INTRODUCTION AND MOTIVATIONTwo of the earliest successful triangular bending elements are still widely used (a) The non-conforming triangle1 of 1965.(b) The subdivided conforming HCT triangle,a also of 1965.Both these elements have the simplest nodal variable vide w, awlax, aw/ay at each of the three vertices.However, the first converges only for certain mesh patterns and the second is nowadays regarded as stiff. Pian's3 concept of hybrid elements applied to plates by Severn and Taylor? Allwood5 and Cornes and Allmane provide an attractive alternative. Allman's twelve degrees of freedom (d.0.f.) element assumes linear variation of bending moments inside the triangle and a cubic representation of the displacement together with a quadratic variation of the normal slope along the element boundaries. In the second model of Allman (this model was earlier formulated by Allwood) the mid-side node is eliminated by imposing a linear variation of the normal slope along the element boundaries. This element has thus w, awlax, aw/ay at each of the corners as d.0.f. so that the connection properties are identical with those of References 1 and 2. But convergence is now guaranteed for any mesh pattern and the element performs consistently better than those of References 1 and 2. However, the usual formulation of hybrid elements is cumbersome. Moreover, with hybrid elements it is not an easy matter to introduce a new type of force (even a uniformly distributed load) or to adapt a bending element to other biharmonic problems (e.g. a hydrodynamic problem). On the other hand, the 'displacement type' elements such as the isoparametric elements7** can be very easily implemented especially if the various element matrices are formed by numerical integration using a shape function subroutine.?
BENDING ELEMENTS WITH DERIVATIVE SMOOTHINGRecently Irons and R a z z a q~e~.~~ have developed two synthetic triangular bending elements by employing 'smoothed' derivatives in place of the 'true' derivatives of the shape function for the corresponding 'strict' displacement elements. For example, the shape functions Nd (describing the transverse displacement w) for the twelve d.0.f. 'conforming displacement triangle' comprise the complete cubic polynomial terms, together with two independent rational functions as given by equation 10.29 of Reference 11. (The rational functions are needed to ensure slope conformity.) The second derivatives N ; of these Ni have singularities at the corners, and as a result a very high order numerical integration is necessary to give convergence. In an effort to produce cheaper but converging elements Irons and Razzaque substitute these true derivatives, N ; by 'pseudo'-derivatives N t " , where N:" are the least squares (best fit) linear polynomial version of N;.
SUMMARYThe main purpose of this paper is to emphasize the necessity of convergence tests in general and the patch test in particular. Following a brief introduction, the patch test is described with examples. This, it is hoped, will help to remove any lingering misunderstanding of this very useful convergence test. Some positive contributions of the test in the formulation and improvement of elements are also discussed.
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