| Polygonal finite elements are a generalization of triangular and quadrilateral finite elements to meshes with n-sided elements (n ≥ 3). The ability to construct conforming approximations on convex and nonconvex shapes provides greater flexibility in mesh generation and render such elements to be a viable and potentially attractive choice in computer modeling and simulation. In this thesis, the development and application of polygonal finite elements to two dimensional elliptic boundary value problems is presented. We use natural neighbor (Laplace) interpolant and mean value coordinates (MVC) to construct C 0(O) conforming interpolants on convex and nonconvex elements, respectively.;We use polygonal finite elements to resolve the issue of hanging nodes on quadtree meshes. For this purpose, quadtree elements are considered as special cases of polygonal elements and Laplace interpolant is used to construct C0 interpolant on quadtree meshes.;We use extended finite element (X-FEM) to model crack growth on polygonal and quadtree meshes. For this purpose, the polygonal interpolants (Laplace and MVC functions) are enriched with the Heaviside and asymptotic near crack-tip field functions. On using extended finite element method in crack growth simulation, the need for remeshing is eliminated while good accuracy on coarse meshes is obtained. |
