روش برنامه ریزی پویا برای یک اصل نوع فرما برای جریان گرما

We consider nonlinear heat conduction satisfying a variational principle of Fermat type in the case of stationary heat flow. We review origins of a physical theory and transform it into a formalism consistent with irreversible thermodynamics, where the theory emerges as a consequence of the theorem of minimum entropy production. Applications of functional equations and the Hamilton–Bellman–Jacobi equation are effective when Bellman’s method of dynamic programming is applied to propagation of thermal rays. Potential functions describing minimum resistance are obtained by analytical and numerical methods. For the latter, approximation schemes are developed. Differences between propagation of thermal and optical rays are discussed and it is shown that while simplest optical rays can be described by Riemmanian geometry, it is rather Finslerian geometry that is valid for thermal rays.

Consider a steady-state heat conduction in a rigid solid. When a thermal field is imposed by fixing the thermal gradient, the flow of thermal energy can be described in terms of ‘thermal rays’, the paths of heat flow determined by the direction of the temperature gradient and nonlinear properties of the conducting medium. When the thermal conductivity changes along the length of a thermal ray, the path along which the ray moves is, in general, curvilinear. Our purpose is prediction of the shapes of thermal rays, regardless of whether their curvilinearity is caused by the thermal inhomogeneity or material inhomogeneity of the medium. Here the thermal rays are shown to travel along paths satisfying the principle of minimum of entropy production which looks at first glance quite different from the well-known Fermat principle of minimum time (minimum optical length) for optical rays. However, taking into account that the minimum of entropy production is associated with the minimum resistivity of the path, it is easy to conclude that the minimum resistivity causes (in the dual problem) the maximum of heat flux through the medium or makes the residence time of heat in the medium as short as possible. This makes the principle for travel of thermal rays quite similar to that for propagation of light [6]. Our purpose is to investigate these phenomena by the method of dynamic programming [1] showing the similarities and differences between the optical and thermal phenomena.