Stay Cartesian, or go natural? A comment on the article 'Supernatural QUAD4: A template formulation' by C. A. Felippa [Comput. Methods Appl. Mech. Engrg., 195 (2006) 5316-5342]

摘要

In a very interesting review and critical study of a five decade long search for an element free of locking and distortion sensitivity, Felippa poses a very intriguing question: "Stay Cartesian, or go natural?" The strategy recommended is: "Stay Cartesian, for the basic stiffness, but go natural for the higher order one". In a series of papers published recently, Rajendran and co-workers have introduced what they call the unsym-metric finite element formulation to improve predictions from severely distorted finite elements. They introduce a special formulation using two separate sets of shape functions, viz., the so-called compatibility (or continuity) enforcing shape functions and the so-called 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. Numerical results from test problems reveal that if the natural (parametric) functions are used as test functions describing the virtual strains but the Cartesian (metric) functions serve as trial functions describing the real stresses (in the context of the virtual work formulation of Strang and Fix [5]), the formulation works surprisingly well because the former satisfy the continuity conditions while the latter ensure that the stress representation during finite element computation can retrieve in a best-fit manner, the actual variation of stress in the metric (Cartesian) space.