Classification of extended finite element method

Abstract

The finite element method is invented for finding numerical approximate solutions of partial differential equations. In the real world, there are many examples such that rapid changes of field variables on surfaces. In many cases, they are regarded as discontinuities, singularities, or high gradients for the modeling. These are found in structures for cracks, dislocations, voids, shear bands, and inclusions. The extended finite element methods can us the accurate approximation of non-smooth solutions. The method constructs an approximation space consisting of mesh-based or mesh-free, enriched functions near discontinuities, singularities, or high gradients and classical finite element method basis functions elsewhere. In this thesis, we introduce strategies for the approximation of non-smooth solutions. First, we conduct an experiment on classical finite element method for the smooth solution problem. After then, we explain the terminologies and examine the structure of the extended finite element methods. Finally, we pick over the linear elastic problem and the Navier-Stokes equation and suggest some methods, respectively.