Will the force method come back

The force method of structural analysis was overtaken by the displacement method in the mid Sixties, and has disappeared from the scene except for few specialized applications. Virtually all existing general-purpose finite element programs are based on the direct stiffness method (DSM) introduced three decades ago by Turner et al. (1956) and Turner (1959). The DSM combines the displacement method of solution with the direct, element-by-element assembly of the stiffness equations. The simplicity and efficiency of the DSM for general applications have not been matched to date; it has the polished "black box" feeling of Lagrange's analytical mechanics. There has been, however, a modest revival of interest in the force method as manifested in recent publications (Kaneko et al., 1982, 1985; Heath et al., 1984; Patnaik, 1986a, 1986b). This activity has been fueled by the hope that the force method may be competitive for a limited but important class of problems, namely those calling for a sequence of analyses of "perturbed" linear or materially-nonlinear structures, provided the perturbations do not affect the discrete equilibrium equations. This situation arises in fully-stressed design and optimization. The main weakness of the conventional force method on the computer is the difficulty of automating the selection of force unknowns that optimize matrix sparseness while maintaining numerical stability. Twenty years ago Fraeijs de Veubeke (1965, p. 83) noted, "A great step forward in the automation of the (self-straining) calculations would be achieved if the computer itself could be taught to investigate the topology of the (connection) matrices and deduce the self-strainings confined to the smallest numbers of elements." Recent developments try to satisfy this goal by taking advantage of more advanced numerical techniques that were known in 1965 (for example, sparse orthogonal factorizations). The purpose of this Note is to give the general formulation of the force method from the standpoint of the field of mathematical programming, and to call attention to links between this formulation and recent efforts in the field of matrix structural analysis. The relationship is noteworthy because great strides have been made in large-scale constrained optimization during the past 15 years while the force method has been neglected. It is hoped that the relations described here may spur further research work as well as development of computer-based applications using the abundant software now available in scientific libraries for linear and nonlinear programming.