## 4 Computational Efficiency Assessment with a 2D Office Problem  
### Outline of the Problem  
This section presents an evaluation of the computational efficiency of 2D room acoustics simulation using the lossy Helmholtz equation-based FEM when using the iterative solver as a linear system solution via a real-scale 2D office problem. The computational efficiency was evaluated by a comparison with the standard lossless Helmholtz equation approach. We also present how room frequency responses change with the air absorption effect. Iterative solvers are well known to be efficient linear system solvers for large-scale problems with vast DOF when they converge rapidly with few iteration numbers. However, the convergence characteristics in the frequency-domain room acoustics simulation, which usually solves the lossless Helmholtz equation, become worse at higher frequencies [54,55]. Therefore, obtaining faster convergence of iterative solvers when solving the lossy Helmholtz equation has a marked effect on computational efficiency in a multi-frequency analysis. It is noteworthy that the present 2D office problem has been used to assess the performance of the dispersion-reduced time-domain FEM, including frequency-dependent impedance boundary conditions and a plane-wave-enriched partition of unity FEM in earlier works [56,57].  
Figure 6 presents the 2D office problem, including weakly absorbing impedance boundary \(\Gamma_{a,weak}\) and a rigid-backed porous sound absorber represented by a frequency-dependent local-reaction impedance boundary \(\Gamma_{a,porous}\). \(\Gamma_{a,weak}\) has a real-valued specific acoustic impedance corresponding to the normal-incidence sound absorption coefficient \(\alpha_{0}=0.1\). For \(\Gamma_{a,porous}\), three glass woods with different densities of 24 kg/m\({}^{3}\) (GW24K), 32 kg/m\({}^{3}\) (GW32K), and 96 kg/m\({}^{3}\) (GW96K) were assumed with a thickness \(L\) of 25 mm. The surface impedance of the glass wood absorbers \(Z_{\text{surf}}\) was computed using the transfer matrix method as \(Z_{\text{surf}}=-\text{j}Z_{\text{c}}\cot(k_{\text{e}}L)\) using the equivalent fluid model. Here, \(Z_{\text{c}}\) and \(k_{\text{e}}\), respectively, denote the characteristic impedance and the complex wave number of the equivalent fluid. The Miki model [58], which uses flow resistivity \(R_{\text{f}}\) as a material property, was used to describe the fluid properties of the glass wood. The \(R_{\text{f}}\) values for GW24K, GW32K, and GW96K are, respectively, 6900 Pa s/m\({}^{2}\), 13,900 Pa s/m\({}^{2}\), and 55,000 Pa s/m\({}^{2}\). The loudspeaker radiating sound wave is modeled by the vibration boundary \(\Gamma_{\text{v}}\) with the vibration acceleration of 1 m\({}^{3}\)/s\({}^{2}\). We computed the room's frequency response from 1 Hz to 6 kHz with 1 Hz intervals. The three atmospheric conditions (a)-(c) used in the preceding section were considered, respectively, for the three sound absorbing conditions with GW24K, GW32K, and GW96K: nine acoustic simulations were performed in all. Spatial discretization was also performed with the dispersion-reduced Q4s with element sizes of 0.01 m \(\times\) 0.01 m. The resulting mesh has 400,720 DOF with a spatial resolution of 5.7 elements per wavelength. All simulations were conducted using a computer (PRIMERGY CX2550/CX2560 M4; Fujitsu Co. Ltd., TYO, JPN) that has two  
Figure 5: Relative errors in attenuation levels for atmospheric condition (a) when using Mesh1–3, respectively, with a spatial resolution of 3.4, 6.8, and 13.6 elements per wavelength.  
processors (Xeon Gold 6154 3.0 GHz, 18 core; Intel Corp., Santa Clara, CA, USA). The computational code was compiled using a Fortran compiler included in the Intel oneAPI HPC Toolkit (Ver. 2021; Intel Corp.). Parallel computation using OpenMP was conducted with 24 threads.