An optimal explicit time stepping scheme for cracks modeled with X-FEM

This paper deals with the numerical modelling of cracks in the dynamic case using X-FEM. More precisely, we are interested in explicit algorithms. We prove that by using a specific lumping technique, the critical time step is exactly the same as if no crack were present. This somewhat improves a previous result for which the critical time step was reduced by a factor of square root of 2 from the case with no crack. The new lumping technique is obtained by using a lumping strategy initially developped to handle elements containing voids. To be precise the results obtained are only valid when the crack is only modeled by the Heaviside enrichment. Note also that the resulting lumped matrix is block diagonal (blocks of size 2 by 2).