Efficient evaluation of the acoustic radiation using multipole expansion

The variational boundary element method (VBEM) is widely used to compute the acoustic radiation of structures. The classical numerical implementation of the VBEM suffers from the computational cost associated with double surface integration. In a previous paper [1], the authors proposed a novel method, based on multipole expansions, to accelerate the double layer potential calculation for structures having a periodic mesh. This technique, while efficient, is still limited by the cost of computing the surface pressure from the double surface potential. This paper presents an acceleration technique, based on multipole expansion, that allies both efficiency and accuracy. Copyright © 1999 John Wiley & Sons, Ltd.     

基于快速多极子边界元法的齿轮箱声场分析

齿轮箱是广泛应用的工程机械零部件,准确地模拟其辐射声场对后续的降噪优化设计有着重要作用。边界元方法非常适合分析此类无限域下的声辐射问题。但传统边界元方法有着计算效率低、内存占用高的缺点。该研究发展了宽频的快速多极子边界元方法,并运用该方法计算了齿轮箱在特定频率下的场点声压以及辐射声场。通过对比商用软件的分析结果,验证了所提快速边界元方法的准确性。此外,运用多核并行计算方法,对计算量较大的扫频分析进行加速计算,最终快速、准确地获取了齿轮箱辐射声场的扫频结果。     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.     

hp Fast multipole boundary element method for 3D acoustics

A fast multipole boundary element method (FMBEM) extended by an adaptive mesh refinement algorithm for solving acoustic problems in three‐dimensional space is presented in this paper. The Collocation method is used, and the Burton–Miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems. Because of the application of the combined integral equation, the developed FMBEM is feasible for all positive wave numbers even up to high frequencies. In order to evaluate the hypersingular integral resulting from the Burton–Miller formulation of the boundary integral equation, an integration technique for arbitrary element order is applied. The fast multipole method combined with an arbitrary order h‐p mesh refinement strategy enables accurate computation of large‐scale systems. Numerical examples substantiate the high accuracy attainable by the developed FMBEM, while requiring only moderate computational effort at the same time. Copyright © 2016 John Wiley & Sons, Ltd.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

A spectral multipole method for efficient solution of large-scale boundary element models in elastostatics

In this paper we introduce a method to reduce the solution cost for Boundary Element (BE) models from O(N³)operations to O(N²logN) operations (where N is the number of elements in the model). Previous attempts to achieve such an improvement in efficiency have been restricted in their applicability to problems with regular geometries defined on a uniform mesh. We have developed the Spectral Multipole Method (SMM) which can be used not only for problems with arbitrary geometries but also with a variety of element types. The memory necessary to store the required influence coefficients for the spectral multipole method is O(N) whereas the memory required for the traditional Boundary Element method is O(N²). We demonstrate the savings in computational speed and fast memory requirements in some numerical examples. We have established that the break-even point for the method can be as low as 500 elements, which implies that the method is not only suitable for extremely large-scale problems, but that it also provides a useful bridge between the small-scale and large-scale problems. We also demonstrate the performance of the multipole algorithm on the solution of large-scale granular assembly models. The large-scale BE capacity provided by this algorithm will not only prove to be useful in large macroscopic models but it will also make it possible to model microscopic damage processes that form the fundamental mechanisms in plastic flow and brittle fracture.     

A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation

A new fast multipole formulation for the hypersingular BIE (HBIE) for 2D elasticity is presented in this paper based on a complex-variable representation of the kernels, similar to the formulation developed earlier for the conventional BIE (CBIE). A dual BIE formulation using a linear combination of the developed CBIE and HBIE is applied to analyze multi-domain problems with thin inclusions or open cracks. Two pre-conditioners for the fast multipole boundary element method (BEM) are devised and their effectiveness and efficiencies in solving large-scale problems are discussed. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM using the dual BIE formulation. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale 2D multi-domain elasticity problems. The method can be applied to study composite materials, functionally-graded materials, and micro-electro-mechanical-systems with coupled fields, all of which often involve thin shapes or thin inclusions.     

A fast multipole boundary element method for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation

A fast multipole boundary element method (FMBEM) for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation is presented in this paper. A multi-tree structure is designed for the multi-domain FMBEM. It results in mismatch of leaves and well separate cells definition in different domains and complicates the implementation of the algorithm, especially for preconditioning. A preconditioner based on boundary blocks is devised for the multi-domain FMBEM and its efficiency in reducing the number of iterations in solving large-scale multi-domain scattering problems is demonstrated. In addition to the analytical moment, another method, based on the anti-symmetry of the moment kernel, is developed to reduce the moment computation further by a factor of two. Frequency sweep analysis of a penetrable sphere shows that the multi-domain FMBEM based on the Burton-Miller formulation can overcome the non-unique solution problem at the fictitious eigenfrequencies. Several other numerical examples are presented to demonstrate the accuracy and efficiency of the developed multi-domain FMBEM for acoustic problems. In spite of the high cost of memory and CPU time for the multi-tree structure in the multi-domain FMBEM, a large BEM model studied with a PC has 0.3 million elements corresponding to 0.6 million unknowns, which clearly shows the potential of the developed FMBEM in solving large-scale multi-domain acoustics problems.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

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.     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A comparison of boundary methods based on inverse variational formulation

The aim of this paper is to review and compare the existing direct boundary methods. Each method is briefly characterised. Similarity of these methods is pointed out by showing their common origin (which is the inverse variational formulation) and much the same way of obtaining final matrices. The accuracy and efficiency of the methods are compared in numerical experiments of two-dimensional Laplace problem.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A shape variational approach for determining potential lines using the boundary element method

A shape optimization (i.e., variational) approach is adopted in this paper to determine potential lines. Solution of the primary potential problem is accomplished by the boundary element method (BEM). To find the position of any potential line a suitably defined objective functional is minimized using structural optimization techniques. In particular, the sensitivity analysis expression of the objective functional with respect to the shape variation of internal curves is derived by employing the material derivative concept. With the internal potential and potential derivative values available through the BEM solution of the potential problem, the correct shapes of the potential contours are found by a shape minimization procedure, After checking the numerical results by a one-dimensional problem, potential contours for two-dimensional example problems are determined and plotted.     

Fast multipole boundary element approaches for acoustic attenuation prediction of reactive silencers

The fast multipole boundary element method (FMBEM) is applied to predict the acoustic attenuation performance of reactive silencers. In order to overcome the difficulty of singular boundaries for the acoustic computation of reactive silencers with internal thin wall structure or/and perforated components, two approaches, the substructure FMBEM (Sub-FMBEM) and mixed-body FMBEM (MB-FMBEM) are proposed, and the theoretical foundations and numerical processes of the both approaches are introduced. The studies demonstrated that the ordering of column vectors and numbering of nodes in the Sub-FMBEM have great influence on the convergence of iteration, and the MB-FMBEM may reduce the number of elements and the computational complexity since it only needs to discretize one side boundary of the thin wall and perforated components and it is not necessary to create the interfaces. The Sub-FMBEM, MB-FMBEM and Sub-BEM are then employed to calculate the transmission loss of reactive silencers with thin wall components and perforated tubes, the computational accuracy and efficiency of the approaches are validated. The data of precomputing time and total iterative computational time demonstrated that, the computational efficiency of Sub-FMBEM will descend as the frequency arising, and the Sub-FMBEM may reveal higher computational efficiency than Sub-BEM only when the number of nodes is big enough.     

A new hybrid boundary node method based on Taylor expansion and the Shepard interpolation method

A novel meshless method based on the Shepard and Taylor interpolation method (STIM) and the hybrid boundary node method (HBNM) is proposed. Based on the Shepard interpolation method and Taylor expansion, the STIM is developed to construct the shape function of the HBNM. In the STIM, the Shepard shape function is used as the basic function, which is the zero‐level shape function, and the high‐power basic functions are constructed through Taylor expansion. Four advantages of the STIM are the interpolation property, the arbitrarily high‐order consistency, the absence of inversion for the whole process of shape function construction, and the low computational expense. These properties are desirable in the implementation of meshless methods. By combining the STIM and the HBNM, a much more effective meshless method is proposed to solve the elasticity problems. Compared with the traditional HBNM, the STIM can improve accuracy because of the use of high‐power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluation of the T-stress and stress intensity factor for a cracked plate in general case using eigenfunction expansion variational method

This paper investigates the T-stress and stress intensity factor for a cracked plate in general case. In the general case, the shape of boundary and the applied loading are arbitrary. The eigenfunction expansion variational method (EEVM) is developed to evaluate the T-stress and stress intensity factor. For the traction boundary value problem, the EEVM is equivalent to the theorem of least potential energy in elasticity. Therefore, the EEVM possesses a clear physical meaning and it does not depend on any boundary collocation scheme. Several numerical examples are presented, which include: (1) a line crack in circular plate and (2) a line crack in rectangular plate. Numerical examination for convergence in an example is carried out.     

基于分布源边界点法的近场声全息实验研究

采用基于分布源边界点法的近场声全息技术对运转状态下的罗茨真空泵进行了噪声源识别和定位研究。实验采用13个声压传感器组成的传声器阵列和1个参考传声器测量全息面上的复声压信息，进而通过基于分布源边界点法的近场声全息技术和Tikhonov正则化方法重建获得声源的表面法向振速。重建结果表明：在测量信号中存在的2个低频噪声是支架在真空泵振动的激励作用下产生共振所致，而非真空泵本身所固有的噪声；而1533．2～1699．2Hz的宽带噪声则是真空泵端盖表面振动所产生，是真空泵所固有的噪声。同时，重建结果给出了各频率处噪声源的具体位置，为有效地进行真空泵的噪声控制提供了依据。