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

The inverse Langevin function is a fundamental part of the statistical chain models used to describe the behavior of polymeric-like materials, appearing also in other fields such as magnetism, molecular dynamics and even biomechanics. In the last four years, several approximants of the inverse Langevin function have been proposed. In most of them, optimization techniques are used to reduce the relative error of previously published approximants to reach orders of magnitude of O ( 10 − 3 % − 10 − 2 % ) . In this paper a new simple and efficient numerical approach to calculate the inverse Langevin function is proposed. Its main feature is the reduction of the relative errors in all the domain x = [ 0 , 1 ) to near machine precision, maintaining function evaluation CPU times similar to those of the most efficient approximants. The method consists in the discretization of the Langevin function, the calculation of the inverse of these discretization points and their interpolation by cubic splines. In order to reproduce the asymptotic behavior of the inverse Langevin function, a rational function is considered only in the asymptotic zone keeping C 1 continuity with the cubic splines. We include customizable Matlab codes to create the spline coefficients, to evaluate the function, and to compare accuracy and efficiency with other published proposals.

Related papers:

title: Effect of the inverse Langevin approximation on the solution of the Fokker–Planck equation of non-linear dilute polymer
abstract: The Langevin function is defined by L ( x ) = coth ( x ) − 1 / x . Its inverse is useful for many applications and especially for polymer science. As the inverse exact expression has no analytic representation, many approximations have been established. The most famous approximation is the one traditionally used for the finitely extensible non-linear elastic (FENE) dumbbell model in which the inverse is approximated by L − 1 ( y ) = 3 y / ( 1 − y 2 ) . Recently Martin Kroger has published a paper entitled ‘Simple, admissible and accurate approximations of the inverse Langevin and Brillouin functions, relevant for strong polymer deformation and flows’ (Kroger, 2015) in which he proposed approximations with very reduced error in relation to the numeric inverse of the Langevin function. The question we aim to analyze in this short communication is: when one uses the traditional approximation rather than the more accurate one proposed by Kroger is that really significant regarding the value of the probability distribution function (PDF) in the frame work of a kinetic theory simulation? If yes when we move to the upper scale by evaluating the value of the stress, can we observe a significant difference? By making some simple 1D simulations in homogeneous extensional flow it is demonstrated in this short communication that the PDF prediction within kinetic theory framework as well as the macroscopic stress value are both affected by the quality of the approximation.

title: Simple, admissible, and accurate approximants of the inverse Langevin and Brillouin functions, relevant for strong polymer deformations and flows
abstract: 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 Pade solution approach that is typically used to obtain coefficients for approximate forms, and recommend a simple rational function L FENE - 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 FENE - 1 ( y ) = 3 y / ( 1 - y 2 ) (50% maximal relative error) or also L Cohen - 1 ( y ) = L FENE - 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.

title: Accuracy of bead-spring chains in strong flows
abstract: We have analyzed the response of bead-spring chain models in strong elongational flow as the amount of polymer represented by a spring is changed. We examined the longest relaxation time of the chains which is used to quantify the strength of the flow in terms of a Weissenberg number. A chain with linear springs can be used to predict the longest relaxation time of the nonlinear chains if the linear spring constant is modified correctly. We used the expansion of the elongational viscosity in the limit of infinite Weissenberg number to investigate the change of the viscosity as the scale of discretization was changed. We showed that the viscosity is less sensitive to the details of the spring force law because the chain is fully extended at very large Weissenberg number. However, the approach to that infinite Weissenberg number response is dependent both on the behavior of the spring force at large force and the behavior at small force. New spring force laws to represent the worm-like chain or the freely jointed chain are correct at both of these limits, while other currently used force laws produce and error. We also investigated the applicability of these expansions to chains including hydrodynamic interactions. Our results suggest that the longest relaxation time may not be the appropriate time scale needed to non-dimensionalize the strain rate in such highly extended states.

title: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials
abstract: Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944,  illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.

title: A micro-macro approach to rubber-like materials—Part I: the non-affine micro-sphere model of rubber elasticity
abstract: The contribution presents a new micro-mechanically based network model for the description of the elastic response of rubbery polymers at large strains and considers details of its numerical implementation. The approach models a rubber-like material based on a micro-structure that can be symbolized by a micro-sphere where the surface represents a continuous distribution of chain orientations in space. Core of the model is a new two-dimensional constitutive setting of the micro-mechanical response of a single polymer chain in a constrained environment defined by two micro-kinematic variables: the stretch of the chain and the contraction of the cross section of a micro-tube that contains the chain. The second key feature is a new non-affine micro-to-macro transition that defines the three-dimensional overall response of the polymer network based on a characteristic homogenization procedure of micro-variables defined on the micro-sphere of space orientations. It determines a stretch fluctuation field on the micro-sphere by a principle of minimum averaged free energy and links the two micro-kinematic variables in a non-affine format to the line-stretch and the area-stretch of the macro-continuum. Hence, the new model describes two superimposed contributions resulting from free chain motions and their topological constraints in an attractive dual geometric structure on both the micro- and the macro-level. Averaging operations on the micro-sphere are directly evaluated by an efficient numerical integration scheme. The overall model contains five effective material parameters obtained from the single chain statistics and properties of the network with clearly identifiable relationships to characteristic phenomena observed in stress–strain experiments. The approach advances features of the affine full network and the eight chain models by a substantial improvement of their modeling capacity. The excellent predictive performance is illustrated by comparative studies with previously developed network models and by fitting of various available experimental data of homogeneous and non-homogeneous tests.

title: The mechanical behavior of skin: Structures and models for the finite element analysis
abstract: Soft biological tissues are complex materials with a large structural variety, with differences in behavior, but with some common characteristics. Skin is an archetypal soft tissue which presents many common characteristics to other soft biological tissues, like being a multilayer collagen-reinforced structure, with nonlinear behavior, anisotropy, viscosity, preconditioning effects, internal stresses and tissue growth and adaptation. Departing from a detailed description of the structures of the skin and the experimental evidence, we herein analyze the different modeling approaches in the literature for the distinct aspects of the skin behavior, with attention to the implementation in finite element codes.

title: Mechanics of Solid Polymers: Theory and Computational Modeling
abstract: Very few polymer mechanics problems are solved with only pen and paper today, and virtually all academic research and industrial work relies heavily on finite element simulations and specialized computer software. Introducing and demonstrating the utility of computational tools and simulations, Mechanics of Solid Polymers provides a modern view of how solid polymers behave, how they can be experimentally characterized, and how to predict their behavior in different load environments. Reflecting the significant progress made in the understanding of polymer behaviour over the last two decades, this book will discuss recent developments and compare them to classical theories. The book shows how best to make use of commercially available finite element software to solve polymer mechanics problems, introducing readers to the current state of the art in predicting failure using a combination of experiment and computational techniques. Case studies and example Matlab code are also included. As industry and academia are increasingly reliant on advanced computational mechanics software to implement sophisticated constitutive models - and authoritative information is hard to find in one place - this book provides engineers with what they need to know to make best use of the technology available. * Helps professionals deploy the latest experimental polymer testing methods to assess suitability for applications* Discusses material models for different polymer types* Shows how to best make use of available finite element software to model polymer behaviour, and includes case studies and example code to help engineers and researchers apply it to their work

title: Taylor expansion of the inverse function with application to the Langevin function
abstract: A Taylor power series is a powerful mathematical tool, which can be used to express an inverse function especially if it is given in an implicit form. This is for example the case for the inverse Langevin function, which is an indispensable ingredient of full-network rubber models. In the present paper, we propose a simple recurrence procedure for calculating Taylor series coefficients of the inverse function. This procedure is based on the Taylor series expansion of the original function and results in a simple recurrence formula. This formula is further applied to the inverse Langevin function. The convergence radius of the resulting series is evaluated. Within this convergence radius the obtained approximation of the inverse Langevin function demonstrates better agreement with the exact solution in comparison to different Padé approximants.

title: New facts concerning the approximation of the inverse Langevin function
abstract: The inverse Langevin function is an integral component for statistically-based network models which describe rubber-like materials. This function is very important from theoretical and practical points of view in variety of scientific fields like rheology, theory of magnetism and polymer science. It cannot be expressed in an explicit form and directly used for analytical manipulation and computer simulation. For these reasons, a lot of researchers have considered this function to provide its optimal approximation. We discuss the latest achievements in our work. We deliver three very important facts about the examined function. We check the hypothesis that an increase in the number of terms of the Taylor series expansion of the inverse Langevin function above 500, does not change its convergence radius. It is true in the light of our detailed analysis. Our achievements are clearly documented in this paper. This analysis is extended up to 3000 terms of the Taylor series expansion, while previous reports include only 500. We verify the statement that the solution based on 115 Taylor series terms shows the best accuracy within the interval [0,0.95]. To be exact, it is true but in a little smaller interval. We find the best new solutions for two intervals [0,0.95] and [0,0.98]. This is the second fact. The last fact is related to rational approximation of this function. We provide a new rational approximation formula, whose maximum relative error is equal to 0.076%. So far, such a high precision was restricted only to very complex approximation formulas. We also include a program written in Mathematica to show application of our new formula on the basis of the known three-chain model.

title: A Padé approximant to the inverse Langevin function
abstract: Application of the methodology of Pade approximants to a Taylor expansion of the inverse Langevin function led to an accurate analytical expression. The approximation, retaining a finite extendibility of the Langevin spring, enables a convenient analysis of experimental data and analytical manipulations of material models.

title: Mechanistic Constitutive Models for Rubber Elasticity and Viscoelasticity
abstract: Physically based models which describe the finite strain behavior of vulcanized rubber are developed. Constitutive laws for elasticity and viscoelasticity are derived by integrating over orientation space the forces due to each individual polymer chain. A novel scheme is presented which effectively approximates these integrals in terms of strain and strain invariants. In addition, the details involving the implementation of such models into a quasi-static large strain finite element formulation are provided. In order to account for the finite extensibility of a molecular chain, Langevin statistics is used to model the chain response. The classical statistical model of rubber assumes that polymer chains interact only at the chemical crosslinks. It is shown that such model when fitted for uniaxial tension data cannot fit compression or equibiaxial data. A model which incorporates the entanglement interactions of surrounding chains, in addition to the finite extensibility of the chains, is shown to give better predictions than the classical model. The technique used for approximating the orientation space integral was applied to both the classical and entanglement models. A viscoelasticity model based on the force equilibration process as described by Doi and Edwards is developed. An assumed form for the transient force in themore » chain is postulated. The resulting stress tensor is composed of an elastic and a viscoelastic portion with the elastic stress given by the proposed entanglement model. In order to improve the simulation of experimental data, it was found necessary to include the effect of unattached or dangling polymer chains in the viscoelasticity model. The viscoelastic effect of such chains is the manifestation of a disengagement process. This disengagement model for unattached polymer chains motivated an empirical model which was very successful in simulating the experimental results considered.« less

title: 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.

title: A simple and accurate approximation of the inverse Langevin function
abstract: The inverse Langevin function cannot be represented in an explicit form and requires an approximation by a series, a non-rational or a rational function as for example by a Padé approximation. In the current paper, an analytical method based on the Padé technique and the multiple point interpolation is presented for the inverse Langevin function. Thus, a new simple and accurate approximation of the inverse Langevin function is obtained. It might be advantageous, for example, for non-Gaussian statistical theory of rubber elasticity where the inverse Langevin function plays an important role.

title: A new based error approach to approximate the inverse langevin function
abstract: In the present paper, we propose a new approximation of the inverse Langevin function. This new approximation is based on a two-step modification of the fractional formula introduced by (Cohen 1991). Our proposal is motivated by the minimization of the error between the Cohen formula and the inverse of the Langevin function. It results in two additional terms adopting a remarkable simple power and polynomial forms. The correction provides an excellent agreement with a maximum relative error equal to 0.046 % (against a maximum error of 4.94 % for the Cohen formula).

title: An error-minimizing approach to inverse Langevin approximations
abstract: The inverse Langevin function is an integral component to network models of rubber elasticity with networks assembled using non-Gaussian descriptions of chain statistics. The non-invertibility of the inverse Langevin often requires the implementation of approximations. A variety of approximant forms have been proposed, including series, rational, and trigonometric divided domain functions. In this work, we develop an error-minimizing framework for determining inverse Langevin approximants. This method can be generalized to approximants of arbitrary form, and the approximants produced through the proposed framework represent the error-minimized forms of the particular base function. We applied the error-minimizing approach to Padé approximants, reducing the average and maximum relative errors admitted by the forms of the approximants. The error-minimization technique was extended to improve standard Padé approximants by way of understanding the error admitted by the specific approximant and using error-correcting functions to minimize the residual relative error. Tailored approximants can also be constructed by appreciating the evaluation domain of the application implementing the inverse Langevin function. Using a non-Gaussian, eight-chain network model of rubber elasticity, we show how specifying locations of zero error and reducing the minimization domain can shrink the associated error of the approximant and eliminate numerical discontinuities in stress calculations at small deformations.

title: Improved approximations for some polymer extension models
abstract: 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.

title: Finite element modeling of human skin using an isotropic, nonlinear elastic constitutive model.
abstract: The collagen network in skin is largely responsible for the nonlinear mechanical stress-strain response of skin. We hypothesize that the force-stretch response of collagen is governed by the entropics of long-chain molecules. We show that a constitutive model derived from the statistical mechanics of long-chain molecules, corresponding to the fibrous collagen network in skin, captures the mechanical response of skin. A connection between the physiologically meaningful parameters of network molecular chain density and free length of collagen fibers and the constitutively significant parameters of initial modulus and limiting stretch is thus established. The relevant constitutive law is shown to have predictive capabilities related to skin histology by replicating in vivo and in vitro experimental results. From finite element simulations, this modeling approach predicts that the collagen network in hypertrophic scars is more dense and the constituent collagen fibers have shorter free lengths than in healthy skin. Additionally, the model is shown to predict that as rat skin ages, collagen network density increases and fiber free length decreases. The importance of knowledge of the in situ stress state for analyzing skin response and validating constitutive laws is also demonstrated.

title: Finite element simulations of orthotropic hyperelasticity
abstract: Given the geometry and boundary conditions that characterize most in vivo tests on biological soft tissue as well as the material heterogeneity that can be present in both in vivo or in vitro tests, it is often necessary to use finite elements to model the response of tissue to a specified loading protocol. The ability of ABAQUS to simulate such tests is explored using a newly developed, microstructurally-based orthotropic hyperelastic constitutive law. Simulations are performed to model simple tests for which there are "exact" solutions in order to verify the implementation of the constitutive law into ABAQUS and more complex boundary value problems are simulated to explore the limits of the computational model.

title: A microstructurally based orthotropic hyperelastic constitutive law
abstract: A constitutive model is developed to characterize a general class of polymer and polymer-like materials that displays hyperelastic orthotropic mechanical behavior. The strain energy function is derived from the entropy change associated with the deformation of constituent macromolecules and the strain energy change associated with the deformation of a representative orthotropic unit cell. The ability of this model to predict nonlinear, orthotropic elastic behavior is examined by comparing the theory to experimental results in the literature. Simulations of more complicated boundary value problems are performed using the finite element method.

title: A constitutive model for finite deformation of amorphous polymers
abstract: The paper introduces a three-dimensional constitutive model for the mechanical behavior of amorphous polymers, thermosets and thermoplastics. The approach is formulated in terms of finite deformations, appropriate for glassy polymers. The rheology of the model consists of a Langevin-type free energy function for the energy storage due to molecular alignment connected in parallel to a Maxwell element with a viscoplastic dashpot. The model proves successful for the constitutive description of glassy polymers over a large range of strain rates. To capture the smooth softening behavior upon yielding is the main purpose of this research. It is reached under consideration of absolute temperature and current strain rate with the proposed evolution law for the viscoplastic dashpot deformation. The rate-dependence of amorphous polymers is reproduced as well as the pressure dependence during different loading scenarios. A fully implicit numerical scheme appropriate for the finite element implementation is presented. The modeling capability of the proposed approach is demonstrated for epoxy, PC and PMMA. The efficiency of the proposed numerical scheme is demonstrated via a necking simulation of a flat PC coupon.

title: Material-symmetries congruency in transversely isotropic and orthotropic hyperelastic materials
abstract: In this work we present the material-symmetries congruency property for anisotropic hyperelastic materials. A transversely isotropic or orthotropic material model holding this property guarantees the recovery of the isotropic behavior when, for example, the volume of reinforcement in a composite material vanishes. A material presenting material-symmetries congruency must show it not only from an analytical, theoretical point of view, but also from a numerical, practical point of view. The former may be obtained from construction of the stored energy function, whereas the latter must be obtained guaranteeing that the parameter-fitting procedure yields material parameters which combined with the specific model results in an isotropic behavior prediction for isotropic materials. We show that some anisotropic models do not present either analytical or numerical material-symmetries congruency.

title: Determination and Finite Element Validation of the WYPIWYG Strain Energy of Superficial Fascia from Experimental Data
abstract: What-You-Prescribe-Is-What-You-Get (WYPIWYG) procedures are a novel and general phenomenological approach to modelling the behavior of soft materials, applicable to biological tissues in particular. For the hyperelastic case, these procedures solve numerically the nonlinear elastic material determination problem. In this paper we show that they can be applied to determine the stored energy density of superficial fascia. In contrast to the usual approach, in such determination no user-prescribed material parameters and no optimization algorithms are employed. The strain energy densities are computed solving the equilibrium equations of the set of experiments. For the case of superficial fascia it is shown that the mechanical behavior derived from such strain energies is capable of reproducing simultaneously the measured load-displacement curves of three experiments to a high accuracy.

title: Extension of the Sussman-Bathe spline-based hyperelastic model to incompressible transversely isotropic materials
abstract: In this paper we extend the Sussman-Bathe spline-based hyperelastic isotropic model to predict the behavior of transversely isotropic isochoric materials. The model is based on an uncoupled decomposition of the stored energy function and a generalization of the inversion formula used by Sussman and Bathe. The present extension introduces some approximations that, in all studied cases, do not yield relevant deviations from the experimental data. The isotropic model results in a particular case of the present formulation. Several possibilities of user-prescribed experimental data are addressed. The model is used to predict experiments of calendered rubber and of biological tissues.

title: Determination of the WYPiWYG strain energy density of skin through finite element analysis of the experiments on circular specimens
abstract: Skin is a biological material which mechanical behavior has large variations depending on the individual and the location of the specimen in that individual, among other factors. Large differences are also encountered in measurements between in vivo and in vitro specimens. Then, optimal characterization of the skin for simulation (for example) of surgical procedures requires that all experiments to characterize the material behavior be performed on the same specimen and in vivo if possible. Recent experiments on circular discs (Groves et al., 2013 ) permit this characterization using a single specimen as we show in this paper, and may constitute a good starting point for ulterior characterization in vivo. However, because in these tests deformations are not homogeneous, the determination of the material behavior is not as direct as with tensile or biaxial tests, so finite element analysis is needed to propose a procedure to determine the material behavior. In this work we perform an analysis of the experiments using finite elements obtaining an insight which permits a very simple iterative procedure to determine the stress-strain behavior of the material and, thereafter, the corresponding What-You-Prescribe-Is-What-You-Get (WYPiWYG) stored energy densities.