## 2 Methodology
### Mesh-Coarsening Criterion  
The error estimator introduced in Section 2.2 quantifies the posterior error calculated in the computational domain based on the current mesh; if the mesh coarsening is carried out at this time, the difference values between the mesh before and after the coarsening is difficult to obtain in advance, so it is necessary to establish a mesh-coarsening criterion to ensure that the additional calculation error introduced by the coarsening process solves the problem or meets the engineering requirements. Coarsening is actually the reverse of refinement, so the method of refinement should be introduced first. There are two main directions of refinement. The first refinement is the simple reduction of the subdivision size (h-refinement), and the second refinement uses the high-order shape functions (p-refinement). It is usually simple and natural for most engineers to use h-refinement, and the method proposed in this paper is also based on h-refinement. If refinement improves computational accuracy, then coarsening as the reverse of refinement certainly adds additional errors that, in the form of energy, can be seen as artificial energy. This artificial energy \(E_{a}\) can be quantified as [39]:  
\[E_{a}=E_{c}-E_{r} \tag{23}\]  
where \(E_{c}\) and \(E_{r}\) are the energy stored in the coarsened mesh and the original refined mesh, respectively. It is assumed that the mesh is unchanged before time \(t^{(n)}\) and coarsening at time \(t^{(n+1)}\); we can obtain that \(E_{c}^{(n)}\) and \(E_{r}^{(n)}\) are equal. So, the artificial energy \(E_{a}\) added at time \(t^{(n+1)}\) can be quantified as  
\[E_{a}=E_{c}^{(n+1)}-E_{r}^{(n+1)}=\Delta E_{c}-\Delta E_{r} \tag{24}\]  
where \(\Delta E_{c}\) and \(\Delta E_{r}\) are the difference of \(E_{c}\) and \(E_{r}\) between time \(t^{(n)}\) and time \(t^{(n+1)}\), respectively. The energy \(E\) stored in the domain \(\Omega\) is  
\[E=\rho c_{\rho}\int_{\Omega}T(x,y,z,t)d\Omega \tag{25}\]  
where \(\rho\) is the density of the material and \(c_{\rho}\) is the specific heat capacity of the material; the unit of temperature needs to be convert to Kelvin, \(K\). And the difference of \(E\) between time \(t^{(n)}\) and time \(t^{(n+1)}\) is  
\[\Delta E=\rho c_{\rho}\int_{\Omega}\Big{(}T\Big{(}x,y,z,t^{(n+1)}\Big{)}-T \Big{(}x,y,z,t^{(n)}\Big{)}\Big{)}d\Omega \tag{26}\]  
Assuming that the mesh is unchanged before time \(t^{(n)}\) and coarsening at time \(t^{(n+1)}\), the temperature at time \(t^{(n)}\) has the following relationship:  
\[T_{c}^{(n)}=T_{r}^{(n)} \tag{27}\]  
Using Equation (26) and substituting Equation (27), Equation (24) can be written as follows:  
\[E_{a}=\rho c_{\rho}\int_{\Omega}\Big{(}T_{c}^{(n+1)}-T_{r}^{(n+1)}\Big{)}d\Omega \tag{28}\]  
The large difference in the energy stored in different time steps and different elements will make it difficult to compare the added artificial energy, so it is necessary to use the relative artificial energy as the coarsening criterion. The relative artificial energy can be written as  
\[\varepsilon_{a}=\frac{E_{a}}{E_{r}}\times 100\%_{o}|\varepsilon_{a}|\leq\varepsilon \tag{29}\]  
where \(\varepsilon\) is the maximum allowable relative artificial energy in the element in order to allow coarsening. Consider a heat-conduction problem in which the temperature field changes gradually stabilize; a very small amount of the value of the relative artificial energy sometimes appears in the early time of the calculation, which leads to an incorrect judgment that the current step satisfies the coarsening criterion.