﻿ 基于改进差分进化算法的超声衰减谱反演计算
 上海理工大学学报  2020, Vol. 42 Issue (4): 332-338 PDF

Inverse calculation of ultrasonic attenuation spectrum based on an improved differential evolution algorithm
JIANG Yu, JIA Nan, SU Mingxu
School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The traditional differential evolution algorithm frequently encounters the problem of premature convergence and low accuracy. By introducing adaptive control variable factors, a differential evolution (DE) algorithm was developed to ensure that the population could be updated continuously and successfully. The individual learning, which is conducive to the evolution of subsequent populations, and the overall inversion yield more accurate results. As a validation, the particle systems obeying three typical distribution functions of Gaussian distribution, R-R distribution and lognormal distribution were numerically simulated. The resultant distribution parameter values $\bar R$ and K yield errors less than 5%, and the deviations of median volume diameter are within ±5% compared with the given distribution. The DE algorithm also reveals obvious stability and noise resistance.
Key words: attenuation spectrum     differential evolution algorithm     inversion     particle size distribution

1 超声衰减谱法及反演优化问题

 ${{{M}}}{[{A_{{n}}},{C_{{n}}},A_{{n}}^{'},C_{{n}}^{'},{B_{{n}}},B_{{n}}^{'}]^{\rm{T}}} = {{e}}$ (1)

 ${\alpha _{\rm{s}}} = - \frac{{3\varphi }}{{2k_{{c}}^2{R^3}}}\sum\limits_{{{n}} = 0}^\infty {(2n + 1){A_n}}$ (2)

 $k = \frac{\omega }{{{c_{\rm{s}}}(\omega )}} + {\rm{j}}{\alpha _{\rm{s}}}(\omega )$ (3)
 ${\left( {\frac{k}{{{k_{\rm{c}}}}}} \right)^2} = 1 + \frac{{3\varphi }}{{{\rm{j}}k_{\rm{c}}^2{R^3}}}\sum\limits_{{{n = }}0}^\infty {(2n + 1){A_n}}$ (4)

 ${{\alpha}} = {{S}} \times {{W}}$ (5)

 ${E_{\rm{SSD}}} = \sum\limits_{{{j}} = 1}^{{N}} {({\alpha _{\rm{m}}} - {\alpha _{\rm{s}}}} {)^2}$ (6)

 $f(R) = \frac{k}{R}{(R/\bar R)^{k - 1}}\exp \left( { - {{\left( {\frac{R}{{\bar R}}} \right)}^k}} \right)$ (7)

 图 1 4种算法反演计算的颗粒粒径分布 Fig. 1 Inversion results by four algorithms for the given particle size distribution
2 差分进化算法

2.1 标准差分进化算法

a. 初始化种群。初始种群 ${\rm{\{ }}{x_i}{\rm{(0)}}|x_{j,i}^{\rm L}$ ${x_{j,i}}{\rm{(0)}}$ $x_{j,i}^{\rm U},i = 1,2, \cdots ,N,j = 1,2, \cdots ,D{\rm{\} }}$ 随机产生

 ${x_{j,i}}{\rm{(0) = }}x_{j,i}^{\rm L} + {\rm{rand}}(0,1)(x_{j,i}^{\rm U} - x_{j,i}^{\rm L})$ (8)

b. 变异操作。通过如下差分策略，产生变异中间体

 ${v_i}(g + 1) = {x_{{r_1}}}(g) + F({x_{{r_2}}}(g) - {x_{{r_3}}}(g))$ (9)
 $i \ne r_1 \ne r_2 \ne r_3$

c. 交叉操作。对第g代种群{xig）}及其变异的中间体{vig+1）}进行个体间的交叉操作

 {u_{j,i}}(g + 1)\! =\! \left\{ \begin{aligned} &{v_{j,i}}(g + 1),\\ &\;\;\;\;{\text{若}}{\rm{rand}(0,1)} \leqslant CR{\text{或者}} j = {j_{{\rm{rand}}}}\\ &{x_{j,i}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{其他}} \end{aligned} \right. (10)

d. 选择操作。差分进化算法采用贪婪算法来选择进入下一代种群的个体。

 ${x_i}(g + 1)\! = \!\left\{ \begin{array}{l} \!\!\!{u_i}(g + 1),\;{\text{若}}f({u_i}(g + 1)) \leqslant f({x_i}{\rm{(g)}})\\ \!\!\!{x_i}{\rm{(g)}},\;\;\;\;\;\;\;\,{\text{其他}} \end{array} \right.$ (11)

2.2 算法改进

DE算法主要有3个控制参数，即种群大小N、缩放因子F和交叉因子CR。缩放因子F控制偏差变量的放大作用，改变搜索的方向，若种群过早收敛，则F应增加，反之亦然。交叉因子CR改变种群多样性，较大的CR会加速收敛。标准DE算法中的FCR为固定值，在算法迭代后期会因其种群个体聚集，造成种群间差异性变小，使算法易陷入局部最优解、出现早熟收敛等问题。因此，在算法中引入自适应控制参数因子加以改进，改进后的缩放因子和交叉因子为

 \begin{aligned}\!\!\!\!\!{F_{i,\;G + 1}} \!=\! \left\{ \begin{array}{l} \!\!\!{F_{\rm{L}}} +{\rm{rand}(0,1)}{F_{\rm{U}}},\;{\text{若}}\;{\rm{rand}(0,1)} \leqslant {\tau _1}\\ \!\!\!{F_{i,\;G}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\qquad{\text{其他}} \end{array} \right.\end{aligned} (12)
 $C{R_{i,\;G + 1}} = \left\{ \begin{array}{l} {\rm{rand}(0,1)}\;\;\;,\;\;\;{\text{若}}\;{\rm{rand}(0,1)} < {\tau _1}\\ C{R_{i,\;G}}\;\;\;,\;\;\;{\text{其他}} \end{array} \right.$ (13)

 图 2 适应值随FU，FL的变化 Fig. 2 Variation of the best fitness value along with FU and FL

 图 3 IDE算法流程 Fig. 3 Flowchart of the improved differential evolution algorithm

a. 确定DE算法参数，随机产生初始种群并计算个体的适应度（fitness）。

b. 判断是否最佳适应度基本不变。若是，则输出最佳个体为最优解；若否，结合自适应控制参数产生缩放因子F、交叉因子CR，进行变异和交叉操作，得到中间种群，在原种群和中间种群选择个体，得到新种群。

c. 若满足停止条件，算法结束；反之，进化代数t=t+1，转步骤b。

3 结果与分析 3.1 数值模拟

3.1.1 单峰分布

3.1.2 双峰分布

 图 4 双峰颗粒粒径分布反演结果 Fig. 4 Inversion results of bimodal particle size distribution
3.2 抗噪性分析

 图 5 在不同噪声幅度下K， $\overline {{R}}$ 值的变化 Fig. 5 Changes of the values of $\overline {{R}}$ and K with different noise magnitudes

 图 6 R-R分布函数加入噪声的反演计算结果 Fig. 6 Inversion results of R-R distribution functions with noise
4 结　论

 [1] 蔡小舒, 苏明旭, 沈建琪. 颗粒粒度测量技术及应用[M]. 北京: 化学工业出版社, 2010. [2] 刘东红, 朱潘炜. 超声检测技术及在食品安全检测中的研究进展[J]. 食品工业科技, 2009, 30(9): 373-376. [3] ABDA F, AZBAID A, ENSMINGER D, et al. Ultrasonic device for real-time sewage velocity and suspended particles concentration measurements[J]. Water Science & Technology, 2009, 60(1): 117-125. [4] 严玉鹏, 唐亚东, 万彪, 等. 颗粒尺寸对纳米氧化物环境行为的影响[J]. 环境科学, 2018, 39(6): 2982-2990. [5] 付加, 祁贵生, 刘有智, 等. 超重力湿法脱除气体中细颗粒物研究[J]. 化学工程, 2015, 43(4): 6-10. DOI:10.3969/j.issn.1005-9954.2015.04.002 [6] MOUGIN P, WILKINSON D, ROBERTS K J, et al. Characterization of particle size and its distribution during the crystallization of organic fine chemical products as measured in situ using ultrasonic attenuation spectroscopy[J]. The Journal of the Acoustical Society of America, 2001, 109(1): 274-282. DOI:10.1121/1.1331113 [7] POVEY M J W. Ultrasound particle sizing: a review[J]. Particuology, 2013, 11(2): 135-147. DOI:10.1016/j.partic.2012.05.010 [8] EPSTEIN P S, CARHART R R. The absorption of sound in suspensions and emulsions. I. Water fog in air[J]. The Journal of the Acoustical Society of America, 1953, 25(3): 553-565. DOI:10.1121/1.1907107 [9] 林春丹, 梁永燊, 张万松, 等. 超声衰减谱法测量含蜡原油中蜡晶粒度[J]. 声学技术, 2013, 32(4): 294-298. [10] 栾志超. 激光粒度仪的无模式数据处理算法研究[D]. 天津: 天津大学, 2005. [11] 苏明旭, 王夕华, 黄有贵, 等. LM优化算法在消光法测粒技术中的应用[J]. 仪器仪表学报, 2005, 26(S1): 63-65. [12] 孔明, 赵军, 李春燕. 自动基线拟合累积量算法纳米颗粒反演[J]. 光电工程, 2009, 36(9): 52-55. DOI:10.3969/j.issn.1003-501X.2009.09.010 [13] 汪雪, 苏明旭, 蔡小舒. 超声衰减谱法颗粒粒径测量中遗传算法参数优化[J]. 上海理工大学学报, 2016, 38(2): 148-153. [14] STORN R, PRICE K. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces[J]. Journal of Global Optimization, 1997, 11(4): 341-359. DOI:10.1023/A:1008202821328 [15] 崔利刚, 邓洁, 王林, 等. 基于改进联合采购及配送模型的RFID投资决策研究[J]. 中国管理科学, 2018, 26(5): 86-97. [16] 周华, 刘昱, 郭谨玮, 等. 基于差分进化算法和相关向量机的车辆油耗预测[J]. 汽车技术, 2018(12): 19-22. [17] 简兆圣, 艾剑良. 差分进化算法在气动力参数辨识中的应用[J]. 复旦学报(自然科学版), 2017, 56(5): 545-550. [18] GREENWOOD M S, PANETTA P D, BAMBERGER J A, et al. System and technique for ultrasonic characterization of settling suspensions: USA, US7140239B2[P]. 2006-11-28.