Graduation Year

2009

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Civil Engineering

Major Professor

Daniel C. Simkins, Jr., Ph.D.

Committee Member

Andrés Tejada-Martinez, Ph.D.

Committee Member

Stanley Kranc, Ph.D., P.E.

Committee Member

Sudeep Sarkar, Ph.D.

Committee Member

David Rabson, Ph.D.

Keywords

meshing, regularity, RKEM, interpolation, surface representation

Abstract

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.

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