# Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods

### Reference:

Carley, M., 2009. Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods. *Mathematical Proceedings of the Royal Irish Academy*, 109 (1), pp. 49-60.

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### Official URL:

http://dx.doi.org/10.3318/PRIA.2008.109.1.49

### Abstract

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: \[ f(x,y,t) = \frac{a(x,y,t)}{(x-t)^{2}+y^{2}} + \frac{b(x,y,t)}{[(x-t)^{2}+y^{2}]^{1/2}} + c(x,y,t)\log[(x-t)^{2}+y^{2}]^{1/2} + d(x,y,t), \] without having to explicitly analyze the singularities of $f(x,y,t)$ or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when $y\equiv0$. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.

### Details

Item Type | Articles |

Creators | Carley, M. |

DOI | 10.3318/PRIA.2008.109.1.49 |

Departments | Faculty of Engineering & Design > Mechanical Engineering |

Research Centres | Aerospace Engineering Research Centre |

Refereed | Yes |

Status | Published |

ID Code | 13956 |

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