A novel data-driven nonlinear solver for solid mechanics using time series forecasting

In this paper, a novel data-driven nonlinear solver (DDNS) for solid mechanics using time series forecasting is first proposed. The key concept behind this work is to modify the starting point of iterations of the modified Riks method (M-R). The modified Riks method starts iterations at the previously converged solution point while the proposed method starts at a predicted point which is very close to the converged solution of the current step. In the prediction phase, the predicted starting point of the current step is simply determined only based on the previously converged solutions and the predictive networks built via group method of data handling (GMDH) known as a self-organizing deep learning method for time series forecasting problems. Then, the correction phase of the modified Riks method is used to obtain the converged solution via an iterative procedure starting at the predicted point. In this work, the training and applying processes of networks are continuously performed during the analysis to predict the starting point of each increment. It is interesting that the present deep learning networks are built with small data in very short time. Especially, the proposed method is not only simple in implementation but also reduces significantly number of iterations and computational cost compared with the conventional modified Riks method. Some benchmark problems on geometrically nonlinear analysis of shells are provided and solved by using isogeometric analysis (IGA) in conjunction with the first-order shear deformation shell theory (FSDT). The high accuracy, efficiency and stability of the proposed method are confirmed.