## 5 Results and Discussion
### Material Properties of the Foamed Specimens  
The properties of the foamed samples are computed using the FE simulations. Here, only a selected volume element (SVE) of each sample in Figure 13 is used for the simulation to improve the computational efficiency. To investigate the thermal properties of the foamed specimen, the thermal conductivities of each specimen are computed. Figure 14 shows the temperature isosurfaces of the heat flows in different directions, and the influence of the pore distributions on the heat transfer can be examined from this temperature contour. In each left, middle and right figure of Figure 14, the isosurfaces in the \(x\), \(y\) and \(z\) directions are presented, respectively. In these figures,  
Figure 11: Pore size distribution of the foamed specimens: (**a**) LithoPore®100, (**b**) LithoPore®200, (**c**) LithoPore®450, (**d**) LithoPore®600. Figure 12: Comparison of the shape factors for the foamed specimens. many fluctuations can be found near the pore regions, particularly the vicinity of the large pores. In addition, the directional heat flow is different in each direction even in the same specimen; this denotes that the spatial distribution of pores in each foamed specimen strongly affects the heat transfer. Figure 14: _Cont._  
Figure 13: Selected volume element (SVE) of the specimen for numerical analysis: (**a**) LithoPore®100, (**b**) LithoPore®200, (**c**) LithoPore®450, (**d**) LithoPore®600 (note: transparent meshes of each figure are the specimens in Figure 9). For the quantitative comparison of the heat flows of the foamed samples, the thermal conductivities are computed by averaging the local heat flux in each sample, as shown in Figure 15. It is confirmed that the computed thermal conductivity values are in good agreement with the properties given by the manufacturer of the LithoPore(r) specimens, which are denoted as the reference in the figure. In Figure 15, the thermal conductivity increases as the specimen density increases. The relationship of the pore characteristics and the thermal conductivity can be examined by comparing Figures 10 and 15. In the LithoPore(r)100 specimen, it is confirmed from \(L_{p}(r)\) that the pores are more clustered for the \(z\) direction than other directions, and the thermal conductivity in that direction is higher than those of the \(x\) and \(y\) directions. Likewise, other specimens show a similar tendency between the probability function and the thermal conductivity; the thermal conductivity is larger as the specimen contains larger pore clusters along that direction because more solid areas can be secured for the direction so that heat can flow. These results confirm that the thermal conductivity of the foamed specimen is strongly affected by the material density, as well as the pore shapes and distributions. The mechanical responses of the SVEs are also investigated in this study. Figure 16 shows the directional modulus and compressive strength of the foamed samples in different directions. The computed mechanical data are also in reasonable agreement with those given by the manufacturer. Both mechanical properties increase as the density of the specimen increases, likewise the thermal  
Figure 14: Temperature isosurfaces of the foamed specimens: (**a**) LithoPore®100, (**b**) LithoPore®200, (**c**) LithoPore®450, (**d**) LithoPore®600 (note: in each figure, the left figure is in the \(x\) direction heat flow, and the middle and right figures are in the \(y\) and \(z\) directions, respectively). Figure 15: Comparison of the thermal conductivity values of the specimens in the \(x\), \(y\) and \(z\) directions. conductivity. In addition, the directional trends of the properties are almost identical to the probabilistic characteristics.