The temperature integral cannot be analytically integrated and many simple closed-form expressions have been proposed to use
in the integral methods. This paper first reviews two types of simple approximation expressions for temperature integral in
literature, i.e. the rational approximations and exponential approximations. Then the relationship of the two types of approximations
is revealed by the aid of a new equation concerning the 1st derivative of the temperature integral. It is found that the exponential approximations are essentially one kind of rational
approximations with the form of h(x)=[x/(Ax+k)]. That is, they share the same assumptions that the temperature integral h(x) can be approximated by x/Ax+k). It is also found that only two of the three parameters in the general formula of exponential approximations are needed
to be determined and the other one is a constant in theory. Though both types of the approximations have close relationship,
the integral methods derived from the exponential approximations are recommended in kinetic analysis.
The paper describes efficient methods to post-process results from the finite element analysis. Amount of data produced by the complex analysis is enormous. However, computer performance and memory are limited and commonly-used software tools do not provide ways to post-process data easily. Therefore, some sort of simplification of data has to be used to lower memory consumption and accelerate data loading. This article describes a procedure that replaces discrete values with a set of continuous functions. Each approximation function can be represented by a small number of parameters that are able to describe the character of resulting data closely enough.
1 Introduction In [ 3 ] Fejes Tóth introduced inscribed triangulations approximating convex surfaces in ℝ 3 optimally and the approximation parameter A 2 (Approximierbarkeit). By a triangulation we shall mean a geometric realization of a
Gal, S. G. , Approximation by Complex Bernstein and Convolution Type Operators , Series on Concrete and Applicable Mathematics, 8. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. xii+337 pp. ISBN: 978
, G., On direct and converse theorems in the theory of weighted polynomial approximation, Math. Z. 126 (1972), 123-134. MR 46 : #7770
On direct and converse theorems in the theory of weighted polynomial approximation