Vibro-acoustic design sensitivity analysis using the wave-based method

Conventional element-based methods, such as the finite element method (FEM) and boundary element method (BEM), require mesh refinements at higher frequencies in order to converge. Therefore, their applications are limited to low frequencies. Compared to element-based methods, the wave-based method (WBM) adopts exact solutions of the governing differential equation instead of simple polynomials to describe the dynamic response variables within the subdomains. As such, the WBM does not require a finer division of subdomains as the frequency increases in order to exhibit high computational efficiency. In this paper, the design sensitivity formulation of a semi-coupled structural-acoustic problem is implemented using the WBM. Here, the results of structural harmonic analyses are imported as the boundary conditions for the acoustic domain, which consists of multiple wave-based subdomains. The cross-sectional area of each beam element is considered as a sizing design variable. Then, the adjoint variable method (AVM) is used to efficiently compute the sensitivity. The adjoint variable is obtained from structural reanalysis using an adjoint load composed of the system matrix and an evaluation of the wave functions of each boundary. The proposed sensitivity formulation is subsequently applied to a two-dimensional (2D) vehicle model. Finally, the sensitivity results are compared to the finite difference sensitivity results, which show good agreement.     

Energy flow analysis and design sensitivity of structural problems at high frequencies

The design sensitivity formulation of an energy finite element method is presented using the direct differentiation and adjoint variable methods. The continuum method is used to derive the design sensitivity equation of the energy flow equation, whereas the discrete method is used to calculate the variation of the coupling relation. For design variables, material property, panel thickness, and structural shape are taken into account, in addition to the structural damping factor. The design variable's effect on the power transfer coefficient is discussed in detail. Even if the system matrix equation is not symmetric, the adjoint problem is solved using the same factorized matrix from response analysis. Design sensitivity results calculated from the proposed method are compared to the finite difference sensitivity results with a good agreement.     

Shape design sensitivity analysis for the radiated noise from the thin-body

Many industrial applications generally use thin-body structures in their design. To calculate the radiated noise from vibrated structure including thin bodies, the conventional boundary element method (BEM) using the Helmholtz integral equation is not an effective resolution. Thus, many researchers have studied to resolve the thin-body problem in various physical fields. No major study in the design sensitivity analysis (DSA) fields for thin-body acoustics, however, has been reported.A continuum-based shape DSA method is presented for the radiated noise from the thin-body. The normal derivative integral equation is employed as an analysis formulation. And, for the acoustic shape design sensitivity formulation, the equation is differentiated directly by using material derivative concept. To solve the normal derivative integral equation, the normal velocities on the surface should be calculated. In the acoustic shape sensitivity formulation, not only the normal velocities on the surface are required but also derivative coefficients of the normal velocities (structural shape design sensitivity) are also required as the input. Hence, the shape design sensitivity of structural velocities on the surface, with respect to the shape change, should be calculated. In this research, the structural shape design sensitivities are also obtained by using a continuum approach. And both a modified interpolation function and the Cauchy principle value are used to regularize the singularities generated from the acoustic shape design sensitivity formulation.A simple annular disk is considered as a numerical example to validate the accuracy and efficiency of the shape design sensitivity equations derived in this research. The commercial BEM code, SYSNOISE, is utilized to confirm the results of the developed in-house code based on a normal derivative integral equation. To validate the calculated design sensitivity results, central finite difference method (FDM) is employed. The error between FDM and the analytical result are less than 3%. This comparison demonstrates that the proposed design sensitivities of the radiated pressure are very accurate.     

小阻尼封闭空间声压级灵敏度分析与优化设计

本文研究了小阻尼界面封闭空间低频声学有限元分析、灵敏度分析和优化设计问题。分别用模态法和直接法计算了封闭空间内声压级响应，并推导了声压级响应对声空间边界形状控制参数的灵敏度分析公式，在此基础上建立了小阻尼空间声学问题的优化模型，同时给出了优化求解算法，并在JIFEX软件中进行了程序实现。本文提出的灵敏度分析和优化设计方法可以使声场的边界布局更为合理，从而达到改进小阻尼界面封闭空间声学性能的目的。数值算例验证了本文提出的灵敏度分析和优化算法的有效性。     

Level set based structural topology optimization for minimizing frequency response

For the purpose of structure vibration reduction, a structural topology optimization for minimizing frequency response is proposed based on the level set method. The objective of the present study is to minimize the frequency response at the specified points or surfaces on the structure with an excitation frequency or a frequency range, subject to the given amount of the material over the admissible design domain. The sensitivity analysis with respect to the structural boundaries is carried out, while the Extended finite element method (X-FEM) is employed for solving the state equation and the adjoint equation. The optimal structure with smooth boundaries is obtained by the level set evolution with advection velocity, derived from the sensitivity analysis and the optimization algorithm. A number of numerical examples, in the frameworks of two-dimension (2D) and three-dimension (3D), are presented to demonstrate the feasibility and effectiveness of the proposed approach.     

A study on the applications of the acoustic design sensitivity analysis of vibrating bodies

The determination of the sensitivity of the acoustical characteristics of vibrating systems with respect to the variation of the design parameters predicting these characteristics is a necessary and important step of the acoustic design and optimization process. Acoustic design sensitivity analysis includes the computation and evaluation of the sensitivity information required for this procedure. In this study, a boundary element code performing the sensitivity analysis of the acoustic pressure by using the matrix sensitivities with respect to different design variables has been developed. The effect of the precision of boundary element discretization on the acoustic pressure sensitivity is examined via this code. The formulation is applied to a multi-source system and the dimension sensitivity analysis of near field pressures of two-dilating-spherical source is performed. The last application is devoted to a real sound source: a washing machine sitting on the floor. Sensitivity of the field pressures to the machine’s dimensions (size), surface velocity and frequency is examined on the bases of the boundary element model of the machine and half-space condition. The impacts of these variables are compared; and a limiting speed for the machine responding both the acoustical and operational requirements is determined.     

Simulative and experimental investigations on pressure-induced structural vibrations of a rear muffler

The periodically blown out exhaust gas of a combustion engine may excite structural vibrations of the exhaust system. In addition to the noise of the orifice, these vibrations contribute to the overall noise radiation of the exhaust system. In this work, the excitation of structural vibrations of a rear muffler via the acoustic path is investigated both in experiments and simulations. In both cases transfer functions from the acoustic pressure at the inlet to the structural deflection on the surface of the rear muffler are determined and compared to each other. For the simulation an FE-FE (finite element) coupling is applied to account for the fluid-structure interaction. To efficiently predict the fluid-structure coupled behavior, a model reduction technique for the finite element method based on the Craig-Bampton method and the Rubin method is presented. In a last step, the sound radiation is evaluated by solving the exterior acoustic problem with the fast multipole boundary element method. For this purpose, the results of the FE computation are used as boundary datum.     

Structural topology optimization on sound radiation at resonance frequencies in thermal environments

Thermal and acoustic environments pose severe challenges to the structural design of hypersonic vehicles.One of them is to find optimal design that exhibits ideal acoustic characteristics in a frequency band,which is discussed in this paper through topology optimization aiming at resonance sound radiation in thermal environments.The sound radiation at resonance frequencies is the main component of response,minimization on which is likely to provide a satisfactory design.A bi-material plate subjected to uniform temperature rise and excited by harmonic loading is studied here.Thermal stress is first evaluated and considered as prestress in the following dynamic analysis;radiated sound power is then calculated through Rayleigh integral.Sensitivity analysis is carried out through adjoint method considering the complicated relationship between stress-induced geometric stiffness and design variables.As the resonance frequency is constantly changing during the optimization,its sensitivity should be considered.It is also noticed that mode switching may occur,so mode tracking technique is employed in this work.Some numerical examples are finally discussed.     

Vibro-acoustic behaviors of an elastically restrained double-panel structure with an acoustic cavity of arbitrary boundary impedance

An analytical study on the vibro-acoustic behaviors of a double-panel structure with an acoustic cavity is presented. Unlike the existing studies, a structural–acoustic coupling model of an elastically restrained double-panel structure with an acoustic cavity having arbitrary impedance on sidewalls around the cavity is developed in which the two dimensional (2D) and three dimensional (3D) modified Fourier series are used to represent the displacement of the panels and the sound pressure inside the cavity, respectively. The unknown expansions coefficients are treated as the generalized coordinates and the Rayleigh–Ritz method is employed to determine displacement and sound pressure solutions based on the energy expressions for the coupled structural–acoustic system. The effectiveness and accuracy of the present model is validated by numerical example and comparison with finite element method (FEM) and existing analytical method, with good agreement achieved. The influence of key parameters on the vibro-acoustic behaviors and sound transmission of the double-panel structure is investigated, including: cavity thickness, boundary conditions, sidewall impedance, and the acoustic medium in the cavity.     

任意阵形水声换能器阵辐射声场计算

本文用边界元的方法计算了阵形任意的水声发射换能器共形阵的辐射声场。在详细研究了边界元方法基本理论的基础上，采用Pro／ENGINEER软件与ANSYS软件相结合来建立换能器阵的边界元模型，再用SYSNOISE软件计算其辐射声场和辐射阻抗。这种声场计算方法可用于发射换能器阵设计阶段对设计结果进行预报。     

Study of the planar optical fiber Bragg grating hydrophone probe with acceleration compensation

Since the probe acceleration response of a planar optical FBG hydrophone is very obvious and has an adverse influence on normal underwater acoustic pressure detection,an acceleration desensitization method employing an acceleration compensation measure is presented here.Results indicate that by utilizing this method,the equivalent pressure output applied by one unit of acceleration(1m·s^-2) is reduced from a range of 2.52-3.26 Pa to 0.0758-0.217 Pa, while the structural sensitivity decreases from 28 fm/Pa to 20 fm/Pa,however,the resonance frequency increases slightly from 6.2 kHz to 6.5 kHz.The increased resonance frequency helps to improve the dynamic frequency characteristics,and while the structural sensitivity is reduced, it can be compensated for by improving the optical FBG dynamic wavelength interrogation sensitivity.The system sensitivity of the optical FBG hydrophone is mainly determined by its wavelength interrogation sensitivity and not the probe structure sensitivity.Therefore,the acceleration compensation method would have broad practical applications.     

透平叶栅三维粘性气动反问题的控制理论方法

将基于控制理论的形状优化设计方法应用于粘性可压流动条件下的透平叶栅三维气动反设计,详细推导了三维N-S方程伴随系统的偏微分方程组及其各类边界条件.讨论了伴随系统的解的适定性条件,并由此给出应用N-S方程进行气动优化的目标函数的选取限制.研究了伴随方程的数值求解技术,给出敏感性导数的最终计算式,结合拟牛顿算法发展了三维透平叶栅粘性反问题的气动设计方法.     

The acoustic modes of a rectangular cavity containing a rigid,incomplete partition

A theoretical and experimental study of the acoustic modes of a rectangular cavity containing a rigid, incomplete partition is presented. The theoretical results are obtained by using finite element techniques. It is shown that component mode synthesis techniques, developed for dynamic structural analysis, can be used for acoustic analysis to increase efficiency. The experimental measurements were made on a Perspex model. Good agreement is shown between theory and experiment.     

拖曳线列阵用光纤水听器的研究

报道了光纤水听器用于拖曳线列阵的研究结果。在保证光纤水听器声相位灵敏度的前提下, 优化设计了光纤水听器的结构,降低了光纤水听器的加速度灵敏度36dB。实验测得在20-1600Hz 频段,该光纤水听器的相位灵敏度为-162．7dB,频段内灵敏度的起伏为±0．7dB,加速度灵敏度小于-30dB。     

二维多流体域耦合声场的光滑有限元解法

针对声学有限元分析中四节点等参单元计算精度低,对网格质量敏感的问题,将光滑有限元法引入到多流体域耦合声场的数值分析中,提出了二维多流体域耦合声场的光滑有限元解法。该方法在Helmholtz控制方程与多流体域耦合界面的声压/质点法向速度连续条件的基础上,得到二维多流体耦合声场的离散控制方程,并采用光滑有限元的分区光滑技术将声学梯度矩阵形函数导数的域内积分转换形函数的域边界积分,避免了雅克比矩阵的计算。以管道二维多流体域耦合内声场为数值分析算例,研究结果表明,与标准有限元相比,对单元尺寸较大或扭曲严重的四边形网格模型,光滑有限元的计算精度更高。因此光滑有限元能很好地应用于大尺寸单元或扭曲严重的网格模型下二维多流体域耦合声场的预测,具有良好的工程应用前景。      

Hybrid finite element method applied to supersonic flutter of an empty or partially liquid-filled truncated conical shell

In this study, aeroelastic analysis of a truncated conical shell subjected to the external supersonic airflow is carried out. The structural model is based on a combination of linear Sanders thin shell theory and the classic finite element method. Linearized first-order potential (piston) theory with the curvature correction term is coupled with the structural model to account for pressure loading. The influence of stress stiffening due to internal or external pressure and axial compression is also taken into account. The fluid-filled effect is considered as a velocity potential variable at each node of the shell elements at the fluid-structure interface in terms of nodal elastic displacements. Aeroelastic equations using the hybrid finite element formulation are derived and solved numerically. The results are validated using numerical and theoretical data available in the literature. The analysis is accomplished for conical shells of different boundary conditions and cone angles. In all cases the conical shell loses its stability through coupled-mode flutter. This proposed hybrid finite element method can be used efficiently for design and analysis of conical shells employed in high speed aircraft structures.     

Directivity calculation with experimental verification for a conformal array of underwater acoustic transducers

The radiation directivity of a complicated conformal array of underwater acoustic transducers is presented based on the boundary ＂element method. It includes the element directivity of each transducer, the natural beam pattern and the controlled beam pattern of the transducer array. At first, the boundary element model of the conformal array is built up, and then the boundary condition is exerted on the model according to the design and environment in which the transducer array is used, and the radiation directivity of the conformal array is calculated. An experiment has been done to measure the directivity in an anechoic water tank. The calculated and the experimental results are compared and analyzed. They are consistent to each other. It is shown that the boundary element method together with the detailed calculations is successful to simulate and predict the radiation directivity of an underwater acoustic transducer array.     

Time-domain analyses of acoustics-structure interactions for piezoelectric transducers.

Piezoelectric transducers coupled with a surrounding medium are analyzed in time domain using the coupled finite element and boundary element method. Three-dimensional solid elements are employed to model the piezoelectric transducer, while the surrounding medium is described by the boundary integral equation and the boundary of the medium is modeled by two-dimensional spatial elements. Verification studies were conducted to evaluate the accuracy and convergence of the present numerical algorithm and they show that the present numerical solutions agree well with the analytical ones. The influence of the surrounding medium on the acoustic field is studied. The interaction between the structure and surrounding medium affects the structure dynamic performances and acoustic pressure distributions significantly. However, in the present study we show that the radiation directivity is insignificantly influenced by the surrounding medium.     

DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION OF AN ENGINE MOUNT SYSTEM USING AN FRF-BASED SUBSTRUCTURING METHOD

The FRF-based substructuring method is one of the most powerful methods in analyzing the responses of complex built-up structures with high modal density. In this paper, a general procedure for the design sensitivity analysis of a vibro-acoustic system has been presented using the FRF-based substructuring formulation. For an acoustic response function, the proposed method gives a parametric design sensitivity expression in terms of the partial derivatives of the connection element properties and the transfer functions of the substructures. The derived noise sensitivity formula is combined with a non-linear programming module to obtain the optimal design for the engine mount system of a passenger car. The objective function is defined as the area of the interior noise graph integrated over a concerned r.p.m. range. The interior noise variations with respect to the dynamic characteristics of the engine mounts and bushings have been calculated using the proposed sensitivity formulation and transferred to a non-linear optimization software. To obtain the FRFs, a finite element analysis was used for the engine mount structures and experimental techniques were used for the trimmed body including the cabin cavity. The optimization based on the sensitivity analysis gives the ideal stiffness of the engine mount and bushings. The resultant interior noise in the passenger car shows that the proposed method is efficient and accurate.     

流体加载下加肋板结构的声辐射特性研究

本文采用有限元和边界元方法对加肋板结构的声辐射进行了计算分析,研究分析了加肋板结构的板厚、板面积、板边长比、肋骨惯性矩和间距、边界条件以及板材和流体介质等对结构辐射声功率的影响,得到了一些有意义的结论,从而为揭示加板结构声辐射规律,降低结构的声辐射提供了一定的依据。