A shape‐free 8‐node plane element unsymmetric analytical trial function method

Abstract

SUMMARYThe unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress-function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourthorder completeness in Cartesian coordinates. The resulting element model, denoted by US-ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US-ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems.

Rotationally invariant distortion resistant finite-elements

“…Before presenting the numerical results of this section, it is worth highlighting the numerical results presented by Cen et al [3]. That paper compares the unsymmetric approach of Rajendran and co-workers [16,17] …”

“…The first analysis in this paper is a comparison of the approaches of Rajendran and co-workers [16,17] (PM), Cen et al [2] (HSF-Q8-15β) and Cen et al [3] (US-ATFQ8) with the PML element proposed in this paper. This plane stress linear elastic cantilever beam problem has been previously analysed by a number of authors [2,3].…”

“…This plane stress linear elastic cantilever beam problem has been previously analysed by a number of authors [2,3]. The beam, of length l 0 = 10m and depth d 0 = 5m, had a Young's modulus of 100Pa and a Poisson's ratio of 0.3.…”

“…The percentage error of the L2 norm of the displacement of node A compared with a reference displacement of 0.054m [2,3] for these elements, along with the published results of [2,3], are presented in Table 1. The PP, PML, HSF-Q8-15β and US-ATFQ8 formulations are invariant to rigid body rotations whereas the PM element is not. The HSF-Q8-15β outperforms the other elements, with the exception of mesh (iii) where the US-ATFQ8 has the lowest error.…”

“…A recent paper by Cen et al [3] proposed an element that combined the unsymmetric approach of Rajendran and co-workers [16,17] with Cen et al's [2] HSF formulation. The element overcame the rotational invariance and interpolation failure problems of the unsymmetric formulation by replacing the metric shape function with basis functions from the Airy stress function solution.…”