A comparison of the finite‐difference and finite‐volume methods for a numerical solution of a hyperbolic thermoelasticity problem utilizing the implicit and explicit schemes

Abstract

A numerical solution for a coupled hyperbolic thermoelasticity problem of the Lord-Shulman type is presented. An infinite layer with free boundaries that are kept at a constant temperature is irradiated by a short laser impulse. In this case the problem can be studied in the one-dimensional formulation. An impact of the laser impulse on a medium is modeled by an internal heat source. The laser impulse intensity decays with a distance from an irradiated surface according to the Bouguer law. The layer is in an unperturbed state at the initial moment of time. The material properties of copper are taken to be the thermo-mechanical properties of the layer.Two approaches for the boundary value problem are considered: the finite-difference method (FDM) and the finite-volume method (FVM). In the first case the system of two second-order PDE is investigated, in the second case a system of integral balance equations and an integral thermal conductivity equation are composed. To integrate these equations both the explicit and implicit schemes are used. The obtained results are compared with each other and with an analytical solution. A numerical error is estimated for different time and space step sizes. The features inherent in various ways of integration of the coupled hyperbolic thermoelastic problem with the thermal excitation are found. K E Y W O R D SEuler implicit scheme, explicit leap-frog scheme, finite-difference method, finite-volume method, tridiagonal matrix algorithm, Lord-Shulman hyperbolic thermoelasticity, Maxwell-Cattaneo equation, relaxation constant, trust region algorithm