Differential correction and arc-length continuation applied to boundary value problems: Examples based on snap-through of circular arches

Abstract

Inspired by the application of differential correction to initial-value problems to find periodic orbits in both autonomous and non-autonomous dynamical systems, here differential correction is applied to boundary-value problems. As a numerical demonstration, the snap-through buckling of circular arches in structural mechanics are selected as examples. Due to the complicated geometrical nonlinearity in such problems, limit points and turning points might exist. In this case, the typical Newton–Raphson method commonly used in numerical algorithms will fail to cross such points. In the current study, an arc-length continuation is introduced to enable the current algorithm to capture complicated load-deflection paths. We present rigorous mathematical derivations inside the current algorithm, for both differential correction and arc-length continuation. To show the accuracy and efficiency of differential correction, we also compare results with the continuation software package COCO. The results obtained by the proposed algorithm and COCO agree well with each other, suggesting the validity and robustness of differential correction for boundary-value problems.