An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

Karami, G.; Derakhshan, Dorna

doi:10.1016/s0955-7997(98)00085-x

Cited by 60 publications

(38 citation statements)

References 12 publications

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“…The singularity is therefore located at (s, t) = (1, 0), and a change of variables w = 1 − s is employed to move the singularity to (w, t) = (0, 0), and the (w, t) domain remaining the unit square. As in Reference [15], it is now convenient to introduce polar co-ordinates w = cos( ) t = sin( ) (17) so that the integral in Equation (16), omitting for now the (Q j ) factor, can be expressed as…”

Section: Adjacent Integrationmentioning

confidence: 99%

“…Adopting the terminology of Reference [17], the third-order derivative of the Green's function will be called supersingular. It is expected that the divergent term behaviour of Galerkin supersingular integrals will be similar to the (manageable) normal derivative hypersingular: the separate coincident and adjacent element integrals are divergent, but the sum of all integrals is finite.…”

Section: Introductionmentioning

confidence: 99%

“…Boundary integral equations having kernel functions beyond hypersingular have not been extensively studied, References [17][18][19][20] may in fact be the entire literature on the subject. The most advanced work is the recent paper by Frangi and Guiggiani [19], which provides a detailed analysis in the context of Kirchoff plate theory, and confirming numerical evaluations.…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of supersingular integrals: second‐order boundary derivatives

Moore

Gray

Kaplan

2006

Numerical Meth Engineering

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SUMMARYThe boundary integral representation of second-order derivatives of the primary function involves secondorder (hypersingular) and third-order (supersingular) derivatives of the Green's function. By defining these highly singular integrals as a difference of boundary limits, interior minus exterior, the limiting values are shown to exist. With a Galerkin formulation, coincident and edge-adjacent supersingular integrals are separately divergent, but the sum is finite, while the individual hypersingular integrals are finite. Moreover, the cancellation of the supersingular divergent terms only requires a continuous interpolation of the surface potential, and there is no continuity requirement on the surface flux. The algorithm is efficient, the non-singular integrals vanish and the singular integrals are computed entirely analytically, and accurate values are obtained for smooth surfaces. However, it is shown that a

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Section: Adjacent Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Evaluation of supersingular integrals: second‐order boundary derivatives

Moore

Gray

Kaplan

2006

Numerical Meth Engineering

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“…Using (36) alone fails to yield unique solutions at certain critical frequencies. To avert this di culty, a second equation is obtained by di erentiating (36) in the normal direction at p as follows:…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Linearly combining (36) and (38) yields unique solutions for all real frequencies when the coupling parameter is selected such that its imaginary part is non-zero [53].…”

Section: Mathematical Formulationmentioning

confidence: 99%

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems

Yang

2001

Numerical Meth Engineering

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SUMMARYThis paper presents the non-singular forms, in a global sense, of two-dimensional Green's boundary formula and its normal derivative. The main advantage of the modiÿed formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element-free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier-Legendre series, together with transforming the integration interval [a; b] to [−1; 1]; the series coe cients are thus to be determined. The hypersingular integral, interpreted in the Hadamard ÿnite-part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands deÿned explicitly when a source point coincides with a ÿeld point. The e ectiveness of the modiÿed formulations is examined by an elliptic cylinder subject to prescribed boundary conditions.The regularization is further applied to acoustic scattering problems. The well-known Burton-Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non-uniqueness problem. A general non-singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made.

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A direct approach for boundary integral equations with high-order singularities

Frangi¹,

Guiggiani²

2000

Int. J. Numer. Meth. Engng.

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An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

Cited by 60 publications

References 12 publications

Evaluation of supersingular integrals: second‐order boundary derivatives

Evaluation of supersingular integrals: second‐order boundary derivatives

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems

A direct approach for boundary integral equations with high-order singularities

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