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.     

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.     

Acoustic problems analysis of 3D solid with small holes by fast multipole boundary face method

In this paper, an adaptive fast multipole boundary face method is introduced to implement acoustic problems analysis of 3D solids with open-end small tubular shaped holes. The fast multipole boundary face method is referred as FMBFM. These holes are modeled by proposed tube elements. The hole is open-end and intersected with the outer surface of the body. The discretization of the surface with circular inclusions is achieved by applying several special triangular elements or quadrilateral elements. In the FMBFM, the boundary integration and field variables approximation are both performed in the parametric space of each boundary face exactly the same as the B-rep data structure in standard solid modeling packages. Numerical examples for acoustic radiation in this paper demonstrated the accuracy, efficiency and validity of this method.     

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 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.     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.     

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 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.     

Application of the multi-level time-harmonic fast multipole BEM to 3-D visco-elastodynamics

This article extends previous work by the authors on the single- and multi-domain time-harmonic elastodynamic multi-level fast multipole BEM formulations to the case of weakly dissipative viscoelastic media. The underlying boundary integral equation and fast multipole formulations are formally identical to that of elastodynamics, except that the wavenumbers are complex-valued due to attenuation. Attention is focused on evaluating the multipole decomposition of the viscoelastodynamic fundamental solution. A damping-dependent modification of the selection rule for the multipole truncation parameter, required by the presence of complex wavenumbers, is proposed. It is empirically adjusted so as to maintain a constant accuracy over the damping range of interest in the approximation of the fundamental solution, and validated on numerical tests focusing on the evaluation of the latter. The proposed modification is then assessed on 3D single-region and multi-region visco-elastodynamic examples for which exact solutions are known. Finally, the multi-region formulation is applied to the problem of a wave propagating in a semi-infinite medium with a lossy semi-spherical inclusion (seismic wave in alluvial basin). These examples involve problem sizes of up to about 3×10⁵ boundary unknowns.     

An iterative boundary element method for Cauchy inverse problems

 We are interested in this paper in recovering an harmonic function from the knowledge of Cauchy data on some part of the boundary. A new inversion method is introduced. It reduces the Cauchy problem resolution to the determination of the resolution of a sequence of well-posed problems. The sequence of these solutions is proved to converge to the Cauchy problem solution. The algorithm is implemented in the framework of boundary elements. Displayed numerical results highlight its accuracy, as well as its robustness to noisy data. Received 6 November 2000     

Uniform bicubic B-splines applied to boundary element formulation for 3-D scalar problems

In this work, uniform bicubic B-spline functions are used to represent the surface geometry and interpolation functions in the formulation of boundary-element method (BEM) for three-dimensional problems. This is done as a natural generalization of cubic B-spline curves, introduced by Cabral et al, for two-dimensional problems. Three-dimensional scalar problems, with particular applications to Laplace and Helmholtz equations, are considered.     

Scalar wave equation by the boundary element method: a D-BEM approach with non-homogeneous initial conditions

This work is concerned with the development of a D-BEM approach to the solution of 2D scalar wave propagation problems. The time-marching process can be accomplished with the use of the Houbolt method, as usual, or with the use of the Newmark method. Special attention was devoted to the development of a procedure that allows for the computation of the initial conditions contributions. In order to verify the applicability of the Newmark method and also the correctness of the expressions concerned with the computation of the initial conditions contributions, four examples are presented and the D-BEM results are compared with the corresponding analytical solutions.     

A low-frequency fast multipole boundary element method based on analytical integration of the hypersingular integral for 3D acoustic problems

A low-frequency fast multipole boundary element method (FMBEM) for 3D acoustic problems is proposed in this paper. The FMBEM adopts the explicit integration of the hypersingular integral in the dual boundary integral equation (BIE) formulation which was developed recently by Matsumoto, Zheng et al. for boundary discretization with constant element. This explicit integration formulation is analytical in nature and cancels out the divergent terms in the limit process. But two types of regular line integrals remain which are usually evaluated numerically using Gaussian quadrature. For these two types of regular line integrals, an accurate and efficient analytical method to evaluate them is developed in the present paper that does not use the Gaussian quadrature. In addition, the numerical instability of the low-frequency FMBEM using the rotation, coaxial translation and rotation back (RCR) decomposing algorithm for higher frequency acoustic problems is reported in this paper. Numerical examples are presented to validate the FMBEM based on the analytical integration of the hypersingular integral. The diagonal form moment which has analytical expression is applied in the upward pass. The improved low-frequency FMBEM delivers an algorithm with efficiency between the low-frequency FMBEM based on the RCR and the diagonal form FMBEM, and can be used for acoustic problems analysis of higher frequency.     

Evolutionary optimization in thermoelastic problems using the boundary element method

 The paper is devoted to application of evolutionary algorithms and the boundary element method to shape optimization of structures for various thermomechanical criteria, inverse problems of finding an optimal distribution of temperature on the boundary and identification of unknown boundary. Design variables are specified by Bezier curves. Several numerical examples of evolutionary computation are presented. Received 6 November 2000     

Application of boundary element method to 3-D problems of coupled thermoelasticity

This paper deals with a new boundary element method for analysis of the quasistatic problems in coupled thermoelasticity. Through some mathematical manipulation of the Navier equation in elasticity, the heat conduction equation is transformed into a simpler form, similar to the uncoupled-type equation with the modified thermal conductivity which shows the coupling effects. This procedure enables us to treat the coupled thermoelastic problems as an uncoupled one, A few examples are computed by the proposed BEM, and the results obtained are compared with the analytical ones available in the literature, whereby the accuracy and versatility of the proposed method are demonstrated.     

Boundary element solution of 3-D wave scatter problems in a poroelastic medium

This study details the development of boundary integral equations suitable for treating problems involving the scatter of a plane harmonic wave by an inclusion embedded in an infinite poroelastic medium. The pore pressure-solid displacement form of the harmonic equations of motion are developed from the form of the equations originally presented by Biot. Fundamental solutions and a generalized reciprocal work relation are developed, and these are used to formulate a solution representation in terms of an integral over the inclusion surface. The corresponding boundary integral equations are developed in a form that is integrable in the usual sense, eliminating the need to evaluate Cauchy principal value integrals. These boundary integral equations are discretized and implemented into a boundary element computer program. The so-called forbidden frequency problem which causes computational difficulties in boundary integral treatments of wave scatter in elastic and acoustic media is shown to be absent in the poroelastic case. Numerical results obtained from the boundary element program are compared with analytical results for some test problems, and the program appears to produce accurate results at moderate frequencies.     

A new boundary element method for resonance of an acoustic enclosure

Abstract

This paper presents a new boundary element formulation in which the eigenvalue appears outside the integral operator, which distinguishes it from the Helmholtz integral equation. Thus, the formation of global matrices need only be assembled once. Since the kernel of the operator used in the new formulation is real‐valued, all calculations can be carried out in a much simpler way in the real domain. The complex acoustic pressure amplitude is considered herein to deivate by a certain amount from a harmonic function. It is an important contribution that an exact relation between the deviator and the complex acoustic pressure amplitude is constructed locally and thus no more approximations are introduced except conventional boundary discretizations. Several examples are given to illustrate the feasibility of an accurate, effective prediction of resonance.     

New variable transformations for evaluating nearly singular integrals in 3D boundary element method

This work presents new variable transformations for accurate evaluation of the nearly singular integrals arising in the 3D boundary element method (BEM). The proposed method is an extension of the variable transformation method in Ref. [4] for 2D BEM to 3D BEM. In this paper, first a new system denoted as $(\alpha ,\beta )$ is introduced compared with the polar coordinate system. So the original transformations in Ref. [4] can be developed to 3D in $(\alpha ,\beta )$ or the polar coordinate system. Then, the new transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, a new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. To perform integration on irregular elements, an adaptive integration scheme combined with the transformations is applied. Numerical examples compared with other methods are presented. The results demonstrate that our method is accurate and effective.     

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

For many engineers and acousticians, the boundary element method (BEM) provides an invaluable tool in the analysis of complex problems. It is particularly well suited for the examination of acoustical problems within large domains. Unsurprisingly, the widespread application of the BEM continues to produce an academic interest in the methodology. New algorithms and techniques are still being proposed, to extend the functionality of the BEM, and to compute the required numerical tasks with greater accuracy and efficiency. However, for a given global error constraint, the actual computational accuracy that is required from the various numerical procedures is not often discussed. Within this context, this paper presents an investigation into the discretisation and computational errors that arise in the BEM for acoustic scattering. First, accurate routines to compute regular, weakly singular, and nearly weakly singular integral kernels are examined. These are then used to illustrate the effect of the requisite boundary discretisation on the global error. The effects of geometric and impedance singularities are also considered. Subsequently, the actual integration accuracy required to maintain a given global error constraint is established. Several regular and irregular scattering examples are investigated, and empirical parameter guidelines are provided.