The direct problem of heat transfer calculation is to solve the temperature distribution in the calculation area when the boundary conditions are known. Correspondingly, the inverse problem of heat transfer calculation is to estimate the heat exchange coefficient on the boundary when the temperature distribution in the calculation area is known. The solution of the heat transfer coefficient of the casting mold interface is a typical inverse problem.

Extrapolation is a method to obtain the temperature change trend of different positions in the mold by fitting the temperature measured by the temperature measuring equipment, and then extrapolate to the mold surface to obtain the mold surface temperature and heat flow. It includes curve fitting and polynomial extrapolation.

Curve fitting method, also known as linear extrapolation method, is a relatively simple method to calculate the interface heat transfer coefficient, which can be used to study the influence of various process factors on the interface heat transfer coefficient. Griffiths et al. And Tillman et al. Used thermocouples to measure the temperature of each measuring point in the casting and mold, and used curve fitting method to obtain the heat transfer coefficient of the casting mold interface. Wang et al. Measured the internal temperature change of metal substrate during solidification process after metal droplet dropping on the surface of metal substrate by experimental means, and calculated the interface heat transfer coefficient by curve fitting method, and obtained the influence law of surface roughness and overheating on the interface heat transfer coefficient.

Polynomial extrapolation method is a method which uses polynomial curve to fit the temperature of each measuring point inside the casting and mold, and extrapolates the surface temperature of the casting and mold, so as to obtain the interface heat transfer coefficient. Brunner et al. Used polynomial extrapolation method to calculate the interfacial heat transfer coefficient between Ni based superalloy and shell in investment casting. Aweda et al. Obtained the heat transfer coefficient of casting die interface in squeeze casting process of industrial pure aluminum by polynomial extrapolation method, and compared it with Beck inverse algorithm. Sun et al. Used polynomial extrapolation method to calculate the heat transfer coefficient of the casting die interface in the squeeze casting process of AM60 magnesium alloy.

Although the extrapolation method is relatively simple to calculate the interface heat transfer coefficient, the calculation error is easy to enlarge and the temperature measurement accuracy is too high.