Taig’s Quadrilateral Elements

Abstract

A Clarification Taig did not use the term isoparametric,in his 1962 report (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). This seminal publication introduced his quadrilateral elements. In 1968,Ergatoudis, Irons, and Zienkiewicz coined (To the best of the author’s knowledge, the use of the term isoparametric thus ensued.):“…for lack of a better name,as the isoparametric quadrilateral.”  (Ergatoudis et al., Int J Solids Struct 4(1):31–42, 1968).In 1974, Ian C. Taig and Bruce Irons were awarded the Von Karman prize, for the introduction of isoparametric element concepts. From §3.3, page 167 of Strang and Fix (An analysis of the finite element method. Prentice-Hall, Inc., Englewood Cliffs, 1973) (Strang and Fix do not credit Taig in their monograph.):“isoparametric means that the same polynomial elements are chosen for coordinate changes for the trial functions themselves;…”

Computation vis-à-vis Calculation To distinguish calculation from computation, we observe that Taig’s interpolants along with numerical quadratures, Li et al. provide a calculation method whereas a general implementation on all quadrilaterals, including convex ones with curved boundary establishes a computational procedure (Li et al., Comput Methods Appl Mech Eng 197(51):4531–4548, 2008).

Triangular to Quadrilateral Finite Elements Ian Taig, in his 1962 report, introduced four-node elements capturing stress/strain profiles beyond Clough’s triangles that are restricted to constant stress/strain distributions. Taig’s intent was to capture bending stresses (spatially linear axial stresses/strains) in beam elements that Taig termed panels. He guaranteed linear displacement on straight boundary edges. This ensured perfect inter-element compatibility. Wilson interpreted Taig’s interpolants as the elements’ natural coordinates (Wilson, Static and dynamic analysis of structures. Computers and Structures, Inc., Berkeley, 2003). Subsequently, researchers analyzed bending in plates and shells, e.g. Zhu et al. (Int J Mod Phys B 19(01n03):687–690, 2005), by overcoming thetoo-stiffconstant stress/strain elements.

Taig’s Interpolations in the Physical (x, y)-FrameAll textbooks and industrial manuals extensively cover the plane quadrilateral elements that Taig introduced.Research and teaching materials, which can be easily accessed on the internet, need no repetition. This chapter analyzes Taig’s formulation in algebraic terms in the interest of brevity and clarity.We can derive that, in general, Taig’s parametric interpolants involve the square root of quadratic expressions in the physical (x, y) frame.For trapezoids Taig’s interpolants become rational polynomials—a quadratic in (x, y) divided by a linear expression in (x, y).Wachspress addressed this issue completely (Wachspress, A rational finite element basis. Academic, New York, 1975; Rational bases and generalized barycentrics: applications to finite elements and graphics. Springer, New York, 2015). Convex, concave, and elements with curved boundaries are treated using projective geometry concepts.

Symbolic Closed-Form ExpressionsTaig in §2.3 of his 1962 report formulated closed-form expressions of element stiffness matrices for a rectangular element that he termed “Rectangular Sheet panel.” He revisited “Triangular Sheet panel”  in §2.4 of his 1962 report and furnished closed-formexpressions for the stiffness matrix. This encouraged the author to analyze all quadrilateral elements using symbolic computational tools.

Taig’s Analysis of Trapezoidal and General Quadrilateral ElementsTaig clearly pointed out the difficulties in determining the element stiffness matrix for non-rectangular elements.He presented stiffness matrices for trapezoidal elements in closed-form (vide pages 59 and 60 of Taig (1962)—“MATRIX IV”). General convex or concave quadrilateral elements were not included in the Report.

Numerical EvaluationTaig, in §6 of his 1962 report,extensively described the steps employed in “Formulation for automatic Computation.” Programs for DEUCE computers (of 1959) were elaborated.

Impact of Taig’s WorkFollowing the spirit of the aforementioned pioneering work of Ian C. Taig, complete Mathematica codes are furnished in this textbook for scalar field problems.However,generalization to deal with elastic elements, where the displacement vector components are coupled through Poisson’s ratio, demanded looking back to Lord Rayleigh’s formulations.