Athanasios G. Polimeridis

Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects

Abstract A stable volume integral equation formulation based on equivalent volumetric currents is presented for modeling electromagnetic scattering of highly inhomogeneous dielectric objects. The proposed formulation is numerically solved by means of Galerkin method of moments on uniform grids, allowing for acceleration of the matrix–vector products associated with the iterative solver with the help of FFT. In addition, the pertinent volume–volume Galerkin inner products are reduced to purely surface–surface integrals with smoother kernels, allowing for highly accurate and fast computation by readily available sophisticated cubatures. Numerical results demonstrate the convergence properties of the algorithm for scatterers with high dielectric contrast. In addition, we describe a road-map for Magnetic Resonance-specific volume integral equations fast solvers based on the proposed algorithm.

DIRECTFN: Fully Numerical Algorithms for High Precision Computation of Singular Integrals in Galerkin SIE Methods

Fully numerical schemes are presented for high precision computations of the four-dimensional integrals arising in Galerkin surface integral equation formulations. More specifically, the focal point of this paper is the singular integrals for coincident, edge adjacent and vertex adjacent planar and curvilinear triangular elements. The proposed method, dubbed as DIRECTFN, utilizes a series of variable transformations, able to cancel both weak (1/R) and strong (1/R2) singularities. In addition, appropriate interchanges in the order of the associated one-dimensional integrations result in further regularization of the overall integrals. The final integrands are analytic functions with respect to all variables involved and, hence, the integrals can be efficiently evaluated by means of simple Gaussian integration. The accuracy and convergence properties of the new schemes are demonstrated by evaluating representative weakly singular and strongly singular integrals over planar and quadratic curvilinear elements.

Fluctuating volume-current formulation of electromagnetic fluctuations in inhomogeneous media: Incandescence and luminescence in arbitrary geometries

We describe a fluctuating volume-current formulation of electromagnetic fluctuations that extends our recent work on heat exchange and Casimir interactions between arbitrarily shaped homogeneous bodies [A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, Phys. Rev. B 88, 054305 (2013)] to situations involving incandescence and luminescence problems, including thermal radiation, heat transfer, Casimir forces, spontaneous emission, fluorescence, and Raman scattering, in inhomogeneous media. Unlike previous scattering formulations based on field and/or surface unknowns, our work exploits powerful techniques from the volume-integral equation (VIE) method, in which electromagnetic scattering is described in terms of volumetric, current unknowns throughout the bodies. The resulting trace formulas (boxed equations) involve products of well-studied VIE matrices and describe power and momentum transfer between objects with spatially varying material properties and fluctuation characteristics. We demonstrate that thanks to the low-rank properties of the associated matrices, these formulas are susceptible to fast-trace computations based on iterative methods, making practical calculations tractable. We apply our techniques to study thermal radiation, heat transfer, and fluorescence in complicated geometries, checking our method against established techniques best suited for homogeneous bodies as well as applying it to obtain predictions of radiation from complex bodies with spatially varying permittivities and/or temperature profiles.

Reduced-Order Models for Electromagnetic Scattering Problems

We consider model-order reduction of systems occurring in electromagnetic scattering problems, where the inputs are current distributions operating in the presence of a scatterer, and the outputs are their corresponding scattered fields. Using the singular-value decomposition (SVD), we formally derive minimal-order models for such systems. We then use a discrete empirical interpolation method (DEIM) to render the minimal-order models more suitable to numerical computation. These models consist of a set of elementary sources and a set of observation points, both interior to the scatterer, and located automatically by the DEIM. A single matrix then maps the values of any incident field at the observation points to the amplitudes of the sources needed to approximate the corresponding scattered field. Similar to a Greens function, these models can be used to quickly analyze the interaction of the scatterer with other nearby scatterers or antennas.

The ultimate signal-to-noise ratio in realistic body models

We compute the ultimate signal‐to‐noise ratio (uSNR) and G‐factor (uGF) in a realistic head model from 0.5 to 21 Tesla.

Fast Electromagnetic Analysis of MRI Transmit RF Coils Based on Accelerated Integral Equation Methods

A fast frequency domain full-wave electromagnetic simulation method is introduced for the analysis of MRI coils loaded with the realistic human body models. The approach is based on integral equation methods decomposed into two domains: 1) the RF coil array and shield, and 2) the human body region where the load is placed. The analysis of multiple coil designs is accelerated by introducing the precomputed magnetic resonance Green functions (MRGFs), which describe how the particular body model used responds to the incident fields from external sources. These MRGFs, which are precomputed once for a given body model, can be combined with any integral equation solver and reused for the analysis of many coil designs. This approach provides a fast, yet comprehensive, analysis of coil designs, including the port S-parameters and the electromagnetic field distribution within the inhomogeneous body. The method solves the full-wave electromagnetic problem for a head array in few minutes, achieving a speed up of over 150 folds with root mean square errors in the electromagnetic field maps smaller than 0.4% when compared to the unaccelerated integral equation-based solver. This enables the characterization of a large number of RF coil designs in a reasonable time, which is a first step toward an automatic optimization of multiple parameters in the design of transmit arrays, as illustrated in this paper, but also receive arrays.

On the Computation of Power in Volume Integral Equation Formulations

We present simple and stable formulas for computing power (including absorbed/radiated, scattered and extinction power) in current-based volume integral equation formulations. The proposed formulas are given in terms of vector-matrix-vector products of quantities found solely in the associated linear system. In addition to their efficiency, the derived expressions can guarantee the positivity of the computed power. We also discuss the application of Poyntings theorem for the case of sources immersed in dissipative materials. The formulas are validated against results obtained both with analytical and numerical methods for scattering and radiation benchmark cases.

Robust J-EFVIE solvers based on purely surface integrals

A simple transformation of the Galerkin inner products arising in the numerical solution of volume integral equation formulations with piece-wise constant functions is presented. Making use of some simple steps, we derive integrals of reduced dimensionality and smoother kernels, amenable to standard cubatures developed for surface integral equation formulations.

The Weighted Averages Method for Semi-Infinite Range Integrals Involving Products of Bessel Functions

An efficient and accurate method, based on the weighted averages (WA) extrapolation technique, is presented for the evaluation of semi-infinite range integrals involving products of Bessel functions of arbitrary order. The method requires splitting the integration interval into a finite and an infinite part. The integral over the first finite part is computed using an adaptive quadrature rule based on Patterson formulas. For the evaluation of the remaining integral, the strongly irregular oscillatory behavior of the product of two Bessel functions is first represented as a sum of two asymptotically simply oscillating functions. Then, by applying the integration-then-summation technique, a sequence of partial integrals is obtained, and its convergence is accelerated with the help of WA. Details and possible complications involved in the method are addressed. Finally, the excellent performance of the proposed method is demonstrated throughout several numerical examples.

Temperature control of thermal radiation from composite bodies

We demonstrate that recent advances in nanoscale thermal transport and temperature manipulation can be brought to bear on the problem of tailoring thermal radiation from wavelength-scale composite bodies. We show that such objects---complicated arrangements of phase-change chalcogenide (