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Full numerical quadrature of weakly singular double surface integrals in Galerkin boundary element methods. (English) Zbl 1211.65030

Summary: When a Galerkin discretization of a boundary integral equation with a weakly singular kernel is performed over triangles, a double surface integral must be evaluated for each pair of them. If these pairs are not contiguous or not coincident, the kernel is regular and a Gauss-Legendre quadrature can be employed. When the pairs have a common edge or a common vertex, then edge and vertex weak singularities appear. If the pairs have both facets coincident, the whole integration domain is weakly singular. D. J. Taylor [IEEE Trans. Antennas Propag. 51, No. 7, 1630–1637 (2001)] proposed a systematic evaluation based on a reordering and partitioning of the integration domain, together with a use of Duffy transformations in order to remove the singularities, in such a way that a Gauss-Legendre quadrature was performed on three coordinates with an analytic integration in the fourth coordinate. Since this scheme is a bit restrictive because it was designed for electromagnetic kernels, a full numerical quadrature is proposed in order to handle kernels with a weak singularity with a general framework. Numerical tests based on modifications of the one proposed by W. Wang and N. Atalla [Commun. Numer. Methods Eng. 13, No. 11, 885–890 (1997; Zbl 0899.65010)] are included.

MSC:

65D32 Numerical quadrature and cubature formulas

Citations:

Zbl 0899.65010

Software:

SERBA; mpi4py
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] París, Boundary Element Method. Fundamentals and Applications (1997)
[2] Hartmann, Introduction to Boundary Elements (1989) · Zbl 0693.73054 · doi:10.1007/978-3-642-48873-3
[3] Power, Boundary Integral Methods in Fluid Mechanics (1995) · Zbl 0815.76001
[4] Kim, Integral equations of second kind for Stokes flow: direct solutions for physical variables and removal of inherent accuracy limitations, Chemical Engineering Communications 82 pp 123– (1989) · doi:10.1080/00986448908940638
[5] Fachinotti, Mecánica Computacional XXIV pp 1104– (2007)
[6] DE, Three-dimensional boundary element computation of potential flow in fractured rock, International Journal for Numerical Methods in Engineering 26 pp 2319– (1988) · Zbl 0664.76126 · doi:10.1002/nme.1620261013
[7] D’Elía, A closed form for low order panel methods, Advances in Engineering Software 31 (5) pp 335– (2000) · Zbl 02178075 · doi:10.1016/S0965-9978(99)00060-5
[8] D’Elía, Iterative solution of panel discretizations for potential flows. The modal/multipolar preconditioning, International Journal for Numerical Methods in Fluids 32 (1) pp 1– (2000) · Zbl 0958.76069 · doi:10.1002/(SICI)1097-0363(20000115)32:1<1::AID-FLD869>3.0.CO;2-8
[9] Paquay S Développement d’une méthodologie de simulation numérique pour les problèmes vibro-acoustiques couplés intérieurs/extérieurs de grandes taille 2002
[10] Wang X Fast Stokes: a fast 3-D fluid simulation program for micro-electro-mechanical systems 2002
[11] Méndez, Effect of geometrical nonlinearity on MEMS thermoelastic damping, Nonlinear Analysis: Real World Applications 10 (3) pp 1579– (2008) · Zbl 1160.74016 · doi:10.1016/j.nonrwa.2008.02.002
[12] Schuhmacher A Sound source reconstruction using inverse sound field calculations 2000
[13] D’Elía, A panel-Fourier method for free surface methods, Journal of Fluids Engineering 122 (2) pp 309– (2000) · doi:10.1115/1.483259
[14] D’Elía, Applied hydrodynamic wave-resistance computation by Fourier transform, Ocean Engineering 29 pp 261– (2002) · doi:10.1016/S0029-8018(00)00073-1
[15] D’Elía, A nonlinear panel method in the time domain for seakeeping flow problems, International Journal of Computational Fluid Dynamics 16 (4) pp 263– (2002) · Zbl 1026.76038 · doi:10.1080/1061856021000025148
[16] 2008 http://www.cimec.org.ar/petscfem
[17] Dalcín, MPI for python: performance improvements and MPI-2 extensions, Journal of Parallel and Distributed Computing 68 (5) pp 655– (2007) · Zbl 06059774 · doi:10.1016/j.jpdc.2007.09.005
[18] Franck, Numerical simulation of the Ahmed vehicle model near-wake, Latin American Applied Research (2009)
[19] Storti, Algebraic Discrete Non-Local (DNL) absorbing boundary condition for the ship wave resistance problem, Journal of Computational Physics 146 (2) pp 570– (1998) · Zbl 0929.76023 · doi:10.1006/jcph.1998.6069
[20] Storti, Dynamic boundary conditions in computational fluid dynamics, Computer Methods in Applied Mechanics and Engineering 197 (13-16) pp 1219– (2008) · Zbl 1159.76354 · doi:10.1016/j.cma.2007.10.014
[21] Storti, Computing ship wave resistance from wave amplitude with the DNL absorbing boundary condition, Computer Methods in Applied Mechanics and Engineering 14 pp 997– (1998) · Zbl 0915.76055
[22] Battaglia, Numerical simulation of transient free surface flows, Journal of Applied Mechanics 73 (6) pp 1017– (2006) · Zbl 1111.74324 · doi:10.1115/1.2198246
[23] Garibaldi, Numerical simulations of the flow around a spinning projectile in subsonic regime, Latin American Applied Research 38 (3) pp 241– (2008)
[24] D’Elía, Numerical simulations of axisymmetric inertial waves in a rotating sphere by finite elements, International Journal of Computational Fluid Dynamics 20 (10) pp 673– (2006) · Zbl 1110.76030 · doi:10.1080/10618560601088301
[25] Storti, Added mass of an oscillating hemisphere at very-low and very-high frequencies, Journal of Fluids Engineering 126 (6) pp 1048– (2004) · doi:10.1115/1.1839932
[26] Eibert, On the calculation of potential integrals for linear source distributions on triangular domains, IEEE Transactions on Antennas and Propagation 43 (12) pp 1499– (1995) · doi:10.1109/8.475946
[27] Sievers, Correction to ’On the calculation of potential integrals for linear source distributions on triangular domains’, IEEE Transactions on Antennas and Propagation 53 (9) pp 3113– (2005) · doi:10.1109/TAP.2005.854549
[28] Burghignoli, Improved quadrature formulas for boundary integral equations with conducting or dielectric edge singularities, IEEE Transactions on Antennas and Propagation 52 (2) pp 373– (2004) · Zbl 1368.78153 · doi:10.1109/TAP.2004.824001
[29] Taylor, Accurate efficient numerical integration of weakly singulars integrals in Galerkin IFIE solutions, IEEE Transactions on Antennas and Propagation 51 (7) pp 1630– (2003) · Zbl 1368.78148 · doi:10.1109/TAP.2003.813623
[30] Taylor, Errata to ’Accurate and efficient numerical integration of weakly singulars integrals in Galerkin IFIE solutions’, IEEE Transactions on Antennas and Propagation 51 (9) pp 2543– (2003) · Zbl 1368.78149 · doi:10.1109/TAP.2003.817014
[31] Duffy, Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM Journal on Numerical Analysis 19 (6) pp 1260– (1982) · Zbl 0493.65011 · doi:10.1137/0719090
[32] Wang, A numerical algorithm for double surface integrals over quadrilaterals with a 1/r singularity, Communications in Numerical Methods in Engineering 13 (11) pp 885– (1997) · Zbl 0899.65010 · doi:10.1002/(SICI)1099-0887(199711)13:11<885::AID-CNM112>3.0.CO;2-D
[33] Berry, A new formulation for the vibrations and sound radiation of fluid-loaded plates with elastic boundary conditions, Journal of the Acoustical Society of America 96 (2) pp 889– (1994) · doi:10.1121/1.410264
[34] Quarteroni, Numerical Mathematics (2000)
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