## 4 Basic Interpolation Rule and Interpolation Functions  
### Basic Interpolation Rule and Interpolation Functions  
As stated above, Equation (15) already satisfies displacement compatibility. However, the equivalent system of forces on the intermediate surface is not guaranteed, and an additional condition has to be introduced that is based on the conservation of energy through the interface [26; 32]. The kinematic boundary condition on a continuous interface, \(\Gamma\), leads to  
\[\delta\mathbf{u}^{(1)}=\delta\mathbf{u}^{(2)} \tag{51}\]  
where \(\delta\mathbf{u}^{(s)}\) is the virtual displacement on the interface of substructure \(s\). For two substructures with non-matching interfaces \(\Gamma^{(s)}\), there is \(\Gamma^{(1)}\
eq\Gamma^{(2)}\), and the virtual displacement vectors \(\delta\mathbf{u}^{(s)}\) on each discrete interface have the relationship  
\[\delta\mathbf{u}_{f}=\mathbf{G}_{u}^{(1)}\delta\mathbf{u}^{(1)}=\mathbf{G}_{u}^{(2)}\delta\bm {u}^{(2)} \tag{52}\]  
where \(\delta\mathbf{u}_{f}\) is the discrete virtual displacement vectors of the intermediate surface. Thus,  
\[\delta\mathbf{u}^{(1)}=\Big{(}\mathbf{G}_{u}^{(1)}\Big{)}^{-1}\mathbf{G}_{u}^{(2)}\delta \mathbf{u}^{(2)} \tag{53}\]  
The virtual work of each side of the interface force \(\mathbf{g}^{(s)}\) acting on interface \(\Gamma^{(s)}\) can be described separately as  
\[\delta w^{(1)}=\mathbf{g}^{(1)\intercal}\delta\mathbf{u}^{(1)} \tag{54}\]  
\[\delta w^{(2)}=\mathbf{g}^{(2)\intercal}\delta\mathbf{u}^{(2)} \tag{55}\]  
Considering Equation (14), Equation (17) and \(\mathbf{B}^{(1)\intercal}\mathbf{\Lambda}^{(1)}=-\mathbf{B}^{(2)\intercal}\mathbf{\Lambda}^{(2)}\), there is  
\[\mathbf{g}^{(1)}=-\mathbf{G}_{f}^{(1)}\Big{(}\mathbf{G}_{f}^{(2)}\Big{)}^{-1}\mathbf{g}^{(2)} \tag{56}\]  
Substituting (53) and (56) into (54) yields  
\[ \begin{split}\delta w^{(1)}&=\left(-\mathbf{G}_{f}^{(1 )}\Big{(}\mathbf{G}_{f}^{(2)}\Big{)}^{-1}\mathbf{g}^{(2)}\right)^{\text{T}}\Big{(}\mathbf{G }_{u}^{(1)}\Big{)}^{-1}\mathbf{G}_{u}^{(2)}\delta\mathbf{u}^{(2)}\\ &=-\mathbf{g}^{(2)\intercal}\bigg{(}\Big{(}\mathbf{G}_{f}^{(2)}\Big{)}^{- \text{T}}\mathbf{G}_{f}^{(1)\intercal}\Big{(}\mathbf{G}_{u}^{(1)}\Big{)}^{-1}\mathbf{G}_{u }^{(2)}\bigg{)}\delta\mathbf{u}^{(2)}\end{split}  \tag{57}\]  
Since there are no external forces, the energy is preserved, and there is  
\[\delta w^{(1)}+\delta w^{(2)}=0 \tag{58}\]Substituting (55) and (57) into (58) yields  
\[g^{(2)\mathrm{T}}\bigg{(}\Big{(}G_{f}^{(2)}\Big{)}^{-\mathrm{T}}G_{f}^{(1) \mathrm{T}}\Big{(}G_{u}^{(1)}\Big{)}^{-1}G_{u}^{(2)}\bigg{)}\delta\mathbf{u}^{(2)}=g ^{(2)\mathrm{T}}\delta\mathbf{u}^{(2)} \tag{59}\]  
Without loss of generality, we assume  
\[\Big{(}G_{f}^{(2)}\Big{)}^{-\mathrm{T}}G_{f}^{(1)\mathrm{T}}\Big{(}G_{u}^{(1) }\Big{)}^{-1}G_{u}^{(2)}=\mathbf{I} \tag{60}\]  
where \(\mathbf{I}\) is the identity matrix, then Equation (59) is satisfied, and thus the energy through the interface is conserved. According to [PERSON]'s work [26], if we set the intermediate surface nodes to coincide with one of the substructure's interfaces, such as \(\Gamma^{(1)}\) shown in Figure 1, then \(G_{u}^{(1)}=\mathbf{I},\ G_{f}^{(1)}=\mathbf{I}\) and  
\[\Big{(}G_{u}^{(2)}\Big{)}^{\mathrm{T}}=G_{f}^{(2)} \tag{61}\]  
Then, the displacement and forces on the interfaces between the two substructures have a brief relationship:  
\[\left\{ \begin{array}{l}\mathbf{u}_{b}^{(1)}=G_{u}^{(2)}\mathbf{u}_{b}^{(2)}\\ -\Big{(}G_{u}^{(2)}\Big{)}^{\mathrm{T}}g^{(1)}=g^{(2)}\end{array} \right.