@article {32695,
	title = {Indirect computation of singular integrals in Helmholtz boundary integral equation},
	journal = {Jisuan Lixue Xuebao/Chinese Journal of Computational MechanicsJisuan Lixue Xuebao/Chinese Journal of Computational MechanicsJisuan Lixue Xuebao/Chinese Journal of Computational Mechanics},
	volume = {36},
	number = {5},
	year = {2019},
	note = {Export Date: 9 November 2022CODEN: JLXIACorrespondence Address: Chen, Y.-Q.; Department of Mechanics and Engineering Science, China; email: chenyq@pku.edu.cnReferences: Burton, A.J., Miller, G.F., The application of integral equation methods to the numerical solution of some exterior boundary-value problems (1971) Proceedings of the Royal Society of London Series A-Mathematical, Physical and Engineering Sciences, 323 (1553), pp. 201-210; Cruse, T.A., A direct formulation and numerical solution of the general transient elastodynamic problem.II (1968) Journal of Mathematical Analysis and Applications, 22 (2), pp. 341-355; Cruse, T.A., Rizzo, F.J., A direct formulation and numerical solution of the general transient elastodynamic problem.I (1968) Journal of Mathematical Analysis and Applications, 22 (1), pp. 244-259; Cruse, T.A., Snow, D.W., Wilson, R.B., Numerical solutions in axisymmetric elasticity (1977) Computers \&amp; Structures, 7 (3), pp. 445-451; Sarihan, V., Mukherjee, S., Axisymmetric viscoplastic deformation by the boundary element method (1982) International Journal of Solids and Structures, 18 (12), pp. 1113-1128; Guiggiani, M., Casalini, P., Direct computation of Cauchy principal value integrals in advanced boundary elements (1987) International Journal for Numerical Methods in Engineering, 24 (9), pp. 1711-1720; Guiggiani, M., Gigante, A., A general algorithm for multidimensional cauchy principal value integrals in the boundary element method (1990) Journal of Applied Mechanics, 57 (4), pp. 906-915; Gao, X.W., Numerical evaluation of two-dimensional singular boundary integrals-Theory and Fortran code (2006) Journal of Computational and Applied Mathe-matics, 188 (1), pp. 44-64; Guiggiani, M., Krishnasamy, G., Rudolphi, T.J., A general algorithm for the numerical-solution of hypersingular boundary integral-equations (1992) Journal of Applied Mechanics, 59 (3), pp. 604-614; Hwang, W.S., Hypersingular boundary integral equations for exterior acoustic problems (1997) The Journal of the Acoustical Society of America, 101 (6), pp. 3336-3342; Li, S.D., Huang, Q.B., An improved form of the hypersingular boundary integral equation for exterior acoustic problems (2010) Engineering Analysis with Boundary Elements, 34 (3), pp. 189-195; Polimeridis, A.G., J{\"a}rvenp{\"a}{\"a}, S., Yl{\"a}-Oijala, P., On the evaluation of hyper-singular double normal deri-vative kernels in surface integral equation methods (2013) Engineering Analysis with Boundary Elements, 37 (2), pp. 205-210; Wang, X.-H., Zheng, X.-S., Qiao, H., Analytical study of hypersinglar integral equations with constant element for 2D Helmholtz problems (2017) Chinese Journal of Computational Physics, 34 (6), pp. 666-672. , 王现辉, 郑兴帅, 乔慧, 等. 二维Helmholtz边界超奇异积分方程解析研究[J]. 计算物理, 2017, 346): 666-672; Tadeu, A., Ant{\'o}nio, J., 3D acoustic wave simulation using BEM formulations: Closed form integration of singular and hypersingular integrals (2012) Engineering Analysis with Boundary Elements, 36 (9), pp. 1389-1396; Wu, H.J., Ye, W.J., Jiang, W.K., A collocation BEM for 3D acoustic problems based on a non-singular Burton-Miller formulation with linear continuous elements (2018) Computer Methods in Applied Mechanics and Engineering, 332, pp. 191-216; Zhao, Z.-G., Huan, Q.-B., Calculation of singular integral for Helmholtz boundary integral equation in acoustics (2004) Chinese Journal of Engineering Mathematics, 21 (5), pp. 779-784. , 赵志高, 黄其柏. Helmholtz声学边界积分方程中奇异积分的计算[J]. 工程数学学报, 2004, 215): 779-784; Skudrzyk, E., The Foundations of Acoustics (1971), Springer-VerlagUR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85074469050\&amp;doi=10.7511\%2fjslx20180910003\&amp;partnerID=40\&amp;md5=e16e58c72688b3ef6aa2317983acf647},
	pages = {576-582},
	publisher = {Editorial Office of Chinese Journal of Computational Mechanics},
	type = {Article},
	abstract = {A new particular solution method is proposed to indirectly calculate the strong singular integrals and free terms in conventional Helmholtz boundary integral equation (CBIE) and hyper-strong singular integrals in Burton-Miller boundary integral equation (BMBIE).For the acoustic problem of interior field,the particular solution satisfying Helmholtz governing equation is given,and the strong singular integral and free terms in CBIE are obtained indirectly.For an exterior field problem,however,calculation of its Cauchy principal value (CPV) for hyper-strong singular integral needs higher-order approximation of the kernel function through Taylor series expansion,which makes numerical implementation quite complex.In this paper,the particular solution satisfying Helmholtz governing equation and Sommerfeld radiation condition is given,and the hyper-strong singular integrals are obtained using a proposed new particular solution method.Also,the CPV of the strongly singular integral for an axisymmetric structure is derived.The high efficiency of the method is demonstrated with axisymmetric examples.The numerical results show that for the interior domain problem,the accuracy obtained by the proposed particular solution method is superior to that of directly calculating the strongly singular integral and the free term coefficient.Furthermore,the particular solution method can avoid calculating the free term with consideration of specific geometric information,and thus is of more general applicability.For an exterior domain problem,both methods provide almost the same accuracy,however,the proposed particular method can avoid expanding the kernel function to higher order and is easier to implement numerically. {\textcopyright} 2019, Editorial Office of Chinese Journal of Computational Mechanics. All right reserved.},
	keywords = {Burton-Miller formulation, Cauchy principal value integral, Hyper singular integral, The characteristic function method},
	isbn = {10074708 (ISSN)},
	author = {Zhou, Q. and Chen, Y. Q.}
}
