In time-stepping schemes, the formulation of the dynamics equations enables the scheme to simultaneously handle the smooth dynamics and non-smooth dynamics. The idea behind these schemes is to consider the dynamics equations as a measure differential inclusion. On the velocity level, they are expressed as:

(5–67)

where is the reaction measure.

In the case :

(5–68)

therefore:

(5–69)

In the Moreau-Jean time stepping method (MJ Time Stepping), unilateral contact is modeled with a Signorini condition at the velocity level, that is a complementarity between the contact impulse and the relative velocity, which ensures impenetrability. In order to handle multiple contacts with potentially changing status, an implicit algorithm is used to integrate the dynamics. As a simplified example, consider and . The discrete form of the equations of motion is:

(5–70)

The smooth terms may be discretized using a θ-method as:

(5–71)

then:

(5–72)

where is the approximation of the impulse over the time step. Note that the acceleration of the system is never explicitly computed because it becomes infinite for impulsive forces. The kinematic equations are discretized as follows for contact :

(5–73)

Finally, Newton's law is implicitly formulated as:

(5–74)

Because the Moreau-Jean time stepping method is formulated in terms of non-smooth velocities, it better handles the acceleration discontinuities that can happen when the geometry is non-smooth. Consequentially, it is well-suited to work with mesh-based contact.