An Automatic Newton‐Raphson Scheme

Abstract

This article presents an automatic Newton‐Raphson method for solving nonlinear finite element equations. It automatically subincrements a series of given coarse load increments so that the local load path error in the displacements is held below a prescribed threshold. The local error is measured by taking the difference between two iterative solutions obtained from the backward Euler method and the SS21 method. By computing both the displacement rates and the displacements this error estimate is obtained cheaply. The performance of the new automatic scheme is compared with the standard Newton‐Raphson scheme, the modified Newton‐Raphson scheme with line search, and two other automatic schemes that are based on explicit Euler methods. Through analyses of a wide variety of problems, it is shown that the automatic Newton‐Raphson scheme is superior to the standard Newton‐Raphson methods in terms of efficiency, robustness, and accuracy.