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 Pade approximants, reducing the average and maximum relative errors admitted by the forms of the approximants. The error-minimization technique was extended to improve standard Pade 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.