Mass lumping strategies for X‐FEM explicit dynamics: Application to crack propagation

Abstract

SUMMARYThis paper deals with the numerical modelling of cracks in the dynamic case using the extended finite element method. 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 developed to handle elements containing voids. To be precise, the results obtained are valid only when the crack is modelled by the Heaviside enrichment. Note also that the resulting lumped matrix is block diagonal (blocks of size 2×2). For constant strain elements (linear simplex elements) the critical time step is not modified when the element is cut. Thanks to the lumped mass matrix, the critical time step never tends to zero. Moreover, the lumping techniques conserve kinetic energy for rigid motions. In addition, tensile stress waves do not propagate through the discontinuity. Hence, the lumping techniques create neither error on kinetic energy conservation for rigid motions nor wave propagation through the crack. Both these techniques will be used in a numerical experiment.

An explicit dynamics extended finite element method. Part 1: Mass lumping for arbitrary enrichment functions

“…We focused our study on cracks, free boundaries and holes modelling with X-FEM. We showed that the proposed lumping formula is a generalization of some lumping techniques that were proposed for discontinuous enrichment for cracks (see [24,25]) and for free boundaries for constant strain elements (see [35]). …”

“…The discrete problem can therefore be written as a classical finite element dynamic problem given in Eqs. (25) to (27).…”

“…The only successful attempts in finding appropriate mass lumping schemes for enrichment that produce non-zero critical time steps are the ones of Menouillard et al [24,25] for dynamic crack growth and Rozycki et al [35] for free boundaries and holes with constant strain elements. However, these two methods have certain drawbacks.…”

Dynamic fracture analysis by explicit solid dynamics and implicit crack propagation

“…[18]. In comparison with implicit analyses, the number of timesteps for dynamic fracture is increased with an explicit time scheme, but the explicit inversion of the diagonal mass matrix makes the computational cost per time step much lower [16].…”

“…The time condition is half of that for a standard FEM solution to resolve the Heaviside function as XFEM crosses an element (discussed in the next section) [18].…”