## 3 Adaptive Subinterval Finite Element Method Based on Dynamic Sensitivity Analysis
### Dynamic-Sensitivity-Based Subinterval Finite Element Analysis Method
Conversely, to predict the lower bound of the displacement response \(u_{r}^{I}\), the subinterval should move in the direction that makes it decrease. Therefore, the new vectors \(\mathbf{x}_{0}^{I}=\big{[}\mathbf{\Xi}_{0,1}(r),\mathbf{\Xi}_{0,2}(r),\cdots,\mathbf{\Xi} _{0,n}(r)\big{]}\) and \(\mathbf{x}_{0}^{II}=[\overline{\mathbf{e}}_{0,1}(r),\overline{\mathbf{e}}_{0, 2}(r),\cdots,\overline{\mathbf{e}}_{0,n}(r)]\) are formed of the \(r\)-th element of the vector sequences \(\mathbf{\Xi}_{0,i}\) and \(\overline{\mathbf{e}}_{0,i}(i=1,2\cdots,n)\), respectively. The superscript \(L\) indicates that this vector is used to determine the lower displacement bounds, and the superscript \(U\) indicates that this vector is used to determine the lower displacement bounds. Figure 2: Possible expansion routes of \(u_{r}(r=1,2,...,d)\) based on dynamic sensitivity analysis. the upper displacement bounds. The meanings of \(L\) and \(U\) are the same in the latter part of the paper. Then, the midpoints of the next subintervals on the expansion routes of the upper and lower bounds of \(u_{r}^{I}\) can be represented, respectively, as follows:  
\[\mathbf{a}_{1}^{L,C}=\mathbf{a}_{0}+\mathbf{x}_{0}^{L}\odot\frac{\mathbf{a}^{R} }{m},\mathbf{a}_{1}^{U,C}=\mathbf{a}_{0}+\mathbf{x}_{0}^{U}\odot\frac{\mathbf{ a}^{R}}{m} \tag{44}\]  
The next subintervals on the expansion routes of the upper and lower bounds of \(u_{r}^{I}\) can be expressed, respectively, as follows:  
\[\mathbf{a}_{1}^{L,I}=\left[\mathbf{a}_{1}^{L,C}-\mathbf{a}^{R}/m\,,\mathbf{a }_{1}^{L,C}+\mathbf{a}^{R}/m\,\right],\mathbf{a}_{1}^{U,I}=\left[\mathbf{a}_{ 1}^{U,C}-\mathbf{a}^{R}/m\,,\mathbf{a}_{1}^{U,C}+\mathbf{a}^{R}/m\,\right] \tag{45}\]  
By solving the interval equilibrium equations for the two subinterval domains \(\mathbf{a}_{1}^{L,I}\) and \(\mathbf{a}_{1}^{U,I}\) using Equations (28)-(31), we can obtain the interval displacement vectors \(\mathbf{u}_{1}^{L,I}\) and \(\mathbf{u}_{1}^{U,I}\) as follows:  
\[\mathbf{u}_{1}^{L,I}=\left[\mathbf{u}_{1}^{L},\overline{\mathbf{u}_{1}^{L}} \right],\mathbf{u}_{1}^{U,I}=\left[\mathbf{u}_{1}^{U},\overline{\mathbf{u}_{ 1}^{U}}\right] \tag{46}\]  
The interval of the \(r\)-th displacement component in the subinterval domains \(\mathbf{a}_{1}^{L,I}\) and \(\mathbf{a}_{1}^{U,I}\) can be expressed as follows:  
\[u_{r,1}^{L,I}=\left[\mathbf{u}_{1}^{L}(r),\overline{\mathbf{u}_{1}^{L}}(r) \right],u_{r,1}^{U,I}=\left[\underline{\mathbf{u}_{1}^{U}}(r),\overline{ \mathbf{u}_{1}^{U}}(r)\right] \tag{47}\]  
Meanwhile, the sensitivity vector sequences \(\mathbf{e}_{1,i}^{L}\) and \(\overline{\mathbf{e}}_{1,i}^{U}(i=1,2\cdots,n)\) in the subintervals \(\mathbf{a}_{1}^{L,I}\) and \(\mathbf{a}_{1}^{U,I}\) domains can be obtained according to Equation (30). \[\mathbf{\underline{e}}_{1,i}^{L}=-sign\Big{(}\Delta\mathbf{u}_{i}\Big{(} \mathbf{a}_{1}^{L,I}\Big{)}\Big{)},\overline{\mathbf{e}}_{1,i}^{U}=sign \Big{(}\Delta u_{i}\Big{(}\mathbf{a}_{1}^{U,I}\Big{)}\Big{)},(i=1,2,\cdots,n) \tag{48}\]  
The \(r\)-th elements of the vector sequences \(\mathbf{\underline{e}}_{1,i}^{L}\) and \(\overline{\mathbf{e}}_{1,i}^{U}(i=1,2\cdots,n)\) are extracted, forming two vectors:  
\[\mathbf{x}_{1}^{L}=[\mathbf{\underline{e}}_{1,1}^{L}(r),\mathbf{ \underline{e}}_{1,2}^{L}(r),\cdots,\mathbf{\underline{e}}_{1,n}^{L}(r)], \tag{49}\] \[\mathbf{x}_{1}^{U}=[\overline{\mathbf{e}}_{1,1}^{U}(r),\overline {\mathbf{e}}_{1,2}^{U}(r),\cdots,\overline{\mathbf{e}}_{1,n}^{U}(r)]\]  
The positions of the next expansion points on the routes of the upper and lower bounds of \(u_{r}^{I}\) can be obtained using these two vectors.