Graduation Year

2011

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Civil and Environmental Engineering

Major Professor

Daniel Simkins, Ph.D.

Committee Member

Andres Tejada-Martinez, Ph.D.

Committee Member

Ryan Toomey, Ph.D.

Committee Member

Autar Kaw, Ph.D.

Committee Member

Nathan Crane, Ph.D.

Committee Member

David Rabson, Ph.D.

Keywords

Numerical analysis, interpolation, weak form, thin plates, continuity

Abstract

The Reproducing Kernel Element Method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker-δ property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This dissertation shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use in previous formulations of incomplete polynomials that are subsequently non-affine invariant. This dissertation lays out the new framework for generating general, symmetric, truly minimal and complete affine invariant global partition polynomials for triangular and tetrahedral elements. It is shown that this new class of reproducing kernel element solves the asymmetry issue that affected previous developed elements. The interpolation capabilities of this new class of reproducing kernel elements is studied in problems of surface representations and in solving problems of bending of thin plates using a Galerkin approach. Optimal convergence rates were observed in the solution of Kirchhoff plate problems with rectangular domains. Furthermore, it is shown that the new proposed two-dimensional elements out perform the previous elements with the addition of only a few internal degrees of freedom.

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