Wayne State University Dissertations

Access Type

Open Access Dissertation

January 2013

Dissertation

Ph.D.

Mathematics

Zhimin Zhang

Fatih Celiker

Abstract

We introduce and analyze discontinuous Galerkin methods

for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking.

We also prove that, when polynomials of degree $k$ are used,

{\em all} the numerical traces superconverge with a rate of order

h 2k+1.

Based on the superconvergent phenomenon and we show how to

post-process them in an element-by-element fashion

to obtain a far better approximation. Indeed, we prove that,

if polynomials of degree k are used, the post-processed

approximation converges with order 2k+1 in the L2-norm throughout the domain. This has to be contrasted with the fact that before post-processing, the approximation converges with order k+1 only. Moreover, we show that this superconvergence property does not deteriorate as the thickness of the arch becomes extremely small.

Since the DG methods suffer from too many degree of freedoms we introduce and analyze a class of hybridizable

discontinuous Galerkin (HDG) methods for Naghdi arches.

The main feature of these methods is that they can be

implemented in an efficient way through a hybridization

procedure which reduces the globally coupled unknowns to

approximations to the transverse and tangential displacement

and bending moment at the element boundaries.

The error analysis of the methods is based on the use

of a projection especially designed to fit the structure

of the numerical traces of the method. This property allows to prove

in a very concise manner that the projection of the errors is

bounded in terms of the distance between the exact solution and its projection.

The study of the influence of the stabilization function

on the approximation is then reduced to the study of how they affect

the approximation properties of the projection in a single element.

Consequently, we prove that HDG methods have the same result as DG methods.

At the end of the thesis, we talk a little bit of shell problems.

COinS