Numerical Method for Solving Nonlinear Heat Transfer Problems in Thin Layers

Abstract

we consider a nonlinear inverse heat conduction problem arising in high-speed, high-temperature technological processes where real-time information about internal temperature fields in thin layers is required. A key feature of this problem is that the thermophysical properties of the interlayer material change depending on its current thermal state.
Mathematically, the problem of calculating temperatures at internal points of the layer is described by a parabolic equation whose coefficients depend on the solution, with initial and boundary conditions reflecting the interlayer’s initial thermal state and the temperature variation near its outer boundary.
We present a new approach to solving the inverse problem based on approximating the coefficient functions and a computational scheme implementing this approach. The scheme uses finite differences and incorporates regularization algorithms to ensure stability. This approach and scheme form the basis of a method for calculating temperature fields in a thin interlayer.
We verified the method through computational experiments, solving the inverse problem and comparing the results with test functions. The results confirm the reliability and sufficient accuracy of the proposed computational scheme.