A robust continuation method to pass limit-point instability

Abstract

In this paper a homotopy map is proposed to pass limit points of snap-through problems encountered in geometrically nonlinear finite element analysis. In the vicinity of such points, the tangent stiffness matrix becomes ill-conditioned, which detrimentally affects the convergence of numerical schemes such as Newton–Raphson method. The proposed method overcomes this problem by tracing a well-conditioned path instead of equilibrium path in the vicinity of critical points. This allows the solution procedure to bypass the critical point without experiencing ill-conditioning. An instance of such a well-conditioned path is constructed for limit points. In particular, starting from the stable (or unstable) configuration, we compute the unstable (or stable) configuration via a robust numerical procedure. Further, since the tangent matrix derivation is consistent with the residual force computation, the quadratic convergence of Newton–Raphson method is retained.