A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction

We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second-order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial-grade virtual prototyping software.