Detection of bifurcation points along a curve traced by a continuation method

The methods for computing the singular points based on approximations of the equilibrium curve by asymptotic extrapolation usually have a limited range of validity. This is because the power series are only built on the local properties of the primary solution branch. This paper suggests ways to improve such a situation: the asymptotic extrapolation is used in the predictor phase of a continuation method and an e:ective parametrization equation is adopted. In this way the description of the equilibrium curve between two consecutive solution points is improved while the local nature of the extrapolation is retained. The bifurcation points computed on the extrapolation are then used to initialize a classical method for the detection of the bifurcation points on the exact equilibrium curve. Moreover the accurate location of the limit points is an immediate consequence of the precision obtained in the description of the solution curve in the predictor–corrector step. Overall the computational cost of the analysis is appreciably reduced. In e:ect a low number of predictor–corrector steps are required to complete the description of the solution curve and most of the computations performed in the detection of bifurcation points are carried out on the extrapolation. The good initialization obtained also makes Newton’s classical method for calculating the bifurcation points on the exact equilibrium curve more reliable with regard to the convergence. Generalization to cases of more bifurcation points within two consecutive solution points is then obtained by a non-linear subspace procedure de>ned on the asymptotic extrapolation. Applications to frame structures and cylindrical shells analysis are presented