Combined boundary integral equations for acoustic scattering‐resonance problems

Steinbach, Olaf; Unger, Gerhard

doi:10.1002/mma.4075

Cited by 19 publications

(12 citation statements)

References 25 publications

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“…where we refer to [8, Sect. From the last theorem and the properties of S(•) it follows that we can apply the convergence results of Theorem 1 to the Galerkin eigenvalue problem (34).…”

Section: Theoremmentioning

confidence: 92%

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Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems

Unger

2021

Partial Differ. Equ. Appl.

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In this paper a convergence analysis of a Galerkin boundary element method for resonance problems arising from the time harmonic Maxwell’s equations is presented. The cavity resonance problem with perfect conducting boundary conditions and the scattering resonance problem for impenetrable and penetrable scatterers are treated. The considered boundary integral formulations of the resonance problems are eigenvalue problems for holomorphic Fredholm operator-valued functions, where the occurring operators satisfy a so-called generalized Gårding’s inequality. The convergence of a conforming Galerkin approximation of this kind of eigenvalue problems is in general only guaranteed if the approximation spaces fulfill special requirements. We use recent abstract results for the convergence of the Galerkin approximation of this kind of eigenvalue problems in order to show that two classical boundary element spaces for Maxwell’s equations, the Raviart–Thomas and the Brezzi–Douglas–Marini boundary element spaces, satisfy these requirements. Numerical examples are presented, which confirm the theoretical results.

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“…where we refer to [8, Sect. From the last theorem and the properties of S(•) it follows that we can apply the convergence results of Theorem 1 to the Galerkin eigenvalue problem (34).…”

Section: Theoremmentioning

confidence: 92%

“…In the next theorem we summarize the approximation properties of V k h which we need for the convergence analysis of the Galerkin eigenvalue problem (34).…”

Section: The Sequel V K H Denotes Either Rt K (T H ) or Bmd K (T H ) We Consider The Following Galerkin Eigenvalue Problem: Findmentioning

confidence: 99%

Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems

Unger

2021

Partial Differ. Equ. Appl.

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“…We next consider the BIEs for the Maxwell equations. We introduce the following potential representation of E via the surface magnetic current m and the electronic current j (see (20)):…”

Section: Bies For the Maxwell Equationsmentioning

confidence: 99%

Boundary Integral Equations for Calculating Complex Eigenvalues of Transmission Problems

Misawa¹,

Niino²,

Nishimura³

2017

SIAM J. Appl. Math.

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Resonance frequencies are complex eigenvalues at which the homogeneous transmission problems have non-trivial solutions. These frequencies are of interest because they affect the behavior of the solutions even when the frequency is real. The resonance frequencies are related to problems for infinite domains which can be solved efficiently with the Boundary Integral Equation Method (BIEM). We thus consider a numerical method of determining resonance frequencies with fast BIEM and the SakuraiSugiura projection Method (SSM). However, BIEM may have fictitious eigenvalues even when one uses Müller or PMCHWT formulations which are known to be resonance free when the frequency is real valued.In this paper, we propose new BIEs for transmission problems with which one can distinguish true and fictitious eigenvalues easily. Specifically, we consider waveguide problems for the Helmholtz equation in 2D and standard scattering problems for Maxwell's equations in 3D. We verify numerically that the proposed BIEs can separate the fictitious eigenvalues from the true ones in these problems. We show that the obtained true complex eigenvalues are related to the behavior of the solution significantly. We also show that the fictitious eigenvalues may affect the accuracy of BIE solutions in standard boundary value problems even when the frequency is real.

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“…Underwater acoustic scattering is also an important issue, such as underwater acoustic scattering signal separation [9], acoustic scattering from underwater elastic objects [10], acoustic scattering by suspended flocculating sediments [11] and acoustic scattering in gassy soft marine sediments [12]. Other problems such as acoustic scattering-resonance [13], acoustic scattering reductions [14], acoustic scattering in nonuniform potential flows [15], designing the bilaminate acoustic cloak [16], control of acoustic absorptions [17], acoustic interaction forces [18] , acoustic radiation forces [19], and scattering effects on an acoustic black hole [20] are also considered.…”

Section: Introductionmentioning

confidence: 99%

Acoustic scattering theory without large-distance asymptotics

Zhou

Dai

2018

J. Phys. Commun.

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In conventional acoustic scattering theory, a large-distance asymptotic approximation is employed. In this approximation, a far-field pattern, an asymptotic approximation of the exact result, is used to describe a scattering process. The information of the distance between the target and the observer, however, is lost in the large-distance asymptotic approximation. In this paper, we provide a rigorous theory of acoustic scattering without the large-distance asymptotic approximation. The acoustic scattering treatmentdeveloped in this paper provides an improved description for the acoustic wave outside the target. Moreover, as examples, we consider acoustic scattering on a rigid sphere and on a nonrigid sphere. We also illustrate the influence of the near target effect on the angular distribution of outgoing waves. It is shown that for long wavelength acoustic scattering, the near target effect must be reckoned in.

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Combined boundary integral equations for acoustic scattering‐resonance problems

Cited by 19 publications

References 25 publications

Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems

Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems

Boundary Integral Equations for Calculating Complex Eigenvalues of Transmission Problems

Acoustic scattering theory without large-distance asymptotics

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