## 4 Validation and Comparison
### Comparison of the Numerical Predictions with Experimental Data  
In what follows, we shall further verify the computational method developed previously by using the experimental data, and three commonly-applied porous materials, i.e., the porous titanium, the porous CoCrMo alloy and the porous tantalum, are considered in this study. As observed experimentally, the shapes of pores of both the porous titanium and the porous CoCrMo alloys are pretty close to spheres [14; 46] (i.e., parameters \(a=b=c\) in Equation (11)), and the radius varies from approximate 300 \(\upmu\)m to 600 \(\upmu\)m for porous titanium [12; 14] and from 0.51 mm to 0.99 mm for porous CoCrMo alloys [46]. The pores of these two porous media roughly satisfy the POS mode defined in the aforementioned Section. Slightly different from the former two materials, the shape of pores constituting the porous tantalum seems more like an ellipsoid [45], and the radii vary in the  
Figure 9: Normalized numerical predictions and theoretical bounds of the isotropic moduli of the porous tantalum and the porous titanium. The red and black lines are the upper and lower bounds of the elastic isotropic moduli; the green line is the numerical predictions obtained by our method. range \([200,400]\)\(\upmu\)m, and the relative positions among different pores are supposed to fulfill the CRS relation defined previously [45]. Hence, the model parameters in Equation (11) are set as \(n=2\) and \((a,b,c)\in[200,400]\)\(\upmu\)m. Sensitivity analysis of the RVE size, as well as the mesh, is then performed by the elastic modulus convergence study, and a cubic domain of side length equal to the value that is fifteen times of the maximum radii of practical pores observed experimentally [45; 46], is employed for the verification and meshed with \(89\times 89\times 89\times 5\) tetrahedral elements so as to generate porous microstructures with the strategy in Section 2 accurately and to obtain the converged numerical evaluations. Figure 10 plots the resulting microstructures of the foregoing porous materials with \(f=0.6\), and the relevant morphological parameters adopted are shown in Figure 11. The overall isotropy of the porous RVE models generated has also been examined by using the test approach together with the error indicators (\(e_{E}\) and \(e_{\kappa}\)) developed in [47] and the necessary and sufficient conditions in [48], and the maximum value of \(e_{E}\) and \(e_{\kappa}\) for all instances under consideration is no larger than 6%. Thus, the macroscopical isotropy of the porous materials consisting of the porous microstructures generated with the computational strategy in this study is approximately guaranteed. The relevant numerically predicted isotropic moduli together with the experimental data in [12; 45; 46] are visualized in Figure 12. Here, all elastic modulus results are normalized with respect to the elastic modulus associated with the lowest porosity of the porous material. Figure 11: Statistical morphological parameters of pores constituting the porous microstructures in Figure 10: **(a)** porous titanium, **(b)** porous CoCrMo alloy and **(c)** porous tantalum. Figure 10: Numerically generated microstructures of the three porous materials with \(f=0.6\): **(a)** porous titanium, **(b)** porous CoCrMo alloy and **(c)** porous tantalum. It is noticed that the elastic moduli of porous media are affected by a number of factors, say the pore size, pore shape, distribution mode and many others, and thus commonly vary from one position to another. Here, the convergence analysis of the number of simulations is carried out to account for the effect ascribed to generating the pore size and position randomly, and Figure 13 visualizes the normalized average elastic moduli together with their upper and lower bounds of the three porous media with different porosities.