The algorithm for solution of contact problems presented in this chapter has been incorporated into DYNAZ. The solution method is applicable to wide range of static and dynamic problems with material and geometric nonlinearities. Although the Lagrange multiplier values are the incremental contact forces based on the solution of the governing equations of equilibrium, they are not used by the algorithm because such a procedure may introduce serious errors of linearization in nonlinear analyses with contact. Instead, the total contact forces are directly evaluated from equilibrium of the applied external loads, inertia forces, damping forces and the nodal point forces equivalent to the current element stresses. Thus, the sole function of the Lagrange multipliers is to enforce that, in each iteration, the incremental displacements of contactor and target surfaces are compatible with each other in the region of contact.
Integral statements generally form the basis of the finite element approximations to problems in elastodynamics. The simplest, and, hence, most popular approach is to perform separate approximations for the two distinct types of independent variables, associated with the temporal and the spatial dimensions. Typically, domain approximation precedes time integration, so that the latter is applied to a set of ordinary differential equations in time. This decomposition leads to what are known as semi-discrete, or direct, time integration methods. A broader class of integration methods resulting in space-time finite element emanate from a priori coupling of temporal and spatial variables in the discretization. In this work, the Newmark family direct time integration technique is used. It should be pointed out, as was also observed in other references ([31], [215]), that the integration of the dynamic response gives better accuracy if performed by using the time integration parameters, 
and 
, which corresponds to the well-known trapezoidal rule. It is reported that these values of the time integration parameters give a dynamic contact solution that fulfill the total energy conservation and the impulse-momentum relationship.