COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Computational Strategies for Curved-side Elements Formulated by
Quadrilateral Area Coordinates (QAC)
Song Cen1,3*, De-Po Song1, Xiao-Ming Chen1, Yu-Qiu Long2
1
School of Aerospace, Tsinghua University, Beijing, 100084 China
Department of Civil Engineering, Tsinghua University, Beijing, 100084 China
3
Failure Mechanics Laboratory, Tsinghua University, Beijing, 100084 China
2
Email: [email protected]
Abstract In order to avoiding the sensitivity problem to mesh distortion which often occurs for a quadrilateral finite
element model, a Quadrilateral Area Coordinate (QAC) method has already been systematically established. But this
new system can not handle curved-side cases. In this paper, two computational strategies for curved-side elements
formulated by QAC are proposed: the first pattern is a generalization of the isoparametric transformation for triangular
elements; and the second pattern is a simply approximate one. Then both schemes are introduced into the eight-node
quadrilateral, QAC element AQ8. Numerical examples demonstrate that the present strategies are effective
complements for QAC system.
Key words: finite element, quadrilateral area coordinate (QAC), mesh distortion, curved-side, AQ8
INTRODUCTION
The high performance finite element models possess important significance for modern computations in engineering
and science. Up to date, many researchers still made great efforts to find an effective way to develop robust elements
with good accuracy [1-3]. Although some achievements have already been obtained, there are still some problems that
are difficult to avoid. For example, the most widely-used quadrilateral isoparametric elements are often sensitive to
various mesh distortion modes.
In order to avoiding the sensitivity problem to mesh distortion for a quadrilateral finite element model, a Quadrilateral
Area Coordinate (QAC) method was established [4,5]. The QAC system has an important advantage: the
transformation between the area coordinates and the Cartesian coordinates is always linear. Thus, the order of the
displacement field expressed by the area coordinates will not vary with the mesh distortion, and as a result, it makes the
element insensitive to mesh distortion. Based on the QAC method, some 4-node [6] and 8-node [7] membrane
elements, 4-node thin [8] and thick [9] plate bending elements, and 4-node shell element [10] were successfully
developed. Compared with those traditional models using isoparametric coordinates, these new elements are more
insensitive to mesh distortion.
Many structures possesses curved boundaries. And it seems to be more physically rational to model them using
elements with curved-sides. Thus, those isoparametric elements with mid-side nodes play important roles when
dealing with these situations. But curved mesh is also a type of mesh distortion. With the curvature of one side
increasing, the accuracy of the element may drop obviously [11]. On the other hand, though the elements by QAC can
exhibit excellent performance in a straight-side mesh (distorted or undistorted), special difficulty may be encountered
if one of element sides is curved. This is because that the present QAC system can not handle curved-side cases
directly.
So in this paper, two computational strategies are proposed for generalizing QAC system to curved-side cases. The
first pattern comes from isoparametric transformation: i) divide each quadrilateral element along its diagonal into two
triangular regions; ii) establish the relationships between the QAC and local triangular area coordinates within each
sub-triangle; iii) perform isoparametric transformation and evaluate stiffness matrices by Hammer numerical
integration (full or reduced) in each sub-triangle; iv) obtain the final element stiffness matrix by sum of the results from
⎯ 705 ⎯
two sub-triangles. The second pattern is an approximate one, which makes some simplification to the Jacobian
determinant during the isoparametric transformation. Then, these two schemes are introduced into the eight-node
quadrilateral, QAC element AQ8 [7]. Two new eight-node membrane QAC elements AQ8-M1 and AQ8-M2, which
can be used for both straight and curved meshes, are successfully constructed. Numerical examples show the new
elements are insensitive to straight and curved mesh distortion, and possess excellent performance in most cases. It
also demonstrates that the present strategies are effective complements of QAC system.
AREA COORDINATES FOR QUADRILATERAL ELEMENTS [4,5]
As shown in Fig. 1, the position of an arbitrary point P within a quadrilateral element 1234 is specified by the area
coordinates L1, L2, L3 and L4, which are defined as:
Figure 1: Definition of the quadrilateral area coordinates Li
Li =
Ai
A
(i = 1,2,3,4)
(1)
where A is the area of the quadrilateral element；Ai (i=1, 2, 3, 4) are the areas of the four triangles constructed by point
P and four element sides 23, 34, 41 and 12, respectively.
L1, L2, L3 and L4 can be expressed in terms of Cartesian coordinates (x, y) as follows:
Li =
1
(ai + bi x + ci y )
2A
(i = 1,2,3,4)
(2)
where
ai = x j yk − xk y j , bi = y j − yk , ci = xk − xi
(i = 1,2,3,4;
j = 2,3,4,1; k = 3,4,1,2)
(3)
and (xi, yi) (i=1,2,3,4) are the Cartesian coordinates of the four corner nodes.
Figure 2: Definition of the four parameters g1, g2, g3 and g4
Four dimensionless shape parameters g1, g2, g3 and g4 to each of the quadrangles, as shown in Fig. 2, are introduced:
g1 =
A′
,
A
g2 =
A′′
,
A
g 3 = 1 − g1 ,
g4 = 1 − g2
(4)
where A′ and A″ are the areas of Δ124 and Δ123, respectively.
It can be showed that gi (i=1, 2, 3, 4) can be expressed by bi and ci (i=1, 2, 3, 4) as follows:
2 Ag i = bk cm − bm ck
(i = 1,2,3,4; k = 3,4,1,2; m = 4,1,2,3)
(5)
It is obvious that any point in a plane problem has two degrees of freedom. Therefore, only two of the coordinates Li
(i=1, 2, 3, 4) are independent. It can be easily shown that Li (i=1, 2, 3, 4) must satisfy the following two equations:
⎯ 706 ⎯
L1 + L2 + L3 + L4 = 1 ,
g 4 g1 L1 − g1 g 2 L2 + g 2 g 3 L3 − g 3 g 4 L4 = 0
(6)
If all sides of a quadrilateral element keep straight, Li (i=1, 2, 3, 4) can also be expressed in terms of the quadrilateral
isoparametric coordinates (ξ, η) as follows:
1
⎧
⎪ L1 = 4 (1 − ξ )[ g 2 (1 − η ) + g 3 (1 + η )]
⎪
1
⎪ L2 = (1 − η )[ g 4 (1 − ξ ) + g 3 (1 + ξ )]
⎪
4
⎨
1
⎪ L3 = (1 + ξ )[ g1 (1 − η ) + g 4 (1 + η )]
4
⎪
⎪ L = 1 (1 + η )[ g (1 − ξ ) + g (1 + ξ )]
1
2
⎪⎩ 4 4
(7)
The transformation of derivatives of first order is:
⎧∂⎫
⎪⎪ ∂x ⎪⎪ 1 ⎡b1 b2
⎨ ∂ ⎬=
⎢
⎪ ⎪ 2 A ⎣c1 c2
⎪⎩ ∂y ⎪⎭
b3
c3
⎧ ∂ ⎫
⎪ ∂L ⎪
⎪ 1⎪
⎪ ∂ ⎪
b4 ⎤ ⎪ ∂L2 ⎪
⎬
⎨
c4 ⎥⎦ ⎪ ∂ ⎪
⎪ ∂L3 ⎪
⎪ ∂ ⎪
⎪
⎪
⎩ ∂L4 ⎭
(8)
Other details about the quadrilateral area coordinates can be seen in references [4] and [5].
FORMULATION OF THE 8-NODE MEMBRANE ELEMENT AQ8 [7]
Figure 3: 8-node quadrilateral membrane element
Reference [7] presented two 8-node membrane elements, AQ8-I and AQ8-II, which constructed by using QAC method
and generalized conforming conditions [12]. As long as all element sides are straight, their displacement fields have
second order completeness in Cartesian coordinates. So they can still keep high accuracy in a extremely distorted mesh
composed of straight-side elements. Since these two elements possess similar performance, only AQ8-II is introduced
here and denoted by AQ8 for simplicity.
Consider an 8-node straight-side serendipity element, as shown in Fig. 3. Nodes 1, 2, 3 and 4 are the corner nodes; and
nodes 5, 6, 7 and 8 are the mid-side nodes of sides 12, 23, 34 and 41, respectively. The coordinates of the eight nodes
are:
node 1: (g2, g4, 0, 0,); node 2: (0, g3, g1, 0); node 3: (0, 0, g4, g2); node 4: (g3, 0, 0, g1);
node 5: ( g 2 , g 3 + g 4 , g1 ,0); node 6: (0, g 3 , g 4 + g1 , g 2 ); node 7: ( g 3 ,0, g 4 , g1 + g 2 ); node 8: ( g 2 + g 3 , g 4 ,0, g1 ).
2
2
2
2
2
2
2
2
2
2
2
2
For corner nodes 1, 2, 3 and 4, the shape functions are as follows
g (g − gk )
1 g − gj
g ( g − gm )
1 g − gi
N i = N i0 − ( − i
) N 4+ i − k j
N 4+ j + k k
N 4+ k − ( + m
) N 4+ m
2
8g m
8g m gi
8gi g j
2
8g j
(i, j , k , m = 1,2,3,4)
⎯ 707 ⎯
(9)
where
N i0 =
L
Li
L
L
(1 − m ) + j (1 − k )
2g j
2gm
gi
gi
(i, j , k , m = 1,2,3,4) ,
(10)
and N4+i (i=1,2,3,4) are the shape functions for mid-side nodes 5, 6, 7 and 8, respectively,
N 4+i =
g + gj
4
Li Lk ( L j − Lm + i
)
gi g j
2
(i, j , k , m = 1,2,3,4)
(11)
Note that above shape functions are cubic both within and along the sides (Li=0, i=1,2,3,4) of the element. So the
displacement fields composed by them satisfy following relaxed conforming conditions, point and the side average
conforming conditions,
⎧(u − u~ ) j = 0
⎨
~
⎩ (v − v ) j = 0
⎧ (u − u~ )ds = 0
⎪∫
(at each node j), ⎨ di
(along each side di)
~
⎪⎩ ∫di (v − v )ds = 0
(12)
Therefore, element AQ8 is a generalized conforming element. Convergence is then guaranteed.
After expressing area coordinates in terms of isoparametric coordinates by equation (7), the element stiffness matrix
[K]e can be written as:
1
[K ] e = ∫
1
∫
−1 −1
[B]T [D][B] t J 4 dξdη
(13)
where [B] is the strain matrix; [D] is the elasticity matrix; t is the thickness of the element; ⎪J⎪4 is the Jacobian
determinant, which is the same as that of the 4-node isoparametric element. As long as the element sides are straight,
exact value of [K]e can be determined by 3×3 Gauss integration scheme. And it is obvious that this computational
strategy is not suitable for curved side cases.
ONE COMPUTATIONAL STRATEGY FOR CURVED SIDE CASES BY QAC METHOD
As shown in Fig. 4, we partition a quadrangle 1234 along its diagonal 24 into two triangular regions, Δ423 and Δ241.
There are two area coordinate system here: one is the QAC system Li(i=1,2,3,4); the other is the triangular area
Figure 4: Partition a quadrangle into two triangular regions
coordinate system L′i (i=1,2,3) within Δ423 and Δ241. These two systems have following relationship
L1 = g 3 L1′ , L2 = g 3 L2′ in Δ423
(14)
and
L3 = g1 L1′ , L4 = g1 L2′ in Δ241
(15)
Thus, the numerical integration of QAC system may be performed in the two triangular regions by Gauss integration
scheme [13].
⎯ 708 ⎯
Let L1 and L2 be two independent variables. From equations (14) and (7), we have
ξ = L1′ =
1
1
L1 , η = L2′ =
L2 , L3 = g 4 − g 4ξ + ( g1 − g 4 )η , L4 = 1 − L1 − L2 − L3 in Δ423.
g3
g3
(16)
Thus, a new differential relationship can be established in Δ423
∂L ∂
∂L ∂
∂L ∂
∂L ∂
∂
∂
∂
∂
∂
) − g4 (
)
= g3 (
−
−
= 1
+ 2
+ 3
+ 4
∂L1 ∂L4
∂L3 ∂L4
∂ξ ∂ξ ∂L1 ∂ξ ∂L2 ∂ξ ∂L3 ∂ξ ∂L4
∂L ∂
∂L ∂
∂L ∂
∂L ∂
∂
∂
∂
∂
∂
) + g2 (
)
= 1
+ 2
+ 3
+ 4
= g3 (
−
−
∂η ∂η ∂L1 ∂η ∂L2 ∂η ∂L3 ∂η ∂L4
∂L2 ∂L3
∂L3 ∂L4
.
(17)
Similarly, let L3 and L4 be two independent variables. From equations (15) and (7), we have
ξ = L1′ =
1
1
L3 , η = L2′ =
L4 , L1 = g 2 − g 2ξ + ( g 3 − g 2 )η , L2 = 1 − L3 − L4 − L1 in Δ241.
g1
g1
(18)
Another new differential relationship can be established in Δ241
∂
∂
∂
∂
∂
== g1 (
−
−
) − g2 (
)
∂ξ
∂L3 ∂L2
∂L1 ∂L2
∂
∂
∂
∂
∂
== g1 (
−
−
) + g4 (
)
∂η
∂L4 ∂L1
∂L1 ∂L2
.
(19)
Based on equations (17) and (19), we can perform coordinate transformation both in Δ423 and Δ241. Then the element
matrix [K]e can be divided into two sub-matrices, and each should be evaluated by triangular Gauss integration in the
corresponding region (Δ423 or Δ241). Finally, by sum of the two sub-matrices, the value of [K]e can be obtained.
The Cartesian coordinates within each sub-triangle can be interpolated by
6
6
i =1
i =1
x = ∑ N i′xi , y = ∑ N i′ y i
(20)
in which (xi, yi) (i=1,…,6) are the coordinates of five element nodes in each sub-triangle and one mid-side node of line
24; N′i (i=1,…,6) are the shape functions of 6-node triangular element T6 and in terms of the triangular area
coordinates L′i (i=1,2,3). Thus, one of the sub-stiffness matrix of each sub-triangular can be written as
[K ′]e =
1
[B ′]T [D][B ′] t J 6 dξdη ,
2 ∫∫A′
(21)
where A′ is the area of each sub-triangle; ⎪J⎪6 is the Jacobian determinant of the 6-node triangular isoparametric
element. We know that there must be a Jacobian inverse J-1 in the sub strain matrix [B′]. In order to keep the
performance of the element in a distorted mesh, we use the Jacobian determinant ⎪J⎪8 of the 8-node isoparametric
element to replace the Jacobian determinant in [B′].
It can be seen that this strategy establishes the coordinate transformation relationship between QAC system and
triangular isoparametric coordinate system. Thus, theoretically, the QAC system can deal with curved-side element
cases following the thought of this section.
The new version of AQ8 by using this strategy is denoted as AQ8-M1.
AN APPROXIMATE STRATEGY FOR CURVED SIDE CASES BY QAC METHOD
In equation (13), the existence of ⎪J⎪4 implies the element sides can not be curved. Here, we simply use the Jacobian
determinant of the 8-node isoparametric element, ⎪J⎪8, to replace ⎪J⎪4. Then, equation (13) can be rewritten as
1
[K ] e = ∫
1
∫
−1 −1
[B]T [D][B] t J 8 dξdη ,
(22)
⎯ 709 ⎯
and it is evaluated by standard quadrilateral Gauss integration. Obviously, when all element sides keep straight, same
exact results will be obtained by equations (13) and (22). But once some sides are curved, equation (22) seems more
rational, though only approximate value can be obtained.
The new version of AQ8 by using this strategy is denoted as AQ8-M2.
CALCULATION OF ELEMENT STRESSES
For element AQ8-M1, the stress matrix employed here is that of standard 8-node isoparametric element. All nodal
stress values of AQ8-M1 and AQ8-M2 are calculated directly by substituting nodal isoparametric coordinates into the
stress matrices. This direct method may be the simplest pattern to obtain stress solutions. But the accuracy is not
always acceptable for most usual isoparametric elements.
NUMERCIAL EXAMPLES
When the mesh is composed of only straight-side elements, both models AQ8-M1 and AQ8-M2 possess similar, even
the same performance with that of AQ8 in reference [7]. They can exactly pass the constant strain/stress patch test,
provide excellent results for displacement and stress solutions in various distorted meshes. What we concern in this
section is their behaviors in a mesh containing cuved-side distortion. Three models are used for all numerical
examples: Q8, standard 8-node isoparametric element with full integration scheme (3×3); AQ8-M1, the frist pattern of
this paper with full integration scheme (7 sample points in each triangular region); AQ8-M2, the frist pattern of this
paper with full integration scheme (3×3). All stress solutions are obtained by the direct method described in previous
section.
Example 1. Constant strain/stress patch test. A small patch is discretized into some arbitrary elements with some
arbitary elements, as shown in Fig. 5. The displacement fields corresponding to the constant strain are:
u = 10 −3 ( x + y / 2) , v = 10 −3 ( y + x / 2)
The exact stress solutions are as follows:
σ x = σ y = 1333.3333 , τ xy = 400
Figure 5: Constant strain/stress patch test
Table 1 Constant strain/stress patch test
Element
Q8
AQ8-M1
AQ8-M2
Element
Q8
AQ8-M1
AQ8-M2
Node 15 (0.20, 0.10)
u(×10-3) v(×10-3)
0.25
0.20
0.25
0.20
0.25
0.20
Node 15 (0.20, 0.10)
u(×10-3) v(×10-3)
0.25
0.20
0.24965 0.19754
0.25037 0.20036
σx
σy
1333.3
1333.3
1333.3
1333.3
1333.3
1333.3
σx
σy
1333.3
1339.0
1316.5
1333.3
1287.7
1353.0
Mesh 1
Node19 (0.12, 0.08)
τxy
u(×10-3) v(×10-3)
400.0
0.1575
0.135
400.0
0.1575
0.135
400.0
0.1575
0.135
Mesh 2
Node19 (0.12, 0.075)
τxy
u(×10-3) v(×10-3)
400.0
0.1575
0.135
411.2
0.1565
0.1331
381.4
0.1616
0.1429
⎯ 710 ⎯
σx
σy
τxy
1333.3
1333.3
1333.3
1333.3
1333.3
1333.3
400.0
400.0
400.0
σx
σy
τxy
1333.3
1319.3
1346.3
1333.3
1339.9
1344.0
400.0
387.9
408.0
The displacements of the boundary nodes (5-12), are the displacement boundary conditions. The results at selected
points are listed in Table1.
From Table 1, it can be seen that for the mesh constructed only by straight lines (Mesh 1), all models can exactly pass
this test. For the mesh including curved line (Mesh 2), though elements AQ8-M1 and AQ8-M2 can not give the exact
solutions, the errors are still in a acceptable range. In fact, if refine the mesh with more elements, these errors will
disappear gradually. Thus, we say both new models can pass weak form patch test [13], which may be beneficial to
higher order problems [6].
Example 2. Constant bending moment for a thin cantilever beam (Fig. 6). The results of the deflections and stresses at
selected points are shown in Table 2.
The Table 2 clearly shows the performance of the three elements in different distorted meshes. It is interesting that the
element AQ8-M2 is insensitive to various mesh distortion.
Figure 6: Constant bending moment test
Table 2 Numerical results at selected location for Example 2
Mesh 1
σx(0,10)
σx(0,0)
v(100,0)×103
Mesh 2
σx(0,10)
σx(0,0)
v(100,0)×103
Mesh 3
σx(0,10)
σx(0,0)
v(100,0)×103
Mesh 4
σx(0,10)
σx(0,0)
v(100,0)×103
Mesh 5
σx(0,10)
σx(0,0)
v(100,0)×103
Mesh 6
σx(0,10)
σx(0,0)
v(100,0)×103
Q8
AQ8-M1
AQ8-M2
120.000
-120.000
-12.000
56.447
-74.863
-2.328
56.766
-64.622
-2.323
49.028
-82.934
-2.075
56.000
-56.000
-1.4718
10.526
-10.526
-1.2474
120.000
-120.000
-12.000
117.71
-115.21
-12.046
90.242
-67.058
-8.054
69.008
-108.09
-7.805
61.134
-60.135
-1.5665
11.621
-12.138
-1.415
120.000
-120.000
-12.000
118.22
114.67
-12.014
115.50
-122.44
-12.493
128.57
-115.81
-12.456
129.96
-129.96
-12.549
111.53
-111.53
-11.601
⎯ 711 ⎯
Exact
120.0
-120.0
-12.0
120.0
-120.0
-12.0
120.0
-120.0
-12.0
120.0
-120.0
-12.0
120.0
-120.0
-12.0
120.0
-120.0
-12.0
Example 3. Thick curving beam (Fig. 7). A cantilever thick curving beam, which is divided by one or two elements,
respectively, is subjected to a transverse force at its tip. The results of the tip vertical deflection vA are shown in Table
3. Better solutions can be obtained by AQ8-M1 and AQ8-M2 than by element Q8.
Figure 7: Bending of a thick curving beam (2 elements)
Table 3 Numerical results of tip deflection for a thick curving beam (Fig. 7)
Elements
vA
One element
AQ8-M1
AQ8-M2
70.9
42.7
Q8
30.2
Q8
77.3
Two elements
AQ8-M1
AQ8-M2
88.2
74.8
Exact
90.1
Example 4. Thin curving beam (Fig. 8). As shown in Fig. 8, a cantilever thin curving beam is subjected to a transverse
force at the tip. And it is also divided by one or two elements. Two thickness-radius ratios, (i) h/R=0.03; and (ii)
h/R=0.006, are considered. The results of the tip displacement, obtained by the elements Q8 and the present elements,
are listed in Table 4. The shape of the elements in this example becomes much narrower.
Figure 8: Bending of a thin curving beam (2 elements)
Table 4 Numerical results of tip deflection for a thin curving beam (Fig. 8)
Elements
Q8
vA
0.0037
vA
0.00015
Elements
Q8
vA
0.145
vA
0.0070
One element
AQ8-M1
AQ8-M2
Q8
h/R=0.03
0.0108
0.3836
0.0356
h/R=0.006
NA
0.3833
0.0015
Three element
AQ8-M1
AQ8-M2
Q8
h/R=0.03
0.453
0.865
0.341
h/R=0.006
0.1276
0.864
0.0214
⎯ 712 ⎯
Two elements
AQ8-M1
AQ8-M2
0.1856
0.743
0.0011
0.743
Four elements
AQ8-M1
AQ8-M2
Exact
1.000
1.000
Exact
0.714
0.915
1.000
0.103
0.913
1.000
CONCLUSIONS
In this paper, two computational strategies are proposed for generalizing QAC system to curved-side cases. The
resulting elements, AQ8-M1 and AQ8-M2, exhibit better performance than the traditional isoparametric element
model in most cases. It has also been demonstrated that the QAC method is an effcient tool for developing simple,
effective and reliable serendipity plane membrane elements.
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.
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