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

Anand, Akash; Reitich, Fernando

doi:10.1121/1.2714919

Cited by 6 publications

(18 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In all examples, the evaluation of multiple scattering iterations is based on the "combined-field" versions of (2.23) and (2.24) (see [10]) which were approximated via the high-order Fourier-based method of [5] (see also [1]). Thus, the numerical approach entails discretizations on the scale of the wavelength and we have, in every case, chosen these so as to guarantee the relevant accuracy (as measured by numerical convergence tests) for meaningful comparisons with our asymptotic results.…”

Section: Numerical Resultsmentioning

confidence: 99%

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

et al. 2009

Self Cite

View full text Add to dashboard Cite

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully threedimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part 123 374 A. Anand et al.I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

show abstract

Section: Numerical Resultsmentioning

confidence: 99%

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

et al. 2009

Self Cite

View full text Add to dashboard Cite

show abstract

“…In recent years, a number of algorithms, including direct and iterative solvers, have been proposed for the solution of Lippmann-Schwinger equation. While we do not review all such contributions, some recent numerical methods include [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Most fast algorithm, among the cited methods, while converging rapidly for smooth scattering media, yield only linear convergence in the presence of discontinuous scattering media.…”

Section: Overviewmentioning

confidence: 99%

“…For instance, schemes introduced in [11,13], provide a fast high-order method for smooth scattering media by means of a Fast Fourier Transform (FFT based pre-corrected trapezoidal rule) but fails to produce high-order accuracy for scattering configurations that contain discontinuous material interfaces. We must note that there do exist fast numerical techniques that fair better in terms of convergence rate while dealing with non-smooth scattering objects, for instance, see [14,15,19,22]. Among these, the approach in [14,15] converges quadratically in the presence of material discontinuity while the algorithm presented in [19], though high-order convergent, is computationally well suited only for thin inhomogeneities.…”

Section: Overviewmentioning

confidence: 99%

“…We must note that there do exist fast numerical techniques that fair better in terms of convergence rate while dealing with non-smooth scattering objects, for instance, see [14,15,19,22]. Among these, the approach in [14,15] converges quadratically in the presence of material discontinuity while the algorithm presented in [19], though high-order convergent, is computationally well suited only for thin inhomogeneities. While a more recent contribution, especially 2 This text focuses on the first component where we introduce an efficient scheme that produces high-order accurate approximations to the volume integral applicable to the Lippmann-Schwinger equation in two dimensions.…”

Section: Overviewmentioning

confidence: 99%

“…In this connection, a mixed trigonometric-polynomial interpolator, introduced in [38], is used, which is known to be an effective strategy when the underlying function has a smooth periodic extension. As shown in [37], the t -integrand is piecewise smooth in (0, t) and (t, 1) and therefore can be integrated to high-order by first breaking the transverse integral in (27) at t = t as…”

Section: Adjacent Interactionmentioning

confidence: 99%

See 2 more Smart Citations

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

Pandey

Anand

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

In recent years, several fast solvers for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nyström scheme [J. Comput. Phys., 311 (2016), 258-274] that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of O(N log N) for an N-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin [J. Comput. Phys., 228(6) (2009), 2152-2174] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde [J. Comput. Phys., 200(2) (2004), 670-694] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nyström solver in [J. Comput. Phys., 311 (2016), 258-274] through a wide range of numerical experiments.designed for a high-order treatment of discontinuous material interfaces [22] does converge rapidly with optimal computational cost, the griding strategy used therein, in certain cases, can restrict the methodology from achieving the theoretical computational complexity. Indeed, as we show later in this text, the scheme in [22] produces far less accurate approximations to the solution when compared to their counterparts obtained through the proposed approach.Apart from these, a couple of fast direct solvers with computational cost O(N 3/2 ) have also been proposed in [20,21] that primarily rely on quad-tree data structure. While these methods offer several advantages, such as, robustness and capability to handle some large scale frequency regime, they are designed only for material properties that are globally smooth.The primary aim of this paper is to provide a framework that allows us to enhance the rate of convergence for those existing O(N log N) solvers that converge rapidly for smooth inhomogeneities but yield low order in the context of discontinuous scattering media, without adversely effecting their asymptotic computational complexity. Indeed, most fast solvers converge slowly in the presence of material discontinuities because evaluation of the integral operator (5) through fast algorithms (e.g., FFT, AIM, FMM, etc.) require them to integrate across the material inte...

show abstract

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

Self Cite

View full text Add to dashboard Cite

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

show abstract

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

Cited by 6 publications

References 36 publications

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

Improved convergence of fast integral equation solvers 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

Contact Info

Product

Resources

About