Simulation of Temperature Effects in Massive Bodies Using the Modified Method of Lines

Abstract

This article presents the main ideas and possibilities of the modified straight line method for modeling temperature effects in massive bodies. The calculation model of a rectangular beam with a defined thermal state is adopted. The procedure for reducing the dimensionality of the original equations using the modified method of straight lines and the Bubnov-Galyorkin-Petrov method for boundary conditions is applied. The reduced equation of thermal conductivity and the reduced equation of heat balance are given. Conclusions are made regarding the addition of reduced second-order equations with components from the boundary conditions on the lateral surfaces, and finally reduced second-order equations in spatial variables are given.
 We are considering a beam of rectangular cross-section, the three dimensions of which are of the same order. The thermal state of such an object is three-dimensional and is therefore described systematically using three-dimensional heat conduction equations. When applying the modified method of straight lines to reduce the dimensionality of the original equations, it is convenient to use them in the form of a system of partial differential equations of the first order in spatial coordinates.
 According to the modified method of straight lines, two systems of straight lines parallel to the coordinate axes Oy and Oz are applied to the beam (a decrease in dimensionality in terms of spatial variables y and z is assumed). The choice of such straight lines is due to the fact that the reduced equations will depend on the variables x and t, that is, they will resemble the one-dimensional equations of the one-dimensional non-stationary problem of thermal conductivity, for which efficient and convenient numerical methods have been developed in the literature of mathematical physics at the moment, to which it is easy to adapt the reduced equation.