## 4 Computational Efficiency Assessment with a 2D Office Problem
### Results and Discussion
In the presented simulation cases, the process consumes more than 95% of the computational time. Figure 9a,c presents the results of the iteration number comparison in both approaches for the cases with three atmospheric conditions (a)-(c) and with the GW24K absorber. The lower part of the figure shows the reduction rate of the iteration number \(R_{\text{iter}}\). Overall, it is  
Figure 8: Comparison of mean SPL reduction \(D\) at three atmospheric conditions for cases with (a) GW24K, (b) GW32K, and (c) GW96K. Figure 7: Comparison of room frequency responses at \(\mathbf{r}=(1,2)\) computed using FEMs based on the lossy and lossless Helmholtz equation approaches for the case with atmospheric condition (a) and with GW24K: (**upper**) frequency response at 1 Hz to 3000 Hz and (**lower**) frequency response at 3 kHz–6 kHz. apparent that there are fewer iteration numbers in the lossy Helmholtz equation approach at a higher frequency range for the three atmospheric conditions. We can find that the reduction rate differs for three atmospheric conditions. The atmospheric condition (a) has the highest iteration number reduction. The reduction rate reaches approximately 60% for high frequencies. In addition, higher reduction can be found at lower frequencies than in either condition (b) or (c). In fact, condition (c) shows the lowest iteration number reduction, but the reduction rate reaches 40% at high frequencies. To explain further how atmospheric conditions influence the reduction rate, Figure 10a and 10c, respectively, show a 1/3 octave band averaged reduction rate at three atmospheric conditions for cases with GW24K, GW32K, and GW96K sound absorbers. In the figure, the error bar represents the standard deviation values. The reduction rates are related clearly to the magnitude of the pure-tone sound attenuation coefficient for frequencies higher than 1 kHz. For frequencies of 1 kHz-2 kHz, the values of pure-tone sound attenuation coefficients are larger in the order of atmospheric conditions (a), (c), and (b) as in Figure 2. This order corresponds to the reduction rate in the iteration number as in Figure 10a,c: higher air absorption engenders fewer iteration quantities of the iterative solver. At 4 kHz, the order in the magnitude of the air absorption changes for conditions (b) and (c), at which the atmospheric condition (b) has larger pure-tone sound attenuation coefficients than those in condition (c). As in Figure 10a,c, the reduction rates in the iteration number are larger for condition (b) than for condition (c). Apparently, the value of the pure-tone sound attenuation coefficient larger than 0.03 has a notable effect on the convergence speed of the iterative solver. In the presented cases, the averaged reduction rate at the 4 kHz band shows a 33-36% computational time reduction for atmospheric condition (a). The computational time reductions for atmospheric conditions (b) and (c), respectively, reached 16-17% and 12-14%. Figure 9: Comparison of iteration numbers in FEMs based on the lossy and lossless Helmholtz equation approaches for cases with the GW24K sound absorber: **(left)** atmospheric condition **(a)**, **(center)** atmospheric condition **(b)**, **(right)** atmospheric condition **(c)**. Lower figures show the reduction rate \(R_{\text{iter}}\). As a reference, Figure 11 presents the convergence history of the iterative solver in FEMs based on the lossy and lossless Helmholtz equations at 4704 Hz for the atmospheric conditions (a) with the GW32K absorber. At the frequency, the maximum reduction rate of 66% is achieved. The convergence speed changed around at the value of the relative residual 2-norm of \(10^{-3}\). The lossy Helmholtz equation-based FEM shows faster decreases in the residual norm. Finally, we further show whether or not the faster convergence holds when using a higher spatial resolution mesh for the atmospheric conditions (a) with the GW24K absorber.