Degree Granting Department
Daniel C. Simkins, Jr., Ph.D.
Andrés Tejada-Martinez, Ph.D.
Stanley Kranc, Ph.D., P.E.
Sudeep Sarkar, Ph.D.
David Rabson, Ph.D.
meshing, regularity, RKEM, interpolation, surface representation
The Reproducing Kernel Element Method (RKEM) is a hybrid between finite elements and meshfree methods that provides shape functions of arbitrary order and continuity yet retains the Kronecker-delta property. To achieve these properties, the underlying mesh must meet certain regularity constraints, unique to RKEM. The aim of this dissertation is to develop a precise definition of these constraints, and a general algorithm for assessing a mesh is developed. This check is a critical step in the use of RKEM in any application.
The general checking algorithm is made more specific to apply to two-dimensional triangular meshes with circular supports and to three-dimensional tetrahedral meshes with spherical supports. The checking algorithm features the output of the uncovered regions that are used to develop a mesh-mending technique for fixing offending meshes. The specific check is used in conjunction with standard quality meshing techniques to produce meshes suitable for use with RKEM.
The RKEM quasi-uniformity definitions enable the use of RKEM in solving Galerkin weak forms as well as in general interpolation applications, such as the representation of geometries. A procedure for determining a RKEM representation of discrete point sets is presented with results for surfaces in three-dimensions. This capability is important to the analysis of geometries such as patient-specific organs or other biological objects.
Scholar Commons Citation
Collier, Nathaniel O., "The Quasi-Uniformity Condition and Three-Dimensional Geometry Representation as it Applies to the Reproducing Kernel Element Method" (2009). Graduate Theses and Dissertations.