# Selection Criterion of Reanalysis Methods for Plane Truss Optimization
## 4 Construction of Selection Criterion
### Method Comparison Study
#### 4.1.1 Unchanged Number of DOFs  
According to Table 1, there are four candidate methods including CA, ICA, Epsilon, and PCG methods when the structural DOFs remain unchanged. Previous studies have suggested that the CA method and the PCG method have certain equivalences, i.e., when the number of basis vectors (s) in the CA method is equal to the iteration steps (\(k\)) in the PCG method, the two methods can achieve the same computational accuracy [16]. Therefore, the CA method and the PCG method can be directly compared at the same level of accuracy, and the calculations of the \(Fr\) and \(Mr\) for the two methods are given as follows:  
\[Fr_{CA/PCG}=\frac{2s^{3}/3+2s^{2}+12 nsb+8 ns+2 ns^{2}-4 nb-2n}{[8 nb+14n+3]k+2n}, \tag{18}\]  
\[Mr_{CA/PCG}=\frac{s^{2}+(n+2)s+(3b+5)n}{2 nb+9n}. \tag{19}\]  
In this section, three cases were selected for the structural DOFs (\(n\)), namely, 20, 100, and 1000, and the half-bandwidth \(b\) was divided into two cases: \(0.2n\) and \(0.4n\). The calculation results are shown in Figure 3.  
It can be observed in Figure 3a that the \(Fr\) values of CA/PCG gradually increase with the increase of \(k\) or \(s\) and gradually converge to around 1.5 with the increase of structural DOFs. Only when the structural DOFs are small, and \(k\) or \(s\) is equal to 1, are the \(Fr\) values of CA/PCG less than 1. However, the accuracy of the two methods is generally low at this time, so the situation where \(k\) or \(s\) is 1 can be ignored. Based on this, it can be seen from a flops perspective that the computational efficiency of the CA method is lower than that of the PCG method. From the \(Mr\) curve in Figure 3b, it can be seen that the values of \(Mr\) for CA/PCG are greater than 1 under different DOFs and \(k\) or \(s\), indicating that the storage space of the CA method is relatively high, which further affects its efficiency. Therefore, the PCG method is more suitable than the CA method for the case of unchanged DOFs. For the Epsilon, PCG, and ICA methods, since the three methods do not have theoretical equivalence, the efficiency indicator of complete analysis (Cholesky factorization) is  
Figure 3: Indicator curves of CA/PCG: (**a**) \(Fr\); (**b**) \(Mr\). used as the numerator for calculation. In order to present the comparison results clearly, a log scale is used for the vertical axis. The relevant results are shown below. From the figures above, it can be observed that the \(\ln(Fr)\) values of the ICA method are the smallest among the three methods. Especially in cases of lower structural DOFs, even with smaller iteration steps, the corresponding \(\ln(Fr)\) values for the ICA method are smaller than 0. This implies that for structures with low DOFs, the computational efficiency of the ICA method is generally lower than that of complete analysis. Compared to the other two methods, the performance of ICA is poor. On the other hand, as shown in Figure 4f, its \(Mr\) values are below 0.7 both for different DOFs and iteration steps, indicating its low performance compared to the other two methods. Therefore, the ICA method is  
As for structures with higher DOFs, as shown in Figure 4a,c, both the Epsilon method and the PCG method have high computational efficiency, especially when there are small changes in high DOF structures (which require fewer basis vectors or iteration steps). Furthermore, when the number of basis vectors in the Epsilon method is less than or equal to 3, its \(\ln(Fr)\) value is significantly higher than that of the PCG method. This indicates that its computational efficiency is higher than that of PCG. However, as the number of basis vectors or iteration steps exceeds three, the difference between the two methods gradually narrows, and the PCG method proves to be more efficient.