Arbitrarily smooth generalized finite element approximations

This paper presents a procedure to build Ck, k arbitrarily large, generalized finite element (FE) shape functions defined on non-structured finite element meshes. The functions have the same support as corresponding global FE Lagrangian shape functions. Meshes with both convex and non-convex clouds (set of elements sharing a vertex node), can be used. The so-called R-functions are used to build Ck FE-based partition of unity functions with non-convex support. A technique to combine C0 Lagrangian FE shape functions with the proposed Ck partition of unity is presented. The technique allows the use of Ck generalized FE shape functions in parts of the computational domain where their high smoothness is required, as in the case of problems with distributional boundary conditions, and the less computationally demanding C0 generalized FE shape functions elsewhere in the domain. A linear elasticity problem with a concentrated moment is solved using the proposed Ck generalized FE method. Higher order distributional boundary conditions can also be handled by the method. A detailed convergence analysis is presented for this class of problems as well as for problems in energy space. The integrability of the functions using standard Gauss–Legendre rules is also investigated.