Projection and Superconvergent Techniques for Adaptive Finite Element Analysis

ABSTRACT

The advances in finite element adaptive analysis have brought
increased attention to the reliability and efficiency of finite element
computations. This thesis addresses two major computation techniques and
their applications for effective and reliable adaptive finite element
analysis. They are investigated in two parts respectively.

In the first part, the projection technique, a conveniently
employed technique for finite element stress interpretations, is
formulated and various different forms of this technique are developed
within the overall projection framework. These projectors have different
characteristics in terms of accuracy and computation efficiency. The
behavior of projected quantities are evaluated in an h-version adaptive
analysis environment. When used to perform energy normed error
estimation, the effectiveness of  projection—based error
estimators are qualified with some adaptive examples. As the emphasis of
this part, the projectors are also employed in  solution mapping process
between related mesh models. With the projection technique used as a
mean to define solution fields on different mesh models,  a modular
rezoning system is developed that is able to perform general purpose
mesh mapping and solution transfer.  Some efficient searching and mesh
mapping techniques are developed and evaluated. This rezoning system
allows easy interfacing with particular applications through a set of
interface operators. It can serve as a handy tool for many finite
element modeling operations such as solution transfer, mesh mapping and
post-processing.

In the second part, a class of superconvergent computation
techniques are developed to extract finite element stress as well as
displacement quantities for linear elastic problems. A significant gain
in  accuracy is achieved through these post-processing operations since
the extracted quantities have a pointwise accuracy comparable to that of
the total strain energy. This research is primarily focused on unified
2D superconvergent extraction formulations with their extensions
to 3D problems  explored and discussed. Although the formulations
cover  stress and displacement quantities both in the interior and on
the boundaries, emphasis is on the extraction of boundary stresses. This
is motivated by the fact that stress quantities are typically of most
engineering interests and stress-critical locations frequently appear at
the boundary. Detailed technical treatments for extracting boundary
stresses are investigated within the finite element analysis environment
to provide useful guidelines for their engineering implementations. The
reliability and effectiveness of these techniques are demonstrated with
a few example problems. The use of these superconvergent techniques for
effective pointwise accuracy control in adaptive finite element analysis
is also explored.