Theoretical And Algorithmic Study On The Inverse Problem Of Heat Conduction In Composite Materials

This paper mainly studies the problem of heat source identification for the steady-state heat conduction equation of double-layer composite materials.The heat source we considered is linearly proportional to the temperature.Through the transformation of polar coordinates and spherical coordinates,we study the problem of heat source identification in two-dimensional radial,non-radial and three-dimensional non-radial.Regarding how to identify the heat source,this paper uses the measured boundary temperature data or the flux data on the boundary to identify the heat source.According to the partial differential equation theory,we construct an analytical solution based on Bessel function theory for the double-layer heat conduction boundary value problem.We then introduce Dirichlet-toNeumman(DN)or Neumman-to-Dirichlet(ND)mapping.By the mapping,we establish the relationship between the source term and the boundary temperature.And the convergence analysis of the DN or ND mapping of the double-layer composite material and the single-layer material is also studied.In order to enable the boundary measurement data to uniquely identify the source item,we establish the uniqueness theorem of identification.Furthermore,we also discuss the source item cannot be uniquely identified by DN or ND mapping when the coefficient of the source and inner radius of the double-layer structure satisfy certain conditions,that is,a non-uniqueness theorem is established.This article is developed in the following three parts: In the first part,the problem of source term identification related to positive linear temperature is studied,and based on the theory of the bessel function and modified bessel function,the convergence analysis,uniqueness and non-uniqueness theorems of two-dimensional radial,two-dimensional non-radial and threedimensional radial are established.In the second part,the theoretical results of the first part are further extended to the problem of source term recognition that is negatively linearly related to temperature.The same convergence analysis,uniqueness and non-uniqueness theorems are established as in the first part.The third part,at the end of this paper,we numerically simulates the identification of source terms in two-dimensional radial.