## 3 Numerical Evaluation
As an example, if one image uses 40 GB of RAM, and the total RAM of the computer system is 192 GB, then the maximum number of usable images is four. When CAF is applied, the extent of performance improvement for the shallow foundation example depends on the number of images and cores per image, as shown in Figure 6. The extent of performance improvement for the deep foundation example was similar to that for the shallow foundation example, as shown in Figure 7. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{\begin{tabular}{c} **PC** \\ **Identification** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Number of** \\ **CPUs** \\ \end{tabular} } & \multirow{3}{*}{**Cores/CPU**} & \multirow{3}{*}{\
\begin{tabular}{c} **CPU Model** \\ **(Clock Speed)** \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} **RAM** \\ **(GB)** \\ \end{tabular} } & \multicolumn{3}{c}{**Runtime for 45 Frequencies**} \\  & & & & & **with Ca1 nc1 Option (Minutes)** \\ \cline{5-8}  & & & & **(G)** & **Shallow** & **Deep** \\ \cline{5-8}  & & & & & **Foundation** & **Foundation** \\ \hline \multirow{3}{*}{PC-1} & \multirow{3}{*}{1} & \multirow{3}{*}{8} & Intel Xeon & & & **Runtime for 45 Frequencies** \\  & & & & (3.9 GHz) & & & \\  & & & Intel i9-9960X & & & & \\  & & & (3.1 GHz) & & & \\  & & & Intel Xeon & & & \\  & 2 & 6 & \(\chi\)5690 & 64 & 47.6 & 141.4 \\  & & & (3.46 GHz) & & & \\  & & & Intel Xeon & & & \\ PC-4 & 2 & 8 & E5-2687w & 192 & 19.2 & 58.5 \\  & & & (3.1 GHz) & & & \\  & & & Intel Xeon & & & \\ PC-5 & 2 & 12 & E5-2687w v4 & 192 & 20.1 & 54.0 \\  & & & (3.0 GHz) & & & \\  & & & Intel Xeon Gold & & & \\ PC-6 & 2 & 16 & 6142 & 384 & 13.6 & 37.0 \\  & & & (2.6 GHz) & & & \\  & & & Intel Xeon Gold & & & \\ PC-7 & 2 & 24 & 6248R & 768 & 12.4 & 34.4 \\  & & & (3.0 GHz) & & & \\  & & & AMD EYC & & & \\ PC-8 & 2 & 32 & 7601 & 384 & 62.2 & 121.3 \\  & & & (2.2 GHz) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of the computer systems used in this study. In general, CAF+OpenMP parallelization performance improves as the number of images increased. However, OpenMP+CAF parallelization performance of converges for Intel CPU systems when the number of cores per image did not exceed four. Moreover, when the optimal image-core number combination for the six dual-CPU computer systems listed in Table 2 was used, the analysis rate for the shallow foundation example improved as the number of cores of the computer system increased, as shown in Figure 8a. In addition, with the implementation of the OpenMP+CAF algorithm as shown in Figure 4, the new KIESSI-3D code analysis rate was between 1.74 and 4.65 times faster than for the existing code, which used only OpenMP. In the case of the deep foundation example, the new KIESSI-3D code analysis rate was between 1.85 and 4.89 times faster than that of the existing code, as shown in Figure 8b. The performance improvement measured by the speedup ratio (\(S\)) in parallel processing programming, as characterized by the parallelization ratio (\(P\)) of codes, can be expressed simply by [PERSON]'s law as follows [27]:  
\[S=\frac{1}{(1-P)+P/N}\text{ or }P=\frac{1-1/S}{1-1/N} \tag{1}\]  
where \(N\) is the number of CPU cores. The extent of performance improvement according to the number of cores and parallelization ratio shown in Equation (1) can be illustrated as shown in Figure 9. Even  
Figure 8: Maximum speedup ratios for OpenMP+CAF and OpenMP using dual-CPU computer systems: (**a**) Shallow foundation example; (**b**) Deep foundation example. if the same number of cores is used, the efficiency of the operation will decrease if the parallelization ratio is low, indicating that the parallelization ratio is an important indicator in parallel processing programming.