COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Studies of 4-node Membrane Element with Analytical Stiffness Matrix
based on the Quadrilateral Area Coordinates (QAC)
Yu Du1*, Song Cen1, 2
1
2
School of Aerospace, Tsinghua University, Beijing, 100084 China
Failure Mechanics Laboratory, Tsinghua University, Beijing, 100084 China
Email: [email protected], [email protected]
Abstract The new Quadrilateral Area Coordinate (QAC) method is a powerful tool to construct 2D finite element
models. Compared with the traditional models using isoparametric coordinates, these new models are less sensitive
to mesh distortion. Theoretically, by using the area integral formulae the analytical stiffness matrix could be obtained,
which may greatly benefit the computation procedure. However, as the derivation of the analytical expression is
relatively complicated, all the current papers still adopt the numerical integration method for computer coding,
which indeed impedes the advantages of the QAC method. So in this paper, by introducing the basic QAC formulae
into software Maple, the analytical expression of the stiffness matrix of the 4-node membrane QAC element AGQ6-I
is obtained for the first time. Then a corresponding FORTRAN subroutine is compiled. Numerical examples show
that the present scheme exhibits excellent performances in both computation efficiency and accuracy when
compared with the traditional isoparametric models using numerical integration scheme.
Key words: finite element, Quadrilateral Area Coordinate (QAC), AGQ6-I, analytical stiffness matrix, computation
efficiency and accuracy
INTRODUCTION
In finite element analysis, the isoparametric coordinate method [1] is widely used. However, many isoparametric
element models are sensitive to mesh distortion [2]. The main reason may be as follows [3]. As the element distorts
from a rectangle to an irregular quadrilateral, the transformation between the isoparametric coordinate system (ξ, η)
and the Cartesian coordinate system (x, y) will change from linear to nonlinear, thus the order of the complete
polynomial of the displacement field in (x, y) will descend greatly. The new Quadrilateral Area Coordinate (QAC)
method [4, 5], which is generalized from the well-known triangular area coordinate method, performs quite well in
overcoming the disadvantage mentioned above. One key reason is that the transformation between (ξ, η) and (x, y) is
always linear. Based on this new system, several new membrane, plate and shell element models have already been
successfully developed [6].
Theoretically, by using the area integral formulae the analytical element stiffness matrix could be obtained, which
may greatly benefit the computation procedure. However, as the derivation of the analytical expression is relatively
complicated, all the current papers still adopt the numerical integration method for computer coding, which indeed
impedes the advantages of the QAC method. So in this paper, by introducing the basic QAC formulae into the
mathematic software Maple [7, 8], the analytical expression of the stiffness matrix of the 4-node membrane QAC
element AGQ6-I [9] is obtained for the first time. Then a corresponding FORTRAN subroutine is compiled.
Numerical examples show that the present scheme exhibits excellent performances in both computation efficiency
and accuracy when compared with the traditional isoparametric models using numerical integration scheme.
AREA COORDINATES FOR QUADRILATERAL ELEMENTS [4, 5]
The definition of QAC, as well as several basic equations, calculus formulae and detailed proof have been presented
in references [4] and [5]. Two equivalent basic integral formulae, which can be applied to evaluate the area integrals
⎯ 714 ⎯
for the arbitrary power function L1m Ln2 L3p Lq4 of area coordinates, are given as follows:
∫∫
A
L1m Ln2 L3p Lq4 dA =
⋅ [g
∫∫
A
m + n +1
3
q
p
∑∑ C
k =0 j =0
L1m Ln2 L3p Lq4 dA =
⋅ [g
n + p +1
4
m! n! p! q!
⋅ 2A ⋅
( m + n + p + q + 2)!
m
q
C
n
n+ p− j
C
k
k+ j
k
2
j
4
g g g
p+q−k − j
1
+g
p + q +1
1
m
n
∑∑ C
k =0 j =0
(1a)
q
m+ q −k
C
p
n+ p− j
C
q
q+ p− j
C
k
k+ j
C
k
k+ j
k
2
j
4
m+n−k − j
3
k
3
j
1
n+ p−k − j
4
g g g
]
m! n! p! q!
⋅ 2A ⋅
( m + n + p + q + 2)!
∑∑ C
k =0 j =0
m
m+ q−k
n
n+ m−k
C
p
p+q− j
C
k
k+ j
k
3
j
1
g g g
m+ q −k − j
2
+g
m + q +1
2
n
p
∑∑ C
k =0 j =0
(1b)
m
m+ n−k
g g g
]
where
C ki =
k!
(k − i )!i!
(2)
and A is the area of the quadrilateral element; gi (i=1, 2, 3, 4) are four dimensionless parameters to each of the
quadrangles (Figure 1):
g1 =
A(Δ124)
A(Δ123)
, g2 =
, g3 = 1 − g1 , g 4 = 1 − g 2
A
A
(3)
Figure 1: Definition of the four parameters g1, g2, g3 and g4
FORMULATION OF THE 4-NODE MEMBRANE ELEMENT AGQ6-I [6]
Reference [9] presented a 4-node membrane element AGQ6-I (Figure 2) by QAC method and the generalized
conforming conditions. The derivatives of its shape functions Ni0 (i=1, 2, 3, 4) in (x, y) are given by:
4
∂N i0 bi bi +1
ξ iη i g i +2
=
+
+
b j ξ jη j [3( L j +1 − L j +3 ) + ( g j +1 − g j +2 )]
∑
∂x
2 A 2 A 2 A(1 + g1 g 3 + g 2 g 4 ) j =1
4
∂N i0 ci ci +1
ξ iη i g i + 2
=
+
+
c j ξ jη j [3( L j +1 − L j +3 ) + ( g j +1 − g j +2 )]
∑
∂y
2 A 2 A 2 A(1 + g1 g 3 + g 2 g 4 ) j =1
(i = 1,2,3,4)
(4)
where
suuuur suuuuuur
bi = y j − yk , ci = xk − x j , i, j , k = 1, 2,3, 4
(5)
and ξi, ηi (i=1, 2, 3, 4) are the isoparametric coordinates of the four element nodes.
The derivatives of the internal parameter shape functions Nλi (i=1, 2) in (x, y) can be obtained:
b
b
∂N λi
= i L i + 2 + i + 2 Li
2A
2A
∂x
c
∂N λi
c
= i Li + 2 + i + 2 Li
2A
2A
∂y
(i = 1,2)
(6)
⎯ 715 ⎯
The element strain matrix [Bq] and [Bλ] are composed of (3) and (4) respectively. After condensation, the element
stiffness matrix can be expressed by:
[K ] e = [K qq ] − [K λq ] T [K λλ ] −1 [K λq ]
(7)
with [K qq ] = ∫∫ [B q ] T [D][B q ]tdA, [K λλ ] = ∫∫ [B λ ] T [D][B λ ]tdA, [K λq ] = ∫∫ [B λ ] T [D][B q ]tdA
A
A
A
(8)
where t is the thickness of the element; [D] is the elasticity matrix.
Figure 2: A 4-node quadrilateral membrane element
DERIVATION OF THE ANALYTICAL ELMENT STIFFNESS MATRIX OF AGQ6-I
In order to obtain the analytical element stiffness matrix of element AGQ6-I, a Maple [7, 8] function,
aaint(expr), is firstly developed to compute the integral for the arbitrary polynomial of area coordinates
analytically. Then, the explicit expressions of the element stiffness matrix are derived as follows: (i) defining [Bq],
[Bλ] and [D]; (ii) integrating the product of various forms [B]T[D][B] in area coordinates by aaint(expr)to
obtain [Kqq], [Kλλ] and [Kλq]. Since the expressions obtained may be very long, the direct integration approach is not
a good selection when evaluating [Kqq] and [Kλq]. Therefore, a simple substitution procedure is adopted here: before
integrating a whole polynomial, substitute its parts containing area coordinates with Maple functions, and then
integrate these functions only. For example, after such substitution, Kqq(1,2) (the component of [Kqq]) can be
rewritten as:
⎛
⎞⎛ 1 1
g32 Iqqxy
⎞
+ miu ⎟
⎜ − y2 x2 + y2 x4 + y4 x2 − y4 x4 +
2 ⎟⎜
⎜
(1 + g1 g3 + g 2 g 4 ) ⎟⎠ ⎝ 2 2 ⎠
⎝
4A
where miu is Poisson’s ratio; Iqqxy contains area coordinates L1, L2, L3, L4, and thus can be integrated by
aaint(expr). By Integrating Kλλ(1,2) with aaint(expr), we obtain:
1 ⎛⎛ 1
1
2
2
2
2
3
⎜ ( g3 ( g1 + g 4 ) + g1 ( g3 + g 2 ) ) ( b1c3 + c1b3 ) + g 3 ( g1 + g 4 g1 + g 4 ) + g1 b1c1
⎜
4 A ⎝ ⎝ 12
6
(
+
1
1 ⎞⎞
⎞⎛ 1
g1 ( g32 + g 2 g3 + g 2 2 ) + g33 b3c3 ⎟ ⎜ miu + ⎟ ⎟
6
2 ⎠⎠
⎠⎝ 2
(
)
The Maple codes of function aaint(expr) are as follows:
aaint:=proc(f)
local ii,co,po,m,n,p,q,z,num,aa;
aa:=expand(f):
z := 0:
co:=coeffs(aa,[L[1],L[2],L[3],L[4]],'powers'):
po:= powers:
co:=convert([co],'list'): po:=convert([po],'list'):
⎯ 716 ⎯
)
num:=nops(po):
m:=array(1..num,[]):n:=array(1..num,[]):
p:=array(1..num,[]):q:=array(1..num,[]):
for ii from 1 to num do
m[ii]:=degree(po[ii],L[1]):
n[ii]:=degree(po[ii],L[2]):
p[ii]:=degree(po[ii],L[3]):
q[ii]:=degree(po[ii],L[4]):
z:=z+INT(m[ii],n[ii],p[ii],q[ii]) * co[ii]:
od
end proc:
where INT(m, n, p, q)is a user-developed function for equation (1).
Furthermore, the components in some matrices possess cycle regularity in area coordinates, which means only a
small part of them need to be evaluated and the rest can be easily deduced by the cycle rules. Thus great effort of
input can be saved. For example, only the first two columns of the 8×8 [Kqq] is coded, and the other columns are
obtained only by cycling the first two ones. However, only symmetric matrix can enjoy such rules, that is, matrices
[Kqq] and [Kλλ] satisfy the circular law, while [Kλq] can not.
A subroutine STIF_ELE_Q_EXP is added to the main program “GCFEM 2000” [10] to generate the analytical
stiffness matrix of the 4-node QAC element AGQ6-I.
NUMERCIAL EXAMPLES
The displacement fields of element AGQ6-I have second order completeness in both Cartesian and area coordinates.
So it performs very well in overcoming the sesitivity problems to mesh distortion [6]. In this section, we are mainly
concerned about the computation efficiency and accuracy of AGQ6-I with analytical stiffness matrix.
1. Example 1: Cantilever beam under transverse force P (Fig. 3). The cantilever beam is subjected to linear
bending under transverse force P. The Young’s modulus E=1500, Poisson’s ratio μ＝0.25, thickness t=1.0.
Compute the plane strain problem with four types of element (Table 1) and record the computation time
(automatically output) under different mesh density (Table 2). The results are compared in Figure 4.
Hardware environment: CPU PIV 2.0G, memory 2.0G .
Software environment: Compaq Visual FORTRAN 6.0.
Figure 3: Cantilever beam under transverse force P
Table 1 Four types of 4-node element and corresponding symbols
Symbols
Element type
AGQ6-I (A)
4-node QAC element using analytical stiffness matrix
AGQ6-I (3×3)
4-node QAC element using 3×3 Gauss integration
Q4 (2×2)
4-node isoparametric element using 2×2 Gauss (full) integration
Q4 (1×1)
4-node isoparametric element using 1×1 Gauss (reduced) integration
⎯ 717 ⎯
Table 2 Computation time of different element types under various grid DOFs (unit: second)
Num. of DOF
AGQ6-I(A)
AGQ6-I(3×3)
Q4 (2×2)
Q4 (1×1)
4242
1.265625
1.304688
2.117188
1.484375
9362
6.015625
6.093750
8.046875
6.562500
16482
18.67969
18.79688
22.44531
19.82812
25602
44.82812
45.03125
50.73438
46.53125
36722
91.84375
92.15625
100.7188
94.40625
49842
168.3484
168.8047
180.5547
172.4375
64962
286.3516
286.9063
301.5781
290.6406
82082
457.1641
457.9532
476.6875
462.9219
101202
693.9531
694.8281
718.2031
701.0000
(a) AGQ6-I(3×3)－AGQ6-I(A)
(b) Q4(2×2)－AGQ6-I(A)
(c) Q4(1×1)－AGQ6-I(A)
(d) AGQ6-I(3×3)－Q4(1×1)
Figure 4: Computation time difference under various mesh density
From Table II and Figure 4, one can conclude that: (1) AGQ6-I(A) is the most efficient model; it is as accurate as
AGQ6-I(3×3), but much faster; (2) the two QAC elements are both faster than Q4 (2×2) while less sensitive to mesh
distortion; (3) comparing with Q4 (1×1), AGQ6-I(A) has obvious advantages in both computation efficiency and
hourglass elimination; (4) however, AGQ6-I(3×3) is faster than Q4(1×1), which is possibly because that FORTRAN
automatically simplifies the computation process of AGQ6-I(3×3).
The analytical stiffness matrix of area coordinate elements should be especially significant to nonlinear problems.
For instance, let’s consider a suppositional moderate scale nonlinear problem. Its solution procedure may use 1000
increments and 10 iterations for each increment, which is equivalent to repeating a solution procedure for a
corresponding linear problem 10,000 times. As the mesh being refined, the computation time of AGQ6-I(A) and
other elements are compared in Figure 5 (neglecting other factors, such as stiffness matrix renewal, etc.).
⎯ 718 ⎯
(a) AGQ6-I(3×3)－AGQ6-I(A)
(b) Q4(2×2)－AGQ6-I(A)
(c) Q4(1×1)－AGQ6-I(A)
Figure 5: Computation time difference in a suppositional nonlinear problem
2. Example 2: Square plate with a circular hole, subjected to uniaxial tension (Fig. 6). The parameters of this
problem are: length of the plate side L=8cm, radius of the circular hole r=1cm, thickness t=0.1cm; distributed load
q=10MPa; Young’s modulus E=2.0×105MPa, Poisson’s ratio μ＝0.3. Because of symmetry, the analysis is performed
on one quarter of the plate; thus the boundary conditions are: u=0, on x=0; v=0, on y=0; distributed load q on y=4cm.
Compute the maximum y-displacement uy along the edge of the plate and the maximum stress concentration factor k
of σy on the edge of the hole with AGQ6-I(A) and Q4 (2×2) (Table I), then compare the results and computation time
under different mesh density (Table III and Figure 7 to 8). Hardware/software environment are the same as those in
previous example.
(a) Geometry, loads and BCs
(b) Mesh density of 10×10, 19×19 and 28×28
Figure 6: A quarter square plate with a circular hole, subjected to uniaxial tension
The convergent solution of uy has been computed by ABAQUS [11]. The results of element Q4 (2×2) are:
uy=2.755×10−6m, k=3.596 by 21806 DOFs (mesh 100×100); and uy=2.755×10-6m, k=3.593 by 45014 DOFs (mesh
144×144). The results of 8-node isoparametric element using 3×3 Gauss integration are: uy=2.755×10−2, k=3.583 by
⎯ 719 ⎯
45014 DOFs (mesh 144×144). So, uy=2.755×10−6m is taken as the FEM convergent solution.
It can be seen that AGQ6-I(A) is obviously more efficient than Q4 (2×2). In addition, when the mesh is relatively
sparse, the displacement solution of AGQ6-I(A) is closer to the FEM convergent solution.
The reason we intently choose Q4 (2×2) is that the isoparametric elements using reduced integration scheme
performs poorly in stress computation. Furthermore, to make computation time comparable, in both approaches
stress is computed directly by their shape functions.
Table III. uy, k solutions and computation time of AGQ6-I(A) and Q4 (2×2)
AGQ6-I(A)
Num. of DOF
Q4 (2×2)
u y /10-6m
k
Time/s
u y /10-6m
k
Time/s
224
2.778
3.621
1.563×10-2
2.718
3.557
7.813×10-2
362
2.772
3.687
4.688×10-2
2.725
3.557
0.1250
552
2.768
3.685
6.250×10-2
2.737
3.591
0.1875
782
2.765
3.671
0.1094
2.743
3.603
0.2969
1680
2.761
3.660
0.3125
2.749
3.606
0.7500
2958
2.758
3.628
0.7813
2.752
3.607
1.563
4544
2.757
3.622
1.656
2.753
3.604
2.891
11418
2.756
3.604
8.953
2.754
3.599
12.16
Figure 8: Convergence rate of uy
Figure 7: Computation time
CONCLUSIONS
In this paper, the analytical stiffness matrix for element AGQ6-I [6], a 4-node membrane element formulated by
QAC method, is evaluated for the first time. Numerical examples show the proposed model possesses better
performance than traditional isoparametric elements for both computation efficiency and accuracy. Altogether with
the results presented in [6], it can be concluded that the element AGQ6-I with the analytical stiffness matrix is an
ideal substitute for usual 4-node isoparametric element models Q4 (compatible) and Q6 (incompatible).
Acknowledgements
The supports of the Natural Science Foundation of China (10502028), the Special Foundation for the Authors of the
Nationwide (China) Excellent Doctoral Dissertation (200242) are gratefully acknowledged.
⎯ 720 ⎯
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⎯ 721 ⎯