Dynamic large deformation contact problems arise in many industrial applications like auto mobile engineering or biomechanics but only few methods exists for their numerical solution, all having their advantages and disadvantages. In this thesis the numerical solution of large deformation contact problems is tackled from an optimisation point of view and an application of this approach within a femoroacetabular impingement analysis is described. In this thesis we use a non-smooth Hamilton principle and Fréchet subdifferential calculus to derive a weak formulation of the problem. The resulting subdifferential inclusion is discretised in time by constructing a contact-stabilised midpoint rule. For the spatial discretisation the state-of- the-art dual mortar method is applied which results in non-convex constrained minimisation problems that have to be solved solved during each time step. For the solution of these problems an inexact filter trust-region method is derived which allows to use inexact linearisations of the non-penetration constraints. This method in combination with fast monotone multigrid method is then shown to be globally convergent.