## 3 Adaptive Subinterval Finite Element Method Based on Dynamic Sensitivity Analysis
### Improved Interval Finite Element Method
\begin{split}&\left(-\mathbf{K}^{C,-1}\sum\limits_{i=1}^{n} \mathbf{K}_{a_{i}}^{R}e_{i}^{I}\right)^{1}=\sum\limits_{i=1}^{n}-\mathbf{K}^{ C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{I},\\ &\left(-\mathbf{K}^{C,-1}\sum\limits_{i=1}^{n}\mathbf{K}_{a_{i} }^{R}e_{i}^{I}\right)^{2}=\sum\limits_{i=1}^{n}\left(-\mathbf{K}^{C,-1} \mathbf{K}_{a_{i}}^{R}e_{i}^{I}\right)^{2}+\sum\limits_{\begin{subarray}{c}i,j=1\\ i\
eq j\end{subarray}}^{n}\left(-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^ {I}\right)\left(-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{j}^{I}\right),\\....\end{split}  \tag{21}\]  
Substituting Equation (21) into Equation (18), we obtain  
\[\mathbf{K}^{-1}\Big{(}\mathbf{a}^{I}\Big{)}=\left[\mathbf{I}+\sum\limits_{i= 1}^{n}\sum\limits_{r=1}^{\infty}\left(-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R} e_{i}^{I}\right)^{r}+\sum\limits_{ \begin{subarray}{c}i,j=1\\ i\
eq j\end{subarray} }^{n}\left(-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{ i}^{I}\right)\left(-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{I}\right)+ \cdots\right]\mathbf{K}^{C,-1} \tag{22}\]  
If condition \(\left\|-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{I}\right\|<1\) is satisfied, the geometric series \(\sum\limits_{i=1}^{n}\sum\limits_{r=1}^{\infty}\left(-\mathbf{K}^{C,-1}\mathbf{ K}_{a_{i}}^{R}e_{i}^{I}\right)^{r}\) of the second term converges and can be calculated using the summation formula. Neglecting the non-geometric series terms after the second term, the inverse matrix of the interval stiffness matrix can be approximated as follows:\[ \begin{split}\mathbf{K}^{-1}\Big{(}\mathbf{a}^{I}\Big{)}& \approx\Bigg{[}\mathbf{I}+\sum_{i=1}^{n}\sum_{r=1}^{\infty}\left(-\mathbf{K}^{ C,-1}K_{a_{i}}^{R}e_{i}^{I}\right)^{r}\Bigg{]}\mathbf{K}^{C,-1}\\ &=\Bigg{[}\mathbf{I}+\sum_{i=1}^{n}-\mathbf{K}^{C,-1}\mathbf{K} _{a_{i}}^{R}e_{i}^{I}\Big{/}\left(\mathbf{I}+\mathbf{K}^{C,-1}K_{a_{i}}^{R}e_{ i}^{I}\right)\Bigg{]}\mathbf{K}^{C,-1}\end{split}  \tag{23}\]  
We write \(-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{I}/\left(\mathbf{I}+\mathbf{K}^ {C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{I}\right)\) in the above equation as \(\mathbf{B}\big{(}a_{i}^{I}\big{)}\), which is clearly a monotonic function about the interval parameter \(a_{i}^{I}\). Thus, \(\mathbf{B}\big{(}a_{i}^{I}\big{)}\) can be expressed in the form of the midpoint and radius as follows:  
\[\mathbf{B}\big{(}a_{i}^{I}\big{)}=\mathbf{B}_{a_{i}}^{C}+\mathbf{B}_{a_{i}}^{ R}e_{i}^{I} \tag{24}\]  
where  
\[\mathbf{B}_{a_{i}}^{C}=\frac{1}{2}\Bigg{(}\frac{-\mathbf{K}^{C,-1}\mathbf{K} _{a_{i}}^{R}}{\mathbf{I}+\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}}+\frac{ \mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}}{I-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i} }^{R}}\Bigg{)},\mathbf{B}_{a_{i}}^{R}=\frac{1}{2}\Bigg{(}\frac{-\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}}{\mathbf{I}+\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R} }-\frac{\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}}{\mathbf{I}-\mathbf{K}^{C,-1} \mathbf{K}_{a_{i}}^{R}}\Bigg{)} \tag{25}\]  
Substituting Equation (25) into Equation (23), we obtain  
\[\mathbf{K}^{-1}\Big{(}\mathbf{a}^{I}\Big{)}=\Bigg{[}\mathbf{I}+\sum_{i=1}^{n} \Big{(}\mathbf{B}_{a_{i}}^{C}+\mathbf{B}_{a_{i}}^{R}e_{i}^{I}\Big{)}\Bigg{]} \mathbf{K}^{C,-1} \tag{26}\]  
Now, we can express the inverse of the interval stiffness matrix in an explicit form, which facilitates the analysis of a structural uncertainty response. Substituting Equations (15) and (26) into Equation (13), we can obtain the expression for the displacement vector as follows:  
\[