Singular boundary method for 2D and 3D acoustic design sensitivity analysis
Introduction
Acoustic design sensitivity analysis is very important for the acoustic optimization, and can provide information on how changes the design parameters affecting the acoustic performance of a given structure [1], [2], [3]. In recent years, great efforts have been made to develop various numerical methods for the acoustic design sensitivity analysis. Among them, the finite element method (FEM) [4], [5], [6], [7] is one of the most widely used approaches. However, this method requires tedious domain meshing which is often computationally costly and sometimes mathematically troublesome for certain classes of problems. In particular, the FEM has to artificially truncate an infinite domain into a finite one with subtle artificial boundary conditions or absorbing layers. This truncation is largely dependent on various trial-error or empirical approaches.
As an alternative approach, the boundary element method (BEM) has a broad application prospect in the acoustic problems due to the boundary-only discretization and semi-analytical property. This scheme approximates the solution of the considered problem by adopting the fundamental solution, which satisfies the governing equation and the boundary condition at infinity, to naturally reduce an infinite domain problem into a finite boundary problem. Smith and Bernhard [8] derived the semi-analytical sensitivity formulation by differentiating the discretized boundary integral equation. Matsumoto et al. [2], [9], Koo et al. [10] and Kane et al. [11] derived the acoustic sensitivity formulations of the BEM with respect to the shape design variables. Zheng et al. [12] proposed the fast multipole BEM formulations of the three-dimensional acoustic shape sensitivity. Chen et al. [13], [14] presented the isogeometric BEM formulations of the acoustic design sensitivity analysis and topology optimization analysis. Despite its great advantages, the BEM has to evaluate weakly singular, strongly singular, or hyper-singular integrals, which is usually a cumbersome and non-trivial task.
In recent years, tremendous efforts have been made to overcome the above shortcomings. Various meshless/meshfree methods [15], [16], [17] have been proposed and applied in acoustic problems, such as the method of fundamental solutions (MFS) [18], [19], the regularized meshless method (RMM) [20], the singular boundary method (SBM) [21], [22], the boundary knot method (BKM) [23], [24], and the element-free Galerkin method [25], [26]. Among them, the BKM uses the non-singular general solution as the basis function, which does not satisfy the boundary condition at infinity. Like the BEM with constant elements, the SBM adopts the fundamental solution, and thus is convenient to simulate exterior acoustic problems. Unlike the BEM, the SBM is mathematically simple, free of integration and mesh, and easy-to program. As a domain-type meshless method with weak form, the element-free Galerkin method has been successfully used to simulate various problems in applied mathematics and mechanics, including acoustics [27] and structural optimizations [28]. This method is theoretically suitable for solving any problem, whether the fundamental solution is available or not. However, the solution of exterior acoustic problems is still troublesome, which is similar to the FEM. The SBM is a semi-analytical and boundary-type meshless collocation method with strong form, and is suggested in solving exterior acoustic problems, since the used fundamental solutions can automatically satisfy Sommerfeld's condition at infinity. Up to now, the SBM has been applied to the numerical simulation of various acoustic problems [21], [22], [29], [30], [31]. For an overview of the state of the art in the theories, algorithms, and software packages, see Ref. [32].
When the SBM is applied to exterior acoustic problems directly, its approximation solution may encounter non-uniqueness issue at eigenfrequencies of a corresponding interior problem. With this regard, Fu et al. [21] introduced the Burton-Miller formulation into the SBM and developed the Burton-Miller-type SBM (BM-SBM). The Burton-Miller formulation [33] approximates the solutions of infinite field problems by a combination of boundary integral equation and its normal derivative, and has been proved to be an efficient method to circumvent the fictitious eigenfrequency problem at all frequencies [34]. Based on this advantage, the Burton-Miller formulation has been widely used in the BEM [35], [36] and certain meshfree methods [21], [37], [38].
This paper aims to establish the numerical framework of the Burton-Miller-type SBM (BM-SBM) for the acoustic design sensitivity analysis. In sight of the problem of exterior acoustics, the Burton-Miller formulation is employed to overcome the fictitious frequency problem. The accurate evaluation of the origin intensity factor (OIF) is vitally important for the SBM, and therefore great efforts have been made to address this problem in recent years. The inverse interpolation technique (IIT) [39], the subtraction and adding-back technique (SAB) [40], and the empirical formula formulas [41], [42], [43] are presented one after another. In the BM-SBM proposed by Fu et al. [21], the OIF of Helmholtz equation is divided into the OIF of Laplace equation and a constant, and then the IIT and the SAB are respectively used to evaluate the OIFs of Laplace equation associated with the Dirichlet and Neumann boundary conditions. A large number of numerical experiments indicate that the IIT is lack of theoretical analysis [32], and is inefficient due to solving the equation twice. For this reason, this study uses the simple, accurate and efficient empirical formula instead of the IIT. In addition, the direct differentiation method is introduced into the BM-SBM formulations for computing the sensitivity of sound pressure with respect to the design variable.
This paper is organized as follows. The basic theory of acoustic design sensitivity analysis is introduced in section 2. The SBM and BM-SBM formulations in the acoustic sensitivity analysis are provided in sections 3 and 4, respectively. In section 5, the OIFs of the SBM and BM-SBM are presented. In Section 6, benchmark numerical examples, including the acoustic sensitivity analysis with the design variable being the wave number and the structure size, are provided to validate the accuracy and efficiency of the proposed method. Finally, some conclusions and remarks are drawn in Section 7.
Section snippets
Acoustic sensitivity
The acoustic sensitivity of the vibrating object is the derivative of the objective function to the design variables. Design variables refer to the wave number, structural shape/size and air density etc. The acoustic sensitivity analysis of the vibrating object can optimize the parameters of the vibrating object, so as to minimize the noise radiation.
The propagation of sound in a homogeneous isotropic medium can be described by the following Helmholtz equation [16]: with
SBM for acoustic design sensitivity
The SBM is a semi-analytical and boundary-type meshless method using singular fundamental solutions. This approach introduces a concept of origin intensity factor (OIF) [39] to address the singularity of fundamental solution, and avoids the fictitious boundary issue in the MFS, as shown in Fig. 1. According to the basic theory of SBM, N source points are firstly placed on the boundary, and the collocation points completely coincide with the source points. The SBM approximates
Burton-Miller-type SBM (BM-SBM) for acoustic design sensitivity
Unlike the traditional SBM, the BM-SBM formulations for the analysis of exterior acoustic field are written as follows where , , , and are the OIFs corresponding to the Burton-Miller-type formulations. The sound pressure at any interior point can be calculated by using the following formula
Origin intensity factors (OIFs)
The accurate and fast estimation of the OIFs is the key of the SBM. In view of this, some effective strategies have been developed in recent years. These methods mainly include the inverse interpolation technique (IIT), the subtraction and adding-back technique (SAB), and the empirical formulas. Among them, the empirical formula is most simple and easy-to-use [41], [42], [43].
For the SBM of acoustic analysis, the OIFs and in Eqs. (4), (5), (13) and (14) can be expressed by [21], [41],
Numerical results and discussions
In this section, benchmark examples are provided to illustrate the accuracy and effectiveness of the proposed scheme. The numerical accuracy is evaluated by using the relative-root-mean-square error (Global error), where represent the M equidistant nodes for numerical differentiation, and indicate exact and numerical solutions of acoustic sensitivity at the design variable . We assume that the air density is
Conclusions
This paper proposed the Burton-Miller-type-SBM (BM-SBM) formulations for the 2D and 3D acoustic sensitivity analysis, based on the direct differentiation method. The introduction of Burton-Miller formula can overcome the problems of non-unique solutions of exterior problems at the fictitious frequencies of the corresponding interior problems. The source intensity factors in the BM-SBM were directly calculated by using the simple and efficient empirical formula. Like the boundary element method,
Acknowledgements
The work described in this paper was supported by the National Natural Science Foundation of China (No. 11802151), the Natural Science Foundation of Shandong Province of China (No. ZR2019BA008), and the China Postdoctoral Science Foundation (Nos. 2019M652315, 2019M650158).
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