Numerical solution of an inverse problem of heat sources reconstruction

• Гренкин Глеб Владимирович

  Gleb V. Grenkin. Vladivostok State University. Vladivostok. Russia

The inverse problem of reconstruction of unknown intensities of spatially distributed heat sources is considered. The spatial distribution of each source is known, and the average temperature of each source is prescribed as additional data. The heat transfer process is considered to be stationary, that is, the simulated fields are independent of time. The steady state of the thermal process is modeled by a system of two elliptic differential equations with boundary conditions of the third kind, describing radiative-conductive heat transfer in a bounded domain. Conductive heat transfer obeys the equation of thermal conductivity, and the diffusion approximation of the radiation transfer equation is used to simulate radiative heat transfer. An iterative method is proposed that calculates successive approximations for the amount of thermal and radiative energy at the sources. The method constructs a sequence of inverse problems solutions for the linear equation of thermal conductivity.
The convergence of the algorithm will be achieved by fulfilling the following property: with an increase in the sum of thermal and radiation energy at all sources, the increase in thermal energy at each source will not exceed the increase in total energy. Though the solution of the inverse problem has not been proven as an unique one, computational experiments do not allow us to identify cases of non-uniqueness of the solution. The practical significance of the method lies in the possibility to implement gradual heating of heat sources up to the prescribed average temperature values without overheating them. Theoretically, the proposed algorithm can serve as a starting point for analyzing
the uniqueness of the inverse problem solution.
Keywords: radiative heat transfer, diffusion approximation, inverse problem, integral overdetermination.