## 2 Methodology
### DOFs Condensation of Transient Heat-Conduction Problem  
Firstly, consider the basic form of the static equations:  
\[[K]\{u\}=\{F\} \tag{1}\]  
The equation can be divided into two groups: the master DOFs, which will be retained, here denoted by the subscript \(m\), and the slave DOFs, which will be removed, here denoted by the subscript \(s\).  
\[\left[\begin{array}{cc}[K_{mm}]&[K_{ms}]\\ [K_{sm}]&[K_{ss}]\end{array}\right]\left\{\begin{array}{c}\{u_{m}\}\\ \{u_{s}\}\end{array}\right\}=\left\{\begin{array}{c}\{F_{m}\}\\ \{F_{s}\}\end{array}\right\} \tag{2}\]  
Define a transformation matrix \([R]\)  
\[\{u\}=\left\{\begin{array}{c}\{u_{m}\}\\ \{u_{s}\}\end{array}\right\}=[R]\{u_{m}\} \tag{3}\]  
Substitute Equation (3) into Equation (1) and multiplying both sides of this equation by a matrix \([R]^{T}\). It can be written in a similar form:  
\[[\hat{K}]\{\hat{u}\}=\left\{\hat{F}\right\} \tag{4}\]  
where:  
\[[\hat{K}]=[R]^{T}[K][R] \tag{5}\]  
\[\{\hat{u}\}=\{u_{m}\} \tag{6}\]  
\[\{\hat{F}\}=[R]^{T}\{F\} \tag{7}\]  
Applying the [PERSON] method for the transient equations subjected to appropriate boundary and initial conditions, the basic form of the transient heat conduction equations can be written as follows [34]:  
\[[C]\{\hat{T}\}+[K]\{T\}=\{F\} \tag{8}\]  
where \([C]\) is the global heat capacitance matrix; \([K]\) is the global conductivity matrix, which contains the heat conduction matrix and the contribution matrix of heat flow boundary to heat conduction matrix; \(\{F\}\) is the thermal force vector, which is contributed to by the heat flow boundary and the heat source; \(\{T\}\) represents the temperature; and \(\{\hat{T}\}\) represents the derivative of temperature with respect to time.  
As suggested by [PERSON] [26], the mass matrix in the basic form of the dynamic equations can be reduced through the transformation matrix given in Equation (3). Similarly, the temperature and the derivative of temperature with respect to time are assumed to have the same transformation relationship in the master and slave degrees of freedom:  
\[\{T\}=\left\{\begin{array}{c}\{T_{m}\}\\ \{T_{s}\}\end{array}\right\}=[R]\{T_{m}\} \tag{9}\]\[\{\hat{T}\}=\left\{\begin{array}{c}\left\{\hat{T}_{m}\right\}\\ \left\{\hat{T}_{s}\right\}\end{array}\right\}=\left[R\right]\{\hat{T}_{m}\} \tag{10}\]  
Substituting Equations (9) and (10) into Equation (8):  
\[\left[\hat{C}\right]\{\hat{T}_{m}\}+\left[\hat{K}\right]\{T_{m}\}=\{\hat{F}\} \tag{11}\]  
where  
\[\left[\hat{C}\right]=[R]^{T}[C][R] \tag{12}\]  
\[\left[\hat{K}\right]=[R]^{T}[K][R] \tag{13}\]  
\[\left\{\hat{F}\right\}=[R]^{T}\{F\} \tag{14}\]  
Using the generalized Newmark method [35], the temperature and the derivative of the temperature with respect to time are as follows, respectively:  
\[T^{(n+1)}=T^{(n)}+\Delta t\hat{T}^{(n)}+\beta\Delta t\Delta\hat{T}^{(n)} \tag{15}\]  
\[\hat{T}^{(n+1)}=\hat{T}^{(n)}+\Delta\hat{T}^{(n)} \tag{16}\]  
where, \(n+1\) and \(n\) represent the time step; \(\Delta t\) represents the time difference between time steps; \(\beta\) is the Newmark parameter; and the parameter generally is 0.5 when using the implicit solution method.  
Substituting Equations (15) and (16) into Equation (11), Equation (11) can be discretized in the time domain as follows:  
\[\left(\left[\hat{C}\right]+\beta\Delta t\big{[}\hat{K}\big{]}\right)\left\{ \Delta\hat{T}_{m}^{(n)}\right\}=\left\{\hat{F}\right\}-\left[\hat{C}\right] \left\{\hat{T}_{m}^{(n)}\right\}-\left[\hat{K}\right]\left(\left\{T_{m}^{(n) }\right\}+\Delta t\left\{\hat{T}_{m}^{(n)}\right\}\right) \tag{17}\]