Parallel multilevel fast multipole method for solving large-scale problems

The Multilevel Fast Multipole Method (FMM) is a well-established method and can be applied to solve electromagnetic (EM) scattering problems. Compared with other traditional methods, it requires less computational time and memory. However, constrained by a single processor's speed and memory limitations, the problem size that can be solved by serial implementation is still relatively small. For a million-unknown target, the computational time on a single processor is intolerable, and memory could be easily exhausted. Parallel-computing technology, which can utilize multiprocessors, provides an efficient way to solve electrically large-scale EM problems. This paper will focus on discussing the parallel methodologies applied to a multilevel FMM code, as well as demonstrating the computational efficiency of the parallel approach.     

A novel implementation of multilevel fast multipole algorithm for higher order Galerkin's method

A new approach is proposed to reduce the memory requirements of the multilevel fast multipole algorithm (MLFMA) when applied to the higher order Galerkin's method. This approach represents higher order basis functions by a set of point sources such that a matrix-vector multiply is equivalent to calculating the fields at a number of points from given current sources at these points. The MLFMA is then applied to calculate the point-to-point interactions. This permits the use of more levels in MLFMA than applying MLFMA to basis-to-basis interactions directly and, thus, reduces the memory requirements significantly.     

A higher order parallelized multilevel fast multipole algorithm for3-D scattering

A higher order multilevel fast multipole algorithm (MLFMA) is presented for solving integral equations of electromagnetic wave scattering by three-dimensional (3-D) conducting objects. This method employs higher order parametric elements to provide accurate modeling of the scatterer's geometry and higher order interpolatory vector basis functions for an accurate representation of the electric current density on the scatterer's surface. This higher order scheme leads to a significant reduction in the mesh density, thus the number of unknowns, without compromising the accuracy of geometry modeling. It is applied to the electric field integral equation (EFIE), the magnetic field integral equation (MFIE), and the combined field integral equation (CFIE), using Galerkin's testing approach. The resultant numerical system of equations is then solved using the MLFMA. Appropriate preconditioning techniques are employed to speedup the MLFMA solution. The proposed method is further implemented on distributed-memory parallel computers to harness the maximum power from presently available machines. Numerical examples are given to demonstrate the accuracy and efficiency of the method as well as the convergence of the higher order scheme     

Boundary conditions for spherical harmonics expansion of Boltzmann equation

A cost effective solution of the Boltzmann equation can be obtained by a spherical harmonics expansion. The Letter introduces improved boundary conditions which account for the fact that the electrons absorbed by a contact may be strongly heated. It is shown that simulations with the new boundary conditions lead to improved results compared to the equilibrium boundary conditions used previously.<>     

An alternative formulation for the fast multipole method

This paper describes an alternative formulation for the fast multipole method based on spherical waves decomposition. It is somewhat simpler to implement than the standard fast multipole method and also better suited at low frequencies. The new formulation is mainly based on a technique for the interpolation of the bistatic radar cross section derived from the Wacker's method for antenna measurements.     

A fast method for designing time-optimal gradient waveforms for arbitrary k-space trajectories

A fast and simple algorithm for designing time-optimal waveforms is presented. The algorithm accepts a given arbitrary multidimensional k-space trajectory as the input and outputs the time-optimal gradient waveform that traverses k-space along that path in minimum time. The algorithm is noniterative, and its run time is independent of the complexity of the curve, i.e., the number of switches between slew-rate limited acceleration, slew-rate limited deceleration, and gradient amplitude limited regions. The key in the method is that the gradient amplitude is designed as a function of arc length along the k-space trajectory, rather than as a function of time. Several trajectory design examples are presented.     

Theory of the circular harmonics expansion method formultiple-optical-fiber system

The circular harmonics expansion method that has been applied to solve the modal characteristics of multiple-core fiber structure is given a formal derivation. The validity of the field expansion expression in the method is rigorously proved by expanding the Green's function for the scalar wave equation in a surface integral equation into circular harmonics. It is found that for coupled round cores only the modified Bessel functions of the second kind are needed in the expansion, and that the modified Bessel functions of the first kind do not appear in the expansion     

Some scattering results computed by surface-integral-equation and hybrid finite-element - boundary-integral techniques, accelerated by the multilevel fast multipole method

Method-of-moments (MoM) solutions of surface integral equations are especially well suited for scattering computations involving metallic objects. Improved modeling flexibility for dielectric (possibly lossy) and mixed dielectric/metallic bodies is obtained by combining a surface-integral-equation formulation, involving electric and magnetic equivalent surface-current densities, with a volumetric finite-element (FE) model of the dielectric regions. This results in the well-known hybrid FEBI (finite-element-boundary-integral) technique. For many years, hybrid FEBI techniques, as well as stand-alone Bl (surface-integral equation, often just termed MoM) techniques, were restricted to relatively small (with respect to a wavelength) geometries. However, with the development of powerful multilevel fast multipole methods/algorithms (MLFMM/MLFMA), it has become possible to compute a larger variety of practical scattering and radiation problems with the hybrid FEBI-MLFMM technique. In this contribution, we give a short review of our hybrid FEBI-MLFMM approach, with a focus on mixed dielectric/metallic geometries and multiple Bl domains. We then present a variety of scattering results for metallic and mixed dielectric/metallic objects, together with comparisons with measured RCS (radar cross section) data. Broadband computations are used to derive high-resolution range (HRR) profiles of several configurations.     

Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid Integral equations in electromagnetics

In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time.     

A far-near field transformation using the fast multipole techniques

We are interested in deducing the coupling between an antenna and some surrounding structures via the only knowledge of the antenna far-field measured in an anechoic room. A new approach based on a multipole expansion of the Green's kernel is considered to evaluate the field in the vicinity of the antenna. Some numerical experiments illustrate the efficiency of the multipole techniques compared to the physical optic approach.     

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

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 ²) to O(N^3/2), where N is the dimensionality of the problem's discretization     

Microwave emission of rough ocean surfaces with full spatialspectrum based on the multilevel expansion method

Microwave emission of ocean surfaces with full spatial spectrum is studied in this paper. For ocean surfaces with full spectrum, the rms height of roughness can be many wavelengths, and the surface size must be chosen to be larger than the longest scale wave in the spectrum. Due to computer resources, it is not straightforward to conduct numerical simulations of emission from rough surfaces with large rms height and size since a large number of unknowns will be involved. In this paper, the multilevel expansion of the sparse matrix canonical grid (SMCG) method, which is available for surfaces with large rms heights, is used to study the emission of one-dimensional (1-D) ocean surfaces. The computational complexity and the memory requirement are still on the order of O(N log (N)) and O (N), respectively, as in the SMCG method. Ocean surfaces with size 1024 wavelengths (21.9 m at 14 GHz) and spatial spectrum bandwidth between 0.858 rads/m (corresponding to the longest scale of 341.3 wavelengths) and 4691.5 rads/m (corresponding to the shortest scale of 1/16 wavelengths), which is rather wide to be regarded as a full spectrum, are studied. The maximum of the electromagnetic wavenumber-surface rms height product is up to 25.18. The surface is modeled as a lossy dielectric surface with large relative permittivity rather than as a perfectly conducting surface, which is often adopted as an approximation in the active remote sensing of ocean surfaces. A relatively high sampling density is used to ensure accuracy. The effects of the low and high portions of the spectrum on the emissivity are studied numerically. Monte Carlo simulation for ocean surfaces is also performed by exploiting the efficiency of the multilevel expansion method and the use of parallel computing techniques. The convergence of the results with respect to the sampling density is also illustrated     

Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling

We present a multilevel fast multipole method (MLFMM) solution for volume integral equations dealing with scattering from arbitrarily shaped inhomogeneous dielectrics. The solution accuracy, convergence, computer time and memory savings of the method are demonstrated. Previous works have employed the MLFMM for impenetrable targets. In this paper, we integrate the MLFMM with the volume integral equation method for scattering by inhomogeneous targets. Of particular importance is the use of curvilinear elements for better volume representation and the use of simple basis functions for ease of parallelization.     

一种类菲涅耳型振荡积分的快速精确算法

通过变量代换和积分路径的复平面变换,将类菲涅耳型振荡积分变换为非振荡型积分,所得积分的计算时间与振 荡频率成反变的关系,使得计算速度变为原来积分速度的几十到上百倍。而且,该方法所用到的仅为数学变换,几乎不 存在误差,很好的解决了类菲涅耳型振荡积分问题。     

A fast multipole method for layered media based on the application of perfectly matched layers - the 2-D case

An efficient fast multipole method (FMM) formalism to model scattering from two-dimensional (2-D) microstrip structures is presented. The technique relies on a mixed potential integral equation (MPIE) formulation and a series expression for the Green functions, based on the use of perfectly matched layers (PML). In this way, a new FMM algorithm is developed to evaluate matrix-vector multiplications arising in the iterative solution of the scattering problem. Novel iteration schemes have been implemented and a computational complexity of order O(N) is achieved. The theory is validated by means of several illustrative, numerical examples. This paper aims at elucidating the PML-FMM-MPIE concept and can be seen as a first step toward a PML based multilevel fast multipole algorithm (MLFMA) for 3-D microstrip structures embedded in layered media.     

Analysis of modal cutoffs of radially inhomogeneous two-coreoptical fibers by circular harmonics expansion method

A scalar theory based on generalization of the circular harmonics expansion method combined with the finite element method is formulated to determine cutoff values for higher order normal modes on the two-core fiber with radially inhomogeneous core index profiles under the weakly guiding approximation. The validity of the field expansion expression in the derived generalized circular harmonics expansion method is proved rigorously by expanding the Green's function for the Laplace equation in a surface integral equation into generalized circular harmonics. Numerical examples are given for the two identical-core cases with power-law core index profiles. Our method is shown to be able to provide exact cutoff values for the touching-core case     

An integrated technique for eliminating harmonics of multilevel inverter with unequal DC sources

For improving the alteration performance of multilevel inverter, selective harmonics elimination methods play a significant role. In this document, an incorporated method is suggested for removing the harmonics of multilevel inverter with uneven DC sources. The suggested incorporated method is the mixture of fuzzy logic intelligent system and particle swarm optimisation (PSO) algorithm. For generating the training information set in terms of switching angle, Fuzzy is one of the synthetic intelligent methods which apply harmonic voltage and harmonic distortion. PSO is one of the swarm intelligence-based optimisation algorithms which is employed for choosing optimal switching angle from the training data set. The suggested hybrid method is executed in MATLAB/Simulink platform. At dissimilar unequal input voltage levels, the performance of the suggested method is checked by cascade H-bridge multilevel inverter. The production of the suggested method is compared with the theoretical effects.     

基于扩展球谐函数的多次散射理论分析

粒子的单次散射在一定程度上描述了光的传播特征。但在实际情况中,单次散射不足以描述传播过程的全貌。因此,我们必须考虑粒子的多次散射效应,以得到对光传播问题的一个完整描述。在本文中,我们首先建立散射传输方程,在求解散射传输方程中引入了辅助函数,然后再利用球谐函数对辅助函数进行扩展,导出了多次散射方程的解。     

Scattering from multiple objects buried beneath two-dimensional random rough surface using the steepest descent fast multipole method

Generalized formulations are presented to analyze the electric field scattered from multiple penetrable shallow objects buried beneath two-dimensional random rough surfaces. These objects could have different materials, shapes, or orientations. In addition, their separation distance may range from a fraction of a wavelength to several wavelengths. The fast algorithm, steepest descent fast multipole method (SDFMM), is used to compute the unknown electric and magnetic surface currents on the rough ground surface and on the buried objects. Parametric investigations are presented to study the effect of the objects proximity, orientations, materials, shapes, the incident waves polarization, and the ground roughness on the scattered fields. A significant interference is observed between the objects when they are separated by less than one free space wavelength. Even when the clutter due to the rough ground is removed, the return from the second object, can be dominating causing a possible false alarm in detecting the target. The results show that the distortion in target signature significantly increases with the increase of both the proximity to a clutter item and the ground roughness.     

A multilevel formulation of the finite-element method forelectromagnetic scattering

Multigrid techniques for three-dimensional (3-D) electromagnetic scattering problems are presented. The numerical representation of the physical problem is accomplished via a finite-element discretization, with nodal basis functions. A total magnetic field formulation with a vector absorbing boundary condition (ABC) is used. The principal features of the multilevel technique are outlined. The basic multigrid algorithms are described and estimations of their computational requirements are derived. The multilevel code is tested with several scattering problems for which analytical solutions exist. The obtained results clearly indicate the stability, accuracy, and efficiency of the multigrid method