T.W. Wu

Boundary Element Acoustics Fundamentals and Computer Codes

Fundamentals of Linear Acoustics The Helmholtz Integral Equation Two-Dimensional Problems Three-Dimensional Problems The Normal-Derivative Integral Equation Indirect Variational Boundary Element Method in Acoustics Acoustic Eigenvalue Analysis by Boundary Element Methods Time Domain Three-Dimensional Analysis Extended Kirchhoff Integral Formulations.

A multidomain boundary element solution for silencer and muffler performance prediction

Abstract The multidomain boundary element technique is used for the analysis of muffler problems in which the acoustic region consists of thin obstacles, such as extended inlet or outlet tubes and internal connecting tubes. In this technique, a muffler is first divided into several subdomains and then the well-known Helmholtz integral equation is applied to each subdomain. Continuity conditions of acoustic pressure and normal velocity are enforced at the interface between two neighboring subdomains. A set of simultaneous linear equations can then be constructed and solved for the boundary unknowns, as well as the interface variables. The boundary element solution is compared with a finite element solution. Excellent agreement between these two solutions is observed in producing the transmission loss curves for different muffler configurations.

The solution of coupled interior/exterior acoustic problems using the boundary element method

The boundary element method (BEM) is used to solve a class of problems in which an interior domain and an exterior domain are coupled by an interface surface, such as when sound propagates along a duct and radiates from the open end into the surrounding space. In the current approach, integral equations for the interior and exterior domains are coupled using continuity conditions at the interface surface between the two domains. The integral equations are reduced to numerical form using second‐order boundary elements. The coupled interior/exterior boundary element method presented here overcomes numerical error and excess computer time problems present when traditional boundary element formulations are applied to this class of problems. The current approach is illustrated using several examples including: the radiation of sound from an open duct, the radiation of sound from a source within a partial enclosure, and the acoustical response of a slotted cavity due to an incident plane wave.

A direct mixed-body boundary element method for packed silencers

Bulk-reacting sound absorbing materials are often used in packed silencers to reduce broadband noise. A bulk-reacting material is characterized by a complex mean density and a complex speed of sound. These two material properties can be measured by the two-cavity method or calculated by empirical formulas. Modeling the entire silencer domain with a bulk-reacting lining will involve two different acoustic media, air and the bulk-reacting material. Traditionally, the interior silencer domain is divided into different zones and a multi-domain boundary element method (BEM) may be applied to solve the problem. However, defining different zones and matching the elements along each interface is tedious, especially when the zones are intricately connected. In this paper, a direct mixed-body boundary element method is used to model a packed silencer without subdividing it into different zones. This is achieved by summing up all the integral equations in different zones and then adding the hypersingular integral equations at interfaces. Several test cases, including a packed expansion chamber with and without an absorbing center bullet, and a parallel baffle silencer, are studied. Numerical results for the prediction of transmission loss (TL) are compared to experimental data.

Boundary element analysis of packed silencers with protective cloth and embedded thin surfaces

Bulk-reacting porous materials are often used as absorptive lining in packed silencers to reduce broadband noise. Modelling the entire silencer domain with a bulk-reacting material will inevitably involve two different acoustic media, air and the bulk-reacting material. A so-called direct mixed-body boundary element method (BEM) has recently been developed to model the two-medium problem in a single-domain fashion. The present paper is an extension of the direct mixed-body BEM to include protective cloth and embedded rigid surfaces. Protective cloth, an absorptive material itself with a higher flow resistivity than the primary lining material, is usually sandwiched between a perforated metal surface and the lining to protect the lining material from any abrasive effect of the grazing flow. Two different approaches are taken to model the protective cloth. One is to approximate sound pressure as a linear function across the cloth thickness and then use the bulk-reacting material properties of the cloth to obtain the transfer impedance. The other is to measure the transfer impedance of the cloth directly by an experimental set-up similar to the two-cavity method. As for an embedded thin surface, it is a rigid thin surface sandwiched between two bulk-reacting linings. Numerical modelling of an embedded thin surface is similar to the modelling of a rigid thin surface in air. Several test cases are given and the BEM results for transmission loss (TL) are verified by experimental TL measurements.

A New Look at the High Frequency Boundary Element and Rayleigh Integral Approximations

This paper revisits the popular Rayleigh integral approximation, and also considers a second approximation, the high frequency boundary element method which is similar to the Rayleigh integral. Both methods are approximations to the boundary integral equation, and can solve problems in a fraction of the time required by the conventional boundary element method. The development of both methods from the Helmholtz integral equation is demonstrated and the differences between the two methods are delineated. Both methods were compared on practical examples including a running engine, gearbox, and construction cab. It was concluded that both methods can reliably predict the sound power for many problems but are inaccurate for sound pressure computations.

Boundary element analysis of packed silencers with a substructuring technique

It is well known that for certain large structures, substructuring can significantly reduce the matrix size and the total computational time. In this paper, a substructuring technique based on the impedance matrix synthesis is used along with the recently developed direct mixed-body boundary element method (BEM) to evaluate the transmission loss (TL) of packed silencers. Due to the single-domain nature of the direct mixed-body BEM, each substructure does not need to be a homogeneous domain. Complex internal components such as extended inlet/outlet tubes, perforated tubes, and thin baffles, as well as bulk-reacting linings, can all be in one single substructure. As such, dividing a large silencer into modular substructures can be done naturally in the longitudinal direction. This completely eliminates the traditional rule of having to construct well defined (no thin bodies) and homogeneous subdomains in the conventional multi-domain BEM. The substructuring technique presented in this paper also has the capability of modeling a catalytic converter in which a catalytic monolith containing a stack of capillary tubes is inserted between two connecting substructures. Several test cases including two parallel-baffle silencers are presented to demonstrate the technique. The BEM predictions for TL are verified by experimental data.

An efficient boundary element algorithm for multi‐frequency acoustical analysis

In this article, a new boundary element algorithm in acoustics is proposed. This new algorithm uses a so‐called ‘‘Green’s function interpolation technique’’ so that it is ideally suited to a multi‐frequency analysis. In the numerical implementation of this new algorithm, the Green’s function used in the boundary integral equation is interpolated by a set of shape functions. Numerical integration is then confined to a frequency‐independent part only, and hence, is carried out only once at the first frequency of a multi‐frequency run. At subsequent frequencies, only the nodal values of the Green’s function have to be re‐calculated for the formation of the boundary element coefficient matrices. Compared to the recently developed ‘‘frequency interpolation technique,’’ this new algorithm requires much less disk space and can be easily extended to half‐space problems and problems with planes of symmetry.

Disk brake squeal prediction using the ABLE algorithm

Abstract Disk brake squeal noise is mainly due to unstable friction-induced vibration. A typical disk brake system includes two pads, a rotor, a caliper and a piston. In order to predict if a disk brake system will generate squeal, the finite element method (FEM) is used to simulate the system. At the contact interfaces between the pads and the rotor, the normal displacement is continuous and Coulombs friction law is applied. Thus, the resulting FEM matrices of the dynamic system become unsymmetric, which will yield complex eigenvalues. Any complex eigenvalue with a positive real part indicates an unstable mode, which may result in squeal. In real-world applications, the FEM model of a disk brake system usually contains tens of thousands of degrees of freedom (d.o.f.s). Therefore any direct eigenvalue solver based on the dense matrix data structure cannot efficiently perform the analysis, mainly due to its huge memory requirement and long computation time. It is well known that the FEM matrices are generally sparse and hence only the non-zeros of the matrices need to be stored for eigenvalue analysis. A recently developed iterative method named ABLE is used in this paper to search for any unstable modes within a certain user-specified frequency range. The complex eigenvalue solver ABLE is based on an adaptive block Lanczos method for sparse unsymmetric matrices. Numerical examples are presented to demonstrate the formulation and the eigenvalues are compared to the results from the component modal synthesis (CMS).

A dual-reciprocity method for acoustic radiation in a subsonic non-uniform flow

Abstract In this paper, the dual-reciprocity boundary-element method is used to model acoustic radiation in a subsonic non-uniform-flow field. The boundary-integral formulation is based on a direct boundary-integral equation developed very recently by the authors for acoustic radiation in a subsonic uniform flow. All the terms due to the non-uniform-flow effect are taken to the right-hand side and treated as source terms. The source terms result in a domain integral in the standard boundary-integral formulation. The dual-reciprocity method is then used to transform the resulting domain integral into boundary integrals. Numerical tests show reasonably good agreement with an analytical soution for a pulsating sphere submerged in a potential-flow field.