Irons’ Patch Test

Abstract

To judge the quality of any element formulation, starting from resolving controversies that surrounded Taig’s four-node element, the patch test (The author failed to locate the first publication where Irons introduced this revolutionary concept of the patch test. In internal documents of Rolls-Royce, Derby, UK, most likely in the early sixties, definitely after the groundwork of Taig (Structural analysis by the matrix displacement method. Tech. rep., British Aircraft Corporation, Warton Aerodrome: English Electric Aviation Limited, Report Number SO 17 based on work performed ca. 1957, 1962), Irons presented the concept. In 1965 (Irons et al., Triangular elements in plate bending–conforming and nonconforming solutions. In: Proceedings of the conference of matrix methods. Wright-Patterson Air Force Base, Ohio, 1965), Irons first authored a paper on plate bending where the issue beyond constant stress elements was unavoidable.) furnished a guideline. Irons thus made the most fundamental contribution in the science of finite elements. Irons’ (co-authored) books, e.g. Irons and Razzaque (Experience with the patch test for convergence of finite elements method. In: Aziz A (ed) Mathematical foundations of the finite element method with application to partial differential equations. Academic, New York, pp 557–587, 1972), Irons and Ahmad (Techniques of finite elements. Ellis Horwood, Chichester, 1980), Irons and Shrive (Finite element primer. Wiley, New York, 1983), bear testament to his tremendous insight in analyzing spatially discretized field problems of mathematical physics. This innovative notion has inspired researchers, for example Anufariev et al. (Exactly equilibrated fields, can they be efficiently used in a posteriory error estimation? In: Physics and mathematics. Scientists notes of the Kazan University, vol 148. Kazan University, Kazan, pp 94–143, 2006), Stewart and Hughes (Comput Methods Appl Mech Eng 158(1):1–22, 1998), over the last 50 years or so (Professor Robert L. Taylor in: http://www.ce.berkeley.edu/projects/feap/example.pdf furnishes explanations with numerical examples.) and no doubt many more interesting papers will appear on this subject.