Fast evaluation of Helmholtz potential on graphics processing units (GPUs)

Li, Shaojing; Livshitz, B.; Lomakin, Vitaliy

doi:10.1016/j.jcp.2010.07.029

Cited by 30 publications

(18 citation statements)

References 32 publications

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“…A GMRES [27] solver with a restart of 80 was used. The solver was implemented on graphics processing units (GPUs) to lead to a fast performance [28], [29]. In all the results, denoted by delta and epsilon are the discretized mesh element size and residual error, respectively.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Quadrilateral Barycentric Basis Functions for Surface Integral Equations

Chang

Lomakin

2013

IEEE Trans. Antennas Propagat.

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A framework for solving surface integral equation on quadrilateral, triangular, and mixed quadrilateral-triangular meshes is presented. The initial meshes are represented in terms of quadrilateral barycentric meshes (QBMs), which are obtained by partitioning each initial quadrilateral and triangle into four and three barycentric quadrilaterals, respectively. Quadrilateral barycentric basis functions (QBBFs) are defined on QBMs. The QBBFs serve a dual role. First, they allow defining primary basis functions (PBFs), which are well suited for representing surface currents on quadrilateral, triangular, and mixed meshes. Second, the QBBFs are used to define dual basis function (DBFs), which are natural for using in conjunction with Calderón multiplicative preconditioners (CMPs). These QBBFs, PBFs, and DBFs result in a substantial reduction in the number of nonzero elements in the sparse projection matrices for PBFs and DBFs as well as reduction of unknowns and quadrature points. When these PBFs and DBFs are used in CMPs, they reduce the number of iterations and eliminate the dense mesh breakdown of the surface electric field integral equation. Numerical examples demonstrate the efficiency of using the introduced QBBFs with associated PBFs and DBFs for solving electromagnetic surface electric field integral equations on quadrilateral, triangular, and mixed meshes.Index Terms-Basis functions, Calderón multiplicative preconditioner (CMP), dual basis functions, electric field integral equation (EFIE), quadrilaterals, testing functions, triangles.

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Section: Numerical Resultsmentioning

confidence: 99%

“…8. The divergence of is (28) where "+" is adopted for and "-" is for with referring to eight barycentric quadrilaterals. Each of the barycentric quadrilaterals contains a charge of value 1, and each of the barycentric quadrilaterals has a charge of value .…”

Section: A Quadrilateral Meshesmentioning

confidence: 99%

Quadrilateral Barycentric Basis Functions for Surface Integral Equations

Chang

Lomakin

2013

IEEE Trans. Antennas Propagat.

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“…Another method for time domain March-On-in-Time acceleration is based on the observation that a specially constructed delay-and amplitude-compensated radiation field of a given group of sources can be represented by an extremely sparse and highly non-uniform space grid over the far-field. 4, 7, 8 A recent Cartesian Nonuniform Grid Time Domain Algorithm (CNGTDA) has been given in [40,47] exploiting the smoothness of the compensated field. This method will be used in the current study.…”

Section: Delay-and Amplitude-compensated Acoustic Field In the Prmentioning

confidence: 99%

“…15,22 The second is the multi-level Cartesian Non-uniform Grid Time Domain algorithm (CNGTDA) based on far-field smoothness of the wave equation kernel. 4,40,47 In both approaches, the computational complexity can be reduced formally from O(N t N 2 ) to as low as O(N t N log 2 N ), where N t is the number of time steps and N is the total number of unknowns. In the present paper, a new, and simplified, algorithm that exploits far field smoothness of the kernel is proposed.…”

Section: Introductionmentioning

confidence: 99%

An efficient solution of time domain boundary integral equations for acoustic scattering and its acceleration by Graphics Processing Units

2013

19th AIAA/CEAS Aeroacoustics Conference

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The present paper is aimed at developing a fast numerical solution of the time domain boundary integral equation (TDBIE) reformulated from the convective wave equation for large scale wave scattering and propagation problems. Historically, numerical solutions of boundary integral equation in the time domain have encountered two major difficulties. The first is the intrinsic numerical instability in the early time domain boundary integral equation formulations. And the second is the formidably high computational cost associated with the direct solution of the time-domain boundary integral equation. In this paper, both issues are addressed. A stable Burton-Miller type formulation is proposed for the time domain boundary integral equation in the presence of a mean flow. A justification for stability through the energy equation associated with the convective wave equation is given. A comparison of the current formulation with a previous one in literature is also offered. The boundary integral equation is solved by a time domain boundary element method (TDBEM), using high-order basis functions and unstructured surface elements. To significantly reduce the computational cost, a Time Domain Propagation and Distribution (TDPD) algorithm is proposed, making use of the delay-and amplitude-compensated field with a mean flow. Implemented in multi-level interactions, the current algorithm shows a computational cost of O(N 1.25 ) per time step where N is the total number of unknowns on surface elements. Furthermore, GPU computing has been utilized to speedup the computation. Numerical aspects of the GPU computing for boundary element solutions are discussed. Comparison with CPU executions is also given. Numerical examples that demonstrate the capabilities of the proposed method are presented.

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“…There are some previous interesting works on FMM implementations on GPGPU architectures. Although neither of those works are focused on the application studied in this paper, it is worth mentioning the approach of the FMM for Laplace kernels presented in [9] and the implementation of the FMM for the rapid evaluation of the acoustic field over a set of observation points due to a set of sources [13].…”

Section: Introductionmentioning

confidence: 99%