A review of extended/generalized finite element methods for material modeling Ted Belytschko, Robert Gracie, Giulio Ventura Extended and generalized finite element methods (XFEM/GFEM) are reviewed with an emphasis on their applications to material science problems, including fracture, dislocations, grain boundaries, and phase interfaces. These methods allow for the modeling of complex geometries and their evolution, especially when combined with level set methods. The methods simplify the modeling of discontinuous phenomena by enabling the finite element mesh to be independent of the geometry of discontinuities. XFEM and GFEM are essentially the same methods, with XFEM being developed for discontinuities like cracks and using local enrichments, while GFEM was introduced in 1995-1996. Both methods can be applied to structured and unstructured meshes, with structured meshes being useful for determining material properties and unstructured meshes for engineering structures. The partition of unity concept is fundamental to XFEM/GFEM, allowing for the enrichment of finite elements or mesh-free approximations. The methods are applied to cracks, dislocations, inclusions, phase boundaries, and grain boundaries. For cracks, the XFEM displacement field is defined using enrichment functions based on asymptotic solutions. The discrete equations for XFEM/GFEM are derived from the principle of virtual work, leading to a system of equations that includes both standard finite element degrees of freedom and enriched degrees of freedom. The methods have been applied to various problems, including cohesive cracks, dislocations, and grain boundaries. For dislocations, the XFEM approach uses enrichment functions based on the Volterra concept, allowing for the modeling of dislocation loops. The methods have also been applied to phase interfaces, where the evolution of interfaces is tracked using level set methods. The level set method allows for the description of complex geometries and the evolution of interfaces, with the crack surface defined by a level set function. Quadrature in XFEM/GFEM is a critical aspect of implementation, as the integrands of the stiffness matrices and force vectors are discontinuous. The methods have been successfully applied to a wide range of problems in material modeling, including fracture, dislocations, grain boundaries, and phase interfaces. The methods offer significant advantages in handling complex geometries and discontinuities, making them a valuable tool in material modeling.A review of extended/generalized finite element methods for material modeling Ted Belytschko, Robert Gracie, Giulio Ventura Extended and generalized finite element methods (XFEM/GFEM) are reviewed with an emphasis on their applications to material science problems, including fracture, dislocations, grain boundaries, and phase interfaces. These methods allow for the modeling of complex geometries and their evolution, especially when combined with level set methods. The methods simplify the modeling of discontinuous phenomena by enabling the finite element mesh to be independent of the geometry of discontinuities. XFEM and GFEM are essentially the same methods, with XFEM being developed for discontinuities like cracks and using local enrichments, while GFEM was introduced in 1995-1996. Both methods can be applied to structured and unstructured meshes, with structured meshes being useful for determining material properties and unstructured meshes for engineering structures. The partition of unity concept is fundamental to XFEM/GFEM, allowing for the enrichment of finite elements or mesh-free approximations. The methods are applied to cracks, dislocations, inclusions, phase boundaries, and grain boundaries. For cracks, the XFEM displacement field is defined using enrichment functions based on asymptotic solutions. The discrete equations for XFEM/GFEM are derived from the principle of virtual work, leading to a system of equations that includes both standard finite element degrees of freedom and enriched degrees of freedom. The methods have been applied to various problems, including cohesive cracks, dislocations, and grain boundaries. For dislocations, the XFEM approach uses enrichment functions based on the Volterra concept, allowing for the modeling of dislocation loops. The methods have also been applied to phase interfaces, where the evolution of interfaces is tracked using level set methods. The level set method allows for the description of complex geometries and the evolution of interfaces, with the crack surface defined by a level set function. Quadrature in XFEM/GFEM is a critical aspect of implementation, as the integrands of the stiffness matrices and force vectors are discontinuous. The methods have been successfully applied to a wide range of problems in material modeling, including fracture, dislocations, grain boundaries, and phase interfaces. The methods offer significant advantages in handling complex geometries and discontinuities, making them a valuable tool in material modeling.