Data Structure and Error Estimation for an Adaptive <i>p</i>-Version Finite Element Method in 2-D and 3-D Solids

Abstract

Automation of finite element analysis based on

a fully adaptive <i>p</i>-refinement procedure

can reduce the magnitude of discretization error

to the desired accuracy with minimum computational cost

and computer resources.

This study aims to 1) develop an efficient

<i>p</i>-refinement procedure with a non-uniform

<i>p</i> analysis capability for solving 2-D and 3-D

elastostatic mechanics problems, and 2) introduce a

stress error estimate.

An element-by-element algorithm that decides the

appropriate order for each element, where element

orders can range from 1 to 8, is described.

Global and element data structures that manage the complex

data generated during the refinement process are introduced.

These data structures are designed to match the concept of

object-oriented programming where data and functions are

managed and organized simultaneously. <P>

The stress error indicator introduced is found to be more

reliable and to converge faster than the error indicator

measured in an energy norm called the residual method.

The use of the stress error indicator results in

approximately 20% fewer degrees of freedom than the

residual method. Agreement of the calculated stress error

values and the stress error indicator values confirms the

convergence of final stresses to the analyst. The error

order of the stress error estimate is postulated to be

one order higher than the error order of the error

estimate using the residual method. The mapping of a

curved boundary element in the working coordinate system

from a square-shape element in the natural coordinate

system results in a significant improvement in the

accuracy of stress results. <P>

Numerical examples demonstrate that refinement using

non-uniform <i>p</i> analysis is superior to uniform

<i>p</i> analysis in the convergence rates of output

stresses or related terms. Non-uniform <i>p</i> analysis

uses approximately 50% to 80% less computational time than

uniform <i>p</i> analysis in solving the selected stress

concentration and stress intensity problems. More

importantly, the non-uniform <i>p</i> refinement procedure

scales the number of equations down by 1/2 to 3/4.

Therefore, a small scale computer can be used to solve

equation systems generated using high order

<i>p</i>-elements. In the calculation of the stress

intensity factor of a semi-elliptical surface crack in a

finite-thickness plate, non-uniform <i>p</i> analysis

used fewer degrees of freedom than a conventional

<i>h</i>-type element analysis found in the literature.