Formulation of THM Governing Equations 51412.4.1 Approximation of Displacement, Pressure, and Temperature Fields 51512.4.2 The X-FEM Spatial Discretization 51712.4.3 The Time Domain Discretization 52012.5 Application of the X-FEM Method to THM Behavior of Porous Media 52112.5.1 A Plate with an Inclined Crack in Thermal Loading 52112.5.2 A Plate with an Edge Crack in Thermal Loading 52212.5.3 An Impermeable Discontinuity in Saturated Porous Media 52412.5.4 An Inclined Fault in Porous Media 527References 533Index 557