Accelerating the nonlinear analysis of hyper-elastic behavior by GMDH-assisted Newton-Raphson scheme

Abstract

This paper presents an accelerated iterative scheme for nonlinear problems. Commonly, analysis of nonlinear behavior is conducted by the Newton-Raphson (NR) method. It is well-known that the number of iterations required depends on the deviation between the “starting point” and the converged solution. In practice, the solution of previous load step is taken as the “starting point”, while the converged solution of the current load step is not known beforehand. Therefore, difficulties or even non-convergence may occur. Recently, it is suggested that a neural network is employed to predict the solution of the current load step. This prediction is then used as the “starting point” for NR scheme. It is expected, that the true converged solution (of current step) is closer to the prediction by neural network than to the solution of previous load step. As a result, the scheme becomes faster due to less iterations. Obviously, any techniques for time-series forecasting can be used. Here, the Group Method of Data Handling (GMDH) is proposed. Loosely speaking, GMDH is a feedforward neural network without backpropagation. Practically, the GMDH-assisted NR scheme should not take longer time than conventional NR scheme. The advantage of GMDH is fast computation; however, the accuracy may be not as high as a network that has backpropagation. Therefore, careful consideration on the construction of GMDH network is needed. In the current work, the performance of GMDH-assisted NR scheme is investigated in analysis of hyper-elastic behavior, which involves both geometrical and material nonlinearity. A study on the influence of activation function on the accuracy is presented. Also, it is found that prediction for incremental displacement (between the current load step and the previous load step) could be better than prediction of displacement of the current load step.