Multilevel Fast Multipole Algorithm Research Articles

A fast algorithm for multiscale electromagnetic problems using interpolative decomposition and multilevel fast multipole algorithm

The interpolative decomposition (ID) is combined with the multilevel fast multipole algorithm (MLFMA), denoted by ID‐MLFMA, to handle multiscale problems. The ID‐MLFMA first generates ID levels by recursively dividing the boxes at the finest MLFMA level into smaller boxes. It is specifically shown that near‐field interactions with respect to the MLFMA, in the form of the matrix vector multiplication (MVM), are efficiently approximated at the ID levels. Meanwhile, computations on far‐field interactions at the MLFMA levels remain unchanged. Only a small portion of matrix entries are required to approximate coupling among well‐separated boxes at the ID levels, and these submatrices can be filled without computing the complete original coupling matrix. It follows that the matrix filling in the ID‐MLFMA becomes much less expensive. The memory consumed is thus greatly reduced and the MVM is accelerated as well. Several factors that may influence the accuracy, efficiency and reliability of the proposed ID‐MLFMA are investigated by numerical experiments. Complex targets are calculated to demonstrate the capability of the ID‐MLFMA algorithm.

Efficient Surface Integral Equation Using Hierarchical Vector Bases for Complex EM Scattering Problems

A set of hierarchical divergence-conforming vector basis functions based on curved triangular patches is presented for method of moment (MoM) solutions of surface integral equations in this paper. The higher order method of combined-field integral equation (CFIE) solves perfect electric conductor electromagnetic scattering problems, the higher order electric-magnetic current combined-field integral equation (JMCFIE) formulations solves dielectric objects, and their combination is able to efficiently analyzes the scattering of hybrid PEC-dielectric objects. The expressions of the divergence- conforming hierarchical basis functions up to order 3.5 are reported in this paper. The multilevel fast multipole algorithm (MLFMA) is then employed to reduce the memory requirements and computational complexity. Numerical experiments indicate that the proposed hierarchical vector basis functions can provide well-conditioned linear system for iterative solution.

Scattering of Gaussian beam by arbitrarily shaped inhomogeneous particles

A hybrid finite element-boundary integral method is applied to characterize the scattering of an arbitrarily incident-focused Gaussian beam by arbitrarily shaped inhomogeneous particles. Specifically, the Davis–Barton fifth-order approximation in combination with rotation Euler angles is used to represent the arbitrarily incident Gaussian beams. The finite element method is employed to formulate the fields in the interior region of the inhomogeneous particle, while the boundary integral equation is applied to represent the fields in the exterior region. The interior and exterior fields are coupled by means of the field continuity conditions. To reduce the computational burden, the frontal method and the multilevel fast multipole algorithm are adopted to solve the resultant matrix equation. Numerical results for differential scattering cross sections of several selected inhomogeneous particles are presented and can be served as further study on this subject.

Inplementation of an Efficient Parallel Multilevel Fast Multipole Algorithm

This paper is concerned with the implementation of the parallel multilevel fast multipole algorithm(MLFMA) for large scale electromagnetics simulation on shared-memory system. The algorithm is implemented on a method of moment discretisation of the electromagnetics scattering problems.The developed procesure is validated by compared to benchmarks defined by Electromagnetics Code Consortium(EMCC) .The procesure can evaluate large problemssuch as electromagnetics scattering of aircraft at high-frequency with up to several millions of unknowns.

Fast Simulation of Array Structures Using T-EPA With Hierarchical LU Decomposition

In this letter, a novel technique that combines the tangential equivalence algorithm (T-EPA) with hierarchical (H-) LU decomposition is proposed to solve the electromagnetic problems of large arrays. The T-EPA is a kind of domain decomposition method based on integral equation. The characteristic basis functions (CBFs) have been used on equivalence surface to reduce the number of unknowns. Multilevel fast multipole algorithm (MLFMA) has also been used to accelerate the matrix-vector multiplication. However, the inversion of dense matrix required in T-EPA is very time-consuming. Here, an efficient H-LU decomposition based on H-matrix framework is proposed to reduce the computational complexity of the inversion of dense matrix. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

Error Analysis of Multilevel Fast Multipole Algorithm for Electromagnetic Scattering Problems

Error analysis of the multilevel fast multipole algorithm is studied for electromagnetic scattering problems. We propose novel error prediction and control methods and verify that the computational error for scattering problems with over one million unknowns can be precisely controlled under desired digits of accuracy. Optimum selection of truncation numbers to minimize computational error also will be discussed.

A fast 2‐D parallel multilevel fast multipole algorithm solver for oblique plane wave incidence

We present a multilevel fast multipole algorithm (MLFMA) implementation to numerically solve Maxwell's equations for large two‐dimensional geometries illuminated by an arbitrary plane wave in three‐dimensional space. The solver's capabilities are augmented by means of an asynchronous and hierarchical parallelization. Its accuracy is demonstrated by comparing the analytical and numerically obtained scattering width of several canonical examples with a size of 700,000 wavelengths.

Fast and accurate solutions of electromagnetics problems involving lossy dielectric objects with the multilevel fast multipole algorithm

Fast and accurate solutions of electromagnetic scattering problems involving lossy dielectric objects are considered. Problems are formulated with two recently developed formulations, namely, the combined-tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCFIE), and solved iteratively using the multilevel fast multipole algorithm (MLFMA). Iterative solutions and accuracy of the results are investigated in detail for diverse geometries, frequencies, and conductivity values. It is demonstrated that CTF solutions are significantly accelerated as the conductivity increases to moderate values and CTF becomes comparable to JMCFIE in terms of efficiency. Considering also the superior accuracy of this formulation, CTF becomes suitable for fast and accurate analysis of scattering problems involving lossy dielectric objects.

Revision of wFMM – A Wideband Fast Multipole Method for the two-dimensional complex Helmholtz equation

The Wideband Fast Multipole Method for the two-dimensional complex Helmholtz equation program is updated. The new version uses significantly less memory than the original version and uses almost constant memory for all the wavenumbers k when the number of particles is given. The CPU time is also improved slightly. Additionally, the memory leak problems and errors from external variables when it is used in an iterative solver are fixed. The new version wFMM and other useful codes are available from the website http://fastmultipole.org/.

Solutions of large-scale electromagnetics problems involving dielectric objects with the parallel multilevel fast multipole algorithm

Fast and accurate solutions of large-scale electromagnetics problems involving homogeneous dielectric objects are considered. Problems are formulated with the electric and magnetic current combined-field integral equation and discretized with the Rao-Wilton-Glisson functions. Solutions are performed iteratively by using the multilevel fast multipole algorithm (MLFMA). For the solution of large-scale problems discretized with millions of unknowns, MLFMA is parallelized on distributed-memory architectures using a rigorous technique, namely, the hierarchical partitioning strategy. Efficiency and accuracy of the developed implementation are demonstrated on very large problems involving as many as 100 million unknowns.

Efficient matrix filling of multilevel simply sparse method via multilevel fast multipole algorithm

[1] In this paper, an improved multilevel simply sparse method (MLSSM) is proposed for solving electromagnetic scattering problems that are formulated using the electric field integral equation approach. Previously, the matrix filling procedure of the conventional MLSSM is based on the adaptive cross approximation (ACA) method. Although the ACA is more efficient than direct filling, it requires a longer filling time for the far-field matrix than that of the multilevel fast multipole algorithm (MLFMA). Three problems with moderate electrical sizes are used to demonstrate that the far-field matrix filling memory of the ACA is also higher than that of the MLFMA. Hence, the MLFMA is utilized to reduce both the far-field matrix filling time and memory of the conventional MLSSM. Since the MLSSM recompresses the far-field interaction matrix of the MLFMA, the matrix-vector multiplication of the proposed method is more efficient than that of the MLFMA. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Multilevel Fast Multipole Algorithm-Based Direct Solution for Analysis of Electromagnetic Problems

In this communication, a multilevel fast multipole algorithm (MLFMA)-based direct method is proposed for solving electromagnetic scattering problems that are formulated using the electric-field integral equation (EFIE) approach. The method is based on the multilevel compressed block decomposition (MLCBD) algorithm. Previously, the matrix filling procedure of the MLCBD is based on the matrix decomposition algorithm-singular value decomposition (MDA-SVD) method. Although the MDA-SVD is more efficient than direct filling, it requires a longer filling time for the far-field matrix than for the MLFMA. The problems are used to demonstrate that the matrix filling memory requirement of the MDA-SVD is also higher than that of the MLFMA. Hence, the MLFMA is utilized to reduce both the matrix filling time and memory of the MLCBD. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Integral Equation Based Domain Decomposition Method for Solving Electromagnetic Wave Scattering From Non-Penetrable Objects

The integral equation (IE) method is commonly utilized to model time-harmonic electromagnetic (EM) problems. One of the greatest challenges in its applications arises in the solution of the resulting ill-conditioned matrix equation. We introduce a new domain decomposition method (DDM) for the IE solution of EM wave scattering from non-penetrable objects. The proposed method is a non-overlapping/non-conformal DDM and it provides a computationally efficient and effective preconditioner for the IE matrix equations. Moreover, the proposed approach is very suitable for dealing with multi-scale electromagnetic problems since each sub-domain has its own characteristics length and will be meshed independently. Furthermore, for each sub-domain, we are free to choose the most effective IE sub-domain solver based on its local geometrical features and electromagnetic characteristics. Additionally, the multilevel fast multi-pole algorithm (MLFMA) is utilized to accelerate the computations of couplings between sub-domains. Numerical results demonstrate that the proposed method yields rapid convergence in the outer Krylov iterative solution process. Finally, simulations of several large-scale examples testify to the effectiveness and robustness of the proposed IE based DDM.

Fast Convergence of Fast Multipole Acceleration Using Dual Basis Function in the Method of Moments for Composite Structures

The dual basis function proposed by Chen and Wilton in 1990 is used to represent the magnetic current for solving electromagnetic (EM) surface integral equations (SIEs) with penetrable materials and the solution process is accelerated with multilevel fast multipole algorithm (MLFMA) for large problems. The MLFMA is a robust accelerator for matrix equation solvers by iterative method, but its convergence rate strongly relies on the conditioning of system matrix. If the MLFMA is based on the method of moments (MoM) matrix in which the electric current is represented with the Rao-Wilton-Glisson (RWG) basis function, then how one represents the magnetic current in electric field integral equation (EFIE) and magnetic field integral equation (MFIE) really matters for the conditioning of system matrix. Though complicated in implementation, the dual basis function is ideal to represent the magnetic current because it is similar to the RWG basis function in properties but approximately orthogonal to it in space. With a simple testing scheme, the resultant system matrix is well-conditioned and the MLFMA acceleration can be rapidly convergent. Numerical examples for EM scattering by large composite objects are presented to demonstrate the robustness of the scheme.

A Scalable Parallel Wideband MLFMA for Efficient Electromagnetic Simulations on Large Scale Clusters

The development of the multilevel fast multipole algorithm (MLFMA) and its multiscale variants have enabled the use of integral equation (IE) based solvers to compute scattering from complicated structures. Development of scalable parallel algorithms, to extend the reach of these solvers, has been a topic of intense research for about a decade. In this paper, we present a new algorithm for parallel implementation of IE solver that is augmented with a wideband MLFMA and scalable on large number of processors. The wideband MLFMA employed here, to handle multiscale problems, is a hybrid combination of the accelerated Cartesian expansion (ACE) and the classical MLFMA. The salient feature of the presented parallel algorithm is that it is implicitly load balanced and exhibits higher performance. This is achieved by developing a strategy to partition the MLFMA tree, and hence the associated computations, in a self-similar fashion among the parallel processors. As detailed in the paper, the algorithm employs both spatial and direction partitioning approaches in a flexible manner to ensure scalable performance. Plethora of results are presented here to exhibit the scalability of this algorithm on 512 and more processors.

Analysis of Electrically Large Problems Using the Augmented EFIE With a Calderón Preconditioner

Calderon preconditioned electric-field integral equation (CP-EFIE) is very efficient in analyzing electromagnetic problems with a moderate electrical size. For the analysis of electrically large problems with closed surfaces, the CP-EFIE, however, suffers from the well-known spurious internal resonance problem because both the EFIE and the Calderon preconditioner are singular at resonant frequencies. In this paper, the resonance problem of the Calderon preconditioner is removed by using a complex wavenumber, and that of the EFIE is eliminated by enforcing the boundary condition n · D = ρs to the EFIE, resulting in the so-called augmented EFIE (AEFIE). It is shown that the proposed Calderon preconditioned AEFIE is a resonant-free formulation, and has a fast convergence rate for an iterative solution. Numerical examples are given to demonstrate the good accuracy and fast convergence of the proposed approach for the analysis of electrically large problems. The multilevel fast multipole algorithm (MLFMA) is employed to reduce the computational and storage complexities of the iterative solution.

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

A novel flexible GMRES with deflated restarted for efficient solution of electromagnetic problems

AbstractTo iterative solve large dense complex linear systems arising from electric field integral equations formulation of electromagnetic scattering problems, multilevel fast multipole algorithm (MLFMA) is built to speed up the matrix‐vector product (MVP) operations. This article presents a novel flexible generalized minimum residual with deflated restarted (FGMRES‐DR), which can greatly improve the convergence of those large dense complex linear systems. Based on physical mechanism of the scattering problems, in inner iteration of FGMRES‐DR, an iterative solution of the near‐field matrix in MLFMA is adopted. In outer iteration, deflation method which is purely algebraic is used to eliminate the influence on convergence because of the smallest eigenvalues. Numerical experiments indicate that FGMRES‐DR is very effective and can reduce the number of MVP compared with the FGMRES in large‐scale electromagnetic problems. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1360–1364, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26019

Simulation of a Luneburg Lens using a broadband Multilevel Fast Multipole Algorithm

[1] In this paper, the full wave simulation of a 2-D Luneburg lens is reported, using the Multilevel Fast Multipole Algorithm (MLFMA). To stabilize the MLFMA at low frequencies, it is augmented with a normalized plane-wave method, yielding a fully broadband solver. To test the proposed method, the Luneburg lens is partitioned into concentric shells with a constant permittivity, resulting in a complex simulation target that consists of multiple embedded dielectric objects. The numerical results are in good agreement with the analytical solutions for both the continuous and discretized lens.

Analysis of Dielectric Photonic-Crystal Problems With MLFMA and Schur-Complement Preconditioners

We present rigorous solutions of electromagnetics problems involving 3-D dielectric photonic crystals (PhCs). Problems are formulated with recently developed surface integral equations and solved iteratively using the multilevel fast multipole algorithm (MLFMA). For efficient solutions, iterations are accelerated via robust Schur-complement preconditioners. We show that complicated PhC structures can be analyzed with unprecedented efficiency and accuracy by an effective solver based on the combined tangential formulation, MLFMA, and Schur-complement preconditioners.