# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 4 Reduction of the Coupling Interface DOFs  
### Formulation  
During the aforementioned coordinate transformations, the component physical coordinates are represented by modal coordinates for both master and slave substructures. The generalized DOFs involving in modal analysis consist of the full dimension of the coupling interface DOFs and a small part of the truncated internal DOFs. The number of the generalized DOFs can still be large when the coupling interface of the substructure is huge and complex. On the one hand, a large number of DOFs involving in modal analysis will lower the computational efficiency; on the other hand, it is impossible to load on each DOF on the connections in a structural testing. Therefore, reducing the coupling interface DOFs of substructures is quite necessary. In this section, the displacement interpolation formulation in FEM is utilized to reduce the coupling interface DOFs. Without loss of generality, a two-dimensional interface, a plane as shown in Figure 2a, is taken as an example to set up interpolation functions and formulations, which can be easily extended to solve three-dimension problems. Dividing the coupling interface into four sub-regions (I \(\sim\) IV), as shown in Figure 2a. The nodes 1\(\sim\)9 is defined as interpolation basic nodes (IBNs), and then the force and displacement of each node \(i\) (Figure 2b) can be described approximately by that of the IBNs by a set of shape functions \(N\left(s,t\right)\)[42]. The displacement or force interpolation formulations of the rectangular element are  
\[ \begin{cases}g_{x}^{i}=\sum_{n=1}^{4}N_{n}(s_{i},t_{i})g_{x}^{n}\\ g_{y}^{i}=\sum_{n=1}^{4}N_{n}(s_{i},t_{i})g_{y}^{n}\quad\text{or}\\ g_{z}^{i}=\sum_{n=1}^{4}N_{n}(s_{i},t_{i})g_{z}^{n}\end{cases} \quad \begin{cases}u _{x}^{i}=\sum_{n=1}^{4}N_{n}(s_{i},t_{i})u_{x}^{n}\\ u_{y}^{i}=\sum_{n=1}^{4}N_{n}(s_{i},t_{i})u_{y}^{n}\\ u_{z}^{i}=\sum_{n=1}^{4}N_{n}(s_{i},t_{i})u_{z}^{n}\end{cases}  \tag{40}\]  
where, \(u^{i}\) and \(g^{i}\) are force and displacement vectors of node \(i\) in the global coordinate system; \(x\), \(y\) and \(z\) are the x, y and z-direction; \(n\) is the number of IBNs in the sub-region that node \(i\) is in; \(N_{n}(s_{i},t_{i})\) is the shape function. Based on the rectangular element theory in FEM, linear interpolation of finite elements is stated more clearly by a natural coordinate system, and interpolation functions can be expressed as follows  
\[ \begin{cases}N_{1}=\dfrac{1}{4}(1-s_{i})(1-t_{i})\\ N_{2}=\dfrac{1}{4}(1+s_{i})(1-t_{i})\\ N_{3}=\dfrac{1}{4}(1+s_{i})(1+t_{i})\\ N_{4}=\dfrac{1}{4}(1-s_{i})(1+t_{i})\end{cases} ,\quad \begin{cases}s_{i}=(x_{i} -X_{0})/m\\ t_{i}=(y_{i}-Y_{0})/n\\ X_{0}=(X_{1}+X_{3})\\ Y_{0}=(Y_{1}+Y_{3})\\ m=|X_{1}-X_{3}|/2\\ n=|Y_{1}-Y_{3}|/2\end{cases}  \tag{41}\]  
where, \((X_{i},Y_{i})\), \((i=0,1,2,3,4)\) are the coordinates of the central node and IBNs in x-y coordinate plane. The nodal displacement or force vector relative to the coupling interface DOFs of the constrained-interface or the free-interface substructure can be obtained from assembling the interpolation transformation of all re-regions. There are  
\[\left\{\mathbf{g}_{f}\right\}=\left[\mathbf{A}\right]\left\{\mathbf{g}_{f}^{n}\right\} \quad\text{or}\quad\left\{\mathbf{u}_{c}^{B}\right\}=\left[\mathbf{B}\right]\left\mathbf{{u}_{c}^{n}\right\} \tag{42}\]  
in which, \(\mathbf{u}^{n}\) and \(\mathbf{g}^{n}\) are the displacement and force vectors relative to the IBNs of the coupling interface; \(\mathbf{A}\) and \(\mathbf{B}\) are transformation matrices that represent the relations between all the interface nodes and the IBNs. Figure 2: The finite element mesh of the interface and the local coordinate of sub-region. (**a**) global coordinate system; (**b**) local coordinate system. \[\left\{ \begin{array}{l}\mathbf{q}_{m}\\ \mathbf{u}_{c}^{n}\end{array} \right\}= \begin{bmatrix}\mathbf{I}_{m}&\mathbf{0}\\ \mathbf{P}_{1}^{r}&\mathbf{P}_{2}^{r}\end{bmatrix} \left\{ \begin{array}{l}\mathbf{q}_{k} \\ \mathbf{q}_{m}\end{array}