Multipole Accelerated Boundary Element Method Research Articles

Kernel-Independent Fast Multipole Boundary Element Solver for Coupled Conduction–Radiation Heat Transfer Problem

The purpose of this paper is to present a kernel-independent fast multipole accelerated boundary element method for the simulation of a system of coupled conduction–radiation heat transfer in three-dimensional participating media. Radiative transfer integral equations are considered as radiation heat transfer model in participating media. For the nonlinear energy equation, Newton–Generalized Minimal Residual (GMRES) iterative scheme is adopted. The loose coupling iterative technique is applied to this strongly coupled system. Then, the separate solutions are both obtained by the kernel-independent fast multipole boundary element method based on the collocation scheme. Numerical examples show that the present fast algorithm is effective and efficient.

Fast multipole accelerated boundary element methods for room acoustics.

Direct and indirect boundary element methods, accelerated via the fast multipole method, are applied to numerical simulation of room acoustics for rooms of volume ∼150 m3 and frequencies up to 5 kHz on a workstation. As the parameter kD (wavenumber times room diameter) is large, stabilization of the previously developed fast algorithms is required for accuracy. A stabilization scheme is one of the key contributions of this paper. The computations are validated using well-known image source solutions for shoebox shaped rooms. Computations for L-shaped rooms are performed to illustrate the ability to capture diffractions. The ability to model in-room baffles and boundary openings (doors/windows) is also demonstrated. The largest case has kD > 1100 with a discretization of size 6 × 106 elements. The performance of different boundary integral formulations was compared, and their rates of convergence using a preconditioned flexible generalized minimal residual method were found to be substantially different. These promising results suggest a path to efficient simulations of room acoustics via high performance boundary element methods.

Differential algebraic fast multipole-accelerated boundary element method for nonlinear beam dynamics in arbitrary enclosures

A novel method is developed to take into account realistic boundary conditions in intense nonlinear beam dynamics. The algorithm consists of three main ingredients: the boundary element method that provides a solution for the discretized reformulation of the Poisson equation as boundary integrals; a novel fast multipole method developed for accurate and efficient computation of Coulomb potentials and forces; and differential algebraic methods, which form the numerical structures that enable and hold together the different components. The fast multipole method, without any modifications, also accelerates the solution of intertwining linear systems of equations for further efficiency enhancements. The resulting algorithm scales linearly with the number of particles $N$, as $m\text{ }\mathrm{log}\text{ }m$ with the number of boundary elements $m$, and, therefore, establishes an accurate and efficient method for intense beam dynamics simulations in arbitrary enclosures. Its performance is illustrated with three different cases and structures of practical interest.

Computationally Efficient Harmonic-Based Reactive Exploration

Although Harmonic Potential Fields constitute a powerful tool for tackling the autonomous robot exploration problem, yet their applicability is limited by the heavy computational load involved in solving the Laplace equation in real time. In this letter, we propose a computationally efficient exploration scheme employing a Fast Multipole accelerated Boundary Element Method, which enjoys both linear complexity w.r.t. the boundary's size as well as linear memory requirements. Furthermore, we devise an adaptive control law for the specified boundary conditions that allows us to tune the robot's behavior without affecting the inherent safety and convergence properties of the underlying potential field. Finally, we validate the performance of the proposed exploration scheme through extensive realistic simulations.

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

This paper presents an efficient algebraic preconditioner to speed up the convergence of Fast Multipole accelerated Boundary Element Methods (FM-BEMs) in the context of time-harmonic 3D wave propagation problems and in particular the case of highly non-uniform discretizations. Such configurations are produced by a recently-developed anisotropic mesh adaptation procedure that is independent of partial differential equation and integral equation. The new preconditioning methodology exploits a complement between fast BEMs by using two nested GMRES algorithms and rapid matrix–vector calculations. The fast inner iterations are evaluated by a coarse hierarchical matrix (H-matrix) representation of the BEM system. These inner iterations produce a preconditioner for FM-BEM solvers. It drastically reduces the number of outer GMRES iterations. Numerical experiments demonstrate significant speedups over non-preconditioned solvers for complex geometries and meshes specifically adapted to capture anisotropic features of a solution, including discontinuities arising from corners and edges.

A fast multipole algorithm for radiative heat transfer in 3D semitransparent media

This paper presents a fast multipole accelerated boundary element method for simulating the radiative heat transfer in 3D semitransparent media. The participating media is absorbing, emitting and isotropically scattering. The proposed implementation adopts the boundary element method based on the collocation scheme to discretize the radiative integral transfer equations. The resulting algebraic system is solved iteratively with GMRES (generalised minimum residual method). A kernel-independent fast multipole method to accelerate the matrix-vector product in each iteration. It is found from three numerical examples that the computational efficiency of the developed fast algorithm is much better than the conventional boundary element method.

Fast multipole accelerated boundary element method for the Helmholtz equation in three dimensions on heterogeneous architectures

Numerical simulations related to human hearing, architectural and underwater acoustics, and multiple scattering require computational solution of the Helmholtz equation in three dimensions with complex shaped boundaries. Boundary element methods (BEM) are among the most accurate and efficient methods used for this purpose. However, solution of high frequency/large domain problems is challenging due to poor scaling of conventional solvers with the frequency and domain size. The use of the fast multipole methods (FMM) resolves many problems related to the scalability [N.A. Gumerov and R. Duraiswami, J. Acoust. Soc. Am. 125(1), 191205, 2009]. Additional accelerations are needed to be able to solve practical problems over the entire range of human audible frequencies and can be provided using graphics processors (GPUs) with multicore CPUs. In this work, we report development and demonstration of the FMM/GPU accelerated BEM for the Helmholtz equation in 3D designed for hybrid CPU/GPU architectures. Innovations related to choices of preconditioners, parallelization strategies, and choice of optimal parameters will be presented. A single PC version of the algorithm shows accelerations of the order of 10 times compared to the BEM accelerated with the FMM alone. Results for computing head related transfer functions and other standard calculations will be provided.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

Fully-Nonlinear Wave-Current-Body Interaction Analysis by a Harmonic Polynomial Cell Method

A new numerical 2D cell method has been proposed by the authors, based on representing the velocity potential in each cell by harmonic polynomials. The method was named the harmonic polynomial cell (HPC) method. The method was later extended to 3D to study potential-flow problems in marine hydrodynamics. With the considered number of unknowns that are typical in marine hydrodynamics, the comparisons with some existing boundary element- based methods, including the fast multipole accelerated boundary element methods, showed that the HPC method is very competitive in terms of both accuracy and efficiency. The HPC method has also been applied to study fully-nonlinear wave-body interactions; for example, sloshing in tanks, nonlinear waves over different sea-bottom topographies, and nonlinear wave diffraction by a bottom-mounted vertical circular cylinder. However, no current effects were considered. In this paper, we study the fully-nonlinear time-domain wave-body interaction considering the current effects. In order to validate and verify the method, a bottom-mounted vertical circular cylinder, which has been studied extensively in the literature, will first be examined. Comparisons are made with the published numerical results and experimental results. As a further application, the HPC method will be used to study multiple bottom-mounted cylinders. An example of the wave diffraction of two bottom-mounted cylinders is also presented.

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green’s tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green’s tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.

2010 非一様B-Splineに基づくアイソジオメトリック要素を用いた2次元Helmholtz方程式に対する高速多重極境界要素法

2010 非一様B-Splineに基づくアイソジオメトリック要素を用いた2次元Helmholtz方程式に対する高速多重極境界要素法

Fast multipole accelerated boundary element method for the Helmholtz equation in acoustic scattering problems

We apply the fast multipole method (FMM) accelerated boundary element method (BEM) for the three-dimensional (3D) Helmholtz equation, and as a result, large-scale acoustic scattering problems involving 400000 elements are solved efficiently. This is an extension of the fast multipole BEM for two-dimensional (2D) acoustic problems developed by authors recently. Some new improvements are obtained. In this new technique, the improved Burton-Miller formulation is employed to overcome non-uniqueness difficulties in the conventional BEM for exterior acoustic problems. The computational efficiency is further improved by adopting the FMM and the block diagonal preconditioner used in the generalized minimum residual method (GMRES) iterative solver to solve the system matrix equation. Numerical results clearly demonstrate the complete reliability and efficiency of the proposed algorithm. It is potentially useful for solving large-scale engineering acoustic scattering problems.

Ascent of Bubbles in Magma Conduits Using Boundary Elements and Particles

Abstract We investigate the use of the Multipole-accelerated Boundary Element Method (BEM) and of the Singularity Method for studying the interaction of many bubbles rising in a volcanic conduit. Observation shows that the expression of volcanic eruption is extremely variable, from slow release of magma to catastrophic explosive manifestation. We investigate the application of the Fast Multipole Method to the solution of (i) the Boundary Element Formulation of the Stokes flow and of (ii) the particle formulation using the Stokeslets, the Green Function of the Stokes flow law, as a particle kernel. We show how these implementations allow for the first time to numerically model in a dynamic setting a very large number of bubbles, i.e few thousands with the BEM models, allowing investigating the feedback between the single bubble deformation and their collective evolution, and few hundred of thousands of bubbles with the particle approach. We illustrate how this method can be used to investigate the intense interaction of a large number of bubbles and suggest a framework for studying the feedback between many bubbles and a complex thermal nonlinear magmatic

The fate of the slabs interacting with a density/viscosity hill in the mid-mantle

In the last two decades it has been proposed several times that a non-monotonic profile might fit the average lower mantle radial viscosity. Most proposed profiles consist in a more or less broad viscosity hill in the middle of the mantle, at a depth roughly between 1200 km and 2000 km. Also many tomographic models display strong signals of the presence of “fast” material lying at mid mantle depths and a recent spectral analysis of seismic tomography shows a very clear transition for degree up to around 16 at a less than 1500 km depth. Finally latest works, both theoretical and experimental, on the high-to-low-spin transition for periclase, have suggested that the high-spin to low-spin transition of Fe++ might lie at the heart of all these observations. To verify the dynamical compatibility between possible mantle profile and observed tomographic images and compare them with possible mineral physics scenarios, such as the spin transition, we employ here a recently developed Fast Multipole-accelerated Boundary Element Method (FMM-BEM), a numerical approach for solving the viscous momentum equation in a global spherical setting, for simulating the interaction of an individual slab with a mid mantle smooth discontinuity in density and viscosity. We have focused on the complexities induced to the behaviour of average and very large plates O (2000–10,000 km), characteristic of the Farallon, Tethys and Pacific plate subducting during the Cenozoic, demonstrating that the a mid mantle density and/or viscosity discontinuity produces a strong alteration of the sinking velocity and an intricate set of slab morphologies. We also employ the Kula–Farallon plate system subducting at 60 Ma as a paradigmatic case, which reveals the best high resolution tomography models and clearly suggests an interaction with a strong and/or denser layer in the mantle. Our 38 models show that a plate might or might not penetrate into the lowest mantle and might stall in the mid lower mantle for long periods, depending on the radial profiles of density and viscosity, within a realistic range (viscosity 1, 10 or 100 times more viscous of the rest of the mantle, and a change of differential density in the range −2% to 2%), of a transitional layer of 200 km or 500 km. We conclude that a layer with high viscosity or negative density would naturally trigger the observed geodynamic snapshot. We finally propose a scenario in which the long time accumulation of depleted slabs in the mid mantle would give rise to a partially chemically stratified mantle, starting from the less prominent high-spin to low-spin contribution on the basis of mantle density and rheology.

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.

Computation of the head-related transfer function via the fast multipole accelerated boundary element method and its spherical harmonic representation

The head-related transfer function (HRTF) is computed using the fast multipole accelerated boundary element method. For efficiency, the HRTF is computed using the reciprocity principle by placing a source at the ear and computing its field. Analysis is presented to modify the boundary value problem accordingly. To compute the HRTF corresponding to different ranges via a single computation, a compact and accurate representation of the HRTF, termed the spherical spectrum, is developed. Computations are reduced to a two stage process, the computation of the spherical spectrum and a subsequent evaluation of the HRTF. This representation allows easy interpolation and range extrapolation of HRTFs. HRTF computations are performed for the range of audible frequencies up to 20 kHz for several models including a sphere, human head models [the Neumann KU-100 ("Fritz") and the Knowles KEMAR ("Kemar") manikins], and head-and-torso model (the Kemar manikin). Comparisons between the different cases are provided. Comparisons with the computational data of other authors and available experimental data are conducted and show satisfactory agreement for the frequencies for which reliable experimental data are available. Results show that, given a good mesh, it is feasible to compute the HRTF over the full audible range on a regular personal computer.

Wideband fast multipole accelerated boundary element methods for the three-dimensional Helmholtz equation.

The development of a fast multipole method (FMM) accelerated iterative solution of the boundary element method (BEM) for the Helmholtz equations in three dimensions is described. The FMM for the Helmholtz equation is significantly different for problems with low and high kD, where k is the wave number and D the domain size, and for large problems, the method must be switched between levels of the hierarchy. The BEM requires several approximate computations: numerical quadrature and approximations of the boundary shapes using elements. These errors must be balanced against approximations introduced by the FMM and the convergence criterion for an iterative solution. These different errors must all be chosen in a way that, on the one hand, excess work is not done and, on the other, that the error achieved by the overall computation is acceptable. Details of translation operators for low and high kD choice of representations and BEM quadrature schemes, all consistent with these approximations, are described. A novel preconditioner using a low accuracy FMM accelerated solver as a right preconditioner is also described. Results of the developed solvers for large boundary value problems with 0.0001⩽kD⩽500 are presented and shown to perform close to theoretical expectations.

A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation

The development of a fast multipole method (FMM) accelerated iterative solution of the boundary element method (BEM) for the Helmholtz equations in three dimensions is described. The FMM for the Helmholtz equation is significantly different for problems with low and high kD (where k is the wavenumber and D the domain size), and for large problems the method must be switched between levels of the hierarchy. The BEM requires several approximate computations (numerical quadrature, approximations of the boundary shapes using elements), and these errors must be balanced against approximations introduced by the FMM and the convergence criterion for iterative solution. These different errors must all be chosen in a way that, on the one hand, excess work is not done and, on the other, that the error achieved by the overall computation is acceptable. Details of translation operators for low and high kD, choice of representations, and BEM quadrature schemes, all consistent with these approximations, are described. A novel preconditioner using a low accuracy FMM accelerated solver as a right preconditioner is also described. Results of the developed solvers for large boundary value problems with 0.0001 less, similarkD less, similar500 are presented and shown to perform close to theoretical expectations.

Fast multipole accelerated boundary element method (FMBEM) for solution of 3D scattering problems

Wideband FMBEM codes are challenging to implement since there are problems at both very low and high frequencies. Substantially different schemes for function representation and translation are efficient for low and high frequency ranges. We present a method which is suitable for solution both high and low frequency problems since it implements a switch between different representations and uses fast translation methods appropriate to each representation. For a high frequency problem the switch between representations may occur at some intermediate levels of hierarchical space subdivision of the FMM. We also present an FMM-based preconditioner used in the flexible GMRES iterative solver for scattering problems and discuss example problems computed in range 0.001<kD<200, where k is the wave number and D is the size of the computational domain. The theoretical and experimental computational complexity of our algorithm is approximately O((kD)3) at large kD with relatively small asymptotic constant and the algorithm allows to solve problems with up to million nodes used for surface discretization in the range tested on a conventional personal computer.

Barrier Properties of Flake-Filled Membranes: Review and Numerical Evaluation

Flake-filled polymer membranes belong to an emerging class of composite materials that show promise as barrier materials in many industrial applications. Though the barrier property enhancement in flake-filled membranes is well known, its dependence on microstructural factors, such as flake spatial distribution, size dispersion, and orientation is not well understood. This study presents a micromechanics-based, direct numerical approach to predicting the barrier improvement factors of flake-filled membranes. Numerical calculations are conducted in two-dimensions for composite membranes containing aligned, impermeable flakes, using the fast multipole-accelerated boundary element method. Theoretical models which predict the barrier properties of flake-filled membranes are reviewed and their predictions compared with the boundary element calculations.