## 1 Introduction  
Vibratory pile driving is a pile installation technique that exploits the continuous periodic load from a vibratory hammer as a driving force. Along with the evolution of the vibratory hammer, a vibratory pile driving with resonance technique springs up in urban areas mainly because this construction approach has advantages in terms of the pile installation speed and the environmental impact [1; 2; 3; 4]. As reported in [4], the driving time for an open-ended steel pile (pile diameter 700 mm and pile length 60 m) is less than 40 min, and the maximum ground surface vibration measured at around 1 m from the pile axis is as low as 13.32 mm/s. The terminology of resonance-free means that the severe vibration influences caused by the resonance phenomenon can be mitigated and avoided during pile installation. The principle of resonance-free technique is that the novel vibratory hammer has a high work frequency as well as generating zero centrifugal force during start-up and shut-down [4; 5; 6]. The advantages of vibratory pile driving with resonance-free technique can contribute to underground constructions that often require a short construction period or encounter a complicated construction environment (Figure 1). To better exploit this novel construction approach, more investigations are still needed. Numerical simulation is a cost-effective method. According to the principle of resonance-free [4; 5; 6] technique, the simulation of resonance-free vibratory pile driving can be treated as conventional vibratory pile driving if the start-up and shut-down processes are able to be neglected. As vibratory pile driving commonly involves soils below the groundwater table, it is critical to consider the hydro-mechanical interactions in numerical studies. However, considering hydro-mechanical interactions in the dynamic analysis is particularly difficult when dealing with the vibratory penetration problem. Most simulations of vibratory pile driving take simplified assumptions for the hydro-mechanical interactions. These assumptions vary from dry condition [7, 8, 9], undrained condition [6, 10, 11], to locally undrained condition (zero soil permeability) [12, 13]. Additionally, although some simulations modelling vibratory pile driving under a partly drained condition (nonzero soil permeability) have been reported, the simulated pile penetrations were zero [14], or very small [15]. To the authors' knowledge, a few simulations have incorporated both the partly drained condition and substantial pile penetration [16, 17, 18, 19]. Certain efforts were made to construct a user-defined element for the dynamic analysis of hydro-mechanical interactions when these simulations were performed in the finite element code ABAQUS. [PERSON] et al. [18] and [PERSON] et al. [19] also carried out several simulations using an inhouse finite-element program NUMGEO, which they developed. Unfortunately, the pile-soil contact might be poorly modelled using ABAQUS as implementations of pile-soil contact were either restricted to a frictionless contact [16, 17] or incapable of calculating the effective contact pressure [19]. Simulations performed using NUMGEO [18, 19] are able to incorporate both frictional contact and effective contact pressure. However, NUMGEO is still under development and not yet available to the public [20]. Moreover, the reviewed studies mainly focused on close-ended piles [16, 17, 18, 19], rarely involving open-ended piles, which are popular in practice [1, 2, 3, 4]. Overall, the simulation of vibratory pile driving in saturated soil is still challenging task. The authors of this paper have been working on modelling vibratory pile driving using the finite difference software FLC3D [6, 11]. There is a build-in procedure available for dynamic analysis of hydro-mechanical interactions in FLC3D [21]. A numerical model is established in this paper for vibratory open-ended pile driving in saturated sand using FLAC3D. A density scaling method is used to increase the computation efficiency of the proposed model.