Finite element analysis for fracture and damage mechanics

This work intends to make a comparative study among different high-order Galerkin methods for the solution of linear and non-linear equations. Initially, the finite element method (FEM) and the discontinuous Galerkin (DG-FEM) are introduced for the linear dynamic elastic equation of motion. It is proposed a local (LFEM) and semi-local (SLFEM) finite element methods based on the combination of FEM and DG-FEM methods. Non-linear phase-field models are specifically addressed due to its wide use in multiphase problems including tumor growth, solidification, fluid separation, fracture and fatigue. The errors of the different Galerkin methods are evaluated for the non-linear phase-field Allen-Cahn (AC) equation. A fracture model which consists of the combination of the motion equation with an Allen-Cahn type equation is introduced and numerically solved using the proposed SLFEM semi-local method.