The CGFFT method with a discontinuous FFT algorithm

Fan, Guo‐Xin; Liu, Qing

doi:10.1002/mop.1079

Cited by 19 publications

(10 citation statements)

References 4 publications

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“…Again, this change in a variable produces a set of quadrature point in τ that do not coincide with the computational grid on the integration patch P k requiring an interpolation strategy for which we utilize the FFT refined polynomial interpolation [29]. The final step in high-order approximation of K sing k [u](x) relates to the the computation of t 2integral in (17). The main difficulty, here, is encountered in the form of a jump discontinuity in the t-derivative of J k (t; x) at t = t x 2 [13].…”

Section: Singular Integrationmentioning

confidence: 99%

“…In contrast, the integral equation approach, where the mathematical formulation directly ensures that the solution satisfies condition (3) by suitably employing the radiating fundamental solution, is free from considerations mentioned above, and consequently, does not require solution strategies to discretize outside the inhomogeneity. We, therefore, base our numerical treatment of the scattering problem on an equivalent integral equation formulation which is given by the Lippmann-Schwinger equation [1,11], (4) u(x) + κ 2 In recent years, a lot of progress has been made toward numerical solution of the Lippmann-Schwinger equation; for example, see [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Most fast numerical schemes among these, though high order accurate for smooth scattering media, exhibit only linear convergence in the presence of material discontinuity [12,15,17,22,26,27].…”

Section: Introductionmentioning

confidence: 99%

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An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

Anand

Pandey

Kumar

et al. 2016

Journal of Computational Physics

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This text proposes a fast, rapidly convergent Nyström method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of highorder quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of O(N log N ) for an N -point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper. arXiv:1509.05063v1 [math.NA] 16 Sep 2015

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Section: Singular Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

Anand

Pandey

Kumar

et al. 2016

Journal of Computational Physics

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“…( ) Figure 1. Arbitrary dielectric object embedded in a cuboid [7]. Journal of Applied Mathematics and Physics where d n is unknown, and q n φ is defined as the rooftop basic function on x,y,z directions respectively, and the superscript q = x,y,z.…”

Section: Theorymentioning

confidence: 99%

“…er, for the simulation of radome, FEM and FDTD usually consume a lot of memory because of its discretization of whole region including the volume which is vacuum [5]. MoM only requires the discretization of dielectric object rather than the whole region, but has to solve the linear equation, which means Journal of Applied Mathematics and Physics that it consumes computational complexity O(N 2 ) if solved iteratively [6] [7].…”

Section: Introductionmentioning

confidence: 99%

The Radar Cross Section Analysis of Radome Based on Conjugate Gradient Fast Fourier Transform

Yu¹,

Gu²,

Liu³

et al. 2020

JAMP

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This paper presents an algorithm for analysis of the dielectric radomes. In this method, the radome is discretized by a regular grid with rooftop basic functions. The Volume Integral Equation (VIE) for 3D dielectric object is transformed to linear system by Galerkin's testing formulation. Furthermore, the linear system is presented by Toeplitz matrix which can be solved by the Conjugate Gradient algorithm combined with Fast Fourier Transform (CG-FFT) iteratively. Also, the algorithm requires less computational complexity and memory. This paper simulates the mono-static Radar Cross Section of dielectric radome by the CG-FFT, which was validated against commercial software FEKO.

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“…Again here, this solution procedure leads to very efficient numerics and is highly accurate for smoothly varying media, but its convergence degrades significantly in the presence of material discontinuities. 17,23 In fact, to our knowledge, only limited attempts have been made toward the development of efficient higher-order integral-equation solvers for volumetric scattering applications. A notable exception is the work of Zhu and Gedney, 24,25 based on the "discontinuous FFT" of Fan and Liu. 23 This latter scheme is based on the accurate evaluation of Fourier coefficients of discontinuous functions ͑through Gaussian quadratures and careful interpolation from and to equispaced grids͒ and, while it can be shown that this improves on the convergence of the CGFFT ͑from first-to second-order accurate, in fact͒, it fails to address the Gibbs phenomenon which arises as the Fourier series is summed.…”

Section: Introductionmentioning

confidence: 99%

An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions

Anand

Reitich

2007

The Journal of the Acoustical Society of America

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This paper presents a high-order accelerated algorithm for the solution of the integral-equation formulation of volumetric scattering problems. The scheme is particularly well suited to the analysis of "thin" structures as they arise in certain applications ͑e.g., material coatings͒; in addition, it is also designed to be used in conjunction with existing low-order FFT-based codes to upgrade their order of accuracy through a suitable treatment of material interfaces. The high-order convergence of the new procedure is attained through a combination of changes of parametric variables ͑to resolve the singularities of the Green function͒ and "partitions of unity" ͑to allow for a simple implementation of spectrally accurate quadratures away from singular points͒. Accelerated evaluations of the interaction between degrees of freedom, on the other hand, are accomplished by incorporating ͑two-face͒ equivalent source approximations on Cartesian grids. A detailed account of the main algorithmic components of the scheme are presented, together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results.

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The CGFFT method with a discontinuous FFT algorithm

Cited by 19 publications

References 4 publications

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

The Radar Cross Section Analysis of Radome Based on Conjugate Gradient Fast Fourier Transform

An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions

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