Signal-Processing Approach to Robust Time-Domain Modeling of Electromagnetic Fields

Abstract

In recent publications^1–9, a new method for integrating partial differential equations describing physical systems has been presented. This method is based on simulating the actual continuous-domain system by means of a discrete-domain system, and this in such a way that the following features hold:

• 1 Preservation of originally existing passivity and incremental passivity, and this in such a way that these properties become available in the multidimensional (MD) sense even though they existed originally only in the one-dimensional (1–D) sense (i.e., with respect to time). As a result, one can achieve not only full stability with respect to the discretization in space and time but also full stability, and, more generally, full robustness with respect to the computational errors that are due to rounding/truncation and overflow corrections and to extraneous sources.This is possible because a multidimensional vector Liapunov function having a sufficiently simple structure is available.

• 2 Preservation of the exclusively local nature of the interconnections and the massive parallelism, which are inherent to all physical systems with finite propagation speed. As a result, for any given fixed time instant to be considered, the computations can be carried out simultaneously, thus fully in parallel, in all the spatial sampling points, and the computations in any of these points require previously computed results only from the immediate neighboring points.

• 3 Arbitrarily changing parameters as well as arbitrary boundary shapes and conditions can be tanken into account in a straightforward manner.

• 4 Discretization is done on the basis of the trapezoidal rule. In order to achieve recursibility (computability), the simulation may not be based on the field variables appearing in the original partial differential equations. Instead, corresponding so-called wave variables should be employed, thus variables of the type occuring in relation with the scattering-matrix formalism. This way, the mechanism involved in the physical system becomes interpretable as an incidence-to-scattering (reflection, transmission) mechanism, i.e. a mechanism exhibiting a cause-to-effect (causality) relationship. The latter in turn gives rise to computational rules that exhibit the sequential nature needed for obtaining an algorithm.

• 5 It appears easiest to apply the method by first representing the system by means of a multidimensional Kirchhoff circuit. From this, the desired algorithm can be derived by applying the standard procedures known from the theory of multidimensions wave digital filters¹⁰, which has originally been developed within the context of digital signal processing. It will be discussed that the approach is applicable without difficulty to systems described by Maxwell’s equations¹¹.