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Class of Accurate Low Order Finite Elements

Fredriksson, Magnus LU (2006)
Abstract
High accuracy and low computational costs are essential properties for efficient



finite element codes. Improvements of the finite element method in order to



satisfy these qualities have been the subject for extensive research activities



ever since its introduction. In this thesis, a class of improved low-order finite



elements is proposed which possesses high accuracy in bending even for coarse



meshes. The elements are insensitive to material incompressibility and they are



all based upon the Hu-Washizu variational principle. A 4-node plane



quadrilateral element as well as a 8-node brick element are proposed... (More)
High accuracy and low computational costs are essential properties for efficient



finite element codes. Improvements of the finite element method in order to



satisfy these qualities have been the subject for extensive research activities



ever since its introduction. In this thesis, a class of improved low-order finite



elements is proposed which possesses high accuracy in bending even for coarse



meshes. The elements are insensitive to material incompressibility and they are



all based upon the Hu-Washizu variational principle. A 4-node plane



quadrilateral element as well as a 8-node brick element are proposed and also a



4-node axisymmetric element is presented.



The plane element was first formulated by the three-field Hu-Washizu principle,



but later it was realized that the same response could be achieved by the



assumed strain method. This reformulation leads, among other things, to gained



computational efficiency. Consequently, the subsequent axisymmetric element and



the brick element were established using the assumed strain method.



The assumed strain



method can be systematically formulated within the framework of the Hu-Washizu



principle which makes the elements variational consistent providing that certain



conditions are fulfilled. Due to its simplicity, the assumed strain method is



highly rewarding in large-scale analysis and since a strain-driven format is



obtained naturally which resembles the standard displacement method,



this method is well suited for nonlinear analysis. For



simplicity, however, only linear problems are considered here.



The approach presented in the appended papers is based upon the fundamental



idea in which the element stiffness matrix is divided into two parts. These



parts are denoted fundamental stiffness and higher-order stiffness. The fundamental stiffness



is the part which guarantees that the element is convergent. This means that the



element fulfills the convergence criterion and it can be shown that all



elements that are proposed



pass the patch test {itshape a priori}. The higher-order stiffness



must restore the correct rank to the total element stiffness matrix. In



addition, the higher-order stiffness is chosen such that it significantly



improves the accuracy. By consistent modifications of the higher-order



stiffness,



the proposed finite elements provide the exact strain energy for pure bending of



regular elements. Sensitivity



to mesh distortion is also effectively reduced by use of dimensionless



parameters



which emerge naturally within this approach which includes use of a coordinate



system aligned with the principal axes of inertia. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Skallerud, Björn, Dept. Structural Engineering, Norwegian University of Science and Technology
organization
publishing date
type
Thesis
publication status
published
subject
keywords
'quadrilateral', 'variational', 'finite element', 'element technology', 'HuWashizu', 'hexahedral', Technological sciences, Teknik
pages
130 pages
publisher
Department of Mechanical Engineering, Lund University
defense location
Room M:B, M-building, Ole Römers väg 1, Faculty of Engineering, Lund University
defense date
2006-06-14 10:15
external identifiers
  • Other:ISRN:LUTFD2/TFHF-06/1035-SE(1-130)
ISBN
91-628-6853-6
language
English
LU publication?
yes
id
e96fef59-f242-465b-91a8-8807b98a7101 (old id 546963)
date added to LUP
2007-09-06 16:22:25
date last changed
2016-09-19 08:45:09
@phdthesis{e96fef59-f242-465b-91a8-8807b98a7101,
  abstract     = {High accuracy and low computational costs are essential properties for efficient<br/><br>
<br/><br>
finite element codes. Improvements of the finite element method in order to<br/><br>
<br/><br>
satisfy these qualities have been the subject for extensive research activities<br/><br>
<br/><br>
ever since its introduction. In this thesis, a class of improved low-order finite<br/><br>
<br/><br>
elements is proposed which possesses high accuracy in bending even for coarse<br/><br>
<br/><br>
meshes. The elements are insensitive to material incompressibility and they are<br/><br>
<br/><br>
all based upon the Hu-Washizu variational principle. A 4-node plane<br/><br>
<br/><br>
quadrilateral element as well as a 8-node brick element are proposed and also a<br/><br>
<br/><br>
4-node axisymmetric element is presented.<br/><br>
<br/><br>
The plane element was first formulated by the three-field Hu-Washizu principle,<br/><br>
<br/><br>
but later it was realized that the same response could be achieved by the<br/><br>
<br/><br>
assumed strain method. This reformulation leads, among other things, to gained<br/><br>
<br/><br>
computational efficiency. Consequently, the subsequent axisymmetric element and<br/><br>
<br/><br>
the brick element were established using the assumed strain method.<br/><br>
<br/><br>
The assumed strain<br/><br>
<br/><br>
method can be systematically formulated within the framework of the Hu-Washizu<br/><br>
<br/><br>
principle which makes the elements variational consistent providing that certain<br/><br>
<br/><br>
conditions are fulfilled. Due to its simplicity, the assumed strain method is<br/><br>
<br/><br>
highly rewarding in large-scale analysis and since a strain-driven format is<br/><br>
<br/><br>
obtained naturally which resembles the standard displacement method,<br/><br>
<br/><br>
this method is well suited for nonlinear analysis. For<br/><br>
<br/><br>
simplicity, however, only linear problems are considered here.<br/><br>
<br/><br>
The approach presented in the appended papers is based upon the fundamental<br/><br>
<br/><br>
idea in which the element stiffness matrix is divided into two parts. These<br/><br>
<br/><br>
parts are denoted fundamental stiffness and higher-order stiffness. The fundamental stiffness<br/><br>
<br/><br>
is the part which guarantees that the element is convergent. This means that the<br/><br>
<br/><br>
element fulfills the convergence criterion and it can be shown that all<br/><br>
<br/><br>
elements that are proposed<br/><br>
<br/><br>
pass the patch test {itshape a priori}. The higher-order stiffness<br/><br>
<br/><br>
must restore the correct rank to the total element stiffness matrix. In<br/><br>
<br/><br>
addition, the higher-order stiffness is chosen such that it significantly<br/><br>
<br/><br>
improves the accuracy. By consistent modifications of the higher-order<br/><br>
<br/><br>
stiffness,<br/><br>
<br/><br>
the proposed finite elements provide the exact strain energy for pure bending of<br/><br>
<br/><br>
regular elements. Sensitivity<br/><br>
<br/><br>
to mesh distortion is also effectively reduced by use of dimensionless<br/><br>
<br/><br>
parameters<br/><br>
<br/><br>
which emerge naturally within this approach which includes use of a coordinate<br/><br>
<br/><br>
system aligned with the principal axes of inertia.},
  author       = {Fredriksson, Magnus},
  isbn         = {91-628-6853-6},
  keyword      = {'quadrilateral','variational','finite element','element technology','HuWashizu','hexahedral',Technological sciences,Teknik},
  language     = {eng},
  pages        = {130},
  publisher    = {Department of Mechanical Engineering, Lund University},
  school       = {Lund University},
  title        = {Class of Accurate Low Order Finite Elements},
  year         = {2006},
}