X-FEM explicit dynamics for constant strain elements to alleviate mesh constraints on internal or external boundaries

Abstract

This paper deals with the use of the extended Finite Element Method (X-FEM) for rapid dynamic problems. To solve the equations of motion, a common technique is the explicit direct integration with a Newmark scheme. Since this temporal scheme is only conditionally stable, the critical time step must be determined. It is generally induced by mesh constraints. The idea of the paper is to weaken constraints on mesh generation algorithms so that the critical time step is as large as possible. Using the X-FEM one allows a non-conformity between mesh and discontinuities such as cracks, holes or interfaces. In a first part, we present a summary about direct integration schemes and about the eXtended Finite Element Method. Then, we focus on the theoretical description of a 1D X-FEM finite element and its generalization to 2D and 3D finite elements. Then, dynamic numerical simulations are shown. They concern structures under impact with holes or external boundaries not exactly matched by the mesh. Comparisons are made with numerical results coming from the ABAQUS software. It shows that developments are satisfactory. We conclude with some outlooks concerning this work.

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

“…Another common use of the X-FEM is for free boundaries and holes as presented in Sukumar et al [37], Daux et al [7] and Rozycki et al [35]. In this case, the displacement approximation for enriched elements contains only an enriched contribution, that is, Eq.…”

“…This expression for the lumped mass is a generalization of the one proposed by Rozycki et al [35] for piecewise linear triangles and tetrahedrons to piecewise bilinear quadrilaterals and trilinear hexahedrons.…”

“…The case of holes and free boundaries has already been studied with explicit time integration schemes for constant strain elements, that is piecewise linear triangles and tetrahedrons, by Rozycki et al [35]. The main difference in this case is that the displacement field contains only an enriched contribution as presented in Eq.…”

“…It was first applied to two-dimensional crack growth simulation by Belytschko and co-workers [4,26], then to three-dimensional crack propagation in Sukumar et al [38], Moës et al [27] and Gravouil et al [14]. It was further extended to many applications such as holes and inclusions in Sukumar et al [37], Daux et al [7] and more recently by Rozycki et al [35]. The use of the X-FEM in dynamic situations was mostly focused on dynamic crack growth modelling.…”

“…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.…”