# Selection Criterion of Reanalysis Methods for Plane Truss Optimization
## 3 Quantitative Analysis of Reanalysis Methods
### Efficiency Study of the Methods  
Various factors affect the efficiency of algorithms, such as hardware conditions, computation scale, and complexity. Therefore, it is difficult to find a universal indicator that accurately reflects computational efficiency. [PERSON] et al. [30] proposed using the number of floating point operations (flops) to measure the computational complexity of algorithms, but results relying solely on this indicator have certain limitations. Thus, this study proposes using the storage space required for input, output, and intermediate variables of an algorithm (referred to as the MS value) as an indicator of the algorithm's reading and writing time, in addition to using the flops. This approach enables a more comprehensive assessment of computational efficiency and assists in the measurement of the computational efficiency of various reanalysis methods. A flop is defined as a single floating point operation that includes one of the four basic arithmetic operations: addition, subtraction, multiplication, or division. For example, if a dot product is performed between two vectors with \(n\) elements each, then the number of flops is \(2n\) because it involves \(n\) multiplications and \(n\) additions. Furthermore, real structural stiffness matrices are generally sparse. In order to save storage space, this study stores the stiffness matrix using a two-dimensional banded format and assumes that a single element occupies one unit of storage space. For instance, if the initial stiffness matrix \(K_{0}\) of size \(n\times n\) has a half-bandwidth of \(b_{0}\), its MS value is \((b_{0}+1)n\). The MS value of other variables is calculated based on their actual size. Using the CA method as an example, the calculation procedure for flops and MS values is illustrated. Firstly, the initial equilibrium equation of a structure with \(n\) DOFs is defined:  
\[K_{0}r_{0}=f_{0}, \tag{2}\]  
where \(K_{0}\) is the initial stiffness matrix of size \(n\times n,f_{0}\) is the initial load vector, and \(r_{0}\) is the initial response of the structure. Due to the symmetry and positive definiteness of \(K_{0}\), it can be decomposed via Cholesky factorization:  
\[K_{0}=U_{0}^{T}U_{0}, \tag{3}\]  
where \(U_{0}\) is the upper triangular matrix. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Modification**} & \multicolumn{2}{c}{**The Number of**} & \multicolumn{2}{c}{**The Number of**} & \multicolumn{2}{c}{**The Number of**} \\  & \multicolumn{2}{c}{**DOFs is Unchanged**} & \multicolumn{2}{c}{**DOFs Is Increased**} & \multicolumn{2}{c}{**DOFs Is Decreased**} \\ \cline{2-6}  & \multirow{2}{*}{**Size Modification**} & **Shape Modification** & **Topology Modification** & **Topology Modification** & **Topology Modification** \\ \hline \multirow{4}{*}{Applicable methods} & CA & CA & CA & Guyan + CA & MPCG \\  & Epsilon & ICA & Epsilon & Guyan + Epsilon & \\ \cline{1-1}  & PCG & Epsilon & PCG & Guyan + PCG & \\ \hline \hline \end{tabular} DOFs: degrees of freedom.  
\end{table} 
Table 1: Qualitative selection of reanalysis methods. Assume that a structural modification causes an incremental stiffness matrix \(\Delta K\), and the modified stiffness matrix is \(K\) (with a half-bandwidth of \(b\)). The loading vector is denoted by \(f\) and the structural response by \(r\). In this study, the changes in loading are not considered, i.e.,\(f=f_{0}\). The new equilibrium equation is given as follows:  
\[Kr=(K_{0}+\Delta K)r=f. \tag{4}\]  
The CA method mainly reduces the computational complexity by constructing a reduced basis vector. The main steps of the CA method [22] in the structural analysis are given as follows:  
1. [noitemsep]
2. Approximate the modified structural response \(r\) using \(s\) linearly independent basis vectors. Define basis vectors \(r_{i}\) and construct a basis vector matrix \(r_{B}\): \[r_{1}=K_{0}^{-1}f,\] (5) \[r_{i}=Br_{i-1}\,\ i=2,3,\cdots s,\] (6) \[r_{B}=[r_{1},r_{2},\cdots,r_{s}],\] (7) where \(B\) is \(-K_{0}^{-1}\Delta K\). 3.