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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 4313—4339 (1997)
CLASSIFICATION OF STRESS MODES IN ASSUMED
STRESS FIELDS OF HYBRID FINITE ELEMENTS
W. FENG1, S. V. HOA1* AND Q. HUANG2
1Department of Mechanical Engineering, Concordia ºniversity, 1455 de Maisonneuve Blvd. ¼., H549,
Montreal, Que., Canada H3G 1M8
2 Shanghai ºniversity, Shanghai, China
ABSTRACT
A classification method is presented to classify stress modes in assumed stress fields of hybrid finite element
based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there
only exist m ("n!r) natural deformation modes in a hybrid finite element which has n degrees of freedom
and r rigid-body modes. For a hybrid element, stress modes in various assumed stress fields proposed by
different researchers can be classified into m stress mode groups corresponding to m natural deformation
modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the flexibility matrix [H] is a diagonal matrix, the classification of stress
modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress
modes that are interchangeable in an assumed stress field and do not cause any kinematic deformation
modes in the element. A necessary and sufficient condition for avoiding kinematic deformation modes in
a hybrid element is also presented. By means of the m classified stress mode groups and the necessary and
sufficient condition, assumed stress fields with the minimum number of stress modes can be constructed and
the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress field can be
constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node
solid element are discussed. ( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
No. of Figures: 0.
No. of Tables: 2.
No. of References: 23.
KEY WORDS: finite element; stress modes; classification
1. DEFINITION
In order to clarify the presentation, it is useful to give the definitions of the stress field, stress
mode, stress matrix, natural deformation mode, rigid-body mode, zero-energy stress mode,
kinematic deformation mode and stress mode group firstly.
Stress field: In the hybrid formulation, a stress field is assumed independently from beginning,
MpN"[P]MbN
(1)
* Correspondence to: S. V. Hoa, Department of Mechanical Engineering, Concordia University, 1455 de Maisonneuve
Blvd. W., H 549 Montreal, Quebec H3G 1M8, Canada
CCC 0029—5981/97/234313—27$17.50
( 1997 John Wiley & Sons, Ltd.
Received 27 December 1995
Revised 8 April 1997
4314
W. FENG, S. V. HOA AND Q. HUANG
For example, for a 2-D, 4-node plane element, one of the assumed stress fields is





b 
1 1 0 0 [x  1 
p 
x 
b
p " 1 [1 0 [y
0
 2
y 
 F 
p 
0 0 1 x
y
xy
b 
5
(2a)
and one of the assumed stress fields for a 3-D, 8-node solid element is









p 
x
p 
y
p 
z "
p 
xy
p 
yz 
p 
zx
1
1
[1 0 0 0 z 0 y
z
1 [1 [1 0 0 0 z x 0 [z
0
y
0
0
0
yz
0
0
x
0
0
0
0
0
zx
0
[x [y 0
0
0
0
0
xy
z
z
0
0
0
1
0
2
0 0 0 0 x y
0
0
0
0
1 0 0 0 0 0
0
0
0
z
0
0
0
0 1 0 0 0 0
0
0
0
x [x
[x
0
0
0
0
0
0
0 0 1 0 0 0
0
0
0
y
[2y
0
0
0


]


b 
1
b 
2
F 
b 
18
0
(2b)
They can be expressed in the form


MpN"[Mp N Mp N . . . Mp N] 
1
2
m


"[P ]MbN
b 
1
b 
2
F 
b 
m
(2c)
In which, the parameters b are the corresponding stress parameters and [P] is the stress matrix.
i
Stress modes: Stress modes are vectors which are functions of the co-ordinates. For example,
vectors Mp N in equation (2c) are stress modes.
i
Stress matrix: An assumed stress matrix consists of several stress modes. In equation (2c), the
matrix [P] is a stress matrix.
Natural deformation modes: Deformation modes in an element that are independent from each
other. In this work, the eigenvectors of an element stiffness matrix are regarded as the natural
deformation modes of the element.
Rigid-body modes: Rigid-body modes are displacement modes in an element that do not
produce deformation energy.
Zero-energy stress modes: Stress modes in the stress matrix of a hybrid element that do not
produce deformation energy. The eigenvalues of the element stiffness matrix corresponding to
these stress modes equal zero. These zero-energy stress modes correspond to rigid-body modes.
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4315
Kinematic deformation modes: Deformation modes in an element corresponding to spurious
zero stiffness may be caused by unsuitable numerical integration technique or unsuitable assumed
stress fields. In this work, it is assumed that suitable numerical integration technique is used. Even
though zero-energy stress modes may not appear in the stress matrix, spurious zero stiffness can
occur. This happens when more than one stress mode is picked from one eligible stress mode
group. This zero stiffness mode is called kinematic deformation mode in this work. The exception
for this case is when the stress modes are interchangeable between different stress mode groups.
This exception is discussed in Section 6.
Stress mode group: A stress mode group contains many stress modes that are interchangeable
in the stress matrix [P] and do not cause kinematic deformation modes. The m stress mode
groups correspond to m natural deformation modes and the zero-energy stress mode group
corresponds to rigid-body modes.
If an element has n degrees of freedom, its nodal displacement vector must consist of
n components. The displacement distribution in the element can be represented by m ("n!r)
natural deformation modes and r rigid-body modes. In this work, it will be shown that the
m natural deformation modes can classify stress modes in various assumed stress fields for
a hybrid element. All existing stress modes are classified into m ("n!r) stress mode groups
corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes in the element.
For a hybrid element, the displacement field is assumed,
MuN"[N]MdN
(3)
where [N] is the shape functions and MdN is a nodal displacement vector. The hybrid element
stiffness matrix [K] in nodal displacement space can be obtained by using Hellinger—Reissner
variational principle,
[K]"[G]T[H]~1[G]
(4)
[H]MbN"[G]MdN
(5)
PV[P]T[S][P] d»
[G]" [P]T[B] d»
P
(6)
and
where
[H]"
V
in which, [S] is the material properties matrix relating stresses to strains, [B] is the geometric
matrix relating strains to displacements, [H] is the flexibility matrix, and [G] is the leverage
matrix.
2. INTRODUCTION
Since the time the hybrid finite element was developed by Pian1 in 1964, various hybrid finite
elements have been proposed.2,3 In the development of hybrid finite elements, a major problem
has obstructed the application of this method. That is the lack of a rational way for deriving
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4316
W. FENG, S. V. HOA AND Q. HUANG
the optimal assumed stress modes. Recently, several partial hybrid finite elements have been
proposed4~6 and the problem concerning the method for deriving assumed stress field stands
out. If an assumed stress field does not contain enough stress modes, the rank of the element
stiffness matrix will be less than the total degrees of deformation freedom and the numerical
solution of the finite element model will not be stable. It is possible to suppress kinematic
deformation modes by adding stress modes of higher-order term, but this cannot guarantee that
all kinematic deformation modes are suppressed. Moreover, each extra term will add more
stiffness7 and overuse of stress modes will cost more computational time because the calculation
of element stiffness matrix requires inversion of the flexibility matrix. Therefore, m ("n!r)
least-order stress modes are considered to be best and are optimal with respect to computer
resources.7,8 In this work, the method to derive this kind of assumed stress field will be given
using the classified stress mode groups and the necessary and sufficient condition for avoiding
kinematic deformation modes in a hybrid element.
Some mathematical basis for the stability of the numerical solution of the finite element model
has been established and a number of approaches for obtaining the optimal stress modes have
been proposed. Fraeijs de Veubeke,9 Pian and Tong10 presented a necessary condition to avoid
kinematic deformation modes,
m@*n!r
(7)
in which m is the total number of stress modes in an assumed stress field, n is the total number of
nodal displacements, and r is the number of rigid-body modes in an element. Brezzi,11 Babuska
et al.12 presented necessary and sufficient conditions for stability and convergence of a hybrid
element. However, it is difficult to use these conditions because of the abstract concept and the
complex analysis involved.
Ahmad and Irons13 suggested use of an eigenvalue technique to assess a hybrid element and to
determine the kinematic modes. Huang14 introduced the concept of natural deformation mode
and natural stress mode and developed a modal analysis technique to find natural stress modes
for hybrid elements. He gave a set of uncoupled stress modes yet without zero-energy stress
modes by means of modal analysis of 3-D, 8-node hybrid element. Using group theory, Punch
and Atluri8,15 tried to solve the zero-energy mode problem. They gave two assumed stress fields
for 2-D, 4-node hybrid element, eight assumed stress fields for 3-D, 8-node hybrid element and
384 assumed stress fields for 3-D, 20-node hybrid element. Pian and Chen16 used the product
MpNTMeN, the deformation energy due to the assumed stresses and displacements, to determine the
necessary assumed stress modes. The assumed stress fields for 2-D, 4-node plane element and 3-D,
8-node solid element were provided.
Pian and Tong17 introduced the internal displacement parameters to relax the stress equilibrium condition and used isoparametric interpolation to construct hybrid element. Pian and Wu18
introduced incompatible displacements to maintain completeness. The initial choice of stress
terms are unconstrained and complete polynomials. The additional displacements are used as
Lagrange multipliers to enforce the stress equilibrium constraint. Chen et al.19,20 constrained the
stress by setting the inner product of the non-constant stress modes with the deformation derived
from the incompatible displacement to zero. Sze21 used orthogonal lower- and higher-order
stress modes to construct hybrid element. It allows the partition of the element stiffness matrix
into a lower- and a higher-order stiffness matrix. When the lower-order stiffness turns out to be
identical to the sub-integrated element, the higher-order stiffness matrix plays the role of
stabilization matrix.
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4317
For the hybrid element, an assumed stress field can be constructed by various approaches.22
One type of hybrid element usually has many different assumed stress fields. However, the
relationship among them has not been investigated although different approaches to assume
stress field of hybrid elements have been proposed. In this work, a classification method is
presented to classify stress modes in various assumed stress fields corresponding to the 2-D,
4-node plane element and the 3-D, 8-node solid element. It will reveal the relationship between
different assumed stress fields and answer the question why there exist kinematic deformation
modes in a hybrid element even when the criterion (m@'n!r) is satisfied. Also, a systematic
procedure is given for constructing assumed stress fields of hybrid elements that contain
minimum number of stress modes and are free from kinematic deformation modes.
3. CLASSIFICATION METHOD
The hybrid formulation based on the Hellinger—Reissner principle relaxes the stress equilibrium
condition. The stress field would satisfy the equilibrium equations only in a variational sense.
Therefore, the stress field may be described in the isoparametric co-ordinate system of the
element, which would make the element less sensitive to mesh distortion. The use of isoparametric
co-ordinates also preserves the invariance of the resulting stiffness matrix. There are many
assumed stress fields for 2-D, 4-node plane element and 3-D, 8-node solid element. For example,
Pian16 proposed an assumed stress field for 2-D, 4-node plane element and another for 3-D,
8-node solid element. Punch and Atluri8,15 gave two assumed stress fields for 2-D, 4-node plane
element, and eight assumed stress fields for 3-D, 8-node solid element. Huang14 presented an
assumed stress field for 3-D, 8-node solid element. Although each of these assumed stress fields
may contain the same number of stress modes, the stress modes in these fields are different.
In order to study the relationship between different stress modes, the concept of natural
deformation modes is used.
A finite element has a finite number of degrees of freedom. For instance, a 2-D, 4-node
displacement element has (n") 8 degrees of freedom, and a 3-D, 8-node displacement element has
(n") 24 degrees of freedom. In an element, there exist (n!r) natural deformation modes and
r rigid-body modes. The displacement distribution in the element can be represented by them.
The equilibrium equation of a displacement element is
[K]MdN"MFN
(8)
If the nodal force vector is proportional to the nodal displacement vector, the equilibrium
equation becomes a eigenvalue equation. It can be expressed as follows:
( [K]!j[I])MdN"0
(9)
[K] is an n]n element stiffness matrix. This equation will give (n!r) non-zero eigenvalues and
r zero eigenvalues, and (n!r) eigenvectors corresponding to the (n!r) non-zero eigenvalues.
The (n!r) eigenvectors MdN (i"1, 2, 3, . . . , m) depend only on the geometry and material
i
properties of the element, and they are unique. If vectors Md N (i"1, 2, . . . , m) are the eigenveci
tors of the stiffness matrix [K], they must satisfy the condition:
Md NTMd N"0, iOj
i
j
(10)
Md NTMd N"1, i"j
i
j
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4318
W. FENG, S. V. HOA AND Q. HUANG
In this work, these eigenvectors are considered to be the natural deformation modes of the
element. They can represent the deformation in the element. The strain modes derived from the
assumed displacement directly also can represent the deformation in the element, but they cannot
classify stress modes in hybrid element. Moreover, the strain modes cannot provide a general way
to derive necessary stress modes for all types of hybrid elements. For example, they can be used to
derive a set of stress modes for conventional hybrid elements, but they cannot be used to derive
stress modes for partial hybrid element.
In the hybrid element, the eigenvectors and eigenvalues of the stiffness matrix will be sensitive
to the assumed stress modes. The eigenvalue examination will give r zero eigenvalues corresponding to rigid-body mode and m ("n!r) non-zero eigenvalues corresponding to natural deformation modes if the assumed stress field is suitable. For a hybrid element, if various stress modes can
be classified, at least m stress mode groups must exist because the stiffness matrix of hybrid
element must have m non-zero eigenvalues, except zero-energy stress mode group. Otherwise, the
hybrid element will contain kinematic deformation modes. On the other hand, no matter how
many stress modes there are in a stress matrix [P], the maximum number of non-zero eigenvalues
of an element stiffness matrix is always equal to or less than m. Therefore, the number of stress
mode groups is also equal to or less than m. Thus, it can be considered that there exist and only
exist m stress mode groups except zero-energy modes. All stress modes in various assumed stress
matrices can be classified into the m stress mode groups corresponding to m natural deformation
modes and the zero-energy stress mode group corresponding to rigid-body modes.
Postulate: There exist and only exist m ("n!r) natural deformation modes in a hybrid
element. All stress modes in assumed stress matrices can be classified into m stress mode groups
corresponding to m natural deformation modes and a zero energy mode group corresponding to
rigid-body modes of the element which has n degrees of freedom and r rigid-body modes.
Based on this postulate, it can be considered that an assumed stress field can be represented by
stress modes in the m stress mode groups related to m natural deformation modes, except
zero-energy stress modes. This can be expressed as follows:
 b1 
b 
n1
n2
nm
m
MpN"[P]MbN" + Mp N, + Mp N, . . . , + Mp N  2  " + [P ]Mb N
l1
l2
lm
i
i
F 
l1/1
l2/1
lm/1
i/1
b 
C
D
(11)
m
where n is the number of stress modes in the ith stress mode group, and [P ] and Mb N
i
i
i
(i"1, 2, . . . , m) are the stress matrices and stress-coefficient vectors related to the ith stress mode
group which corresponds to the ith natural deformation mode. They are
C
ni
[P ]" M0N . . . M0N, + Mp N, M0N . . . M0N
li
i
li/1
D
(12)
and
Mb N"[0 . . . 0 b 0 . . . 0]T
(13)
i
i
The summation sign in the matrix [P] is not necessary to be carried out. One can have only one
stress mode or many. Many stress modes will increase the size of stress matrix [P], and this is not
desirable. The stress mode which belongs to the ith stress mode group can be expressed in the
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4319
form
Mp N"[P]Mb N
i
i
(14)
Therefore, the vector Mb N represents the ith stress mode group which corresponds to the natural
i
deformation mode Md N (i"1, 2, . . . , m). Using equation (5), we have
i
[H]Mb N"[G]Md N
i
i
(15)
If the stress matrix [P] does not contain any stress mode which belongs to the ith stress mode
group, the value of b in the vector Mb N should be equal to zero. Then, we can add a new stress
i
i
mode into the stress matrix [P]. The new stress mode will be classified by m natural deformation
modes. Corresponding to the ith natural deformation mode Md N, the condition to check whether
i
the new stress mode belongs to the ith stress mode group can be expressed in the form,
b "0 if new stress mode does not belong to ith stress mode group
i
b O0 if new stress mode belongs to ith stress mode group
i
Using equations (9) (10) and (15), the eigenvalues are obtained as follows:
j "Md NT [K]Md N"Mb NT[H]Mb N
i
i
i
i
i
(16)
Because all of the diagonal elements in the flexibility matrix [H] may not be equal to zero, the
classification condition above becomes
j "0 if new stress mode does not belong to ith stress mode group
i
j O0 if new stress mode belongs to ith stress mode group
i
Before classifying stress modes, one can find a number of initial stress modes since there are
many ways to derive assumed stress matrices for a hybrid element. For example, Pian and Chen16
used the product MpNTMeN to determine the necessary assumed stress modes, and gave an assumed
stress matrix for 2-D, 4-node plane element and another for 3-D, 8-node solid element. Punch and
Atluri8,15 used group theory to obtain two assumed stress matrices for 2-D, 4-node hybrid
element, eight assumed stress matrices for 3-D, 8-node hybrid element, and 384 assumed stress
matrices for 3-D, 20-node hybrid element. One can also derive an assumed stress matrix using
iso-function method.23 In the stress matrix derived using iso-function method, the number of the
stress modes is larger than m ("n!r).
In order to present a systematic procedure for classifying stress modes and constructing
assumed stress fields, the iso-function method23 is used to derive initial stress modes to be
classified in this work. This is because the hybrid element constructed by the iso-function stress
matrix has the same eigenvalues and eigenvectors as conventional displacement element.
Also, the method using iso-function is straightforward and can be followed easily. After obtaining initial stress modes, one can use eigenvalue examination to find m representative stress
modes that represent m stress mode groups corresponding to m natural deformation modes. The
stress matrix consisted of the m representative stress modes is a optimal stress matrix. Then, all
existing stress modes can be classified into m#1 stress mode groups. Its detail is presented as
follows.
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4320
W. FENG, S. V. HOA AND Q. HUANG
3.1. Determination of optimal stress matrix from the stress matrix derived by iso-function method
Step 1. Derive an initial stress matrix [P] by iso-function method. The number of initial
*40
stress modes in the matrix is always larger than m ("n!r). In order to select m necessary stress
modes, these initial stress modes have to be classified into (n!r) stress mode groups.
Step 2. Select stress modes in the order from low-order term to high-order term. Now select
a stress mode from the existing stress matrix [P] and form an assumed stress matrix [P ]. The
*40
1
element stiffness matrix [K] corresponding to stress matrix [P ] can be obtained by using
1
equations (4) and (6). If the eigenvalue examination gives a non-zero eigenvalue, the stress mode is
a non-zero-energy stress mode; otherwise, it is a zero-energy stress mode. Repeating the eigenvalue examination to check whether a stress mode is a zero- energy stress mode for all stress
modes in the existing stress matrix [P] .
*40
Take all zero-energy stress modes out and keep non-zero-energy stress modes in the matrix
[P] . All zero-energy stress modes form a zero-energy stress mode group.
*40
Step 3. Take a non-zero-energy stress mode from the existing stress matrix [P] and form an
*40
assumed stress matrix [P ]. The stress mode Mp N is the representative stress mode which
1
1
represents group 1 of stress modes.
Step 4. Add another stress mode selected in the existing stress matrix [P] into the assumed
*40
stress matrix [P ] and form a new stress matrix [P ],
1
2
[P ]"[Mp N Mp N]
2
1
2
(17)
Step 5. The eigenvalue examination gives the eigenvalues of the stiffness matrix. If there is only
one non-zero eigenvalue, continue to step 6. If there are two non-zero eigenvalues, go to step 7.
Step 6. In this case, the added stress mode belongs to group 1 of stress modes. Take the second
stress mode out and put it in group 1 of stress modes. Then, go back to step 4.
Step 7. The two stress modes belong to two different groups of stress modes. The second stress
mode Mp N is the representative stress mode which represents group 2 of stress modes.
2
Step 8. Add another stress mode selected from the matrix [P] into the assumed stress matrix
*40
[P ] and form a new stress matrix [P ],
2
3
[P ] " [Mp N Mp N Mp N]
3
1
2
3
(18)
Step 9. The element stiffness matrix [K] and its eigenvalues are calculated. If there are only
two non-zero eigenvalues, continue to step 10. If there are three non-zero eigenvalues, go to
step 11.
Step 10. In this case, the new stress mode Mp N belongs to one of the two stress mode groups.
3
Construct the matrices [P@ ] and [P@@ ] as follows:
2
2
[P@ ]"[Mp N Mp N] or [PA ]"[Mp N Mp N]
2
2
3
2
1
3
(19)
If the stiffness matrix corresponding to the stress matrix [P@ ] has two non-zero eigenvalues, the
2
stress mode Mp N belongs to the group 2 of stress modes. Otherwise, the stress mode Mp N belongs
3
3
to the group 1 of stress modes. Put the stress mode Mp N into the corresponding stress mode
3
group, and go back to step 8.
Step 11. In this case, the three stress modes belong to three different stress mode groups. The
added stress mode Mp N is the representative stress mode which represents group 3 of stress
3
modes.
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4321
Step 12. Add one more stress mode selected from the matrix [P] into the matrix [P ] and
*40
3
form a new stress matrix [P ],
4
[P ]"[Mp N Mp N Mp N Mp N]
4
1
2
3
4
(20)
and so on. Repeating the same process until m representative stress modes that represent m stress
mode groups are obtained. The m("n!r) representative stress modes correspond to m natural
deformation modes and form a optimal stress matrix [P] from the existing stress matrix [P] .
015
*40
Classification of other stress modes
Step 13. After m representative stress modes are obtained, other initial stress modes that
remain in the existing stress matrix [P] can be classified into the m stress mode groups. Many
*40
other stress modes derived by different methods also can be classified into the m stress mode
groups corresponding to m natural deformation modes and the zero-energy stress mode group
corresponding to rigid-body modes.
Based on the optimal stress matrix [P] , any remaining stress mode in [P] can be classified
015
*40
by using it to replace each and every stress mode in the matrix [P] in order. Once the
015
eigenvalue examination results in m non-zero eigenvalues, the representative stress mode which is
replaced and the remaining stress mode which replaces the stress mode in [P] belong to the
015
same stress mode group. Put the remaining stress mode into the corresponding stress mode group
and recover the optimal stress matrix [P] . Then, classify another remaining stress mode.
015
Step 14. Repeating the same process until all remaining stress modes are classified. Thus, all
existing stress modes are classified into m#1 different mode groups. Every stress mode group
contains many interchangeable stress modes. For a stress mode derived by other method,
if eigenvalue examination always gives m!1 non-zero eigenvalues when this stress mode
replaces each and every stress mode in the matrix [P] , this stress mode is a zero- energy
015
stress mode.
4. ILLUSTRATION FOR THE CLASSIFICATION OF STRESS MODES
As an illustration for the above procedure, the stress modes presented in References 8, 14—16 and
those derived by iso-function method are classified.
4.1. 2-D, 4-node plane hybrid element
4.1.1. Determination of optimal stress matrix from the existing stress matrix derived by isofunction method. The 2-D, 4-node plane element has (n") 8 degrees of freedom and (r")
3 rigid-body modes. So it has (m"n!r") 5 natural deformation modes. Firstly, an assumed
stress matrix can be derived from the assumed displacement field of the element by the isofunction method.23
1 0 0 x y 0
[P ]" 0 1 0 0
I
0 0 1 0
( 1997 John Wiley & Sons, Ltd.
0 0
0
0 x y 0
0
0 0
(21)
0 x y
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4322
W. FENG, S. V. HOA AND Q. HUANG
The number of stress modes in the stress matrix is larger than m ("5). The eigenvalue examination indicates that the eigenvalues and eigenvectors of the hybrid element stiffness matrix
constructed by the assumed stress matrix [P ] are the same as that of displacement element
I
stiffness matrix. Here, we take the stress modes in the stress matrix as initial stress modes to be
classified. There are nine stress modes in the matrix [P ],
I
1
0
0
Mp N" 0  , Mp N" 1  ,
1
2
 
 
0
0




Mp N" 0 
3
 
1


x

Mp N" 0
4

0


,




Mp N"
5


y
0

,




Mp N"
6


0
0

Mp N" y
7

0


,


0

Mp N" 0
8

x


,




Mp N"
9


0

x
0


(22)
0

0

y
The stress matrix derived by iso-function method contains a few unnecessary stress modes. It
will save computation time for calculating element stiffness matrix if the number of the stress
modes can be reduced to m ("n!r). In order to do it, the initial stress modes in the existing
stress matrix have to be classified into m stress mode groups. First of all, one must find
m representative stress modes corresponding to m natural deformation modes. Following steps
2—12 in the procedure of the classification method given in the above section, one can obtain five
representative stress modes Mp p p p p N corresponding to (m") of 5 natural deformation
1 2 3 5 6
modes and the zero-energy stress modes Mp N and Mp N corresponding to rigid-body modes. The
4
7
eigenvalues of the stiffness matrix related to Mp p p p p N are not equal to zero, and the
1 2 3 5 6
eigenvalue of stiffness matrix related to Mp N or Mp N is equal to zero. The 5 representative stress
4
7
modes form an optimal stress matrix [P ] from the existing stress matrix [P ],
II
I
1 0 0 y 0
[P ]"[p p p p p ]" 0 1 0 0 x
II
1 2 3 5 6
0 0 1 0 0
(23)
The stress matrix is the same as that given by Pian.16
4.1.2. Classification of other stress modes. After obtaining the optimal stress matrix, one can
classify stress modes in the existing stress matrix [P ] into (m#1") 6 stress mode groups by
I
following steps 13 and 14 in the procedure,
Tension mode (Group 1): Mp N
1
Tension mode (Group 2): Mp N
2
Shear mode (Group 3): Mp N
3
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4323
Bending mode (Group 4): Mp N, Mp N
5
8
Bending mode (Group 5): Mp N, Mp N
6
9
Zero-energy stress mode (Group 6): Mp N, Mp N
4
7
The first five stress mode groups correspond to five natural deformation modes and the zeroenergy stress mode group corresponds to rigid-body modes.
There are many methods to derive initial stress modes. For example, in the two assumed stress
matrices derived by means of group theory8,15 for the same finite element, there are four stress
modes that are different from stress modes Mp N—Mp N:
1
9


Mp N"
10


1
1
0

,


 1

Mp N" [1
11

 0


,


 0

Mp N" [y
12

 x


,


 !x 


Mp N" 0 
13


 y 
(24)
Moreover, one may want to introduce some stress modes of high-order terms into the assumed
stress matrix [P] in order to describe special stress distribution in a local region of a structure to
be solved. For example,
 x2

Mp N" 0
14

 0


,


 0

Mp N" x2
15

 0


,


 0

Mp N" 0
16

 x2





 y2

Mp N" 0
17

 0


,


 0

Mp N" y2
18

 0


,


 0

Mp N" 0
19

 y2





 xy

Mp N" 0
20

 0


,


 0

Mp N" xy
21

 0


,




Mp N"
22


(25)


0 

xy 
0
According to the steps 13 and 14 in the procedure of classification method, these new stress modes
Mp N—Mp N can also be classified into the 6 stress mode groups above,
10
22
Tension mode (Group 1): Mp N, Mp N, Mp N, Mp N
1
10
14
17
Tension mode (Group 2): Mp N, Mp N, Mp N, Mp N
2
11
15
18
Shear mode (Group 3): Mp N, Mp N, Mp N
3
16
19
Bending mode (Group 4): Mp N, Mp N, Mp N
5
8
12
Bending mode (Group 5): Mp N, Mp N, Mp N
6
9
13
Zero-energy stress mode (Group 6): Mp N, Mp N, Mp N, Mp N, Mp N
4
7
20
21
22
More high-order stress modes can be classified into the 6 stress mode groups above by using
the classification method. If the flexibility matrix [H] is a diagonal matrix, the classification of the
stress modes is unique (see Section 6).
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4324
W. FENG, S. V. HOA AND Q. HUANG
4.2. 3-D, 8-node solid hybrid element
4.2.1. Determination of optimal stress matrix from the existing stress matrix derived by isofunction method. The 3-D, 8-node solid element has (n") 24 degrees of freedom and (r")
6 rigid-body modes. So it has (m"n!r") 18 natural deformation modes. By means of
iso-function method,23 an initial stress matrix [P] can be derived from the assumed displacement
field of the element as follows:
z
1 0 0 0 0 0 x
0
0
0
0
0
y 0 0 0 0 0
0 1 0 0 0 0
0
x
0
0
0
0
0 y 0 0 0 0 0
0 0 1 0 0 0
[P] "
ISO
0 0 0 1 0 0
0
0
x
0
0
0
0 0 y 0 0 0 0 0
0
0
0
x
0
0
0 0 0 y 0 0 0 0 0
0 0 0 0 1 0
0
0
0
0
x
0
0 0 0 0 y 0 0 0 0 0
0 0 0 0 0 1
0
0
0
0
0
x
0 0 0 0 0 y
0 0 0 0 0
z
0 0 0 0
z
0 0 0
z
0 0
z
0
0 0 0 0 0
z
xy
0
0
0
0
yz
0
0
0
0
zx
0
0
0
0
0
xy
0
0
0
0
yz
0
0
0
0
zx
0
0
0
0
0
xy
0
0
0
0
yz
0
0
0
0
zx
0
0
0
0
0
0
0
0
0
0
yz
0
0
0
0
zx
0
0
0
0
xy
0
0
0
0
0
0
0
0
0
0
zx
0
0
0
0
xy
0
0
0
0
yz
0
0
0
0
0
(26)
There are 39 stress modes to be classified in the stress matrix. The number of stress modes is
larger than m ("18). The eigenvalue examination shows that the eigenvalues and eigenvectors of
the hybrid element stiffness matrix corresponding to the assumed stress matrix [P] are the
ISO
same as that of displacement element stiffness matrix. We take the stress modes in the matrix
[P] as initial stress modes to be classified. The 39 stress modes in the matrix [P] are
ISO
ISO
numbered as follows:
1

0
0
Mp p p p p p N"
1 2 3 4 5 6
0
0

0
0 0 0 0 0

1 0 0 0 0
0 1 0 0 0

0 0 1 0 0
0 0 0 1 0

0 0 0 0 1
x 0

0 x
0 0
Mp p p p p p N"
7 8 9 10 11 12
0 0
0 0

0 0
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
0

0 0 0 0
x 0 0 0

0 x 0 0
0 0 x 0

0 0 0 x
0
0
(27)
0
(28)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4325
y

0
0
Mp p p p p p N"
13 14 15 16 17 18
0
0

0
0 0 0 0 0

y 0 0 0 0
0 y 0 0 0

0 0 y 0 0
0 0 0 y 0

0 0 0 0 y
(29)
z

0
0
Mp p p p p p N"
19 20 21 22 23 24
0
0

0
0 0 0 0 0

z 0 0 0 0
0 z 0 0 0

0 0 z 0 0
0 0 0 z 0

0 0 0 0 z
(30)
 xy 0

 0 xy
 0
0
Mp p p p p N"
25 26 27 28 29
0
 0
 0
0

0
 0
0 

0
0
0 
xy 0
0 

0
0
0 
0 xy 0 

0
0 xy 
0
0
(31)
 yz 0 0 0 0 


 0 yz 0 0 0 
 0 0 yz 0 0 
Mp p p p p N"

30 31 32 33 34
 0 0 0 yz 0 
 0 0 0 0 0 


 0 0 0 0 yz 
(32)
 zx 0

 0 zx
 0 0
Mp p p p p N"
35 36 37 38 39
 0 0
 0 0

 0 0
(33)
0 

0 0 0 
zx 0 0 

0 zx 0 
0 0 zx 

0 0 0 
0
0
In order to reduce the number of stress modes in the assumed stress matrix [P] , these stress
ISO
modes have to be classified one by one in the order from low-order term to high-order term.
Following steps 2—12 in the procedure of the classification, one can obtain (m") 18 representative
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4326
W. FENG, S. V. HOA AND Q. HUANG
stress modes Mp p p p p p p p p p p p p p p p p p N corresponding to 18
1 2 3 4 5 6 8 9 11 13 15 18 19 20 22 27 30 36
natural deformation modes. These representative stress modes form a optimal stress matrix [P ]
1
from the existing stress matrix [P] as follows:
*40
[P ]"[p p p p p p p p p p p p p p p p p p ]
1
1 2 3 4 5 6 13 20 9 19 8 15 22 11 18 30 36 27
1 0 0 0 0 0 y 0
0 1 0 0 0 0 0
z
0 z
0 0
0
0 yz
0
0
0 0 x 0 0
0
0
0
zx
0
0
0 0 1 0 0 0 0 0 x 0
0
y 0
0
0
0
0
xy
0 0 0 1 0 0 0 0
0 0
0
0
z
0
0
0
0
0
0 0 0 0 1 0 0 0
0 0
0
0 0 x 0
0
0
0
0 0 0 0 0 1 0 0
0 0
0
0 0
y
0
0
0
"
0
(34)
This stress matrix is the same as that proposed by Pian.16
4.2.2. Classification of other stress modes. Following steps 13 and 14 in the procedure, other
stress modes that remain in the existing stress matrix [P] can be classified into m#1 ("19)
*40
stress mode groups as follows:
Tension and compressive modes (3 groups): [Mp N , Mp N , Mp N ]
1 G1
2 G2
3 G3
Pure shear modes (3 groups): [Mp N , Mp N , Mp N ]
4 G4
5 G5
6 G6
Bending modes (6 groups): [Mp p N , Mp p N , Mp p N ,
8 16 G7
9 24 G8
13 10 G9
Mp p N , Mp p N , Mp p N ]
15 23 G10
19 12 G11
20 17 G12
Torsion modes (3 groups): [Mp N , Mp N , Mp N ]
11 G13
18 G14
22 G15
Saddle modes (3 groups): [Mp p p N , Mp p p N , Mp p p N ]
29 30 38 G16
28 33 36 G17
27 34 39 G18
Zero-energy stress modes (1 group): [Mp , p , p , p , p , p , p , p , p N ]
7 14 21 25 26 31 32 35 37 G19
The first 18 stress mode groups correspond to (m"n!r") 18 natural deformation modes
and the last group corresponds to rigid-body modes. There are many other ways to derive initial
stress modes. For example, in the assumed stress matrix presented in Reference 14, there are 12
stress modes that are different from stress modes Mp N!Mp N. These stress modes can be
1
39
expressed as follows:
Tension and compressive modes:
1
 1 
 [1 
 




1
 [1 
 [1 
1
 0 
 2 
Mp N"  , Mp N"
 , Mp N"

40
41
42
0
 0 
 0 
0
 0 
 0 
 




0
 0 
 0 
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
(35)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4327
Symmetric bending modes:
z
 
z
0
Mp N"  ,
43
0
0
 
0
0
y
 
 
x
0
x
y
Mp N"  , Mp N" 
44
45
0
0
0
0
 
 
0
0
(36)
 0 


 x 
 [x 
Mp N"
,
47
 0 
 0 


 0 
(37)
Anti-symmetric bending modes:
 z 


 [z 
 0 
Mp N"
,
46
 0 
 0 


 0 
 y 


 0 
 !y 
Mp N"

48
 0 
 0 


 0 
Torsion modes:
0
 0 
 0 
 




0
 0 
 0 
0
 0 
 0 
Mp N"  , Mp N"
, Mp N"

49
50
51
z
 z 
 z 
x
 [x 
 x 
 




y
 0 
 !2y 
(38)
In the stress matrices derived by means of the symmetric group theory,8,15 there are 11 stress
modes that are different from stress modes Mp N—Mp N. They can be expressed in the form.
1
51
Tension and compressive mode:
 0 


 1 
 [1 
Mp N"

52
 0 
 0 


 0 
( 1997 John Wiley & Sons, Ltd.
(39)
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4328
W. FENG, S. V. HOA AND Q. HUANG
Torsion modes:
 0 


 0 
 0 
Mp N"

53
 0 
 [x 


 y 
(40)
Bending modes:
 2x
0
0 


2y
0 
 0
 0
0
2z 
Mp p p N"

54 55 56
0 
 [y [x
 0
[z [y 


0
[x 
 [z
 0
0
0 


0
0 
 0
 0
0
0 
and Mp p p N"

57 58 59
x
0 
 y
 0 [z [y 


x 
 [z 0
(41)
Saddle modes:


0
0
0


0
0
0




0
0
0
Mp p p N"

60 61 62
[2yz x2#y2 
 [2xz
 y2#z2 [2xy
[2xz 


 [2xy x2#z2 [2yz 
(42)
Other stress modes may be also needed in an assumed stress matrix in order to describe special
stress distribution in a local region of a structure to be analysed. For instance,
Bending modes:
z
 
z
z
Mp N"  ,
63
0
0
 
0
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
x
y
 
 
x
y
x
y
Mp N" , Mp N" 
64
65
0
0
0
0
 
 
0
0
(43)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4329
Saddle modes:
 yz 
 
 yz 
 yz 
Mp N" ,
66
 0 
 0 
 
 0 
 zx 
 xy 
 


 zx 
 xy 
 zx 
 xy 
Mp N" , Mp N"

67
68
 0 
 0 
 0 
 0 
 


 0 
 0 
(44)
 z2 
 
 z2 
 0 
Mp N" 
69
 0 
 0 
 
 0 
(45)
Tension and compressive mode:
According to steps 13 and 14 in the proposed procedure of classification, the 30 new stress
modes Mp N—Mp N can be classified into different stress mode groups above as follows:
40
69
Tension and compressive modes (3 groups): [Mp , p , p N , Mp , p N , Mp , p , p N ]
1 40 69 G1 2 41 G2 3 42 52 G3
Pure shear modes (3 groups): [Mp N , Mp N , Mp N ]
4 G4
5 G5
6 G6
Bending modes (6 groups): [Mp , p , p , p , p N , Mp , p , p , p N ,
8 16 44 54 64 G7
9 24 47 57 G8
Mp , p , p , p , p N , Mp , p , p , p N ,
13 10 45 55 65 G9
15 23 48 58 G10
Mp , p , p , p , p N , Mp , p , p , p N ]
19 12 43 56 63 G11
20 17 46 59 G12
Torsion modes (3 groups): [Mp , p N , Mp , p N , Mp , p , p N ]
11 49 G13
18 50 G14
22 51 53 G15
Saddle modes (3 groups): [Mp , p , p , p , p N , Mp , p , p , p , p N ,
29 30 38 66 60 G16
28 33 36 67 61 G17
Mp , p , p , p , p N ]
27 34 39 68 62 G18
Zero-energy stress modes (1 group): [Mp , p , p , p , p , p , p , p , p N ]
7 14 21 25 26 31 32 35 37 G19
More stress modes can be classified into the stress mode groups above. If the flexibility matrix
[H] is a diagonal matrix, the stress modes are uncoupled and the classification of the stress modes
is unique (see Section 6). Otherwise, some stress modes may appear in more than one group.
5. CONSTRUCTION OF ASSUMED STRESS MATRICES
As shown above, by means of the proposed procedure for the classification of stress mode, stress
modes can be classified into m ("n!r) stress mode groups corresponding to m natural
deformation modes and a zero-energy mode group corresponding to rigid-body modes. Each
natural deformation mode is related to a stress mode group except zero-energy mode group, and
each stress mode group may contain many different stress modes that are interchangeable in the
stress matrix [P].
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4330
W. FENG, S. V. HOA AND Q. HUANG
The classification of stress modes reveals the relationship among the different stress modes that
are used in the different stress matrices for any type of a hybrid element proposed by different
researchers. In order to avoid kinematic deformation mode, the stress matrix [P] must contain
m stress modes at least. No matter how many stress modes there are in the stress matrix [P], the
order of the stiffness matrix is equal to or less than m. Therefore, m stress modes are necessary and
sufficient to form a stress matrix for avoiding kinematic deformation modes in the hybrid element.
Moreover, in view of the classification of stress modes, the m stress modes in the stress matrix [P]
must come from m different stress mode groups except zero-energy mode group. Thus, for
a hybrid element to be free from kinematic deformation mode, we have
¹he necessary and sufficient condition:
The number of stress modes in an assumed stress matrix must be equal to or more than
m ("n!r) and at least m stress modes in the stress matrix [P] must be chosen from
m different stress mode groups corresponding to m natural deformation modes of an
element which has n degrees of freedom and r rigid-body modes.
In this statement, the necessary condition is that the number of stress modes for a hybrid
element must be equal to or more than m ("n!r). It was presented by Veubeke9 and Pian.10 The
sufficient condition is that the stress matrix [P] must contain m stress modes chosen from
m different stress mode groups corresponding to m natural deformation modes. This condition
explains why in some examples there exist kinematic deformation modes even when the necessary
condition (m@'n!r) is satisfied. In these examples, the stress modes in the stress matrix [P] do
not come from m different stress mode groups except the zero-energy mode group.
For a hybrid element, overuse of stress modes will result in over-rigid element,7 and will cost
more computational time because the calculation of element stiffness matrix requires an inversion
of the flexibility matrix [H]. Therefore, an assumed stress field, its stress matrix contains
m ("n!r) least-order stress modes and its resulting finite element is free from kinematic
deformation modes, is considered to be best and is optimal with respect to computer resources.7,8
By means of the m classified stress mode groups and the necessary and sufficient condition, this
kind of stress matrices can be constructed. Furthermore, it is convenient to construct an assumed
stress matrix according to the problem to be solved because there are many stress modes in every
stress mode group for choice. The procedure of constructing stress matrix is presented as follows:
1. Using the iso-function method,23 one can derive a number of initial stress modes to be
classified.
2. One may put the initial stress modes one by one into stress matrix [P] in the order from
low-order term to high-order term. By means of the classification method, one can obtain
m representative stress modes corresponding to m natural deformation modes. These
representative stress modes form a optimal stress matrix [P] from the existing stress
015
matrix [P] .
*40
3. One may obtain other initial stress modes derived by different methods. Following the steps
13 and 14 in the procedure of the classification, one can classify all initial stress modes into
m#1 different stress mode groups.
4. By means of the m#1 classified stress mode groups and the necessary and sufficient
condition above, many stress matrices [P] can be constructed according to the problem to
be solved. It is necessary to choose one stress mode at least from each group except the
zero-energy mode group in order to avoid kinematic deformation modes.
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
4331
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
The necessary steps have been illustrated in the section above. The following gives some
examples to illustrate the procedure for constructing a stress matrix [P] which has minimum
number of stress modes.
5.1. 2-D, 4-node plane element
By means of the m#1 stress mode groups classified above and the necessary and sufficient
condition for avoiding kinematic deformation modes, one can choose one stress mode from each
stress mode group except zero-energy mode group to form a stress matrix. For example,
1
1
0
[x
0
[P ]"[p p p p p ]" 1 [1 0 [y
III
10 11 3 12 13
0 0 1 x
0
(46)
y
and
1
0 y
1
0
[P ]"[p p p p p ]" 1 [1 0 0 x
IV
10 11 3 7 5
0 0 1 0 0
(47)
Five stress modes in each stress matrix come from five different stress mode groups corresponding to five natural deformation modes. The two stress matrices are the same as that proposed by
Atluri.8,15 More stress matrices can be constructed on purpose. For example,
1
0 y [x
1
[P ]"[p p p p p ]" 1 [1 0 0
V
10 11 3 7 13
0 0 1 0
0
(48)
y
and
x2
[P ]"[p p p p p ]" 0
VI
14 11 3 12 13
0
1
0
0
[1 0 [y
0
1
x
[x
0
(49)
y
The eigenvalue examination shows that the hybrid element constructed by [P ]—[P ] are free
I
VI
from kinematic deformation modes as shown in Table I. In the last column of the table, the
Table I. Eigenvalues of element stiffness matrices (2-D, 4-node plane element,
v"0·3)
[P ]
I
0·4945
0·4945
0·7692
0·7692
1·4290
[P ]
II
[P ]
III
[P ]
IV
[P ]
V
[P ]
VI
Disp.
0·3333
0·3333
0·7692
0·7692
1·4290
0·09259
0·09259
0·7692
0·7692
1·4290
0·3333
0·3333
0·7692
0·7692
1·4290
0·09259
0·3333
0·7692
0·7692
1·4290
0·09259
0·09259
0·7692
0·7692
1·4290
0·4945
0·4945
0·7692
0·7692
1·4290
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4332
W. FENG, S. V. HOA AND Q. HUANG
eigenvalues of displacement element stiffness matrix are given. If stress modes in a stress matrix
[P] come from m ((m) stress mode groups, the hybrid element will have kinematic deformation
1
modes even if the number of stress modes is larger than m. This is why a hybrid element contains
kinematic deformation modes when the necessary condition (m@'n!r) is satisfied. A stress
matrix [P] must have m stress modes corresponding to m natural deformation modes of an
element.
5.2. 3-D, 8-node solid element
Using the same way as 2-D case, one can choose m stress modes from m classified stress mode
groups except zero-energy mode group above to form the eight stress matrices [P ]—[P ]
2
9
proposed by Atluri et al.8,15 as follows:
[P ]"[p p p p p p p p p p p p p p p p p p ]
2
40 41 42 4 5 6 49 50 53 54 55 56 57 58 59 60 61 62
1
1
0
0 0 0 0
0
0
2x
0
0
0
0
0
0
0
0
1
0 0 0 0
0
0
0
2y
0
0
0
0
0
0
0
[1 0 0 0 0
0
0
0
0
2z
0
0
0
0
0
0
z
0
0
y
x
0
[2xz
[2yz
x2#y2
[2xy
[2xz
x2#z2
[2yz
1 [1
1
0
0
0
0
1 0 0 z
0
0
0
0 1 0 x [x [x
0
0
0
0 0 1 y
"
0
y
[y [x
[z [y
0
[z
[z [y y2#z2
0
[x [z
0
x
0
[2xy
(50)
[P ]"[p p p p p p p p p p p p p p p p p p ]
3
40 41 42 4 5 6 49 50 53 54 55 56 57 58 59 27 36 30
1
0
0 0 0 0
0
0
2x
0
0
0
0
0
0
0
yz
1
0 0 0 0
0
0
0
2y
0
0
0
0
0
xz
0
[1 0 0 0 0
0
0
0
0
2z
0
0
0
xy
0
0
z
0
0
y
x
0
0
0
0
0
0
0
0
1
1 [1
1
0
0
0
0
1 0 0 z
0
0
0
0 1 0 x [x [x
0
0
0
0 0 1 y
"
!y [x
y
0
0
[z [y
[z
0
p
53
p
53
p
53
p
53
p
53
p
53
p
54
p
54
p
45
p
45
p
45
p
45
0
[x [z
[z [y
0
x
0
0
p
p
46
p
46
p
59
p
59
p
46
p
46
p
60
p
27
p
60
p
27
p
60
p
27
p
61
p
36
p
61
p
36
p
61
p
36
(51)
and
[P ]"[p
4
40
[P ]"[p
5
40
[P ]"[p
6
40
[P ]"[p
7
40
[P ]"[p
8
40
[P ]"[p
9
40
p
p
p
p
p
p
41
41
41
41
41
41
p
42
p
42
p
42
p
42
p
42
p
42
p p
4 5
p p
4 5
p p
4 5
p p
4 5
p p
4 5
p p
4 5
p
6
p
6
p
6
p
6
p
6
p
6
p
49
p
49
p
49
p
49
p
49
p
49
p
p
p
p
p
p
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
50
50
50
50
50
50
p
55
p
55
p
44
p
44
p
44
p
44
p
56
p
56
p
43
p
43
p
43
p
43
p
48
p
48
p
57
p
57
p
48
p
48
p
p
p
p
p
47
47
58
58
47
47
p ]
62
p ]
30
p ]
62 (52)
p ]
30
p ]
62
p ]
30
( 1997 John Wiley & Sons, Ltd.
4333
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
The assumed stress matrix given by Huang14 also can be formed by means of the same way,
[P ]"[p p p p p p p p p p p p p p p p p p ]
10
40 41 42 4 5 6 43 44 45 46 47 48 49 50 51 27 36 30
0
y
0 0
0
yz 0 0
1 1 [1 0 0 0 z 0 y z
1 [1 [1 0 0 0 z x 0 [z
x
0
0
0
0
zx
0
[x !y 0
0
0
0
0
xy
z
z
0
0
0
0
1
0
2
0 0 0 0 x y
0
0
0
0
1 0 0 0 0 0
0
0
0
z
0
0
0
0 1 0 0 0 0
0
0
0
x [x
[x
0
0
0
0
0
0
0 0 1 0 0 0
0
0
0
y
[2y
0
0
0
"
0
(53)
Moreover, many stress matrices [P] can be constructed on purpose. Three new stress matrices
are given as follows:
[P* ]"[p p p p p p p p p p p p p p p p p p ]
1
40 41 42 4 5 6 63 64 65 46 47 48 49 50 51 66 67 68
0
y 0 0
0
yz xz xy
1 1 [1 0 0 0 z x y z
1 [1 [1 0 0 0 z x y [z
x
0
0
0
yz xz xy
[x [y 0
0
0
yz xz xy
z
z
0
0
0
0
1
0
2
0 0 0 z x y
0
0
0
0
1 0 0 0 0 0
0
0
0
z
0
0
0
0 1 0 0 0 0
0
0
0
x [x
[x
0
0
0
0
0
0
0 0 1 0 0 0
0
0
0
y
[2y
0
0
0
"
0
(54)
[P*]"[p p p p p p p p p p p p p p p p p p ]
2
40 41 42 4 5 6 43 44 45 57 58 59 22 11 18 30 36 27
1 0 0 0 0 0 z 0 y
0
0
0
0 0 0 yz
0
0
0 1 0 0 0 0 z x 0
0
0
0
0 0 0
0
xz
0
0 0 1 0 0 0 0 x y
0
0
0
0 0 0
0
0
xy
0 0 0 1 0 0 0 0 0
y
x
0
z 0 0
0
0
0
0 0 0 0 1 0 0 0 0
0
[z [y 0 x 0
0
0
0
0
0
0
(55)
"
0 0 0 0 0 1 0 0 0 [z
x
0
0 0 y
[P*]"[p p p p p p p p p p p p p p p p p p ]
3
69 41 42 4 5 6 63 64 65 46 47 48 49 50 51 66 67 68
z2
1
[1 0 0 0 z x y
z
z2 [1 [1 0 0 0 z x y [z
0
y
0
0
0
yz xz xy
x
0
0
0
0
yz zx xy
[x [y 0
0
0
yz xz xy
z
z
0
0
0
0
0
2
0 0 0 z x y
0
0
0
0
1 0 0 0 0 0
0
0
0
z
0
0
0
0 1 0 0 0 0
0
0
0
x [x
[x
0
0
0
0
0
0
0 0 1 0 0 0
0
0
0
y
[2y
0
0
0
"
( 1997 John Wiley & Sons, Ltd.
0
(56)
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4334
W. FENG, S. V. HOA AND Q. HUANG
Table II. Eigenvalues of element stiffness matrices (3-D, 8-node solid element,
v"0·3)
[P ]
2
0·07123
0·07123
0·07123
0·1282
0·1282
0·1282
0·1282
0·1282
0·07246
0·07246
0·07246
0·5128
0·7692
0·7692
0·7692
0·7692
0·7692
2·5000
[P ]
4
[P ] [P ] [P ]
1
9
10
[P ], [P* ]
7
2
[P* ]
1
[P* ]
3
0·07123
0·07123
0·07123
0·2564
0·2564
0·2564
0·1282
0·1282
0·07264
0·07264
0·07264
0·5128
0·7692
0·7692
0·7692
0·7692
0·7692
2·5000
0·1111
0·1111
0·1111
0·2564
0·2564
0·2564
0·1282
0·1282
0·4762
0·4762
0·4762
0·5128
0·7692
0·7692
0·7692
0·7692
0·7692
2·5000
0·1111
0·1111
0·1111
0·1282
0·1282
0·1282
0·1282
0·1282
0·4762
0·4762
0·4762
0·5128
0·7692
0·7692
0·7692
0·7692
0·7692
2·5000
0·09259
0·09259
0·09259
0·2564
0·2564
0·2564
0·1282
0·1282
0·5556
0·5556
0·5556
0·5128
0·7692
0·7692
0·7692
0·7692
0·7692
2·5000
0·09259
0·09259
0·09259
0·2564
0·2564
0·2564
0·1282
0·1282
0·5556
0·5556
0·5556
0·5128
0·7692
0·7692
0·7692
0·7692
0·7692
0·8065
The results of eigenvalue examination are given in Table II. It shows that each of the stiffness
matrices constructed by the assumed stress matrices [P ]—[P ], [P* ], [P* ] and [P* ] has
1
10
1
2
3
m non-zero eigenvalues. The resulting hybrid elements do not have any kinematic deformation
modes.
More assumed stress matrices can also be constructed by means of this method. If one stress
mode group is missed except the zero-energy mode group in the process of choosing stress modes,
the hybrid element will contain kinematic deformation modes. In the previous work, it is
proposed to suppress kinematic deformation modes by adding stress modes of high-order term.
Actually, it cannot guarantee that all kinematic deformation modes are suppressed. If the
high-order stress modes do not belong to the stress mode groups which are missed in the
construction of the assumed stress matrix except the zero-energy mode group, adding stress
modes of high-order term cannot improve the hybrid element any more. Moreover, overuse of
stress modes will result in over-rigid elements.7 Therefore, an ideal situation is to choose
m("n!r) least-order stress modes, but with the suppression of all kinematic deformation modes.
Thus, an assumed stress matrix [P] can be constructed by choosing m stress modes from m stress
mode groups that correspond to m natural deformation modes.
6. UNIQUENESS OF STRESS MODE GROUPS
When stress modes are classified, it is observed that if the flexibility matrix [H] is a diagonal
matrix, the classification of stress modes is unique; otherwise, some stress modes may be
interchangeable between two stress mode groups. For example, the stress modes Mp N and Mp N
1
2
for 2-D, 4-node plane hybrid element may be interchanged between group 1 and group 2. This
makes the flexibility matrix [H] not diagonal when the stress matrix [P] consists of (Mp N, Mp N,
1
2
Mp N, Mp N, Mp N). Therefore, the first two stress mode groups for 4-node plane element may
3
5
6
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4335
become
Tension mode (Group 1): Mp N (or Mp N), Mp N, Mp N, Mp N
1
2
10
14
17
Tension mode (Group 2): Mp N (or Mp N), Mp N, Mp N, Mp N
2
1
11
15
18
However, the stress modes Mp N and Mp N for the plane element cannot be interchanged between
10
11
the two groups because the matrix [H] is diagonal when the stress matrix [P] consists of (Mp N,
10
Mp N, Mp N, Mp N, Mp N). In fact, if the flexibility matrix [H] is a diagonal matrix, the stress modes
11
3
5
6
that form the stress matrix [P] are a set of uncoupled stress modes. It has been proved by
Huang14 that if the matrix [H] is a diagonal matrix, the stiffness matrix satisfies the superposition
principle:
m
[K]" + [K ]
i
i/1
(57)
where
[K ]"[G ]T[H ]~1[G ]
i
i
i
i
Pv [Pi ]T[S][Pi ] d»
[G ]" [P ]T[B] d»
i
Pv i
[H ]"
i
(58)
and
[K]"[G]T[H]~1[G]
Pv [P]T[S][P] d»
[G]" [P]T[B] d»
Pv
[H]"
(59)
in which
[P]"[Mp N Mp N Mp N . . . Mp N]
1
2
3
m
[P ]"[M0N M0N . . . Mp N . . . M0N]
i
i
(60)
 G1 
G 
[G]" 2 
 ... 
G 
m
Therefore, the elastic energy of the element is decomposable if the flexibility matrix [H] is
a diagonal matrix.
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4336
W. FENG, S. V. HOA AND Q. HUANG
6.1. Classification condition of stress modes
A hybrid element stiffness matrix [K] can be formulated using equations (59) and (60). Its
eigenvalues and eigenvectors are calculated from equation (9). The eigenvectors Md N (i"1,
i
2, . . . , m) satisfy the condition (10). Using equations (9) and (10), the eigenvalue equation is
changed to
j "Md NT[K] Md N
i
i
i
(61)
For any stress mode Mp N among m stress modes Mp , p , . . . , p N, the stiffness matrix [K ] can
j
1 2
m
j
be derived using equations (58) and (60). Corresponding to the ith natural deformation mode, we
have
j "Md NT[K ]Md N
i
i
j
i
(62)
According to the classification condition of stress modes, if the stress mode Mp N belongs to the
j
ith stress mode group corresponding to the natural deformation mode Md N, the eigenvalue j is
i
i
a non-zero value; otherwise, the eigenvalue j equals zero. This condition can be expressed in the
i
form
Md NT[K ]Md N"0,
i
j
i
Md NT[K ]Md N"j ,
i
j
i
i
iOj
(63)
i"j
If the stress mode Mp N is a zero-energy stress mode, all eigenvalue j (i"1, 2, . . . , m) equal zero.
j
i
¹heorem 1. If and only if the flexibility matrix [H] is a diagonal matrix, the eigenvalues obtained
from separate mode equations
( [K ]!j[I])MdN"0, i"1, 2, 3, . . . , m
i
(64)
are equal to the eigenvalues obtained from the total equation
( [K]!j[I])MdN"0
(65)
In which, the matrices [K ] and [K] are defined in equations (58) and (59). ¹his was stated as
i
a postulate in Reference.14
Proof. From the equations (61), (59) and (15), we have
j"Md NT[K]Md N"Mb NT[H]Mb N
i
i
i
i
(66)
Because the matrix [H] is a diagonal matrix, we have
m
[H]" + [H ]
j
j/1
(67)
Thus, using equations (12), (13) and (58), the eigenvalue of the matrix [K] is
m
j" + Mb NT[H ]Mb N"Mb NT[H ]Mb N
i
j
i
i
i
i
j/1
Int. J. Numer. Meth. Engng., 40, 4313—4339 (1997)
(68)
( 1997 John Wiley & Sons, Ltd.
CLASSIFICATION OF STRESS MODES IN ASSUMED STRESS FIELDS
4337
Furthermore, using equations (15) and (63), we obtain
j"Mb NT[H ]Mb N"Md NT[K ]Md N"j K
(69)
i
i
i
i
i i
i
¹heorem 2. If the flexibility matrix [H] is a diagonal matrix, the classification of m stress modes
is unique.
Proof. If a stress mode among m stress modes that form the stress matrix [P] appears in more
than one stress mode group, one of the m stress mode groups must contain two stress modes.
Assume that the stress modes Mp N and Mp N belong to the ith stress mode group Mb N correspondi
j
i
ing to the natural deformation modes Md N. Thus, we have
i
j "Md NT[K ]Md N and j "Md NT [K ]Md N
(70)
ii
i
i
i
ij
i
j
i
Corresponding to the natural deformation mode Md N, we can obtain the eigenvalue of the
i
stiffness matrix [K] formulated by m stress modes as follows:
j"Md NT [K]Md N
(71)
i
i
Because the flexibility matrix [H] is diagonal, the stiffness matrix satisfies the superposition
principle. From equations (57) and (70), we obtain
m
j"Md NT [K]Md N" + Md NT [K ]Md N"j #j
i
i
i
k
i
ii
ij
k/1
using Theorem 1, we have
j"j
ii
From the equations (72) and (73), we obtain
(72)
(73)
j "0
(74)
ij
According to the condition of classification, the stress modes Mp N does not belong to the ith
j
stress mode group. The stress mode group Mb N only contains Mp N. Therefore, the stress modes
i
i
Mp N cannot appear in two stress mode groups Mb N and Mb N. Thus, if the matrix [H] is diagonal,
j
i
j
the classification of m stress modes is unique.
K
7. CONCLUSION
A new method for classifying stress modes in assumed stress matrices is presented. It is assumed
that there are m("n!r) natural deformation modes of an element which has n degrees of
freedom and r rigid-body modes. For any type of hybrid element, all stress modes in various stress
matrices derived by different methods can be classified into m stress mode groups corresponding
to m natural deformation modes and a zero-energy mode group corresponding to rigid-body
modes. If the flexibility matrix [H] is diagonal, the deformation energy of the element is
decomposable and the classification of stress modes is unique. The necessary and sufficient
condition for avoiding kinematic deformation modes is that an assumed stress matrix [P]
must contain m stress modes chosen from m different stress mode groups, except zero-energy
( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
4338
W. FENG, S. V. HOA AND Q. HUANG
mode group. The reason of the existence of kinematic deformation modes when the criterion
(m@'n!r) is satisfied is that the stress modes in the assumed stress matrix [P] are not
chosen from m different stress mode groups corresponding to m natural deformation modes.
The classification method can be applied to any type of hybrid elements and be used for two
purposes:
1. to determine the optimal stress matrix from the existing stress matrix [P] or any other
*40
stress matrix [P] derived using other method, and classify stress modes into m different
stress mode groups;
2. to construct many new assumed stress matrices by using minimum number of stress modes
according to the problems to be analysed. These stress matrices are without zero-energy
stress modes, and the resulting element stiffness matrices are free from kinematic deformation modes.
The classification of stress modes reveals the relationship among the different assumed stress
fields for any type of hybrid element proposed by different researchers. An assumed stress matrix
[P], which consists of m("n!r) least-order stress modes and results in the element stiffness
matrix without kinematic deformation modes, is considered to be best and is optimal with respect
to computer resources because overuse of stress modes will result in over-rigid element and cost
more computational time.
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( 1997 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng., 40, 4313— 4339 (1997)
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