Conservation Properties of the Trapezoidal Rule for Linear Transient Electromagnetics

Abstract

The Time Domain Finite Element Method (TDFEM) has been used extensively to solve transient electromagnetic radiation and scattering problems. But in most implementations so far, vector basis functions have been used to discretize the field variables. In multiphysics simulations that involve coupling the electromagnetic equations with structural or fluid flow equations, nodal finite elements can provide a unified data structure for a monolithic coupled formulation. With such multiphysics simulations in view, in this work we develop a time-stepping strategy to model electromagnetic radiation and scattering within the nodal finite element framework. Although conservation of energy is well-known, we show in this work that there are additional quantities that are also conserved in the absence of loading. We then show that the developed time-stepping strategy (which is closely related to the trapezoidal rule that is widely used for solving linear hyperbolic problems) mimics these continuum conservation properties either exactly or to a very good approximation. Thus, the developed numerical strategy can be said to be ‘unconditionally stable’ (from an energy perspective) allowing the use of arbitrarily large time-steps. The developed method uses standard elements with Lagrange interpolation functions and standard Gaussian quadrature. We demonstrate the high accuracy and robustness of the developed method for solving both interior and exterior domain radiation problems, and for finding the scattered field from conducting and dielectric bodies.