On the numerical stability and mass‐lumping schemes for explicit enriched meshfree methods

Abstract

SUMMARYMeshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method.It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass-lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node.

Computational Methods for Fracture in Brittle and Quasi-Brittle Solids: State-of-the-Art Review and Future Perspectives

“…Note that, efficient mass lumping schemes have meanwhile been proposed for XFEM [171,172] (and XEFG [173]). (iv) Based on the full interpolation basis of the overlapping elements, the Hansbo-XFEM can straightforwardly integrate enhanced techniques that are more difficult to integrate in an enriched XFEM formulation [174].…”

Concurrent multiscale modeling of three dimensional crack and dislocation propagation

“…In the continuum sub-domain, the displacement field is approximated by the extended finite element method (XFEM). In this approximation, the displacement field is decomposed into a continuous part and a discontinuous part as in [9,25]:…”

“…The bridging scale method of Wagner and Liu [8] can eliminate reflection of unauthorized elastic waves on the atom/continuum transition area. In this coupling method the continuum domain is modeled using finite elements or meshfree discretization [9]. Likewise, in the bridge area the nodes should placed in the atom positions.…”

Thermomechanical Simulation of Friction Stir Welding of Aluminum Using an Adaptive Element‐Free Galerkin Method

“…However, when using the Finite Element Method (FEM) problems occur for three dimensional (3D) remeshing. Alternative approaches in this regard are meshless methods such as the Element Free Galerkin Method (EFG) [4,6] among many others. As a results, in this study we exploit an rh-adaptive meshless method to model the FSW process.…”