Class of Accurate Low Order Finite Elements
(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:
https://lup.lub.lu.se/record/546963
- author
- Fredriksson, Magnus LU
- supervisor
- opponent
-
- Professor Skallerud, Björn, Dept. Structural Engineering, Norwegian University of Science and Technology
- organization
- publishing date
- 2006
- 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 10:15:00
- 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
- 2016-04-04 11:18:56
- date last changed
- 2018-11-21 21:04:02
@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}},
keywords = {{'quadrilateral'; 'variational'; 'finite element'; 'element technology'; 'HuWashizu'; 'hexahedral'; Technological sciences; Teknik}},
language = {{eng}},
publisher = {{Department of Mechanical Engineering, Lund University}},
school = {{Lund University}},
title = {{Class of Accurate Low Order Finite Elements}},
year = {{2006}},
}