# A Form-Finding Method for Branching Structures Based on Dynamic Relaxation
## 3 Characteristics of the Element-Clustered Method  
In this section, a simple example is given to illustrate the influence of the setting of element clusters on the form found results and the implementation process of the method. Figure 1a shows a branching model, which is composed of the trunk and two-level branches. The elements in the model are numbered. The upper nodes of the model are connected to a board and these nodes are fixed. A vertical downward load is applied to the bottom node to simulate a common physical experiment for determining the shape of branching structures. In the model, different element clusters are used to obtain different shape schemes, and the results are compared with those obtained by the traditional dynamic relaxation method. Figure 1: Shape schemes of different element clusters. (**a**) initial flexible model; (**b**) form-found shape of traditional DR method; (**c**,**d**) form-found shapes of element-clustered method. The form found shape shown in Figure 1b is obtained by the traditional dynamic relaxation method. In this process, the compressive elastic modulus of the element is not considered but only the tensile elastic modulus is considered, because we assume that the model is composed of cables that can only bear tensile axial forces. When the model reaches an equilibrium state, the internal forces of the elements are shown in Table 1. It is easy to observe that there are elements whose internal forces are much smaller than the others in the second-level branches. Figure 1c shows the equilibrium shape of the model when elements in the same branching level are clustered. In this example, the elements are assigned to three clusters. As shown in Figure 1c, different color represents a different cluster and the elements with the same color are in the same cluster. The axial forces of the elements in the same cluster are almost equal in scheme II according to Table 1 data, and there are no slack elements in the model. To further illustrate the formation of element clusters, the clustering matrix **B** corresponding to Figure 1c is given. Element 1 is in cluster 1, elements 2-4 are in cluster 2, and elements 5-13 are in cluster 3. According to Equation (7), the clustering matrix **B** is  
\[\textbf{B}=\left[ \begin{array}{cccccccccccc}1&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&1&1&1&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&1&1&1&1&1&1&1&1\end{array} \right] \tag{16}\]  
In the form-found result shown in Figure 1d, all elements are assigned to five element clusters. There are three element clusters in the second-level branches, each cluster contains three interconnected elements. The three elements in the first-level branch form the fourth cluster. The trunk acts as a cluster alone. As shown in Figure 1d, same as the previous figure, the elements are organized by color and each color type represents a cluster. From Table 1, the axial forces of elements in the same cluster are almost equal, and there are no slack elements in the model in scheme III. In scheme I, there are slack elements in the equilibrium model. When the structure is determined according to this shape, there will be inefficient members in the structure. In schemes II and III, the automatic adjustment of the model eliminates all slack elements by introducing element clusters. In scheme II, the elements 5-13 (second-level branches) are in the same cluster, it determines that the axial forces of these elements must be equal in the equilibrium model. Different from scheme II, in scheme III the elements 5-13 are assigned to three clusters, which results in the force-bearing states of these elements are different from that of scheme II, so the two form-found shapes are different. The resultant shape is determined by forces and vice versa.