http://ab-initio.mit.edu/~oskooi/papers/Oskooi10.pdf

Meep: A flexible free-software package for electromagnetic simulations by the FDTD method

The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are reviewed. The number of modes required, block-diagonal preconditioner, near singularity extraction, and the choice of initial guesses are discussed to apply the MLFMA to calculating electromagnetic scattering by large complex objects. Using these techniques, we can solve the problem of electromagnetic scattering by large complex three-dimensional (3-D) objects such as an aircraft (VFY218) on a small computer.

Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects

A practical and complete, but not rigorous, exposition of the fact multiple method (FMM) is provided. The FMM provides an efficient mechanism for the numerical convolution of the Green's function for the Helmholtz equation with a source distribution and can be used to radically accelerate the iterative solution of boundary-integral equations. In the simple single-stage form presented here, it reduces the computational complexity of the convolution from O(N/sup 2/) to O(N/sup 3/2/), where N is the dimensionality of the problem's discretization. >

The fast multipole method for the wave equation: a pedestrian prescription

The use of liquid metals based on gallium for soft and stretchable electronics is discussed. This emerging class of electronics is motivated, in part, by the new opportunities that arise from devices that have mechanical properties similar to those encountered in the human experience, such as skin, tissue, textiles, and clothing. These types of electronics (e.g., wearable or implantable electronics, sensors for soft robotics, e-skin) must operate during deformation. Liquid metals are compelling materials for these applications because, in principle, they are infinitely deformable while retaining metallic conductivity. Liquid metals have been used for stretchable wires and interconnects, reconfigurable antennas, soft sensors, self-healing circuits, and conformal electrodes. In contrast to Hg, liquid metals based on gallium have low toxicity and essentially no vapor pressure and are therefore considered safe to handle. Whereas most liquids bead up to minimize surface energy, the presence of a surface oxide on these metals makes it possible to pattern them into useful shapes using a variety of techniques, including fluidic injection and 3D printing. In addition to forming excellent conductors, these metals can be used actively to form memory devices, sensors, and diodes that are completely built from soft materials. The properties of these materials, their applications within soft and stretchable electronics, and future opportunities and challenges are considered.

Stretchable and Soft Electronics using Liquid Metals.

Prior to abour 1990, the modeling of electromagnetic engineering systems was primarily implemented using solution techniques for the sinusoidal steady-state Maxwell's equations. Before about 1960, the principal approaches in this area involved closed-form and infinite-series analytical solutions, with numerical results from these analyses obtained using mechanical calculators. After 1960, the increasing availability of programmable electronic digital computers permitted such frequency-domain approaches to rise markedly in sophistication. Researchers were able to take advantage of the capabilities afforded by powerful new high-level programming languages such as Fortran, rapid random-access storage of large arrags of numbers, and computational speeds that were orders of magnitude faster than possible with mechanical calculators. In this period, the principal computational approaches for Maxwell's equations included the high-frequency asymptotic methods of Keller (1962) as well as Kouyoumjian and Pathak (1974) and the integral equation techniques of Harrington (1968) .

9 – Computational Electromagnetics: The Finite-Difference Time-Domain Method

The interpolation errors of the higher order bivariate Lagrange polynomial interpolation based on the rectangular, right and equilateral triangular interpolations are measured by using the maximum and root-mean-square (RMS) errors. The error distributions of above three kinds of interpolations are analyzed to find the regions having the smallest interpolation error. Both analytical and numerical results show that the right triangular interpolation is the most efficient interpolation method. Although both the maximum and RMS errors of the right triangular interpolation are larger than that of the rectangular interpolation, the number of data points used in the triangular interpolation is up to 50% less than that used in the rectangular one. On the other hand, the equilateral triangular interpolation using the regions inside a big triangle as interpolation area is proved to be the most accurate interpolation method. The interpolation errors of the equilateral triangular are almost half of that of the rectangular interpolation. In addition, the right triangular, equilateral triangular and rectangular interpolations for the third and fourth orders are applied to accelerate the calculation of doubly periodic Green’s function (PGF).

/pdf/error-analysis-of-higher-order-bivariate-lagrange-and-56zzujgbze.pdf

Error Analysis of Higher Order Bivariate Lagrange and Triangular Interpolations in Electromagnetics

This communication proposes a novel approach to evaluate the doubly periodic Green's function for layered medium in matrix-friendly formulation. Several techniques, including factorization of the Green's function, generalized pencil of function method and high order Taylor expansion, are employed in a delicate manner to derive the high order asymptotic expressions, which are then evaluated by new fast convergent series derived from Ewald transformation. This approach delivers fast and highly accurate results as well as high order convergence, and also allows fast frequency sweep for calculating Brillouin diagram in eigenvalue problem and for normal incidence in scattering problem.

Accurate and Efficient Computation of Layered Medium Doubly Periodic Green's Function in Matrix-Friendly Formulation

The advent of fast algorithms in computational electromagnetics has permitted the solution of integral equations with an unprecedented number of unknowns. This is the consequence of the development of the fast multipole algorithms (FMA) and the multilevel fast multipole algorithms (MLFMA). Such algorithms allow a matrix-vector multiplication to be performed in O(NlogN) operations or less for many scattering problems. Moreover, the memory requirements of these methods are O(NlogN), or almost matrix free. Using the fast matrix-vector multiplications in an iterative solver, problems for integral equations involving millions of unknowns have been solved previously. One of the most important mathematical formulas in FMA is the addition theorem. In the numerical implementation of the addition theorem, the infinite series should be truncated. The error analysis for the truncation error in the scalar Green's functions has been done by many researchers. In this paper, the error analysis for the truncation error in the multipole expansion of vector Green's functions is given.

Error analysis for the truncation of multipole expansion of vector Green's functions

Several surface integral formulations for electromagnetic interactions with homogeneous material bodies are studied numerically. Included are the well-known Muller and PMCHWT formulations and two previously unreported formulations. One of the two formulations introduced gives a simple form of the Neumann series for low-contrast material bodies. The integral formulations are discretized to matrix equations using the method of moments. The condition numbers of the matrices are evaluated using several definitions of norms. The number of iterations in the CG method is also presented for each formulation.

Numerical study of surface integral formulations for homogeneous bodies

In this paper, we simulate some of the benchmark problems proposed by the World Federation of Nondestructive Evaluation Centers (WFNDEC) using a full wave simulation model based on accurate solutions to the boundary integral equations for ultrasonic scattering. Much of the previous work on modeling these problems relied on the Kirch-hoff approximation to find the scattered fields from defects. Here we instead use a numerical method, called the Nystrom method, for finding the scattered fields more accurately by solving the boundary integral equations of scattering. We compare our model’s predictions with both measurements and Kirchhoff approximation based models. We expect the presented results to serve as a validation of our model as well as a comparison between the Kirchhoff approximation and the Nystrom method.

Jiming Song

Papers

Error Analysis of Higher Order Bivariate Lagrange and Triangular Interpolations in Electromagnetics

Accurate and Efficient Computation of Layered Medium Doubly Periodic Green's Function in Matrix-Friendly Formulation

Error analysis for the truncation of multipole expansion of vector Green's functions

Numerical study of surface integral formulations for homogeneous bodies

Full wave modeling of ultrasonic NDE benchmark problems using Nyström method