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Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems. (English) Zbl 1425.65202
Summary: A boundary element method (BEM), based on non-uniform rational B-splines (NURBS), is used to find solutions to three-dimensional wave scattering problems governed by the Helmholtz equation. The method is extended in a partition-of-unity sense, multiplying the NURBS functions by families of plane waves; this method is called the eXtended Isogeometric Boundary Element Method (XIBEM).{
}In this paper, the collocation XIBEM formulation is described and numerical results are given. The numerical results are compared against closed-form or converged solutions. Comparisons are made against the conventional boundary element method and the non-enriched isogeometric BEM (IGABEM).{
}When compared to non-enriched boundary element simulations, using XIBEM significantly reduces the number of degrees of freedom required to obtain a solution of a given error; thus, with a fixed computational resource, problems of a shorter wavelength can be solved.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65D17 Computer-aided design (modeling of curves and surfaces)
74S15 Boundary element methods applied to problems in solid mechanics
76Q05 Hydro- and aero-acoustics
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[1] Banaugh, R. P.; Goldsmith, W., Diffraction of steady acoustic waves by surfaces of arbitrary shape, J. Acoust. Soc. Am., 35, 1590-1601, (1963) · Zbl 0134.44704
[2] Copley, L. G., Fundamental results concerning integral representations in acoustic radiation, J. Acoust. Soc. Am., 44, 28-32, (1963) · Zbl 0162.57204
[3] Schenck, H. A., Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Am., 44, 41-58, (1968) · Zbl 0187.50302
[4] Burton, A. J.; Miller, G. F., The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proc. R. Soc. Lond. Ser. A., 323, 201-210, (1971) · Zbl 0235.65080
[5] Bettess, P., Short-wave scattering: problems and techniques, Philos. Trans. R. Soc. Lon. Ser. A Math. Phys. Eng. Sci., 362, 421-443, (2004) · Zbl 1076.78004
[6] Melenk, J. M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289-314, (1996) · Zbl 0881.65099
[7] Bériot, H.; Perrey-Debain, E.; Ben Tahar, M.; Vayssade, C., Plane wave basis in Galerkin BEM for bidimensional wave scattering, Eng. Anal. Bound. Elem., 34, 130-143, (2010) · Zbl 1244.76032
[8] Perrey-Debain, E.; Laghrouche, O.; Bettess, P.; Trevelyan, J., Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362, 561-577, (2004) · Zbl 1073.78016
[9] Massimi, P.; Tezaur, R.; Farhat, C., A discontinuous enrichment method for three-dimensional multiscale harmonic wave propagation problems in multi-fluid and fluid-solid media, Internat. J. Numer. Methods Engrg., 76, 400-425, (2008) · Zbl 1195.74292
[10] Kovalevsky, L.; Ladevèze, P.; Riou, H.; Bonnet, M., The variational theory of complex rays for three-dimensional Helmholtz problems, J. Comput. Acoust., 20, 1250021, (2012) · Zbl 1360.76261
[11] Luostari, T.; Huttunen, T.; Monk, P., Error estimates for the ultra weak variational formulation in linear elasticity, ESAIM Math. Model. Numer. Anal., 47, 183-211, (2013) · Zbl 1457.74025
[12] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[13] Cabral, J. J.S. P.; Wrobel, L. C.; Brebbia, C. A., A BEM formulation using B-splines: I-uniform blending functions, Eng. Anal. Bound. Elem., 7, 136-144, (1990)
[14] Cabral, J. J.S. P.; Wrobel, L. C.; Brebbia, C. A., A BEM formulation using B-splines: II-multiple knots and non-uniform blending functions, Eng. Anal. Bound. Elem., 8, 51-55, (1991)
[15] Politis, C.; Ginnis, A. I.; Kaklis, P. D.; Belibassakis, K.; Feurer, C., An isogeometric BEM for exterior potential-flow problems in the plane, (2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, SPM ’09, (2009), ACM New York, NY, USA), 349-354
[16] Kang, L.; Qian, X., Isogeometric analysis and shape optimization via boundary integral, Comput. Aided Design, 43, 1427-1437, (2011)
[17] Simpson, R. N.; Bordas, S. P.A.; Trevelyan, J.; Rabczuk, T., A two-dimensional isogeometric boundary element method for elastostatic analysis, Comput. Methods Appl. Mech. Engrg., 209-212, 87-100, (2012) · Zbl 1243.74193
[18] Simpson, R. N.; Bordas, S. P.A.; Lian, H.; Trevelyan, J., An isogeometric boundary element method for elastostatic analysis: 2D implementation aspects, Comput. Struct., 118, 2-12, (2013)
[19] Takahashi, T.; Matsumoto, T., An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions, Eng. Anal. Bound. Elem., 36, 1766-1775, (2012) · Zbl 1351.74138
[20] Scott, M. A.; Simpson, R. N.; Evans, J. A.; Lipton, S.; Bordas, S. P.A.; Hughes, T. J.R.; Sederberg, T. W., Isogeometric boundary element analysis using unstructured T-splines, Comput. Methods Appl. Mech. Engrg., 254, 197-221, (2013) · Zbl 1297.74156
[21] Belibassakisa, K. A.; Gerostathisb, K. V.; Kostasb, Th P.; Politis, C. G.; Kaklisa, P. D.; Ginnisa, A. I.; Feurera, C., A BEM-isogeometric method for the ship wave-resistance problem, Ocean Eng., 60, 53-67, (2013)
[22] Heltai, L.; Arroyo, M.; DeSimone, A., Nonsingular isogeometric boundary element method for Stokes flows in 3D, Comput. Methods Appl. Mech. Engrg., 268, 514-539, (2014) · Zbl 1295.76022
[23] Simpson, R. N.; Scott, M. A.; Taus, M.; Thomas, D. C.; Lian, H., Acoustic isogeometric boundary element analysis, Comput. Methods Appl. Mech. Engrg., (2013) · Zbl 1296.65175
[24] Peake, M. J.; Trevelyan, J.; Coates, G., Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems, Comput. Methods Appl. Mech. Engrg., 259, 93-102, (2013) · Zbl 1286.65176
[25] Wrobel, L. C., The boundary element method, Vol. I: Applications in Thermo-fluids and Acoustics, (2002), John Wiley & Sons, Ltd. Chichester, UK
[26] Liu, Y., Fast multipole boundary element method: theory and applications in engineering, (2009), Cambridge University Press
[27] Rêgo Silva, J. J.; Wrobel, L. C.; Telles, J. C.F., A new family of continuous/discontinuous three-dimensional boundary elements with application to acoustic wave propagation, Internat. J. Numer. Methods Engrg., 36, 1661-1679, (1993) · Zbl 0772.76041
[28] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J. A.; Hughes, T. J.R.; Lipton, S.; Scott, M. A.; Sederberg, T. W., Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Engrg., 199, 229-263, (2010) · Zbl 1227.74123
[29] Piegl, L.; Tiller, W., The NURBS book, (1997), Springer-Verlag · Zbl 0868.68106
[30] Borden, M. J.; Scott, M. A.; Evans, J. A.; Hughes, T. J.R., Isogeometric finite element data structures based on Bézier extraction of NURBS, Internat. J. Numer. Methods Engrg., 87, 15-47, (2011) · Zbl 1242.74097
[31] H. Bériot, E. Perrey-Debain, M. Ben Tahar, C. Vayssade, On a Galerkin wave boundary element formulation for scattering by non-smooth obstacles, in: Proceedings of 8th International Conference on Mathematical and Numerical Aspects of Waves, pp. 400-402. · Zbl 1244.76032
[32] Peake, M. J.; Trevelyan, J.; Coates, G., The equal spacing of \(n\) points on a sphere with application to partition-of-unity wave diffraction problems, Eng. Anal. Bound. Elem., 40, 114-122, (2014) · Zbl 1297.76146
[33] Peake, M. J.; Trevelyan, J.; Coates, G., Novel basis functions for the partition of unity boundary element method for Helmholtz problems, Internat. J. Numer. Methods Engrg., 93, 905-918, (2013) · Zbl 1352.65593
[34] Diwan, G. C.; Trevelyan, J.; Coates, G., A comparison of techniques for overcoming non-uniqueness of boundary integral equations for the collocation partition of unity method in two-dimensional acoustic scattering, Internat. J. Numer. Methods Engrg., 96, 645-664, (2013) · Zbl 1352.76105
[35] Morse, P. M.; Feshbach, H., Methods of theoretical physics: part II, (1953), McGraw-Hill · Zbl 0051.40603
[36] Perrey-Debain, E.; Trevelyan, J.; Bettess, P., Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications, J. Sound Vib., 261, 839-858, (2003) · Zbl 1237.74107
[37] Kondapalli, P. S.; Shippy, D. J.; Fairweather, G., Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions, J. Acoust. Soc. Am., 91, 1844-1854, (1992) · Zbl 0722.76073
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