## 6 Simulation of a Realistic Wall
### Mesh Construction  
In the used approximation, no heat flows and no physical quantities change in the \(y\) direction, which is perpendicular to the surface of Figure 9. We use \(\Delta y_{i}=0.2\) m in order to obtain a realistic problem. The other two coordinates are in the unit interval, \((x,z)\in[0,\,1]\times[0,\,1]\), thus the surface area of the meshes is 1 m\({}^{2}\). Two kinds of meshes were constructed: the equidistant mesh has square-shaped cells, and the non-equidistant mesh has rectangular cells. We set \(N_{x}=100\) and \(N_{x}=100\) for the number of cells in the \(x\) and in the z direction; therefore, \(N=N_{x}N_{z}=10\),000 is the total number of cells. The non-equidistant mesh contains high cells on the upper side and low ones on the lower side of the wall, as well as the cells being wide on the left side of the wall and narrow on the right. The width and height decrease gradually both in the \(x\) and \(z\) directions, consecutively, following a geometric series. The sum of the first \(n+1\) terms of a geometric sequence, up to the term \(r^{n}\), (\(r\
eq 1\)), is  
\[\sum_{k=0}^{n}ar^{k}=a\left(\frac{1-r^{n+1}}{1-r}\right)=a+ar+ar^{2}+ar^{3}+ \ldots\ldots+ar^{n}. \tag{37}\]  
where \(n=N_{x}-1=N_{z}-1\). The values \(r=0.98\) and \(a=0.0234\) are used, which give \(\Delta x_{1}=0.0234\) and \(\Delta z_{1}=0.0234\) on the left and the upper sides, respectively, while \(\Delta z_{N_{2}}=\Delta x_{N_{x}}=0.98^{99}\cdot\Delta x_{1}=0.00317\) on the right and lower sides. The obtained meshes are shown in Figure 10.  
In the case of the wall's surface, we apply only the equidistant grid. However, in the case of the cross section of the wall with an insulator, we apply both kinds of mesh: the equidistant and the non-equidistant. For programming simplicity, in a cross section, bricks always make up the left half of the cells, whereas the insulator (containing the thermal \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\rho\) (\(k\mathbf{g}\cdot\mathbf{m}^{-3}\)) & \(c(\mathbf{J}\cdot\mathbf{kg}^{-1}\cdot\mathbf{K}^{-1}\)) & \(k(\mathbf{W}\cdot\mathbf{m}^{-1}\cdot\mathbf{K}^{-1}\)) \\ \hline \hline Brick & 1900 & 840 & 0.73 \\ Rigid Polyurethane Foam & 320 & 1400 & 0.023 \\ Steel Beam & 7800 & 840 & 16.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the applied materials [2]. Figure 10: (a) Equidistant mesh. (b) Gradual change in the \(x\) and \(z\) directions. bridge) makes up the right half. This means that, in the equidistant case, the thermal bridge has the same thickness and volume as the insulator, but the thickness of the insulator is smaller (0.269 m) in the non-equidistant case, as shown in Figure 9B. The horizontal position of the thermal bridge is between \(x=0.5\) m and \(x=1\) m for the equidistant mesh and between \(x=0.735\) m and \(x=1\) m for the non-equidistant one. The height of the thermal bridge is two cells (2 cm) in the \(z\) direction, i.e., 0.02 m, while it is vertically positioned in rows number 20 and 21 from z = 0.37 m to z = 0.39 m.  
As in our previous work, we apply a resistance-capacitance-type model [38] of heat conduction, where it is necessary to switch to cell variables. The cell temperature is defined as the temperature in the middle of the cell. We calculate the heat capacity of the cells using the formula \(C_{i}=c_{i}\rho_{i}\Delta x_{i}\Delta y\), while for the approximate formula for the thermal resistance in the \(x\)-direction, we use \(R_{i,j+1}\approx\frac{\Delta x_{i}}{k_{i,j+1}\Delta x}\). Here, \(Ax=\Delta y\Delta z_{i}\) is the area of the cell surface orthogonal to \(x\). Therefore, the horizontal and vertical resistances in the surface simulation case (with homogenous material and uniform mesh) can be expressed as  
\[R_{i,i+1}\approx\frac{\Delta x}{k_{i}\Delta z\Delta y}\text{ and }R_{i,i+N_{x}} \approx\frac{\Delta z}{k_{i}\Delta x\Delta y_{i}},\]  
respectively, where the cell with the second subscript \(i+N_{x}\) is just below the cell \(i\).