Industrial Mathematics On The Heat Conduction Equation Application Of Inverse Problem And Its Numerical Solution,

In this dissertation, we mainly discuss on several inverse problems which arise from the industrial applications and their requirements.In Chapter 1, we introduce the steel producing process including the steel making, continuous casting, hot dip galvanizing and electro-galvanizing procedures, which are the industrial backgrounds of our mathematical problem. After that, our main achievements of the research are presented, along with the brief of the works done by other scholars in related areas. We make the comparison between these works and introduce our advantages and advances. Besides, we also provide the basic knowledge of the Stefan - Boltzmann law and its derivative conditions.In Chapter 2, we introduce the inverse problem of detecting the corrosion on inner surface of the molten steel containers, the information we need for this problem are the temperature measurement on the outer surface of the container. At first, we present the related forward problem with its corresponding theories and the numerical experiments. Then we provide the mathematical formulation of the related inverse problem, as well as its theoretic results and the reconstruction methods. The numerical examples we offer show that our schemes and techniques of both forward and inverse problem are stable and applicable.In Chapter 3, we introduce the inverse problem which arises from the requirements of quality control system in the steel sheet production process. The observation data of surface temperature supported by the thermal cameras are used to reconstruct the physical coefficients (in Stefan - Boltzmann condition) of the steel sheet surface, which can be used to find the defects on products. Similarly, we present the formulation of the forward problem first, with the corresponding numerical simulations. Then the inverse problem is introduced, followed with essential theories, its reconstruction methods as well as the regularization methods. As shown by our numerical results, the methods we offered are effective which can be taken into consideration for applications.In Chapter 4, we get a brief view on our entire research, presenting some related problem which could be considered for future works. In this dissertation, we have provided some firm theories for the forward problems under complicated areas or with hard conditions, and presented the stable schemes and applicable reconstruction methods for the inverse problems without the initial data, which is the greatest contribution of our dissertation.