A direct formulation of finite element models in mechanics

This paper offers a simple valid alternative approach to the formulation of finite element models. Using two‐dimensional steady state heat conduction as a representative problem, the paper formalizes and extends a direct approach, normally restricted to constant strain triangular (CST) elements, to the construction of general finite element models. The approach is illustrated with the conventional 4, 8, and 9 noded quadrilateral elements. The sufficient conditions for convergence are explained in straightforward physical terms. The direct approach is based on the dual roles of shape functions, and does not require the use of variational methods or functional analysis. The formulation is shown to be equivalent to a weak form of Galerkin which makes explicit use of three forms of residual. This approach is intended to enable engineering students to gain a sound understanding of basic finite element methods with a minimum of mathematics.