## 3 Adaptive Subinterval Finite Element Method Based on Dynamic Sensitivity Analysis
### Improved Interval Finite Element Method
\begin{split}\mathbf{u}\Big{(}\mathbf{e}^{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}\Bigg{(} \mathbf{f}^{C}+\sum_{j=1}^{n}\mathbf{f}_{a_{j}}^{R}e_{j}^{I}\Bigg{)}\\ &=\mathbf{K}^{C,-1}\mathbf{f}^{C}+\Bigg{(}\sum_{i=1}^{n}\mathbf{ B}_{a_{i}}^{C}\Bigg{)}\mathbf{K}^{C,-1}\mathbf{f}^{C}+\sum_{i=1}^{n}\mathbf{B}_{a_{i}} ^{R}\mathbf{K}^{C,-1}\mathbf{f}^{C}e_{i}^{I}+\sum_{j=1}^{n}\mathbf{K}^{C,-1} \mathbf{f}_{a_{j}}^{R}e_{j}^{I}+\\ &\sum_{j=1}^{n}\Bigg{(}\sum_{i=1}^{n}\mathbf{B}_{a_{i}}^{C} \Bigg{)}\mathbf{K}^{C,-1}\mathbf{f}_{a_{j}}^{R}e_{j}^{I}+\sum_{i=1}^{n}\sum_{j =1}^{n}\mathbf{B}_{a_{i}}^{R}\mathbf{K}^{C,-1}\mathbf{f}_{a_{j}}^{R}e_{i}^{I} e_{j}^{I}\end{split}  \tag{27}\]  
The following symbols are introduced to simplify the writing of Equation (27)  
\[ \begin{split}\mathbf{u}^{C}&=\mathbf{K}^{C,-1} \mathbf{f}^{C}+\Bigg{(}\sum_{i=1}^{n}\mathbf{B}_{a_{i}}^{C}\Bigg{)}\mathbf{K}^{ C,-1}\mathbf{f}^{C},\\ \Delta\mathbf{u}_{i}&=\mathbf{B}_{a_{i}}^{R}\mathbf{ K}^{C,-1}\mathbf{f}^{C}+\mathbf{K}^{C,-1}\mathbf{f}_{a_{j}}^{R}+\Bigg{(}\sum_{j=1}^{n} \mathbf{B}_{a_{j}}^{C}\Bigg{)}\mathbf{K}^{C,-1}\mathbf{f}_{a_{j}}^{R},\\ \Delta\mathbf{u}_{i,j}&=\mathbf{B}_{a_{i}}^{R} \mathbf{K}^{C,-1}\mathbf{f}_{a_{j}}^{R}\end{split}  \tag{28}\]  
where \(\mathbf{u}^{C}\) denotes the midpoint of the interval displacement vector \(\mathbf{u}\); \(\Delta\mathbf{u}_{i}\) denotes the deviation radius of the first-order deviation due to the parameter \(a_{i}^{I}\); \(\Delta\mathbf{u}_{i,j}\) is the second-order deviation determined by the interval parameters \(a_{i}^{I}\) and \(a_{j}^{I}\) together. Then, the interval displacement vector can be written as follows:  
\[\mathbf{u}\Big{(}\mathbf{e}^{I}\Big{)}=\mathbf{u}^{C}+\sum_{i=1}^{n}\Delta \mathbf{u}_{i}\mathbf{e}_{i}^{I}+\sum_{i=1}^{n}\sum_{j=1}^{n}\Delta\mathbf{u}_ {i,j}\mathbf{e}_{i}^{I}\mathbf{e}_{j}^{I} \tag{29}\]Now, define the vectors \(\underline{\mathbf{e}_{i}}\) and \(\overline{\mathbf{e}_{i}}\) as follows:  
\[\underline{\mathbf{e}_{i}}=-sign(\Delta\mathbf{u}_{i}),\overline{\mathbf{e}_{i} }=sign(\Delta\mathbf{u}_{i}),(i=1,2,\cdots,n) \tag{30}\]  
where \(\mathbf{e}_{i}\) and \(\overline{\mathbf{e}_{i}}\) correspond to the lower and upper boundaries of the structural response, respectively. Then, substituting \(\mathbf{e}_{i}\) and \(\overline{\mathbf{e}_{i}}\) into Equation (29), the upper and lower bounds of the displacement response can be expressed according to IIA as follows:  
\[ \begin{split}\underline{\mathbf{u}}&=\mathbf{u}^{C }+\sum_{i=1}^{n}\Delta\mathbf{u}_{i}\odot\underline{\mathbf{e}_{i}}+\ \sum_{i=1}^{n}\sum_{j=1}^{n}\Delta\mathbf{u}_{i,j}\odot\underline{\mathbf{e}_{i }}\odot\underline{\mathbf{e}_{j}},\\ \overline{\mathbf{u}}&=\mathbf{u}^{C}+\sum_{i=1}^{n }\Delta\mathbf{u}_{i}\odot\overline{\mathbf{e}_{i}}+\ \sum_{i=1}^{n}\sum_{j=1}^{n}\Delta\mathbf{u}_{i,j}\odot\overline{\mathbf{e}_{ i}}\odot\overline{\mathbf{e}_{j}}\end{split}  \tag{31}\]  
The interval finite element method proposed in this paper retains a portion of the higher order terms when solving the inverse matrix of the stiffness matrix and considers the second-order deviation terms when solving the displacements. As a result, it is able to obtain more accurate displacement response bounds for the structure than conventional methods such as the first-order truncated Neumann-series-based method [38; 39].