1998
DOI: 10.1002/(sici)1097-0207(19980415)41:7<1255::aid-nme333>3.0.co;2-n
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Tangential derivative of singular boundary integrals with respect to the position of collocation points

Abstract: This paper investigates the evaluation of the sensitivity, with respect to tangential perturbations of the singular point, of boundary integrals having either weak or strong singularity. Both scalar potential and elastic problems are considered. A proper deÿnition of the derivative of a strongly singular integral with respect to singular point perturbations should accommodate the concomitant perturbation of the vanishing exclusion neighbourhood involved in the limiting process used in the deÿnition of the inte… Show more

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Cited by 9 publications
(6 citation statements)
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“…Let S : x 3 = f (x 1 ) in R 3 be a smooth surface periodically modulated in the coordinate x 1 with the period Λ and uniform in the x 2 direction. The periodical interface S with the unit normal vector ν divides the space into two semi-infinite homogeneous domains Ω (1) and Ω (2) , where the materials are characterized by the constant relative permittivities ε (1) = ε (2) , ε (1) ∈ R and ε (2) ∈ C, Re (ε (2) ) > 0, Im (ε (2) ) 0 and the relative permeabilities µ (1) = µ (2) = 1 (both the materials are supposed to be magnetically neutral), see Figure 1.…”
Section: Formulation Of Problemmentioning
confidence: 99%
See 1 more Smart Citation
“…Let S : x 3 = f (x 1 ) in R 3 be a smooth surface periodically modulated in the coordinate x 1 with the period Λ and uniform in the x 2 direction. The periodical interface S with the unit normal vector ν divides the space into two semi-infinite homogeneous domains Ω (1) and Ω (2) , where the materials are characterized by the constant relative permittivities ε (1) = ε (2) , ε (1) ∈ R and ε (2) ∈ C, Re (ε (2) ) > 0, Im (ε (2) ) 0 and the relative permeabilities µ (1) = µ (2) = 1 (both the materials are supposed to be magnetically neutral), see Figure 1.…”
Section: Formulation Of Problemmentioning
confidence: 99%
“…During the last thirty years, numerous works treating the optical diffraction in periodical structures have been published (see [1]) and references therein. One of the relatively new approaches is based on the Boundary Integral Equations (BIE) [2], [5]. In this article, we present specific integral formulation of the boundary problem for the system of the Maxwell equations based on the tangential vector fields and propose its numerical implementation.…”
Section: Introductionmentioning
confidence: 99%
“…However, there have arisen new computational problems involving propagating surfaces (see, e.g., [21]), which require formulae for temporal derivative of singular and hypersingular integrals. Such formulae are also needed for the sensitivity analysis applied to error estimation of the boundary element method [8]. In the case of non-singular integrals, the differentiation rules were given in [19].…”
Section: Introductionmentioning
confidence: 99%
“…In the case of non-singular integrals, the differentiation rules were given in [19]. They have been employed in [8] for obtaining the derivative of a singular integral over a surface of a 3D domain when points of the surface move in such a way that the initial domain stays globally unchanged (the changes in positions of the surface points occur in the tangential direction). This case is of prime significance when studying how the change of the position of a collocation point influences the value of a singular integral.…”
Section: Introductionmentioning
confidence: 99%
“…Estimators of nodal‐sensitivity type were presented by Guiggiani 24, Bonnet and Guiggiani 25 and Paulino et al 26. The error is estimated as the derivative of the solution with respect to the movement, in the tangential direction, of the collocation points.…”
Section: Introductionmentioning
confidence: 99%