An explicit formula for calculating the validity range for vector Padé approximants in the ANM algorithm

Abstract

In this work, we propose an analytical formula that allows the determination of the validity range of a vectorial Padé approximant. The purpose of this formula is to reduce the computation time required for this determination. Indeed, as the search for this domain is done, generally by applying the dichotomy method [1] to the relative error between two consecutive approximants, functions of a parameter ”a”, one needs a considerable computation time. The technique used is to approach this error by its truncated Taylor development to order 3 with respect to the parameter ”a” and to give the explicit form of its root which characterizes an approximation of the validity range.
Recall that in the Asymptotic Numerical Methods (ANM) [1], the vectorial Padé approximants are constructed from the solution of a partial differential equation written in the form of a vector series developed with respect to the parameter ”a”. The main objective of these approximants is to widen the validity range of this series.
The choice of the Padé approximant used in this work is a special case of a class of approximants that appears from a generalization of the definition of the scalar Padé approximant to the vector case [2]. In this choice, the coefficients of the Padé approximant are those which minimize the norm of the difference of this approximant and the series representation whose calculation is deduced from the orthogonalization of the vectors of the series. This corresponds to the closest approximant of the vector series.
A comparison between this new validity range and that determined numerically by the dichotomy method is made on examples of structural mechanics. The examples studied are those of buckling of the shells where the structures are discretized by the Finite Element Method (FEM) using the DKT18 element.