# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 4 Physical Modelling
### Implementations of BEM  
The numerical implementation of BEM and other approaches in solving the wave equation is supported by several solvers. For instance, the Acoustic Module of COMSOL Multiphysics(r) [106], available starting at version 5.3a to the latest version 6.3, supportsBEM applications in acoustics. COMSOL is a proprietary software. Other proprietary tools supporting BEM for acoustics include FastBEMAcoustic(r) [107], suited for parallel computing and high-frequency simulations, and Ansys(r) [108]. Open-source software is also available, such as Mesh2 HRTF [109], version 1.x, with its BEM core NumCalc. In [95], an overview of the numerical methods behind NumCalc is given. Many practical suggestions for implementing an effective simulation are shown throughout the paper. Furthermore, this paper contains examples of the accuracy of NumCalc in solving some benchmark problems; it is also useful for estimating the computational resources (CPU and memory) required in such simulations. Finally, the cited paper contains information concerning specific topics, e.g., the specific version of FMM and the implementation of frequency-dependent boundary conditions with frequency-dependent admittance. The references [101,110] contain information about the computational pipeline of Mesh2 HRTF from pre- to post-processing stages through numerical simulation. Reference [111] contains an introduction to the theoretical background of Mesh2 HRTF and a description of the software package. Another open-source tool for solving boundary problems in homogeneous media, like the Kirchhoff-Helmholtz Equation (14), is BEM++ [112]. This software is a C++ library that can be used with Python scripts (latest version Python 3.13.0). The main citation of the Bempp library is provided by [113], while an introduction to solving boundary problems with BEM++ can be found in [94]. Many available solvers make use of a modified formulation of the problem (17) proposed by [PERSON] and [PERSON] [114] to prevent the possible degeneration of the boundary operators (15) at frequencies corresponding to resonances of the internal problem defined by \(\Sigma\). These irregular frequencies may lead to the non-uniqueness in the solution and, even when the solution is unique, poor conditioning near these frequencies. The authors of [115] present a Helmholtz equation solver, making use of [PERSON] and [PERSON]'s formulation to avoid problems arising from spurious resonances. In the following, we discuss some applications of BEM to scattering problems in acoustics, and, more specifically, to HRTF computation. In [105], FMBEM is used for HRTF generation over a wide frequency range. This work shows that, with FMBEM, it is possible to overcome the computational limitations at high frequencies due to the increased number of elements required to resolve short wavelengths. In [99], FMBEM is applied for the computation of HRTFs on different models (a rigid sphere, the head models Neumann KU-100 and Kemar and the torso and head model Kemar mannequin), all in the audible range up to 20 kHz. The computation is organised into two steps: the computation of the spherical harmonic representation of the HRTF Equations (26) and (27) and the subsequent HRTF evaluation. FMBEM is also used in [116] to numerically compute the HRTF of a dummy head in a wide range of frequencies between 20 Hz and 20 kHz. The aim of this study is to show how the FMBEM can be effectively used in this kind of problem, with emphasis on computational efficiency. In order to reduce the number of calls to the iterative solver, the authors make use of singular value decomposition (SVD) to express the excitation vectors related to the different positions of the source as linear combinations of a limited number of basis vectors. In [117], BEM is used to generate the HRTF of the standard dummy head BHead210, which is designed with the typical anatomical characteristics of Chinese people, which may differ from those of Western people.