In a recent paper' the convergence of various finite element models was studied using the following procedure :
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A finite element molecule is obtained algebraically for a representative nodal point i which involves the values of the function at surrounding nodes along with parameters characterizing the geometry of the element.
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The values of the function at surrounding nodes are rewritten in terms of the values of the derivatives at node i by introducing appropriate Taylor-series expansions of the function about i. The resulting equation contains only the values of the function and its derivatives at i, characteristic dimensions of the elements and load terms.
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Through a limiting process, element dimensions are allowed to become arbitrarily small. 4. Overall convergence of the model is based on the criterion that the differential expressions obtained after the limiting process of Step 3 should coincide with the governing differential equation for the problem at hand at each node i.
Among the conclusions reached by following this procedure is that results obtained by representing certain curved structures by straight-line elements will not converge to results obtained from 'exact' solutions of the differential equations for curved members.
It is the purpose of this note to point out several pitfalls inherent in the convergence test outlined above.
Consider, for example, the fourth-order differential equation governing the deflection o(x) of a simply supported beam subjected to a load p(x):
Suppose that the beam is subjected to a centrally located concentrated load P[i.e. p(x) = PS We now attempt to obtain an approximate solution d to equation (1) by assuming, for example, (xa m .