Multilevel fast multipole method based on a potential formulation for 3D electromagnetic scattering problems

It is suitable to solve the acoustic problems by using the fast multipole boundary element method (FMBEM), since the FMBEM can accelerate the matrix-vector multiplication dramatically by reducing the CPU time and memory of conventional boundary element method to O(N log2N) and O(N) respectively. We propose a 3D acoustic FMBEM based on Burton-Miller formulation in this paper. A new adaptive algorithm is applied to the diagonal form FMBEM, and a new proposed analytical moment formulation is used in the moment computation. Both of them further improve the efficiency of FMBEM. Acoustic scattering of soft sphere at resonant frequency is investigated to validate the accuracy of solution using Burton-Miller formulation. Comparisons of solution to the multi-radiating spheres problem with the one solved by Bapat's program demonstrate the accuracy and the efficiency of our algorithm in solving large-scale acoustic problems. In the end, we use our algorithm to analyze the inner sound filed of a car and dolphin acoustic scattering.

A new numerical method called the half multilevel fast multipole algorithm basis method (MLFMABM) for highly efficient and accurate Radar Cross Section (RCS) calculations of complex discontinuous characters is presented. The proposed method is used to estimate the RCS of a multi-discontinuous character plate based on a single-discontinuous character plate and its corresponding plate with no discontinuous character. It is shown that MLFMABM can achieve high efficiency and precision.

In R. Mueller et. al. (2006) the design of a UHF direction finding antenna with optimised radar cross section (RCS) was introduced. In the present paper the monostatic RCS of the antenna is characterised up to 18 GHz. Measured results of the RCS are compared to simulated results which are obtained using the new commercially available multilevel fast multipole method (MLFMM) ( U. Jakobus and J. van Tonder, 2005).

Finite Element Method (FEM1) is pervasively used in most of 3D product design analysis, in which Computer Aided Design (CAD) models need to be converted in to mesh models first and then enriched with some material features and boundary conditions data, etc. The interaction between CAD models and FEM models is intensive. Boundary Element Method (BEM) has been expected to be advantageous in large-scale problems in recent years owing to its reduction of the dimensionality and its reduced complexity in mesh generation. However, the BEM application has so far been limited to relatively small problems due to the memory and computational complexity for matrix buildup are O(N2). The fast multipole BEM (FMBEM) combined with BEM and fast multipole method (FMM) can overcome the defect of the traditional BEM, and provides an effective method to solve the large-scale problem. Combining GPU parallel computing with FMBEM can further improve its efficiency significantly. Based on the three-dimensional elastic mechanics problems, the parallelisms of the multipole moment (ME), multipole moment to multipole moment (M2M) translation, multipole moment to local expansion (M2L) translation, local expansion to local expansion (L2L) translation and near-field direct calculation were analyzed respectively according to the characteristics of the FMM, and the parallel strategies under CUDA were presented in this paper. Three main major parts are included herein: (1) FMBEM theory in 3D elastostatics, (2) the parallel FMBEM algorithm using CUDA, and (3) comparison the GPU parallel FMBEM with BEM, FEM and FMBEM respectively by engineering examples. Numerical example results show the 3D elastostatics GPU FMBEM not only can speed up the boundary element calculation process, but also save memory which can be effective to solve the large-scale engineering problems.

Quadcopter is very popular as a consumer electronics, but recently it is also being used on military purpose. To the defence side, detecting the quadcopter is a first priority, and to the offense side, reducing its detectability is important, and both are related to its radar cross section (RCS). We simulate the RCS of a typical quadcopter using the multi-level fast multi-pole method (MLFMM), and then try to reduce its radar cross section by modifying its external shape into a disk type. The results indicate such modification indeed can reduce its RCS to -10 dBsm when the radar wave is incident at horizontal direction, but less effective at other radar wave incident direction.

This study presents the numerical results for the aerodynamic and electromagnetic performances of an S-duct caret intake. Using the multilevel fast multipole method (MLFMM) to solve Maxwell equations, the current on the intake surface is calculated, and the radar cross-section (RCS) is analyzed. Moreover, the intake flow field is numerically investigated using the SST k–ω turbulence model to solve the Reynolds-averaged Navier–Stokes equations. Compared to a straight intake, for an S-duct caret intake, the average RCS is lower by 7.65 dB, and the maximum RCS difference value caused by the blade rotation is lower by 6.75 dB. However, the flow capacity deteriorates when the total pressure recovery coefficient decreases by 0.004. Based on the analysis of the aerodynamic and electromagnetic characteristics of different intakes, a double S-duct intake is designed. Compared to a traditional S-duct intake, for the novel intake after model parameter modification, the average RCS is lower by 0.05 dB, and the total pressure distortion (TPD) is lower by 0.18. The analysis of the effects of different boundary-layer bleed systems shows that the symmetrical layout adversely affects the aerodynamic and electromagnetic performances of the S-duct intake, but the unilateral partial layout is beneficial, whose TPD is lower by 0.04 and average RCS is higher by −2.17 dB compared to a straight intake.

The shape and design characteristics of a wing have a strong influence on the radar cross-section of a military aircraft. Little information regarding the airfoil design characteristics on the radar cross-section (RCS) is available in the open literature. This paper studies the change in RCS when the wing with different profiles is subjected to various trailing edge flap deflections. The effect of thickness of the airfoil on the RCS is studied by using NACA 0009, NACA 0012, and NACA 0015 as the wing cross-section shape. A wing with a NACA 0012 airfoil was subjected to various flap deflections to study the change in RCS pattern. The significant variation in RCS and increase in number of lobes was observed as the amount of flap deflection increases. In addition, a comparison of the RCS was done using the three airfoil profiles. RCS dip was highest for a thinner wing, i.e., NACA 0009 and lowest for a thicker wing, i.e., NACA 0015 and for all three wings RCS dip occurs at around 260° observation angle when the flap deflects to 45° in the downward direction. Multilevel Fast Multipole Method (MLFMM), a full-wave technique available in the commercial software FEKO, was used to conduct this study.

This work studies the use of simplified geometric models to represent vehicles for Integrated Sensing and Communication (ISAC) systems. Four simplified models: a simple cuboid, a curved cuboid, two curved cuboids, and a modified two curved cuboid, are designed based on the dimensions of a real car (VW Sharan). Their bistatic radar cross sections (RCS) are simulated using the multilevel fast multipole method (MLFMM) in Ansys HFSS and compared to that of the full-scale vehicle model using cross-correlation and median of absolute error (MedAE) metrics. Among the proposed models, the modified two curved cuboid shows the closest resemblance to the scattering response of the VW Sharan.

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.

An adaptive dual-information FMBEM for 3D elasticity and its GPU implementation

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.

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.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

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)