| This dissertation presents the design, the implementation, and the capabilities of the Generalized Finite Element Method (GFEM), which is a direct generalization of the standard Finite Element Method (SFEM, or FEM), endowed with the following additional capabilities: (1)Geometrical Flexibility . The construction of the standard FEM solution employs a FE mesh which is a discretization (a decomposition) of the problem domain into a set of simple nonoverlapping subdomains, e.g., triangles and/or quadrilaterals with straight and/or curved edges, which are not too distorted, and must satisfy various adjacency rules (e.g., 1-to-1, 1-to-2, or 1-to-n, connections between neighboring elements). The construction of such a FE mesh for complex domains is sometimes practically impossible. The GFEM does away with this requirement, and constructs the GFEM approximate solution by employing GFEM meshes which can be partially, or totally independent of the geometry of the problem domain, the precise geometry of which enters into the approximation in terms of element integration meshes. (2) Hybrid Capability . The standard FEM constructs the basis of the approximation by piecing together specially constructed polynomial element shape functions, which do not include any information about the problem which is being solved. The GFEM approach is to build the basis using the standard mapped polynomial FE bases on the employed mesh, and in addition, to enrich the basis by employing special handbook functions, which reflect known information about the problem which is being solved. These handbook functions are added only in the neighborhood of the features to which they correspond, e.g., the comers, cracks etc.; In this dissertation, the GFEM is presented for the Laplacian in polygonal domains with straight or curved edges, which may or may not include a large number of voids and cracks. The goal of the dissertation is to show that a properly designed GFEM can make possible the accurate solution of difficult engineering problems, which cannot be practically solved by the FEM. |
