Fast Multipole Burton-Miller Boundary Element Method for Two and Three-Dimensional Acoustic Scattering

Wolf, William R.; Lele, Sanjiva K.

doi:10.5028/jatm.2012.04021812

Abstract:

A multistage adaptive fast multipole method is used to accelerate the matrix-vector products arising from the Burton-Miller boundary integral equations, which are formed in a boundary element method. The present study considers the scattering of acoustic waves, generated by localized sources from bodies with rigid surfaces. Details on the implementation of a multistage adaptive fast multipole method are described for two and three-dimensional formulations. The code is verified through the solution of well documented test cases. The fast multipole method is tested for acoustic scattering problems of single and multiple bodies, and a discussion is provided on the performance of the method. Results for engineering problems with complex geometries, such as a multi-element wing, are presented in order to assess the implemented capability.

Keywords:
Fast Multipole Method; Boundary Element Method; Burton-Miller Formulation; Acoustic Scattering

Full text available only in PDF format.

REFERENCES

Brandão, M. P., 1987, "Improper Integrais in Theoretical Aerodynamics: The Problem Revisited," AIAA Journal, Vol. 25, pp. 1258-1260.
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 of London, Series A, Vol. 323, pp. 201-210.
Carrier, J. et al, 1988, "A Fast Adaptive Multipole Algorithm for Particle Simulations," SIAM Journal on Scientific and Statistical Computing, Vol. 9, No. 4, pp. 669-686.
Chen, J. T., Chen, K. H., 2003, "Applications of the Dual Integral Formulation in Conjunction with Fast Multipole Method in Large-Scale Problems for 2D Exterior Acoustics," Engineering Analysis with Boundary Elements, Vol. 28, pp. 685-709.
Cheng, H., et al, 1999, "A Fast Adaptive Multipole Algorithm in Three Dimensions," Journal of Computational Physics, Vol. 155, pp. 468-498.
Cheng, H., et al, 2006, "An Adaptive Fast Solver for the Modified Helmholtz Equation in Two Dimensions," Journal of Computational Physics, Vol. 211, pp. 616-637.
Chien, C. C., et al, 1990, "An Effective Method for Solving the Hypersingular Integral Equations in 3-D Acoustics," Journal of Acoustic Society of America, Vol. 88, pp. 918-937.
Darve, E., 2000, "The Fast Multipole Method I: Error Analysis and Asymptotic Complexity," SIAM Journal on Numerical Analysis, Vol. 38, No. 1, pp. 98-128.
Darve, E., Have P., 2004, "Efficient Fast Multipole Method for Low-Frequency Scattering," Journal of Computational Physics, Vol. 197, pp. 341-363.
Delnevo, A., et al, 2005, "Numerical Methods: Fast Multipole Method for Shielding Effects," Proceedings of the 11th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2005-2971, pp. 1-9.
Dunn, M. V., Tinetti, A. F., 2004, "Aeroacoustic Scattering Via the Equivalent Source Method," Proceedings of the 10th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2004-2937, pp. 1-11.
Epton, M. A., Dembart, B., 1995, "Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations," SIAM Journal on Scientific Computing, Vol. 16, No. 4, pp. 865-897.
Geng, P., et al, 1996, "Massive Parallel Computation for Acoustical Scattering Problems Using Boundary Element Methods," Journal of Sound and Vibration, Vol. 191, pp. 145-165.
Greengard, L., Rokhlin, V., 1987, "A Fast Algorithm for Particle Simulations," Journal of Computational Physics, Vol. 73, pp. 325-348.
Gumerov, N. A., Duraiswami, R., 2003, "Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation," SIAM Journal on Scientific Computing, Vol. 25, No. 4, pp. 1344-1381.
Jones, D. S., 1974, "Integral Equations for the Exterior Acoustic Problem," The Quartely Journal of Mechanics and Applied Mathematics, Vol. 27, pp. 129-142.
Labreuche, C., 1998, "A Convergence Theorem for the Fast Multipole Method for 2 Dimensional Scattering Problems," Mathematics of Computation, Vol. 67, No. 222, pp. 553-591.
Li, S., Huang, Q., 2011, "A Fast Multipole Boundary Element Method Based on the Improved Burton-Miller Formulation for Three-Dimensional Acoustic Problems," Engineering Analysis with Boundary Elements, Vol. 35, pp. 719-728.
Liu, Y. J., Rizzo, F. J., 1992, "A Weakly Singular Form of the Hypersingular Boundary Integral Equation Applied to 3D Acoustic Wave Problems," Computer Methods in Applied Mechanics and Engineering, Vol. 96, pp. 271-287.
Messiah, A., 1981, "Quantum Mechanics," Vol. 2, North Holland.
Moir, I. R. M., 1994, "Measurements on a Two-Dimensional Aerofoil with High-Lift Devices," A Selection of Experimental Test Cases for the Validation of CFD Codes, AGARD Report No. 303, Vol. 1 and 2.
Morris, R J., 1995a, "The Scattering of Sound from a Spatially Distributed Axisymmetric Cylindrical Source by a Circular Cylinder," Journal of Acoustic Society of America, Vol. 97, pp. 2651-2656.
Morris, P. J., 1995b, "Scattering of Sound from a Spatially Distributed, Spherically Symmetric Source by a Sphere," Journal of Acoustic Society of America, Vol. 98, pp. 3536-3539.
Nishimura, N., 2002, "Fast Multipole Accelerated Boundary Integral Equation Methods," Applied Mechanics Reviews, Vol. 55, No. 4, pp. 299-324.
Piaszczyk, C. M., Klosner, J. M., 1984, "Acoustic Radiation from Vibrating Surface at Characteristic Frequencies," Journal of Acoustic Society of America, Vol. 75, pp. 363-375.
Rokhlin, V., 1990, "Rapid Solution of Integral Equations of Scattering Theory in Two Dimensions," Journal of Computational Physics, Vol. 86, pp. 414-439.
Rokhlin, V., 1983, "Rapid Solution of Integral Equations of Classical Potential Theory," Journal of Computational Physics, Vol. 60, pp. 187-207.
Sakuma, T., Yasuda, Y., 2002, "Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part I: Setup and Validation," Acta Acústica - Acústica, Vol. 88, pp. 513-525.
Schenck, H. A., 1968, "Improved Integral Formulation for Acoustic Radiation Problems," Journal of Acoustic Society of America, Vol. 44, pp. 41-58.
Seybert, A. F., Rengarajan, T. K., 1987, "The Use of CHIEF to Obtain Unique Solutions for Acoustic Radiation Using Boundary Integral Equations," Journal of Acoustic Society of America, Vol. 81, pp. 1299-1306.
Shen, L., Liu, Y. J., 2007, "An Adaptive Fast Multipole Boundary Element Method for Three-Dimensional Acoustic Wave Problems Based on the Burton-Miller Formulation," Computational Mechanics, Vol. 40, pp. 461-472.
Sonneveld, P., 1989, "CGS, a Fast Lanczos-type Solver for Nonsymmetric Linear Systems," SIAM Journal on Scientific and Statistical Computing, Vol. 10, pp. 36-52.
Tinetti, A. F., Dunn, M. V, 2005, "Aeroacoustic Noise Piediction Using the Fast Scattering Code," Proceedings of the 11th AIAA/ CEAS Aeroacoustics Conference, AIAA Paper 2005-3061, pp. 1-12.
Tinetti, A. F., et al, 2007, "Curved Duct Noise Prediction Using the Fast Scattering Code," Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2007-3528, pp. 1-15.
Wolf, W. R., Lele, S. K., 2010, "Fast Acoustic Scattering Simulations with Non-Uniform Potential Flow Effects," Proceedings of the 16th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2010-3712, pp. 1-12.
Wolf, W. R., Lele, S. K., 2009, "Assessment of Fast Multipole Method Formulations for Acoustic Scattering," Proceedings of the 15th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2009-3229, pp. 1-19.
Wu, H., et al, 2011, "Diagonal Form Fast Multipole Boundary Element Method for 2D Acoustic Problems Based on Burton-Miller Boundary Integral Equation Formulation and its Applications," Applied Mathematics and Mechanics, Vol. 32, pp. 981-996.
Wu, H., et al, 2012, "A Fast Multipole Boundary Element Method for 3D Multi-Domain Acoustic Scattering Problems Based on the Burton-Miller Formulation," Engineering Analysis with Boundary Elements, Vol. 36, pp. 779-788.
Yoshida, K, 2001, "Applications of Fast Multipole Method to Boundary Integral Equation Method," PhD. Thesis, Department of Global Environment Engineering, Kyoto University, Japan.
Yasuda, Y., Sakuma, T., 2002, "Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part II: Examination of Numerical Items," Acta Acústica-Acústica, Vol. 88, pp. 513-525.

Publication Dates

Publication in this collection
Apr-Jun 2012

History

Received
26 Mar 2012
Accepted
26 Apr 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[1] Brandão, M. P., 1987, "Improper Integrais in Theoretical Aerodynamics: The Problem Revisited," AIAA Journal, Vol. 25, pp. 1258-1260.

[2] 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 of London, Series A, Vol. 323, pp. 201-210.

[3] Carrier, J. et al, 1988, "A Fast Adaptive Multipole Algorithm for Particle Simulations," SIAM Journal on Scientific and Statistical Computing, Vol. 9, No. 4, pp. 669-686.

[4] Chen, J. T., Chen, K. H., 2003, "Applications of the Dual Integral Formulation in Conjunction with Fast Multipole Method in Large-Scale Problems for 2D Exterior Acoustics," Engineering Analysis with Boundary Elements, Vol. 28, pp. 685-709.

[5] Cheng, H., et al, 1999, "A Fast Adaptive Multipole Algorithm in Three Dimensions," Journal of Computational Physics, Vol. 155, pp. 468-498.

[6] Cheng, H., et al, 2006, "An Adaptive Fast Solver for the Modified Helmholtz Equation in Two Dimensions," Journal of Computational Physics, Vol. 211, pp. 616-637.

[7] Chien, C. C., et al, 1990, "An Effective Method for Solving the Hypersingular Integral Equations in 3-D Acoustics," Journal of Acoustic Society of America, Vol. 88, pp. 918-937.

[8] Darve, E., 2000, "The Fast Multipole Method I: Error Analysis and Asymptotic Complexity," SIAM Journal on Numerical Analysis, Vol. 38, No. 1, pp. 98-128.

[9] Darve, E., Have P., 2004, "Efficient Fast Multipole Method for Low-Frequency Scattering," Journal of Computational Physics, Vol. 197, pp. 341-363.

[10] Delnevo, A., et al, 2005, "Numerical Methods: Fast Multipole Method for Shielding Effects," Proceedings of the 11th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2005-2971, pp. 1-9.

[11] Dunn, M. V., Tinetti, A. F., 2004, "Aeroacoustic Scattering Via the Equivalent Source Method," Proceedings of the 10th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2004-2937, pp. 1-11.

[12] Epton, M. A., Dembart, B., 1995, "Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations," SIAM Journal on Scientific Computing, Vol. 16, No. 4, pp. 865-897.

[13] Geng, P., et al, 1996, "Massive Parallel Computation for Acoustical Scattering Problems Using Boundary Element Methods," Journal of Sound and Vibration, Vol. 191, pp. 145-165.

[14] Greengard, L., Rokhlin, V., 1987, "A Fast Algorithm for Particle Simulations," Journal of Computational Physics, Vol. 73, pp. 325-348.

[15] Gumerov, N. A., Duraiswami, R., 2003, "Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation," SIAM Journal on Scientific Computing, Vol. 25, No. 4, pp. 1344-1381.

[16] Jones, D. S., 1974, "Integral Equations for the Exterior Acoustic Problem," The Quartely Journal of Mechanics and Applied Mathematics, Vol. 27, pp. 129-142.

[17] Labreuche, C., 1998, "A Convergence Theorem for the Fast Multipole Method for 2 Dimensional Scattering Problems," Mathematics of Computation, Vol. 67, No. 222, pp. 553-591.

[18] Li, S., Huang, Q., 2011, "A Fast Multipole Boundary Element Method Based on the Improved Burton-Miller Formulation for Three-Dimensional Acoustic Problems," Engineering Analysis with Boundary Elements, Vol. 35, pp. 719-728.

[19] Liu, Y. J., Rizzo, F. J., 1992, "A Weakly Singular Form of the Hypersingular Boundary Integral Equation Applied to 3D Acoustic Wave Problems," Computer Methods in Applied Mechanics and Engineering, Vol. 96, pp. 271-287.

[20] Messiah, A., 1981, "Quantum Mechanics," Vol. 2, North Holland.

[21] Moir, I. R. M., 1994, "Measurements on a Two-Dimensional Aerofoil with High-Lift Devices," A Selection of Experimental Test Cases for the Validation of CFD Codes, AGARD Report No. 303, Vol. 1 and 2.

[22] Morris, R J., 1995a, "The Scattering of Sound from a Spatially Distributed Axisymmetric Cylindrical Source by a Circular Cylinder," Journal of Acoustic Society of America, Vol. 97, pp. 2651-2656.

[23] Morris, P. J., 1995b, "Scattering of Sound from a Spatially Distributed, Spherically Symmetric Source by a Sphere," Journal of Acoustic Society of America, Vol. 98, pp. 3536-3539.

[24] Nishimura, N., 2002, "Fast Multipole Accelerated Boundary Integral Equation Methods," Applied Mechanics Reviews, Vol. 55, No. 4, pp. 299-324.

[25] Piaszczyk, C. M., Klosner, J. M., 1984, "Acoustic Radiation from Vibrating Surface at Characteristic Frequencies," Journal of Acoustic Society of America, Vol. 75, pp. 363-375.

[26] Rokhlin, V., 1990, "Rapid Solution of Integral Equations of Scattering Theory in Two Dimensions," Journal of Computational Physics, Vol. 86, pp. 414-439.

[27] Rokhlin, V., 1983, "Rapid Solution of Integral Equations of Classical Potential Theory," Journal of Computational Physics, Vol. 60, pp. 187-207.

[28] Sakuma, T., Yasuda, Y., 2002, "Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part I: Setup and Validation," Acta Acústica - Acústica, Vol. 88, pp. 513-525.

[29] Schenck, H. A., 1968, "Improved Integral Formulation for Acoustic Radiation Problems," Journal of Acoustic Society of America, Vol. 44, pp. 41-58.

[30] Seybert, A. F., Rengarajan, T. K., 1987, "The Use of CHIEF to Obtain Unique Solutions for Acoustic Radiation Using Boundary Integral Equations," Journal of Acoustic Society of America, Vol. 81, pp. 1299-1306.

[31] Shen, L., Liu, Y. J., 2007, "An Adaptive Fast Multipole Boundary Element Method for Three-Dimensional Acoustic Wave Problems Based on the Burton-Miller Formulation," Computational Mechanics, Vol. 40, pp. 461-472.

[32] Sonneveld, P., 1989, "CGS, a Fast Lanczos-type Solver for Nonsymmetric Linear Systems," SIAM Journal on Scientific and Statistical Computing, Vol. 10, pp. 36-52.

[33] Tinetti, A. F., Dunn, M. V, 2005, "Aeroacoustic Noise Piediction Using the Fast Scattering Code," Proceedings of the 11th AIAA/ CEAS Aeroacoustics Conference, AIAA Paper 2005-3061, pp. 1-12.

[34] Tinetti, A. F., et al, 2007, "Curved Duct Noise Prediction Using the Fast Scattering Code," Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2007-3528, pp. 1-15.

[35] Wolf, W. R., Lele, S. K., 2010, "Fast Acoustic Scattering Simulations with Non-Uniform Potential Flow Effects," Proceedings of the 16th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2010-3712, pp. 1-12.

[36] Wolf, W. R., Lele, S. K., 2009, "Assessment of Fast Multipole Method Formulations for Acoustic Scattering," Proceedings of the 15th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2009-3229, pp. 1-19.

[37] Wu, H., et al, 2011, "Diagonal Form Fast Multipole Boundary Element Method for 2D Acoustic Problems Based on Burton-Miller Boundary Integral Equation Formulation and its Applications," Applied Mathematics and Mechanics, Vol. 32, pp. 981-996.

[38] Wu, H., et al, 2012, "A Fast Multipole Boundary Element Method for 3D Multi-Domain Acoustic Scattering Problems Based on the Burton-Miller Formulation," Engineering Analysis with Boundary Elements, Vol. 36, pp. 779-788.

[39] Yoshida, K, 2001, "Applications of Fast Multipole Method to Boundary Integral Equation Method," PhD. Thesis, Department of Global Environment Engineering, Kyoto University, Japan.

[40] Yasuda, Y., Sakuma, T., 2002, "Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part II: Examination of Numerical Items," Acta Acústica-Acústica, Vol. 88, pp. 513-525.

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