An explicit time-integrator with singular mass for non-smooth dynamics

Abstract

This article addresses the simulation of non-smooth dynamics problems with unilateral contact constraints between rigid and deformable bodies. It proposes a modified CD-Lagrange scheme with a singular mass matrix. This scheme is explicit, and based on a contact condition on velocity. The formulation is designed for a 1D impact problem between deformable and rigid body with unilateral constraint. The singular mass matrix allows to get a more accurate energy balance on the discrete system, especially during non-smooth events. An extension is presented then for the 3D cases. Its implementation is easy, and fully compatible with large deformations or non-linear materials. Indeed it consists only in adding a numerical parameter for each contact node. The energy balance for the singular 3D formulation is improved compared to the consistent one.

Appendix: a parallel between the contact law and the velocity contact condition of Moreau-Jean

The contact law described by Eqs. (10),  (11) is close from the Moreau–Jean velocity conditions [20, 28]. With a mass contact node, these conditions express as:

$$\begin{aligned}&\text {If }\mathbf {g}_{n+1}>0,&\widetilde{\mathbf {r}}_{n+\frac{3}{2}}=0 \end{aligned}$$

(35)

$$\begin{aligned}&\text {If }\mathbf {g}_{n+1}\leqslant 0,&0 \leqslant \widetilde{\mathbf {r}}_{n+\frac{3}{2}} \perp \dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}} \geqslant 0 \end{aligned}$$

(36)

The Eq. (35) describes the free of contact state, and the Eq. (36) describes the active contact. $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}$, the contact impulse, acts as a Lagrange multiplier. It imposes $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}}$, the contact node velocity, if it does not respect the contact conditions:

• If $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}} \leqslant 0$, the beam tends to penetrate into rigid frontier. The dynamic gives a $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}>0$ to stop the contact node. It compensates both the internal stress of skin and the inertia of contact node. The skin is in compression.

• If $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}} > 0$, the beam tends to leave the rigid frontier. $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}$ is set to zero because the system already meets the contact conditions. The skin is no more constrained.

The velocity contact law (10), (11) for singular mass matrix follows the same principles as the Moreau–Jean’s condition  (35), (36). But the Moreau–Jean’s condition have access to $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}}$ through the dynamic before to set $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}$. At the release time-step , $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}}>0$ and so $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}$ is directly set to zero. For the velocity contact law, $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}}$ is computed once $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}$ is known. This leads to a release “late” of one time-step: at release, $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}\leqslant 0$ but already fixed and $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}}>0$. This temporary violation on positivity of contact stresses is necessary as no dynamical link exists between $\widetilde{\mathbf {r}}_{n+\frac{3}{2}}$ and $\dot{\widetilde{\mathbf{u }}}_{n+\frac{3}{2}}$.