## 4 Numerical Examples
### Example 3  
The third example under investigation is the modeling of two-dimensional isotropic stationary heat conduction problems in composite materials with a rectangular domain [40], as shown in Figure 1. The thermal conductivities in layer 1 and layer 2 are 1 \(W/\)m K and 2 \(W/\)m K, respectively. For a two-dimensional domain, \(\Omega\), enclosed by a rectangular boundary, the governing equation is expressed as follows:  
\[\
abla(k(\mathbf{x})\
abla u(\mathbf{x}))=0\text{ in }\Omega \tag{38}\]  
where \(k\) denotes thermal conductivity. The two-dimensional object boundary under consideration is defined as  
\[\Gamma_{1}=\{\,(x_{1},y_{1})|-1\leq x_{1}\leq 1,\ 0\leq y_{1}\leq 1\} \tag{39}\]  
\[\Gamma_{2}=\{\,(x_{2},y_{2})|-1\leq x_{2}\leq 1,\ 0\leq y_{2}\leq-1\} \tag{40}\]  
For the heat conduction equation, the analytical solution can be obtained as  
\[u(x,y)=e^{-k_{1}/k_{2}(x^{2}-y^{2})}\cos\left(2k_{1}/k_{2}xy\right) \tag{41}\]  
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 12. For modeling the heat conduction problem in composite materials with a rectangular domain, the DDM [37] is adopted so that the domain boundary can be divide into several subdomains. Figure 1 shows that the two-dimensional object boundary is divided into several sub-boundaries: \(\Gamma_{1}\), \(\Gamma_{2}\), \(\Gamma_{3}\),..., \(\Gamma_{8}\). The Dirichlet boundary condition is imposed on the sub-boundaries using the analytical solution as shown in Equation (41) for the problem. At the interface, the sub-boundaries, including \(\Gamma_{2}\) and  
Figure 11: Comparison of temperature distribution with the analytical solution.  
\(\Gamma_{6}\), should satisfy the flux conservation and the continuity of temperature. Therefore, the boundary conditions are given as follows:  
\[u|_{\Gamma_{2}}=\left.u\right|_{\Gamma_{6}},\ \left.\frac{\partial u}{ \partial n}\right|_{\Gamma_{2}}=\left.\frac{\partial u}{\partial n}\right|_{ \Gamma_{6}} \tag{42}\]  
In this example, we utilized the hybrid boundary-type meshless method for modeling heat conduction problems in composite materials with a rectangular domain. The reported CPU time is 1.67 seconds using the computer with Intel Core i7-7700 CPU at 3.60 GHz and the equipment is manufactured by the ASUSteK Computer Inc., Taipei, Taiwan. To validate the accuracy of the proposed method, we collocated 1200 inner points inside the domain uniformly to obtain the solution of the temperature. Figure 13 demonstrates the comparison of the computed results with those from the analytical solution. It is found that the results agree very well with each other.  
To evaluate the accuracy of the proposed method, the absolute error and relative error of the computed results with the analytical solution were calculated as shown in Figure 14. We obtained highly accurate numerical solutions in the order of \(10^{-11}\) for this problem. The singular boundary method [40] has been used to model the same problem. As shown in Table 1, it was found that the accuracy of the singular boundary method for this example only reached to the order of \(10^{-3}\).  
Figure 12: Illustration of the collocation of boundary and source points.  
Figure 14: _Cont._  
Figure 13: The comparison of temperature distribution with the analytical solution.