Approximation of the inverse Langevin function revisited

Abstract

The main purpose of this paper is to provide an easy-to-use approximation formula for the inverse Langevin function. The mathematical complexity of this function makes it unfeasible for an analytical manipulation and inconvenient for computer simulation. This situation has motivated a series of papers directed on its approximation. The best known solution is given by Cohen. It is used in a lot of statistically based models of rubber-like materials. The formula is derived from rounded Padé approximation [3/2]. The main idea of the presented approach in this paper relies on improvement of the precision of approximation formula for the inverse Langevin function by using multipoint Padé approximation method. We focused our study strongly on obtaining a simple and accurate approximation. It is assumed that the proposed approximation formula may be considered a useful tool for verification of the results obtained in other ways. Our results are supported by investigating several numerical examples. The paper also presents a few applications of computer software named Mathematica which can be used to calculate symbolically one point Padé approximants and numerically multipoint Padé approximants. Using this software, we showed also how to compute higher order derivatives of the inverse function in a simple and elegant way. This issue was discussed by Itskov et al.

A simple and accurate approximation of the inverse Langevin function

“…In comparison to the recently proposed approximation by Jedynak (2015), its maximal relative error with respect to the original inverse Langevin function is slightly higher. However, our formula appears to be the exactest one among all other Padé approximations around the pole y = 1.…”

“…2. Accordingly, the maximal error of the proposed approximation is about 2.8 % and thus slightly over that one by Jedynak (2015). However, for y ≥ 0.76 and thus around the pole, our formula appears to be the exactest one among all other Padé approximations.…”

“…In the context of the statistical theory of rubber elasticity, this area corresponds to the chain extensibility limit and is thus very important. In comparison to Jedynak (2015), the proposed approximation (22) requires also one division less and is thus numerically slightly faster, which can be critical for a large number of repeating calculations as for example within the finite element procedure. The numerical efficiency becomes especially pronounced by contrast with very complex but more accurate non-rational approximations by Bergström (1999) and recently proposed by Nguessong et al (2014).…”

Mechanochromic response of pyrene functionalized nanocomposite hydrogels

“…where N is the number of statistical links in a chain between crosslinks, chain = 1 3 ( 2 + 2 ) is the chain stretching in uniaxial extension, is a network parameter associated with the mechanical stiffness, and L −1 (x) ≈ x 3−x 2 1−x 2 is the inverse Langevin function [51]. The functions C 1 and C 2 are related to the geometric constraints imposed on the network and depend on the stretch .…”

Padé approximants for large-amplitude oscillatory shear flow

“…We call each ratio the [x,y] Padé approximant, where x is the order of the numerator, and y, of the denominator. Curiously, when the starting function is itself a truncated series, one or more of the Padé approximants may represent the full series more accurately than the starting function (and sometimes, much more accurately), and rheologists sometimes exploit this useful property (Cohen 1991;Jedynak 2015).…”