| This dissertation presents two novel contributions to the fields of isogeometric analysis and p-version finite elements. First, we present a framework for geometrically exact volumetric mesh generation. By leveraging ideas from both traditional mesh generation as well as isogeometric analysis, we develop a framework for volumetric mesh generation using rational Bernstein--Bezier discretizations. Within this framework, we provide a set of easily verifiable sufficient conditions for guaranteeing that a mesh will be geometrically exact. Second, we develop a complete theory of mesh quality for these rational Bernstein--B'{e}zier elements. From this, we derive a set of easily computable mesh quality metrics for verifying that a rational Bernstein--Bezier discretization will be analysis suitable. |
