Near-field Matrix Research Articles

Electromagnetic Scattering From Multiple Moving Targets Above a Rough Surface

Aimed at solving composite scattering from multiple moving targets above a rough surface, a hybrid method combining the multilevel fast multipole algorithm (MLFMA) and Kirchhoff approximation (KA) is presented. The technology combining the MLFMA with the ray-propagation fast multipole algorithm (RPFMA) with the fast far-field approximation (FAFFA) is applied to accelerate coupling matrix filling. There are two main bottlenecks in this problem. The first is that with moving targets; specifically, we need to refill the impedance matrix and recompute composite scattering, and the second is solving the tremendous coupling matrix between multiple targets and a rough surface. To solve the first bottleneck, the hybrid MLFMA-KA based on the domain decomposition method (DDM) is used, with the target region solved by the MLFMA and the surface region solved by the KA. Additionally, the impedance matrix on each target is divided into the near-field matrix and far-field matrix on the same target, the far-field matrix between different targets and between targets with a rough surface. With moving targets, the impedance matrix on the same target remains unchanged and only needs to be solved once, and the far-field matrix between different targets and between targets with a rough surface needs to be recomputed. To solve the second bottleneck, the MLFMA-RPFMA-FAFFA is used to accelerate matrix vector multiplication between the targets and rough surfaces. Compared with the MLFMA, the proposed method is proven to be efficient and accurate. In addition, electromagnetic scattering from multiple moving targets above the surface are analyzed.

Fast Power Series Solution of Large 3-D Electrodynamic Integral Equation for PEC Scatterers

This paper presents a new fast power series solution method to solve the Hierarchal Method of Moment (MoM) matrix for a large complex, perfectly electric conducting (PEC) 3D structures. The proposed power series solution converges in just two (2) iterations which is faster than the conventional fast solver–based iterative solution. The method is purely algebraic in nature and, as such applicable to existing conventional methods. The method uses regular fast solver Hierarchal Matrix (H-Matrix) and can also be applied to Multilevel Fast Multipole Method Algorithm (MLFMA). In the proposed method, we use the scaling of the symmetric near-field matrix to develop a diagonally dominant overall matrix to enable a power series solution. Left and right block scaling coefficients are required for scaling near-field blocks to diagonal blocks using Schur’s complement method. However, only the right-hand scaling coefficients are computed for symmetric near-field matrix leading to saving of computation time and memory. Due to symmetric property, the left side-block scaling coefficients are just the transpose of the right-scaling blocks. Next, the near-field blocks are replaced by scaled near-field diagonal blocks. Now the scaled near-field blocks in combination with far-field and scaling coefficients are subjected to power series solution terminating after only two terms. As all the operations are performed on the near-field blocks, the complexity of scaling coefficient computation is retained as O(N)O⁢(N). The power series solution only involves the matrix-vector product of the far-field, scaling coefficients blocks, and inverse of scaled near-field blocks. Hence, the solution cost remains O(NlogN)O⁢(N⁢l⁢o⁢g⁢N). Several numerical results are presented to validate the efficiency and robustness of the proposed numerical method.

Fast Analysis of Broadband Electromagnetic Scattering Characteristics of Electrically Large Targets using Precorrected Fast Fourier Transform Algorithm based on Near Field Matrix Interpolation Method

In this paper, a new method is proposed to analyze the broadband electromagnetic characteristics of electrically large targets by combining the precorrected-FFT algorithm (P-FFT) with the near-field matrix interpolation technique. The proposed method uses the precorrected-FFT algorithm to reduce the storage and accelerate the matrix vector product of the far field. In order to make the precorrected-FFT algorithm can calculate the broadband characteristics of electrically large targets more quickly, the matrix interpolation method is used to interpolate the near-field matrix of the precorrected-FFT algorithm to improve the efficiency of calculation. The numerical results obtained validate the proposed method and its implementation in terms of accuracy and runtime performance.

Fast Computation by MLFMM-FFT with NURBS in Large Volumetric Dielectric Structures

A refinement for the computation of the rigorous part of the multi-level fast multipole method (MLFMM) of analyzing volumetric objects is presented. A scheme based on the fast Fourier technique (FFT) is proposed with the objective of reducing the computational resources required to accurately analyze large homogeneous and non-homogeneous dielectric volumes. In order to reduce the memory requirements, the storage of the near-field terms of the method of moments (MoM) matrix is performed only for the positions corresponding to a parallelepiped with the size of the level 1 block of the MLFMM, computed with the vacuum permittivity, taking advantage of the Toeplitz symmetry present in regular hexahedral meshes. The FFT avoids applying the near-field MoM matrix in the iterative solution process. The application of this approach results in huge improvements in terms of memory usage, but also a speeds up the iterative solution process because the use of three-dimensional (3D) FFTs is very efficient for computing convolutions when the number of unknowns of the problems becomes very large as happens in volumetric problems. We also propose a new approach for the numerical treatment of the transition of the dielectric permittivity between different dielectrics or between a dielectric and a free space. To validate the computation technique, the radar cross section (RCS) of several dielectric bodies is computed using the classical MLFMM approach and it is compared with the presented FFT-based-MLFMM solution. The results demonstrate that the efficient memory and computation time usage of the proposed approach.

Massively Parallel Discontinuous Galerkin Surface Integral Equation Method for Solving Large-Scale Electromagnetic Scattering Problems

In this communication, we present flexible and efficient solutions of large-scale electromagnetic scattering problems using the parallel discontinuous Galerkin (DG) surface integral equation (SIE) method. A simplified formulation of the conventional DG is used by removing the interior penalty term, thereby avoiding computational burden of the contour integral in the conventional DG. It is demonstrated numerically that the simplification causes convergence deficiency and changes the accuracy convergence rate of the solution for objects with sharp corners. The former is overcome by constructing preconditioners using the near-field matrix of the whole region, and the latter is proved to have a neglectable influence on the accuracy of the final solutions as long as normal mesh density requirement is satisfied. The well-scaling ternary parallelization approach of the multilevel fast multipole algorithm (MLFMA) is incorporated into the DG domain decomposition framework to reduce the complexity of the solution and strength its capability for large-scale problems, with the distribution of the near-field-related calculation adjusted to adapt the nonuniform meshes for a better workload balance. Numerical results are included to validate the accuracy and demonstrate the scalability and versatility of the proposed method. In addition, we demonstrated the effectiveness of our algorithm by solving a high-definition complicated aircraft model with inlets and exhausts involving over one billion unknowns.

A Discontinuous Galerkin Surface Integral Solution for Scattering From Homogeneous Objects With High Dielectric Constant

The discontinuous Galerkin (DG) method for homogeneous bodies has been studied and shown to be an efficient tool for multiscale homogeneous bodies. However, the slow convergence of DG with the block diagonal preconditioner (BDP) is still observed in solving high contrast homogeneous bodies. An efficient preconditioning approach is designed for the DG method in this communication by using the sparsing approach on the near-field matrix of the whole region. The iteration convergence speed of the DG method is improved while the computing resources for constructing the preconditioner are effectively reduced. Numerical experiments demonstrate the capability of the presented DG method for multiscale homogeneous bodies, especially for those with a high dielectric constant.

Improved Algebraic Preconditioning for Multiscale Electromagnetic Problems

An improved scheme for the algebraic preconditioners is proposed in this letter to accelerate the iterative convergence rate of the multiscale electromagnetic problems, in which the overcrowded leaf boxes of the multilevel fast multipole algorithm are recursively bisected with the application of the multilevel accelerated Cartesian expansion algorithm to maintain the computational efficiency. Due to a lack of some strong couplings measured by the relative distance between the testing/basis functions, the algebraic preconditioners constructed by filtering the near-field interaction matrix of the fast algorithm are consequently ineffective. To alleviate the adverse effect of the multiscale-size boxes, the proposed scheme constructs the preconditioners on higher levels rather than the leaf level of the related fast algorithm, thus speeding up the iterative convergence rate significantly. Moreover, the rank-revealing adaptive cross approximation algorithm is first introduced to compute and store the enlarged computational domain of the preconditioner. As a result, there is only a modest increase in the computational cost. Numerical experiments have been performed to demonstrate the effectiveness and efficiency of the proposed method.

Fitting Green’s Function FFT Acceleration Applied to Anisotropic Dielectric Scattering Problems

A volume integral equation based fast algorithm using the Fast Fourier Transform of fitting Green’s function (FG-FFT) is proposed in this paper for analysis of electromagnetic scattering from 3D anisotropic dielectric objects. For the anisotropic VIE model, geometric discretization is still implemented by tetrahedron cells and the Schaubert-Wilton-Glisson (SWG) basis functions are also used to represent the electric flux density vectors. Compared with other Fast Fourier Transform based fast methods, using fitting Green’s function technique has higher accuracy and can be applied to a relatively coarse grid, so the Fast Fourier Transform of fitting Green’s function is selected to accelerate anisotropic dielectric model of volume integral equation for solving electromagnetic scattering problems. Besides, the near-field matrix elements in this method are used to construct preconditioner, which has been proved to be effective. At last, several representative numerical experiments proved the validity and efficiency of the proposed method.

The MLFMA Equipped with a Hybrid Tree Structure for the Multiscale EM Scattering

We present an efficient strategy for reducing the memory requirement for the near-field matrix in the multilevel fast multipole algorithm (MLFMA) for solving multiscale electromagnetic (EM) scattering problems. A multiscale problem can obviously lower the storage efficiency of the MLFMA for the near-field matrix. This paper focuses on overcoming this shortcoming to a certain extent. A hybrid tree structure for the MLFMA that possesses two kinds of bottom-layer boxes with different edge sizes will be built to significantly reduce the memory requirement for the near-field matrix in the multiscale case compared with the single-tree-structure technique. Several numerical examples are provided to demonstrate the efficiency of the proposed scheme in the multiscale EM scattering.

Modified compressed block decomposition preconditioner for electromagnetic problems

A modified compressed block decomposition (CBD) preconditioner combined with the multilevel fast multipole algorithm (MLFMA) is proposed for efficiently solving large dense complex linear systems that arise in electromagnetic problems.The modified preconditioner utilizes some improvements to improve the CBD preconditioner which is constructed from the available sparse near-field matrix. On the basis of the modified preconditioner, an efficient matrix–vector multiplication is implemented. Accordingly, the modified preconditioner is comparable with the unpreconditioned generalized minimal residual algorithm (GMRES) in terms of the number of iterations and the total solution time. Remarkably, the modified preconditioner provides a reduction close to one order of magnitude in both the number of iterations and the total solution time. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26149

Multiresolution preconditioner combined with matrix decomposition – singular value decomposition algorithm for fast analysis of electromagnetic scatterers with dense discretisations

A combination of a multiresolution (MR) preconditioner and matrix decomposition algorithm – singular value decomposition (MDA-SVD) algorithm is proposed for the analysis of densely discretised electromagnetic (EM) scatterers. In realistic EM scattering problems, the discretisation may be very dense to accurately describe the geometric shape of targets. However, dense discretisations will generate two major drawbacks: (i) the badly conditioned method of moments (MoM) matrix which makes iterative solvers difficult to converge; (ii) the dense near-field MoM matrix for which the computational cost is large, even multilevel fast multipole algorithm (MLFMA) is applied. To remedy the first shortcoming of dense discretisations, an MR preconditioner can be applied to improve the condition of MoM matrices. To reduce the computational cost in solving the MoM matrix equation, an MDA-SVD algorithm can be applied to accelerate the matrix−vector multiplication. Compared with the traditional MLFMA, the MDA-SVD algorithm has no limit on the size of octree boxes which makes the MDA-SVD algorithm suitable for the analysis of dense discretisations. Therefore an MR preconditioner is combined with the MDA-SVD algorithm for fast analysis of dense discretisations. Numerical results indicate that the combination of the MR preconditioner and the MDA-SVD algorithm is efficient.

Data-sparse LU-decomposition preconditioning combined with multilevel fast multipole method for electromagnetic scattering problems

Owing to the robust of the multilevel fast multipole method (MLFMM), large dense linear systems, arising from discretised electric field integral equations of electromagnetic scattering problems, are generally solved by a Krylov iterative solver. An efficient preconditioning technique based on hierarchical LU-decomposition (ℋ-LU) is proposed to accelerate the convergence rate of the Krylov iterations. The ℋ-LU preconditioner is constructed from the available sparse near-field matrix and provides a data-sparse way to approximate its LU-factors, which can be computed and stored in ℋ-matrix arithmetic with almost linear complexity. The accuracy of the approximation is adjustable that will guarantee a bounded number of iterations. Some means are introduced to further lower the computational cost of the ℋ-LU. Numerical examples are provided to illustrate that the resulting ℋ-LU preconditioner is effective and flexible with MLFMM to reduce both the iteration number and the overall simulation time significantly.

Schur Complement Preconditioners for Surface Integral-Equation Formulations of Dielectric Problems Solved with the Multilevel Fast Multipole Algorithm

Surface integral-equation methods accelerated with the multilevel fast multipole algorithm (MLFMA) provide a suitable mechanism for electromagnetic analysis of real-life dielectric problems. Unlike the perfect-electric-conductor case, discretizations of surface formulations of dielectric problems yield $2 \times 2$ partitioned linear systems. Among various surface formulations, the combined tangential formulation (CTF) is the closest to the category of first-kind integral equations, and hence it yields the most accurate results, particularly when the dielectric constant is high and/or the dielectric problem involves sharp edges and corners. However, matrix equations of CTF are highly ill-conditioned, and their iterative solutions require powerful preconditioners for convergence. Second-kind surface integral-equation formulations yield better conditioned systems, but their conditionings significantly degrade when real-life problems include high dielectric constants. In this paper, for the first time in the context of surface integral-equation methods of dielectric objects, we propose Schur complement preconditioners to increase their robustness and efficiency. First, we approximate the dense system matrix by a sparse near-field matrix, which is formed naturally by MLFMA. The Schur complement preconditioning requires approximate solutions of systems involving the (1,1) partition and the Schur complement. We approximate the inverse of the (1,1) partition with a sparse approximate inverse (SAI) based on the Frobenius norm minimization. For the Schur complement, we first approximate it via incomplete sparse matrix-matrix multiplications, and then we generate its approximate inverse with the same SAI technique. Numerical experiments on sphere, lens, and photonic crystal problems demonstrate the effectiveness of the proposed preconditioners. In particular, the results for the photonic crystal problem, which has both surface singularity and a high dielectric constant, shows that accurate CTF solutions for such problems can be obtained even faster than with second-kind integral equation formulations, with the acceleration provided by the proposed Schur complement preconditioners.

Extended near field preconditioner for the analysis of large problems using the nested-FMM-FFT algorithm

We present here an extension of the near-field matrix iterative preconditioner applied in the context of the Nested FMM-FFT algorithm, which is a bilevel variation of the original FMM-FFT. The new preconditioner benefits from the nested scheme to extend the information of the near-field matrix, then allowing to achieve a closer approximation to the complete EM problem. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:430–433, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25709

Iterative Near-Field Preconditioner for the Multilevel Fast Multipole Algorithm

For iterative solutions of large and difficult integral-equation problems in computational electromagnetics using the multilevel fast multipole algorithm (MLFMA), preconditioners are usually built from the available sparse near-field matrix. The exact solution of the near-field system for the preconditioning operation is infeasible because the LU factors lose their sparsity during the factorization. To prevent this, incomplete factors or approximate inverses can be generated so that the sparsity is preserved, but at the expense of losing some information stored in the near-field matrix. As an alternative strategy, the entire near-field matrix can be used in an iterative solver for preconditioning purposes. This can be accomplished with low cost and complexity since Krylov subspace solvers merely require matrix-vector multiplications and the near-field matrix is sparse. Therefore, the preconditioning solution can be obtained by another iterative process, nested in the outer solver, provided that the outer Krylov subspace solver is flexible. With this strategy, we propose using the iterative solution of the near-field system as a preconditioner for the original system, which is also solved iteratively. Furthermore, we use a fixed preconditioner obtained from the near-field matrix as a preconditioner to the inner iterative solver. MLFMA solutions of several model problems establish the effectiveness of the proposed nested iterative near-field preconditioner, allowing us to report the efficient solution of electric-field and combined-field integral-equation problems involving difficult geometries and millions of unknowns.

Accelerating the Multilevel Fast Multipole Algorithm with the Sparse-Approximate-Inverse (SAI) Preconditioning

With the help of the multilevel fast multipole algorithm, integral-equation methods can be used to solve real-life electromagnetics problems both accurately and efficiently. Increasing problem dimensions, on the other hand, necessitate effective parallel preconditioners with low setup costs. In this paper, we consider sparse approximate inverses generated from the sparse near-field part of the dense coefficient matrix. In particular, we analyze pattern selection strategies that can make efficient use of the block structure of the near-field matrix, and we propose a load-balancing method to obtain high scalability during the setup. We also present some implementation details, which reduce the computational cost of the setup phase. In conclusion, for the open-surface problems that are modeled by the electric-field integral equation, we have been able to solve ill-conditioned linear systems involving millions of unknowns with moderate computational requirements. For closed-surface problems that can be modeled by the combined-field integral equation, we reduce the solution times significantly compared to the commonly used block-diagonal preconditioner.

Optimal Panel-Clustering in the Presence of Anisotropic Mesh Refinement

In this paper we consider the numerical solution of discrete boundary integral equations on polyhedral surfaces in three dimensions. When the solution contains typical edge singularities, highly stretched meshes are preferred to uniform meshes, since they reduce the number of degrees of freedom needed to obtain a fixed accuracy. The classical panel-clustering method can still be applied in the presence of such highly stretched meshes. However, we will show that the savings in computation time and storage become suboptimal because the nearfield matrix arising in the panel-clustering algorithm is no longer as sparse as it is in the case of uniform meshes. Hence, a natural question arises as to whether a new enhanced panel-clustering algorithm can be designed which performs efficiently even in the presence of highly stretched meshes. The main result of this paper is to formulate such an enhanced version of the panel-clustering algorithm. The key features of the algorithm are (i) the employment of partial analytic integration in the direction of stretching, yielding a new kernel function on a one-dimensional manifold where the influence of high aspect ratios in the stretched elements is removed, and (ii) the introduction of a generalized admissibility condition with respect to the partially integrated kernel, which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible. In the context of a model problem, we prove that our algorithm yields an accurate (up to the discretization error) matrix-vector multiplication which requires ${\cal O}(N log^k N)$ operations, where $N$ is the number of degrees of freedom and $k$ is small and independent of the aspect ratio. The generalized admissibility condition can be viewed as an addition to the classical method which may be useful in general when stretched meshes are present. We also have performed a numerical experiment which shows that the sparsity of the nearfield matrix for the enhanced panel-clustering method is not negatively affected by stretched elements, and the method will perform optimally.

Bistatic Scattering and Emissivities of Lossy Dielectric Surfaces With Exponential Correlation Functions

Bistatic scattering and emissivities of surfaces with exponential correlation functions are studied numerically for 2-D geometries in a numerical Maxwell model with 2-D simulations. Surfaces with exponential correlation functions are important for the active and passive microwave remote sensing of land surfaces. Because of the fine-scale features with large slopes of such surfaces, numerical accuracy, which is particularly important for the calculation of emissivity in passive remote sensing, is ensured by a variety of procedures in this paper. The rooftop function and Galerkin's method with numerical integration of near-field impedance matrix elements are used. Cubic spline interpolation is employed to connect knots on random rough surfaces. Numerical accuracy convergence tests are performed for numerical solutions of Maxwell equations by varying the number of points from 13 to 103 points per wavelength in the dielectric medium corresponding to 50-400 points per free wavelength. Surface lengths of up to 100 and 200 free wavelengths and root mean square heights of up to 0.4 and 0.8 free wavelengths, respectively, are used at 5 and 10 GHz to capture all the essential features. Because of the large number of surface unknowns (up to 80 000), the multilevel UV method is further used to accelerate the matrix equation solver. Numerical results are illustrated for both bistatic scattering and emissivities as functions of frequencies and incidence and scattering angles for cases of interests in microwave remote sensing. Comparisons are made with the second-order small perturbation method and Kirchhoff's approximation to reestablish the regimes of validity of these methods

Perturbed incomplete ILU preconditioner for efficient solution of electric field integral equations

To efficiently solve large, dense, complex linear systems that arise in the electric field integral equation (EFIE) formulation of electromagnetic scattering problems, a new modified incomplete LU (ILU) preconditioner is developed and used in the context of the generalised minimal residual iterative method accelerated with the multilevel fast multipole method. The key idea is to perturb the near-field impedance matrix of EFIE with the principle value term of the magnetic field integral equation operator before constructing ILU preconditioners. Numerical experiments indicate that this new perturbation technique is very effective with the ILU preconditioner and the resulted ILU preconditioner can reduce both the iteration number and the computational time substantially.

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

We consider the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE). For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. The electromagnetic scattering problem is here solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix. In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations in the solution of the resulting system of equations.