The generalized finite element method with global-local enrichment functions

The main feature of partition of unity methods such as hp-cloud and generalized or extended finite element methods is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. Linear combination of partition of unity shape functions can reproduce exactly any enrichment function and thus their approximation properties are preserved. Due to this reproducing property, the generalized finite element method has been applied to many classes of problems where a priori knowledge about the solution exists such as modeling of cracks, inclusions and microstructures. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems.;In this work, we propose a procedure to build enrichment functions to overcome this limitation. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. This procedure allows the use of a coarse and fixed global mesh for any configuration of local features and this enables efficient solution of problems with multiple local features. It is also appealing for problems with localized nonlinearities since computationally intensive nonlinear iterations can be performed on coarse global meshes. The parallel computation of local solutions can be straightforwardly implemented and large problems can be efficiently solved in massively parallel machines with this approach.