Determination of travel directions in path-following methods

Many variants of continuation methods have been put forward in the last decade to trace solution paths in nonlinear static structural problems. Since ‘forward’ or ‘backward’ directions at a point along the solution path could not be identified without further reference to the whole solution procedure, solution paths may be incorrectly followed. To this end, determination of the correct travel directions in every aspect of continuation methods is discussed in this paper. In addition, a new criterion used to determine the sign of the loading parameter increment in the predictor phase of the nonconsistent arc-length method has been proposed, which may avoid a misleading travel direction of the continuation method due to inappropriate initial guesses. The new criterion can be incorporated into both direct and iterative solvers, and has proved to be very successful for the test problems considered.