Consistent linearization for path following methods in nonlinear FE analysis

Abstract This paper is focussed on path following methods which are derived from consistent linearizations. The linearization procedure leads to some well-known constraint equations—like the constant arc length in the load-displacement space—and to different formulations than those given in the literature. A full Newton scheme for the unknown quantities (displacements and load parameter) can be formulated. A comparison of the derived algorithms with other path following methods is included to show advantages and limits of the methods. Using the linearization technique together with scaling a family of path following methods is introduced. Here, the scaling bypasses physical inconsistencies associated with mixed quantities like displacements and rotations in the global vector of the unknowns. Several possible scaling procedures are derived from a unified formulation. A discussion of these methods by means of numerical examples shows that up to now the choice of the scaling procedure is problem-dependent. If the arc-length methods are combined with a modified Newton method, an enhancement of the algorithms is achieved by line search techniques. Here, a simple but efficient line search was implemented and compared with a numerical relaxation technique. Both methods improve the convergence rate considerably.