## 2 Methodology  
Representative volume elements (RVEs) of HONNs (see Figure 1a,b) and CNHONNs (see Figure 1c-f) are chosen for MD simulations to investigate the mechanical behavior of the RVE in the in-plane (\(y\) axis) and out-of-plane (\(z\) axis) direction. The \(x\), \(y\) and \(z\) axes correspond to [010], [100], [001] directions, respectively. The lattice constant of Ni is 3.52 A and the size of RVE is 140.8 A \(\times\) 140.8 A (as shown in Figure 1a), with a through cylindrical pore in the center and a quarter cylindrical pore in each corner which have identical radius with central pore (as shown in Figure 1b). In order to investigate the influence of relative density, 7 classes of samples with a center cylindrical pore of different radii are built and the simulated system contains 112,720-175,440 atoms, as listed in Table 1. The relative density \(\rho_{\rm r}\) of the samples is calculated by \(\rho_{\rm r}=M_{\rm H}/M_{\rm B}\), where \(M_{\rm B}\) is the sum of atomic mass of the bulk Ni and \(M_{\rm H}\) is the sum of atomic mass of HONNs. And its porosity can be calculated simply by (1\(-\rho_{\rm r}\)). In order to investigate the influence of CNTs, we take the 4 th group of samples as the example by adding different mass fractions of CNTs (as shown in Figure 1c-f). In previous published studies, both zigzag [40] and armchair [36; 37; 38; 39; 41; 42; 43] CNTs were used to strengthen different types of material. And the (4, 4) CNT [36; 37; 42; 43] was often used to simulate the mechanical properties of CNT/metal composites. When investigating the strengthening effect, the present work is limited to a smaller diameter of CNT, that is, (4, 4) CNT. The effect of the diameter and chirality of the CNTs will be the topic of further study. The locations of CNTs are in the center of ligament throughout \(z\) axis and distribute symmetrically around the central pore (as shown in Figure 1f). The mass fractions of CNTs \(\omega_{\rm CNT}\) is calculated by \(\omega_{\rm CNT}=M_{\rm C}/(M_{\rm C}+M_{\rm H})\), where \(M_{\rm C}\) is the sum of atomic mass of the CNT. The \(\omega_{\rm CNT}\) in our simulations are 1.28\(\%\) (Figure 1c), 2.58\(\%\) (Figure 1d), 3.89\(\%\) (Figure 1e) and 5.22\(\%\) (Figure 1f) respectively. The rate of weight reduction due to the addition of CNT can be calculated by \((M_{\rm H}\)-\(M_{\rm CNH})/M_{\rm H}\) and they are 5.82\(\%\) (Figure 1c), 1.16\(\%\) (Figure 1d), 1.75\(\%\) (Figure 1e) and 2.33\(\%\) (Figure 1f) respectively in our simulations, where \(M_{\rm CNH}\) is the sum of atomic mass of CNHONNs. Atomic images are visualized via Ovito [44]. MD simulations are performed by means of the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [45]. The Ni-Ni interaction is commonly described by an embedded atom method (EAM) potential [46] and the binding interaction of C-C is described by the widely used adaptive intermolecular reactive empirical bond order (AIREBO) potential [47]. The Lennard-Jones (LJ) 12-6 potential is adopted to simulate interatomic forces between C and Ni atoms and long rang van der Waals (vdW) interaction between C atoms. The LJ parameters \(\varepsilon_{\text{Ni-C}}\) and \(\sigma_{\text{Ni-C}}\) are determined by Lorentz-Berthelot (LB) mixing rule [48]. The value of \(\varepsilon_{\text{Ni-C}}\) is estimated from geometric average (\(\varepsilon_{\text{Ni-C}}=\sqrt{\varepsilon_{\text{Ni}}\varepsilon_{\text{C}}}\)) between \(\varepsilon_{\text{Ni}}\) and \(\varepsilon_{\text{C}}\) and the value of \(\sigma_{\text{Ni-C}}\) is estimated from the arithmetic average ((\(\sigma_{\text{Ni}}+\sigma_{\text{C}}\))/2) between \(\sigma_{\text{Ni}}\) and \(\sigma_{\text{C}}\). For non-bonded C atoms [PERSON] and [PERSON] [49] recommended the LJ parameters \(\varepsilon_{\text{C}}=0.00296\) ev and \(\sigma_{\text{C}}=3.407\) A and [PERSON] et al. [46] suggested the LJ parameters \(\varepsilon_{\text{Ni}}=0.219877\) ev and \(\sigma_{\text{Ni}}=2.30\) A.