تجزیه و تحلیل حساسیت طراحی ارتعاشی آکوستیک با استفاده از روش مبتنی بر موج

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.

The most commonly used numerical prediction techniques for structural–acoustic problems are deterministic element-based methods such as the finite element method (FEM) [1] and [2] and the boundary element method (BEM) [3] and [4]. These methods use locally supported simple polynomials as shape functions within the elements. Since these methods have no restrictions regarding the geometric complexity of the considered problem, they are powerful for analyzing a generally shaped model. For example, in the FEM the entire domain must be divided into small elements; the system matrices are composed of real, banded, symmetric, and frequency independent coefficients, such that efficient storage and solution schemes are possible. However, unbounded problems can only be tackled after the introduction of an additional artificial boundary condition, thereby making the FEM not inherently suited for the prediction of unbounded acoustic problems. In contrast, the BEM requires only the boundary discretization of the considered problem, such that the number of element of boundary element (BE) models is smaller than the corresponding finite element (FE) models. It can also handle unbounded problems, because the Sommerfeld radiation condition is inherently satisfied due to the Green functions applied. However, BE matrix coefficients are complex, fully populated, asymmetric, and frequency dependent, thereby making the model construction and solution processes more complicated than for the FEM. Furthermore, the drawback of element-based methods is that the number of elements and subsequent computational efforts increase as the wavelengths shorten at higher frequencies in order to maintain a reasonable prediction accuracy. As such, the prediction results at higher frequencies are substantially affected by interpolation and pollution errors [5]; thus, the practical applicability of element-based methods is limited to low frequencies. The statistical energy analysis (SEA) [6] is the most widely used method for predicting system response at higher frequencies, using the power flow between subsystems. It is used to estimate the averaged energy response functions of individual subsystems. This statistical approach is computationally efficient, and can be used to predict the dynamic response under uncertainty pertaining to manufacturing tolerances, fabrication imperfections, and the dynamic properties of joints between components. However SEA is not valid for mid-low frequency analyses due to its insufficient modal density for obtaining reliable results [7]. To make up for this weakness, a hybrid method combining the finite element analysis (FEA) and SEA were developed [8] and [9]. However, compatibility at the joint between the SEA and FEA variables must be considered in order to maintain accuracy. Furthermore, major drawbacks for SEA are that the spatial variation of dynamic responses within a subsystem cannot be represented, and that design variables do not explicitly appear in the governing equation. Therefore, SEA analyses are not generally preferred for use in a design sensitivity analysis (DSA); Ref. [10] gives an overview of other, non-SEA based mid-frequency prediction techniques. One of them is the wave-based method (WBM). The WBM is a deterministic prediction technique based on an indirect Trefftz approach [11] and [12]. Dynamic response variables are described by a set of wave functions, which are the exact solutions of the governing differential equations. In the WBM, the main processes of model construction and solution are to determine the contribution factors to make wave functions satisfy the boundary and continuity conditions. Compared to conventional element-based methods, no fine discretization of the domain is necessary, only multiple convex subdomains are required for its convergence—which are independent of the frequencies. As a result, the WBM yields a small number of subdomains and exhibits a high computational efficiency. The efficiency of the WBM at higher frequencies has been shown for various applications in several publications like in Refs. [10] and [13]. In those publications, the WBM has been shown to be more efficient as compared to commonly used element-based methods. Moreover, every field variable can be represented via a combination of corresponding wave functions and contribution factors, thereby leading to a regular sensitivity equation form. In this paper, the DSA of a structural–acoustic semi-coupled problem using the WBM is presented. The DSA is the essential part in the gradient-based optimization. There are many published works conducted for DSA and optimization in the structural–acoustic problem. Ma and Hagiwara [14] and [15] and Choi et al. [16] performed DSA using FEM on the structural–acoustic coupled problem. However, it requires excessive number of 3-D elements predicting the acoustic field variable using FEM. To avoid the discretization of air domain, BEM replaces FEM in the acoustic analysis. Salagame et al. [17] derived analytical sensitivity of acoustic power radiated from plates using analytic method. Wang and Lee [18] developed global acoustic DSA by employing continuum sensitivity. The chain rule regarding the continuum structural sensitivity with acoustic sensitivity was used to obtain accurate and efficient sensitivity results. Lee et al. [19] introduced the topology optimization on boundary element model using genetic algorithm. Kim et al. [20] presented the design sensitivity analysis of structural–acoustic problems using sequential adjoint variable method (AVM). The adjoint variable is calculated from the structural adjoint reanalysis. Note that the earlier works are the element-based DSA of a structural–acoustic system, whereas there are few papers pertaining to DSA using the WBM. Recently, a DSA for a mid-frequency analysis of a coupled vibro-acoustic problem was developed [21], though it adopts a direct differentiation method (DDM) to obtain the sensitivity. DDM directly solves for the design dependency of a state variable, and then computes the sensitivity using a chain rule. However, the AVM constructs an adjoint problem that solves for the adjoint variable, which contains all implicitly dependent terms. Hence, the AVM is preferable when the number of design variables is larger than active constraints [22]. To this end, most engineering problems are focused on optimizing a particular objective function of a model consisting of many design variables; therefore, the AVM is considered much more efficient than the DDM. In this paper, the structural dynamic response is obtained using the FEM, and the results are imported as boundary conditions for the wave-based acoustic model. The cross-sectional area of each structural element is considered as a sizing design variable. The adjoint variable is then obtained from a structural reanalysis using an adjoint load comprised of the wave-based system matrix and an evaluation of the wave functions of each boundary location. The sensitivity formulation proposed in this paper is subsequently applied to a generally shaped two-dimensional (2D) vehicle model, and finally the results are compared to finite difference sensitivity results, with good agreement obtained.

The DSA of a structural–acoustic semi-coupled problem was presented using the WBM. The computation of the acoustic analysis was performed efficiently using the WBM, which also resulted in enhanced computation algorithm in the DSA. For more efficient and accurate sensitivity analyses, the AVM was implemented with an adjoint equation. In the process of finding the adjoint load, the wave-based system matrix and wave function evaluations at field points and boundary locations were manipulated. Using the original and adjoint responses of the structural model, the design sensitivity of each design variable was then formulated in a sesqui-linear form comprised of an explicit derivative equation of the original system. The accuracy of the AVM was subsequently validated through a comparison with the sensitivity results from the FDM. As a result, it was found that the adjoint variable method significantly reduced computational costs compared with the DDM as the number of the design variables was increased. These sensitivity results can be helpful for model updates in gradient-based optimizations. In this paper, the 2D vehicle cavity model is considered for the validation of design sensitivity analysis of interior acoustic problem. There are only velocity boundary conditions on the model. To deal with more practical problems, the exterior sound pressure of 3D cavity should be considered in the future. Also impedance boundary conditions should be considered with velocity boundary conditions to give acoustic damping on the cavity. Thus far the FEM was used for the structural analysis for the sizing DSA. If the shape design parameters such as the length or area of the structural model are considered, it would be good to use the WBM for finding both of structural and acoustic responses without the FEM. This is because the structural model does not need to be divided into several elements for configuring the design variables in sizing DSA. Then the analysis and sensitivity formulations are processed with less time consumption, and DSA considering design parameters of WBM model would be possible.