Bivariate rational approximations of the general temperature integral

TL;DR

This paper develops highly accurate bivariate rational approximations for the general temperature integral, which arises in non-isothermal material analysis involving the Arrhenius equation with temperature-dependent frequency factors.

Contribution

It introduces a novel method for approximating the general temperature integral using rational functions optimized via quasiconvex minimization, surpassing existing methods in accuracy.

Findings

More accurate approximations than existing literature

Utilizes quasiconvex optimization and bisection method

Applicable to non-isothermal material analysis

Abstract

The non-isothermal analysis of materials with the application of the Arrhenius equation involves temperature integration. If the frequency factor in the Arrhenius equation depends on temperature with a power-law relationship, the integral is known as the general temperature integral. This integral which has no analytical solution is estimated by the approximation functions with different accuracies. In this article, the rational approximations of the integral were obtained based on the minimization of the maximal deviation of bivariate functions. Mathematically, these problems belong to the class of quasiconvex optimization and can be solved using the bisection method. The approximations obtained in this study are more accurate than all approximates available in the literature.