An error-minimizing approach to inverse Langevin approximations

Marchi, Benjamin C.; Arruda, Ellen M.

doi:10.1007/s00397-015-0880-9

Cited by 27 publications

(43 citation statements)

References 26 publications

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“…In Figure 4 it can be observed that all approximants employ similar computational times, except the Nguessong et al [17] and the Marchi and Arruda [18] approximants. The reason seems to be that the efficiency of power products with integer exponents is about four times better than those using real exponents, as it can easily be checked in Matlab.…”

Section: Computational Efficiencymentioning

confidence: 86%

“…However, we consider that the error spectrum in the full domain gives a better picture because as seen below, achievable machine errors are also different in different parts of the domain. In Figure 2 we show a comparison of the present proposal using n = 10 5 and x r = 0.98 with the approximants of Kröger [2], Petrosyan [19], Nguessong et al [17], Marchi and Arruda [18], and Jedynak [11]. The formulae used for these approximants are given in the references and in the code in Appendix B.…”

Section: Load-type Accuracymentioning

confidence: 99%

“…The order of magnitude of the error has not improved substantially with the error minimizing techniques. With this optimization methods the order of magnitude of the relative error is O (10 −2 − 10 −3 %) [11,17,18].…”

Section: Introductionmentioning

confidence: 99%

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A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

Benítez

Montáns

2018

Journal of Non-Newtonian Fluid Mechanics

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Section: Computational Efficiencymentioning

confidence: 86%

Section: Load-type Accuracymentioning

confidence: 99%

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A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

Benítez

Montáns

2018

Journal of Non-Newtonian Fluid Mechanics

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“…This high accuracy will be at the cost of the simplicity of the expression. Such accurate approximations, with more than one correction steps, have been obtained for the invers Langevin function previously 31,32 .…”

Section: Appendixmentioning

confidence: 96%

Improved approximations for some polymer extension models

Petrosyan

2016

Rheol Acta

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We propose approximations for force-extension dependencies for the freely jointed chain (FJC) and worm-like chain (WLC) models as well as for extension-force dependence for the WLC model. Proposed expressions show less than 1% relative error in the useful range of the corresponding variables. These results can be applied for fitting force-extension curves obtained in molecular force spectroscopy experiments. Particularly, they can be useful for cases where one has geometries of springs in series and/or in parallel where particular combination of expressions should be used for fitting the data. All approximations have been obtained following the same procedure of determining the asymptotes and then reducing the relative error of that expression by adding an appropriate term obtained from fitting its absolute error. * rafayel.petrosyan@alumni.ethz.ch 2

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“…Cohen (1991) derived an approximation based upon a [3/2] Padé approximant of the inverse Langevin function which has been widely used and is fairly accurate over the whole domain of definition. Many more accurate approximations have been devised, see for example, Treloar (1975), Puso (1994), Nguessong et al (2014), Darabi & Itskov (2015), Kröger (2015), Marchi & Arruda (2015), Rickaby & Scott (2015) and Jedynak (2015Jedynak ( , 2017. However, rather than finding further approximations to the inverse Langevin function, we emphasize in this paper how to find the approximate positions of its singularities in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

On the complex singularities of the inverse Langevin function

Rickaby¹,

Scott

2018

IMA Journal of Applied Mathematics

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We study the inverse Langevin function L −1 (x) because of its importance in modelling limited-stretch elasticity where the stress and strain energy become infinite as a certain maximum strain is approached, modelled here by x → 1. The only real singularities of the inverse Langevin function L −1 (x) are two simple poles at x = ±1 and we see how to remove their effects either multiplicatively or additively. In addition, we find that L −1 (x) has an infinity of complex singularities. Examination of the Taylor series about the origin of L −1 (x) shows that the four complex singularities nearest the origin are equidistant from the origin and have the same strength; we develop a new algorithm for finding these four complex singularities. Graphical illustration seems to point to these complex singularities being of a square root nature. An exact analysis then proves these are square root branch points. which has a removable singularity at y = 0 and is defined on the domain −∞ < y < ∞. All the above models, however, involve the inverse Langevin function defined byThese are both odd functions though on physical grounds y and x may be restricted to be positive. It is the property L −1 (x) → ∞ as x → 1 that makes the inverse Langevin function so useful in modelling limited-stretch elasticity. The Arruda & Boyce [1] eight-chain model has proved to be the most successful of these models, both theoretically and experimentally. However, Beatty [2] has shown that quite remarkably the Arruda & Boyce [1] stress response holds in general in isotropic nonlinear elasticity for a full network model of arbitrarily orientated molecular chains. Thus the eight-chain cell structure is unnecessary and so the inverse Langevin function has an important role to play in the theory of the finite isotropic elasticity of rubber and rubber-like materials.The inverse Langevin function cannot be expressed in closed form and so many approximations to it have been devised and applied to many models of rubber elasticity. Perhaps the simplest is to consider the Taylor series but this does not converge over the whole domain of definition −1 < x < 1 of the inverse Langevin function, see [12] and [11]. Cohen [4] derived an approximation based upon a [3/2] Padé approximant of the inverse Langevin function which has been widely used and is fairly accurate over the whole domain of definition. Many more accurate approximations have been devised, see for example, [5,14,15,16,18,20,21,23,24]. However, rather than finding further approximations to the inverse Langevin function, we emphasise in this paper how to find the approximate positions of its singularities in the complex plane. We are also able to find exactly the positions and nature of these singularities.This paper is structured as follows. In Section 2 we give a brief account of non-linear isotropic elasticity theory as applied to the limited-stretch theory of elasticity of, for example, [1] and [2]. In Section 3 we define and discuss the Langevin and inverse Langevin functions, giving Taylor series fo...

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An error-minimizing approach to inverse Langevin approximations

Cited by 27 publications

References 26 publications

A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

Improved approximations for some polymer extension models

On the complex singularities of the inverse Langevin function

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