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An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal’s theorem. (English) Zbl 1352.74158
Summary: Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal’s theorem: any 4-node, 8-DOF membrane element will either lock in in-plane bending or fail to pass a $$C_{0}$$ patch test when the element’s shape is an isosceles trapezoid. In this paper, a 4-node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4-node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates ($$x$$,$$y$$) and the second form of quadrilateral area coordinates (QACM-II) ($$S$$,$$T$$) are applied together. The resulting element US-ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first-order patch test and the second-order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM-II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well-known contradiction defined by MacNeal’s theorem.

##### MSC:
 74K15 Membranes 74S05 Finite element methods applied to problems in solid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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