On generation of lumped mass matrices in partition of unity based methods

Abstract

SummaryThe partition of unity based methods, such as the extended finite element method and the numerical manifold method, are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials. In the dynamic analysis of cracked bodies using an explicit time integration algorithm, as a result, huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions. Observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force, we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses, where we inversely utilize the ‘from local to global’ skill. The proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix, but has more excellent dynamic property than the latter. Meanwhile, the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes — a very excellent dynamic property. Copyright © 2017 John Wiley & Sons, Ltd.