Semi-analytic solution of three-dimensional temperature distribution in multilayered materials based on explicit frequency response functions

Multilayer coatings have been widely using in a wide range of applications, including industrial, biological and electrical areas, and the thermal distribution in a multilayered material is of great interest. In present paper, the frequency response functions (FRFs) of temperature field under unit point heat flux are derived through thermal conduction equation. The unknown coefficients in the FRFs are assembled in a linear system of matrix equations according to the heat input and continuity condition of heat flux and temperature at each interface; then the coefficients are solved and expressed recursively. Based on the closed-form solution of FRFs, a fast semi-analytical method (SAM) is developed to solve the three-dimensional steady state heat conduction in arbitrary multilayered materials, and there are no limits on the number or the thickness of layers. The temperature fields under different kinds of heat flux in multilayered coatings are studied. Moving heat flux and convection on the surface are also considered.