AIMx: An Extended Adaptive Integral Method for the Fast Electromagnetic Modeling of Complex Structures

Conference of 42nd Annual Review of Progress in Quantitative Nondestructive Evaluation, QNDE 2015, Incorporating the 6th European-American Workshop on Reliability of NDE ; Conference Date: 26 July 2015 Through 31 July 2015; Conference Code:119404

The surface integral equation method is one of the most popular numerical methods for the electromagnetic analysis of composite metallic and dielectric structures. In this paper, the authors seek the "optimal" surface integral equation formulations for closed metallic, homogeneous and piecewise homogeneous dielectric and composite objects by studying the properties of the integral equations. The goal is to develop formulations in which all unknown quantities are well tested in the Galerkin method and the discretized matrix equation has good spectral properties leading to rapidly converging iterative solutions. The results show that by deriving the equations so that they map the currents onto itself and are directly tested in the same space as the unknowns are expanded can significantly improve conditioning of the matrix equation.

An efficient algorithm for electromagnetic wave scattering from rough dielectric surfaces is being developed for the simulation of bistatic scattering and emission from ocean surface. The algorithm is bused on a single magnetic field integral equation (SMFIE) and the surface is discretized using Rao-Wilton-Glisson (RWG) triangular basis function. The new feature of the algorithm is the application of the adaptive integral method (AIM) with SMFIE for speeding up the calculation. The new method will enable accurate simulations over large ocean surfaces, which have been until recently prohibited by the lack of computer power and method efficiency. Although the solver is applicable to a wide range of practical problems, the main goal of this work is to simulate the bistatic scattering and emission from a large area, with respect to wavelength, of ocean surface, size of which is made possible by the efficiency of the new method. The surface is illuminated with a Gaussian beam so that the edges of the surface do not contribute to the results significantly. In this paper we present a solution to the rough surface scattering using SMFIE with AIM, so that the computation can he speeded up with fast Fourier transform (FFT)

A key quantity in the design of plasmonic antennas and metasurfaces, as well as metamaterials, is the electrodynamic polarizability of a single scattering building block. In particular, in the current merging of plasmonics and metamaterials, subwavelength scatterers are judged by their ability to present a large, generally anisotropic electric and magnetic polarizability, as well as a bi-anisotropic magnetoelectric polarizability. This bi-anisotropic response, whereby a magnetic dipole is induced through electric driving, and vice versa, is strongly linked to the optical activity and chiral response of plasmonic metamolecules. We present two distinct methods to retrieve the polarizibility tensor from electrodynamic simulations. As a basis for both, we use the surface integral equation (SIE) method to solve for the scattering response of arbitrary objects exactly. In the first retrieval method, we project scattered fields onto vector spherical harmonics with the aid of an exact discrete spherical harmonic Fourier transform on the unit sphere. In the second, we take the effective current distributions generated by SIE as a basis to calculate dipole moments. We verify that the first approach holds for scatterers of any size, while the second is only approximately correct for small scatterers. We present benchmark calculations, revisiting the zero-forward scattering paradox of Kerker et al (1983 J. Opt. Soc. Am. 73 765–7) and Alù and Engheta (2010 J. Nanophoton. 4 041590), relevant in dielectric scattering cancelation and sensor cloaking designs. Finally, we report the polarizability tensor of split rings, and show that split rings will strongly influence the emission of dipolar single emitters. In the context of plasmon-enhanced emission, split rings can imbue their large magnetic dipole moment on the emission of simple electric dipole emitters. We present a split ring antenna array design that is capable of converting the emission of a single linear dipole emitter in forward and backward beams of directional emission of opposite handedness. This design can, for instance, find application in the spin angular momentum encoding of quantum information.

The surface integral equation (SIE) method is used for the computational study of radiation torque on arbitrarily shaped homogeneous particles. The Multilevel Fast Multipole Algorithm (MLFMA) is employed to reduce memory requirements and improve the capability of SIE. The resultant matrix equations are solved iteratively to obtain equivalent electric and magnetic currents. Then, radiation torque is computed using the vector flux of the pseudotensor over a spherical surface tightly enclosing the particle. We use, therefore, the analytical electromagnetic field expression for incident waves in the near region, instead of the far-field approximation. This avoids the error which may be caused when describing the incident beam. The numerical results of three kinds of non-spherical particles are presented to illustrate the validity and capability of the developed method. It is shown that our method can be applied to predict, in the rigorous sense, the torque from a beam of any shape on a particle of complex configuration with a size parameter as large as 650. The radiation torques on large ellipsoids are exemplified to show the performance of the method and to study the influence that different aspect ratios have on the results. Then, the code is used for the calculation of radiation torque on objects of complex shape including a biconcave cell-like particle and a motor with a non-smooth surface.

Surface-integral-equation (SIE) approaches are presented for the accurate solution of different problems in computational electromagnetics. First, an efficient message passing interface (MPI)/OpenMP parallel implementation of the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) is presented for the solution of large-scale conducting bodies. Second, looking for the extension of these rigorous approaches to new demanded areas, the SIE method is successfully applied to the solution of left-handed metamaterials and plasmonic nanostructures.

A surface integral equation method with multibranch Rao–Wilton–Glisson (MB-RWG) basis functions for scattering from dielectric objects is proposed based on the electric and magnetic current combined field integral equations (JMCFIE). The MB-RWGs are defined at the boundaries between dense and coarse meshes and can keep the current continuity. Thus, the JMCFIE with MB-RWGs allows different surfaces of the dielectric models to be independently discretized by triangles with different sizes. Compared with the conventional JMCFIE, the JMCFIE with MB-RWGs increases the flexibility of mesh generation, which is attractive for multiscale dielectric models. Furthermore, the iterative convergence of JMCFIE with MB-RWGs is similar to that of conventional JMCFIE. Numerical examples are presented to demonstrate the advantages of the proposed method.

Recently, scattering from 3D composite conducting and dielectric object receives much attention. Traditionally, FEM-BI method is applied to solve the problem. But absorption boundary condition (ABC) is needed in FEM-BI method. In this paper, we use two combinations of surface integral equation to solve the RCS of 3D composite conducting and dielectric object. Fast multipole method (FMM) is used. The problem is formulated in terms of a set of coupled integral equations involving equivalent electric and magnetic surface currents based on the equivalence theorem. The conducting structures and the dielectric materials are modeled by planar triangular patches, RWG basis and Galerkin method are used. The fast multipole method is applied to accelerate the computation of matrix-vector multiplication. The computational complexity and storage requirement is O(N/sup 1.5/), respectively. Numerical results are given for various structures and compared with other available data. The numerical results show that the present method has satisfying accuracy.

The surface integral equation (SIE) method is utilized to characterize plasmonic waveguide made of two parallel chains of silver nanowires with radius of 25nm fed by a V-shaped funnel at a working wavelength of 600nm. The efficiency of energy transport along the waveguide due to surface plasmonic coupling is investigated for different dimensions and shapes. The opening angle of the V-shaped funnel region for optimum light capturing is included in the investigation as well. A long plasmonic double-chain waveguide of length ~3.3mum has been analyzed and optimized.

Like the fast multipole method (FMM), the adaptive integral method (AlM) is a fast integral method that has O(N/sup /spl alpha//logN), /spl alpha/<1.5 computational complexity and O(N/sup /spl alpha//), /spl alpha/<1.5 memory requirements. AlM achieves its memory and complexity reduction by mapping the original edge-basis functions onto a regular grid and then using the Toeplitz property of the Green's function along with the fast Fourier transform (FFT) for a fast execution of the matrix-vector products in the iterative solver. So far, application of AIM has been restricted to metallic structures. In this paper, we extend AIM to modeling scattering by three dimensional multi-layered surfaces. For validation RCS results are given for layered and coated spheres and for composite wing-like structures.

Non-local boundary conditions and surface integral equations in electromagnetic scattering theory

To calculate scattering of a vegetation field consisting of a large number of plants, an improved fast version of the hybrid method (FHM) combining fast multiple scattering theory (FMST) and a numerical electromagnetic approach has been developed. In the HM, we use a library of T-matrices and the Foldy–Lax (FL) equation to calculate the electromagnetic field interactions outside of the cylinders enclosing the individual plants. To improve CPU time and memory requirements, the FL equation is solved by a triplet of fast Fourier transforms (FFTs) consisting of two FFTs solving 2-D FFTs of spatial distribution of plants and one 1-D FFT for solving the transformation of the order of cylindrical waves in the translation addition theorem. The triple FFTs minimize the CPU time and the memory requirement for computing translation addition matrix, which has been the bottleneck of large-scale simulations in the HM. To account for nonperiodic distribution of scatterers in a corn field, the premultiplication and the postmultiplication processes were applied to the FHM. The proposed method is validated at L-band (1.4 GHz) for 100 corn plants of height 1.25 m occupying an area of 9.54 by 9.54 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{2}$ </tex-math></inline-formula> . On a standard laptop, the CPU time is less than 6 min and the memory consumption is less than 710 Mbytes. The CPU time and the memory requirements are orders of magnitude more efficient than those required for solving the problem with a commercial software or using the previous version of the HM.

Magnetic Confinement Fusion (MCF) devices are characterized by complex geometries with both thick and thin components. Integral equation (IE) methods, including Volume Integral Equations (VIE) and Surface Integral Equations (SIE), are particularly advantageous for solving eddy-current problems because they eliminate the need for air/vacuum discretization. This paper explores hierarchical matrices (ℋ-matrices) to reduce the computational complexity of the coupled VIE-SIE formulation, enabling the efficient analysis of large-scale eddy current problems in the conductive structures of MCF devices.

The multilayer surface integral equations (SIEs) for electromagnetic scattering by infinitely long magnetodielectric cylinders with an arbitrary number of layers are derived and solved by the spectral integral method (SIM). Singularity subtraction for the 2-D Green's function is used to enhance the computation accuracy and achieve exponential convergence. The final matrix equation after discretization is formed in the Fourier spectral domain rather than the spatial domain, which greatly expedites the SIM solution by accelerating the convolution via the fast fourier transform (FFT) algorithm. A recursive method is proposed to solve the spectral integral equations instead of using an iterative method to lower the computation complexity. Numerical examples for ordinary multilayer cylinders and invisibility cloak cylinders are presented to validate the SIM results by comparing the total fields, scattered fields, and radar cross section (RCS) to analytical solutions or finite-element simulations. They verify that the recursive solution has a complexity of O(MN log N) for an M-layer cylinder with N discrete points on each interface. Meanwhile, the SIM outperforms the analytical method because only the 0th-order and 1st-order special functions (Bessel functions and Hankel functions) are used in the SIM but higher-order functions are necessary for the analytical method to maintain the accuracy.

The increasing demand for accurate and computationally efficient modeling of light-matter interactions in stratified media has driven significant advancements in numerical techniques. The surface integral equation (SIE) method provides a robust framework for analyzing electromagnetic scattering problems, offering distinctive advantages in terms of accuracy and computational efficiency. However, the application of SIEs to layered environments is hindered by various computational challenges, particularly those associated with the complexity of Green's functions in stratified media.