## 2 Methodology
### Error Estimator  
For heat-conduction problems, there are locations with large or small temperature gradients, the former of which require a refined mesh and the latter of which can use a coarse mesh. However, it is usually difficult to obtain the laws before calculation. Moreover, the temperature gradient changes over time for transient heat-conduction problems, and regions with a large temperature gradient may be smaller after a number of time steps. At this time, the initial refined mesh, which has a large number of DOFs, will have a large computational cost, so it is important to reduce the DOFs without affecting the computational accuracy too much. In fact, it is necessary to obtain the global error of the whole computed domain before considering DOFs condensation. If the error exceeds the prespecified error limit, it is usually recommended that the mesh needs be refined to achieve a better calculation result, and conversely if the error is small the mesh can be coarsened. The error is calculated from the approximate solution of the computational mesh and the analytical solution of the partial differential equation, but the analytical solution of the practical engineering is usually not available (obviously, there is no need to perform a finite element calculation if the analytical solution has already been obtained). Therefore, an error estimator is needed to estimate the global error of the current calculation result. [PERSON] and [PERSON] [36] first proposed a simple error estimator for the linear elastic static problem, and then [PERSON] and [PERSON] [37] derived an error estimator for heat-conduction problems that use the most commonly used energy norm and apply the boundary conditions results. It can be defined as \[\|e\|^{2}=\int_{\Omega}\
abla(T-\bar{T})^{T}k\
abla(T-\bar{T})d\Omega+\int_{\Gamma_ {q}}(T-\bar{T})h(T-\bar{T})d\Gamma_{q} \tag{18}\]  
where \(T\) is the temperature calculation value and \(\bar{T}\) is the temperature exact value, respectively; \(k\) is the thermal conductivity coefficient; \(h\) is the convective heat-transfer coefficient; \(\Omega\) is the computational domain; and \(\Gamma_{q}\) is the heat flow boundary. Similarly, the exact energy norm considered to be a measure of the total heat dissipation in the whole domain can be calculated as  
\[\|q\|^{2}=\int_{\Omega}\
abla T^{T}k\
abla Td\Omega+\int_{\Gamma_{q}}ThTd\Gamma _{q} \tag{19}\]  
A relative error measure can be defined as  
\[\eta=\frac{\|e\|}{\|q\|}\times 100\%\leq\bar{\eta} \tag{20}\]  
where \(\bar{\eta}\) is a prespecified error limit for the global relative error and the relative error is bounded by it. [PERSON] and [PERSON] [36] use the smoothed values of stresses instead of the exact values to calculate the error norm. Similarly, [PERSON] and [PERSON] [38] applied the smoothing procedure to heat-transfer problems. Using the smoothing procedure, the computation field in Equation (18) can be replaced with the smoothed values. The smoothed temperature gradient at any location within the element can be interpolated by the smoothed temperature gradient of the nodes and the shape functions. It can be calculated as:  
\[\frac{\partial T^{*}}{\partial x_{i}}=\sum_{j=1}^{n}N_{j}\frac{\partial\bar{T }_{j}}{\partial x_{i}} \tag{21}\]  
where \(n\) is the number of nodes per element, \(N_{j}\) are the shape functions, \(\frac{\partial\bar{T}}{\partial x_{i}}\) are the smoothed temperature gradient of the nodes, and \(\frac{\partial\bar{T}^{*}}{\partial x_{i}}\) are the values of the smoothed temperature gradient an any point in the element, respectively. In the case of transient heat conduction, the precondition for mesh coarsening is not only that the error of a certain step is below the prespecified error limit but also to ensure that the error is below the tolerance value for a long time. [PERSON] and [PERSON] [37] proposed the variation of the error with respect to time to determine the trend of error over time, and to ensure that the mesh can be coarsened during this time.