The generalized finite element method

This paper describes a pilot design and implementation of the generalized f inite element method (GFEM), as a direct extension of the standard finite e lement method (SFEM, or FEM), which makes possible the accurate solution of engineering problems in complex domains which may be practically impossibl e to solve using the FEM. The development of the GFEM is illustrated for th e Laplacian in two space dimensions in domains which may include several hu ndreds of voids, and/or cracks, for which the construction of meshes used b y the FEM is practically impossible. The two main capabilities are: (1) It can construct the approximation using meshes which may overlap parr, or all . of the domain boundary. (2) It can incorporate into the approximation han dbook functions, which are known analytically, or are generated numerically , and approximate well the solution of the boundary value problem in the ne ighborhood of corner points, voids, cracks, etc. The main tool is a special integration algorithm, which we call the Fast Remeshing approach, which is robust and works for any domain with arbitrary complexity. The incorporati on of the handbook functions into the GFEM is done by employing the partiti on of unity method (PUM). The presented formulations and implementations ca n be easily extended to the multimaterial medium where the voids are replac ed by inclusions of various shapes and sizes. and to the case of the elasti city problem. This work can also be understood as a pilot study for the fea sibility and demonstration of the capabilities of the GFEM, which is needed before analogous implementations are attempted in the three-dimensional an d nonlinear cases, which are the cases of main interest for future work. (C ) 2001 Elsevier Science B.V. All rights reserved.