Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n+1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called `serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform `a priori' error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.Comment: 24 page

Unstructured mesh generation for mesh improvement techniques and contour meshing.

This thesis will investigate surface mesh generation and develop ideas to improve the quality of surface meshes that are currently produced. Surface geometries are represented by a CAD definition, but the CAD definition does not necessarily guarantee that the surface geometry is acceptable for mesh generation. CAD geometries will often contain a number of detailed features which will need to be improved by processes such as CAD repair before mesh generation can take place. Even then the geometries can still contain problems in the features such as, small sliver surface patches and sliver edges. These features cause major difficulties when meshed, as they generate small distorted elements. Here we will look to improve the meshes by merging together neighboring surface patches to create a super patch and then generate the mesh on this one surface. The merging of surfaces is controlled by the angle between surface patches. Another method that will be investigated involves the re-meshing of the geometry based on a prescribed metric. In addition to looking at this problem of CAD representation we will also look at the growing area of medical imaging. Here we will look to produce a 3D mesh from a set of contours. From this the mesh produced will be remeshed using the previous ideas to produce a mesh that can be used for analysis

Simplex Control Methods for Robust Convergence of Small Unmanned Aircraft Flight Trajectories in the Constrained Urban Environment

Constrained optimal control problems for Small Unmanned Aircraft Systems (SUAS) have long suffered from excessive computation times caused by a combination of constraint modeling techniques, the quality of the initial path solution provided to the optimal control solver, and improperly defining the bounds on system state variables, ultimately preventing implementation into real-time, on-board systems. In this research, a new hybrid approach is examined for real-time path planning of SUAS. During autonomous flight, a SUAS is tasked to traverse from one target region to a second target region while avoiding hard constraints consisting of building structures of an urban environment. Feasible path solutions are determined through highly constrained spaces, investigating narrow corridors, visiting multiple waypoints, and minimizing incursions to keep-out regions. These issues are addressed herein with a new approach by triangulating the search space in two-dimensions, or using a tetrahedron discretization in three-dimensions to define a polygonal search corridor free of constraints while alleviating the dependency of problem specific parameters by translating the problem to barycentric coordinates. Within this connected simplex construct, trajectories are solved using direct orthogonal collocation methods while leveraging navigation mesh techniques developed for fast geometric path planning solutions. To illustrate two-dimensional flight trajectories, sample results are applied to flight through downtown Chicago at an altitude of 600 feet above ground level. The three-dimensional problem is examined for feasibility by applying the methodology to a small scale problem. Computation and objective times are reported to illustrate the design implications for real-time optimal control systems, with results showing 86% reduction in computation time over traditional methods