有限元法（FEM）是目前求解偏微分方程最流行的方法之一，被广泛应用到航空与航天工程、汽车工业、机械工程、土木工程，生物力学、材料科学、地质力学等领域。扩展有限元法（XFEM）的出现克服了FEM一些缺点使得可以在有限单元内引入任意的非连续性，本教材的目的是将XFEM在求解连续介质以及结构的线性与非线性问题时的理论与应用展现给更多的读者。

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
3.3 Level Set Method
3.3.1 Numerical Implementation of LSM
3.3.2 Coupling the LSM with X-FEM
3.4 Fast Marching Method
3.4.1 Coupling the FMM with X-FEM
3.5 X-FEM Enrichment Functions
3.5.1 Bimaterials, Voids, and Inclusions
3.5.2 Strong Discontinuities and Crack Interfaces
3.5.3 Brittle Cracks
3.5.4 Cohesive Cracks
3.5.5 Plastic Fracture Mechanics
3.5.6 Multiple Cracks
3.5.7 Fracture in Bimaterial Problems
3.5.8 Polycrystalline Microstructure
3.5.9 Dislocations
3.5.10 Shear Band Localization
4 Blending Elements
4.1 Introduction
4.2 Convergence Analysis in the X-FEM
4.3 Ill-Conditioning in the X-FEM Method
4.3.1 One-Dimensional Problem with Material Interface
4.4 Blending Strategies in X-FEM
4.5 Enhanced Strain Method
4.5.1 An Enhanced Strain Blending Element for the Ramp Enrichment Function
4.5.2 An Enhanced Strain Blending Element for Asymptotic
Enrichment Functions
4.6 The Hierarchical Method
4.6.1 A Hierarchical Blending Element for Discontinuous Gradient Enrichment
4.6.2 A Hierarchical Blending Element for Crack Tip Asymptotic Enrichments
4.7 The Cutoff Function Method
4. 7.1 The Weighted Function Blending Method
4.7.2 A Variant of the Cutoff Function Method
4.8 A DG X-FEM Method
……
5 Large X-FEM Deformation
6 Contact Friction Modeling with X-FEM
7 Linear Fracture Mechanics with the X-FEM Technique
8 Cohesive Crack Growth with the X-FEM Technique
9 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM
10 X-FEM Modeling of Saturated/Semi-Saturated Porous Media
11 Hydraulic Fracturing in Multi-Phase Porous Media with X-FEM