Adaptive step size selection for use with the continuation method

An adaptive procedure for selecting the step size when incremental or continuation methods are used to solve sets of non-linear equations is presented. The increment size is limited by requiring the corrective iteration procedure employed to reduce the drifting error to be within a contractive boundary at each level. The usefulness of the procedure is extended by the development of a set of conditions for detecting impending divergence of the corrective iteration process. These conditions, used in conjunction with the step size selection procedure permit the continuation of the solution through highly non-linear regions and also provide a simple means of isolating a limit point, if one exists. Two additional benefits of the procedure are an effective convergence criterion for terminating the iteration process and a simple means for switching between the Newton-Rephson and modified Newton-Raphson iteration procedures. The paper concludes with a number of example problems, three are hyperelastic bodies at finite strain and the final example is the large displacement analysis of an elastic beam. The results illustrate that the procedure is computationally inexpensive, using only information normally obtained in a non-linear analysis, flexible in the sense that a dense or sparse distribution of points along the total solution curve may be obtained, and effective, normally requiring less total computational effort than a constant step size procedure.