Simulation of crashworthiness problems with implicit time integration methods for non-linear dynamics

When studying impact problems, time integration of the equations of evolution occurs in the non-linear range. Usually, explicit algorithms are used in such a context. Nevertheless, due to its lack of stability in the non-linear range, and its limitation in the time step size, an implicit scheme could advantageously be used. The most widely used implicit algorithm is the Newmark algorithm. Nevertheless, with this algorithm used in the non-linear range, the conservation of the energy is no longer satisfied. To avoid divergence due to the numerical instabilities, numerical damping was introduced, leading to the generalized- methods. But these schemes can exhibit instabilities in the non-linear range too. Therefore a new family of integration algorithms for problems in structural dynamics has appeared that satisfies the mechanical laws of conservation (i.e. conservation of linear momentum, angular momentum and total energy) and that remains stable in the non-linear range.