## 3 The Integrated Shape Optimization Method for Branch-Supported Free-Form Surface Structure
### Establishment of the Shape Optimization Method for Branch-Supported Free-Form Surface Structure  
The Taylor expansion of the structural strain energy \(C(\textbf{P})\) at design variables **P** is given by:  
\[C(\textbf{P}+\Delta\textbf{P})=C(\textbf{P})+\sum_{i=1}^{n}\Delta P_{i}\frac{ \partial C(\textbf{P})}{\partial P_{i}}+\frac{1}{2}\left(\sum_{i=1}^{n}\Delta P _{i}\frac{\partial}{\partial P_{i}}\right)^{2}C(\textbf{P})+R_{2} \tag{7}\]  
where design variables **P** = (\(P_{1}\), \(P_{2}\), \(\ldots\), \(P_{n}\))T, \(\Delta\textbf{P}\) represents the variation in design variables **P**, and \(R_{2}\) is the Taylor series remainder term.  
Let  
\[\Delta P_{i}=-\lambda\cdot\frac{\partial C(\textbf{P})}{\partial P_{i}} \tag{8}\]  
When \(\lambda\) is a sufficiently small positive real number, substituting Equation (8) into Equation (7) and neglecting higher-order terms, we obtain:  
\[C(\textbf{P}+\Delta\textbf{P})\doteq C(\textbf{P})-\sum_{i=1}^{n}\lambda\cdot \left(\frac{\partial C(\textbf{P})}{\partial P_{i}}\right)^{2} \tag{9}\]  
Equation (9) shows that the structural strain energy can be reduced if the design variables are slightly adjusted in the negative direction of the strain energy gradient, i.e.,  
\[C(\textbf{P}+\Delta\textbf{P})\leq C(\textbf{P}) \tag{10}\]  
By adjusting the design variables according to Equation (11) until the strain sensitivity approaches zero (i.e., \(\partial C(\textbf{P})/\partial P_{i}\) = 0 (\(i\) = 1, 2, \(\ldots\), \(n\))). If further calculations are performed, the design variables will oscillate slowly around the extremum point, and the overall shape of the structure will remain almost unchanged. A reasonable structure with strain energy at a local minimum is obtained.  
\[P_{i}^{k+1}=P_{i}^{k}-\lambda\cdot\frac{\partial C\left(\textbf{P}^{k}\right) }{\partial P_{i}^{k}} \tag{11}\]  
where \(P_{i}^{k}\) and \(P_{i}^{k+1}\) represent the \(i\)-th design variable of the structure for the \(k\)-th and \((k+1)\)-th iterations, respectively, and \(\lambda\) represents the iteration step size.  
From Equation (11), it can be observed that the variation of the design variable is jointly determined by the iteration step size and the strain sensitivity. The iteration step size is an empirical parameter. For the branch-supported free-form surface structure, due to the different dimensionalities of the three types of design variables, the same iteration step size may result in differences in the order of magnitude of the adjusted physical distances. In order to ensure the validity of Equation (10) (i.e., the variations should be sufficiently small in magnitude), we propose to set different gradient reference values, denoted as \(\textbf{V}_{\text{R}}\) = (\(V_{\text{R}}^{\text{S}}\), \(V_{\text{R}}^{\text{O}}\), \(V_{\text{R}}^{\text{B}}\)), for different types of design variables. The size of \(\textbf{V}_{\text{R}}\) is determined as follows:In terms of the data points of the free-form surface:  
\[V_{\rm R}^{\rm S}=\max\left(\left|\frac{\partial{\rm C}}{\partial{\bf P}_{i}^{ \rm S}}\right|\right) \tag{12}\]  
In terms of the intersection points between the branches and free-form curved roof:  
\[V_{\rm R}^{\rm O}=\max\left(\left|\frac{\partial s}{\partial{\bf P}_{i}^{\rm O} }\cdot\frac{\partial{\rm C}}{\partial{\bf P}_{i}^{\rm O}}\right|\right) \tag{13}\]  
where, the expression \"\(\partial s/\partial{\bf P}_{i}^{\rm O}\)\" represents the partial derivative of arc length \"\(s\)\" with respect to \({\bf P}_{i}^{\rm O}\).  
In terms of the non-intersection points of the branching structure:  
\[V_{\rm R}^{\rm B}=\max\left(\left|\frac{\partial{\rm C}}{\partial{\bf P}_{i}^ {\rm B}}\right|\right) \tag{14}\]  
Furthermore, introduce a sufficiently small positive real number \(\varepsilon\) to determine \(\lambda\):  
\[\lambda=\frac{\varepsilon}{{\bf V}_{\rm R}} \tag{15}\]  
where, \(\varepsilon\) represents the maximum physical spatial variation of the design variables along their respective tuning directions allowed based on experience.