Researching progress of generalized finite element method

Generalized finite element method is the extension of conventional finite element method. Based on the partition of unity method, it improves the approximation accuracy of the finite element method or achieves the special approximation to particular problems by introducing the generalized degrees of freedom and by re-interpolating the nodal degrees of freedom. From the profound study on constructing the shape functions of the generalized finite element method, arbitrarily complex problems with internal features (e. g. void, inclusion and crack) and external features (e. g. re-entrant, corner and edge) can be expected to solve by the simple and domain independent mesh. The essential ideas and corresponding strategies, including treatment of the linear dependency and boundary conditions, capture of the local approximation functions, numerical integration techniques, are introduced in details. The features and connections are analyzed as compared with the extended finite element method and the finite cover method. The progress of the generalized finite element method is reviewed, and then current practical applications are summarized.