## 3 Numerical Evaluation  
In order to evaluate the applicability and performance of the KIESSI-3D with CAF incorporated, two SSI analysis problems were selected as representative of real nuclear power plant structures supported by a shallow and a deep foundation, as shown in Figure 5. In addition, eight computer systems as shown in Table 2 were used for the evaluation of various computer environments. Two are single-CPU systems and six are dual-CPU systems. Analysis rates were compared by number of images (or number of coarrays) and cores used in the individual images. In Figures 6 and 7, the number of images and cores per image can be labeled as ca\(\{n_{1}\}\)nc\(\{n_{2}\}\), where \(n_{1}\) is the number of images, \(n_{2}\) is the number of cores per image, and the total number of cores used for the analysis is \(n_{1}\times n_{2}\). For instance, for ca2 nc4 the total number of cores used for analysis is 8. It should be noted here that the number of single-frequency solution processes in each coarray is the number of frequency points used in these analyses (NF in Figures 3 and 4) divided by the number of images. Forty-five frequency points were used in the soil-structure interaction analyses (i.e., NF = 45). Thus, the number of single-frequency solution processes in each image is 22 or 23 for ca2 nc\(\{n_{2}\}\), 11 or 12 for ca4 nc\(\{n_{2}\}\), and so on. Figure 5: Numerical examples for evaluating the performance of the proposed CAF parallelization technique for frequency domain soil–structure interaction analysis: (**a**) Shallow foundation (number of nodes = 22,690); (**b**) Deep foundation (number of nodes = 45,137). Figure 6: Speedup ratios for OpenMP and OpenMP+CAF for the shallow foundation example using various multi-core computer systems (abbreviation: ca[\(n_{1}\)]nc[\(n_{2}\) ]; \(n_{1}\) = number of coarrays; \(n_{2}\) = number of cores/image): (**a**) Single-CPU computer systems; (**b**) Dual-CPU computer systems. Figure 7: Speedup ratios of OpenMP and OpenMP+CAF for the deep foundation example using various multi-core computer systems (abbreviation: ca[\(n_{1}\)]nc[\(n_{2}\) ]; \(n_{1}\) = number of coarrays; \(n_{2}\) = number of cores/image): (**a**) Single-CPU computer systems; (**b**) Dual-CPU computer systems. The speedup ratios corresponding to the cores used in the soil-structure interaction analysis for a shallow foundation are summarized in Figure 6. As shown in Figure 6a, the result of using only one image (ca1 nc\(\{n_{2}\}\)) is the case where only the OpenMP functions were used, without using the CAF functions. In Figure 6b, it can be seen that in the cases where the six dual-CPU systems are used, the performance of the OpenMP operation is not improved; rather, it degrades even when a large number of cores (larger than the number of cores for a single CPU) are used. This performance degradation can be attributed to factors such as the overhead due to communication between the CPUs. In addition, scalability, which can be defined as the ability to retain performance levels when adding additional processors, decreases when the number of cores per image exceeds four, even in sections where the performance increases with the increase in the number of cores. Based on these results, optimal performance can be expected when the number of cores per image does not exceed four. The numbers of images and cores per image depend on the resources of the computer system (number of cores, RAM capacity) as well as the size of the problem (RAM capacity necessary for one image). As an example, in the case of a 12-core PC, various image-core number combinations (ca2 nc6, ca4 nc3, ca6 nc2, ca12 nc1, etc.) are possible. Notably, since the size of the internal memory (RAM) used by each image is the same, if the number of images increases then the amount of RAM used will also increase. That is, the number of images is determined by the size of the problem (the size of RAM used by one image) and total size of the RAM available.