## 3 The Integrated Shape Optimization Method for Branch-Supported Free-Form Surface Structure
### The Sensitivity of Branch-Supported Free-Form Surface Structure  
Assuming that the loads \({\bf F}\) applied to the nodes of branch-supported free-form surface structure remain unchanged during the adjustment of the design variables (i.e., the loads are independent of the design variables), the partial derivative of the strain energy \(C\) with respect to the design variable \({\bf P}\) can be determined using the chain rule as given in Equation (16):  
\[\frac{\partial{\rm C}}{\partial{\bf P}_{i}}=\sum_{j}\frac{\partial{\rm C}}{ \partial{\bf X}_{j}}\cdot\frac{\partial{\bf X}_{j}}{\partial{\bf P}_{i}} \tag{16}\]  
where \({\bf X}_{j}\) represents the \(j\)-th node in the structural finite element model, \({\bf P}_{i}\) represents the \(i\)-th design variable in the structural geometric model, the expression \"\(\partial{\rm C}/\partial{\bf X}_{j}\)\" represents the partial derivative of the strain energy \(C\) with respect to \({\bf X}_{j}\), and \"\(\partial{\bf X}_{j}/\partial{\bf P}_{i}\)\" represents the partial derivative of \({\bf X}_{j}\) with respect to \({\bf P}_{i}\). 2.1 The Partial Derivative of Strain Energy \({\rm C}\) with Respect to Structural Nodes \({\rm X}\)  
According to the finite element stiffness equation, we have:  
\[{\bf K}{\bf U}={\bf F} \tag{17}\]  
where \({\bf K}\) is the global stiffness matrix of the structure, \({\bf U}\) is the column vector of nodal displacements, and \({\bf F}\) is the column vector of nodal loads. Differentiating both sides of Equation (17) with respect to \({\bf X}_{i}\) yields:  
\[\frac{\partial{\bf U}}{\partial{\bf X}_{i}}=-{\bf K}^{-1}\frac{\partial{\bf K }}{\partial{\bf X}_{i}}{\bf U} \tag{18}\]  
The expression for the strain energy \(C\) can be given as:  
\[C=\frac{1}{2}{\bf F}^{\rm T}\cdot{\bf U} \tag{19}\]Differentiating both sides of Equation (19) with respect to \(\mathbf{X}_{i}\) yields:  
\[\frac{\partial C}{\partial\mathbf{X}_{i}}=\frac{1}{2}\mathbf{F}^{\mathrm{T}} \cdot\frac{\partial\mathbf{U}}{\partial\mathbf{X}_{i}} \tag{20}\]  
Substituting Equations (17) and (18) into Equation (20), we obtain:  
\[\frac{\partial C}{\partial\mathbf{X}_{i}}=-\frac{1}{2}\mathbf{U}^{\mathrm{T}} \frac{\partial\mathbf{K}}{\partial\mathbf{X}_{i}}\mathbf{U} \tag{21}\]  
2.2 The Partial Derivative of Structural Nodes \(\mathbf{X}\) with Respect to Design Variables \(\mathrm{P}\)  
According to Section 2.3, the design variables \(\mathbf{P}\) of branch-supported free-form surface structure consist of three parts: the data points of surface (\(\mathbf{P}^{\mathrm{S}}\)), the intersection points between branches and surface (\(\mathbf{P}^{\mathrm{O}}\)), and the non-intersection points of branches (\(\mathbf{P}^{\mathrm{B}}\)). Similarly, the nodes \(\mathbf{X}\) in the structural finite element model can also be divided into three types based on their positions: the nodes exclusively belonging to shell elements (\(\mathbf{X}^{\mathrm{S}}\)), the intersection nodes between shells and beams (\(\mathbf{X}^{\mathrm{O}}\)), and the nodes exclusively belonging to beam elements (\(\mathbf{X}^{\mathrm{B}}\)), i.e., \(\mathbf{X}=\{\mathbf{X}^{\mathrm{S}},\mathbf{X}^{\mathrm{O}},\mathbf{X}^{ \mathrm{B}}\}^{\mathrm{T}}\). The partial derivative \(\partial\mathbf{X}/\partial\mathbf{P}\) can be calculated based on the influence area of the design variables \(\mathbf{P}\). * For the design variable \(\mathbf{P}^{\mathrm{S}}_{j,k}\in\{\mathbf{P}^{\mathrm{S}}\}\), moving \(\mathbf{P}^{\mathrm{S}}_{j,k}\) only affects the surface shape, i.e., \(\{\mathbf{X}^{\mathrm{S}},\mathbf{X}^{\mathrm{O}}\}\).