Journal of Computational Acoustics

Theory and algorithm for the acoustic boundary element solver will be described. The solver represents the core of the package AField (acronym for acoustical field) and can be used for getting fast and stable solutions for huge as well as for small boundary elements models on PCs under Windows. The boundary element solver includes an early proposed GMRES iterative linear system solver (which can be used for big and huge models) and several traditional linear system solvers (for small and medium models). Implementation of the Burthon and Miller regularization method for constant boundary elements will be discussed. Approximations in proposed boundary element solver and their influence on solution errors are investigated. Results for several models are included. Several possibilities of Afield package are illustrated.

This paper reviews a number of techniques developed to overcome the well-known nonuniqueness problem in boundary integral formulations of acoustic radiation. Although the problem has received much attention, comparative studies are hardly known in this field. Furthermore, the problem has often been studied using an unsuitable example, namely a simple radiating sphere. In this case, often the addition of one collocation point in the centre of the sphere suffices to remove the nonuniqueness problem for a large range of wavenumbers. In contrast to the radiating sphere, the radiating cat's eye structure is considered in this paper. Solution of the discretized ordinary Kirchhoff–Helmholtz integral equation, also known as the Surface Helmholtz Equation, reveals a large number of so-called irregular frequencies, i.e. frequencies where the BEM fails. The paper compares the performance of different methods in alleviating this failure. The CHIEF method and its variation due to Rosen et al. are found to encounter difficulties at high frequencies. A much better performance is obtained by combining the Kirchhoff–Helmholtz integral equation with its normal derivative. In particular the method of Burton and Miller and a modification of it which avoids evaluating the hypersingular operator at nonsmooth points are tested. Both methods seem to provide reliable solutions. The modified method encounters minor failures in the frequency response function at a geometric singularity, although performing surprisingly well in many cases. More tests need to be carried out to assess fully the effectiveness of this method which allows easy use of continuous quadratic elements. However, it is the Burton and Miller formulation which appears to be the most reliable for acoustic radiation analysis. The use of CHIEF and its variations cannot be recommended.

The fast multipole boundary element method (FMBEM) is an advanced BEM that leads to drastic reduction of processing time and memory requirements in a large-scale steady-state sound field analysis. In the FMBEM, hierarchical cell structure is employed to apply multipole expansion in multiple levels, and the setting of the hierarchical cell structure considerably affects the computational efficiency of the FMBEM. This paper deals with effective settings of hierarchical cell structure for taking full advantage of the FMBEM. A numerical study with objects of different shapes with the same DOF shows that both the computational complexity and the memory requirements with the FMBEM were greater for 1D-shaped objects than for 2D- or 3D-shaped ones, without a special setting of hierarchical cell structure for each problem. An effective setting for 1D-shaped objects is derived through theoretical and numerical studies, where special considerations are given to the arrangement of the cell structure and the treatment of translation coefficients between cells. This setting allows for efficient calculations not dependent on the shape of an analyzed object. A simple method to arrange hierarchical cell structure is proposed, which realizes the derived setting for arbitrarily-shaped problems.

The fast multipole boundary element method (FMBEM) is an advanced BEM, with which both the operation count and the memory requirements are O(N^alog^b N) for large-scale problems, where N is the degree of freedom (DOF), a ≥ 1 and b ≥ 0. In this paper, an efficient technique for analyses of plane-symmetric sound fields in the acoustic FMBEM is proposed. Half-space sound fields where an infinite rigid plane exists are typical cases of these fields. When one plane of symmetry is assumed, the number of elements and cells required for the FMBEM with this technique are half of those for the FMBEM used in a naive manner. In consequence, this technique reduces both the computational complexity and the memory requirements for the FMBEM almost by half. The technique is validated with respect to accuracy and efficiency through numerical study.

The fast multipole method is employed to accelerate the Galerkin boundary element method for acoustics. Important parameters as the nearfield size and the expansion length are discussed and numerical examples are provided. An approximate inverse preconditioner and a preconditioner based on spectral properties of the boundary integral operators are presented. For the example of sound radiation from a brake disk, both approaches prove suitable to reduce the iteration count and to improve the efficiency of the method.

The prediction of the spatial mean-square pressure distribution within enclosed high-frequency broadband sound fields is computationally intensive if determined on a frequency-by-frequency basis. Recently an energy-intensity boundary element method (EIBEM) has been formally developed. This method employs uncorrelated broadband directional energy sources to expeditiously predict such pressure distributions. The source directivity accounts for local correlation effects and specular reflection. The method is applicable to high modal density fields, but not restricted to the usual low-absorption, diffuse, and quasi-uniform assumptions. The approach can accommodate fully specular reflection, or any combination of diffuse and specular reflection. This boundary element method differs from the classical version in that element size is large compared to an acoustic wavelength and equations are not solved on a frequency-by-frequency basis. In the earlier EIBEM, the source strength and directivity associated with the energy sources, distributed over enclosure boundaries, were determined in an iterative manner and the directivity was limited to three terms of a Fourier expansion. Here, the original method is improved by eliminating the iteration and allowing for an unlimited number of terms in the Fourier expansion of the directivity function. For verification, the improved EIBEM is compared to experimental measurements and exact analytical solutions; excellent agreement is obtained.

In applying the boundary element method to an acoustic interior problem modeled as a single domain, even a small change of the cavity geometry affects the whole system matrix. Thus, calculating sensitivities for each intermediate shape is performed independently during iterations for an optimum shape. By utilizing the sub-domain modularization capability of the MBEM, however, the matrix inversion task is performed only once for the initial shape during whole iterations. This can reduce the total computational costs of finding an optimum shape design and the computational efficiency increases as the number of iterations increases. A demonstration example is given by a two-dimensional automotive interior cavity.

A boundary integral equation is described for the prediction of acoustic propagation from a monofrequency coherent line source in a cutting with impedance boundary conditions onto surrounding flat impedance ground. The problem is stated as a boundary value problem for the Helmholtz equation and is subsequently reformulated as a system of boundary integral equations via Green's theorem. The numerical solution of the coupled boundary integral equations by a simple boundary element method is then described. Predictions of A-weighted insertion losses for a traffic noise spectrum are made illustrating the effects of depth of the cutting and the profile of the associated noise barrier.

We consider the problem of recovering surface vibrations from acoustic pressure measurements taken in the interior or the exterior of a region. We give two formulations of the problem. One is based on a representation of the pressure as layer potentials and the other is based on approximation by a class of specific solutions to the Helmholtz equation. Boundary element methods are developed to approximate the integral operators. A conjugate gradient algorithm based on Lanczos bidiagonalization is applied to compute regularized solutions to the ill-conditioned equations in the presence of measurement errors. Numerical examples which compare the two formulations are presented.

The self-developed Boundary Element code BEMCUP-3D solves structural-dynamic and acoustic problems as well as fluid-structure-interaction-phenomena in the frequency domain. Attainable outputs of this program are e.g. the system matrices. The inverse acoustic problem (sound source identification) is considered without inversion of matrices. The envelope surface (measurement surface), which encloses the entire arbitrarily shaped sound source, is treated as an exterior problem. The Dirichlet data on this surface or boundary are given from the proper sound pressure distribution of the sound source, which is also an exterior problem. This ensures that the corresponding velocity values on the measurement surface are exactly the same for both problems. Next the region between the sound source (an arbitrarily vibrating structure) and the envelope surface (a measurement surface in experimental investigations) is treated as an interior problem. The boundary conditions on the outer surface (measurement surface) for this problem are of Dirichlet type and the already available Neumann data, the sound pressure and the velocity distributions. An algorithm makes sure that after solving the unknown sound pressure and velocity values of the sound source are situated in the solving vector. Simple sound sources enable to investigate the stability, an optimal shape and an optimal position of the measurement surface.

Numerical methods for computing the three-dimensional pressure field in a flat fluid channel bounded either by a rigid boundary, an elastic semi-infinite medium or by a layer of sediment, subjected to incoherent line sources are presented. After verification Greens functions are incorporated in a Boundary Element Method (BEM) code that simulates the pressure variation inside the fluid channel in the vicinity of a rigid or elastic inclusion, avoiding the discretization of the fluid and solid channel interfaces. After the verification of the solution, the models developed are then used to simulate the pressure variation within the fluid channel in the presence of infinitely long rigid and elastic inclusions of differing sizes, when the channel is struck by a spatially-sinusoidal harmonic pressure line load. The results are then compared with those obtained when the channel floor is assumed to be rigid. Time domain results are given by means of inverse Fourier transforms, to help understand how the mechanical properties of the channel floor may affect the variation of the pressure field within the channel.

Green's functions for elastic solids with random properties are usually derived by means of the perturbation method. This paper deals with a new approach that has the potential to deal with a large variability of random shear modulus based on the transformation of a polynomial chaos. The deterministic Green's functions for stresses and displacements and the principal values in the boundary integrals caused by a pressure load on the surface are nonlinear transformations of the random variables. A series of transformations of the polynomial chaos is used to transform significant parts of the equation. The first operation is a projection of the log normal distributed shear modulus to a series of Hermite's polynomials based on a Gaussian variable. The second operation is the determination of an arbitrary potential of the wave velocity. The last operation, similar to the first one, consists in the determination of an exponential function depending on the inverse of the wave velocity. These operations, together with multiplications and summations, transform the complete relation from the random shear modulus to Green's functions and principal values. The inversion of the system matrix is already derived for the random finite element approach. The operations are independent of the specific problem and can be applied to almost all acoustic media and similar nonlinear problems.