Probabilistic finite-difference time-domain simulations using stochastic electromagnetic macro-models

To enable an efficient stochastic-based design optimization methodology for multi-scale structures of electrical devices and systems, we propose the infusion of stochastic modeling with the electromagnetic macro-model in the finite-difference time-domain (FDTD) method. We provide a methodology for the efficient generation and utilization of the stochastic macro-model for the purpose of time-domain analysis in FDTD. The methodology quantifies the impact of uncertainty, manifested as random material and structural variations in the design and manufacturing process of the realized system, on the stochastic system's electromagnetic performance.;In the current state of the art, and in absence of the stochastic macromodel, it is necessary to perform repeated discretization of the large deterministic domain for every random variation in the small fine-featured stochastic domain. The development of the stochastic macro-model eliminates the need for repeated discretization of the overall structure for every such random variation. Indeed, only a single FDTD grid needs to be developed for the deterministic portion of the overall structure irrespective of the realization generated by a specific choice of the random parameters in the domains exhibiting statistical variability; thus, the macro-model results in significant computational savings by eliminating operations pertaining to repeated discretization of the deterministic domain for each variation in the stochastic domain.;In essence, the macro-model is a state-space representation of the discretized Maxwell's equations which encapsulates a certain fine-featured region of a multi-scale structure, in the FDTD grid. To enhance the computational efficiency of this state-space abstraction layer we apply a modified embodiment of the model order reduction (MOR) technique, known as enhanced nodal order reduction (ENOR), to minimize the internal degrees of freedom of the abstraction layer while maintaining sufficient engineering accuracy in the system response. ENOR provides a passive reduced order admittance boundary condition model which expedites the efficient computation of field quantities in the fine-featured regions.;To enable cost-effective and high-accuracy FDTD simulations involving design and manufacturing variations of disparate spatial scales, the method of sub-gridding is utilized, with the stochastic macro-model implemented in the sub-gridded region. To enable the insertion of the high spatial-resolution stochastic macro-model inside the FDTD grid, a class of isotropic spatial filters is developed to suppress the spurious noise waves which are generated by the discrete wave-impedance mismatch at the boundary of the macro-model. To this end, we develop a class of spatial filter operators that: (a) are straightforward to design and implement within the existing Yee style FDTD explicit time-stepping scheme; (b) do not require complicated spatial/temporal interpolation in the field update equations; (c) are able to accommodate broadband electromagnetic sources; (d) exhibit spatial isotropy in their suppression of the spurious numerical re ections while preserving the pertinent portions of the signal's power spectral density.