# A Sigmoidal and Distance Combined Transformation Method for Nearly Singular Integral on Asymmetric Patch

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**x**and the source point

**y**in physical space, and ${\mathsf{\Gamma}}_{e}$ represents the integral patch or element. The power l is usually considered as the order of the singularity. The integral in Equation (1) is regular if the source point

**y**is located outside the integral patch. The integral becomes singular when the source point is located inside the integral patch. Many classic numerical quadrature formulas, including the well-known Gaussian quadrature formula and the Newton–Cotes quadrature formula, could be applied to accurately compute the regular integral.

## 2. Statement of the Problem

**x**

^{p}represents the nearest point in the patch to the source point

**y**.

**n**

^{p}denotes the out-normal of

**x**

^{p}. The integrand in 3D BEM is usually of the following form:

**y**is located very close to the patch. The traditional Gaussian quadrature method usually leads to unacceptable results in the computation of the nearly singular integral.

## 3. The Distance Transformation and the Sigmoidal Transformation

**x**and the center

**x**as defined in the introduction are expanded into the Taylor series. The superscript p denotes variables related to the center

^{p}**x**. The subscripts i and k denote the components in physical space. $\xi $,$\eta $ represent the two parametric coordinates of

^{p}**x**in $\left(\xi ,\eta \right)$ space.

**y**and

**x**can be expressed in $\left(\rho ,\theta \right)$ space as

#### 3.1. The Distance Transformation

#### 3.2. The Sigmoidal Transformation

**x**

^{p}is located very closely to the boundary of the integral patch. To further eliminate this near-singularity, we introduce a sigmoidal transformation for that the variable $\theta $.

## 4. Numerical Examples

#### 4.1. Numerical Integral With Integrand $\frac{1}{r}$

#### 4.2. Numerical Integral With Integrand $\frac{1}{{r}^{2}}$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chen, S.; Liu, Y. A unified boundary element method for the analysis of sound and shell-like structure interactions. I. Formulation and verification. J. Acoust. Soc. Am.
**1999**, 106, 1247–1254. [Google Scholar] [CrossRef] - Zhang, J.; Qin, X.; Han, X.; Li, G. A boundary face method for potential problems in three dimensions. Int. J. Numer. Methods Eng.
**2009**, 80, 320–337. [Google Scholar] [CrossRef] - Xie, G.; Zhou, F.; Zhang, D.; Wen, X.; Li, H. A novel triangular boundary crack front element for 3D crack problems based on 8-node serendipity element. Eng. Anal. Bound. Elem.
**2019**, 105, 296–302. [Google Scholar] [CrossRef] - Xie, G.; Zhou, F.; Li, H.; Wen, X.; Meng, F. A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method. Front. Mech. Eng.
**2019**, 14, 332–341. [Google Scholar] [CrossRef] - Feng, S.Z.; Han, X. A novel multi-grid based reanalysis approach for efficient prediction of fatigue crack propagation. Comput. Methods Appl. Mech. Eng.
**2019**, 353, 107–122. [Google Scholar] [CrossRef] - Wei, Z.; Feng, J.; Ghalandari, M.; Maleki, A.; Abdelmalek, Z. Numerical Modeling of Sloshing Frequencies in Tanks with Structure Using New Presented DQM-BEM Technique. Symmetry
**2020**, 12, 655. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.; Shu, X.; Trevelyan, J.; Lin, W.; Chai, P. A solution approach for contact problems based on the dual interpolation boundary face method. Appl. Math. Model.
**2019**, 70, 643–658. [Google Scholar] [CrossRef] - Shu, X.; Zhang, J.; Han, L.; Dong, Y. A surface-to-surface scheme for 3D contact problems by boundary face method. Eng. Anal. Bound. Elem.
**2016**, 70, 23–30. [Google Scholar] [CrossRef] [Green Version] - Mossaiby, F.; Bazrpach, M.; Shojaei, A. Extending the method of exponential basis functions to problems with singularities. Eng. Comput.
**2015**, 32, 406–423. [Google Scholar] [CrossRef] - Bazazzadeh, S.; Shojaei, A.; Zaccariotto, M.; Galvanetto, U. Application of the peridynamic differential operator to the solution of sloshing problems in tanks. Eng. Comput.
**2019**, 36, 45–83. [Google Scholar] [CrossRef] - Qin, X.; Zhang, J.; Xie, G.; Zhou, F.; Li, G. A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. J. Comput. Appl. Math.
**2011**, 235, 4174–4186. [Google Scholar] [CrossRef] [Green Version] - Wang, Q.; Zhou, W.; Cheng, Y.; Ma, G.; Chang, X. A line integration method for the treatment of 3D domain integrals and accelerated by the fast multipole method in the BEM. Comput. Mech.
**2016**, 59, 1–14. [Google Scholar] [CrossRef] - Zhang, J.M.; Chi, B.T.; Singh, K.M. A binary-tree element subdivision method for evaluation of nearly singular domain integrals with continuous or discontinuous kernel. J. Comput. Appl. Math.
**2019**, 362, 22–40. [Google Scholar] [CrossRef] - Zhang, J.; Wang, P.; Lu, C.; Dong, Y. A spherical element subdivision method for the numerical evaluation of nearly singular integrals in 3D BEM. Eng. Comput.
**2017**, 34, 2074–2087. [Google Scholar] [CrossRef] - Johnston, P.R.; Elliott, D. A sinh transformation for evaluating nearly singular boundary element integrals. Int. J. Numer. Methods Eng.
**2005**, 62, 564–578. [Google Scholar] [CrossRef] - Johnston, B.M.; Johnston, P.R.; Elliott, D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. Int. J. Numer. Methods Eng.
**2007**, 69, 1460–1479. [Google Scholar] [CrossRef] - Xie, G.; Zhang, D.; Zhang, J.; Meng, F.; Du, W.; Wen, X. Implementation of sinh method in integration space for boundary integrals with near singularity in potential problems. Front. Mech. Eng.
**2016**, 11, 412–422. [Google Scholar] [CrossRef] - Ma, H.; Kamiya, N. Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. Eng. Anal. Bound. Elem.
**2002**, 26, 329–339. [Google Scholar] [CrossRef] - Ma, H.; Kamiya, N. A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two-and three-dimensional elasticity. Comput. Mech.
**2002**, 29, 277–288. [Google Scholar] [CrossRef] - Lv, J.H.; Jiao, Y.Y.; Feng, X.T.; Wriggers, P.; Zhuang, X.Y.; Rabczuk, T. A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities. Int. J. Numer. Methods Eng.
**2019**, 118, 38–60. [Google Scholar] [CrossRef] - Qin, X.; Zhang, J.; Li, G.; Sheng, X.; Song, Q.; Mu, D. An element implementation of the boundary face method for 3D potential problems. Eng. Anal. Bound. Elem.
**2010**, 34, 934–943. [Google Scholar] [CrossRef] - Elliott, D. The Cruciform Crack Problem and Sigmoidal Transformations. Math. Methods Appl. Sci.
**1997**, 20, 121–132. [Google Scholar] [CrossRef] - Johnston, P.R. Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals. Int. J. Numer. Methods Eng.
**1999**, 45, 1333–1348. [Google Scholar] [CrossRef] - Zhou, F.; Wang, W.; Xie, G.; Liao, H.; Cao, Y. The Distance-Sinh combined transformation for near-singularity cancelation based on the generalized Duffy normalization. Eng. Anal. Bound. Elem.
**2019**, 108, 108–114. [Google Scholar] [CrossRef]

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.9, −0.9) | $0.01/\sqrt{A}$ | 4.2289068121035 | 4.2289067166042 (2.258 × 10^{−8}) | 4.2288490197253 (1.367 × 10^{−8}) |

$0.001/\sqrt{A}$ | 4.428204210893 | 4.4282041147709 (2.171 × 10^{−8}) | 4.4281464181109 (1.305 × 10^{−5}) | |

$0.0001/\sqrt{A}$ | 4.450547238332 | 4.4505471439829 (2.12 × 10^{−8}) | 4.4504894473208 (1.299 × 10^{−5}) | |

$0.00001/\sqrt{A}$ | 4.4528064198506 | 4.4528063239112 (2.155 × 10^{−8}) | 4.4527486272491 (1.298 × 10^{−5}) |

**Table 2.**Results for the case of the second nearest point (−0.5, 0.5) and the integrand $\frac{1}{r}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.5, 0.5) | $0.01/\sqrt{A}$ | 6.0813473610944 | 6.0813473607766 (5.225 × 10^{−11}) | 6.0813473610825 (1.955 × 10^{−12}) |

$0.001/\sqrt{A}$ | 6.3012224169857 | 6.3012224170493 (1.01 × 10^{−11}) | 6.3012224173312 (5.482 × 10^{−11}) | |

$0.0001/\sqrt{A}$ | 6.3237786229076 | 6.3237786231514 (3.855 × 10^{−11}) | 6.3237786234336 (8.317 × 10^{−11}) | |

$0.00001/\sqrt{A}$ | 6.3260399369998 | 6.326039936616 (6.068 × 10^{−11}) | 6.3260399368983 (1.605 × 10^{−11}) |

**Table 3.**Results for the case of the third nearest point (−0.9, 0.1) and the integrand $\frac{1}{r}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.9, 0.1) | $0.01/\sqrt{A}$ | 5.2691932954958 | 5.2691933018071 (1.198 × 10^{−9}) | 5.2691856054382 (1.459 × 10^{−6}) |

$0.001/\sqrt{A}$ | 5.4780628589679 | 5.478062864997 (1.101 × 10^{−9}) | 5.4780551691103 (1.404 × 10^{−6}) | |

$0.0001/\sqrt{A}$ | 5.5005050978374 | 5.5005051048003 (1.266 × 10^{−9}) | 5.5004974089089 (1.398 × 10^{−6}) | |

$0.00001/\sqrt{A}$ | 5.502765271843 | 5.5027652773982 (1.01 × 10^{−9}) | 5.5027575815068 (1.398 × 10^{−6}) |

**Table 4.**Results for the case of the fourth nearest point (−0.98, 0.9) and the integrand $\frac{1}{r}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.98, 0.9) | $0.01/\sqrt{A}$ | 3.8917255375904 | 3.8917235422117 (5.127 × 10^{−7}) | 3.8896240278628 (0.00054) |

$0.001/\sqrt{A}$ | 4.0532815787312 | 4.053279590575 (4.905 × 10^{−7}) | 4.0511800685558 (0.0005185) | |

$0.0001/\sqrt{A}$ | 4.0750213786998 | 4.0750193923585 (4.874 × 10^{−7}) | 4.0729198703437 (0.0005157) | |

$0.00001/\sqrt{A}$ | 4.0772744765221 | 4.0772724885125 (4.876 × 10^{−7}) | 4.0751729664998 (0.0005154) |

**Table 5.**Results for the case of the nearest point (−0.9, −0.9) and the integrand $\frac{1}{{r}^{2}}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.9, −0.9) | $0.01/\sqrt{A}$ | 13.015827662368 | 13.015827632251 (2.314 × 10^{−9}) | 13.015820800269 (5.272 × 10^{−7}) |

$0.001/\sqrt{A}$ | 27.27020393192 | 27.27020273636 (4.384 × 10^{−8}) | 27.270195897312 (2.946 × 10^{−7}) | |

$0.0001/\sqrt{A}$ | 41.735509126402 | 41.735518653687 (2.283 × 10^{−7}) | 41.735511815265 (6.443 × 10^{−8}) | |

$0.00001/\sqrt{A}$ | 56.203055300579 | 56.203013521208 (7.434 × 10^{−7}) | 56.203006682186 (8.65 × 10^{−7}) |

**Table 6.**Results for the case of the second nearest point (−0.5, 0.5) and the integrand $\frac{1}{{r}^{2}}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.5, 0.5) | $0.01/\sqrt{A}$ | 19.469535296955 | 19.469535289059 (4.055 × 10^{−10}) | 19.469535288481 (4.353 × 10^{−10}) |

$0.001/\sqrt{A}$ | 33.927476616435 | 33.927476006764 (1.797 × 10^{−8}) | 33.927476005970 (1.799 × 10^{−8}) | |

$0.0001/\sqrt{A}$ | 48.394948935260 | 48.394966004226 (3.527 × 10^{−7}) | 48.394966003915 (3.527 × 10^{−7}) | |

$0.00001/\sqrt{A}$ | 62.862516795008 | 62.862414899525 (1.621 × 10^{−6}) | 62.862414899703 (1.621 × 10^{−6}) |

**Table 7.**Results for the case of the second nearest point (−0.9, 0.1) and the integrand $\frac{1}{{r}^{2}}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.9, 0.1) | $0.01/\sqrt{A}$ | 16.226537779634 | 16.226537809600 (1.847 × 10^{−9}) | 16.226536399831 (8.503 × 10^{−8}) |

$0.001/\sqrt{A}$ | 30.575099766199 | 30.575098840856 (3.026 × 10^{−8}) | 30.575097469037 (7.513 × 10^{−8}) | |

$0.0001/\sqrt{A}$ | 45.041409946084 | 45.041425764178 (3.512 × 10^{−7}) | 45.041424391664 (3.207 × 10^{−7}) | |

$0.00001/\sqrt{A}$ | 59.508966177007 | 59.508887524201 (1.322 × 10^{−6}) | 59.508886152009 (1.345 × 10^{−6}) |

**Table 8.**Results for the case of the second nearest point (−0.98, 0.9) and the integrand $\frac{1}{{r}^{2}}$.

Nearest Point | Distance | Referred Result | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.98, 0.9) | $0.01/\sqrt{A}$ | 10.118196699929 | 10.11819591766 (7.731 × 10^{−8}) | 10.117472488011 (7.158 × 10^{−5}) |

$0.001/\sqrt{A}$ | 23.036432410893 | 23.036430691757 (7.463 × 10^{−8}) | 23.035707019533 (3.149 × 10^{−5}) | |

$0.0001/\sqrt{A}$ | 37.472632150961 | 37.472640949233 (2.348 × 10^{−7}) | 37.471917264908 (1.908 × 10^{−5}) | |

$0.00001/\sqrt{A}$ | 51.939882717469 | 51.939831194738 (9.92 × 10^{−7}) | 51.939107505899 (1.493 × 10^{−5}) |

**Table 9.**Comparisons of time consumption in the computation of the nearly singular integral with integrand $\frac{1}{{r}^{2}}$.

Nearest Point | Distance | Element Subdivision | Presented Method | Single Distance Transformation |
---|---|---|---|---|

(−0.98, 0.9) | $0.1/\sqrt{A}$ | 114,956 ms | 6312 ms | 6051 ms |

$0.01/\sqrt{A}$ | 364,591 ms | 6485 ms | 6231 ms | |

$0.001/\sqrt{A}$ | 648,553 ms | 6551 ms | 6308 ms |

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## Share and Cite

**MDPI and ACS Style**

Yu, J.; Lei, Z.; Yao, Q.; Zhou, F.
A Sigmoidal and Distance Combined Transformation Method for Nearly Singular Integral on Asymmetric Patch. *Symmetry* **2020**, *12*, 983.
https://doi.org/10.3390/sym12060983

**AMA Style**

Yu J, Lei Z, Yao Q, Zhou F.
A Sigmoidal and Distance Combined Transformation Method for Nearly Singular Integral on Asymmetric Patch. *Symmetry*. 2020; 12(6):983.
https://doi.org/10.3390/sym12060983

**Chicago/Turabian Style**

Yu, Jianghong, Zhengbao Lei, Qishui Yao, and Fenglin Zhou.
2020. "A Sigmoidal and Distance Combined Transformation Method for Nearly Singular Integral on Asymmetric Patch" *Symmetry* 12, no. 6: 983.
https://doi.org/10.3390/sym12060983