## 4 Numerical Examples
### Example 4  
The fourth example under investigation is the modeling of two-dimensional isotropic stationary heat conduction problems of composite materials in a square domain with a hole [40], as depicted in Figure 15. The thermal conductivities in layer 1 and layer 2 are 1 \(W\)/m K and 2 \(W\)/m K, respectively. In the first layer, two squares with sides f 2 and 20 around the boundary domain, respectively and the second layer is composed of two squares with sides of 20 and 60 around the boundary domain. For a two-dimensional domain, \(\Omega\), enclosed by a boundary with a square domain, the governing equation is shown in Equation (38). The analytical solution can be obtained as  
\[u(x,y)=k_{1}/k_{2}(x^{2}-y^{2}) \tag{43}\]  
where \(k_{1}\) and \(k_{2}\) are thermal conductivities in layer 1 and layer 2, respectively. The boundary and source points are uniformly distributed on the boundary, as demonstrated in Figure 16. For modeling the heat conduction problem with composite materials, the DDM [37] was adopted so that the domain boundary can be divided into several subdomains. Figure 15 shows that the two-dimensional object boundary is divided into several sub-boundaries: \(\Gamma_{1}\), \(\Gamma_{2}\), \(\Gamma_{3}\),..., \(\Gamma_{16}\). The Dirichlet boundary condition is imposed on the sub-boundaries using the analytical solution for the problem, as shown in Equation (43). At the interface, the sub-boundaries, including \(\Gamma_{5}\), \(\Gamma_{6}\), \(\Gamma_{7}\), \(\Gamma_{8}\), \(\Gamma_{9}\), \(\Gamma_{10}\), \(\Gamma_{11}\) and \(\Gamma_{12}\), should satisfy the flux conservation and the continuity of temperature. Therefore, the boundary conditions are given as follows. \[u\big{|}_{\Gamma_{5}}=\left.u\big{|}_{\Gamma_{9}^{\prime}},\left.\frac{\partial u }{\partial n}\right|_{\Gamma_{5}}=\left.\frac{\partial u}{\partial n}\right|_ {\Gamma_{9}},\left.u\big{|}_{\Gamma_{6}}=\left.u\big{|}_{\Gamma_{10}^{\prime}},\left.\frac{\partial u}{\partial n}\right|_{\Gamma_{6}}=\left.\frac{\partial u }{\partial n}\right|_{\Gamma_{10}}\right.\right. \tag{44}\]  
\[u\big{|}_{\Gamma_{7}}=\left.u\big{|}_{\Gamma_{11}^{\prime}},\left.\frac{\partial u }{\partial n}\right|_{\Gamma_{7}}=\left.\frac{\partial u}{\partial n}\right|_ {\Gamma_{11}},\left.u\big{|}_{\Gamma_{8}}=\left.u\big{|}_{\Gamma_{12}^{\prime }},\left.\frac{\partial u}{\partial n}\right|_{\Gamma_{8}}=\left.\frac{\partial u }{\partial n}\right|_{\Gamma_{12}}\right.\right. \tag{45}\] \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**This Study**} & \multicolumn{2}{c}{**Singular Boundary Method**} \\ \cline{2-5}  & **1 st Layer** & **2 nd Layer** & **1 st Layer** & **2 nd Layer** \\ \hline Maximum absolute error with the analytical solution & \(2.74\times 10^{-11}\) & \(3.25\times 10^{-11}\) & \(10^{-3}\) & \(10^{-3}\) \\ Maximum relative error with the analytical solution & \(3.90\times 10^{-11}\) & \(5.76\times 10^{-11}\) & \(10^{-3}\) & \(10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of computed results with those from references. Figure 14: Accuracy of the proposed method (with the comparison of the analytical solution). (**a**) Absolute error in the first layer, (**b**) absolute error in the second layer, (**c**) relative error in the first layer, and (**d**) relative error in the second layer. The reported CPU time was 18.00 s using the Intel Core i7-7700 CPU at 3.60 GHz. To examine the accuracy of the proposed method, we collocated 5445 inner points inside the domain uniformly to obtain the solution of the temperature. The computed temperature distribution of the inner points is depicted in Figure 17. It was found that the computed results agree very well with those from the previous results. Figure 16: Illustration of the collocation of boundary and source points. Figure 15: The configuration of the boundary condition. analytical solution.