Non-dimensional FEM analysis of a functionally graded thermopiezoelectric rod subjected to a moving heat source

A functionally graded one-dimensional piezoelectric rod excited by a moving heat source is considered. Its dynamic multifield response under this type of thermal load is investigated in the context of coupled Lord-Shulman thermopiezoelectricity theory.In the literature, the transient response of piezoelectric materials to a moving heat source is extensively studied, but in most cases the solution is obtained using Laplace transform. In this paper we propose a completely numerical approach to solve the problem.The original initial boundary value problem is transformed into a variational one. The variational problem is then rewritten in a non-dimensional form using appropriate variable substitutions. For space discretization the finite element method is used and for time discretization we utilize a hybrid time integration scheme based on Newmark scheme for hyperbolic equations and generalized trapezoidal rule for parabolic equations. The approximation of Dirac delta function by means of Gaussian probability density function is used to simulate the moving heat source.The developed numerical scheme is validated against the benchmark solutions available in the literature and our results show great agreement with them. Besides, we empirically analyze the energy conservation properties of proposed time integration scheme and verify spatial and temporal convergence of the obtained solutions. Finally, we study the influence of non-homogeneity index, thermal relaxation time and moving heat source velocity on the solutions of the problem.