The material point method (MPM), which is known to be advantageous in modeling large deformation and contact problems, is utilized in this study to simulate displacement pile installation in sand. A modified Bardenhagen’s contact algorithm is proposed that calculates the node-to-surface distances between the deformable soil mass and the rigid pile surface more effectively, along with a smoothing factor for the change of momenta between the contacting bodies. The modified algorithm is shown to produce more accurate and stable contact results. Three tests, namely, the rolling cylinder, the penetrating wedge, and the strip footing, are first conducted to demonstrate the validity of the modified algorithm. It is then applied to analyze the installation of a closed-ended pile, employing a state-dependent Mohr–Coulomb model to capture the sand’s state-dependent shearing behavior. Detailed analyses of the stress and deformation fields developed around the pile during steady penetration reveal quantitative agreement between the MPM predictions and published experiments, indicating that the approach holds promise for use in more sophisticated analyses designed to improve the understanding of industrially driven piles.