GEOMETRICAL HYPOTHESES AND PHYSICAL INADEQUACY OF SERENDIPITY FINITE ELEMENTS

Abstract

The physical inadequacy of standard serendipity finite elements (FEs) is referred to as the "paradox of gravitational repulsion," i.e., negative nodal loads from a uniform mass force. Professor O. Zenkiewicz who, together with Ergatoudis and Irons discovered serendipitous FEs, was the first to notice this peculiarity. He believed that this drawback could not be eliminated. This conclusion is shared by most experts who use the matrix method of constructing basis functions. By analyzing Taylor's non-matrix method, it is possible to find the causes of negative loads in the angular nodes of the FEs. This is a sort of "revenge" of the mathematical model for idealization. The standard algorithm of Taylor incorporates geometric hypotheses of forming linear intermediate (non-angular) basis surfaces. In cases where a reconciliation between geometric hypotheses and physical adequacy of serendipity FE models is required, it is better to avoid matrix analysis. It is important to find a constructive way to reduce the average application of intermediate basis surfaces. Such surfaces, under the Lagrange interpolation hypothesis, are suitable for the Taylor method and can successfully replace traditional conoids. Conoids can be preserved if one turns to trigonometric directions. The popularity of trigonometric bases in the FEM is growing, and the potential of the elegant Taylor method has not yet been exhausted. Regarding the Q12 cubature, it should be noted that it is efficient and gives accurate results even when only four (or three) nodes out of twelve are "working". We propose doubly convex surfaces with negative curvature instead of linear surfaces (conoids) with zero Gauss curvature. The effectiveness of such a modification of the Taylor method is illustrated by examples of serendipity FEs: Q8 (biquadratic), Q12 (bicubic), and mixed Q10 (quadratic-cubic). The bases of alternative FEs and the integral and local characteristics of the new models are presented.