The BEM in Acoustics

Chapter 1
Introduction

Over recent decades, the boundary element method ( BEM ) ( boundary-element-method.com ) has received much attention from researchers and has become an important technique in the computational solution of a number of physical problems. In common with the better-known finite element method (FEM) and finite difference method (FDM), the boundary element method is essentially a method for solving partial differential equations (PDEs) and can only be employed when the physical problem can be expressed as such. As with the other methods mentioned, the boundary element method is a numerical method ( numerical-methods.com ) and hence it is an important subject of research amongst the numerical analysis community. However, the potential advantages of the BEM have seemed so considerable that the strongest impetus behind its development has come from the engineering community, in its enthusiasm to obtain flexible and efficient computer-based solutions to a range of engineering problems. The boundary element method has found application in such diverse topics as stress analysis, potential flow , fracture mechanics, electromagnetics and acoustics (the subject of this text).

Acoustics is an important branch of physical science. An acoustic field can exist in a fluid domain such as air or water, the two most important acoustic media. The linear wave equation forms an acceptable model in many fluids and it is often used in the cases of air and water media. In many physical situations the acoustic field is periodic, and has the outcome of reducing the wave equation to a sequence of Helmholtz equations by a Fourier decomposition with one Helmholtz equation for each sample frequency which as a particular form of digital signal processing . In the methods of this text, solutions of such acoustic problems are obtained through the consideration of the individual Helmholtz problems.
The Helmholtz equation governing a range of classes of domains may be solved by the boundary element method. Hence the BEM has received attention from engineers that are interested in applications such as the sound output of a loudspeaker, the noise from a radiating source such as an engine and the interior acoustic modes of an enclosure such as a vehicle interior. The method is equally applicable in underwater acoustics and can be used to model the scattering effect of an obstruction in the ocean or to determine the acoustic field surrounding a sonar transducer.

As well as being a subject in Physics, acoustics is important in many branches of engineering. In electrical engineering acoustics is important in audio and sonar, for example in the design of loudspeakers and sonar transducers. In mechanical engineering the noise from vehicles and machines is important factor to consider. In civil engineering the acoustics of rooms, studios and auditoria is of interest as well as the noise from roads and railways.

Here the development of the boundary element method for a range of acoustic domains and conditions is presented so that the software can be applied across the range outlined above. Fortran subroutines that implement the methods and a set of test problems are available. A chapter is devoted to each of the interior, exterior and modal acoustic problems. The codes implement the boundary element method in its simplest form. The boundaries are approximated by the straight lines for two dimensional problems, planar triangles for three-dimensional problems and by truncated cones in axisymmetric three-dimensional problems. The boundary functions are approximated by a constant on each segment or panel of the boundaries. Specifically, the boundary elements are C0; constant on each panel and discontinuous on the edges of the panels.

The subroutines have been designed to allow as much flexibility as possible whilst minimizing their complexity. The parameter list in each main subroutine is stated in the text to enable the reader to appreciate the way in which a call to the subroutine can be included in a Fortran program. Toward the end of Chapters 4, 5 and 6 test problems are applied to each subroutine, to demonstrate how to use the routine and to validate the methods. Results from relevant applications are given to demonstrate the usefulness of the methods in the computer-aided analysis of engineering problems .


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