Numerical and analytical modeling of heat equation in current-carrying conductors using the heat equation implemented using Finite-JAX

Abstract

Current-carrying conductors inevitably experience resistive heating due to the material's finite electrical conductivity. The resulting temperature distribution within the wire has essential implications for structural integrity, efficiency, and long-term reliability of electronic and power systems. In this work, we model the steady-state heat distribution in a current-carrying wire using the classical heat conduction equation. In a two-dimensional formulation, heat transport is considered both along and across the conductor. The governing partial differential equation is discretized using finite-difference methods implemented in Finite-JAX, with appropriate initial and boundary conditions, including the Dirichlet condition relevant to practical scenarios. Time integration is performed using the Euler explicit scheme, and stability constraints are systematically examined. To assess the accuracy of the numerical approach, we compare the computed temperature fields with the exact analytical solution of the heat equation for canonical geometry. Results show that the numerical prediction converges toward the analytical solution, with error norms decreasing at the expected order of accuracy. This study demonstrates how the heat equation provides a rigorous mathematical foundation for modeling resistive heating in conductors. The errors between the numerical and analytical solutions are 1.981 K, 0.8975 K, and 0.7917 K, corresponding to the L-infinity, L1, and L2 norms. For tensor computations, TPUs deliver the highest performance, surpassing GPUs and CPUs.