On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation

Kreβ, R.; Spassov, W. T.

doi:10.1007/bf01400919

Cited by 75 publications

(4 citation statements)

References 9 publications

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“…Let us rewrite the right-hand side of the above identity. The operator Ŝε nn is normal, as proven in [8]; moreover, its (non-normalized) eigenfunctions are {e ikθn(x) } k∈Z , and eigenvalues are given by [22,21],…”

Section: Galerkin Foldy-lax Modelmentioning

confidence: 93%

A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles

Kachanovska

2024

Multiscale Model. Simul.

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In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem valid when the size of the particles tends to zero. We present a systematic approach to constructing such models, based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. We prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments.

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Section: Galerkin Foldy-lax Modelmentioning

confidence: 93%

A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles

Kachanovska

2024

Multiscale Model. Simul.

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“…The argument quickly shifted from introducing µ in order to avoid the potential catastrophic errors in the original equations of Section 4.3.1 to considering an optimal value. For several decades, based originally on the works of Kress [395][396][397][398] and the further works of Amini [399], there has been a strong recommendation, with supporting research, that µ = i k is reasonably close to the 'optimal' choice, as least for the simple boundaries, like spheres. Obviously i k is unsuitable for low wavenumbers, as the parameter would be large, violating the condition |µ| ∞, and the second equation would dominate and a cap on its value is recommended, for example,…”

Section: The Coupling Parametermentioning

confidence: 99%

The Boundary Element Method in Acoustics: A Survey

Kirkup

2019

Applied Sciences

108

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The boundary element method (BEM) in the context of acoustics or Helmholtz problems is reviewed in this paper. The basis of the BEM is initially developed for Laplace’s equation. The boundary integral equation formulations for the standard interior and exterior acoustic problems are stated and the boundary element methods are derived through collocation. It is shown how interior modal analysis can be carried out via the boundary element method. Further extensions in the BEM in acoustics are also reviewed, including half-space problems and modelling the acoustic field surrounding thin screens. Current research in linking the boundary element method to other methods in order to solve coupled vibro-acoustic and aero-acoustic problems and methods for solving inverse problems via the BEM are surveyed. Applications of the BEM in each area of acoustics are referenced. The computational complexity of the problem is considered and methods for improving its general efficiency are reviewed. The significant maintenance issues of the standard exterior acoustic solution are considered, in particular the weighting parameter in combined formulations such as Burton and Miller’s equation. The commonality of the integral operators across formulations and hence the potential for development of a software library approach is emphasised.

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“…There is no general theory concerning the optimal value of η that yields the lowest condition number. Some discussions on this topic can be found in [27] and [25].…”

Section: Galerkin Discretizationmentioning

confidence: 99%

Modified fast multipole algorithm using well‐conditioned electric field integral equation for electromagnetic scattering problem

Chen¹,

Wang²,

Chen³

2011

Micro & Optical Tech Letters

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Conventional fast multipole algorithm is one of the most widely used integral equation method for its high efficiency. Unfortunately, it always come across a problem named subwavelength breakdown when the finest group size is smaller than 0.2 wavelength. In this article, singular value decomposition method is used to modify the translation operator of fast multipole algorithm. The finest group size will be smaller without subwavelength breakdown by this method, which leads to much reduction of the memory requirement. Furthermore, the time consumption is also reduced by applying the well-conditioned electric field integral equation. Numerical results prove the validity and efficiency of this method.ABSTRACT: Design of a high gain cavity backed slot antenna is presented with relatively small size. The large ground plane of the original design is cut 75%. Mushroom cells and ground plane orientations have been employed to improve the antenna gain. An 18.75-dB maximum gain is obtained with an average gain of 17.3 dB in the entire operating band.

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On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation

Cited by 75 publications

References 9 publications

A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles

A New Class of Uniformly Stable Time-Domain Foldy–Lax Models for Scattering by Small Particles. Acoustic Sound-Soft Scattering by Circles

The Boundary Element Method in Acoustics: A Survey

Modified fast multipole algorithm using well‐conditioned electric field integral equation for electromagnetic scattering problem

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