# Selection Criterion of Reanalysis Methods for Plane Truss Optimization
## 2 Qualitative Classification of Structural Reanalysis Methods  
The qualitative classification of reanalysis methods is primarily achieved by analyzing the effects of different types of structural modifications (size, shape, or topology) on the structure. This analysis is to establish the correspondence between the reanalysis methods and the applicable modification types. In this section, CA, ICA, PCG, Epsilon [29], and their variant methods oriented towards changes in DOFs are selected as research objects. The improved PCG method is selected for the reduction of DOFs in the structure, whereas the combinations of the Guyan method and other methods are used for the increase of DOFs. The STSA algorithm is used for structural generation and optimization in this study. It integrates shape grammar and structural evaluation into an improved simulated annealing algorithm, forming a discrete structural optimization method. The shape grammar defines the basic rules for structural size, shape, and topology transformations. Each rule acts dynamically on the initial structure based on probabilities, achieving free transformation of the structural configuration. The workflow is shown in Figure 1[28]. According to the principles of the STSA algorithm, structural modifications only involve one of size, shape, or topology transformation at a time. Therefore, determining the effects of a single modification on the structure will be beneficial for the qualitative classification of reanalysis methods. In this paper, the effects of a single modification are divided according to the number of modifications to the elements in the stiffness matrix and according to whether the modifications are local or global in the structure. The size transformation of a structure involves modifying the cross-section of a single member. The resulting incremental stiffness matrix, \(\Delta K\), as shown in Equation (1), contains  
Figure 1: Flow chart of STSA. non-zero elements only at the corresponding nodes \(i\) and \(j\), whereas all the other elements are zero. This indicates that the impact of the cross-section changes in a single member of the structure is relatively limited, especially when the total number of nodes increases. Overall, the effect of size transformation in the structural optimization process is relatively small and can be considered as a local small modification. \[\Delta K=\frac{E\Delta A}{l_{ij}}\left[\begin{matrix}\cdots&i&\cdots&j&\cdots\\ &&&&&\vdots\\ && \begin{bmatrix}c^{2}&sc\\ sc&s^{2}\end{bmatrix} &\cdots& \begin{bmatrix}-c^{2}&-sc\\ -sc&-s^{2}\end{bmatrix} &&i\\ &&\vdots&\ddots&\vdots&\vdots\\ && \begin{bmatrix}-c^{2}&-sc\\ -sc&-s^{2}\end{bmatrix} &\cdots& \begin{bmatrix}c^{2}&sc\\ sc&s^{2}\end{bmatrix} &&j\\ &&&&&\vdots\end{bmatrix}, \tag{1}\]  
where \(E\) is the elasticity modulus, \(L_{ij}\) is the length of a member, and \(\Delta A\) represents the changes in the cross-section. Shape transformation of the structure refers to the modification of a single node's position, assuming that there are \(m\) nodes connected to the modified node. The resulting non-zero element count in the incremental stiffness matrix is \(12m+4\). The size of this modification is closely related to the number of connected nodes. In the structural design process, the maximum number of connections for a single node is generally limited due to the consideration of subsequent construction difficulties. In this study, the maximum number of connections for a single node is 5. Overall, the incremental stiffness matrix caused by the changes of node positions exhibits a nonlinear relationship with the design variable (node coordinates). At the same time, its rank is also relatively high. Compared with the size transformation, shape transformation brings relatively greater effects and can be considered as a local moderate change. Topology transformation includes three cases. The first one is a topology modification where the DOFs remain unchanged, as shown in Figure 2a.