Effect of the temporal discretization of contact forces on dynamic contact simulations using singular non-standard quadrature rules for the mass matrix

Spurious Lagrange Multipliers oscillations present a significant challenge in implicit contact dynamics simulations. To mitigate these oscillations, (Hager and Wohlmuth 2007) proposed non-standard quadrature rules for the mass matrix, developing the $Q_0$ and $Q_1$ modified mass matrices. This work introduces and evaluates an extension of the $Q_1$ mass matrix to quadratic elements. Additionally, we investigate the influence of the parameter ${\alpha }_c$, which governs the temporal discretization of contact forces, on the performance of both consistent and modified mass matrices. Numerical results demonstrate that the proposed formulation effectively preserves the oscillation-suppressing capabilities of the $Q_1$ mass matrix to quadratic elements. Furthermore, it was observed that for ${\alpha }_c=0$ the consistent mass matrix displays good results, while for ${\alpha }_c={\alpha }_f$ the modified mass matrices showcase the best results.