Simple, admissible, and accurate approximants of the inverse Langevin and Brillouin functions, relevant for strong polymer deformations and flows

M Kroger, JOURNAL OF NON-NEWTONIAN FLUID MECHANICS, 223, 77-87 (2015).

DOI: 10.1016/j.jnnfm.2015.05.007

Approximants to the inverse Langevin and Brillouin functions appear in diverse contexts such as polymer science, molecular dynamics simulations, turbulence modeling, magnetism, theory of rubber. The exact inverses have no analytic representations, and are typically not implemented in software distributions. Various approximants for the inverse Langevin function L-1 had been proposed in the literature. After proving asymptotic features of the inverse functions, that had apparently been overlooked in the past, we use these properties to revisit this field of ongoing research. It turns out that only a subset of existing approximations obeys the relationships. Here we are able to derive improved (or 'corrected') approximations analytically. We disqualify the classical Fade solution approach that is typically used to obtain coefficients for approximate forms, and recommend a simple rational function L-FINE(-1)(y)/(1 + y(2)/2) that has a maximum relative error of 1-2 orders of magnitude smaller compared with the usually employed approximations L-FINE(-1)(y) = 3y/(1 - y(2)) (50% maximal relative error) or also L-cohen(-1)(y) = L-FINE(-1)(y) (1 - y(2)/3) (4.94%), while it is exactly as efficiently implemented and convenient for analytic (both force and energy) calculations. A number of applications is worked out. Moreover is the strategy proposed in this manuscript general and can be equally applied to fitting problems when asymptotic features are known. (C) 2015 The Author. Published by Elsevier B.V.

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