Acoustic Boundary Element Solver.
Dr Dan J. O’Boy
http://www.djoboy.co.uk
R1
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Revision 1
Copyright ©2008
Dr Dan J O’Boy
4 The Green
Exton
Oakham
Rutland
LE158AP
United Kingdom
enquires@djoboy.co.uk
There are several different methods available for the prediction of the acoustic field in either an enclosed interior space or an exterior domain. These methods are usually designed to support experimental measurements or hypothetical predictions. The main prediction tools for the prediction of the acoustic field include theoretical models, finite elements methods (FE), ray methods / geometrical acoustics approximations, statistical energy analysis, variational energy methods or the boundary element method. The main features leading to advantages when the methods are compared are provided in this section.
Finite element (FE) methods are the most widely used, due to the efficient and simple integration of computer aided design tools in the engineering workplace. As computer aided engineering design usually involves the creation of CAD drawings of the object, it is easy to take those drawings and incorporate them into the acoustic analysis. Finite element tools require the discretisation of a full volume and subsequent meshing of that volume. Although it produces extremely good solutions, it can tend to be complex, requiring high computing power and for higher frequency prediction, accurate geometry. The computing requirement increases with the upper analysis frequency and size of the geometry.
The statistical energy method was designed to produce approximate acoustic prediction for segmented rooms using diffuse field assumption. Due to the averaging process, the prediction can be error prone, however, the method is fast and efficient. As diffuse theory is assumed leading to average energy flow between compartments, the method is not suitable for low frequencies, yet this then means that the design process is tolerant to variations in geometry.
Ray tracing / ray theory or geometrical acoustics approximations are extremely useful for enclosed domains, bounded exterior spaces or rooms where diffuse theory could be applied. Each acoustic wave is represented as a ray, tracing a predictable path through space, with the incident and reflection angles from a surface taken into account. Damping can be applied at the reflective surface through a variable impedance boundary condition. As ray theory has a lower frequency limit, it is usually used for high frequency studies of room acoustics and reverberation times. The computational time varies depending on the level of complexity of the room geometry, as small changes in room angles can lead to large changes in the ray transmission paths, hence the importance of overall diffuse field assumption.
Theoretical models are used in studies to gain an understanding of the physical sound transmission processes that can occur. By idealising a geometry using simple geometrical primitives or through using significant assumptions, it may be possible to reduce the complexity of the problem down to a level where a simple theoretical model can provide insight into the problem, order of magnitude calculations and trend analysis in a parametric study. As these models are often much simpler than reality, they often are far less computationally expensive or time consuming, making them a popular first choice for initial investigations.
The boundary element is closely related to the finite element method and exploits the principle that in acoustic predictions, it is often the case that the governing transmission medium is nondispersive and isotropic. The method shares some similarities to the finite element method in that it can utilise information from a computer aided engineering process, however the key advantage is that rather than discretising and meshing the volume, only the surfaces need treating. This leads to a reduction in computational effort and memory for a given problem. As the surfaces only need meshing, it is relatively easy to join dissimilar domains through common boundary conditions. The boundary element method requires the use of a fundamental solution governing the propagation of sound in free space, applied as a weighting function to every node on the surface mesh. This fundamental solution takes the form of a Green's function found in theoretical analysis. This usage of ideal propagation solutions provides the potential for extremely accurate solutions.
Greek notation 
Explanation 
_{} 
Vector of source amplitudes of the fundamental solutions 
_{} 
Surface 
_{} 
Error 
_{} 
Nondimensional coordinate 
_{} 
Wavelength 
_{} 
Unknown amplitude of the fluxes of the fundamental solutions 
_{} 
Density 
_{} 
Unknown amplitude of the fundamental solution 
_{} 
Frequency [rad/sec] 
_{} 
Domain 
Table 3 Notation for Greek notation
English notation 
Explanation 
_{} 
Matrix of boundary integrals for each source element to each observer element 
BEM 
Boundary element method 
CAD 
Computer aided design 
CAE 
Computer Aided Engineering 
C++ 
Compiler language 
_{} 
Vector of prescribed boundary conditions for each element 
FE 
Finite element 
IREF 
Reference Intensity 
PREF 
Reference Pressure 
WREF 
Reference Power 
_{} 
Constant 
_{} 
Speed of sound 
_{} 
Element number 
_{} 
Frequency [Hz] 
_{} 

_{} 

_{} 
Hankel function 
_{} 
Imaginary number 
_{} 
Jacobian 
_{} 
Wavenumber 
_{} 
Node number 
_{} 
Flux of pressure 
_{} 
Distance from source element to observer element 
_{} 
Variable (Pressure) 
_{} 
Fundamental solution (pressure) 
_{} 
Fundamental solution (Normal pressure gradient) 
_{} 
Weighting factor 
_{} 
Coordinates in Cartesian space 
Table 4 Notation for English notation
Acoustic: The medium in which a sound wave may propagate is an acoustic medium. A problem relating to sound (pressure perturbations in the audible frequency range) is considered an acoustic problem. Typically, this medium cannot sustain shear waves, only longitudinal waves (compression).
Speed of sound: It is assumed that in the elastic acoustic medium, the distance travelled by a sound wave in a given time does not vary, this is the speed of sound. For dry air at a temperature of twenty degrees Centigrade, this equates to approximately 343m/s.
Surface: The surface is a boundary between interior and exterior solution domains. The surface must be a continuous function in both displacement and slope (the method can fail should sharp corners exist in the geometry).
Node: The surface is discretised into a series of evenly spaced points. Each point on the surface is defined by three coordinates _{} where the join between nodes are assumed to be a linear segment. It is important that when discretising the surface, the nodes are spaced evenly across the whole domain, with the spacing dictated by the maximum excitation frequency of interest. Thus for exterior calculations, there should be at least four nodes per acoustic wavelength to avoid aliasing, while for interior domains the recommended minimum number rises to eight nodes per wavelength.
The spacing between nodes does not need to be the same, only that the macrospacing is similar.
Figure 1 A node characterised by three independent coordinates
Wavelength: The acoustic frequency, _{} is related to the acoustic wavelength _{} by the speed of sound _{}. It is assumed that the acoustic medium is an isotropic material where shear forces cannot be sustained.
Element: The nodes are just points in Cartesian space, however, by joining three or more unique nodes together a linear flat surface is created. This nonparabolic surface can be defined by three distinct nodes for a triangular element or four for a quadrilateral element. Through a regular distribution of node points, the elements are evenly distributed and spaced over the entire surface of the geometry. Elements may share common nodes, and it is important that in the definition of the nodes creating all elements that the normal directions are consistent. This is shown on the figure as the node connections all pointing clockwise or anticlockwise.
Figure 2 A triangular element characterised by three independent nodes, and a quadrilateral element characterized by four. Note that the direction normal of all elements must be consistent.
The density of the mesh is equivalent to the number of elements for a given surface area. The mesh should be smooth and the density approximately constant.
Excitation frequency: In order that a nonzero sound pressure field is calculated, the nontrivial solution, the geometric domain must contain at least one sound source, otherwise the only sound field that can exist is silence. This monopole source is located at an arbitrary position. The boundary element calculation routine calculates a predicted sound field for each discrete excitation frequency, up to a finite upper limit. There are particular numerical problems when attempting to calculate the steady state acoustic pressure deflection (the frequency equal to zero Hertz 0Hz). In order for the calculation to converge to a satisfactory solution, it is required that the zero frequency instead be replaced by a relatively small value, for example 0.01Hz.
Boundary condition: Each element has an associated boundary condition which is an initial condition required to be able to solve the problem. The types of physical condition which can be applied to the element include a prescribed velocity or pressure, including complex impedance. These are proportional to the fundamental solution.
In the input file, each element has an associated boundary condition. In the code, these are denoted boundary condition types 1 or 2 or 3.
The pressure on an observer element can be written in terms of a fundamental solution on a source element a distance r away. This is a type 1 boundary condition (pressure).
As will be described in the following sections, the velocity applied to the element (type 2) can be written in terms of a fundamental solution, as can the pressure, from this observer element to a source element a distance r away. The normal velocity can be applied to the element to represent a solid surface, or prescribed flow condition. The amplitude of the fundamental solution provided by the source element yielding the appropriate boundary condition on the observer element is denoted Q.
_{}
{Note: Determine if it is normal velocity only or magnitude of velocity which is zero. Difference is rx/r type term but in the unit normal direction of the element.}
The alternative boundary condition is a (type 3) impedance condition, where the complex value of normal pressure to normal velocity is specified. Although this can also be written in terms of a fundamental solution, it may be shown that these terms cancel.
_{}
Fundamental solution: The boundary element method relies on the use of fundamental solutions to be applied to each node, as a weighting factor. In the case of acoustic predictions, the fundamental solutions take the form of the Green's function solution to the Helmholtz equation. This represents the propagation of a wave through free space and can be applied in either twodimensional or threedimensional forms (the twodimensional form utilises Hankel functions rather than exponential functions). Further details are provided in the section on monopole sound fields.
Helmholtz fundamental solutions (monopoles) 


_{} 
_{} 
2D 
_{} 
_{} 
3D 
_{} 
_{} 
Table 5 Fundamental Solutions for acoustic wave propagation in free space
These Green’s functions represent how a wave would propagate through the free space, between the source position and observer location a distance r away. It is important to understand that the fundamental solution can be directly linked to the parameters of interest for acousticians, such as fluid velocity or pressure.
To take an example, a sphere of radius _{} pulsating with a frequency _{} and a radial velocity _{} at the surface generates a spherical wave. The oscillatory volume flow at the surface has an amplitude _{} that is given by the monopole strength _{}. In the limiting case of a monopole point source, the wavelength dimension _{} we obtain for the acoustic pressure _{}. In the code listed below, the source strength is user provided as a complex constant B.
Therefore, both the velocity V and the pressure p can be written in terms of this fundamental solution.
_{}
The intensity can also be related to the acoustic pressure and velocity through the analogy with the power dissipation of electrical circuits.
_{}, where the overbar denotes the complex conjugate is taken.
Data recovery mesh: The boundary element method starts with a known boundary condition on a geometrical surface, and the unknown distribution of fundamental solutions across the domain which yield that boundary condition is formulated in terms of a matrix problem. Once this distribution is known, the sound pressure can be obtained everywhere in the domain. Data recovery points are the locations where the user is interested in knowing these resulting sound pressures, intensities or acoustic fluid velocities. Although it is described as a mesh for graphical interpretive purposes, it can equally be defined as a set of single unconnected points. In designing the data recovery mesh or locations of the recovery points, it may be of interest to obtain the acoustic pressure close to the solid geometry. In order to avoid singularities in the matrix solution, it is necessary to locate each data recovery point at least 1/4 of an acoustic wavelength away from the solid geometry element or node.
The sound field from multiple monopoles: A monopole is defined as a solution to the Helmholtz equation which yields a sound field which propagates omnidirectionally. For a threedimensional field, this results in an omnidirectional field where the amplitude falls away as a function of the inverse of the radius. This is illustrated in Figure 3 for a monopole originating on the left hand side of the screen and shown propagating to the right hand side of the diagram.
When several monopoles are situated in a domain, it is possible to obtain very complicated sound fields through linear superposition leading to regions of high sound pressures from constructive interference and regions of low sound pressure from waves destructively interfering. This is most easily illustrated using a large number of monopoles situated close to each other, in a baffled piston arrangement. At lower frequencies, as seen in Figure 4, the monopoles are spaced within an acoustic wavelength of each other such that the overall sound field is omnidirectional. However at higher frequencies, as also shown in Figure 4, the spacing between each monopole becomes more significant compared to the acoustic wavelength and the phase change between each monopole results in a more complicated sound field, with lobe patterns. This is the primary reason why loudspeaker packages comprise many different speaker sizes, each providing sound in a specified frequency range. In the boundary element method, we assume that we know the boundary conditions applied to the surface of a geometry and require knowledge of the amplitude and phase of a distribution of fundamental solutions on each node that satisfies this boundary condition.
Figure 3 Omnidirectional sound pressure propagating from a monopole fundamental solution.
Figure 4 Directional sound field propagating from a distribution of monopoles at low and high frequencies.