## 3 Adaptive Subinterval Finite Element Method Based on Dynamic Sensitivity Analysis
### Improved Interval Finite Element Method  
The interval equilibrium equation of Equation (10) is rewritten as an explicit equation:  
\[\mathbf{u}^{I}=\mathbf{K}^{-1}\left(\mathbf{a}^{I}\right)\mathbf{f}\left( \mathbf{a}^{I}\right) \tag{13}\]  
By using the first-order Taylor series expansion, the stiffness matrix and load vector of the uncertain structure can be expressed as follows:  
\[\mathbf{K}\left(\mathbf{a}^{I}\right)\approx\mathbf{K}\left(\mathbf{a}^{C} \right)+\sum_{i=1}^{n}\frac{\partial\mathbf{K}}{\partial a_{i}}\left(\mathbf{ a}^{C}\right)\Delta a_{i}^{I}=\mathbf{K}\left(\mathbf{a}^{C}\right)+\sum_{i=1}^{n} \frac{\partial\mathbf{K}}{\partial a_{i}}\left(\mathbf{a}^{C}\right)\Delta a_ {i}e_{i}^{I}=\mathbf{K}^{C}+\sum_{i=1}^{n}\mathbf{K}_{a_{i}}^{R}e_{i}^{I} \tag{14}\]  
\[\mathbf{f}\left(\mathbf{a}^{I}\right)\approx\mathbf{f}\left(\mathbf{a}^{C} \right)+\sum_{i=1}^{n}\frac{\partial\mathbf{f}}{\partial a_{i}}\left(\mathbf{ a}^{C}\right)\Delta a_{i}^{I}=\mathbf{f}\left(\mathbf{a}^{C}\right)+\sum_{i=1}^{n} \frac{\partial\mathbf{f}}{\partial a_{i}}\left(\mathbf{a}^{C}\right)\Delta a _{i}e_{i}^{I}=\mathbf{f}^{C}+\sum_{i=1}^{n}f_{a_{i}}^{R}e_{i}^{I} \tag{15}\]  
where \(\mathbf{K}^{C}\) and \(\mathbf{f}^{C}\) denote midpoints of \(\mathbf{K}\left(\mathbf{a}^{I}\right)\) and \(\mathbf{f}\left(\mathbf{a}^{I}\right)\), and \(\sum\limits_{i=1}^{n}\mathbf{K}_{a_{i}}^{R}e_{i}^{I}\) and \(\sum\limits_{i=1}^{n}\mathbf{f}_{a_{i}}^{R}e_{i}^{I}\) denote deviation intervals of \(\mathbf{K}\left(\mathbf{a}^{I}\right)\) and \(\mathbf{f}\left(\mathbf{a}^{I}\right)\). The Neumann series expansion [37] is a method used to compute the inverse of a matrix. For an arbitrary matrix \(\mathbf{H}\in\mathbb{R}^{n\times n}\), \(\rho(\mathbf{H})\) denotes its spectral radius and \(\mathbf{I}\in\mathbb{R}^{n\times n}\) denotes a unit matrix of the same size as \(\mathbf{H}\). If \(\rho(\mathbf{H})<1\), then \(\left(\mathbf{I}-\mathbf{H}\right)^{-1}\) exists:  
\[\left(\mathbf{I}-\mathbf{H}\right)^{-1}=\sum_{r=0}^{\infty}\mathbf{H}^{r} \tag{16}\]  
The uncertainty matrix \(\mathbf{M}\) is written in the form of the matrix \(\mathbf{M}^{C}+\Delta\mathbf{M}^{I}\), whose inverse matrix \(\mathbf{M}^{-1}\) can be expanded using the Neumann series as follows:\[ \begin{split}\mathbf{M}^{-1}&=\left(\mathbf{M}^{C}+\Delta \mathbf{M}^{I}\right)^{-1}\\ &=\left\{\mathbf{M}^{C}\Big{[}\mathbf{I}+\mathbf{M}^{C-1}\Delta \mathbf{M}^{I}\Big{]}\right\}^{-1}\\ &=\left\{\mathbf{I}+\sum\limits_{r=1}^{\infty}\left[-\mathbf{M} ^{C,-1}\Delta\mathbf{M}^{I}\right]^{r}\right\}\mathbf{M}^{C-1}\end{split}  \tag{17}\]  
where \(r\) is the order of the Neumann expansion and \(\mathbf{M}^{C,-1}=\left(\mathbf{M}^{C}\right)^{-1}\) is the inverse matrix of \(\mathbf{M}^{C}\). Using the above method, the inverse matrix of the interval stiffness matrix can be expanded as follows:  
\[ \begin{split}\mathbf{K}^{-1}\Big{(}\mathbf{a}^{I}\Big{)}& =\left(\mathbf{K}^{C}+\sum\limits_{i=1}^{n}\mathbf{K}_{a_{i}}^{R} e_{i}^{I}\right)^{-1}\\ &=\left[\mathbf{I}+\sum\limits_{r=1}^{\infty}\left(-\mathbf{K}^ {C,-1}\sum\limits_{i=1}^{n}\mathbf{K}_{a_{i}}^{R}e_{i}^{I}\right)^{r}\right] \mathbf{K}^{C,-1}\end{split}  \tag{18}\]  
The convergence condition for Equation (18) is as follows:  
\[\rho\left(\sum\limits_{i=1}^{n}\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{ I}\right)\leq\left\|\sum\limits_{i=1}^{n}\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{ I}\right\|_{2}<1 \tag{19}\]  
A sufficient and unnecessary condition for the convergence of the Neumann series expansion is as follows:  
\[\sum\limits_{i=1}^{n}\left\|\mathbf{K}^{C,-1}\mathbf{K}_{a_{i}}^{R}e_{i}^{I} \right\|_{2}<1 \tag{20}\]  
For the term \(\left(-\mathbf{K}^{C,-1}\sum\limits_{i=1}^{n}\mathbf{K}_{a_{i}}^{R}e_{i}^{I} \right)^{r}\) in Equation (18), we can convert it to the following explicit form:  
\[