# Potential of Room Acoustic Solver with Plane-Wave Enriched Finite Element Method
## 1 Introduction
### Partition of Unity Finite Element Method for Acoustic Problems  
As a numerical method with such potential, acoustic numerical methods based on the Partition of Unity FEM [29] (PUFEM) have been formulated and examined in some studies [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. Two papers [36; 37] present demonstrations of PUFEM on 2D car interior analyses at high frequencies including porous absorbing materials modeled respectively using an equivalent fluid model [41; 42; 43; 44] and poroelastic material model [45]. A very recent study [35] examined a 2D plane wave-scattering problem by which PUFEM can significantly reduce the DOF of the FEM model at high frequencies. The study also includes demonstration of partition of unity isogeometric analysis of 2D car interior sound field analysis at 20 kHz. The ability of acoustic PUFEM or PU-based method derives from enriching the approximation of sound fields by incorporating the general solution of Helmholtz equation into finite element approximation. For example, when using plane waves as the general solution, the sound pressure at a nodal point is expressed by the superposition of plane waves propagating various directions. It incorporates into local finite element approximation via the partition of the unity property. Additionally, sound fields are approximated up to high frequencies using \(q\)-refinement, which is a refinement technique by which plane waves propagating in various directions are added at nodal points of a fixed mesh gradually with increasing frequency. The most notable feature of acoustic PUFEM is that it obviates re-meshing according to frequencies, which is necessary for conventional FEM. Actually, PUFEM analysis can use elements of many times greater length than the wavelength of analyzed frequency, whereas conventional linear FEM must use an element size satisfying less than one-tenth of the wavelength of the analyzed frequency, as described above. Some closely related approaches with PUFEM exist. A review article presents additional details [46].  
The PUFEM feature is apparently favorable for room acoustic simulations. In addition, the plane wave enrichment is applicable to any FEM mesh. That characteristic is useful to improve the existing FEM code performance. However, acoustic PUFEM is still developing. Various aspects remain to be studied for application to practical room acoustic simulations. In most earlier works [35; 36; 37], the potential of PUFEM for practical applications has been presented on 2D acoustic analysis in a small car having an area of less than 3 m\({}^{2}\), without quantitative comparison with reference solutions. Additionally, those studies examine only pure tone analyses at high frequencies. Therefore, from the perspective of room acoustic applications, it remains unclear how PUFEM will perform robustly and accurately for problems of calculating wideband frequency responses in larger interior sound fields such as architectural spaces. This is an extremely important aspect because room acoustic simulations specifically address large room models and require wideband frequency components with fine frequency resolution especially for calculating room impulse responses. For reference, such an evaluation using frequency-domain methods can be found in earlier reports of the literature [16; 23; 24]. This study is the first attempt at revealing the PUFEM performance for wideband frequency response analysis in room acoustic problems with quantitative evaluation in accuracy. The architectural spaces treated here have 13-20 times larger dimensions than earlier studies.