Several Treatments on Nonconforming Element Failed in the Strict Patch Test

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.