A simple extrapolated predictor for overcoming the starting and tracking issues in the arc-length method for nonlinear structural mechanics

TL;DR

This paper introduces a straightforward extrapolated predictor for the arc-length method in nonlinear structural mechanics, improving path tracking efficiency and accuracy without complex techniques or small step sizes.

Contribution

It proposes a simple, cost-effective extrapolated predictor for the arc-length method that enhances equilibrium path computation in nonlinear structural problems.

Findings

Successfully computes complex equilibrium paths in various structural models

Achieves excellent agreement with reference solutions

Does not require very small load increments

Abstract

This paper presents a simplified implementation of the arc-length method for computing the equilibrium paths of nonlinear structural mechanics problems using the finite element method. In the proposed technique, the predictor is computed by extrapolating the solutions from two previously converged load steps. The extrapolation is a linear combination of the previous solutions; therefore, it is simple and inexpensive. Additionally, the proposed extrapolated predictor also serves as a means for identifying the forward movement along the equilibrium path without the need for any sophisticated techniques commonly employed for explicit tracking. The ability of the proposed technique to successfully compute complex equilibrium paths in static structural mechanics problems is demonstrated using seven numerical examples involving truss, beam-column and shell models. The computed numerical results are in excellent agreement with the reference solutions. The present approach does not require prohibitively small increments for its success.