Series Preface Preface 1 Introduction 1.1 Introduction 1.2 An Enriched Finite Element Method 1.3 A Review on X-FEM: Development and Applications 1.3.1 Coupling X-FEM with the Level-Set Method 1.3.2 Linear Elastic Fracture Mechanics (LEFM) 1.3.3 Cohesive Fracture Mechanics 1.3.4 Composite Materials and Material Inhomogeneities 1.3.5 Plasticity, Damage, and Fatigue Problems 1.3.6 Shear Band Localization 1.3.7 Fluid-Structure Interaction 1.3.8 Fluid Flow in Fractured Porous Media 1.3.9 Fluid Flow and Fluid Mechanics Problems 1.3.10 Phase Transition and Solidification 1.3.11 Thermal and Thermo-Mechanical Problems 1.3.12 Plates and Shells 1.3.13 Contact Problems 1.3.14 Topology Optimization 1.3.15 Piezoelectric and Magneto-Electroelastic Problems 1.3.16 Multi-Scale Modeling 2 Extended Finite Element Formulation 2.1 Introduction 2.2 The Partition of Unity Finite Element Method 2.3 The Enrichment of Approximation Space 2.3.1 Intrinsic Enrichment 2.3.2 Extrinsic Enrichment 2.4 The Basis of X-FEM Approximation 2.4.1 The Signed Distance Function 2.4.2 The Heaviside Function 2.5 Blending Elements 2.6 Governing Equation of a Body with Discontinuity 2.6.1 The Divergence Theorem for Discontinuous Problems 2.6.2 The Weak form of Governing Equation 2.7 The X-FEM Discretization of Governing Equation 2.7.1 Numerical Implementation of X-FEM Formulation 2. 7.2 Numerical Integration Algorithm 2.8 Application of X-FEM in Weak and Strong Discontinuities 2.8.1 Modeling an Elastic Bar with a Strong Discontinuity 2.8.2 Modeling an Elastic Bar with a Weak Discontinuity 2.8.3 Modeling an Elastic Plate with a Crack Interface at its Center 2.8.4 Modeling an Elastic Plate with a Material Interface at its Center 2.9 Higher Order X-FEM 2.10 Implementation of X-FEM with Higher Order Elements 2.10.1 Higher Order X-FEM Modeling of a Plate with a Material Interface 2.10.2 Higher Order X-FEM Modeling of a Plate with a Curved Crack Interface 3 Enrichment Elements 3.1 Introduction 3.2 Tracking Moving Boundaries