汤渭霖 , 范军 , 马忠成 . 水下目标声散射 [M ] . 北京 : 科学出版社 , 2018 .

TANG W L , FAN J , MA Z C . Acoustic scattering of underwater targets [M ] . Beijing : Science Press , 2018 . (in Chinese)

ZAMPOLLI M , TESEI A , JENSEN F B , et al . A computationally efficient finite element model with perfectly matched layers applied to scattering from axially symmetric objects [J ] . Journal of the Acoustical Society of America , 2007 , 122 ( 3 ): 1472 - 1485 . A frequency-domain finite-element (FE) technique for computing the radiation and scattering from axially symmetric fluid-loaded structures subject to a nonsymmetric forcing field is presented. The Berenger perfectly matched layer (PML), applied directly at the fluid-structure interface, makes it possible to emulate the Sommerfeld radiation condition using FE meshes of minimal size. For those cases where the acoustic field is computed over a band of frequencies, the meshing process is simplified by the use of a wavelength-dependent rescaling of the PML coordinates. Quantitative geometry discretization guidelines are obtained from a priori estimates of small-scale structural wavelengths, which dominate the acoustic field at low to mid frequencies. One particularly useful feature of the PML is that it can be applied across the interface between different fluids. This makes it possible to use the present tool to solve problems where the radiating or scattering objects are located inside a layered fluid medium. The proposed technique is verified by comparison with analytical solutions and with validated numerical models. The solutions presented show close agreement for a set of test problems ranging from scattering to underwater propagation.

GONZALEZ J D , LAVIA E F , BLANC S , et al . Boundary element method to analyze acoustic scattering from a coupled swimbladder-fish body configuration [J ] . Journal of Sound and Vibration , 2020 , 486 : 115609 . DOI: 10.1016/j.jsv.2020.115609 http://doi.org/10.1016/j.jsv.2020.115609 https://linkinghub.elsevier.com/retrieve/pii/S0022460X20304405 https://linkinghub.elsevier.com/retrieve/pii/S0022460X20304405

龚家元 , 安俊英 , 马力 , 等 . 边界元奇异与近奇异数值积分方法及其应用于大规模声学问题 [J ] . 声学学报 , 2016 , 41 : 768 - 775 .

GONG J Y , AN J Y , MA L , et al . Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems [J ] . Acta Acustica , 2016 , 41 : 768 - 775 . (in Chinese)

陈鑫 , 罗祎 , 李爱华 . 水下弹性角反射器声散射特性 [J ] . 兵工学报 , 2018 , 39 : 2236 - 2242 . DOI: 10.3969/j.issn.1000-1093.2018.11.018 http://doi.org/10.3969/j.issn.1000-1093.2018.11.018 计算水下弹性角反射器声散射问题，需要考虑声波多次散射以及结构和水的流固耦合作用。针对现有水下弹性角反射器声散射特性分析方法的不足，利用结构有限元耦合流体间接边界元法对水下弹性角反射器的远场散射声场进行仿真计算。分析了入射波频率为5.0～20.0 kHz范围内，平板材料、厚度、入射角度及角反射器类型等因素对目标强度的影响及其散射机理。结果表明：水下弹性角反射器的散射声场具有很强的频率特性，且随材料、平板厚度和入射波频率的变化而改变；多次散射波对角反射器的反向散射声场具有重要贡献，仿真数据和实测数据基本一致。进一步提出了角反射器结构设计和优化方法，为水下无源声反射器的论证与设计提供了理论参考依据。

CHEN X , LUO Y , LI A H . Acoustic scattering characteristics of underwater elastic corner reflectors [J ] . Acta Armamentarii , 2018 , 39 : 2236 - 2242 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2018.11.018 http://doi.org/10.3969/j.issn.1000-1093.2018.11.018 The multiple scattering of sound waves and the fluid-structure interaction should be taken into account to calculate the acoustic scattering of underwater elastic corner reflectors. The scattering acoustic field of underwater elastic corner reflectors in far field is simulated by using structure finite element method (FEM)/fluid indirect coupled boundary element method (IBEM). The influences of plate material and thickness, incident angle and the type of corner reflector on target strength and the scattering mechanism are analyzed when the incident wave frequency varies from 5.0 kHz to 20.0 kHz. The results show that the scattering acoustic field of underwater elastic corner reflectors has a strong frequency characteristic. It varies with plate material and thickness,and incident wave frequency. The multiple scattering of sound waves has important contributions to the backscattering acoustic field of corner reflectors. The simulated results are in good agreement with the experimental results. The structure design and optimization method of corner reflectors are presented. Key

GANESH M , HAWKINS S C . A numerically stable T-matrix method for acoustic scattering by nonspherical particles with large aspect ratios and size parameters [J ] . Journal of the Acoustical Society of America , 2022 , 151 ( 3 ): 1978 - 1988 . DOI: 10.1121/10.0009679 http://doi.org/10.1121/10.0009679 We consider a two-part method for computing the acoustic scattering T-matrix of a three dimensional particle. The first part involves accurately computing the far fields by solving a number of particular scattering problems. The second part calculates the T-matrix from these far fields using the Fourier transform over the sphere. The two-part method was first introduced in Ganesh and Hawkins [J. Comput. Appl. Math. 234, 1702-1709]. The focus of this work is to demonstrate the numerical stability and physical correctness of the two-part method for scattering by nonspherical particles with large aspect ratios and size parameters that are at the upper limit of numerical stability for the current state-of-the-art algorithm. The numerical stability of the method is attributed to elimination of the Hankel functions by working with the far field. The numerical experiments use our recently developed open-source software package (TMATROM3) that implements the two-part method.

ZHAO C , ZHANG T , HOU G X . Finite-difference time-domain modeling for underwater acoustic scattering applications based on immersed boundary method [J ] . Applied Acoustics , 2022 , 193 : 108764 . DOI: 10.1016/j.apacoust.2022.108764 http://doi.org/10.1016/j.apacoust.2022.108764 https://linkinghub.elsevier.com/retrieve/pii/S0003682X22001384 https://linkinghub.elsevier.com/retrieve/pii/S0003682X22001384

范军 , 汤渭霖 , 卓琳凯 . 声呐目标回声特性预报的板块元方法 [J ] . 船舶力学 , 2012 , 16 ( 1/2 ): 171 - 180 .

FAN J , TANG W L , ZHUO L K . Planar elements method for forecasting the echo characteristics from sonar targetsAcoustic scattering of underwater targets [J ] . Journal of Ship Mechanics , 2012 , 16 ( 1/2 ): 171 - 180 . (in Chinese)

陈文剑 , 孙辉 . 计算水下凹面目标散射声场的声束弹跳法 [J ] . 声学学报 , 2013 , 38 : 147 - 152 .

CHEN W J , SUN H . A shooting and bouncing beams method for calculating the acoustic scattering field of concave targets [J ] . Acta Acustica , 2013 , 38 : 147 - 152 . (in Chinese)

PENG Z L , WANG B , FAN J . Simulation and experimental studies on acoustic scattering characteristics of surface targets [J ] . Applied Acoustics , 2018 , 137 : 140 - 147 . DOI: 10.1016/j.apacoust.2018.02.014 http://doi.org/10.1016/j.apacoust.2018.02.014 https://linkinghub.elsevier.com/retrieve/pii/S0003682X1630456X https://linkinghub.elsevier.com/retrieve/pii/S0003682X1630456X

刘成元 , 张明敏 , 程广利 , 等 . 一种改进的板块元目标回声计算方法 [J ] . 海军工程大学学报 , 2008 : 25 - 27 , 31.

LIU C Y , ZHANG M M , CHENG G L , et al . Improved planar element method for computing target echo [J ] . Journal of naval university of engineering , 2008 : 25 - 27 , 31. (in Chinese)

PIGNIER N J , O’REILLY C J , BOIJ S . A Kirchhoff approximation-based numerical method to compute multiple acoustic scattering of a moving source [J ] . Applied Acoustics , 2015 , 96 : 108 - 117 . DOI: 10.1016/j.apacoust.2015.03.016 http://doi.org/10.1016/j.apacoust.2015.03.016 https://linkinghub.elsevier.com/retrieve/pii/S0003682X15001012 https://linkinghub.elsevier.com/retrieve/pii/S0003682X15001012

HIPTMAIR R , MOIOLA A , PERUGIA I . Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes [J ] . Applied Numerical Mathematics , 2014 , 79 : 79 - 91 . DOI: 10.1016/j.apnum.2012.12.004 http://doi.org/10.1016/j.apnum.2012.12.004 https://linkinghub.elsevier.com/retrieve/pii/S0168927413000044 https://linkinghub.elsevier.com/retrieve/pii/S0168927413000044

CHAILLAT S , GROTH S P , LOSEILLE A . Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods [J ] . Journal of Computational Physics , 2018 , 372 : 473 - 499 . DOI: 10.1016/j.jcp.2018.06.048 http://doi.org/10.1016/j.jcp.2018.06.048 https://linkinghub.elsevier.com/retrieve/pii/S0021999118304261 https://linkinghub.elsevier.com/retrieve/pii/S0021999118304261

ABAWI A T . Kirchhoff scattering from non-penetrable targets modeled as an assembly of triangular facets [J ] . Journal of the Acoustical Society of America , 2016 , 140 ( 3 ): 1878 - 1886 . Frequency and time domain solutions for the scattering of acoustic waves from an arbitrarily shaped target using the Kirchhoff approximation are developed. In this method, the scattering amplitude is analytically evaluated on a single triangle and scattering from a triangularly facetted target is computed by coherently summing the contributions from all the triangles that make up its surface. In the frequency domain, the solution is expressed in terms of regular (non-singular) functions, which only require the knowledge of the directions of the incident and scattered fields, the edge vectors for the triangles and position vectors to one of their vertices. To derive representations using regular functions in the time domain, the scattered signal is expressed by different expressions for various limiting cases. The frequency domain solution is validated by comparing its results to the solutions of problems for which the Kirchhoff approximation has analytic solutions. In order of increasing complexity, they include the square plate, the circular plate, the finite cylinder and the sphere. The time domain solution is validated by comparing it to the time domain solution of the Kirchhoff approximation for a rigid sphere.

WANG B , WANG W H , FAN J , et al . Modeling of bistatic scattering from an underwater non-penetrable target using a Kirchhoff approximation method [J ] . Defence Technology , 2022 , 18 ( 7 ): 1097 - 1106 . DOI: 10.1016/j.dt.2022.04.008 http://doi.org/10.1016/j.dt.2022.04.008 A numerical triangulation and transformation into the time domain of a Kirchhoff approximation (KA) method is proposed for the modeling of bistatic scattering from an underwater non-penetrable target. The time domain solution in this approximation can be split up into two parts: the solution of reflected field, contributing around the specular direction, and the solution of shadow radiation, contributing around the forward direction. An average solution in the time domain satisfying the reciprocity principle is presented. The solution is expressed in terms of non-singular functions. The proposed method is validated against a normal mode method for bistatic scattering from a rigid sphere. Moreover, the reflected and shadow highlights on the surface of the sphere are shown to verify the integration surface of the reflected field and shadow radiation. It is also tested against a finite element method and an experiment involving a scaled Benchmark Target Strength Simulation Submarine model. The time-angle bistatic spectra for the model are evaluated by the direct and transformed average solutions of KA, and the former accelerates its speed of calculation. The results are good, and show that this method can be used to predict the bistatic scattered field of a non-penetrable target. © 2022 China Ordnance Society

WU T W , WAN G C . Numerical modeling of acoustic radiation and scattering from thin bodies using a Cauchy principal integral equation [J ] . The Journal of the Acoustical Society of America , 1992 , 92 ( 5 ): 2900 - 2906 . DOI: 10.1121/1.404375 http://doi.org/10.1121/1.404375 https://pubs.aip.org/jasa/article/92/5/2900/843583/Numerical-modeling-of-acoustic-radiation-and https://pubs.aip.org/jasa/article/92/5/2900/843583/Numerical-modeling-of-acoustic-radiation-and In this paper, the Helmholtz integral equation and its normal derivative are used for the numerical solution of acoustic radiation and scattering from thin bodies. A regularized form of the normal derivative integral equation, originally derived by Maue, is adopted to calculate the pressure jump across the thin body. This regularized normal derivative integral equation converges in the Cauchy principal value sense rather than only in the finite-part sense. The Cauchy principal value integral can be further transformed into an integral that converges in the normal sense. The C0 continuous isoparametric elements are used in the formulation so that the numerical model is compatible with other analysis tools such as the finite element method. Collocation points are placed inside each element to insure a unique normal direction and the C1 continuity condition. The pressure jump is forced to be zero at the knife edge where the 1/(r)1/2 singularity of the tangential velocities is modeled by the quarter-point technique. Numerical examples on scattering from a circular disk and an open cylindrical shell are given to verify the formulation.

FOOTE K G , FRANCIS D T . Comparing Kirchhoff-approximation and boundary-element models for computing gadoid target strengths [J ] . Journal of the Acoustical Society of America , 2002 , 111 ( 4 ): 1644 - 1654 . To establish the validity of the boundary-element method (BEM) for modeling scattering by swimbladder-bearing fish, the BEM is exercised in several ways. In a computation of backscattering by a 50-mm-diam spherical void in sea water at the four frequencies 38.1, 49.6, 68.4, and 120.4 kHz, agreement with the analytical solution is excellent. In computations of target strength as a function of tilt angle for each of 15 surface-adapted gadoids for which the swimbladders were earlier mapped, BEM results are in close agreement with Kirchhoff-approximation-model results at each of the same four frequencies. When averaged with respect to various tilt angle distributions and combined by regression analysis, the two models yield similar results. Comparisons with corresponding values derived from measured target strength functions of the same 15 gadoid specimens are fair, especially for the tilt angle distribution with the greatest standard deviation, namely 16 degrees.

VENÅS J V , KVAMSDAL T . Isogeometric boundary element method for acoustic scattering by a submarine [J ] . Computer Methods in Applied Mechanics and Engineering , 2020 , 359 : 112670 . DOI: 10.1016/j.cma.2019.112670 http://doi.org/10.1016/j.cma.2019.112670 https://linkinghub.elsevier.com/retrieve/pii/S0045782519305559 https://linkinghub.elsevier.com/retrieve/pii/S0045782519305559

RATHOD H T , SHAJEDUL KARIM M . An explicit integration scheme based on recursion for the curved triangular finite elements [J ] . Computers & Structures , 2002 , 80 ( 1 ): 43 - 76 . DOI: 10.1016/S0045-7949(01)00156-0 http://doi.org/10.1016/S0045-7949(01)00156-0 https://linkinghub.elsevier.com/retrieve/pii/S0045794901001560 https://linkinghub.elsevier.com/retrieve/pii/S0045794901001560

陈云飞 , 李桂娟 , 王振山 , 等 . 水中目标回波亮点统计特征研究 [J ] . 物理学报 , 2013 , 62 ( 8 ): 284 - 294 .

CHEN Y F , LI G J , WANG Z S , et al . Statistical feature of underwater target echo highlight [J ] . Acta Physica Sinica , 2013 , 62 ( 8 ): 284 - 294 . (in Chinese)

NELL C W , GILROY L E . An improved BASIS model for the BeTSSi submarine: TR 2003-199 [R ] . Cumberland St, Ottawa , Canada : Defence Research and Development Canada , 2003 .