J. M. Bértolo

MLFMA-FFT PARALLEL ALGORITHM FOR THE SO- LUTION OF LARGE-SCALE PROBLEMS IN ELECTRO- MAGNETICS (INVITED PAPER)

MLFMA-FFT PARALLEL ALGORITHM FOR THE SO-LUTION OF LARGE-SCALE PROBLEMS IN ELECTRO-MAGNETICS (INVITED PAPER)J. M. TaboadaDepartment Tecnolog¶‡as de los Computadores y de lasComunicaciones, Escuela Polit¶ecnicaUniversidad de ExtremaduraC¶aceres 10071, SpainM. G. Araujo¶ and J. M. B¶ertoloDepartment Teor¶‡a do Sinal e Comunicaci¶ons, E.T.S.E.Telecomunicaci¶onUniversidade de VigoVigo (Pontevedra) 36310, SpainL. LandesaDepartment Tecnolog¶‡as de los Computadores y de lasComunicaciones, Escuela Polit¶ecnicaUniversidad de ExtremaduraC¶aceres 10071, SpainF. Obelleiro and J. L. RodriguezDepartment Teor¶‡a do Sinal e Comunicaci¶ons, E.T.S.E.Telecomunicaci¶onUniversidade de VigoVigo (Pontevedra) 36310, SpainAbstract|An e–cient hybrid MPI/OpenMP parallel implementationof an innovative approach that combines the Fast Fourier Transform(FFT) and Multilevel Fast Multipole Algorithm (MLFMA) has beensuccessfully used to solve an electromagnetic problem involving 620millions of unknowns. The MLFMA-FFT method can deal withextremely large problems due to its high scalability and its reducedcomputational complexity. The former is provided by the use of the

SUPERCOMPUTER AWARE APPROACH FOR THE SOLUTION OF CHALLENGING ELECTROMAGNETIC PROBLEMS

Abstract|It is a proven fact that The Fast Fourier Transform(FFT) extension of the conventional Fast Multipole Method (FMM)reduces the matrix vector product (MVP) complexity and preservesthe propensity for parallel scaling of the single level FMM. In thispaper, an e–cient parallel strategy of a nested variation of the FMM-FFT algorithm that reduces the memory requirements is presented.The solution provided by this parallel implementation for a challengingproblem with more than 0.5 billion unknowns has constituted the worldrecord in computational electromagnetics (CEM) at the beginning of2009.1. INTRODUCTIONRecent years have seen an increasing efiort in the development of fastand e–cient electromagnetic solutions with a reduced computationalcost regarding the conventional Method of Moments. Among others,the Fast Multipole Method (FMM) [1] and its multilevel version, theMLFMA [2,3] have constituted one of the most important advances inthat context.This development of fast electromagnetic solvers has gone handin hand with the constant advances in computer technology. Dueto this simultaneous growth, overcoming the limits in the scalabilityof the available codes became a priority in order to take advantageof the large amount of computational resources and capabilities thatare available in modern High Performance Computer (HPC) systems.For this reason, works focused on the parallelization improvement ofthe Multilevel Fast Multipole Algorithm (MLFMA) [4{13] have gainedinterest in last years.Besides, the FMM-Fast Fourier Transform (FMM-FFT) deservesbe taken into account as an alternative to beneflt from massivelyparallel distributed computers. This variation of the single-level FMMwas flrst proposed in [14] as an acceleration technique applied to almostplanar surfaces. Later on, a parallelized implementation was applied togeneral three-dimensional geometries [15]. The method uses the FFTto speedup the translation stage resulting in a dramatic reduction ofthe matrix-vector product (MVP) time requirement with respect tothe FMM. Although in general the FMM-FFT is not algorithmically ase–cient as the MLFMA, it has the advantage of preserving the naturalparallel scaling propensity of the single-level FMM in the spectral (

General purpose software package for electromagnetics engineering education

This article presents a general‐purpose educational software applied to electromagnetics engineering education. The program presented allows the students to obtain the RCS, current density, and near field distributions of perfectly electric conducting (PEC) canonical geometries being illuminated by arbitrary incident plane waves.

GEOMETRY BASED PRECONDITIONER FOR RADIATION PROBLEMS INVOLVING WIRE AND SURFACE BASIS FUNCTIONS

An innovative preconditioner has been developed in this work. It signiflcantly improves the convergence of the iterative solvers applied to electromagnetic radiation problems by a renormalization of the matrix equation. The preconditioner balances the disparities in terms of magnitude and units caused by the strong self-coupling of the antennas, the non-uniformity of the meshes and also by the coexistence of wire and surface basis functions. It can be easily integrated into difierent electromagnetic solvers with a negligible impact on the computational cost on account of its simple implementation.

Analysis of 0.5 billion unknowns using a parallel FMM-FFT solver

The last decade witnessed a great effort in the development of fast and efficient algorithms to reduce the computational cost of the MoM [1]. Among others, perhaps the most important advance in this area has been the fast multipole method (FMM) introduced by Rokhlin [2], and later on its multilevel version, namely the MLFMA [3], [4]. Simultaneously, the sustained growth in computer technology has lead to the availability of computer clusters and multi-core processors with very large memory and computational capabilities. Concurrent advances in these two areas have motivated the development of codes for massively parallel distributed computers, enabling faster, more efficient and accurate numerical solutions in computational electromagnetics. In this context, the parallelization of the MLFMA became an objective of great interest, and it has been handled by a lot of research groups (see e.g. [5]–[17]); unfortunately, they were faced with the poor scalability and load-unbalance characteristics of this algorithm, both being key questions in large distributed memory supercomputers. While some success has been demonstrated, the parallel scalability of the MLFMA has been limited to only 16 to 64 processors, which contrasts with the thousands of processors that are available in modern high performance computing (HPC) systems.

MLFMA-FFT algorithm for the solution of challenging problems in electromagnetics

The development of fast and efficient algorithms to reduce the computational cost of the method of moments (MoM) has received a great attention in recent years. Among others, one of the most important advances was the development of the fast multipole method (FMM) [1] and its multilevel version, the MLFMA [2]. The FMM reduces the computational complexity from O(N2) -using an iterative resolution of the MoM-, to O(N3/2), and the multilevel versions have achieved O(N log N). So, while substantially more difficult to implement, the MLFMM has become the choice when solving large-scale electromagnetics scattering problems.

Parallel FMM-FFT solver for the analysis of hundreds of millions of unknowns

An efficient parallel implementation of the fast multipole method (FMM) combined with the fast fourier transform (FFT) is presented. The good scaling properties of the FMM-FFT, combined with a careful parallelization strategy, has shown to be very effective when using large parallel high performance supercomputers. For the case of very large problems, with hundreds of millions of unknowns, a nested scheme has been derived that further reduces the memory consumption. A challenging problem with more than 0.5 billion unknowns has been solved using this implementation, which demonstrates the ability of the algorithm to take advantage of the availability of supercomputers for the analysis of large, leading-edge electromagnetic problems.

Geometrically based preconditioner for the Fast Multipole Method using rooftop basis functions and Galerkin testing procedure

The EFIE formulation has been adopted in the FMM-Fast Fourier Transform (FFT) code [7], using a restarted GMRES solver. The first tested geometry consists on a hollow cylinder (1.6λ radius, 3.2λ height) and two λ/2-length dipoles modeled employing wire basis functions. These dipoles are set at a λ/4 distance from the cylinder wall spaced 90°. The estimation of the mutual admittance provided by a reference Method of Moments (MoM) code, non-preconditioned FMM-FFT and preconditioned FMM-FFT approaches is shown in Figure 1. The non-preconditioned result comes up slowly to the reference solution without reaching it after the allowed number of iterations. A better agreement is achieved in a few iterations with the preconditioned solution. In order to give more insight on the preconditioner behavior, it has been evaluated an error measure of the current coefficients regarding to MoM solution. With an analogous aim than for rw in (3), this error has been weighted by L to express all the terms in Amperes, independently of the basis function type. It has been obtained an error of 1% with the preconditioned method, better than the 6% for the non-preconditioned solution. [htpb] Both original and weighted residues of equation (3) are plotted in Figure 2. It illustrates clearly the chaotic behavior of the original residue in contrast to the smooth curves obtained using the weighting procedure. It may be appreciated that a better convergence is obtained for the preconditioned method and that the weighted residue is more correlated with the actual situation reflected in Figure 1.