Padé approximants applied to a non-linear finite element solution strategy

Abstract

The present work deals with an asymptotic numerical method, based on Pade approximants. The expected advantage of this method is twofold. Firstly, it reduces the computational costs. Secondly, the automatization of the continuation process becomes easier, since the step-length can be determined a posteriori. So far, this method has only been applied to DKT elements. Here it is applied to other types of elements, namely truss elements and ®nite rotation non-linear shell elements. It will be shown that diculties arise when this method is applied to ®nite rotation shell elements.

High‐order predictor–corrector algorithms

“…In any case, the expansion parameter is the one given by formula (4). However, we have limited ourselves to corrections with a ÿxed Reynolds number in uid mechanics (12) whereas the calculus of thin shells has been done with a corrector using the arc-length procedure (14), (15).…”

“…These algorithms depend on a lot of parameters: ÿrst the orders of truncation of the series N pred and N corr , second the control parameters 1 , 2 and 3 , third the choice of the homotopy transformation (formulae (12) or (14) and (15)), and last the choice of the matrices that are actually triangulated during the prediction and the correction. We now focus on the latter point, by deÿning three basic strategies of combining a predictor and a corrector.…”

“…Moreover, it has been proposed to replace the truncated series by rational fractions, that are known in the literature under the name of Padà e approximants, see References [11][12][13][14]. It is tedious to associate a continuation method with these rational representations because of the poles that can appear in the denominator of the rational approximations.…”