## 3 Numerical Example  
### Example 1  
In order to verify that the adaptive DOFs condensation algorithm can effectively reduce the computational cost of the transient temperature field with a small loss of accuracy, the first example is a two-dimensional transient heat-transfer problem without a heat source, whose computational domain is unchanged [40]. In this paper, we calculate the results of different numbers of elements with and without adaptive DOFs condensation and record the computational time. The calculation results are then compared with the analytical solutions to show the changes in computational accuracy. Suppose that there is an infinitely long steel block with cross-section that is \(200\times 200\) mm, a heat-conduction coefficient of steel \(k=25\) W/(m\(\cdot\)\({}^{\circ}\)C), a density of steel \(\rho=7850\) kg/m\({}^{3}\), specific heat \(c_{p}=500\) J/(kg\(\cdot\)\({}^{\circ}\)C), and the surface heat-transfer coefficient between the steel and the external medium \(h=2000\) W/(m\({}^{2}\)\({}^{\circ}\)C). The initial temperature of steel \(T_{0}=860\)\({}^{\circ}\)  
Figure 2: Flow diagram of ADOF-FEM. and the temperature of the external medium \(T_{a}=20\)\({}^{\circ}\)C. The coordinate origin is in the center of the cross-section. The analytical solutions of temperature in different regions of steel block can be obtained by the separation of variables method (SVM) and [PERSON]'s multiplication theorem. The specific expression of the analytical equation of temperature can be found in the paper. In this paper, we take the internal position coordinates (0,0), (25,0), (50,0), and (75,0) and the surface position coordinate (100,0) of the steel block as the analysis positions and then use the meshes with different numbers of elements to calculate the FEM results. The initial coarse mesh is divided evenly into square elements, each element is 10 mm long, and the mesh contains a total of 400 elements and 441 nodes. On the basis of the coarse mesh, it is refined 1 and 2 times, and the refined meshes, which are named refined1 and refined2, contain 1600 elements and 6400 elements, respectively. The FEM time step is 1 s, and a total of 3600 steps are calculated. In order to show the advantages of ADOF-FEM in using the refined model to calculate the transient temperature, we first measured the finite element computational time with and without adaptive DOFs. In this paper, Intel MKL PARDISO is used as the solution method of the FEM equation. The total computational time mainly includes assembling the global conductivity matrix, setting DOFs, matrix factorization, and solving the equation. Table 1 shows the computational time of Example 1 with and without adaptive DOFs for different numbers of elements. As can be seen from Table 1, the main computational cost is matrix factorization, followed by solving the equation and assembling the global conductivity matrix (the more numerous the elements, the higher the computational cost). When ADOF-FEM is used, the interpolation relationship between master and slave DOFs needs to be judged when assembling the global conductivity matrix, and DOFs needs to be set several times in the whole calculation process, so the computational cost of these two parts increases. However, because of the size of the equation, that is, because the number of DOFs can be reduced gradually based on the calculation results, the matrix factorization time can be saved effectively, which accounts for about 80% of the total time. As shown in Figure 3, it shows how the number of DOFs changes over time when ADOF-FEM is used. The computational domain of example 1 is invariant, so the DOFs of Example 1 are gradually condensed when using ADOF-FEM. When the time reaches half of the simulation time, the master DOFs used for calculation are those on the boundary and the lowest level DOFs in the coarse mesh. The process of DOFs change is similar to that demonstrated in Figure 1b, which basically reaches the limit of what can be condensed, so the number of DOFs no longer changes.