**44**, 3, pp. 475–492, 2019

**10.24425/aoa.2019.129263**

### Isogeometric Fast Multipole Boundary Element Method Based on Burton-Miller Formulation for 3D Acoustic Problems

**Keywords**: isogeometric analysis; boundary element method; Burton-Miller method; acoustic scattering; fast multipole method

**Full Text:**PDF

#### References

Bai Y., Dong C. Y., Liu Z.Y. (2015), Effective elastic properties and stress states of doubly periodic array of inclusions with complex shapes by isogeometric boundary element method, Composite Structures, 128, 54–69.

Bazilevs Y., Calo V., Hughes T.J.R., Zhang Y.J. (2008), Isogeometric fluid-structure interaction: Theory, algorithms, and computations, Computational Mechanics, 43, 1, 3–37.

Bordas S., Lian H.J., Simpson R. (2013), Stress analysis without meshing: Isogeometric boundaryelement method, Engineering and Computational Mechanics, 166, 88–99.

Buffa A., Sangalli G., Vazquez R. (2014), Isogeometric methods for computational electromagnetics: Bspline and T-spline discretizations, Journal of Computational Physics, 257, 1291–1320.

Burton A.J., Miller G.F. (1971), The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 323, 1553, 201–210.

Chen L., Zhao W., Liu C., Chen H. (2017a), 2D structural acoustic analysis using the FEM/FMBEM with different coupled element types, Archives of Acoustics, 42, 1, 37–48.

Chen L., Chen H., Zheng C., Marburg S. (2016a), Structural-acoustic sensitivity analysis of radiated sound power using a finite element/ discontinuous fast multipole boundary element scheme: Radiated sound power sensitivity, International Journal for Numerical Methods in Fluids, 82, 12, 858–878.

Chen L., Liu L., Zhao W., Chen H. (2016b), 2D acoustic design sensitivity analysis based on adjoint variable method using different types of boundary elements, Acoustics Australia, 44, 343–357.

Chen L., Marburg S., Chen H., Zhang H., Gao H. (2017b), An adjoint operator approach for sensitivity analysis of radiated sound power in fully coupled structural-acoustic systems, Journal of Computational Acoustics, 25, 01, 1750003.

Chen L., Liu C., Zhao W., Liu L. (2018), An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution, Computer Methods in Applied Mechanics and Engineering, 336, 507–532.

Cho S., Ha S. (2009), Isogeometric shape design optimization: exact geometry and enhanced sensitivity, Structural and Multidisciplinary Optimization, 38, 1, 53–70.

Coifman R.R., Rokhlin V., Wandzura S. (1993), The fast multipole method for the wave equation: a pedestrian prescription, IEEE Antennas and Propagation Magazine, 35, 3, 7–12.

Coox L., Atak O., Vandepitte D., Desmet W. (2017), An isogeometric indirect boundary element method for solving acoustic problems in open-boundary domains, Computer Methods in Applied Mechanics and Engineering, 316, 186–208.

De Luycker E., Benson D.J., Belytschko T., Bazilevs Y., Hsu M. (2011), X-FEM in isogeometric analysis for linear fracture mechanics, International Journal for Numerical Methods in Engineering, 87, 541–565.

do Rˆego Silva J J. (1994), Acoustic and Elastic Wave Scattering Using Boundary Elements, Computational Mechanics Publications, Southampton.

Guiggiani M., Krishnasamy G., Rudolphi T.J., Rizzo F.J. (1992), A general algorithm for the numerical solution of hypersingular boundary integral equations, Journal of Applied Mechanics, 59, 5, 604–614.

Hughes T.J., Cottrell J.A., Bazilevs Y. (2005), Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 3941, 4135–4195.

Kostas K.V., Ginnis A.I., Politis C.G., Kaklis P.D. (2015), Ship-hull shape optimization with a T-spline based BEM-isogeometric solver, Computer Methods in Applied Mechanics and Engineering, 284, 611–622.

Kostas K.V., Ginnis A.I., Politis C.G., Kaklis P.D. (2017), Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver, Computer-Aided Design, 82, 1, 79–87.

Kupradze V.D. (1965), Potential methods in the theory of elasticity, Jerusalem: Israel Programme for Scientific Translation.

Lee S., Yoon M., Cho S. (2017), Isogeometric topological shape optimization using dual evolution with boundary integral equation and level sets, Computer-Aided Design, 82, 88–99.

Li K., Qian X. (2011), Isogeometric analysis and shape optimization via boundary integral, Computer-Aided Design, 43, 11, 1427–1437.

Lian H., Kerfriden P., Bordas S. (2017), Shape optimization directly from cad: an isogeometric boundary element approach using T-splines, Computer Methods in Applied Mechanics and Engineering, 317, 1–41.

Liu Y.J., Nishimura N. (2006), The fast multipole boundary element method for potential problems, Engineering Analysis with Boundary Elements, 30, 5, 371– 381.

Liu C., Chen L., Zhao W., Chen H. (2017), Shape optimization of sound barrier using an isogeometric fast multipole boundary element method in two dimensions, Engineering Analysis with Boundary Elements, 85, 142–157.

Liu Z., Majeed M., Cirak F., Simpson R.N. (2018), Isogeometric FEM-BEM coupled structuralacoustic analysis of shells using subdivision surfaces, International Journal for Numerical Methods in Engineering, 113, 9, 1507–1530.

Manh N.D., Evgrafov A., Gersborg A.R., Gravesen J. (2011), Isogeometric shape optimization of vibrating membranes, Computer Methods in Applied Mechanics and Engineering, 200, 1343–1353.

Marburg S. (2016), The Burton and Miller method: unlocking another mystery of its coupling parameter, Journal of Computational Acoustics, 23, 01, 1550016.

Marburg S., Amini S. (2005), Cat’s eye radiation with boundary elements: comparative study on treatment of irregular frequencies, Journal of Computational Acoustics, 13, 01, 21–45.

Marburg S., Schneider S. (2003), Influence of element types on numeric error for acoustic boundary elements, Journal of Computational Acoustics, 11, 03, 363–386.

Marussig B., Zechner J., Beer G., Fries T. (2003), Fast isogeometric boundary element method based on independent field approximation, Computer Methods in Applied Mechanics and Engineering, 284, 458–488.

Matsumoto T., Tanaka M., Yamada Y. (1995), Design sensitivity analysis of steady-state acoustic problems using boundary integral equation formulation, Jsme International Journal Series C-mechanical Systems Machine Elements and Manufacturing, 38, 1, 9–16.

Merz S., Kessissoglou N., Kinns R., Marburg S. (2010), Minimisation of the sound power radiated by a submarine through optimisation of its resonance changer, Journal of Sound and Vibration, 329, 8, 980– 993.

Nagy A.P., Abdalla M.M., Gurdal Z. (2010), Isogeometric sizing and shape optimisation of beam structures, Computer Methods in Applied Mechanics and Engineering, 199, 1216–1230.

Nguyen B.H., Tran H.D., Anitescu C., Zhuang X., Rabczuk T. (2016), An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems, Computer Methods in Applied Mechanics and Engineering, 306, 252–275.

Peake M.J., Trevelyan J., Coates G. (2013), Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems, Computer Methods in Applied Mechanics and Engineering, 259, 93–102.

Peake M.J., Trevelyan J., Coates G. (2015), Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems, Computer Methods in Applied Mechanics and Engineering, 284, 762–780.

Peng X., Atroshchenko E., Kerfriden P., Bordas S. (2017a), Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth, Computer Methods in Applied Mechanics and Engineering, 316, 151–185.

Peng X., Atroshchenko E., Kerfriden P., Bordas S. (2017b), Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment, International Journal of Fracture, 204, 1, 55–78.

Peters H., Kinns R., Kessissoglou N. (2014), Effects of internal mass distribution and its isolation on the acoustic characteristics of a submerged hull, Journal of Sound and Vibration, 333, 6, 1684–1697.

Rokhlin V. (1990), Rapid solution of integral equations of scattering theory in two dimensions, Journal of Computational Physics, 86, 2, 414–439.

Schenck H.A. (1968), Improved integral formulation for acoustic radiation problems, Journal of Acoustic Society of America, 44, 41–58.

Scott M.A. et al. (2013), Isogeometric boundary element analysis using unstructured T-splines, Computer Methods in Applied Mechanics and Engineering, 254, 197–221.

Simpson R.N., Bordas S., Lian H., Trevelyan J. (2013), An isogeometric boundary element method for elastostatic analysis: 2D implementation aspects, Computers & Structures, 118, 2–12.

Simpson R.N., Liu Z. (2016), Acceleration of isogeometric boundary element analysis through a blackbox fast multipole method, Engineering Analysis with Boundary Elements, 66, 168–182.

Simpson R.N., Scott M.A., Taus M., Thomas D.C., Lian H. (2014), Acoustic isogeometric boundary element analysis, Computer Methods in Applied Mechanics and Engineering, 269, 265–290.

Takahashi T., Matsumoto T. (2012), An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions, Engineering Analysis With Boundary Elements, 36, 12, 1766–1775.

Taus M., Rodin G.J., Hughes T.J. (2016), Isogeometric analysis of boundary integral equations: Highorder collocation methods for the singular and hypersingular equations, Mathematical Models and Methods in Applied Sciences, 26, 08, 1447–1480.

Wall W.A., Frenzel M., Cyron C.J. (2008), Isogeometric structural shape optimization, Computer Methods in Applied Mechanics and Engineering, 197, 2976–2988.

Wolf W., Lele S.K. (2011), Wideband fast multipole boundary element method: Application to acoustic scattering from aerodynamic bodies, International Journal for Numerical Methods in Fluids, 67, 12, 2108–2129.

Zhao W., Chen L., Chen H., Marburg S. (2019), Topology optimization of exterior acoustic-structure interaction systems using the coupled FEM-BEM method, International Journal for Numerical Methods in Engineering, 59, 59, 1–28.

Zheng C., Chen H., Gao H., Du L. (2015), Is the Burton-Miller formulation really free of fictitious eigen frequencies?, Engineering Analysis with Boundary Elements, 59, 59, 43–51.

Zheng C., Matsumoto T., Takahashi T., Chen H. Analysis with Boundary Elements, 36, 3, 361–371.

Zieniuk E. (2003), Bezier curves in the modification of boundary integral equations (BIE) for potential boundary-values problems, International Journal of Solids and Structures, 40, 9, 2301–2320.

Zieniuk E., Szerszen K. (2014), The PIES for solving 3D potential problems with domains bounded by rectangular Bezier patches, Engineering Computations, 31, 4, 791–809.

DOI: 10.24425/aoa.2019.129263

Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN)