# Optimization of Pile Driver Frame Based on Sensitivity Analysis
## 3 Theoretical Foundations of Sensitivity Analysis  
We refer to the gradient of variation of the structural or product performance values with respect to the design variables as sensitivity. The design variables in this paper are the thickness of each plate among the large body structure mentioned above. Changes in the design variables cause changes in the product performance values, such as the mass, strength, and stiffness of the structure. When analyzing the structural stiffness, the gradient of the stiffness variation to the value of the thickness (\(d\)) of each plate of the large body is the sensitivity [18]. Mathematically, if \(F(d)\) is continuous and derivable, the first-order sensitivity S of this function when Equation (1) exists is:  
\[\frac{\partial F(d)}{\partial d_{j}} \tag{1}\]  
where: \(F(d)\) is the performance value of the large body structure, \(d_{j}\) is the design variable of the large body structure. Finite element analysis theory emphasizes that the general structural static finite element analysis has:  
\[\left[K\right]_{n\times n}\{u\}_{n\times 1}\{F\}_{n\times 1}=\left\{F\right\}_{n \times 1} \tag{2}\]  
where: \(\left\{F\right\}_{n\times 1}\) denotes the external load vector of the structure, \(\left\{u\right\}_{n\times 1}\) is the displacement vector of the structure. \(\left[K\right]_{n\times n}\) is the total stiffness matrix of the finite element model; \(n\) is the degrees of freedom of the structure. \(\left[F_{d}\right]_{n\times 1}\) is the partial derivative of \(\left\{F\right\}_{n\times 1}\) with respect to the design variables \(d\). \(\left[u_{d}\right]_{n\times 1}\) is the partial derivative of \(\left\{u\right\}_{n\times 1}\) with respect to the design variables \(d\). \(\left[K_{d}\right]_{n\times n}\) is the derivative matrix of \(\left[K\right]_{n\times n}\) to \(d\), and the derivative of (2) with respect to \(d\) is obtained:  
\[\left[K_{d}\right]_{n\times n}\{u\}_{n\times 1}+\left\{K\right\}_{n\times n} \{u_{d}\}_{n\times 1}=\left\{F_{d}\right\}_{n\times 1} \tag{3}\]  
Figure 7: Large body skeleton stress cloud and skeleton displacement cloud. (**a**) Large body skeleton stress cloud; (**b**) large body skeleton displacement cloud. If the vector \(\left\{F\right\}_{n\times 1}\) is equal to 0, then there is:  
\[\left[F_{\mathcal{A}}\right]_{n\times 1}=\left\{0\right\}_{n\times 1} \tag{4}\]  
\[\left[u_{\mathcal{A}}\right]_{n\times 1}=\left\{K\right\}_{n\times n}^{-1} \left\{K_{\mathcal{A}}\right\}_{n\times n}\left\{u\right\}_{n\times 1} \tag{5}\]  
where: \(\left[K\right]_{n\times n}^{-1}\) is the inverse matrix of the total stiffness matrix of the structural finite element model. The total stiffness matrix of the structural finite element model is obtained after summing the corresponding matrices of each cell, which is:  
\[\left[K\right]_{n\times n}=\sum\left[K_{\mathcal{A}}^{i}\right]_{n\times n} \tag{6}\]  
where: \(i\) is the cell number; \(\left[K\right]_{n\times n}\) is the expanded cell stiffness matrix. The derivative of the formula of (6) to \(d\) is obtained:  
\[\left[K_{\mathcal{A}}\right]_{n\times n}=\sum i\left[K_{\mathcal{A}}^{i} \right]_{n\times n} \tag{7}\]  
Equation (5) can be replaced by:  
\[\left[u_{\mathcal{A}}\right]_{n\times 1}=-\left[K\right]_{n\times n}^{-1} \sum\left[K_{\mathcal{A}}^{i}\right]_{ni\times ni}\left\{u\prime\right\}_{ni \times 1} \tag{8}\]  
where: \(\left[K_{\mathcal{A}}^{i}\right]_{ni\times ni}\) is the cell stiffness derivative matrix without order expansion, \(ni\) is the cell degrees of freedom. \(\left\{u\prime\right\}_{ni\times 1}\) is the component of the corresponding unit \(i\) in \(\left\{u\right\}_{n\times 1}\), which is the vector of the individual unit displacements. Then, the formula of (8) is required to calculate the sensitivity of the structural displacement vector \(\left\{u\right\}_{n\times 1}\) to the design variable d. In the optimization analysis of the large body skeleton structure, the large body finite element model is composed of plate and shell units.