## 3 Results  
### Penetration Force  
Figure 4a showed the simulation model and the maximum force of each specimen penetrating different soils. Specimen A got the maximum force for all the types of soil. For the other specimens, the resistance was lower than that of specimen A in varying degrees, indicating the bionic curves have ability in reducing resistance during penetration, and the bionic specimens designed by different teeth curves positions have different performances in one soil. Moreover, the specimens C, B and D obtained the lowest force under soils 1, 2, and 3, respectively. Take specimen A into the basis, the maximum force of optimal specimen was reduced by 26.15%, 22.68%, and 25.86%, respectively. Thus, it is believed the bionic specimens C, B, and D had the lowest penetration force in penetrating soils 1, 2, and 3, respectively, which should be selected to conduct the further analysis.  
From Figure 4b-d, the depth-force curves during the penetration using specimen A and C in soil 1, specimen A and B in soil 2, and specimen A and D in soil 3 in the simulation is compared with that in the experiments. It proves that the depth-force curve compared well, which can verify that the  
Figure 4: (**a**) The max force of experiments and; The depth-force curves of (**b**) specimen A and C penetrating soil 1, (**c**) specimen A and B penetrating soil 2, and (**d**) specimen A and D penetrating soil 3 in experimental and simulation results.  
experimental tests and numerical simulations were in close agreement. The increase tendency of force curves was similar in soil 1 and soil 2, while it had distinct difference when comparing with soil 3.  
In order to analysis the mechanism of the reduction in different soils, the force equations of the specimens with maximum and minimum final force in three soils were fitted by MATLAB. During the penetration process, the curve of force in soil 1 and soil 2 can be described by \(\mathrm{y}=\mathrm{ab}^{\mathrm{x}}\), and the correlation was listed in the Equations (13)-(16). The R-Square of the equations are higher than 0.95, which accord with the accuracy.  
\[\mathrm{Y_{A1}} =0.1458\times 1.158^{\mathrm{x}}\;(\mathrm{R-Square}\;=\;0.9987) \tag{13}\] \[\mathrm{Y_{C1}} =0.5716\times 1.119^{\mathrm{x}}\;(\mathrm{R-Square}\;=\;0.9991)\] (14) \[\mathrm{Y_{A2}} =0.07495\times 1.165^{\mathrm{x}}\;(\mathrm{R-Square}\;=\;0.9902)\] (15) \[\mathrm{Y_{B2}} =0.4986\times 1.116^{\mathrm{x}}\;(\mathrm{R-Square}\;=\;0.9634) \tag{16}\]  
Indeed, \(\mathrm{b}\) is the main parameter that can represent the increase rate of the force. As \(\mathrm{b}\) in Equations (6) and (8) was smaller compared with Equations (13) and (15), respectively, which means it can slow down the force increase rate, in that way the reduction of penetration force can be enhanced in the end. Moreover, the larger a in Equations (14) and (16) counteracted the influence of \(\mathrm{b}\), which lead to the phenomenon that the force difference at the beginning was small.  
When looking at how the specimens penetrated the soil 3, the force-depth curves can be described by \(\mathrm{y}=\mathrm{cx}-\mathrm{d}\) as shown in Equations (17) and (18). The R-Square of the equations are also higher than 0.95, which accord with the accuracy. It is illustrated the force was increased linearly, while for soil 1 and soil 2 the increase method was index.  
\[\mathrm{Y_{A3}} =1.528\mathrm{x}-6.154 \tag{17}\] \[\mathrm{Y_{D3}} =0.9915\mathrm{x}-3.531 \tag{18}\]  
The specimen \(\mathrm{D}\) got the smaller \(\mathrm{c}\), which is the slope of the function, representing the increase rate of force. Same as the soil 1 and soil 2, the mechanism of the force reduction when using the bionic outline was because the bionic outline can reduce the \(\mathrm{c}\) and slow down the force increase rate.