روش پهنای باند سریع عنصر مرزی چند قطبی برای تجزیه و تحلیل حساسیت شکلی آکوستیکی سه بعدی بر اساس روش تمایز مستقیم

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton–Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.

Acoustic shape sensitivity analysis is an important step of acoustic design and optimization processes. It can provide information on how the geometry change affects the acoustic performance of the given structure. So far, many sensitivity analysis methods have been proposed based on the finite element method (FEM) [1], [2] and [3] and the boundary element method (BEM) [4], [5], [6], [7], [8], [9] and [10]. But due to the convenience in remeshing compared with the FEM as well as relatively good accuracy of the solutions on the boundary and especially the incomparable superiority in solving infinite or semi-infinite acoustic problems, the BEM has been very widely used in acoustic sensitivity analyses [11] and [12]. Matsumoto et al. [5] and Koo et al. [6] derived two different acoustic sensitivity boundary integral equations with respect to shape design variables. Kim and Dong [7] presented a shape design sensitivity formulation for structural-acoustic problems using sequential finite element and boundary element methods. However, in these researches, the authors started from the conventional boundary integral equation (CBIE) but did not consider the fictitious eigenfrequency problem of it in solving exterior acoustic wave problems. Smith and Bernhard [8] proposed a semi-analytical sensitivity formulation and resolved the fictitious eigenfrequency problem by using the CHIEF method [13]. Arai et al. [9] derived an analytical shape sensitivity equation based on a modified Burton–Miller formulation [14]. However, though the BEM reduces the problem dimensionality by one, the conventional boundary element method (CBEM) produces a fully populated, non-symmetric and sometimes ill-conditioned coefficient matrix which leads to increased storage requirements and machine time in comparison with domain methods such as the FEM and the finite difference method (FDM). This well-known drawback had constrained the extensive usage of the BEM in large-scale engineering applications for several decades. For instance, as for a problem involving N degrees of freedom, a direct solver, such as the Gauss elimination method requires the storage of O(N2)O(N2) and the solution cost of O(N3)O(N3). Even worse in the shape sensitivity analysis by the direct differentiation method, more integral evaluations for each pair of boundary elements are needed. While, use of iterative solvers, such as the GMRES [15], does not reduce the storage requirements but can reduce the solution cost to O(MN2)O(MN2), where M is the number of iterations required, and O(N2)O(N2) per iteration cost arising from the dense matrix–vector product. This is still quite expensive for large-scale problems. In order to further improve the efficiency and reduce the storage requirements of the BEM with iterative solvers, various acceleration techniques, such as the fast multipole method (FMM) [16], the fast wavelet transforms [17], the precorrected-FFT [18], and the H-matrices [19], have been proposed to accelerate the matrix–vector product. Among these methods, the FMM seems to be one of the most widely accepted methods in the fast BEM community. Moreover, it has also been acclaimed as one of the top 10 algorithms of the 20th century [20]. The FMM allows the matrix–vector product to be performed to a given precision in O(N) operations and reduces the storage requirements to O(N) as well, for instance, for potential problems or low-frequency acoustic wave problems. This method was first introduced by Greengard and Rokhlin [16], and then intensively studied and extended to the solution of problems arising from the Laplace, Helmholtz, Maxwell, and other equations [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] and [38]. A comprehensive review can be found in [39]. This paper promotes the applications of fast multipole boundary element method (FMBEM) in three dimensional acoustic shape sensitivity analyses. In order to tackle the fictitious eigenfrequency problem when solving exterior acoustic problems, the Burton–Miller method [40] is employed in this study. The Burton-Miller method is a linear combination of the CBIE and the normal derivative boundary integral equation (NDBIE), and has been proved to yield unique solutions for all frequencies if the coupling constant of the two equations is chosen properly [40] and [41]. However, the most difficult part in implementing this approach is that the NDBIE is a hypersingular type involving a double normal derivative of the fundamental solution. Although various singularity subtration techniques have been proposed to evaluate such hypersingular terms, most of them are still cumbersome and require extremely complicated numerical procedure in general [5], [6], [9], [42], [43], [44], [45], [46], [47] and [48]. Worse is the case when it comes to the FMBEM, for instance when the formulation is regularized by using the fundamental solution of Laplace's equation, multipole expansion formulations and other translation formulations have to be implemented not only for the fundamental solution and its derivatives of the Helmholtz equation but also for those of Laplace's equation. However, the constant element is used to discretize the problem boundary in this study, and in this case all the hypersingular boundary integrals can be evaluated explicitly and directly. Hence the computational process is more efficient than that of any other singularity subtraction technique [49], [50] and [51]. This paper is organized as follows. The BEM and FMM formulations for the boundary integral equations for the sensitivity coefficients derived based on the direct differentiation method are introduced in Section 2. The wideband FMM algorithm is presented in Section 3. Section 4 gives three numerical examples to demonstrate the validity and efficiency of the method. Section 5 concludes the paper with further discussions.

A wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis has been presented in this paper. The sensitivity equations are obtained by the direct differentiation method. The Burton–Miller method is employed to conquer the fictitious eigenfrequency problem which is also observed in the computation for solving exterior acoustic sensitivity problems based on the conventional sensitivity boundary integral formulation. Numerical examples clearly demonstrate the potential of FMBEM approach for solving large-scale acoustic shape sensitivity problems. Further studies can be carried out for more complicated and practical problems by the developed code. In the numerical analysis, because the constant triangular element is used to discretize the model, the strongly singular and hypersingular boundary integrals existing in the sensitivity formulations can be evaluated explicitly and directly. Therefore, no additional multipole expansion formulas that are needed in the boundary integral representations based on regularization are required in the present formulation. Although, this paper is focused only on the acoustic shape sensitivity analysis, the approach can also be applied to solve other types of sensitivity problems, such as potential, elastostatic and elastodynamic problems. Moreover, this paper implements the fast multipole boundary element shape sensitivity analysis based on the direct differentiation method, so that the approach is very powerful when there are few design variables. However, if there are large number of design variables, the adjoint variable method is advantageous to use, and it is very easy to extend the present work to the adjoint variable method.