## 3 An Improved Objective Function Based on SVDM
### Selection of Key Parameters in Improved Objective Function
#### 3.3.4 The Gradient of the Improved Objective Function  
The gradient of the objective function might be used to increase the efficiency of optimization. In Equation (8), the first derivative of the FRF \(h_{ai}(\mathbf{\mu},\overline{\omega_{i}})\) with respect to the damage factor can be obtained by differentiation of Equations (6) and (7). Then the gradient expression of the objective function is as shown in Equation (10):  
\[\frac{df(\mathbf{\mu})}{d\mu_{s}}=\alpha_{\varphi}\sum\limits_{i=1}^{n_{\varphi}} \|1-\frac{\mathbf{A}_{i,s}}{\mathbf{B}_{i}}-\frac{\mathbf{A}_{i}\cdot\mathbf{B}_{i,s}}{\mathbf{B}_ {i}^{2}}\|-\alpha_{\omega}\sum\limits_{l=1}^{n_{\omega}}\sum\limits_{\alpha=1} ^{n_{s}}2h_{ai}(\mathbf{\mu},\overline{\omega_{i}})\frac{dh_{ai}(\mathbf{\mu},\overline {\omega_{l}})}{d\mu_{s}} \tag{10}\]  
where \(\mathbf{A}_{i},\mathbf{B}_{i},\mathbf{A}_{i,s},\text{ and }B_{i,s}\) can be calculated by Equation (11).  
\[\left\{\begin{array}{l}\mathbf{A}_{i}=\sum\limits_{\alpha=1}^{n_{s}}h_{ai}(\mathbf{ \mu},\overline{\omega}_{i})\overline{\varphi}_{ai}\\ \mathbf{B}_{i}=\sqrt{\sum\limits_{\alpha=1}^{n_{s}}h_{ai}^{2}(\mathbf{\mu},\overline{ \omega_{i}})}\;\times\sum\limits_{\alpha=1}^{n_{s}}\overline{\varphi}_{ai}^{2} \\ \mathbf{A}_{i,s}=\sum\limits_{\alpha=1}^{n_{s}}\frac{dh_{ai}(\mathbf{\mu},\overline{ \omega_{i}})}{d\mu_{s}}\overline{\varphi}_{ai}\\ \mathbf{B}_{i,s}=\sum\limits_{\alpha=1}^{n_{s}}h_{ai}(\mathbf{\mu},\overline{\omega_{ i}})\frac{dh_{ai}(\mathbf{\mu},\overline{\omega_{i}})}{d\mu_{s}}\times\sum\limits_{ \alpha=1}^{n_{s}}\overline{\varphi}_{ai}^{2}/\mathbf{B}_{i}\end{array}\right. \tag{11}\]  
For the calculation of the gradient of the objective function, it is crucial to find the first derivative of the FRF with respect to the damage factor, i.e., \(\frac{dh_{ai}(\mathbf{\mu},\overline{\omega_{i}})}{d\mu_{s}}\). The first derivative is applied to both sides of Equation (6), and Equation (12) is obtained.  
\[\frac{dh_{ai}(\omega)}{d\mu_{s}}=\sum\limits_{i,j}D_{aij}(\omega)\frac{d\gamma _{ij}^{0}(\omega)}{d\mu_{s}} \tag{12}\]The first derivative with respect to the damage factor is then applied to both sides of Equation (7), which yields Equation (13).  
\[(1-\delta_{\text{is}})\gamma_{lj}^{L}(\omega)=\frac{d^{\gamma 0}_{lj}(\omega)}{d\mu_{s}}-(1-\delta_{\text{is}})\times\sum_{kJ}D_{ijkl}(\omega )\frac{d\gamma_{kl}^{0}(\omega)}{d\mu_{s}} \tag{13}\]  
where \(i,k,s=1,2,\cdots,n_{d}\); and \(\delta_{\text{is}}\) is Kronecker's delta:  
\[\delta_{\text{is}}=\begin{cases}1&i=s\\ 0&i\
eq s\end{cases} \tag{14}\]  
Equation (13) is a linear equation that can be solved to obtain the first derivative \(\frac{d\gamma_{lj}^{0}(\omega)}{d\mu_{s}}\) of the substructural virtual distortion with respect to the damage factor. Then, the first derivative \(\frac{d\mu_{s}(\mathbf{\mu}\overline{\mathbf{\sigma}}_{lj})}{d\mu_{s}}\) of the FRF with respect to the damage factor can be obtained by substituting \(\frac{d\gamma_{lj}^{0}(\omega)}{d\mu_{s}}\) into Equation (12). In this way, the entire gradient of the objective function can be successively obtained.