Hierarchical Electromagnetics
EmCAD – Cloud-native EM Simulator for Circuit-Level Modeling
Prospective Extension to Nonlinear Materials
In its current form, EmCAD handles a wide class of dispersive materials — including Debye, Lorentz, and Drude models — by replacing direct capacitive elements with circuital realizations of the chosen dispersion law. This replacement occurs prior to model reduction, preserving causality and passivity by construction.
A future extension aims to accommodate nonlinear materials, such as those exhibiting Kerr-type effects, where permittivity depends on the local electric field amplitude. While the substitution of capacitive elements with nonlinear circuit equivalents is conceptually straightforward, it introduces challenges for the MOR process — which is designed for linear systems.
1. Local Nonlinearity Handling via MOR Exclusion
One practical solution arises when the nonlinear region is confined to a small, localized subdomain. In this case, the subcircuit associated with that region can be excluded from the MOR stage and retained in its full, uncompressed form. This preserves accuracy and allows the rest of the system — assumed to be linear — to benefit from hierarchical compression.
2. Nonlinearity in Homogeneous Large Domains via Separated E-H Subbasis
A more general approach — applicable even to large, homogeneous nonlinear domains — would require a fundamental change in the MOR strategy. Currently, each resonant mode used in the reduced basis combines electric (direct) and magnetic (inverted) components. However, by doubling the basis size and constructing a set of basis vectors where electric and magnetic components are fully separated, one could decouple the influence of permittivity and permeability in the reduced model.
In this formulation, direct capacitive elements (linked to permittivity) would be projected only onto the purely electric basis vectors, while inverted capacitive elements (linked to permeability) would be projected onto the magnetic ones. The resulting reduced circuit would retain a clean structural separation between ε- and μ-dependent components, allowing the post-MOR substitution of nonlinear models for permittivity (and potentially permeability) without interfering with the linear compression process.