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A new diagonal form fast multipole boundary element method for solving acoustic Helmholtz equation

Li Shan-De Huang Qi-Bai Li Tian-Yun

A new diagonal form fast multipole boundary element method for solving acoustic Helmholtz equation

Li Shan-De, Huang Qi-Bai, Li Tian-Yun
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  • The conventional exterior acoustic Helmholtz boundary integral equation is prohibitively expensive for solving large-scale engineering problems. In order to effectively overcome this problem, the fast multipole method is introduced to the BIE, and accelerates the iterative solution for the system matrix equation. Due to using the diagonal form multipole expansions of the fundamental solution in the BIE, the computational efficiency of the new fast multipole boundary element method (FMBEM) is improved significantly compared to the conventional BEM. Both the computational complexity and memory requirement of the FMBEM are drastically reduced to O(N), where N is the number of degrees of freedom (DOFs). Numerical examples including a large submarine model with more than 420000 DOFs demonstrate the accuracy and efficiency of the FMBEM, and clearly show the advantage of the new algorithm for solving the large-scale acoustic problems. The developed FMBEM would be potential for engineering applications.
      Corresponding author: Li Shan-De, lishande@gmail.com ; Huang Qi-Bai, lishande@gmail.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No.51175195).
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    Li X G,Dai B D,Wang L H 2010 Chin.Phys.B 19 120202

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    Yoshida K,Nishimura N,Kobayashi S 2001 Eng.Anal.Bound.Elem.25 239

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    Liu Y J 2006 Int.J.Numer.Methods Eng.65 863

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    Tong M S,Chew W C 2009 J.Comput.Phys.228 921

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    Nishimura N 2002 Appl.Mech.Rev.55 299

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    Rokhlin V 1993 Appl.Comput.Harmon.Anal.1 82

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    Epton M,Dembart B 1995 SIAM J.Sci.Comput.16 865

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    Sakuma T,Yasuda Y 2002 Acust.Acta Acust.89 28

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    Bapat M S,Shen L,Liu Y J 2009 Eng.Anal.Bound.Elem.33 1113

    [30]

    Kropinski M C,Quaife B D 2011 J.Comput.Phys.230 425

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    Li S D,Huang Q B 2011 Comput.Meth.Appl.Mech.Eng.200 1333

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    Li S D,Huang Q B 2010 Eng.Anal.Bound.Elem.34 89

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    Burton A J,Miller G F 1971 Proc.R.Soc.London.A 323 201

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    Abramowita M,Stegun I A 1964 Handbook of Mathematical Function (New York:Dover)

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    Jakob C R,Alpert B K 1997 J.Comput.Phys.136 580

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  • [1]

    Ciskowski R D,Brebbia C A 1991 Boundary Element Methods in Acoustics (Southampton:Elsevier)

    [2]

    Mao Y J,Qi D T 2009 Acta Phys.Sin.58 6764 (in Chinese) [毛义军,祁大同 2009 物理学报 58 6764]

    [3]
    [4]

    Li X G,Dai B D,Wang L H 2010 Chin.Phys.B 19 120202

    [5]
    [6]
    [7]

    Zhang H Y,Yu J B 2011 Chin.Phys.B 20 094301

    [8]

    Saad Y,Schultz M H 1986 SIAM J.Sci.Statist.Comput.7 856

    [9]
    [10]

    Rokhlin V 1985 J.Comput.Phys.60 187

    [11]
    [12]
    [13]

    Greengard L,Rokhlin V 1987 J.Comput.Phys.73 325

    [14]

    Yoshida K,Nishimura N,Kobayashi S 2001 Eng.Anal.Bound.Elem.25 239

    [15]
    [16]
    [17]

    Liu Y J 2006 Int.J.Numer.Methods Eng.65 863

    [18]
    [19]

    Tong M S,Chew W C 2009 J.Comput.Phys.228 921

    [20]

    Nishimura N 2002 Appl.Mech.Rev.55 299

    [21]
    [22]

    Rokhlin V 1993 Appl.Comput.Harmon.Anal.1 82

    [23]
    [24]
    [25]

    Epton M,Dembart B 1995 SIAM J.Sci.Comput.16 865

    [26]
    [27]

    Sakuma T,Yasuda Y 2002 Acust.Acta Acust.89 28

    [28]
    [29]

    Bapat M S,Shen L,Liu Y J 2009 Eng.Anal.Bound.Elem.33 1113

    [30]

    Kropinski M C,Quaife B D 2011 J.Comput.Phys.230 425

    [31]
    [32]

    Li S D,Huang Q B 2011 Comput.Meth.Appl.Mech.Eng.200 1333

    [33]
    [34]

    Li S D,Huang Q B 2010 Eng.Anal.Bound.Elem.34 89

    [35]
    [36]

    Burton A J,Miller G F 1971 Proc.R.Soc.London.A 323 201

    [37]
    [38]
    [39]

    Abramowita M,Stegun I A 1964 Handbook of Mathematical Function (New York:Dover)

    [40]
    [41]

    Jakob C R,Alpert B K 1997 J.Comput.Phys.136 580

    [42]
    [43]

    Chen K,Harris P J 2001 Appl.Numer.Math.36 475

    [44]

    Chien C C,Rajiyah H,Atluri S N 1990 J.Acoust.Soc.Am.88 918 064301-7

    [45]
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  • Received Date:  27 September 2011
  • Accepted Date:  04 November 2011
  • Published Online:  20 March 2012

A new diagonal form fast multipole boundary element method for solving acoustic Helmholtz equation

    Corresponding author: Li Shan-De, lishande@gmail.com
    Corresponding author: Huang Qi-Bai, lishande@gmail.com
  • 1. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China;
  • 2. Department of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No.51175195).

Abstract: The conventional exterior acoustic Helmholtz boundary integral equation is prohibitively expensive for solving large-scale engineering problems. In order to effectively overcome this problem, the fast multipole method is introduced to the BIE, and accelerates the iterative solution for the system matrix equation. Due to using the diagonal form multipole expansions of the fundamental solution in the BIE, the computational efficiency of the new fast multipole boundary element method (FMBEM) is improved significantly compared to the conventional BEM. Both the computational complexity and memory requirement of the FMBEM are drastically reduced to O(N), where N is the number of degrees of freedom (DOFs). Numerical examples including a large submarine model with more than 420000 DOFs demonstrate the accuracy and efficiency of the FMBEM, and clearly show the advantage of the new algorithm for solving the large-scale acoustic problems. The developed FMBEM would be potential for engineering applications.

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