## 3 Adaptive Subinterval Finite Element Method Based on Dynamic Sensitivity Analysis
### Dynamic-Sensitivity-Based Subinterval Finite Element Analysis Method
By repeating the above process \(m-1\) times, a series midpoints points \(\mathbf{a}_{t}^{L,C}\) and \(\mathbf{a}_{t}^{U,C}\) of subintervals \(\mathbf{a}_{t}^{L,I}\) and \(\mathbf{a}_{t}^{U,I}\) (\(t=1,2\cdots,m-1\)) can be determined:  
\[\mathbf{a}_{t}^{L,C}=\mathbf{a}_{t-1}^{L,C}+\mathbf{x}_{t-1}^{L}\odot\frac{ \mathbf{a}^{R}}{m},\mathbf{a}_{t}^{U,C}=\mathbf{a}_{t-1}^{U,C}+\mathbf{x}_{t- 1}^{U}\odot\frac{\mathbf{a}^{R}}{m} \tag{50}\]  
In subintervals \(\mathbf{a}_{t}^{L,I}\) and \(\mathbf{a}_{t}^{U,I}(t=1,2\cdots,m-1)\), the equilibrium equations are solved using Equations (28)-(31), respectively. The interval \(u_{r,t}^{L,I}\) in subinterval domains \(\mathbf{a}_{t}^{L,I}\) and \(u_{r,t}^{U,I}\) in subinterval domains \(\mathbf{a}_{t}^{U,I}\) of the displacement component \(u_{r}\) can be obtained. Due to  
Figure 2:0 Figure 2:1 Figure 2:2Thus,  
\[ \begin{split}& u_{r,2}^{I}\cap u_{r,1}^{L,I}\cap u_{r,2}^{L,I}\cap...\cap u_{r,m-1}^{L,I}\
eq\varnothing,\\ & u_{r,0}^{I}\cap u_{r,1}^{U,I}\cap u_{r,2}^{U,I}\cap...\cap u_{ r,m-1}^{U,I}\
eq\varnothing\\ \end{split}  \tag{52}\]  
Via a union operation, the lower bound of \(u_{r}\Big{(}\big{(}\mathbf{a}_{0}^{I},\mathbf{a}_{1}^{L,I},\mathbf{a}_{2}^{ L,I},\cdots,\mathbf{a}_{m-1}^{L,I}\big{)}\Big{)}\) and upper bound of \(u_{r}\Big{(}\Big{(}\mathbf{a}_{0}^{I},\mathbf{a}_{1}^{U,I},\mathbf{a}_{2}^{U, I},\cdots,\mathbf{a}_{m-1}^{U,I}\big{)}\Big{)}\) can be represented as follows:  
\[ \begin{split}&\underline{u}_{r}\Big{(}\Big{(}\mathbf{a}_{0}^{I}, \mathbf{a}_{1}^{L,I},\mathbf{a}_{2}^{L,I},\cdots,\mathbf{a}_{m-1}^{L,I}\big{)} \Big{)}=\min_{t=0,1,\ldots,m-1}\cup\limits_{t,r}\Big{(}u_{r,t}^{L,I}\Big{)}, \\ &\overline{u}_{r}\Big{(}\Big{(}\mathbf{a}_{0}^{I},\mathbf{a}_{1 }^{U,I},\mathbf{a}_{2}^{U,I},\cdots,\mathbf{a}_{m-1}^{U,I}\big{)}\Big{)}=\max \limits_{t=0,1,\ldots,m-1}\cup\limits_{t,r,t}\Big{(}u_{r,t}^{U,I}\Big{)}\end{split}  \tag{53}\]  
Theoretically, if the expansion routes are not chosen, there are a total of \(m^{n}\) possible combinations of subintervals. According to Equation (50), \(\underline{u}_{r}\Big{(}\mathbf{a}_{t}^{L,I}\Big{)}\) moves in the decreasing direction of \(u_{r}^{I}\), and \(\overline{u}_{r}\Big{(}\mathbf{a}_{t}^{U,I}\Big{)}\) moves in the increasing direction of \(u_{r}^{I}\). Thus, based on dynamic sensitivity analysis, we can obtain the following relationship:  
\[ \begin{split}\underline{u}_{r}\Big{(}\mathbf{a}^{I}\Big{)}& =\min_{t_{1}=1,2,\cdots,m}\Big{(}\underline{u}_{r}\Big{(}\Big{(} a_{1,t_{1}}^{I},...,a_{t_{1},t_{r}}^{I},...,a_{n,t_{n}}^{I}\Big{)}\Big{)}\Big{)}\\ &=\underline{u}_{r}\Big{(}\Big{(}\mathbf{a}_{0}^{I},\mathbf{a}_{1 }^{L,I},\mathbf{a}_{2}^{L,I},\cdots,\mathbf{a}_{m-1}^{L,I}\Big{)}\Big{)}=\min \limits_{t=0,1,\ldots,m-1}\cup\limits_{t,r,t}\Big{(}u_{r,t}^{L,I}\Big{)}\end{split}  \tag{54}\]  
Similarly,  
\[ \begin{split}\overline{u}_{r}\Big{(}\mathbf{a}^{I}\Big{)}& =\max_{t_{1}=1,2,\cdots,m}\Big{(}\overline{u}_{r}\Big{(}\Big{(} a_{1,t_{1}}^{I},...,a_{t,t_{n}}^{I},...,a_{n,t_{n}}^{I}\Big{)}\Big{)}\Big{)}\\ &=\overline{u}_{r}\Big{(}\Big{(}\mathbf{a}_{0}^{I},\mathbf{a}_{1 }^{U,I},\mathbf{a}_{2}^{U,I},\cdots,\mathbf{a}_{m-1}^{U,I}\Big{)}\Big{)}=\max \limits_{t=0,1,\ldots,m-1}\cup\limits_{t,r,t}\Big{(}u_{r,t}^{U,I}\Big{)}\end{split}  \tag{55}\]  
By employing the aforementioned method, the lower and upper bounds \(\overline{u}_{r}\) and \(\underline{u}\), \((r=1,2\cdots,d)\) for each displacement component are obtained, and they are combined to form the bounds of displacement vector. It is important to note that there is no need to repeat the calculation for the same expansion point on different expansion routes. If interval finite element analysis has already been performed within a certain subinterval domain, the previous results can be directly utilized.