Accelerated IE-GSTC Solver for Large-Scale Metasurface Field Scattering Problems Using Fast Multipole Method (FMM)

Abstract: In this paper, a half space adaptive fast multipole boundary facemethod (FMBFM) is presented for solving the three-dimensional half space ex-terior acoustic problems. In the presented method, the Burton-Miller equationbased on the conventional boundary integral equation (CBIE) and its hyper-singularboundary integral equation (HBIE) is used to deal with the fictitious eigenfrequen-cies problem. The half space Green’s function is employed, thus the tree structurein the fast multipole method can be used only for the real domain. The higher orderelements and an adaptive tree structure are used to improve the efficiency of theFMBFM. This half space adaptive FMBFM for half space acoustic problems is anextension of the adaptive FMBFM for full space acoustic problems developed bythe authors. Numerical examples for half space acoustic problems in this paperdemonstrate the efficiency and validity of this method.Keywords: fast multipole boundary face method, Burton-Miller equation, acous-tic problems, half space, modified hyper-singular boundary integral equation1 IntroductionAcoustic problems in half space are a major subject in engineering. Many researchworks have been done on this aspect (Seybert et al., 1988; Secgin et al., 2010; Caoet al., 2010). The half space Green’s function (Seybert et al., 1988) can be appliedby adding the source solution at the image point to the original full-space Green’sfunction for dealing with half space problems. Sound source localization and iden-tification of a refrigerator were accomplished by using the half space boundaryelement method (BEM) (Secgin et al., 2010). Using the half space Green’s func-tion, the discretization of the rigid infinite plane is removed and only the boundaries

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

Fast multipole boundary element method using the binary tree structure with tight bounds: application to a calculation of an electrostatic force for the manipulation of a metal micro particle ☆

Though computational molecular dynamics is an effective tool for nano-scale phenomenon analysis, computational costs associated with its computer simulation are extremely high. There are two major computational steps associated with computer simulation of dynamics of molecular structures. They are calculation of interatomic forces and formation and solution of the equations of motion. Currently, these two computational steps are treated separately in most commonly-used methods. For example, Fast Multipole Method (FMM) and Cell Multipole Method (CMM) have been used for calculation of interatomic forces, and Cartesian Coordinate Method (CCM) and Internal Coordinate Molecular Dynamics Method (ICMD) have been created for the formation and solution of equations of motion of an atomistic molecular system. In this paper, a new procedure is presented through a proper integration between multibody molecular algorithms (MMA) and fast multipole methods to improve computational efficiency for computer simulation of the dynamical behaviors of multibody molecular structures in polymers and biopolymers. For the computational costs associated with interatomic forces, a fast multipole method is used to calculate the interatomic forces due to the potentials. For the computational costs associated with formation and solution of equations of motion, a multibody molecular algorithm developed by the author in his previous work will be utilized to integrate with fast multipole methods. The algorithm significantly improves computational efficiency when comparing with its counterpart procedures. The fast multipole method begins by scaling all atoms into a box with coordinate ranges to ensure numerical stability of subsequent operations. The parent box is then divided into half in the direction of each Cartesian axis and each child box is then subdivided to form a computational family tree. The flow of calculations is carried out along the tree structure with five passes. The fast multipole method has been improved and modified to achieve better effectiveness and higher efficiency since it was created. The multibody molecular algorithm starts with numbering subsets, forming bond graph, and developing three computing passes along the tree structure of an atomistic molecular system. Computing data flows in the fast multipole method and the multibody molecular algorithm will properly line up with the parent-child recursive relationship along the configuration of the tree structure due to linear recursive natures of both fast multipole method and multibody molecular algorithm. Then the time spent on the recursive simulation passes in the fast multipole method for computing forces may overlap with the time spent on the three recursive computational passes in the multibody molecular algorithm for forming and solving equations of motion.

Fast multipole singular boundary method for Stokes flow problems

Standard numerical models in acoustics rely on the isentropic Helmholtz equation. Its derivation assumes adiabatic and reversible, i.e., dissipation-free, wave propagation. Sound waves in fluids are, however, subject to viscous and thermal losses. These losses originate from viscous friction and heat conduction, leading to the formation of acoustic boundary layers. Considering these effects becomes significant in setups with acoustic cavities of similar dimension as the boundary layers. Recently, boundary element methods (BEM) accounting for the viscothermal dissipation have been proposed. These methods limit the discretization to the surface of the fluid domain and require significantly fewer degrees of freedom than comparable finite element models. However, the BEM coefficient matrices are fully populated, resulting in high computational costs and storage requirements. This study develops a new BEM formulation for acoustics in viscothermal fluids that uses the fast multipole method - a low-rank approximation technique based on a hierarchical subdivision of the computational domain - to alleviate this shortcoming. It is shown that the fast viscothermal BEM improves the algorithmic complexity over the conventional formulation, reducing both the memory requirements and solution time. The results indicate good scalability of the method making it feasible for larger applications such as acoustic metamaterials.

The fast multipole method (FMM) is useful for solving electromagnetic scattering problems through integral equations. Some asymptotic laws are given for the truncation of the series involved in the FMM. Those laws were rigorously proved in Carayol (2002). For the first time, the accuracy of the widely used empirical formulas was studied mathematically. Our results also give some elements to understand the difference of FMM error due to the configuration of points in FMM boxes (connected to the term u/spl I.cap//spl middot/ v/spl I.cap/). We also derived in Carayol some consequences on the number of quadrature points needed for the numerical integrations in the FMM.

A modified dual-level fast multipole boundary element method based on the Burton–Miller formulation for large-scale three-dimensional sound field analysis

A numerical solution of the boundary integral equation for the exterior Helmholtz problem in three dimensions, leads to the solution of a dense linear system. In order to have a well conditioned system, we consider the Despres integral equations (see J. Electromag. Waves and Appl., vol.13, p.1553-68, 1999). An iterative method is given by Despres to solve this system. If Niter is the number of the iterations, the complexity of this resolution is of order Niter k/sup 4/, where k is the wave number. In order to speed up the iterative solution of the system, we have considered the coupling of two methods, the microlocal discretization method and the fast multipole method (FMM). The microlocal discretization method of T. Abboud, J.-C. Nedelec and B. Zhou (1995), enables one to consider a new system whose size is of order k/sup 2/3//spl times/k/sup 2/3/ instead of k/sup 2//spl times/k/sup 2/ for convex geometries. However, due to the geometrical approximation of the surface, the fine mesh of the standard case is still considered. Another method, the fast multipole method, is one of the most efficient and robust methods used to speed up the calculation of matrix-vector products, with a cost of order k/sup 3/ instead of k/sup 4/ for the one level FMM. In this paper, a coupling of these two methods is presented. It enables one to reduce the CPU time very efficiently for large wave number, with a complexity of order k/sup 3/+Niter k/sup 4/3/.

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N 2) to O(N 1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N 1.5. The computational complexity of the algorithm is further reduced to O(N 4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.

Animportant and computationally demanding part of molecular dynamicssimulations is the calculation of long-range electrostatic interactions.Today, the prevalent method to compute these interactions is particlemesh Ewald (PME). The PME implementation in the GROMACS moleculardynamics package is extremely fast on individual GPU nodes. However,for large scale multinode parallel simulations, PME becomes the mainscaling bottleneck as it requires all-to-all communication betweenthe nodes; as a consequence, the number of exchanged messages scalesquadratically with the number of involved nodes in that communicationstep. To enable efficient and scalable biomolecular simulations onfuture exascale supercomputers, clearly a method with a better scalingproperty is required. The fast multipole method (FMM) is such a method.As a first step on the path to exascale, we have implemented a performance-optimized,highly efficient GPU FMM and integrated it into GROMACS as an alternativeto PME. For a fair performance comparison between FMM and PME, wefirst assessed the accuracies of the methods for various sets of inputparameters. With parameters yielding similar accuracies for both methods,we determined the performance of GROMACS with FMM and compared itto PME for exemplary benchmark systems. We found that FMM with a multipoleorder of 8 yields electrostatic forces that are as accurate as PMEwith standard parameters. Further, for typical mixed-precision simulationsettings, FMM does not lead to an increased energy drift with multipoleorders of 8 or larger. Whereas an ≈50 000 atom simulationsystem with our FMM reaches only about a third of the performancewith PME, for systems with large dimensions and inhomogeneous particledistribution, e.g., aerosol systems with water droplets floating ina vacuum, FMM substantially outperforms PME already on a single node.

Boundary element method is one of the most powerful tools to solve partial differential equations. However, it takes a lot of computational time when applied to a problem with a large number of unknowns. In this paper, we adopted the binary tree structure with tight bounds and the downward pass for that structure for the fast multipole method to reduce the number of M2L translations. The fast multipole boundary element method using the binary tree structure with tight bounds is developed and the analyses of an electrostatic force acting on a matal micro particle above a complex structure to confirm the effectiveness of this method.

The fast multipole methods are used for solving a scalar acoustic or vector electromagnetic wave equation by integral equation methods with a large number of unknowns. In this paper a new method is presented for performing interpolation and anterpolation in both spherical coordinates $\theta$ and $\phi$ by FFT in the three-dimensional (3-D) multilevel fast multipole algorithm (MLFMA). The key idea is to approximate functions on the unit sphere by truncated Fourier series in two variables rather than by the usual spherical harmonics. The proposed method is exact in interpolating and anterpolating and has the high numerical efficiency of FFT. The method is numerically compared to the method of performing interpolation and anterpolation using Lagrangian interpolation, which presently is probably the fastest method for those operations, and the results suggest that the proposed new method is equally or more efficient, depending on the desired accuracy.

In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). We implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.

This paper presents flexible inner-outer Krylov subspace methods, which are implemented using the fast multipole method (FMM) for solving scattering problems with mixed dielectric and conducting object. The flexible Krylov subspace methods refer to a class of methods that accept variable preconditioning. To obtain the maximum efficiency of the inner-outer methods, it is desirable to compute the inner iterations with the least possible effort. Hence, generally, inaccurate matrix-vector multiplication (MVM) is performed in the inner solver within a short computation time. This is realized by using a particular feature of the multipole techniques. The accuracy and computational cost of the FMM can be controlled by appropriately selecting the truncation number, which indicates the number of multipoles used to express far-field interactions. On the basis of the abovementioned fact, we construct a less-accurate but much cheaper version of the FMM by intentionally setting the truncation number to a sufficiently low value, and then use it for the computation of inaccurate MVM in the inner solver. However, there exists no definite rule for determining the suitable level of accuracy for the FMM within the inner solver. The main focus of this study is to clarify the relationship between the overall efficiency of the flexible inner-outer Krylov solver and the accuracy of the FMM within the inner solver. Numerical experiments reveal that there exits an optimal accuracy level for the FMM within the inner solver, and that a moderately accurate FMM operator serves as the optimal preconditioner.