EI、Scopus 收录
中文核心期刊
Yaoming Zhang, Cuilian Sun, Yan Gu. The evaluation of nearly singular integrals in the boundary integral equations with variable transformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(2): 207-214. DOI: 10.6052/0459-1879-2008-2-2007-123
Citation: Yaoming Zhang, Cuilian Sun, Yan Gu. The evaluation of nearly singular integrals in the boundary integral equations with variable transformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(2): 207-214. DOI: 10.6052/0459-1879-2008-2-2007-123

The evaluation of nearly singular integrals in the boundary integral equations with variable transformation

  • Received Date: March 13, 2007
  • Revised Date: August 06, 2007
  • The numerical solution of boundary value problemsusing boundary integral equations demands the accurate computationof the integral of the kernels, which occur as the nearly singularintegrals when the collocation point is close to the element ofintegration but not on the element in boundary element method(BEM). Such integrals are difficult to compute by standardquadrature procedures, since the integrand varies very rapidlywithin the integration interval, more rapidly the closer thecollocation point is to the integration element. Practice showsthat we can even obtain the results of superconvergence for thecomputed point far enough from the boundary; however, usingstandard quadrature procedures, which neglect the pathologicalbehavior of the integrand as the computed point approaches theintegration element, will lead to a degeneracy of accuracy of thesolution, even no accuracy, which is the so-called ``boundarylayer effect''. To avoid the ``boundary layer effect'', the accuratecomputation of the nearly singular boundary integrals would bemore crucial to some of the engineering problems, such as thecrack-like and thin or shell-like structure problems.The importance of the accurate evaluation of nearlysingular integrals is considered to be next to the singularboundary integrals in BEM, and great attentions have beenattracted and many numerical techniques have been proposed for itin recent years. These developed methods can be divided on thewhole into two categories: ``indirct algorithms'' and ``directalgorithms'', which have obtained varying degree of success, butthe problem of the nearly singular integrals has not beencompletely resolved so far. In this paper, a new efficienttransformation is proposed based on a new idea of transformationwith variables. The proposed transformation can remove the nearlysingularity efficiently by smoothing out the rapid variations ofthe integrand of nearly singular integrals, and improve theaccuracy of numerical results of nearly singular integrals greatlywithout increasing the computational effort. Numerical examples ofpotential problem with their satisfactory results in both curvedand straight elements are presented, showing encouragingly thehigh efficiency and stability of the suggested approach, even whenthe internal point is very close to the boundary. The suggestedalgorithm is general and can be applied to other problems in BEM.
  • Related Articles

    [1]Wei Chang, Fan Yuchen, Zhou Yongqing, Liu Xin, Li Chi, Wang Heyang. SOLVING UNSTEADY PARTIAL DIFFERENTIAL EQUATIONS USING TIME-WEIGHTED PHYSICS-INFORMED NEURAL NETWORK[J]. Chinese Journal of Theoretical and Applied Mechanics, 2025, 57(3): 755-766. DOI: 10.6052/0459-1879-24-289
    [2]Chen Haolong, Tang Xinyue, Wang Runhua, Zhou Huanlin, Liu Zhanli. SOLVING MULTI-MEDIA NONLINEAR TRANSIENT HEAT CONDUCTION PROBLEM BASED ON PHYSICS-INFORMED NEURAL NETWORKS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2025, 57(1): 89-102. DOI: 10.6052/0459-1879-24-337
    [3]Zhang Linghai, Zhou Bin, Luo Yi, Feng Jun. AN ADAPTIVE COLLOCATION POINT ALGORITHM FOR PHYSICS-INFORMED NEURAL NETWORKS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(10): 3069-3083. DOI: 10.6052/0459-1879-24-244
    [4]Wei Chang, Fan Yuchen, Zhou Yongqing, Zhang Chaoqun, Liu Xin, Wang Heyang. SELF-REGRESSIVE PHYSICS-INFORMED NEURAL NETWORK BASED ON RUNGE-KUTTA METHOD FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(8): 2482-2493. DOI: 10.6052/0459-1879-24-106
    [5]Pan Xiaoguo, Wang Kai, Deng Weixin. ACCELERATING CONVERGENCE ALGORITHM FOR PHYSICS-INFORMED NEURAL NETWORKS BASED ON NTK THEORY AND MODIFIED CAUSALITY[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(7): 1943-1958. DOI: 10.6052/0459-1879-24-087
    [6]Guo Yuan, Fu Zhuojia, Min Jian, Liu Xiaoting, Zhao Haitao. CURRICULUM-TRANSFER-LEARNING BASED PHYSICS-INFORMED NEURAL NETWORKS FOR LONG-TIME SIMULATION OF NONLINEAR WAVE PROPAGATION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(3): 763-773. DOI: 10.6052/0459-1879-23-457
    [7]Feng Tangsijie, Liang Wei. THE BUCKLING ANALYSIS OF THIN-WALLED STRUCTURES BASED ON PHYSICS-INFORMED NEURAL NETWORKS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(11): 2539-2553. DOI: 10.6052/0459-1879-23-277
    [8]Wei Chang, Fan Yuchen, Zhou Yongqing, Liu Xin, Zhang Chaoqun, Wang Heyang. MULTI-OUTPUT PHYSICS-INFORMED NEURAL NETWORKS MODEL BASED ON THE RUNGE-KUTTA METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2405-2416. DOI: 10.6052/0459-1879-23-299
    [9]Song Jiahao, Cao Wenbo, Zhang Weiwei. FD-PINN: FREQUENCY DOMAIN PHYSICS-INFORMED NEURAL NETWORK[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(5): 1195-1205. DOI: 10.6052/0459-1879-23-169
    [10]Fan Xincheng, Ye Zuyang, Huang Shibing, Cheng Aiping. STUDY ON CONNECTIVITY AND ENTROPY SCALE OF THREE-DIMENSIONAL FRACTURE NETWORK[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(3): 792-804. DOI: 10.6052/0459-1879-22-579
  • Cited by

    Periodical cited type(3)

    1. 姚军,王通,孙致学,孙海,黄朝琴. 通用嵌入式离散裂缝模型热-流-固耦合大规模并行数值模拟技术. 石油学报. 2025(03): 574-587 .
    2. 计秉玉,张文彪,何应付,段太忠,刘合. 油藏地质建模与数值模拟一体化内涵及发展趋势. 石油学报. 2024(07): 1152-1162 .
    3. 曹琳,张新龙,刘伟,尹彦君,盛广龙,赵辉. 油藏开发非均相智能调驱评价及模拟预测方法. 长江大学学报(自然科学版). 2024(05): 82-93 .

    Other cited types(1)

Catalog

    Article Metrics

    Article views (2075) PDF downloads (789) Cited by(4)
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return