Higher-order polygonal finite element analysis of nearly-incompressible isotropic elastic materials

In the first part of this thesis, we present a stable higher-order polygonal finite element method for modeling nearly-incompressible isotropic materials. Our method is based on applying the discontinuous Petrov-Galerkin methodology on a hybridized version of the ultraweak formulation of linear elasticity. As a result, the unknown degrees of freedom are defined only on the skeleton of the mesh (interface variables) and have a symmetric positive-definite coefficient matrix. The performance and convergence of the method is demonstrated with numerical examples.In the second part of the thesis, we present a heuristic algorithm that generates coarsened non-uniform hexahedral meshes with higher resolution close to selected regions in the domain of 3D micro-CT and micro-MR images. Applying a coarsening step on areas of the problem domain in order to reduce the computational cost is inevitable; however, on the other hand, it is desirable to have a fine mesh in regions containing small vital features. This algorithm takes as input a very fine micro-CT data set, the location of the regions containing delicate geometrical details, and the coarsening factor and produces a ready-to-use graded mesh