Analytical integration of weakly singular integrals in boundary element analysis of Helmholtz and advection–diffusion equations

Singh, Krishna M.; Tanaka, Masataka

doi:10.1016/s0045-7825(99)00316-3

Cited by 14 publications

(14 citation statements)

References 17 publications

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“…These include analytical evaluation [41,42], division of the interval about the singularity [43], subtraction of the singularity [42], special quadrature routines [44], and the use of non-linear coordinate transformations [45][46][47][48][49]. Each method has its own particular advantages and disadvantages depending on the type and order of the boundary elements used, and the form of the singular integral.…”

Section: Computation Of Weakly Singular Integralsmentioning

confidence: 99%

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

Treeby

2009

Engineering Analysis with Boundary Elements

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a b s t r a c tFor many engineers and acousticians, the boundary element method (BEM) provides an invaluable tool in the analysis of complex problems. It is particularly well suited for the examination of acoustical problems within large domains. Unsurprisingly, the widespread application of the BEM continues to produce an academic interest in the methodology. New algorithms and techniques are still being proposed, to extend the functionality of the BEM, and to compute the required numerical tasks with greater accuracy and efficiency. However, for a given global error constraint, the actual computational accuracy that is required from the various numerical procedures is not often discussed. Within this context, this paper presents an investigation into the discretisation and computational errors that arise in the BEM for acoustic scattering. First, accurate routines to compute regular, weakly singular, and nearly weakly singular integral kernels are examined. These are then used to illustrate the effect of the requisite boundary discretisation on the global error. The effects of geometric and impedance singularities are also considered. Subsequently, the actual integration accuracy required to maintain a given global error constraint is established. Several regular and irregular scattering examples are investigated, and empirical parameter guidelines are provided.

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Section: Computation Of Weakly Singular Integralsmentioning

confidence: 99%

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

Treeby

2009

Engineering Analysis with Boundary Elements

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“…However, let us note that for the integral of this choice, K0RT, closed form analytical formula [30] as well as Chebyshev expansion based approximation [31] are available, the latter being no more expensive than evaluation of K 0 (t). Further, in SSM or SSNT implementation based on K0RT, we can also exploit the fact that the integral a 0 K 0 (t) dt is simply a constant (= =2) for large arguments a [24], which would be the case for large values of .…”

Section: Modiÿed Helmholtz Equationmentioning

confidence: 99%

On non‐linear transformations for accurate numerical evaluation of weakly singular boundary integrals

Singh

Tanaka

2001

Numerical Meth Engineering

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SUMMARYThis paper presents a study of the performance of the non-linear co-ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non-linear polynomial transformations is presented for two-dimensional problems. E ectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more e ciently handled with transformations valid for end-point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non-linear transformation has been proposed. This composite approach is found to be more accurate, e cient and robust than the singularity subtraction method and the non-linear transformation methods.

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“…Many methods for numerical evaluation of singular integrals have been developed. These include analytical evaluation methods [1,2], recursive subdivision of the integration interval [3], subtraction of singularity [4,5], specialized quadrature formulas [6][7][8][9][10][11], and non-linear co-ordinate transformation techniques . Among them we are interested in the non-linear co-ordinate transformation techniques which are known to be efficient and easy to use in adaptive approaches.…”

Section: Introductionmentioning

confidence: 99%

“…where 0 min{J (1) (1), J (2) (−1)}/4J (1) (1). See Reference [13] for definitions of J (1) (1), J (2) …”

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confidence: 99%

A generalized non‐linear transformation for evaluating singular integrals

Yun

2005

Numerical Meth Engineering

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SUMMARYIn this paper, we propose a function which, according to a parameter included therein, generates a new sigmoidal transformation and converges to the well-known Sato polynomial transformation. Following well-established procedures in the literature, we employ the present transformation in the numerical evaluation of weakly singular integrals, Cauchy principal value integrals and Hadamard finite-part integrals.By several numerical examples it is shown that the present transformation is available for all kinds of singular integrals mentioned above and, in some cases, gives better results compared with those of traditional non-linear transformations.

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Analytical integration of weakly singular integrals in boundary element analysis of Helmholtz and advection–diffusion equations

Cited by 14 publications

References 17 publications

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

On non‐linear transformations for accurate numerical evaluation of weakly singular boundary integrals

A generalized non‐linear transformation for evaluating singular integrals

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