Optimum/adaptive incremental sequence in nonlinear analysis

Abstract

An approach to improve the accuracy of the incremental solutions to a nonlinear problem, through a strategy to control the size of the increment, based on stationary of an argumented energy functional, is presented. The problem of control of an optimum step size in the incremental theory is formulated for a fixed number of increments. The variables in this argumented functional are: (i) the incremental displacement vector, (ii) the scalar parameters λ _i which characterize the size of each of the increments, i = 1,..., N, and (iii) a Lagrange multiplier μ which enforces the constraint that the sum of all the normalized increments, i. e., Σλ_i is equal to 1. The optimality condition provides us a rigorous approach which gives rise to an iterative procedure because of nonlinearity of the stationary condition. If the number of increments is not prescribed, a noniterative procedure can be obtained, where the incremental sequence is controlled adaptively with less computational effort. The extension of the proposed method to non-selfadjoint problems, where a potential energy function does not exist, is also discussed. Numerical examples demonstrate the remarkable improvement in the accuracy of the solution by optimizing the incremental sequence, as well as the effectiveness of the adaptive control procedure proposed.