A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

Abstract

SUMMARYAn 8-node quadrilateral plane ÿnite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter-as well as intra-element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement ÿeld. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement ÿeld under all types of admissible mesh distortions. In this respect, the proposed 8-node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8=9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement ÿeld, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8=9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2 × 2 numerical integration.

Rotationally invariant distortion resistant finite-elements

“…The unsymmetric distortion resistant formulations proposed in the literature are based on using a polynomial test and global metric trial basis [12,16,17,18,8]. However, it is well known that these global metric derivatives produce element formulations that are dependent on the orientation of the element.…”

“…These components can then be directed into the appropriate locations within (16). The key point to notice from (17), is that the deformation gradient is evaluated using the derivatives of the trial shape functions. The remainder of the finite deformation framework follows the approach of Coombs [4], albeit with a total, rather than updated, description of motion.…”

“…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].…”

“…Rajendran and Liew [17] presented an unsymmetric 8-noded quadrilateral element, reporting that this element could exhibit immunity to any kind of mesh distortion under a quadratic displacement field. They named their new element the US-QUAD8 (unsymmetric quadrilateral element with eight nodes).…”

“…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.…”