A Perfectly Matched Layer for the Nonlinear Dispersive Finite-Element Time-Domain Method

Abstract

A novel implementation of a Perfectly Matched Layer (PML) is presented for the truncation of Finite-Element Time-Domain (FETD) meshes containing electrically complex materials, exhibiting any combination of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Based on the complex coordinate stretching formulation of the PML, the presented technique yields an artificial absorbing layer whose matching condition is independent of material parameters. Moreover, by virtue of only modifying spatial derivatives, the incorporation of the PML into existing solvers for complex media is simple and straightforward. The resulting Material Independent PML (MIPML) is incorporated into a nonlinear dispersive method for the vector wave equation, which leverages the z-transform and Newton-Raphson techniques to yield an implementation free from recursive convolutions, auxiliary differential equations, and linearizations. This permits the unprecedented truncation and attenuation of nonlinear phenomena, such as spatial and temporal solitons, within the FETD method.

Energy-Stable Time-Domain Finite Element Methods for the 3D Nonlinear Maxwell's Equations

“…We conclude from the computational experiments presented in this paper that the proposed novel TDFEMs for the full system of nonlinear Maxwell's equation in 3D conserve the energy (at semi-discrete and fully discrete levels), are unconditionally stable (no Courant-Friedrichs-Lewy condition), computationally efficient (one Newton iteration per time step) and figure out the fields (quantities) directly, in contrast to many existing methods (SVEA, BPM, the electric field formulation, the magnetic field formulation, A − φ method, operator form, magnetic vector potential, decoupled schemes and A-Formulation). In particular, our proposed semi-discrete and fully discrete methods could replace the existing 1D [43] and 2D [45], [46] schemes to 3D, and [39], [42]. Moreover our proposed methods are intermediate results for the theoretical and computational development of energy conserving time-domain discontinuous methods for 3D nonlinear problems in Optics and Photonics.…”

“…Therefore, the proposed methods are unconditionally stable in contrast to many existing methods [13], [26], [43]. Moreover, in the proposed Euler-type time discretization scheme the Newton iterations reduce to a single step at each time step while many existing methods require several Newton iterations at each time step [37]- [39], [42], [43], [45], [46]. Therefore our scheme is computationally more efficient than the a lot of existing methods.…”

“…The parameters are: time step size t = 10 −9 , χ (1) = 2.2, and χ (3) directly [15], [18]- [25], [28], [29], [60] or solve the problem only in 1D and 2D, e.g. [26], [37]- [46]. Those methods do not solve the system of nonlinear Maxwell's equations in Optics and Photonics directly and may cause spurious solutions [26], [50].…”

“…In addition, the fully discrete formulations proposed in [43] are only conditionally stable. The TDFEM for nonlinear problems in Electromagnetics proposed in [37]- [39], [42], [45], [46] may also fail to satisfy an energy stability property. We conclude from the computational experiments presented in this paper that the proposed novel TDFEMs for the full system of nonlinear Maxwell's equation in 3D conserve the energy (at semi-discrete and fully discrete levels), are unconditionally stable (no Courant-Friedrichs-Lewy condition), computationally efficient (one Newton iteration per time step) and figure out the fields (quantities) directly, in contrast to many existing methods (SVEA, BPM, the electric field formulation, the magnetic field formulation, A − φ method, operator form, magnetic vector potential, decoupled schemes and A-Formulation).…”

“…For the linear situation the papers [31]- [34] can be mentioned as examples, for dispersive media we refer to [35], [36]. For the system of Maxwell's equations with material nonlinearities, there are still very few rigorous analysis, error estimates, and simulation results using time domain finite element methods ((TDFEM/TDdG) available [26], [37]- [46]. In the paper [43] a higher-order discontinuous Galerkin method is used to discretize the problem in space.…”