Multilevel Fast Multipole Method for Higher Order Discretizations

The Multilevel Fast Multipole Method (MLFMM) allows for a reduced computational complexity when solving electromagnetic scattering problems. Combining this with the reduced number of unknowns provided by Higher-Order discretizations has proven to be a difficult task, with the general conclusion being that going above 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> order is not worthwhile. In this paper, we challenge this conclusion, providing results that demonstrate the potential performance gains with Higher-Order MLFMM and showing some modifications to the traditional MLFMM that can benefit both Higher-Order and standard discretizations.

A combination of the multilevel fast multipole method (MLFMM) and boundary element method (BEM) can solve large scale photonics problems of arbitrary geometry. Here, MLFMM-BEM algorithm based on a scalar and vector potential formulation, instead of the more conventional electric and magnetic field formulations, is described. The method can deal with multiple lossy or lossless dielectric objects of arbitrary geometry, be they nested, in contact, or dispersed. Several examples are used to demonstrate that this method is able to efficiently handle 3D photonic scatterers involving large numbers of unknowns. Absorption, scattering, and extinction efficiencies of gold nanoparticle spheres, calculated by the MLFMM, are compared with Mie's theory. MLFMM calculations of the bistatic radar cross section (RCS) of a gold sphere near the plasmon resonance and of a silica coated gold sphere are also compared with Mie theory predictions. Finally, the bistatic RCS of a nanoparticle gold-silver heterodimer calculated with MLFMM is compared with unmodified BEM calculations.

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.

To study wave scattering from random lossy dielectric rough surfaces with large permittivities using the method of moments, a dense grid is needed for accurate results. A dense grid requires more CPU and memory. The physics‐based two‐grid (PBTG) method can reduce both CPU and memory requirements. In this paper, the PBTG is used in conjunction with the multilevel fast multipole method (FMM) to solve wave scattering from one‐dimensional random lossy dielectric rough surfaces. The proposed algorithm has the computational complexities of O(Ndg) for near‐field interactions and O(Ncg) for nonnear‐field interactions, where Ndg and Ncg are the number of sampling points on the dense and coarse grids, respectively. Using the proposed algorithm, wave scattering from Gaussian and non‐Gaussian rough surfaces is investigated and illustrated. Special emphasis is put on checking the accuracy of the algorithm and energy conservation.

The MLFMM (Multilevel Fast Multipole Method) is an alternative formulation of the technology behind the MoM (Method of Moment) and is applicable to much larger structures than the MoM, making full-wave current-based solutions of electrically large structures a possibility. This fact implies that it can be applied to most large models that were previously treated with the MoM without having to change the mesh. In this paper we discuss the development and implementation of volumetric integral equations for both dielectric and magnetically permeable materials using curvilinear hexahedral elements. Both piecewise constant and higher order basis functions will be examined in the context of volumetric multilevel fast multipole method implementation. Comparisons with corresponding finite element-boundary integral solutions will also be presented and compared in terms of accuracy and efficiency.

Complex scattering problems are solved with the multilevel fast multipole method (MLFMM). Geometries include metallic structures (wires and surfaces) as well as dielectric bodies (volume and surface equivalence principle). Validation and application examples show the large savings in memory and run-time of the MLFMM as compared to the standard method of moments (MoM). Limitations regarding dielectric bodies are addressed, which are encountered when attempting to model highly lossy dielectric bodies with a high permittivity.

This paper describes the efficient parallisation of the multilevel fast multipole method (MLFMM) for distributed memory systems. Data is distributed between the processes to obtain a good load balance and to minimise the number of communications. The total runtime efficiency is highly dependent on the efficiency of the matrix-vector-product, as well as the computation of the preconditioner. (4 pages)

The application of a parallelized multilevel fast multipole method (MLFMM) is described and illustrated through examples for the solution of large scale and complex EMC problems. Furthermore, to reduce the modeling complexity of such EMC problems, we propose a combination of field with network theory to split large problems into smaller sub-problems which can be analyzed individually and then again assembled together by means of network theory.

Numerical computations of electromagnetic radiation and scattering problems for arbitrarily shaped and electrically large objects in the same environment are preferably performed by combining the method of moments (MoM) with the uniform geometrical theory of diffraction (UTD). Composite metallic/dielectric objects with arbitrary shape are handled efficiently with the finite element boundary integral (FEBI) method. For large scale problems, the MoM solution is accelerated by the multilevel level fast multipole method (MLFMM). On the other hand, electrically large objects with relatively simple shape are treated sufficiently with ray-based high-frequency methods, like geometrical optics (GO) and UTD. However, until now MoM-UTD hybrid approaches have been restricted to conventional MoM formulations with only the electric field integral equation (EFIE), resulting in limited modeling capabilities. The same restrictions also exist in the hybrid FEBI-UTD method. The paper presents MLFMM-UTD hybridization for the combined field integral equation (CFIE), in order to extend further the applicability of the FEBI-UTD method. The resulting modeling technique is referred to as the FEBI-MLFMM-UTD method. The MLFMM-UTD hybridization is performed in the translation procedure on the various MLFMM levels, by taking into account additional contributions received at the testing groups due to UTD objects. Due to different ray directions for outgoing and incoming waves, appropriate interpolation and anterpolation procedures must be applied to consider the UTD contributions.

Coupling of a multilevel fast multipole method and a microlocal discretization for the 3-D integral equations of electromagnetism

An IPO (Iterative Physical Optics) scheme is proposed to analyze some important radome RF characteristics at high operating frequencies. We implement the proposed IPO scheme and simulate for an elliptically shaped radome at different gimbal angles. The results are verified with the MLFMM (Multi-Level Fast Multipole Method) by a general 3D EM (Electro magnetic) simulation software. Using the IPO method, the imagelobes can be accurately predicted while the first-order PO method cannot. Also, the transmission loss and boresight errors are computed for the radome.

a Design Concept of Piecewise Variable Wall Thickness of Radome Is Proposed to Simulate the Radome which Has Variable Wall. the Paper Calculates the Far Field of a Medium-Size Radome in the Case of Piecewise Variable Thickness by Three Methods as Follows: Method of Moment (MOM), Multilevel Fast Multipole Method (MLFMM) and Physical Optics&amp; the Method of Moments (PO/MOM) Respectively. after Comparing the Results, we Find that the PO/MOM Method Have the Superiority in Simulation of Radome’s Electromagnetic because it’s More Accuracy and Less Memory Consuming than the other Two Methods. Also it Proves the Feasibility of the Design Concept of Piecewise Variable Thickness for Radome.

Fast sweeping methods are efficient Gauss---Seidel iterative numerical schemes originally designed for solving static Hamilton---Jacobi equations. Recently, these methods have been applied to solve hyperbolic conservation laws with source terms. In this paper, we propose Lax---Friedrichs fast sweeping multigrid methods which allow even more efficient calculations of viscosity solutions of stationary hyperbolic problems. Due to the choice of Lax---Friedrichs numerical fluxes, general problems can be solved without difficult inversion. High order discretization, e.g., WENO finite difference method, can be incorporated to achieve high order accuracy. On the other hand, multigrid methods, which have been widely used to solve elliptic equations, can speed up the computation by smoothing errors of low frequencies on coarse meshes. We modify the classical multigrid method with regard to properties of viscous solutions to hyperbolic conservation equations by introducing WENO interpolation between levels of mesh grids. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving singularities of the viscosity solutions.

This paper presents a summary of a new version of the electromagnetic solver MONURBS based on method of moments (MoM). The tool includes a friendly graphical user interface (GUI) and provides low and high frequency analysis of arbitrary metallic or dielectric 3D structures. The code has been parallelized and improved by using multilevel fast multipole method (MLFMM), characteristic basis functions method (CBFM) and periodic method of moments (PMOM). The novel implementation of the optimizer allows robust and efficient electromagnetic optimization for a large number of optimization variables and goals.

A large circular loop antenna is used to excite the TE 41 mode at operation frequency of 9 GHz inside a circular waveguide section to launch a vortex beam. The effect of the antenna location inside the waveguide on its effective input impedance is modeled using transmission-line relations and optimal location for matching is determined. The analytical model results are confirmed using multi-level fast multipole method (MLFMM) full-wave simulations.