A multilevel algorithm is presented for analyzing scattering from electrically large surfaces. The algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics. The algorithm permits a fast matrix-vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks, with each describing the interaction between distant scatterers. The multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast Fourier transform (FFT) and that only relies on well-known algebraic techniques. The computational complexity and the memory requirements of the proposed algorithm are O(N log/sup 2/ N).

A multilevel matrix decomposition algorithm for analyzing scattering from large structures

Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements.

Fast solution methods in electromagnetics

The problem of electromagnetic scattering by a three-dimensional dielectric object can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free space Green's function and the contrast source over the domain of interest. A weak form of the integral equation for the relevant unknown quantity is obtained by testing it with appropriate testing functions. The vector potential is then expanded in a sequence of the appropriate expansion functions and the grad-div operator is integrated analytically over the scattering object domain only. A weak form of the singular Green's function has been used by introducing its spherical mean. As a result, the spatial convolution can be carried out numerically using a trapezoidal integration rule. This method shows excellent numerical performance. >

/pdf/the-three-dimensional-weak-form-of-the-conjugate-gradient-2dtfqhz35v.pdf

The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems

In this paper, an algorithm for computation of the scattered fields from three dimensional inhomogeneous dielectric scatterers is presented. In this method, the Galerkin's testing formulation of an integral equation for 3D electromagnetic (EM) fields is represented by a multi-input and multi-output linear system with known kernels. On a regular grid with rooftop basis functions, the kernels are discretized and accurately evaluated. Furthermore they are represented by Toeplitz matrices which dramatically reduces the storage and computational complexity in solving for the scattered fields. Also, the kernels are independent of the scattering configuration and the incident waves. For a given frequency, they can be evaluated once and for all. The biconjugate gradient (BCG) algorithm combined with fast Fourier transform (FFT) is applied to solve the discrete linear system iteratively. The memory required for this algorithm is of order N, and the computational complexity of the BCG process costs order N log N op...

A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems

A numerical method is presented to compute electromagnetic fields inside a 2 mm high resolution, anatomically detailed model of a human head for high-frequency magnetic resonance imaging (MRI) applications. The method uses the biconjugate gradient algorithm in combination with the fast Fourier transform to solve a matrix equation resulting from the discretization of an integrodifferential equation representing the original physical problem. Given the current distribution in an MRI coil, the method can compute both the electric field (thus the specific energy absorption rate (SAR)) and the magnetic field, also known as the field. Results for the SAR and field distribution, excited by a linear and a quadrature birdcage coil, are calculated and presented at 64 MHz, 128 MHz and 256 MHz, corresponding to the operating frequencies of the 1.5 T, 3 T and 6 T MRI systems. It is shown that compared with that at 64 MHz, the SAR at 128 MHz is increased by a factor over 5 and the SAR at 256 MHz is increased by a factor over 10, assuming the same current strength in the coil. Furthermore, compared with the linear excitation, the average SAR for the quadrature excitation is reduced by a factor over 2 and the maximum SAR is reduced by a factor over 3. It is also shown that the field at high frequencies exhibits a strong inhomogeneity, which is attributed to dielectric resonance.

https://iopscience.iop.org/article/10.1088/0031-9155/41/12/011/pdf

Computation of electromagnetic fields for high-frequency magnetic resonance imaging applications

An accurate and efficient numerical method is presented for solving many differential-integral equations arising from electromagnetic scattering theory. It uses the discrete Fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations. As a consequence, this method is extremely simple to implement, uses less computer memory than comparable methods, and yields accurate predictions. The differential-integral equation is recast into a periodic form conducive to application of the discrete Fourier convolution theorem. The differential operators are approximated by appropriate finite-difference and discrete-convolution operators. All these quantities are computed by using the fast Fourier transform. An approximate solution is obtained by using the conjugate gradient method. Results are compared to experimental data or analytical solutions for a 3 lambda *3 lambda metal plate (where lambda is the wavelength), a homogeneous and a layered infinite circular dielectric cylinder, and a dielectric sphere. The accuracy of the method is further illustrated by comparing predictions with independent measurements by R.A. Ross (1966) on a 2 lambda *1 lambda metal plate at grazing incidence. In all cases, agreement is excellent. >

The discrete Fourier transform method of solving differential-integral equations in scattering theory

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The discrete Fourier transform method of solving differential-integral equations in scattering theory