Abstract
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.
Highlights
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 [1], the rectangular thin plate bending element ACM proposed by Melosh [2], and the triangular thin plate bending element BCIZ proposed by Bazeley et al [3]
The 4-node quadrilateral plane membrane element AGQ6-I was formulated by the quadrilateral area coordinate methods [17]
The second approach is the constant stress multiplier method proposed by Pian and Wu [12], and can make a nonconforming element pass the strict patch test
Summary
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 [1], the rectangular thin plate bending element ACM proposed by Melosh [2], and the triangular thin plate bending element BCIZ proposed by Bazeley et al [3]. 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 [10], QP6 by Wachspress [11], NQ6 by Pian and Wu [12], the Mathematical Problems in Engineering generalized conforming element GC-Q6 by Long and Huang [13], the quasiconforming element QC6 by Chen and Tang [14], the hybrid-stress element P-S by Pian and Sumihara [15], and the Hamilton hybrid-stress element HH4 by Cen et al [16] All these elements can pass the strict form of patch test and possess much better performance than usual 4-node conforming isoparametric element Q4. 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
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