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

In this paper, we present a new ANM continuation algorithm with a predictor based on a new Pade 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 Pade approximants in the ANM algorithm. In previous works [1, 2, 4, 5], the Pade 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 b(i)(M) of the Pade approximants can be chosen in an arbitrary manner. For this purpose, we propose a new choice of vectorial Pade approximants {U-p(M)} at order which minimizes the relative error between two consecutive vectorial Pade approximants {U-p(M)} at order M and {U-p(M-1)} at order M - 1. This minimization has been made by a judicious choice of the coefficients bMi of the Pade {U(M)p(}) at order M as a function of coefficients b(i)(M-1) of the Pade approximants {U-p(M-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 Pade approximants is presented.