## 3 Numerical Example
### Example 2  
The objective of this paper is to simulate the temperature field in the construction process of asphalt concrete, so the computational domain of transient heat transfer should be considered to change with time. Example 2 verifies the applicability of ADOF-FEM in the  
Figure 4: Relative error at typical positions of Example 1: (**a**) relative error at position (50,0). (**b**) Relative error at position (75,0). variable computational domain where the variable computational domain is simulated by the birth-death element method. More specifically, the parts have been constructed to participate in the calculation as birth elements, while the parts that have not been constructed are defined as death elements that do not participate in the calculation. With the progress of construction, the death elements are gradually transformed into the birth elements. Example 2 has four layers of asphalt concrete, and the height and width of each layer are 30 cm and 60 cm, respectively. The parameters of the material are as follows: the heat-conduction coefficient of asphalt concrete \(k=1.5\) W/ (m \(\cdot\) \(\circ\) C), the density of asphalt concrete \(\rho=2280\) kg/m\({}^{3}\), the specific heat \(c_{\rho}=1075\) J/ (kg \(\cdot\) \(\circ\) C), and the surface heat-transfer coefficient between the asphalt concrete and the air outside \(h=17.22\) W/ (m\({}^{2}\cdot\) \(\circ\) C). The initial temperature of each layer of asphalt concrete \(T_{0}=150\)\(\circ\)C, and the temperature of the air outside is \(T_{a}=20\)\(\circ\)C. The boundary conditions of the model are as follows: the upper surface and the two sides are the heat dissipation boundary (the third type of boundary conditions), and the bottom surface is the adiabatic boundary. The calculation condition is that each layer dissipates heat for 12 h, and then the latter layer is started to construct, that is, the construction interval is 12 h.  
The FEM mesh of Example 2 is shown in the Figure 5. The initial coarse mesh contains a total of 288 elements and 325 nodes, and the initial computational mesh which contains 4608 elements and 4753 nodes, is refined two times on the basis of the coarse mesh. The computational time domain includes the heat dissipation for 12 h after the construction of the first three layers and the overall heat dissipation for 120 h after all the construction where the FEM time step is fixed as 1 min. In fact, the construction of the latter layer will have a great influence on the temperature of the previous layer. In order to avoid the large error, Example 2 is required to condense the DOFs of the previous layer (except for the last layer) only after the construction of the latter layer is completed. Figure 6 shows the FEM results of the temperature field based on the initial computational mesh, and Figure 7 shows the FEM results with adaptive DOFs. Figure 8 shows the distribution of master and slave DOFs of the model when ADOF-FEM is used. It can be seen that the temperature field in the construction process of asphalt concrete is actually the heat transfer from the new layer to the old layers. The points with a large temperature gradient are basically distributed in the new layer and then move down slowly with time. At the same time, the value of the temperature gradient also gradually decreases with time, and this is also reflected in the change of the condensation range of DOFs. The results  
Figure 5: The FEM mesh of Example 2. Figure 9 compares the changes of the DOFs of the model over time steps with or without using ADOF-FEM and shows their total computational time. The computational domain of Example 2 is variable, so each time the computational domain is increased some new DOFs are added, while the DOFs condensation proceeds in parallel. So, the number of DOFs is first increased, then slowly reduced, and then increased again. When the computational domain is no longer increasing, the number of DOFs in Example 2 changes similarly to Example 1, and eventually, the number of DOFs no longer changes.