Lumped mass finite element implementation of continuum theories with micro‐inertia

Continuum theories can be equipped with additional inertia terms to make them capable of capturing wave dispersion effects observed in micro‐structured materials. Such terms, often called micro‐inertia, are convenient and straightforward extensions of classical continuum theories. Furthermore, the critical time step is usually increased via the inclusion of micro‐inertia. However, the drawback exists that standard finite element discretisation leads to mass matrices that cannot be lumped without losing the micro‐inertia terms. In this paper, we will develop a solution algorithm based on Neumann expansions by which this disadvantage is avoided altogether. The micro‐inertia terms are translated into modifications of the residual force vector, so that the system matrix is the usual lumped mass matrix and all advantages of explicit time integration are maintained. The numerical stability of the algorithm and its effect on the dispersive properties of the model are studied in detail. Numerical examples are used to illustrate the various aspects of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.