Fast Multipole Acceleration Research Articles

Fast and Accurate EEG/MEG BEM-Based Forward Problem Solution for High-Resolution Head Models.

A BEM (boundary element method) based approach is developed to accurately solve an EEG/MEG forward problem for a modern high-resolution head model in approximately 60 seconds using a common workstation. The method utilizes a charge-based BEM with fast multipole acceleration (BEM-FMM) and a "smart" mesh pre-refinement (called b-refinement) close to the singular source(s). No costly matrix-filling or direct solution steps typical for the standard BEM are required; the method generates on-skin voltages as well as MEG magnetic fields for high-resolution head models in approximately 60 seconds after initial model assembly. The method is verified both theoretically and experimentally.

An adaptive h-refinement method for the boundary element fast multipole method for quasi-static electromagnetic modeling

Objective. In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems. Approach. We propose, describe, and systematically investigate an AMR method using the boundary element method with fast multipole acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy. The implemented AMR method’s accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a ‘silver-standard’ solution found by subsequent 4:1 global refinement of the adaptively-refined model. Main results. Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models. Significance. This error has especially important implications for TES dosing prediction—where the stimulation strength plays a central role—and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.

Integral equation methods for the Morse-Ingard equations

We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nyström method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.

Laplace Green's functions for infinite ground planes with local roughness

The Green's functions for the Laplace equation satisfying the Dirichlet and Neumann boundary conditions on the upper side of the infinite plane with a circular hole are introduced and studied. These functions enable solutions of the boundary value problems in domains where the hole is closed by an arbitrary mesh (locally rough surfaces). The developed approach accounts for arbitrary positive and negative ground elevations inside the domain of interest, which is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply to other domains where the Laplace or Poisson equations govern. Integral and series representations of the Green's functions are provided. Using these Green's functions, an efficient computational technique based on the boundary element method with fast multipole acceleration is developed. A numerical study of some benchmark problems is presented.

FMM Accelaration of 2D Well-Conditioned T-Matrix Method for Multiple Scatterers

The T-Matrix method for a multitude of two-dimensional dielectric targets can be well-conditioned based on the ideas on Analytical Regularization Method. A peculiar implementation of Fast Multipole Acceleration for this method is revisited. The advantages of working with linear algebraic equations of the second kind instead of the first kind are underlined in terms of reduction in the number of iterations for the iterative solver for a specific residual error.

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

AbstractWe present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green's functions. We describe fast solvers for the two‐dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed “near” part plus a small number of auxiliary sources that represent the (smooth) remaining “far” contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank‐1 or rank‐3 correction and a Schur complement, leaves awell‐conditionedsquare system that can be solved iteratively using fast multipole acceleration plus a low‐rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension‐ and PDE‐independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close‐to‐touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against high‐accuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 104randomly located inclusions per unit cell. © 2018 Wiley Periodicals, Inc.

Compressive Sensing Based Multilevel Fast Multipole Acceleration for Fast Scattering Center Extraction and ISAR Imaging.

In recent years, Compressive Sensing (CS) theory has been very popular in the data sensing and process area as it utilizes the sparsity and measurement matrix to reconstruct the compressible signal from limited samples successfully. In this paper, CS is introduced into an efficient numerical method, multilevel fast multipole acceleration (MLFMA), for the electromagnetic (EM) scattering problem over a wide incident angle. This allows composition of a new kind of incident wave, which obtains efficient and reliable data for scattering centers extraction with low complexity. The resulting data from CS-based MLFMA are processed for ISAR) imaging. Simulation results show the received data for ISAR imaging from MLFMA with CS can outperform the data from MLFMA, which achieves a similar quality of ISAR imaging. Additionally, the computation complexity is improved by CS through the reduced matrix computation for fewer incident waves. It makes ISAR imaging using real data feasible and meaningful.

An adaptive fast multipole accelerated Poisson solver for complex geometries

We present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition. The volume integral is computed by applying the FMM on a square box that encloses the domain of interest. For the sake of efficiency and convergence acceleration, we first extend the source distribution (the right-hand side in the Poisson equation) to the enclosing box as a C0 function using a fast, boundary integral-based method. We demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and, as a result, to an extremely efficient “black box” fast solver.

Integral Equation Methods for Elastance and Mobility Problems in Two Dimensions

We present new integral representations in two dimensions for the elastance problem in electrostatics and the mobility problem in Stokes flow. These representations lead to resonance-free Fredholm integral equations of the second kind and well conditioned linear systems upon discretization. By coupling our integral equations with high order quadrature and fast multipole acceleration, large-scale problems can be solved with only modest computing resources. We also discuss some applications of these boundary value problems in applied physics.

Regularized conjugate gradient method with fast multipole acceleration for wave scattering from 1D fractal rough surface

Based on the integral equation method and regularization theory, the regularized conjugate gradient method (RCGM) is presented for studying wave scattering from a one-dimensional rough surface with tapered wave incidence. This study is limited to scattering of acoustic waves from sound soft surfaces or horizontally polarized electromagnetic waves from perfectly conducting surfaces. The problem is formulated by using the integral equation approach. Compared with plane wave incidence, there is a remainder on the right side of the integral equation. To solve the surface integral equation, especially when the rough surface is very large in terms of the wavelength, we treat the large-scale matrix computation problem which is generated by the fast multipole algorithm using the RCGM. A strategy for the selection of the regularization parameter is obtained by estimating the double integral as a perturbed right-hand side of the integral equation. Energy conservation is considered simultaneously in the process of designing stopping criteria of the RCGM. By comparison of the numerical results of bistatic scattering coefficient (BSC) between RCGM and method of moments (MOM), the validity and accuracy of RCGM is verified. Numerical examples are also given to show the relationship between the scattering characteristic and surface roughness of the one-dimensional fractal rough surfaces.

FAST ANALYSIS OF ELECTRICALLY LARGE RADOME IN MILLIMETER WAVE BAND WITH FAST MULTIPOLE ACCELERATION

Radome has strong efiects on the radiation performances of the antenna in millimeter wave band. In this paper, the aperture integration-surface integration (AI-SI) method is adopted to analyze the electrically large antenna-radome system. The fast multipole method (FMM) is proposed to accelerate the aperture integration and inner surface integration in the AI-SI method. An electrically large antenna-radome system at W band is analyzed and measured. The radiation patterns of the system calculated using the AI-SI method with and without the fast multipole acceleration and the measured patterns are compared. The calculated patterns agree very well with each other, and both have the same agreement with the experimental results. However, the computational time of the proposed analysis with the fast multipole acceleration is reduced signiflcantly.

A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations

In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green’s function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green’s function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green’s function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.

Characterizing scattering coefficients numerically via the fast multipole accelerated boundary element method.

Various panels are used in acoustical installations to provide desired characteristics to spaces to be used for listening. These panels have complex shapes with geometrical features that have sizes corresponding to wavelengths of sounds of interest. The complex interaction of acoustical waves with these shapes is what gives these surfaces their desirable properties. Experimental characterization of the acoustical properties of these surfaces under random and specular incidence is relatively time consuming. An alternate procedure is to numerically simulate the scattering behavior, and then computing the coefficients of interest from the simulation. A significant obstacle to such computations is the time taken for simulation. Fast multipole acceleration of boundary element methods [Gumerov & Duraiswami, J. Acoust. Soc. Am. 125 (2009)] is a promising approach to speeding up computations. We report on the application of this method to the computation of various scattering coefficients.

An Integrated Imaging and BEM for Fast Simulation of Freeform Objects

AbstractProposed in this paper is a boundary-type approach as a way to increase efficiency of reverse engineering simulation of complex freeform digital models acquired from scanning. The developed method utilizes surface discretization nature of the boundary element method (BEM), and eliminates time-consuming NURBS surface/solid generation and volume discretization needed in domain-type simulation. Unstructured as-scanned image data is imported and regularized into a well-behaved BEM mesh in the computation. The developed BEM achieves a near O(N) computational complexity, by coupling the conventional BEM (of O(N3)) with the fast multipole acceleration. Numerical examples are presented to demonstrate its effectiveness and efficiency. Preliminary results show that the developed BEM can be a promising tool for many reverse engineering simulation applications involving highly complex organic shapes and large-scale models, e.g., the ones acquired from medical imaging.

A parallel domain decomposition boundary element method approach for the solution of large-scale transient heat conduction problems

This paper develops a parallel domain decomposition Laplace transform BEM algorithm for the solution of large-scale transient heat conduction problems. In order to tackle large problems the original domain is decomposed into a number of sub-domains, and a Laplace transform method is utilized to avoid time marching. A procedure is described which provides a good initial guess for the domain interface temperatures, and an iteration procedure is carried out to satisfy continuity of temperature and heat flux at the domain interfaces. The decomposition procedure significantly reduces the size of any single problem for the BEM, which significantly reduces the overall storage and computational burden and renders the application of the BEM to large transient conduction problems on modest computational platforms. The procedure is readily implemented in parallel and applicable to 3D problems. Moreover, as the approach described herein readily allows adaptation and integration of traditional BEM codes, it is expected that the domain decomposition approach coupled to parallel implementation should prove very competitive to alternatives proposed in the literature such as fast multipole acceleration methods that require a complete re-write of traditional BEM codes.

Fast multipole acceleration of the MEG/EEG boundary element method

The accurate solution of the forward electrostatic problem is an essential first step before solving the inverse problem of magneto- and electroencephalography (MEG/EEG). The symmetric Galerkin boundary element method is accurate but cannot be used for very large problems because of its computational complexity and memory requirements. We describe a fast multipole-based acceleration for the symmetric boundary element method (BEM). It creates a hierarchical structure of the elements and approximates far interactions using spherical harmonics expansions. The accelerated method is shown to be as accurate as the direct method, yet for large problems it is both faster and more economical in terms of memory consumption.

Fast Multipole Acceleration of a MoM Code for the Solution of Composed Metallic/Dielectric Scattering Problems

Abstract. An existing method of moments (MoM) code for the solution of complex scattering bodies has been accelerated by means of a multilevel fast multipole method (MLFMM). We demonstrate the usage of this technique both for metallic structures (wires and surfaces) and for dielectric bodies (volume and surface equivalence principle). Aspects like the effect of the type of integral equation, preconditioning schemes, or iterative solution techniques are discussed. But also limitations are addressed, which are encountered when for instance attempting to model highly lossy dielectric bodies with a high permittivity. Several validation and application examples demonstrate the usefulness of this method, both with regard to the obtained accuracy, but also with respect to the potential saving in memory and run-time as compared to a standard MoM formulation.

Application of the fast multipole method to hybrid finite element–boundary element models

This paper focuses on the acceleration of the hybrid finite element–boundary element analysis of 2D eddy current problems by means of the fast multipole method. An adaptive truncation scheme for the expansion of the 2D Laplace Green function is proposed. A linear time harmonic test case is considered. The results obtained with the hybrid model, with and without fast multipole acceleration, agree well with those obtained with a finite element model. The computational cost of the three calculations is compared and discussed. The proposed adaptive truncation scheme significantly contributes to the computation time savings achieved with the fast multipole method, particularly when dealing with moderate sized problems.

Fast Multipole Acceleration of the Hybrid Finite-Element/Boundary-Element Analysis of 3-D Eddy-Current Problems

This paper deals with the acceleration of the hybrid finite-element/boundary-element analysis of three-dimensional (3-D) eddy-current problems by means of the fast multipole method. An adaptive truncation scheme for the multipole expansion of the 3-D Laplace Green function is proposed. As an application example, the TEAM workshop problem 28 is considered. The computational cost without and with fast multipole acceleration is discussed.

Application of the fast multipole method to the 2D finite element‐boundary element analysis of electromechanical devices

The present paper deals with the fast multipole acceleration of the 2D finite element‐boundary element modelling of electromechanical devices. It is shown that the fast multipole method, usually applied to large 3D problems, can also lead to a reduction in computational time when dealing with relatively small 2D problems, provided that an adaptive truncation scheme for the expansion of the 2D Laplace Green function is used. As an application example, the 2D hybrid modelling of a linear actuator is studied, taking into account saturation, the voltage supply and the mechanical equation. The computational cost without and with fast multipole acceleration is discussed for both the linear and nonlinear case.