Abstract
In the fields of continuous casting and the roll stepped cooling, the heat transfer coefficient is piecewise linear. However, few papers discuss the solution of the backward heat conduction problem in this situation. Therefore, the aim of this paper is to solve the backward heat conduction problem, which has the piecewise linear heat transfer coefficient. Firstly, the ill-posed of this problem is discussed and the truncated regularized optimization scheme is introduced to solve this problem. Secondly, because the regularization parameter is the key factor for the regularization method, this paper presents an improved method for choosing the regularization parameter to reduce the iterative number and proves the fourth-order convergence of this method. Furthermore, the numerical simulation experiments show that, compared with other methods, the improved method of fourth-order convergence effectively reduces the iterative number. Finally, the truncated regularized optimization scheme is used to estimate the initial temperature, and the results of numerical simulation experiments illustrate that the inverse values match the exact values very well.
Highlights
There has been extensive research on the backward heat conduction problem
In order to reduce the iterative number, this paper presents an improved method of fourth-order convergence, the detail process of which is given in the following
The inverse results recovered by the truncated regularized optimization scheme are shown in Figure 3 ( δ = 0.01 )
Summary
There has been extensive research on the backward heat conduction problem. Wang et al [1]. A number of studies [11,12,13,14,15,16] have proposed the stability and application of regularization method for the backward heat conduction problem. The backward heat conduction problem (1) from the end temperature u(x, T ) to the whole temperature u(x, t) is severely ill-posed because the solving process considers the measuring error and the computational error. For this ill-posed problem, recovering the temperature of slab from the measuring data is very difficult.
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