For nonconforming finite elements, it has been proved that the models whose convergence is controlled only by the weak form of patch tests will exhibit much better performance in complicated stress states than those which can pass the strict patch tests. However, just because the former cannot provide the exact solutions for the patch tests of constant stress states with a very coarse mesh (strict patch test), their usability is doubted by many researchers. In this paper, the non-conforming plane 4-node membrane element AGQ6-I, which was formulated by the quadrilateral area coordinate method and cannot pass the strict patch tests, was modified by three different techniques, including the special numerical integration scheme, the constant stress multiplier method, and the orthogonal condition of energy. Three resulting new elements, denoted by AGQ6M-I, AGQ6M-II, and AGQ6M, can pass the strict patch test. And among them, element AGQ6M is the best one. The original model AGQ6-I and the new model AGQ6M can be treated as the replacements of the well-known models Q6 and QM6, respectively.
In order to overcome the over-stiffness problem existing in conforming finite elements and improve their performances in regular and distorted meshes, the appearance of the nonconforming elements seems to be inevitable, such as the famous 4-node quadrilateral plane element Q6 proposed by Wilson et al. [
The patch test proposed by Irons et al. [
Many efforts have been made for developing 4-node nonconforming plane elements without any problem in convergence, such as the element QM6 proposed by Taylor et al. [
In this paper, three treatments were tried to improve the convergence of the element AGQ6-I, including the special numerical integration scheme, the constant stress multiplier method, and the orthogonal condition of energy. Three resulting new elements, denoted by AGQ6M-I, AGQ6M-II, and AGQ6M, respectively, can pass the strict patch test. Numerical examples show that, among the three new models, the element AGQ6M is the best one. The original model AGQ6-I and the new model AGQ6M can be treated as the replacements of the well-known models Q6 and QM6, respectively.
The 4-node quadrilateral plane membrane element AGQ6-I was formulated by the quadrilateral area coordinate methods [
For a 4-node quadrilateral plane element, the degrees of freedom can be expressed by the nodal displacement vector:
Then, the low-order displacement fields can be assumed as
Then, the corresponding strain matrix can be expressed by
The additional displacements are assumed to be
Finally, the element stiffness matrix of the element can be expressed by
The resulting element model is the element AGQ6-I. It can be easily shown that the values of the additional displacement fields given by (
Unfortunately, as a nonconforming element like Wilson’s Q6, element AGQ6-I cannot give the exact solutions for strict patch test either. Although the results can converge to the exact solutions by subdividing the mesh (weak patch test), further improvements on its convergence are valuable.
In order to make the element Q6 [
In order to employ the previous technique to modify element AGQ6-I, the quadrilateral area coordinates can be rewritten in terms of the isoparametric coordinates:
It is obvious that the additional strain matrix
The constants in the additional strain matrix
The new model is named as AGQ6 M-I.
The second approach is the constant stress multiplier method proposed by Pian and Wu [
The final element stiffness matrix can be derived by usual static condense procedure. The resulting new element is denoted by AGQ6 M-II and can also pass the strict form of patch test.
In order to ensure the element to pass the strict patch test, the strain energy under constant stress state caused by the nonconforming strains is required to be zero:
By eliminating the constant factor in (
Thus, four constants
Substitution of (
The constant strain/stress patch test using irregular mesh is shown in Figure
Patch test.
This example was proposed by Cook et al. [
Displacement at point C of Cook’s beam.
Element |
| ||
---|---|---|---|
2 × 2 | 4 × 4 | 8 × 8 | |
Q4 | 11.80 | 18.29 | 22.08 |
Q6 | 22.94 | 23.48 | 23.80 |
QM6 | 21.05 | 23.02 | — |
AGQ6-I | 23.06 | 23.68 | 23.87 |
AGQ6M-I | 20.86 | 23.00 | 23.69 |
AGQ6M-II | 11.75 | 18.28 | 22.08 |
AGQ6M | 20.74 | 22.99 | 23.69 |
| |||
Reference value | 23.96 |
Cook’s skew beam.
Compared with the other elements, the accuracy of AQ6 M-II is even poor as Q4 element, while AGQ6 M-I and AGQ6 M are as accurate as QM6.
The cantilever beam, as shown in Figure
The deflections at point A of a cantilever beam.
Element type | Load (a) | Load (b) | ||
---|---|---|---|---|
|
|
|
| |
Q4 | 45.7 | −1761 | 50.7 | −2448 |
Q6 | 98.4 | −2428 | 100.4 | −3354 |
QM6 | 96.1 | −2497 | 98.0 | −3235 |
QC6 | 96.1 | −2439 | 98.1 | −3339 |
NQ6 | 96.1 | −2439 | 98.0 | −3294 |
AGQ6-I | 100.0 | −3000 | 102.0 | −4151 |
AGQ6M-I | 91.9 | — | 94.5 | — |
AGQ6M-II | 44.4 | — | 49.4 | — |
AGQ6M | 96.0 | −3015 | 97.9 | −4135 |
| ||||
Exact | 100 | −3000 | 102.6 | −4050 |
Cantilever beam with five irregular elements.
From the results listed in Table
Consider the thin beams presented in Figure
MacNeal’s thin beam.
There are two loading cases under consideration: pure bending and transverse linear bending. The Young’s modulus of the beam
The results of the tip deflection are shown in Table
The normalized deflections at the free end of MacNeal’s beam.
Element type | Load |
Load |
||||
---|---|---|---|---|---|---|
Mesh (a) | Mesh (b) | Mesh (c) | Mesh (a) | Mesh (b) | Mesh (c) | |
Q4 | 0.093 | 0.035 | 0.003 | 0.093 | 0.031 | 0.022 |
Q6 | 0.993 | 0.677 | 0.106 | 1.000 | 0.759 | 0.093 |
QM6 | 0.993 | 0.623 | 0.044 | 1.000 | 0.722 | 0.037 |
AGQ6-I | 0.993 | 0.994 | 0.994 | 1.000 | 1.000 | 1.000 |
AGQ6M-I | 0.993 | 0.631 | 0.050 | 1.000 | 0.722 | 0.044 |
AGQ6M-II | 0.093 | 0.034 | 0.027 | 0.093 | 0.031 | 0.022 |
AGQ6M | 0.993 | 0.632 | 0.051 | 1.000 | 0.726 | 0.046 |
| ||||||
Exact | −0.1081 | −0.0054 |
(1) both AGQ6 M-II and Q4 suffer from locking problems for the three kinds of mesh distortions ((a) length-width ratio distortion, (b) parallelogram distortion, and (c) trapezoidal distortion) of three different meshes.
(2) As a nonconforming element, Q6 still cannot overcome the locking problem for the trapezoidal mesh. Benefited from the area coordinate methods, AGQ6-I is locking free from each kind of mesh.
(3) All those elements which can pass the strict patch test are locked in the trapezoidal mesh distortion. This verified the theory proposed in [
The cantilever beam shown in Figure
Results of the tip deflection of a cantilever beam subjected to a pure bending
|
0 | 0.5 | 1 | 2 | 3 | 4 | 4.9 |
---|---|---|---|---|---|---|---|
Q4 | 28.0 | 21.0 | 14.1 | 9.7 | 8.3 | 7.2 | 6.2 |
Q6 | 100 | 78.0 | 56.1 | 42.5 | 41.5 | 44.2 | 47.4 |
QM6 | 100 | 80.9 | 62.7 | 54.4 | 53.6 | 51.2 | 46.8 |
AGQ6-I | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
AGQ6M-I | 100 | 80.8 | 55.4 | 24.1 | 12.9 | 9.4 | 7.6 |
AGQ6M-II | 28.0 | 21.2 | 14.1 | 9.2 | 7.3 | 5.9 | 4.9 |
AGQ6M | 100 | 83.8 | 66.5 | 60.1 | 61.4 | 60.3 | 56.0 |
| |||||||
Exact | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
Cantilever beam divided by two elements with distorted parameter
The accuracy of element Q4 and AGQ6 M-II is the poorest. Their relative error reaches 72% when
In order to make the nonconforming element AGQ6-I pass the strict patch test, three treatments are used and tested. These methods are special numerical integration scheme used in QM6, constant stress multiplier method, and the orthogonal condition of energy. Numerical results of numerical examples show that each method can achieve the goal, but the performance for complicated stress problem of new elements will degenerate inevitably. Among the three treatments, the constant stress multiplier method made the element so rigid that the accuracy will deteriorate even in rectangular meshes. The special numerical integration scheme can make element QM6 exhibit good performance, but it is not always suitable for all other elements. The orthogonal condition of energy can effectively improve the compatibility of element AGQ6-I without losing too much accuracy.
The authors would like to acknowledge the financial supports of the National Natural Science Foundation of China (11272181), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20120002110080), and the National Basic Research Program of China (Project no. 2010CB832701).