﻿ 穿孔消声器声学计算的快速多极混体边界元法
«上一篇
 文章快速检索 高级检索

 哈尔滨工程大学学报  2017, Vol. 38 Issue (8): 1247-1253  DOI: 10.11990/jheu.201606040 0

### 引用本文

YANG Liang, JI Zhenlin. Acoustic computation of perforated silencers by fast multi-pole mixed-body boundary element method[J]. Journal of Harbin Engineering University, 2017, 38(8), 1247-1253. DOI: 10.11990/jheu.201606040.

### 文章历史

Acoustic computation of perforated silencers by fast multi-pole mixed-body boundary element method
YANG Liang, JI Zhenlin
School of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: To improve computation efficiency and extend the computation frequency range, the mixed-body boundary element method (MBEM) was combined with the fast multi-pole algorithm to evaluate the acoustic attenuation performance of perforated silencers. The correction of fast MBEM was validated by comparing the measuring result of transmission loss with the computation result of traditional MBEM. The fast multi-pole mixed-body boundary element method (FMMBEM) may efficiently save computational time for large-scale acoustic problems without affecting accuracy. The FMMBEM was then employed to predict the transmission loss of perforated silencers. Predicted results showed that the locations of inlet and outlet tubes may affect the acoustic attenuation characteristics of the two-pass perforated tube silencer at higher frequencies, while the perforation on bulkhead(s) may improve the acoustic attenuation performance of three-pass perforated tube silencer in the middle frequency range.
Key words: perforated silencer    transmission loss    mixed body boundary element method    fast multi pole algorithm    computation efficiency

1 计算方法 1.1 混体边界元方法

 $\begin{array}{*{20}{c}} {\int_R {\left( {p\left( {{r_Q}} \right)\frac{{\partial G\left( {{r_p},{r_Q}} \right)}}{{\partial {n_Q}}} + {\rm{j}}kz{v_n}\left( {{r_Q}} \right)G\left( {{r_p},{r_Q}} \right)} \right){\rm{d}}S} + }\\ {\int_{T + P} {\frac{{\partial G\left( {{r_p},{r_Q}} \right)}}{{\partial {n_Q}}}\left( {{p^ + } - {p^ - }} \right){\rm{d}}S} = } \end{array}$ (1)
 $\left\{ \begin{array}{l} 0.5p\left( {{r_p}} \right),p \in R\\ 0.5\left[ {{p^ + }\left( {{r_p}} \right) + {p^ - }\left( {{r_p}} \right)} \right],p \in T + P \end{array} \right.$ (2)
 $\begin{array}{*{20}{c}} {\int_R {\left( {p\left( {{r_Q}} \right)\frac{{{\partial ^2}G\left( {{r_p},{r_Q}} \right)}}{{\partial {n_Q}\partial {n_p}}} + jkz{v_n}\left( {{r_Q}} \right)\frac{{\partial G\left( {{r_p},{r_Q}} \right)}}{{\partial {n_p}}}} \right){\rm{d}}S} + }\\ {\int_{T + P} {\frac{{{\partial ^2}G\left( {{r_p},{r_Q}} \right)}}{{\partial {n_Q}\partial {n_p}}}\left( {{p^ + } - {p^ - }} \right){\rm{d}}S} = } \end{array}$ (3)
 $\left\{ \begin{array}{l} 0,p \in T\\ \frac{{{\rm{j}}{k_0}}}{\zeta }\left[ {{p^ + }\left( {{r_p}} \right) - {p^ - }\left( {{r_p}} \right)} \right],p \in P \end{array} \right.$ (4)

 $\left[ {\begin{array}{*{20}{c}} {E + B + C}&I\\ {E' + B' + C'}&{I'} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} p\\ {\Delta {p_{T + P}}} \end{array}} \right] = {\rm{j}}\rho \omega \left[ {\begin{array}{*{20}{c}} {{v_n}}&0\\ 0&{{v_n}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} A\\ {A'} \end{array}} \right]$ (5)

 ${E_{ij}} = \frac{1}{2}{\delta _{ij}},\;\;\;{{E'}_{ij}} = \frac{{ik}}{\zeta }{\delta _{ij}},$
 ${B_{ij}} = \int\limits_R {\frac{{\partial G}}{{\partial n}}{\rm{d}}{S_Q}} ,\;\;\;{{B'}_{ij}} = \int\limits_R {\frac{{{\partial ^2}G}}{{\partial n\partial {n_P}}}{\rm{d}}{S_Q}} ,$
 ${C_{ij}} = \frac{{ik}}{z}\int\limits_{{R_2}} {G{\rm{d}}{S_Q}} ,\;\;\;{{C'}_{ij}} = \frac{{ik}}{z}\int\limits_{{R_2}} {\frac{{\partial G}}{{\partial {n_P}}}{\rm{d}}{S_Q}} ,$
 ${A_{ij}} = - \int\limits_{{R_1}} {G{\rm{d}}{S_Q}} ,\;\;\;{{A'}_{ij}} = - \int\limits_{{R_1}} {\frac{{\partial G}}{{\partial {n_P}}}{\rm{d}}{S_Q}} ,$
 ${I_{ij}} = \int\limits_P {\frac{{\partial G}}{{\partial n}}{\rm{d}}{S_Q}} ,\;\;\;{{I'}_{ij}} = \int\limits_p {\frac{{{\partial ^2}G}}{{\partial n\partial {n_P}}}{\rm{d}}{S_Q}}$

 图 1 直通穿孔管消声器 Fig.1 Straight-through perforated tube silencer
1.2 快速多极混体边界元方法

 $G = \frac{{{\rm{j}}k}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {{E_{PL}}\left( {\mathit{\boldsymbol{\hat s}}} \right){T_{LM}}\left( {\mathit{\boldsymbol{\hat s}}} \right){E_{MQ}}\left( {\mathit{\boldsymbol{\hat s}}} \right){\rm{d}}\mathit{\boldsymbol{\hat s}}}$ (6)

 ${E_{PL}}\left( {\hat s} \right) = \exp \left( {{\rm{j}}k\mathit{\boldsymbol{\hat s}} \cdot {\mathit{\boldsymbol{r}}_{PL}}} \right)$ (7)
 ${T_{LM}}\left( {\hat s} \right) = \sum\limits_{l = 0}^\infty {{{\rm{j}}^l}\left( {2l + 1} \right)h_l^{\left( 1 \right)}\left( {k{r_{LM}}} \right){P_l}\left( {\mathit{\boldsymbol{\hat s}} \cdot {{\mathit{\boldsymbol{\hat r}}}_{LM}}} \right)}$ (8)

 $\frac{{\partial G}}{{\partial n}} = \frac{{{{\left( {{\rm{j}}k} \right)}^2}}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {{E_{PL}}\left( {\mathit{\boldsymbol{\hat s}}} \right){T_{LM}}\left( {\hat s} \right){E_{MQ}}\left( {\mathit{\boldsymbol{\hat s}}} \right)\left( {\mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{\hat s}}} \right){\rm{d}}\mathit{\boldsymbol{\hat s}}}$ (9)

 $\begin{array}{*{20}{c}} {\frac{{{\partial ^2}G}}{{\partial n\partial {n_P}}} = \frac{{{{\left( {{\rm{j}}k} \right)}^3}}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {\left( {{\mathit{\boldsymbol{n}}_P} \cdot \mathit{\boldsymbol{\hat s}}} \right){E_{PL}}\left( {\mathit{\boldsymbol{\hat s}}} \right){T_{LM}}\left( {\mathit{\boldsymbol{\hat s}}} \right) \cdot } }\\ {{E_{MQ}}\left( {\mathit{\boldsymbol{\hat s}}} \right)\left( {\mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{\hat s}}} \right){\rm{d}}\hat s} \end{array}$ (10)

 $G = \frac{{{\rm{j}}k}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {{E_{P{\lambda _{mL}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)Z_{{\lambda _{mL}}{\lambda _{m'L}}}^I\left( {\mathit{\boldsymbol{\hat s}}} \right){E_{{\lambda _{m'L}}Q}}\left( {\mathit{\boldsymbol{\hat s}}} \right){\rm{d}}\mathit{\boldsymbol{\hat s}}}$ (11)
 $\frac{{\partial G}}{{\partial n}} = \frac{{{{\left( {jk} \right)}^2}}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {{E_{P{\lambda _{mL}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)Z_{{\lambda _{mL}}{\lambda _{m'L}}}^I\left( {\mathit{\boldsymbol{\hat s}}} \right){E_{{\lambda _{m'L}}Q}}\left( {\mathit{\boldsymbol{\hat s}}} \right)\left( {\mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{\hat s}}} \right){\rm{d}}\mathit{\boldsymbol{\hat s}}}$ (12)

 $\begin{array}{*{20}{c}} {\frac{{{\partial ^2}G}}{{\partial n\partial {n_P}}} = \frac{{{{\left( {{\rm{j}}k} \right)}^3}}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {\left( {{\mathit{\boldsymbol{n}}_P} \cdot \mathit{\boldsymbol{\hat s}}} \right){E_{P{\lambda _{{m_L}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)Z_{{\lambda _{{m_L}}}{\lambda _{{{m'}_L}}}}^I\left( {\mathit{\boldsymbol{\hat s}}} \right) \cdot } }\\ {{E_{{\lambda _{{{m'}_L}}}Q}}\left( {\mathit{\boldsymbol{\hat s}}} \right)\left( {\mathit{\boldsymbol{n}} \cdot \mathit{\boldsymbol{\hat s}}} \right){\rm{d}}\hat s} \end{array}$ (13)

 $Z_{{\lambda _{{m_L}}}{\lambda _{{{m'}_L}}}}^I\left( {\mathit{\boldsymbol{\hat s}}} \right) = \prod\limits_{l = I}^{L - 1} {{E_{{\lambda _{{m_{l + 1}}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right){T_{{\lambda _{{m_I}}}{\lambda _{{{m'}_I}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)} \prod\limits_{l = 1}^{L - 1} {{E_{{\lambda _{{{m'}_l}}}{\lambda _{{{m'}_{l + 1}}}}}}}$ (14)

 $\left[ {\begin{array}{*{20}{c}} {{A_{ij}}}\\ {{B_{ij}}}\\ {{C_{ij}}}\\ {{I_{ij}}} \end{array}} \right] = \frac{{{\rm{j}}k}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {{E_{i{\lambda _{{m_L}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)Z_{{\lambda _{{m_L}}}{\lambda _{{{m'}_L}}}}^I\left( {\mathit{\boldsymbol{\hat s}}} \right)} \left[ {\begin{array}{*{20}{c}} {{\alpha _{{\lambda _{m_L^{'j}}}}}}\\ {{\beta _{{\lambda _{m_L^{'j}}}}}}\\ {{\gamma _{{\lambda _{m_L^{'j}}}}}}\\ {{\eta _{{\lambda _{m_L^{'j}}}}}} \end{array}} \right]{\rm{d}}\mathit{\boldsymbol{\hat s}}$ (15)
 $\left[ {\begin{array}{*{20}{c}} {{{A'}_{ij}}}\\ {{{B'}_{ij}}}\\ {{{C'}_{ij}}}\\ {{{I'}_{ij}}} \end{array}} \right] = \frac{{ - {k^2}}}{{16{{\rm{ \mathsf{ π} }}^2}}}\oint {{{\left( {{\mathit{\boldsymbol{n}}_P} \cdot \mathit{\boldsymbol{\hat s}}} \right)}_{i{\lambda _{{m_L}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)Z_{{\lambda _{{m_L}}}{\lambda _{{{m'}_L}}}}^I\left( {\mathit{\boldsymbol{\hat s}}} \right)} \left[ {\begin{array}{*{20}{c}} {{\alpha _{{\lambda _{m_L^{'j}}}}}}\\ {{\beta _{{\lambda _{m_L^{'j}}}}}}\\ {{\gamma _{{\lambda _{m_L^{'j}}}}}}\\ {{\eta _{{\lambda _{m_L^{'j}}}}}} \end{array}} \right]{\rm{d}}\mathit{\boldsymbol{\hat s}}$ (16)

 ${\alpha _{{\lambda _{m_L^{'j}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right) = \int_{{R_1}} {{E_{{\lambda _{m_L^{'Q}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right){\rm{d}}{S_Q}}$ (17)
 ${\beta _{{\lambda _{m_L^{'j}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right) = {\rm{i}}k\int_{{R_1}} {{E_{{\lambda _{m_L^{'Q}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)\left( {{\mathit{\boldsymbol{n}}_Q} \cdot \mathit{\boldsymbol{\hat s}}} \right){\rm{d}}{S_Q}}$ (18)
 ${\gamma _{{\lambda _{m_L^{'j}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right) = \frac{{{\rm{j}}k}}{z}\int_{{R_2}} {{E_{{\lambda _{m_L^{'Q}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right){\rm{d}}{S_Q}}$ (19)
 ${\eta _{{\lambda _{m_L^{'j}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right) = {\rm{i}}k\int_P {{E_{{\lambda _{m_L^{'Q}}}}}\left( {\mathit{\boldsymbol{\hat s}}} \right)\left( {{\mathit{\boldsymbol{n}}_Q} \cdot \mathit{\boldsymbol{\hat s}}} \right){\rm{d}}{S_Q}}$ (20)
 图 2 分级结构及通用交互组 Fig.2 Hierarchical cell structure and common interaction cell

1) 远场区域矩阵矢量积计算过程。

① 在最低级L，计算所有组mL在每个球面积分插值点$\hat s$nL的聚合系数ξmL，公式为

 $\begin{array}{l} {\xi _{{m_L}}} = \left( {\hat s_n^L} \right) = \sum\limits_{j \in {B_{{m_L}}}} {} \\ \left[ {\begin{array}{*{20}{c}} {\left( {{\beta _{{\lambda _{mL}}j}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right) + {\gamma _{{\lambda _{mL}}j}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right) + {\eta _{{\lambda _{mL}}j}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right)} \right){p_j}}\\ {{\alpha _{{\lambda _{mL}}j}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right){v_j}} \end{array}} \right] \end{array}$ (21)

② 在接下来的较高级l(l=L－1，L－2，…，2) 中，继续计算所有组在积分插值点$\mathit{\boldsymbol{\hat s}}$nl的上传系数ξml，公式为

 ${\xi _{{m_l}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right) = \sum\limits_{{m_{l + 1}} \in {C_{{m_l}}}} {{E_{{\lambda _{{m_l}}}{\lambda _{{m_{l + 1}}}}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right)} \cdot \sum\limits_{n = 1}^{2{{\left( {N_c^{l + 1}} \right)}^2}} {{W_{n'n}}{\xi _{{m_{l + 1}}}}\left( {\mathit{\boldsymbol{\hat s}}_n^{l + 1}} \right)}$ (22)

③ 在每一级l(l=2，3，…，L)，计算该级所有组在积分插值点$\mathit{\boldsymbol{\hat s}}$nl的交互系数τml，公式为

 ${\tau _{{m_l}}}\left( {\mathit{\boldsymbol{\hat s}}_n^l} \right) = \sum\limits_{{{m'}_l} \in {F_{{m_l}}}} {{T_{{\lambda _{{m_l}}}{\lambda _{{{m'}_l}}}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right){\xi _{{{m'}_l}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right)}$ (23)

④ 在接下来的较高级l(l=2，3，…，L－1)，计算所有组在积分插值点$\mathit{\boldsymbol{\hat s}}$nl+1的下传系数ζml，对积分点引入伴随插值方法，该系数的计算公式为

 $\begin{array}{l} {\zeta _{{m_{l + 1}}}}\left( {\mathit{\boldsymbol{\hat s}}_n^{l + 1}} \right) = \sum\limits_{n' = 1}^{2{{\left( {N_c^l} \right)}^2}} {\frac{{\omega _{n'}^l}}{{\omega _n^{l + 1}}}{W_{n'n}}{E_{{\lambda _{{m_l} + 1}}{\lambda _{{m_l}}}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right) \cdot } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{\zeta _{{m_l}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right) + {\tau _{{m_l}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^l} \right)} \right) \end{array}$ (24)

 ${\zeta _{{m_2}}}\left( {\mathit{\boldsymbol{\hat s}}_{n'}^2} \right) = 0$ (25)

⑤ 在最低级L，计算每个节点i的远场影响系数ϕF, i，对应式(15)，计算公式为

 ${\phi _{F,i}} = \frac{{{\rm{j}}k}}{{16{{\rm{ \mathsf{ π} }}^2}}}\sum\limits_{n = 1}^{2{{\left( {N_c^L} \right)}^2}} {\omega _n^L{E_{i{\lambda _{{m_L}}}}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right) \cdot \left( {{\zeta _{{m_L}}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right) + {\tau _{{m_l}}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right)} \right)}$ (26)

 $\begin{array}{l} {\psi _{F,i}} = \frac{{ - {k^2}}}{{16{{\rm{ \mathsf{ π} }}^2}}}\sum\limits_{n = 1}^{2{{\left( {N_c^L} \right)}^2}} {\omega _n^L\left( {{\mathit{\boldsymbol{n}}_i} \cdot \mathit{\boldsymbol{\hat s}}_n^L} \right){E_{i{\lambda _{{m_L}}}}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right) \cdot } \\ \;\;\;\;\;\;\;\;\;\left( {{\zeta _{{m_L}}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right) + {\tau _{{m_l}}}\left( {\mathit{\boldsymbol{\hat s}}_n^L} \right)} \right) \end{array}$ (27)

2) 近场区域矩阵矢量积计算过程。

 ${\phi _{N,i}} = \sum\limits_{{{m'}_L} \in {N_{{m_L}}}} {\sum\limits_{j \in {B_{{{m'}_L}}}} {\left[ {\begin{array}{*{20}{c}} {\left( {{E_{ij}} + {B_{ij}} + {C_{ij}}} \right){p_j} + {I_{ij}}\Delta {p_j}}\\ {{A_{ij}}{v_j}} \end{array}} \right]} }$ (28)

 ${\psi _{N,i}} = \sum\limits_{{{m'}_L} \in {N_{{m_L}}}} {\sum\limits_{j \in {B_{{{m'}_L}}}} {\left[ {\begin{array}{*{20}{c}} {\left( {{{E'}_{ij}} + {{B'}_{ij}} + {{C'}_{ij}}} \right){p_j} + {{I'}_{ij}}\Delta {p_j}}\\ {{{A''}_{ij}}{v_j}} \end{array}} \right]} }$ (29)

2 算例验证及计算效率分析

 $\zeta = \left[ {R + {\rm{j}}X} \right]/\phi$ (30)

RX的计算表达式：

 $R = \left( {1 + {t_w}/{d_h}} \right)\sqrt {8k\mu /{z_0}}$ (31)
 $X = k\left( {{t_w} + \alpha {d_h}} \right)$ (32)

 $\alpha =\frac{4}{{{\text{ }\!\!\pi\!\!\text{ }}^{2}}}\frac{1}{{{\left( \xi \eta \right)}^{0.5}}}\sum\limits_{m=0}^{\infty }{\sum\limits_{n=0}^{\infty }{{}^{'}{{\varepsilon }_{mn}}\frac{\text{J}_{1}^{2}\left( \text{ }\!\!\pi\!\!\text{ }\sqrt{{{\left( m\xi \right)}^{2}}+{{\left( n\eta \right)}^{2}}} \right)}{{{\left( {{m}^{2}}\left( h/b \right)+{{n}^{2}}\left( b/h \right) \right)}^{1.5}}}}}$ (33)

 图 3 直通穿孔管消声器计算结果比较 Fig.3 Transmission loss prediction of straight-through perforated silencer

3 计算结果分析

 图 4 两通穿孔管消声器 Fig.4 Two-pass perforated tube silencer
 图 5 两通穿孔管消声器传递损失计算结果 Fig.5 TL comparison of two-pass perforated tube silencer

 图 6 三通穿孔管消声器 Fig.6 Three-pass perforated tube silencer
 图 7 三通穿孔管消声器计算结果比较 Fig.7 Transmission loss prediction results of three-pass perforated tube silencer

 图 8 穿孔挡板对消声器传递损失的影响 Fig.8 Effect of perforated baffle on silencer TL

 图 9 挡板穿孔率对三通穿孔消声器传递损失的影响 Fig.9 Effect of baffle porosity on TL of three-pass perforated tube silencer

 图 10 挡板孔径对三通穿孔消声器传递损失的影响 Fig.10 Effect of baffle hole diameter on TL of three-pass perforated tube silencer

 图 11 膨胀腔长度对三通穿孔消声器传递损失的影响 Fig.11 Effect of inlet/outlet expansion chamber length on TL of three-pass perforated tube silencer
4 结论

1) 快速多极混体边界元方法在能够保证计算精度的情况下，对大尺度高频问题能够有效地减少计算时间，在实际应用中，可以根据研究问题的情况选择适当的方法进行计算。

2) 当进口位于轴线处时，消声器的声学性能更好，出口穿孔管的位置变化对计算结果的影响应该给予考虑，而当进口位于非轴线处时，进出口穿孔管距离的变化对传递损失影响很小，基本可以忽略不计。

3) 使用穿孔板代替三通穿孔消声器中的刚性隔板能改善其中低频的声学性能，减小穿孔板穿孔孔径、增大穿孔率以及进出口膨胀腔的长度会使传递损失曲线向高频方向移动。

 [1] SELAMET A, EASWARAN V, FALKOWSKI A G. Three-pass mufflers with uniform perforations[J]. The journal of the acoustical society of America, 1999, 105(3): 1548-1562. DOI:10.1121/1.426694 (0) [2] SELAMET A, XU M B, LEE I J, et al. Analytical approach for sound attenuation in perforated dissipative silencers with inlet/outlet extensions[J]. The journal of the acoustical society of America, 2005, 117(4): 2078-2089. DOI:10.1121/1.1867884 (0) [3] JI Z L. Acoustic attenuation characteristics of straight-through perforated tube silencers and resonators[J]. Journal of computational acoustics, 2008, 16(03): 361-379. DOI:10.1142/S0218396X08003622 (0) [4] JI Z L. Boundary element acoustic analysis of hybrid expansion chamber silencers with perforated facing[J]. Engineering analysis with boundary elements, 2010, 34(7): 690-696. DOI:10.1016/j.enganabound.2010.02.006 (0) [5] JI Z L, SELAMET A. Boundary element analysis of three-pass perforated duct mufflers[J]. Noise control engineering journal, 2000, 48(5): 151-156. DOI:10.3397/1.2827962 (0) [6] FANG Z, JI Z L. Finite element analysis of transversal modes and acoustic attenuation characteristics of perforated tube silencers[J]. Noise control engineering journal, 2012, 60(3): 340-349. DOI:10.3397/1.3701000 (0) [7] WU T W, CHENG C Y R, ZHANG P. A direct mixed-body boundary element method for packed silencers[J]. The Journal of the acoustical society of America, 2002, 111(6): 2566-2572. DOI:10.1121/1.1476920 (0) [8] WU T W, WAN G C. Muffler performance studies using a direct mixed-body boundary element method and a three-point method for evaluating transmission loss[J]. Journal of vibration and acoustics, 1996, 118(3): 479-484. DOI:10.1115/1.2888209 (0) [9] SAKUMA T, YASUDA Y. Fast multipole boundary element method for large-scale steady-state sound field analysis. Part Ⅰ:setup and validation[J]. Acta acustica united with acustica, 2002, 88(4): 513-525. (0) [10] WU H, LIU Y, JIANG W. A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems[J]. Engineering analysis with boundary elements, 2013, 37(2): 309-318. DOI:10.1016/j.enganabound.2012.09.011 (0) [11] LI S, HUANG Q. A fast multipole boundary element method based on the improved Burton-Miller formulation for three-dimensional acoustic problems[J]. Engineering analysis with boundary elements, 2011, 35(5): 719-728. DOI:10.1016/j.enganabound.2010.12.004 (0) [12] 王雪仁, 季振林. 快速多极子声学边界元法及其应用研究[J]. 哈尔滨工程大学学报, 2007, 28(7): 752-757. WANG Xueren, JI Zhenlin. Fast multipole acoustic BEM and its application[J]. Journal of Harbin Engineering University, 2007, 28(7): 752-757. (0) [13] 吴海军, 蒋伟康, 鲁文波. 三维声学多层快速多极子边界元及其应用[J]. 物理学报, 2012, 61(5): 54301-054301. WU Haijun, JIANG Weikang, LU Wenbo. Multilevel fast multipole boundary element method for 3D acoustic problems and its applications[J]. Acta physica sinica, 2012, 61(5): 54301-054301. DOI:10.7498/aps.61.054301 (0) [14] JI Z L, FANG Z. On the acoustic impedance of perforates and its application to acoustic attenuation predictions for perforated tube silencers[C]//Proceedings of INTER-NOISE. Osaka, 2011(5):2803-2813. (0)