# The Boundary Element Method in Acoustics: A Survey
## 6 Areas of Discussion
### Efficiency
#### 6.1.2 The Frequency Factor, Multi-Frequency Methods and Wave Boundary Elements  
Typically, in vibration and acoustics, the time-dependent signals are resolved into frequency components. For example, for air-acoustics, the audible range for human being is up to 20 kHz and typically his could be resolved into components with a 10 Hz resolution; multiplying the computational cost, considered in the previous Sub-subsection, by around 2000. The prospect of solving acoustic problems over the full frequency range has led to another set of techniques focus on reducing the computational burden by solving a range of frequencies together and these are usually termed multi-frequency methods [297; 348; 349; 350; 351; 352; 353]. However, with the nature of an acoustic field, originating typically from structural vibration, is such that structural and acoustic resonances, and the driving profile, dominate the total acoustic response. From a computational point of view, effort should therefore be concentrated on these 'peak' frequencies. Hence, this adaption returns us to the standard mono-frequency problem, or applying the multifrequency method, centred on each peak.  
Moreover, in general the frequency of observation is matched in the acoustic and vibratory properties. In practice, as the frequency increases, a higher elemental resolution is required and hence \(n=O(k)\) for two-dimensional and axisymmetric problems and \(n=O\!\left(k^{2}\right)\) in three dimensions. Although this seems to imply a revision of the mesh at every frequency, it is more likely that one mesh that is suitable for the highest frequency is used throughout the range, or separate meshes are applied in significant sub-ranges of the frequency range, in order that the burden of meshing is proportionate within the overall project.  
A review of the actual number of elements required to capture the solution is provided by Marburg [356; 357; 358]. In general, for the simple elements that are often used, it is considered that 6-10 elements per wavelength is a reasonable guideline. Obviously, higher-order boundary element methods [100; 359; 360; 361; 362; 363] would often return the same accuracy with fewer elements. There is significant interest in isogeometric elements, in which the boundary and the boundary functions are modelled with the same basis functions (typically splines), so that the BEM can be more easily integrated with computer-aided design [364; 365; 366; 367; 368; 369; 370; 371]. For air acoustics, at high frequencies reaching 20 kHz, the wavelength is less than 2 cm. For example, even for a 10 cm cube, the number of elements required to model the high frequencies is in the tens of thousands and for a 1m cube, over a million elements would be required. A variation on the boundary element method has arisen for developing sinusoidal basis functions or wave boundary elements, in order to more accurately model the acoustic functions and the term PUBEM or partition of unity BEM is often used to identify such methods [267; 371; 372; 373; 374], which has similarity with the application of the Treffetz method [292; 293; 294].