On the use of Padé approximant in the Asymptotic Numerical Method ANM to compute the post-buckling of shells

Abstract

In this paper, we present a new ANM continuation algorithm with a predictor based on a new Padé approximant and without the use of a correction process. The ANM is a numerical method to obtain the solution of a nonlinear problem as a succession of branches [1–7]. Each branch is represented by a vectorial series which is obtained by inverting only one tangent stiffness matrix. The series representation can be replaced by a rational representation which reduces the number of branches necessary to obtain the entire branch of desired solution. In this work, we discuss the use of a new vectorial Padé approximants in the ANM algorithm. In previous works [1, 2, 4, 5], the Padé approximants have been introduced after an orthonormalization of the terms of the vectorial series. In a recent article [8], we have demonstrated that the coefficients biM of the Padé approximants can be chosen in an arbitrary manner. For this purpose, we propose a new choice of vectorial Padé approximants {UpM} at order M which minimizes the relative error between two consecutive vectorial Padé approximants {UpM} at order M and {UpM−1} at order M − 1. This minimization has been made by a judicious choice of the coefficients biM of the Padé {UpM} at order M as a function of coefficients biM−1 of the Padé approximants {UpM−1} at order M − 1 obtained by the classical process of Gram-Schmidt orthonormalization. A comparison of the obtained results with those computed by the use of classical Padé approximants is presented.