# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 4 Physical Modelling
### Other Approaches
This enables them to use the Kirchhoff-Helmholtz integral representation of the sound pressure field (14) to propagate the sound field in space. In [132], a special form of the FEM tailored to the solution of the wave equation, the ultra-weak variational formulation (UWVF), is implemented. This method is especially efficient at high frequencies and it enables the authors to extend the computation over the entire frequency range. It is also applied to the computation of HRTFs of a standard head-torso mannequin. In [133], the authors apply and validate a FEM model for HRTF computation in the near field up to a frequency of 6 kHz. The computation domain is delimited with a triangular head mesh and a triangulated outer sphere. Volumes are meshed with tetrahedral elements. The requirement of 6 elements per wavelength leads to a number of about 1.4 million of elements. The main objective of the paper is to prove that FEM can be efficiently used in HRTF computation. Furthermore, the method allows for investigating features like diffractive region determination, which is possible with other methods. In [134], the authors propose and validate two computational tools based on finite element methods for generating distance-dependent HRTFs. Experimental measurements of distance-dependent HRTFs are rare due to the complexity of experimental procedures. Therefore, computer simulations can help to fill this gap. The method proves to be computationally efficient and accurate over a frequency range of 0.2 to 20 kHz. Furthermore, the approach is robust against mesh quality reduction. The tool is versatile enough to incorporate bio-acoustic attributes such as the middle ear wall reflections. In [135], the authors face the problem of the high costs in computing HRTFs with FEM due to geometrical complexity, fine spatial discretisation, and densely sampled frequencies. They apply a model order reduction (MOR) based on the Krylov subspace [136] to a large-scale FEM, obtaining a considerable reduction in the computation time while preserving accuracy below a frequency of 6813 Hz. The MOR reduces the complexity of large-scale models by constructing accurate surrogate models of lower complexity. Generally speaking, the MOR is based on detecting patterns in the solution and restricting the solution to these patterns. Finally, some examples of geometrical approaches that are only indirectly associated with the solution of the wave equation are reported in the following. Acoustic ray tracing is based on the fact that, for wavelengths shorter than the size of the spatial inhomogeneities, the sound propagates in rays following geometrical laws; this is common in both acoustics and optics. This makes the use of computer graphics approaches in computing impulse responses at the location of a receiver when a pulse is sent from a source possible. In [137], the authors use acoustical ray tracing to simulate the HRIR of a standardised head described through a high-resolution mesh. Their approach is also able to include effects due to diffraction. The model is able to predict notches in the HRTF due to pinna reflections. In [138], the authors use ray tracing to correct an HRTF for taking into account torso reflections. HRTF computation usually considers the head and torso as a rigid body, because taking into account the relative motion of the two would be too costly. The authors use a simplified model of a head, neck and torso and optimize the parameters in such a way that the reflections lags computed with ray tracing fit those from dynamical HRTF measures. These time lags can be computed efficiently and are thus well suited for real-time applications. To conclude this section, we mention some simplified models, in which the head is approximated with simple geometrical shapes like spheres and ellipsoids. The most extreme simplification consists of considering a pair of ears without a head. In this model, the HRTF is reduced to the pure, frequency-independent ITD and ILD generated by the propagation of the sound from the source.