Burkay Donderici

Metamaterial Blueprints for Reflectionless Waveguide Bends

A derivation of metamaterial blueprints for the reflectionless (perfectly matched) guiding of electromagnetic waves through waveguide bends is performed. The sensitivity of the response with respect to small perturbations in the associated constitutive tensors is examined.

Mixed Finite-Element Time-Domain Method for Transient Maxwell Equations in Doubly Dispersive Media

We describe a mixed finite-element time-domain algorithm to solve transient Maxwell equations in inhomogeneous and doubly dispersive linear media where both the permittivity and permeability are functions of frequency. The mixed finite-element time-domain algorithm is based on the simultaneous use of both electric and magnetic field as state variables with a mix of edge (Whitney 1-form) and face (Whitney 2-form) elements for discretization of the coupled first-order Maxwell curl equations. The constitutive relations are decoupled from the curl equations and cast in terms of (auxiliary) ordinary differential equations involving time derivatives. Permittivity and permeability dispersion models considered here are quite general and recover Lorentz, Debye, and Drude models as special cases. The present finite-element time-domain algorithm also incorporates the perfectly matched layer absorbing boundary conditions in a natural way.

Improved FDTD subgridding algorithms via digital filtering and domain overriding

In numerical simulations of Maxwells equations for problems with disparate geometric scales, it is often advantageous to use grids of varying densities over different portions of the computational domain. In simulations involving structured finite-difference time-domain (FDTD) grids, this strategy is often referred as subgridding (SG). Although SG can lead to major computational savings, it is known to cause instabilities, spurious reflections, and other accuracy problems. In this paper, we introduce two strategies to combat these problems. First, we present an overlapped SG (OSG) approach combined with digital filters (in space). OSG can recover standard SG (SSG) schemes but it is based upon a more general, explicit separation between interpolation/decimation operations and the FDTD field update itself. This allows for a better classification of errors associated with the subgrid interface. More importantly, digital filters and phase matching techniques can be then employed to combat those errors. Second, we introduce SG with a domain overriding (SG-DO) strategy, consisting of overlapped (sub)grid regions that contain auxiliary (buffer) subdomains with perfectly matched layers (PML) to allow explicit control on the reflection and transmission properties at SG interfaces. We provide two-dimensional (2-D) numerical examples showing that residual errors from 2-D SG-DO FDTD simulations can be significantly reduced when compared to SSG schemes.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

Symmetric source implementation for the ADI-FDTD method

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method has been introduced to overcome the Courant limit present in the standard FDTD scheme. However, the ADI-FDTD scheme may produce small asymmetries in the field distributions according to the source implementation. In this communication, we compare different source implementation strategies and show that ADI-FDTD scheme has no asymmetry errors up to the numerical noise level, and can be implemented most accurately if the excitation is directly incorporated within the tridiagonal matrix and if the time discretization of the source is done appropriately within each full time step.

Robust computation of dipole electromagnetic fields in arbitrarily anisotropic, planar-stratified environments.

We develop a general-purpose formulation, based on two-dimensional spectral integrals, for computing electromagnetic fields produced by arbitrarily oriented dipoles in planar-stratified environments, where each layer may exhibit arbitrary and independent anisotropy in both its (complex) permittivity and permeability tensors. Among the salient features of our formulation are (i) computation of eigenmodes (characteristic plane waves) supported in arbitrarily anisotropic media in a numerically robust fashion, (ii) implementation of an hp-adaptive refinement for the numerical integration to evaluate the radiation and weakly evanescent spectra contributions, and (iii) development of an adaptive extension of an integral convergence acceleration technique to compute the strongly evanescent spectrum contribution. While other semianalytic techniques exist to solve this problem, none have full applicability to media exhibiting arbitrary double anisotropies in each layer, where one must account for the whole range of possible phenomena (e.g., mode coupling at interfaces and nonreciprocal mode propagation). Brute-force numerical methods can tackle this problem but only at a much higher computational cost. The present formulation provides an efficient and robust technique for field computation in arbitrary planar-stratified environments. We demonstrate the formulation for a number of problems related to geophysical exploration.

Domain-overriding and digital filtering for 3-D FDTD subgridded simulations

We report on an extension of finite-difference time-domain (FDTD) subgridding (SG) algorithms incorporating digital filters and domain-overriding to three-dimensional (3-D) simulations and to problems involving materials traversing the SG interfaces. We show that significant improvements in accuracy can be obtained for these cases as well

Complex-plane generalization of scalar Levin transforms: A robust, rapidly convergent method to compute potentials and fields in multi-layered media

Abstract We propose the complex-plane generalization of a powerful algebraic sequence acceleration algorithm, the method of weighted averages (MWA), to guarantee exponential-cum-algebraic convergence of Fourier and Fourier–Hankel (F–H) integral transforms. This “complex-plane” MWA, effected via a linear-path detour in the complex plane, results in rapid, absolute convergence of field and potential solutions in multi-layered environments regardless of the source-observer geometry and anisotropy/loss of the media present. In this work, we first introduce a new integration path used to evaluate the field contribution arising from the radiation spectra. Subsequently, we (1) exhibit the foundational relations behind the complex-plane extension to a general Levin-type sequence convergence accelerator, (2) specialize this analysis to one member of the Levin transform family (the MWA), (3) address and circumvent restrictions, arising for two-dimensional integrals associated with wave dynamics problems, through minimal complex-plane detour restrictions and a novel partition of the integration domain, (4) develop and compare two formulations based on standard/real-axis MWA variants, and (5) present validation results and convergence characteristics for one of these two formulations.

Stable evaluation of Green's functions in cylindrically stratified regions with uniaxial anisotropic layers

We present a robust algorithm for the computation of electromagnetic fields radiated by point sources (Hertzian dipoles) in cylindrically stratified media where each layer may exhibit material properties (permittivity, permeability, and conductivity) with uniaxial anisotropy. Analytical expressions are obtained based on the spectral representation of the tensor Greens function based on cylindrical Bessel and Hankel eigenfunctions, and extended for layered uniaxial media. Due to the poor scaling of these eigenfunctions for extreme arguments and/or orders, direct numerical evaluation of such expressions can produce numerical instability, i.e., underflow, overflow, and/or round-off errors under finite precision arithmetic. To circumvent these problems, we develop a numerically stable formulation through suitable rescaling of various expressions involved in the computational chain, to yield a robust algorithm for all parameter ranges. Numerical results are presented to illustrate the robustness of the formulation including cases of practical interest.

Computation of potentials from current electrodes in cylindrically stratified media

We present an efficient and robust semi-analytical formulation to compute the electric potential due to arbitrary-located point electrodes in three-dimensional cylindrically stratified media, where the radial thickness and the medium resistivity of each cylindrical layer can vary by many orders of magnitude. A basic roadblock for robust potential computations in such scenarios is the poor scaling of modified-Bessel functions used for computation of the semi-analytical solution, for extreme arguments and/or orders. To accommodate this, we construct a set of rescaled versions of modified-Bessel functions, which avoids underflows and overflows in finite precision arithmetic, and minimizes round-off errors. In addition, several extrapolation methods are applied and compared to expedite the numerical evaluation of the (otherwise slowly convergent) associated Sommerfeld-type integrals. The proposed algorithm is verified in a number of scenarios relevant to geophysical exploration, but the general formulation presented is also applicable to other problems governed by Poisson equation such as Newtonian gravity, heat flow, and potential flow in fluid mechanics, involving cylindrically stratified environments.