The Generalized Finite Element Method - Improving Finite Elements Through Meshless Technology

The Generalized Finite Element Method (GFEM) presented in this paper combines and extends the best features of the finite element method with the help of meshless formulations based on the Partition of Unity Method. Although an input finite element mesh is used by the proposed method, the requirements on the quality of this mesh are significantly relaxed. The main technique presented in this paper, "element clustering", allows the combination of neighboring elements, and generate mathematically correct and smooth solution approximation for such clusters. The paper shows how this can be used to effectively hide mesh defects internal to each cluster, and also allows for effective coarsening of the solution to reduce the computational cost and memory requirements in exchange for the solution accuracy. In particular, the proposed GFEM can correctly and efficiently deal with: (i) severely distorted or elements with large aspect ratio; (ii) elements with negative Jacobian (inverted inside–out); (iii) large number of small elements; (iv) meshes consisting of several sub-domains with mismatched interfaces. Under such relaxed requirements for an acceptable mesh, and for correctly defined geometries, today’s automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user. In addition, the method is fully applicable to all existing finite element algorithms (e.g. non–linear, time–dependent) and is also fully hp-adaptive.