Singularities of continuation of wave fields

Kyurkchan, A. G.; Sternin, B. Yu.; Shatalov, V. E.

doi:10.1070/pu1996v039n12abeh000184

Cited by 57 publications

(16 citation statements)

References 7 publications

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“…(5), where the solution error (2norm of the difference between the left and the right hand side of the linear system) is plotted before and after quasidiagonalization. It should be emphasized that the foci of this ellipse lie at a distance equal to 0.31 from the origin, and therefore the MAS system is not capable of solving the scattering problem for very small auxiliary surfaces not enclosing these two points [7]. Hence, the deterioration of the condition number for small a/b, as seen in Figs.…”

Section: Numerical Resultsmentioning

confidence: 96%

“…In general terms, the singularities of the scattered field should be enclosed by the auxiliary surface as tightly as possible. Although the exact location of these singularities is unknown for arbitrary geometries, it was shown in [7] that the radius of the smallest possible circle, or sphere (in 2D and 3D respectively), surrounding them all, can be determined analytically. An auxiliary surface, conformal to the scatterer surface, circumscribing this circle/sphere is expected to yield optimal results, i.e.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient Preconditioning of the Method of Auxiliary Sources (MAS) for Cylindrical Scatterers of Quasi- Circular Cross-Section

Anastassiu¹,

Avdikos²,

Vouldis³

2008

TOEEJ

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Two different forms of a preconditioning process (i.e. standard preconditioning and quasi-diagonalization) are presented, in conjunction with the Method of Auxiliary Sources (MAS), when the latter is applied to a specific class of two-dimensional scattering problems. The method enhances the efficiency of MAS, when the linear system becomes illconditioned, due to distancing of the auxiliary surface from the outer boundary. If the cross-sectional boundary is geometrically close to a circle, it is proven that the MAS matrix becomes quasi-circulant, as intuition dictates. By exploiting the properties of the exactly circulant matrix, pertaining to the original circular configuration, the perturbed system is transformed to a quasi-diagonal one, whose inversion is a numerically stable operation.

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Section: Numerical Resultsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Efficient Preconditioning of the Method of Auxiliary Sources (MAS) for Cylindrical Scatterers of Quasi- Circular Cross-Section

Anastassiu¹,

Avdikos²,

Vouldis³

2008

TOEEJ

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“…The ideas behind this work were inspired in part by our recent investigations of applications of multivariable complex analysis to diffraction problems [4,5,6,7,8], in part by our work on coordinate equations [1,3,2,9,10] and in part by the work of Sternin and his co-authors [11,12,13], in which the analytical continuation of wave fields is considered.…”

Section: Introductionmentioning

confidence: 99%

Analytical continuation of two-dimensional wave fields

Assier,

Shanin

2020

Preprint

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Wave fields obeying the 2D Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green's integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and utilised by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.

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“…Here we choose surfaces for which the Rayleigh hypothesis holds that is surfaces for which the Rayleigh expansion (20) is uniformly convergent up to the surface itself cf. [Petit and Cadilhac, 1966;Millar, 1969;Kyurkchan et al, 1996]. In that case the Rayleigh series can be differentiated with respect to the normal and the results from the Rayleigh series and the integral formulation described in section 1.…”

Section: Dirichlet To Neumann Mapmentioning

confidence: 99%

Electromagnetic Scattering from Multiple Scale Geometries

Caponi¹,

Sei²

2002

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Singularities of continuation of wave fields

Cited by 57 publications

References 7 publications

Efficient Preconditioning of the Method of Auxiliary Sources (MAS) for Cylindrical Scatterers of Quasi- Circular Cross-Section

Efficient Preconditioning of the Method of Auxiliary Sources (MAS) for Cylindrical Scatterers of Quasi- Circular Cross-Section

Analytical continuation of two-dimensional wave fields

Electromagnetic Scattering from Multiple Scale Geometries

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