# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.