Computation of High-Order Electromagnetic Field Derivatives with FDTD and the Complex-Step Derivative Approximation

TL;DR

This work addresses the need for accurate, high-order derivatives of electromagnetic fields with respect to material and geometric design parameters in full-wave simulations. It introduces MCSD-FDTD, a framework that embeds multi-complex step derivatives into FDTD by perturbing materials and geometry with imaginary steps, allowing real parts to solve the unperturbed problem and imaginary parts to yield derivatives of any order. The authors develop an $n$-complex arithmetic class, formulate Jacobian and Hessian computations, and present iterative methods to reduce overhead, validating the approach on cavity and 3-D microstrip-filter problems. The results show that MCSD-FDTD achieves high-accuracy derivatives with robust performance against numerical dispersion and outperforms conventional finite-difference sensitivity analyses, enabling efficient parametric modeling and uncertainty quantification in electromagnetics.

Abstract

This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD) simulation. The proposed method is based on embedding the complex-step derivative (CSD) approximation into the standard FDTD update equations. Being finite-difference free, CSD provides accurate derivative approximations even for very small perturbations of the design parameters, unlike finite-difference approximations that are prone to subtractive cancellation errors. The availability of accurate approximations of field derivatives with respect to design parameters enables studies such as sensitivity analysis of multiple objective functions (as derivatives of those can be derived from field derivatives via the chain rule), uncertainty quantification, as well as multi-parametric modeling and optimization of electromagnetic structures. The theory, FDTD implementation and applications of this technique are presented.

Figures (9)

• Figure 1: Absolute error in the approximation of the analytical value of (a) $\partial b / \partial l$ in (\ref{['example_an']}) and (b) $\partial^2 b / \partial l^2$ in (\ref{['secondorder']}) for forward, backward, and centered finite-differences as well as the complex-step derivative (CSD) approximation.
• Figure 2: Example of a microstrip circuit: field derivatives with respect to the stub length $l$ and the dielectric permittivity, $\varepsilon_r$, of the substrate are sought for. Dark grey areas indicate the microstrip segments; light grey areas correspond to the substrate.
• Figure 3: FDTD mesh for the computation of field derivatives with respect to the length $l$ of the microstrip stub in Fig. \ref{['msexample']}.
• Figure 4: Comparison of analytical solution and MCSD-FDTD for (a) the electric field and (b) its second-order derivative with respect to the dimensions of a rectangular metallic cavity, at a sampling point within the cavity, in the time-domain.
• Figure 5: Relative error, according to (\ref{['errornorm']}), of the numerical sensitivity $\partial^2 E / \partial a \partial b$ of the electric field inside a rectangular cavity, computed via FDTD with central finite differences (CFD), as well as the multi-complex step derivative (MCSD) approximation. In (a), the Yee cell size $\Delta$ is set at $1$ mm and step-size $h$ varies from $5\times10^{-4}$ to $10^{-5}$. In (b), the cell size varies from $2$ to $0.4$ mm, and the relative error of the electric field $E$ computed by real-valued FDTD is included.