A novel step-length adjustment method for adaptive path-following in geometrically nonlinear problems of solid mechanics is proposed in this paper. The core idea is how to predict the critical points on the equilibrium path properly, and then distribute the points on the path accordingly. We characterize the equilibrium path employing a scalar stiffness parameter. Here, we show how the stiffness parameter could be used to detect the critical points on the path. Then, we employ time-series forecasting based on group method of data handling (GMDH) to predict the stiffness parameter which in fact leads to the prediction of critical points. To adaptively adjust the steplength to the predicted critical point, we present a simple formulation to distribute the points on the equilibrium path. In this paper, a novel adaptive path-following algorithm is presented to trace and predict the path efficiently. The proposed algorithm, minimum residual displacement method (MRDM), arc length method (ALM) and generalized displacement method (GDM) are comparatively investigated for the geometrically nonlinear analysis of structures in both continuum and discrete problems (truss and cylindrical shell). Selected examples present large fluctuations in stiffness. We demonstrate with the numerical examples that the trained neural network can properly predict the critical points on the path. The results also illustrate that our adaptive path-following method is able to distribute converged points properly and trace the equilibrium path with a significantly reduced number of points compared to the analyses with the uniform step-length.

Keywords