# A Form-Finding Method for Branching Structures Based on Dynamic Relaxation
###### Abstract  
Branching structure is often used as a supporting structure of the grid shell due to its geometrical and force-transferring features, and the rationality of its shape is very important. The \"physical\" and \"numerical\" hanging models can be used for the joint form-finding of the branching structure and free-form grid shell. However, slack elements may exist in the equilibrium model which corresponds to the inefficient members in the form-found branching structure. To solve this problem, a form-finding method of branching structure based on dynamic relaxation is proposed in this study. The proposed method clusters the elements of the branching model and equalizes the axial forces of the elements in the same cluster, in other words, there are no slack elements in the equilibrium branching model. This method overcomes the defect that the equilibrium branching model may have slack elements and needs many manual adjustments during the procedure of determining the rational shape of a branching structure, and effectively prevents the inefficient members existing in the form-found structure. Numerical examples are provided to demonstrate the characteristics of the proposed method and its effectiveness is verified as well. branching structure; element cluster; dynamic relaxation; form finding; numerical inverse hanging method +
Footnote †: journal: Applied Sciences  
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Tt. [PERSON], [PERSON], [PERSON], Y. A Form-Finding Method for Branching Structures Based on Dynamic Relaxation. _Appl. Sci._**2021**, _11_, 7165. [[https://doi.org/10.3390/appl1157165](https://doi.org/10.3390/appl1157165)]([https://doi.org/10.3390/appl1157165](https://doi.org/10.3390/appl1157165))  
Academic Editor:  
[PERSON]  
Received: 24 June 2021  
Accepted: 30 July 2021  
Submitted: 3 August 2021  
**Problems:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. to generate \"natural\" shapes of branching structures with curved branches [11; 12]. [PERSON] et al. proposed the double element method and applied it to the shape-finding analysis of branching structure [13]. In addition to the above-mentioned numerical methods for branching structure, other methods that can simulate the physical model experiment also can be used to determine the reasonable shape of branching structure, such as dynamic relaxation method [14; 15], vector form intrinsic finite element method [16; 17; 18], and generalized inverse method [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. The branching structure is often used as the supporting structure of the free-form latticed shell. To determine the reasonable shape of the whole structure, it is necessary to consider both the upper latticed shell and the lower branching structure. The physical and computational hanging upside-down models can be applied to the form-finding of both branching structure and free-form latticed shell. However, when the flexible model becomes equilibrium, there may be slack parts in the equilibrium branching model and these slack parts correspond to the inefficient members of the form-found branching structure. The slack parts in the model are difficult to avoid and therefore designers need to manually adjust the model many times to prevent the generation of the inefficient members in form-found branching structure. The adjustment is time-consuming and error-prone when establishing the model. Dynamic relaxation method is a frequently used form-finding method. In this study, the element cluster is introduced into the dynamic relaxation method to solve the problem causing by the slack elements of the equilibrium branching model when a computational hanging model is used to determine the shapes of branching structures. The remainder of this study is as follows. The illustration of the numerical implementation of the method is presented in Section 2. Then, a simple example is given to illustrate the characteristics and implementation in Section 3.