Effective quadrature rules for quadratic solid isopatametric finite elements

In a recent note, 1 Irons demonstrated several integration formulae for use with solid isoparametric finite elements. The use of different formulae enabled variations in accuracy and running times to be achieved, and clearly one aims for the cheapest rule for a given degree of accuracy. One of the most successful finite elements is the 20-node isoparametric solid element, and this has been used frequently in the CEGB. This note is mainly concerned with this particular element. The usual integrating rule used is the Gauss 3 x 3 x 3 rule, with 27 points per element. In certain circum­ stances, namely shell-type structures subjected to bending modes, the use of a 2 x 2 x 2 rule (8 points per element) has been shown to give good results with rapid convergence in a manner very similar to the results of Zienkiewicz and co-workers using a quadratic thick shell element. 1 However, this reduced rule does not always give satisfactory answers in membrane modes or in shells with solid attachments, and so, as with solid problems, alternative economies in integration techniques are desirable. The two most 0conomical rules giving the same order of accuracy as the Gauss 3 x 3 x 3 rule appear to be the 14 point rule mentioned in Reference I, and given originally by Hammer and Stroud5 and the slightly cheaper, slightly less accurate, 13 point rule given originally by Stroud. Details of the 14 point rule may be found in Reference 1 and are not repeated here. The 13 point rule is defined by the follow­ ing co-ordinates (in the double unit cube) and weighting coefficients: