Fast multipole accelerated indirect boundary elements for the Helmholtz equation

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.

Abstract: In the boundary element method, treatment of all the possible singular integrals is very important for the correctness and accuracy of the solutions. Generally, the directional derivative of a weakly singular integral is computed by an integral in the sense of Cauchy principle value if the directional derivative of the weakly singular integral kernel is strongly singular or in the sense of Hadamard finite part integral if the the directional derivative of the weakly singular integral kernel is hypersingular. We will try to discover how the strongly singular and hypersingular integrals are generated and propose a method to avoid the appearance of such kind of strongly singular and hypersingular integrals. Here, using some examples, we will show that with an ’exact derivation’ the directional derivative of a weakly singular integral will still be a weakly singular integral or a normal integral. That is none strongly or hypersingular integrals are generated in such a process. And therefore, Cauchy principle value and Hadamard finite part integral are not indispensable. The result will help us to form a new approach to overcome the so-called strongly and hypersingular integrals occurring in the boundary element method.

The project consisted of two main tasks. Each task had several specific aims. The three main objectives of Task 1 were: Investigation of the flat beam technique to produce high transverse emittance ratio longitudinally-compressed bunches, Exploration of a new chicane-like transverse-to-longitudinal phase space exchanger for precise pulse shaping and emittance exchange, Applications of flat-beam with shaped current profile(s) to dielectric wakefield acceleration in an asymmetric waveguide. The outcome of our research was to performed details simulations of flat-beam generation and compression. Some preliminary experiment on flat-beam generation and compression were conducted at the FAST-IOTA injector in parallel to commissioning activities. Likewise, the longitudinal phase-space transformation was not fully explored; we participated in experiments conducted at the Argonne Wakefield Accelerator and especially introduced a simple transverse-shaping technique. The three main objectives of Task 2 were: Development of the Fast Multipole-Accelerated Boundary Element Method, Formulation of a new dynamic aperture enlarging method, Applications to nonlinear integrable optics. We developed a new method, termed the Fast Multipole Accelerated Boundary Element Method. In beam dynamics and accelerator physics it contributes to the understanding and advancement of the intensity frontier. Applications are widespread, ranging from discovery science to energy production by accelerator driven systems. The numerical methods we developed might find applications outside accelerator and beam physics, with potential reach as far as engineering and applied mathematics. The most immediate impact is the enhanced understanding of novel accelerators such as the Integrable Optics Test Accelerator (IOTA) at Fermilab, and accurate and efficient simulations of ultra-cold high -performance electron sources. We made significant progress toward dynamic aperture enlargement of nonlinear accelerator systems. Applications to nonlinear integrable optics will push the intensity frontier even further than previously thought to be possible.

The determination of the acoustic sensitivity characteristics with respect to the variation of the design parameters can provide a method for low-noise design of mechanical structure objectively and quantitatively.Based on the boundary element method, the expressions of sensitivities of the acoustic pressure with respect to changes in frequency and surface vibration velocity are presented.The sensitivities of coefficient matrix are computed by differentiation directly,instead of finite difference approximation. Using displaced points method,the singular integral can be obtained fast and correctly.Numerical sensitivity results obtained by the boundary element method are compared to analytical solutions,and the validation and availability of the present method are proved. These acoustic sensitivity analysis methods can lay a solid foundation for the low-noise design of mechanical structure.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

A numerical study of 2D electrothermal flow using boundary element method

The interaction of acoustic waves with submerged structures remains one of the most difficult and challenging problems in underwater acoustics. Many techniques such as coupled Boundary Element (BE)/Finite Element (FE) or coupled Infinite Element (IE)/Finite Element approximations have evolved. In this paper, we focus on the steady-state formulation only, and study a general coupled hp-adaptive BE/FE method. The work is devoted to a study on viscoelastic scattering problems in linear acoustic medium by means of hp boundary and finite element approximations. A particular emphasis is placed on an a-posteriori error estimation for the viscoelastic scattering problems.

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

Kussmaul's (1969) formulation of acoustic scattering problems involves the superposition of a double-layer distribution and a simple-layer distribution on the boundary.This ensured the existence of a unique solution for all wave numbers of the incoming wave. However,the boundary integral equation involves a highly singular kernel which proves difficult to treat numerically. An alternative approach is to place the double-layer distribution on an interior boundary similar (similarly situated) to the given boundary.This avoids the singularity problems and enables excellent numerical results to be achieved.Some numerical results are presented. INTRODUCTION Problems of hard acoustic scattering may be formulated in terms of boundary integral equations.Direct formulations are based upon the Helmholtz formula;indirect formulations utilise layer potentials.This is a Neumann boundary-value problem which has a unique solution subject to regular behaviour at infinity.However the relevant classical integral equations,whether direct or indirect,fail at wave numbers corresponding with the eigenfrequencies of the interior Dirichlet problem.In this paper we show that a relatively simple, nonsingular,supplement to the classical indirect integral equation ensures a a unique solution at all wave numbers. An immediate consequence is that meaningful numerical solutions can be achieved,for a wide range of wavenumbers,by computationally effective boundary-element methods (BEMs). Some results for scattering by a hard sphere are provided.For completeness we mention some recent papers presenting alternative resolutions of the difficulty. Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 282 Boundary Elements Rego Silva et al [1] Numerical Implementation of a Hypersingular BEM formulation for Acoustic Radiation Problems,following the approach of Burton and Miller [2].Here the hypersingular integral is evaluated numerically by an algorithm of Guiggiani et al [3] building essentially upon Hadamard's finite-part integration. Liu and Rizzo [4] A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,again following the approach of Burton and Miller. However here the hypersingular integral is weakened by a regularisation technique,which enables it to be evaluated numerically by straightforward methods. Wu et al [8] Vectorization and parallelization of the acoustic boundary element code BEMAP on the IBM ES/3090 VF,exploits the supercomputing facilities to improve the CHIEF method originated by Schenck [15]. Kirkup and Henwood [5] Computational Solution of Acoustic Radiation Problems by Kussmaul's Boundary Element Method,adopting Kussmaul's representation [6] of the wave function as the superposition of a simple-layer and a double-layer potential generated from sources on the boundary.This formulation also involves a hypersingular integral,which is evaluated numerically by the method of Terai [7],an early exploitation of Hadamard's finite part integration. Kussmaul's approach may be modified by locating the double-layer sources on an interior boundary,similar and similarly situated to the given boundary.This eliminates the hypersingular integral and at the same time provides a unique solution at all wave numbers. Brief details are given in the following section. KUSSMAUL FORMULATION Any solution, ,of the Helmholtz equation existing on the infinite domain B exterior to a closed (simply connected) surface dB (fig. 1) of order 0(r~') as /•co can generally be represented by a simple-layer potential dq ; q € dB , p € B+dB (1) dB where o(q) is the source density at q and dq is the area element at q ; also Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 283 /-',-'» ; r=|f-g| (2) where k is the wave number associated with , defined by

Time-domain boundary element method in viscoelasticity with application to a spherical cavity

A high-order boundary element method (BEM) for the analysis of the steady two-dimensional full-potential transonic equation is presented. The use of a high-order (piecewise cubic on the boundary and piecewise bi-cubic in the field) numerical formulation, the main novelty of the present work, is important in two respects: first, the convergence of the solution as h vanishes is faster than the zeroth-order (the piecewise constant) one, yielding more accurate results with coarser grid resolutions. In addition, in supercritical flows, the derivation of the velocity field from the high-order representation for potential gives, in the vicinity of the shock, a sharper discontinuity, and allows for an in-depth analysis of the shock properties. Both aspects are analyzed in the present paper through applications to steady, two-dimensional, full-potential flows. All the results obtained using the present method are validated through comparison to other computational fluid dynamics (CFD) solutions of the full-potential flows and, when applicable, Euler equations. A comparison to a zeroth-order BEM, based on the same integral formulation is also included.

Fictitious eigenfrequencies in the BEM for interior acoustic problems

An isogeometric Burton-Miller method for the transmission loss optimization with application to mufflers with internal extended tubes

A fast direct boundary element method for 3D acoustic problems based on hierarchical matrices

Research Article Variational boundary element acoustic modelling over mixed quadrilateral-triangular element meshes Ahlem Alia 1 *, Mhamed Souli 1, Fouad Erchiqui 2 1LML, Boulevard Paul Langevin, Cite Scientifique, 59655 Villeneuve d'Ascq Cedex, France 2Universite du Quebec en Abitibi-Temiscamingue, 445 boul. de l'Universite, Rouyn-Noranda QC, Que., Canada J9X5E4 email: Ahlem Alia (Ahlem.Alia@ed.univ-lille1.fr) *Correspondence to Ahlem Alia, LML, Boulevard Paul Langevin, Cite Scientifique, 59655 Villeneuve d'Ascq Cedex, France Keywords variational indirect boundary element method • singularity • triangles • polar co-ordinate transformation Abstract The variational indirect boundary element method is widely used in many acoustical problems such as simulation of scattering and radiation phenomena. Although it has many advantageous features in dealing with those problems, it suffers from the singularity problem when the double surface integral is done over the same element. In this paper, a straightforward technique for computing the singularity 1/R over triangular element is presented. It is based on a generalized polar co-ordinates transformation to transform the triangular master element into a square one. The singularity factor is taken as weight function. Consequently, a single straightforward implementation without any special treatment of singularity is possible for meshes involving both quadrilateral and triangular elements of higher order. The efficiency of the proposed method is demonstrated for some numerical examples.