## 3 Results
### First Order and Total Sensitivity Coefficients  
The GSA results for the main effects and total effects are shown in Figure 5. In Figure 5a the first-order effects of the four design variables bar width \(d\), bar height \(h\), bar spacing \(s\) and plane angle \(\varphi\) are plotted for all output quantities. It can be seen that even the highest effect (bar width \(d\) on tortuosity \(\alpha_{\infty}\)) is rather low (less than 0.50). This indicates that strong interaction effects are present and that the underlying model function found by the ANN is nonlinear. Apart from the finding of an expected nonlinearity, the results in Figure 5a are not easy to interpret. However, it can be stated that the bar width and bar spacing have the largest influence on the flow resistivity, and the bar width has a strong effect on tortuosity. Both findings seem reasonable: the influence of the bar width and bar spacing on the flow resistivity can be explained with a blocking of the open area, thus blocking the fluid that flows through the material. An increase in bar width should increase the flow resistivity, while an increase in the spacing should reduce the flow resistivity. However, both effects might compensate each other when happening simultaneously. This interaction indeed is found using the second-order effects and is discussed in the following Section 3.2. Moreover, the main effect of the bar width on tortuosity seems explainable, too: as the bar width increases while the other geometry parameters remain unchanged (first order effect), the streamline of a fluid particle becomes longer in contrast with a small bar width. This is illustrated in Figure 6.  
Figure 6 schematically shows a cut through two specimen with different bar widths. The specimens are cut perpendicular to the bars in the first, third and fifth layer (accordingly: parallel to the bars in the second and fourth layer). A plane angle of 90\({}^{\circ}\) is assumed,  
Figure 5: First order and total sensitivity coefficients of the microscale geometry parameters of the porous material with respect to the JCA model parameters. Results reported after \(1\cdot 10^{6}\) Monte Carlo runs (Quasi Monte Carlo with Sobol sequence). The Greek letters denote the JCA model parameters, i.e., \(\Xi\): flow resistivity, \(\Phi\): porosity, \(\alpha_{\infty}\): tortuosity, \(\Lambda\), \(\Lambda^{\prime}\): viscous and thermal characteristic length, respectively. (**a**) First-order sensitivity coefficients. (**b**) Total sensitivity coefficients. i.e., the bars are perpendicular from one layer to the next. The bar spacing and bar height are identical in Figure 6a,b, respectively. In blue, a possible fluid streamline is shown. It can be seen that the path is longer in Figure 6a; thus, an increase in the bar width increases the tortuosity. A similar description can be found in [59]. There, the authors conduct flow computations using the Lattice-Boltzmann method on several generic porous structures, including a U-shaped channel. From the computed flow fields, the tortuosity is estimated. The authors show that the tortuosity of such U-shaped channel scales approximately with \(\alpha_{\infty}\sim 1+\frac{2\cdot b}{L}\) with \(b\) denoting the depth of the U-shape and \(L\) being the overall length of the channel in the flow direction. Since in this study, the sensitivity is investigated, a comparison between the cases should employ the derivative of the given formula with respect to \(b\), i.e., \(\frac{\partial\alpha_{\infty}}{\partial b}\sim\frac{2}{L}\). Thus, the tortuosity scales positively and linearly with the depth of the U-shaped channel. Applying the geometry description used in this study, the depth of the U-shaped \(b\) is the analogue to the bar width \(d\), the length of the channel \(L\) in the direction of propagation is analogue to the bar height \(h\). Thus, an increase in bar width \(d\) results in an increased tortuosity. In summary, the observed influence of the bar width on the tortuosity seems plausible.