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传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势, 具有良好的工程应用前景.
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关键词:
- 快速多极方法 /
- 边界元法 /
- 对角形式 /
- Helmholtz方程
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.-
Keywords:
- fast multipole method /
- boundary element method /
- diagonal form /
- Helmholtz equation
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[30] Kropinski M C,Quaife B D 2011 J.Comput.Phys.230 425
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[45] -
[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 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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