## 1 Introduction  
It is crucial to analyze the characterization of the forces on moving objects in granular matter in various engineering fields such as geotechnics [1], road engineering [2; 3], machine learning [4], chemical sciences [5], and biological engineering [6]. It is the reason why researching drag force in detail plays a key role in understanding the dynamics of granular materials. Analysis of the optimal nose shape [7; 8; 9] is significantly more intricate than the conventional investigation of penetration using prescribed shape data [10; 11; 12; 13; 14]. In general, the conundrum of determining the optimal shape of the nose of an axisymmetric rigid projectile can only be solved using numerical methodologies. The key to studying the optimal nose shape is the way friction is accounted for. Research has demonstrated that while the effect of friction affecting the ultimate penetration depth of a projectile is noticeable, its influence on determining the optimal nose shape is minimal across all scenarios [7]. A closed-form solution to the optimal nose shape can be obtained using simplified constitutive relationships when friction is not considered [9]. In a word, an oval or conical shape is considered the optimal nose shape for a one-time penetration. However, projectile tips are easily worn or broken under extensive repetitive work. Considering their practical engineering uses, such as in pile drivers, we should also consider other shapes. Previous studies have proposed that the drag force on an object penetrating into a granular material vertically is almost universally separated into depth and velocity dependent terms, so that \(F=F_{\rm Z}+F_{\rm V}\)[15; 16; 17; 18; 19; 20]. When \(V<\sqrt{2 gd_{\rm p}}\) (here, \(g\) is the acceleration of gravity and \(d_{\rm p}\) is mean diameter of the granular media), the drag force is independent of the object's speed [21] and has been found to be linear in depth, which has different proposed forms according to the state of impact and the intruder shape [22; 23]. For a cylinder \(F_{Z}=\alpha\mu\rho_{\mathrm{p}}gd_{\mathrm{i}}^{2}Z\), \(\alpha\) is a constant which is found to be 20-30 [24], \(\mu\) is an internal friction coefficient equal to the tangent of the repose angle, \(\rho_{\mathrm{p}}\) is the material density of the particles, \(d_{\mathrm{i}}\) is the diameter of the intruder, and \(Z\) is the vertical penetration distance. When \(V>\sqrt{2 gd_{\mathrm{p}}}\), the peak drag force is observed to be proportional to the square of the velocity only for a short period of time at the beginning of penetration, and then, at larger depths, all force profiles fall on the line of the former experiments (\(V<\sqrt{2 gd_{\mathrm{p}}}\)) [25; 26]. In addition, in order to avoid changes in the contact area between the intruder and the granular media at larger depths, a cylindrical intruder was placed horizontally in the media, which is different from the experiments of [PERSON], L.K. [20]. With the further acceleration of penetration velocities, three different regimes, from the quasi-static (force independent of the velocity) to the viscous (force linearly related to velocity), and then to the inertial (force related to the square of velocity), are exhibited [27]. However, it has not been investigated whether there are still significant turning points when the contact area between the intruder and the granular media gradually increases (e.g., in a pile driver). In the meantime, many researchers [28; 29] have assessed the sensitivity of flow characteristics to interparticle friction. The initial solid volume fraction was varied by adjusting the interparticle friction coefficients during compression [30], setting \(\mu\) to 0.1, 0.25, 0.5, 0.75, or 1.0. The range of \(\mu\) values (0.12-0.35) was given in experimental particle-particle friction tests [31]. Many DEM simulations using spherical or elliptical particles have set higher values (\(\mu\geq 0.5\)) [32; 33; 34; 35; 36] to explore the influence of interparticle friction on critical-state behavior.