Lötstedt [6] seems to be the first to have set up the contact problem as a linear complementarity problem. His model has been at the origin of most of the work done in the area, but it did not provide good resolution of problems including Coulomb friction. He has also produced or referenced examples for which the model either does not have a solution or has multiple solutions.
Baraff [1] developed an algorithm (related to Dantzig's principal pivoting algorithm) that could solve a very large class of problems. He was also the first to approach arbitrarily shaped contacts - using an implicit definition for the surfaces and solving an optimization problem. He also pointed out the difficulties in dealing with arbitrarily shaped contact regions. Trinkle et al. [8] have produced an enhanced complementarity model which seems very promising, especially for rolling friction. For an extended discussion of the literature in the area see [8].
In what follows, we will analyze only the frictionless case, as being the ones for which we have performed the simulations. Also we will not discuss the collision modelling techniques.