A method for creating a class of triangular $C^{1}$ finite elements Finite elements providing a $C^{1}$ continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order, such as beam and plate bending and deformation of strain-gradient-dependent materials. Although a few $C^{1}$ elements have been presented in the literature, their development has largely been heuristic, rather than the result of a rational design to a predetermined set of desirable element properties. Therefore, a general procedure for developing $C^{1}$ elements with particular desired properties is still lacking.par This paper presents a methodology by which $C^{1}$ elements, such as the TUBA3 element proposed by {it J. H. Argyris} et al.[“The TUBA family of plate elements for the matrix displacement method”, Aeronautical Journal of the Royal Aeronautical Societ, 72, No. 692, 701--709 (1968)], can be constructed. In this method (which, to the best of our knowledge, is the first one of its kind), a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry, the location of the nodes and the possible types of nodal DOFs. A set of necessary conditions is then imposed to obtain appropriate interpolations. Generic procedures are presented, which determine whether a given potential member of the element class meets the necessary conditions. The behaviour of the resulting elements is checked numerically using a benchmark problem in strain-gradient elasticity.

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