Nonlinear transient analysis of structures has been of increasing interest to engineers by virtue of their interest in minimizing human and property damage resulting from the catastrophic failure of such structures under crash or seismic conditions. Complexities of the structural configuration and its equally complex transient response in the presence of material inelasticity make finite element modeling of such structures a very natural and plausible recourse. For solution two distinct approaches exist: 1) the vector approach and 2) the scalar approach. In the former, a mathematical model is derived on the basis of the principle of virtual work and reduces to a system of nonlinear secondorder differential equations in time. In the latter approach, a potential function associated with the energy of the model is introduced, minimization of which yields the desired equilibrium configuration. In both approaches a temporal finite-difference scheme is utilized to effectively eliminate time as a variable. In the scalar approach the problem is then reduced to a well known problem in mathematical programming; namely, the unconstrained minimization of a nonlinear function of several variables . The scalar approach, although used by previous investigators for nonlinear analysis, was, with the exception of Ref. 1, restricted to static conditions. The algorithm of Ref. 1 had difficulties in converging to correct solutions because of inherent element formulation deficiencies and the use of highly expensive and inefficient finite-difference operations for gradients, besides being restricted to stringer and frame element models. As a result, no meaningful results using energy minimization were obtained. The present formulation overcomes such limitations using analytically derived gradients, an extension of the element library coupled with consistent element formulations, and the best current variable metric update formula for use in unconstrained minimization.2. © 1979 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
